Intermediate Physics for Medicine and Biology

The Intermediate Physics for Medicine and Biology Tourist

2012-01-27T06:31:00.011-05:00

Over the Christmas break I was browsing through the Guidebook for the Scientific Traveler: Visiting Physics and Chemistry Sites Across America, and it got me to wondering what sites a reader of the 4th edition of Intermediate Physics for Medicine and Biology might want to visit. Apparently having too much time on my hands, I devised a trip through the United States for our readers. (Perhaps I’ll prepare an international edition later.) The trip starts and ends in Rochester, Michigan, where I work, but the path consists of a large circle and you can begin anywhere. I have not visited all these places, but I know enough about them to suspect you would enjoy them all. Tell me if I have forgotten any important sites. Happy travels!

Oakland University in Rochester, Michigan. OU is home to Intermediate Physics for Medicine and Biology (IPMB) coauthor Brad Roth, in the Department of Physics. Here Roth collaborated with Russ Hobbie to prepare the 4th edition of IPMB.

The University of Chicago in Chicago, Illinois. The elementary charge (the magnitude of the charge of an electron, mentioned in IPMB in Chapter 3 and many times later) was first measured accurately by Robert Millikan at the University of Chicago using his famous oil drop experiment. The American Physical Society has an initiative to present commemorative plaques at important sites in the history of physics. Be sure to visit the Millikan plaque. You can see the original equipment used by Millikan at Chicago’s wonderful Museum of Science and Industry. Chapter 1 of IPMB cites the book Powers of Ten by Phillip and Phylis Morrison and the office of C. & R. Eames. The book is centered on a couple picnicking in Chicago, near Soldier Field and the Shedd Aquarium. Be sure to stop there, with your copy of Powers of Ten in hand.

Morrison, Illinois. Robert Millikan was born in Morrison, and a downtown park in this small town about 120 miles west of Chicago bears his name (although there is no sign or marker to indicate it). IPMB coauthor Brad Roth grew up in Morrison.

University of Minnesota, in Minneapolis, Minnesota. IPMB’s author Russ Hobbie worked at the University of Minnesota for years, and remains an emeritus faculty member in the Department of Physics and Astronomy. Stop by a visit him in nearby Saint Paul. While in Minneapolis, be sure to visit the Bakken Museum, perhaps the only museum in the country dedicated entirely to electricity and magnetism, and especially bioelectricity and biomagnetism, as discussed in Chapters 6-9 of IPMB. Earl Bakken was one of the founders of the medical device company Medtronic. Stop by at Medtronic's nearby Mounds View Bakken Education Center.

University of California Berkeley, in Berkeley, California. The cyclotron, crucial for nuclear medicine (see Chapter 17 of IPMB), was invented by Ernest Lawrence at UC Berkeley. See the APS plaque commemorating this invention. Material from a cyclotron in Lawrence’s lab led to the discovery of technetium, an element with no stable isotopes that is widely used in nuclear medicine imaging and is discussed at length in Chapter 17 of IPMB. While in the San Francisco area, visit Stanford University where Felix Bloch performed his pioneering experiments in nuclear magnetic resonance (Chapter 18, IPMB), and where Mark Denny has his Biomechanics Laboratory (Denny’s book Air and Water is cited often in IPMB). Don’t forget to visit the Exploratorium.

California Institute of Technology, in Pasadena, California. Carl Anderson discovered positrons while working at CalTech (see the APS plaque for Anderson's discovery). Positrons are used in positron emission tomography (PET) imaging (Chapter 17, IPMB). Also from Cal Tech in Richard Feynman, whose Lectures on Physics are cited in IPMB.

Washington University in St Louis, in St Louis, Missouri. Arthur Compton performed his groundbreaking experiments on Compton Scattering (Chapter 15, IPMB) at Washington University. See the APS plaque commemorating his work. Their biomedical engineering department now is home to many leading researchers in cardiac electrophysiology, including post doc Debbie Janks, a reader and often a commenter on the IPMB facebook group and blog.

Vanderbilt University, in Nashville, Tennessee. IPMB coauthor Brad Roth attended graduate school at Vanderbilt, working with John Wikswo in the Department of Physics and Astronomy. There, they measured the magnetic field of a single nerve axon, as described in Chapter 8 of IPMB. IPMB author Russ Hobbie was a Visiting Professor at Vanderbilt in 1999. Max Delbruck, an early biological physicist who contributed to our understanding of genetics, performed many of his Nobel Prize winning experiments at Vanderbilt.

Duke University, in Durham, North Carolina. Duke’s Department of Biomedical Engineering has been the home of many leaders in bioelectricity and cardiac electrophysiology, including Robert Plonsey, whose books are often cited often in IPMB. The Duke Biology Department is home to Steven Vogel, author of Life in Moving Fluids, another book cited in IPMB. Be sure to find the statue of former Duke physiologist Knut Schmidt-Nielsen studying a camel, which graces the Duke campus.

National Institutes of Health, in Bethesda, Maryland. No tour of biomedical facilities in the United States would be complete without stopping at the NIH campus in Bethesda. Be sure to visit the Stetton Museum of Medical Research in Building 10: the Warren Grant Magnuson Clinical Center. Stop by Building 13 and see where IPMB coauthor Brad Roth worked on transcranial magnetic stimulation (IPMB, Chapter 8) and where his friend Peter Basser invented MRI Diffusion Tensor Imaging (IPMB, Chapter 18) (Peter’s office is still there; stop by and say hi). You could spend a week visiting all the historic medical research sites in the Washington DC area.

Yale University, in New Haven, Connecticut. Visit Yale and walk the path of the early American physicist Josiah Williard Gibbs, whose work on chemical thermodynamics is discussed in Chapter 3 of IPMB, including the Gibbs Free Energy. See the APS plaque commemorating Gibbs’ work.

Framingham, Massachusetts. Visit the town that contributed more to uncovering the diseases of the heart than any other, through the Framingham Heart Study. Framingham is one of the few locations mentioned explicitly in IPMB, in Chapter 2.

Harvard University, in Cambridge, Massachusetts. Edward Purcell performed his early experiments on nuclear magnetic resonance at Harvard, which resulted in the Nobel Prize. He is also author of a beloved paper cited in IPMB, Life at Low Reynolds Number. Visit the site of the Harvard cyclotron, where IPMB author Russ Hobbie was a graduate student, and where Allan Cormack worked on the mathematical methods underlying computed tomography (IPMB, Chapter 16). Visit the nearby Massachusetts Institute of Technology Museum, containing a collection of artifacts related to science and technology (IPMB author Russ Hobbie obtained his undergraduate degree from MIT). While near Boston, visit the Museum of Science, especially their Theater of Electricity.

Woods Hole Marine Biological Laboratories, in Woods Hole, Massachusetts. At Woods Hole, Kenneth Cole developed the voltage clamp method (Chapter 6, IPMB), which played an important role in the discovery of how nerves conduct action potentials. Stop by the Visitors Center and take a tour.

Oakland University, in Rochester Michigan. Back to the starting point. Be sure to stop by Brad Roth’s office (166 Hannah Hall) and see his collection of all four editions of IPMB sitting on his bookshelf.

Radiation Risks from Medical Imaging Procedures

2012-01-20T07:04:00.006-05:00

On December 13, 2011 the American Association of Physicists in Medicine issued a position statement (PP 25-A) about radiation risks from medical imaging procedures. It is brief, and I will quote it in its entirety:

“The American Association of Physicists in Medicine (AAPM) acknowledges that medical imaging procedures should be appropriate and conducted at the lowest radiation dose consistent with acquisition of the desired information. Discussion of risks related to radiation dose from medical imaging procedures should be accompanied by acknowledgement of the benefits of the procedures. Risks of medical imaging at effective doses below 50 mSv for single procedures or 100 mSv for multiple procedures over short time periods are too low to be detectable and may be nonexistent. Predictions of hypothetical cancer incidence and deaths in patient populations exposed to such low doses are highly speculative and should be discouraged. These predictions are harmful because they lead to sensationalistic articles in the public media that cause some patients and parents to refuse medical imaging procedures, placing them at substantial risk by not receiving the clinical benefits of the prescribed procedures.

AAPM members continually strive to improve medical imaging by lowering radiation levels and maximizing benefits of imaging procedures involving ionizing radiation.”

News articles discussing this position statement appeared on the Inside Science and Physics Central websites.

The 4th edition of Intermediate Physics for Medicine and Biology discusses the risk of radiation in Section 16.13. Dose is the energy deposited by radiation in tissue per unit mass, and its unit of a gray is equal to one joule per kilogram. A sievert is also a J/kg, but it differs from a gray in that it includes a weighting factor that measures the relative biological effectiveness of the radiation, and is used to measure the equivalent dose (although often, including in the remainder of this blog entry, people get a little sloppy and just say “dose” when they really mean “equivalent dose”). A sievert is a rather large dose of radiation, and when discussing medical imaging or background radiation exposure, scientists often use the millisievert (mSv).

Table 16.7 of Intermediate Physics for Medicine and Biology lists typical radiation doses for many medical imaging procedures. For example, a simple chest X ray has a dose of about 0.06 mSv, and a CT scan is 1-10 mSv. The average radiation dose from all natural (background) sources is given in Table 16.6 as 3 mSv per year (primarily from exposure to radon gas). A pilot logging 1000 hours in the air per year receives on the order of 7 mSv annually.

Perhaps the most interesting sentence in the AAPM position statement is “Risks of medical imaging at effective doses below 50 mSv for single procedures or 100 mSv for multiple procedures over short time periods are too low to be detectable and may be nonexistent.” To me, the phrase “may be nonexistent” seems to cast doubt on the linear nonthreshold model often used when discussing the risk of low-dose radiation. Russ Hobbie and I discuss this model in Intermediate Physics for Medicine and Biology.

“In dealing with radiation to the population at large, or to populations of radiation workers, the policy of the various regulatory agencies has been to adopt the linear-nonthreshold (LNT) model to extrapolate from what is known about the excess risk of cancer at moderately high doses and high dose rates, to low doses, including those below natural background.”

We also consider other ideas, such as a threshold model for radiation effects and even hormesis, the idea that very low doses of radiation may be beneficial. The controversy over the biological effects of low-dose radiation is fascinating, but as best I can tell the validity of each of these models remains uncertain; getting accurate data when measuring tiny effects is difficult. I assume this is what motivates the word “may” in the phrase “may be nonexistent” from the position statement (although, I hasten to add, I have no inside information about the intent of the authors of the position statement—I’m just guessing). In our book, Russ and I come to a conclusion that is fairly consistent with the AAPM position statement.

“Some investigators feel that there is evidence for a threshold dose, and that the LNT model overestimates the risk [Kathren (1996); Kondo (1993); Cohen (2002)]. Mossman (2001) argues against hormesis but agrees that the LNT model has led to ‘enormous problems in radiation protection practice’ and unwarranted fears about radiation”

Although I find the AAPM position statement to have a slightly condescending tone, I applaud it primarily as an antidote for those “unwarranted fears about radiation”. My impression is that many in the general public have a fear of the word radiation that borders on the irrational, stemming from a lack of knowledge about the basic physics governing how radiation interacts with tissue, and a poor understanding of risk analysis. I hope the AAPM position statement (and, immodestly, our textbook) helps change those concerns from irrational fears to reasoned and fact-based assessment. I would not discourage analysis of public safety, but I definitely encourage an intelligent and scientific analysis.

Open Access

2012-01-13T06:46:00.004-05:00

The journal Medical Physics is one of the leading publications in the field of physics applied to medicine. Recently, many articles in Medical Physics have become free to everyone (open access) (see the editorial here). This is great news to those readers of the 4th edition of Intermediate Physics for Medicine and Biology who do not have a personal or institutional subscription to Medical Physics. Some of the articles that can now be downloaded for free are the ever-popular Point/Counterpoint debates, Review papers, Award papers, and something called the “Editor’s Picks”. Also available free are the special 50th anniversary articles published as part of the celebration of half a century of contributions by the American Association of Physicists in Medicine in 2008. Several of these were cited by Russ Hobbie and me in our American Journal of Physics Resource Letter MP-2: Medical Physics (Volume 77, Pages 967-978, 2009). To access this wealth of free material, just go to the home page of the Medical Physics website and click on the Open Access Tab.

Open Access publishing is becoming more common, and has been championed by many leading scientists, such as former NIH director and Nobel laureate Harold Varmus (listen to Varmus talk about open access here). Nevertheless, the topic is hotly debated. For instance, see the Point/Counterpoint discussion in the November 2005 issue of Medical Physics, titled "Results of Publicly Funded Scientific Research Should Be Immediately Available Without Cost to the Public." Additional debate can be found in the journal Nature, and at physicsworld.com.

Open Access to journal articles should benefit readers of Intermediate Physics for Medicine and Biology, because it will allow those readers immediate access to cutting-edge papers that otherwise would require a journal subscription. Another source of open access papers is BioMed Central:

"BioMed Central is an independent publishing house committed to providing immediate open access to peer-reviewed biomedical research. All original research articles published by BioMed Central are made freely and permanently accessible online immediately upon publication. BioMed Central views open access to research as essential in order to ensure the rapid and efficient communication of research findings."

BioMed Central journals that will be of interest to readers of Intermediate Physics for Medicine and Biology are BMC Medical Physics, Biomedical Engineering Online, and Radiation Oncology.

A third source of papers is the Public Library of Science. Specific journals are PLoS One (the flagship journal, covering all areas of science), PLoS Medicine, PLoS Biology, and especially PLoS Computational Biology. Also of interest is PLoS Blogs.

The Open Access movement continues, slowly but steadily, to remake scientific publication. There are now hundreds of Open Access journals. Even some of the most prestigious leading publishers are getting into the act: the American Physical Society recently initiated the open access, all on-line journal Physical Review X to go along with its other Physical Review journals.

In the spirit of Open Access, I'm pleased to announce that the 4th edition of Intermediate Physics for Medicine and Biology will now be given away, free of cha........ Just kidding. Maybe someday the Open Access movement will reach to textbooks, but not yet. At least this blog is free. ;)

Destiny of the Republic

2012-01-06T07:00:00.004-05:00

Regular readers of this blog know that I am in the habit of listening to audio books while I take my dog Suki on her daily walks. My tastes lean toward science, history, and biography, and I always keep a watch out for biological or medical physics in these books. Over the Christmas break, I listened to Destiny of the Republic: A Tale of Madness, Medicine and the Murder of a President, by Candice Millard, about the assassination of President James Garfield in 1881, shot by madman Charles Guiteau.

The book tells the fascinating story of Garfield’s nomination at the Republican National Convention in 1880, back in a time when conventions were less choreographed and predictable than they are today. Garfield nominated his fellow Ohioan John Sherman (General William Tecumseh Sherman’s brother), who was running against Senator James Blaine and former president Grant. After many ballots in which no nominee obtained a majority, the delegates turned to Garfield as their compromise choice. After being chosen the Republican nominee, he defeated Democrat and former Civil War general Winfield Scott Hancock in the general election.

A few months after being sworn in, Garfield was shot by Guiteau, who had applied for a job in the new administration but had been turned down. The bullet did not kill Garfield immediately, and he lingered on for weeks. At this point, medical physics enters the story through one of the book’s subplots about the career of Alexander Graham Bell, inventor of the telephone. Millard tells the tale of how Bell set up one of his early telephones for demonstration at the 1876 Centennial Exposition, but was ignored until a chance meeting with his acquaintance, Emperor Pedro II of Brazil, who drew attention to Bell’s display. Upon hearing that the President had been shot, Bell quickly invented a metal detector with the goal of locating the bullet still lodged in Garfield’s abdomen. The detector is based on the principle of electromagnetic induction, discussed in Section 8.6 of the 4th edition of Intermediate Physics for Medicine and Biology. A changing magnetic field induces eddy currents in a nearby conductor. These eddy currents produce their own magnetic field, which is then detected. Essentially, the device monitored changes in the inductance of the metal detector caused by the bullet. Such metal detectors are now common, particularly for nonmedical uses such as searching for metal objects buried shallowly in the ground. At the time, the device was rather novel. Michael Faraday (and, independently, Joseph Henry) had discovered electromagnetic induction in 1831, and Maxwell’s equations summarizing electromagnetic theory were formulated by James Maxwell in 1861, only twenty years before Garfield’s assassination. Being a champion of medical and biological physics, I wish I could say that Bell’s invention saved the president’s life, or at least had a positive effect during his treatment. Unfortunately, it did not, in part because of interference from metal springs in the mattress Garfield laid on, but mainly because the primary physician caring for Garfield, Dr. Willard Bliss, insisted that Bell only search the right side of the body where he believed the bullet was located, when in fact it was on the unexplored left side.

Another issue discussed in the book is the development of antiseptic methods in medicine, pioneered by Joseph Lister in the 1860s. Apparently the direct damage caused by the bullet was not life-threatening, and Millard suggests that if Garfield had received no treatment whatsoever for his wounds, he would have likely survived. Unfortunately, the doctors of that era, being skeptical or hostile to Lister’s new ideas, probed Garfield’s wound with various non-sterile instruments, including their fingers. Garfield died of an infection, possibly caused by these actions.

I enjoyed Millard’s book, and came away with a greater respect for President Garfield. Bell’s metal detector was used to locate bullets in injured soldiers throughout the rest of the 19th century, until X rays became the dominant method for finding foreign objects. It is an early example of the application of electricity and magnetism to medicine.

Click here to listen to Candice Millard speak about her book.

Wilhelm Roentgen

2011-12-30T06:57:00.005-05:00

The medical use of X rays is one of the main topics discussed in the 4th edition of Intermediate Physics for Medicine and Biology. However, Russ Hobbie and I don’t say much about the discoverer of X rays, Wilhelm Roentgen (1845-1923). Let me be more precise: we never mention Roentgen at all, despite his winning the first ever Nobel Prize in Physics in 1901. We do refer to the unit bearing his name, but in an almost disparaging way:

“Problem 8 The obsolete unit, the roentgen (R), is defined as 2.08 x 10⁹ ion pairs produced in 0.001 293 g of dry air. (This is 1 cm³ of dry air at standard temperature and pressure.) Show that if the average energy required to produce an ion pair in air is 33.7 eV (an old value), then 1 R corresponds to an absorbed does of 8.69 x 10^-3 Gy and that 1 R is equivalent to 2.58 x 10^-4 C kg^-1.”

Roentgen’s story is told in Asimov’s Biographical Encyclopedia of Science and Technology. (My daughter gave me a copy of this book for Christmas this year; Thanks, Kathy!)

“…The great moment that lifted Roentgen out of mere competence and made him immortal came in the autumn of 1895 when he was head of the department of physics at the University of Wurzburg in Bavaria. He was working on cathode rays and repeating some of the experiments of Lenard and Crookes. He was particularly interested in the luminescence these rays set up in certain chemicals.

In order to observe the faint luminescence, he darkened the room and enclosed the cathode ray tube in thin black cardboard. On November 5, 1895, he set the enclosed cathode ray tube into action and a flash of light that did not come from the tube caught his eye. He looked up and quite a distance from the tube he noted that a sheet of paper coated with barium platinocyanide was glowing. It was one of the luminescent substances, but it was luminescing now even though the cathode rays, blocked off by the cardboard, could not possibly be reaching it.

He turned off the tube; the coated paper darkened. He turned it on again; it glowed. He walked into the next room with the coated paper, closed the door, and pulled down the blinds. The paper continued to glow while the tube was in operation…

For seven weeks he experimented furiously and then, finally, on December 28, 1895 [116 years ago this week], submitted his first paper, in which he not only announced the discovery but reported all the fundamental properties of X rays...

The first public lecture on the new phenomenon was given by Roentgen on January 23, 1896. When he had finished talking, he called for a volunteer, and Kolliker, almost eighty years old at the time, stepped up. An X-ray photograph was taken of this hand—which shows the bones in beautiful shape for an octogenarian. There was wild applause, and interest in X rays swept over Europe and America.”

You can learn more about X rays in Chapter 15 (Interaction of Photons and Charged Particles with Matter) and Chapter 16 (Medical Use of X Rays) in Intermediate Physics for Medicine and Biology.

Poisson's Ratio

2011-12-23T06:53:00.005-05:00

One of the many new problems that Russ Hobbie and I added to the 4th Edition of Intermediate Physics for Medicine and Biology deals with Poisson's Ratio. From Chapter 1:

“Problem 25 Figure 1.20, showing a rod subject to a force along its length, is a simplification. Actually, the cross-sectional area of the rod shrinks as the rod lengthens. Let the axial strain and stress be along the z axis. They are related to Eq. 1.25, s_z = E ε_z. The lateral strains ε_x and ε_y are related to s_z by s_z = - (E/ν) ε_x = -(E/ν) ε_y, where ν is called the 'Poisson’s ratio' of the material.
(a) Use the result of Problem 13 to relate E and ν to the fractional change in volume ΔV/V.
(b) The change in volume caused by hydrostatic pressure is the sum of the volume changes caused by axial stresses in all three directions. Relate Poisson’s ratio to the compressibility.
(c) What value of ν corresponds to an incompressible material?
(d) For an isotropic material, -1 ≤ ν ≤ 0.5. How would a material with negative ν behave?
Elliott et al. (2002) measured Poisson’s ratio for articular (joint) cartilage under tension and found 1 ≤ ν ≤ 2. This large value is possible because cartilage is anisotropic: Its properties depend on direction.”

The citation is to a paper by Dawn Elliott, Daria Narmoneva and Lori Setton, Direct Measurement of the Poisson’s Ratio of Human Patella Cartilage in Tension, in the Journal of Biomechanical Engineering, Volume 124, Pages 223-228, 2002. (Apologies to Dr. Narmoneva, whose name was misspelled in our book. It is now corrected in the errata, available at the book website.)

As hinted at in our homework problem, a particularly fascinating type of material has negative Poisson's Ratio. Some foams expand laterally, rather than contract, when you stretch them; see Roderic Lakes, Foam Structures with a Negative Poisson’s Ratio, Science, Volume 235, Pages 1038-1040, 1987. A model for such a material is shown in this video. Lakes’ website contains much interesting information about Poisson’s ratio. For instance, cork has a Poisson ratio of nearly zero, making it ideal for stopping wine bottles.

Simeon Denis Poisson (1781-1840) was a French mathematician and physicist whose name appears several times in Intermediate Physics for Medicine and Biology. Besides Poisson’s ratio, in Chapter 9 Russ and I present the Poisson Equation in electrostatics, and its extension the Poisson-Boltzmann Equation governing the electric field in salt water. Appendix J reviews the Poisson Probability Distribution. Finally, Poisson appeared in this blog before, albeit as something of a scientific villain, in the story of Poisson’s spot. Poisson is one of the 72 names appearing on the Eiffel Tower.

Gadolinium

2011-12-16T06:20:00.004-05:00

While school children know the most famous elements listed in the periodic table (for example hydrogen, oxygen, and carbon), even many scientists are unfamiliar with those rare earth elements down at the bottom of the table, listed under the generic label of lanthanides. But one of these, gadolinium (Gd, element 64), has become crucial for modern medicine because of its use as a contrast agent during magnetic resonance imaging. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss gadolinium in Chapter 18.

“Differences in relaxation time are easily detected in an image. Different tissues have different relaxation times. A contrast agent containing gadolinium (Gd³⁺), which is strongly paramagnetic, is often used in magnetic resonance imaging. It is combined with many of the same pharmaceuticals used with ^99mTc, and it reduces the relaxation time of nearby nuclei.”

In 1999, Peter Caravan and his coworkers published a major review article about the uses of gadolinium in imaging, which has been cited over 1500 times (Gadolinium(III) Chelates as MRI Contrast Agents: Structure, Dynamics, and Applications, Chemical Reviews, Volume 99, Pages 2293-2352). The review is well written, and I reproduce the introduction below.

“Gadolinium, an obscure lanthanide element buried in the middle of the periodic table, has in the course of a decade become commonplace in medical diagnostics.
Like platinum in cancer therapeutics and technetium in cardiac scanning, the unique magnetic properties of the gadolinium(III) ion placed it right in the middle of a revolutionary development in medicine: magnetic resonance imaging (MRI). While
it is odd enough to place patients in large superconducting magnets and noisily pulse water protons in their tissues with radio waves, it is odder still to inject into their veins a gram of this potentially toxic metal ion which swiftly floats among the water
molecules, tickling them magnetically.

The successful penetration of gadolinium(III) chelates into radiologic practice and medicine as a whole can be measured in many ways. Since the approval of [Gd(DTPA)(H2O)]^2- in 1988, it can be estimated that over 30 metric tons of gadolinium have been administered to millions of patients worldwide. Currently, approximately 30% of MRI exams include the use of contrast agents, and this is projected to increase as new agents and applications arise; Table 1 lists agents currently approved or in clinical trials. In the rushed world of modern medicine, radiologists, technicians, and nurses often refrain from calling the agents by their brand names, preferring instead the affectionate 'gado.' They trust this clear, odorless 'magnetic light', one of the safest class of drugs ever developed. Aside from the cost ($50-80/bottle), asking the nurse to 'Give him some gado' is as easy as starting a saline drip or obtaining a blood sample.

Gadolinium is also finding a place in medical research. When one of us reviewed the field in its infancy, in 1987, only 39 papers could be found for that year in a Medline search for 'gado-' and MRI. Ten years later over 600 references appear each year. And as MRI becomes relied upon by different specialties, 'gado' is becoming known by neurologists, cardiologists, urologists, opthamologists, and others in search of new ways to visualize functional changes in the body.

While other types of MRI contrast agents have been approved, namely an iron particle-based agent and a manganese(II) chelate, gadolinium(III) remains the dominant starting material. The reasons for this include the direction of MRI development and the nature of Gd chelates.”

In Section 18.12 about Functional MRI, Russ and I again mention gadolinium.

"Magnetic resonance imaging provides excellent structural information. Various contrast agents can provide information about physiologic function. For example, various contrast agents containing gadolinium are injected intravenously. They leak through a damaged blood-tissue barrier and accumulate in the damaged region. At small concentrations T1 is shortened."

Here at Oakland University, several of our Biomedical Sciences: Medical Physics PhD students study brain injury using this method. See, for instance, the dissertation Magnetic Resonance Imaging Investigations of Ischemic Stroke, Intracerebral Hemorrhage and Blood-Brain Barrier Pathology by Karki, Kishor, 2009.

The Cyclotron

2011-12-09T06:28:00.004-05:00

The 4th edition of Intermediate Physics for Medicine and Biology has its own Facebook group, and any readers of this blog who use Facebook are welcome to join. One nice feature of Facebook is that is encourages comments, such as the recent one that asked “Why isn't there a chapter or a subchapter in the textbook 'Intermediate physics for medicine and biology' that refers to the fundamental concepts of the cyclotron and the betatron and how are they used in medicine?” This is a good question, because undoubtedly cyclotrons are important in nuclear medicine. I can’t do anything to change the 4th edition of our book, but this blog provides an opportunity to address such comments, and to try out possible text for a 5th edition.

Although the term does not appear in the index (oops…), the cyclotron is mentioned in Intermediate Physics for Medicine and Biology at the end of Section 17.9 (Radiopharmaceuticals and Tracers).

“Other common isotopes are ²⁰¹Tl, ⁶⁷Ga, and ¹²³I. Thallium, produced in a cyclotron, is chemically similar to potassium and is used in heart studies, though it is being replaced by ^99mTc-sestamibi and ^99mTc-tetrofosmin. Gallium is used to image infections and tumors. Iodine is also produced in a cyclotron and is used for thyroid studies.”

Cyclotrons are again mentioned in Section 17.14 (Positron Emission Tomography)

“Positron emitters are short-lived, and it is necessary to have a cyclotron for producing them in or near the hospital. This is proving to be less of a problem than initially imagined. Commercial cyclotron facilities deliver isotopes to a number of nearby hospitals. Patterson and Mosley (2005) found that 97% of the people in the United States live within 75 miles of a clinical PET facility.”

(Note: on page 513 of our book, we omitted the word “emission” from the phrase “positron emission tomography” in the title of the Patterson and Mosley paper; again, oops…)

Perhaps the best place in Intermediate Physics for Medicine and Biology to discuss cyclotrons would be after Section 8.1 (The Magnetic Force on a Moving Charge). Below is some sample text that serves as a brief introduction to cyclotrons.

8.1 ½ The Cyclotron

One important application of magnetic forces in medicine is the cyclotron. Many hospitals have a cyclotron for the production of radiopharmaceuticals, or for the generation of positron emitting nuclei for use in Positron Emission Tomography (PET) imaging (see Chapter 17).

Consider a particle of charge q and mass m, moving with speed v in a direction perpendicular to a magnetic field B. The magnetic force will bend the path of the particle into a circle. Newton’s second law states that the mass times the centripetal acceleration, v²/r, is equal to the magnetic force

m v²/r = q v B . (8.4a)

The speed is equal to circumference of the circle, 2 π r, divided by the period of the orbit, T. Substituting this expression for v into Eq. 8.4a and simplifying, we find

T = 2 π m/(q B) . (8.4b)

In a cyclotron particles orbit at the cyclotron frequency, f = 1/T. Because the magnetic force is perpendicular to the motion, it does not increase the particles’ speed or energy. To do that, the particles are subjected periodically to an electric field that must change direction with the cyclotron frequency so that it is always accelerating, and not decelerating, the particles. This would be difficult if not for the fortuitous disappearance of both v and r from Eq. 8.4b, so that the cyclotron frequency only depends on the charge-to-mass ratio of the particles and the magnetic field, but not on their energy.

Typically, protons are accelerated in a magnetic field of about 1 T, resulting in a cyclotron frequency of approximately 15 MHz. Each orbit raises the potential of the proton by about 100 kV, and it must circulate enough times to raise its total energy to at least 10 MeV so that it can overcome the electrostatic repulsion of the target nucleus and cause nuclear reactions. For example, the high-energy protons may be incident on a target of ¹⁸O (a rare but stable isotope of oxygen), initiating a nuclear reaction that results in the production of ¹⁸F, an important positron emitter used in PET studies.

Since Intermediate Physics for Medicine and Biology is not a history book, I didn’t mention the interesting history of the cyclotron, which was invented by Ernest Lawrence in the early 1930s, for which he received the Nobel Prize in Physics in 1939. The American Institute of Physics Center for the History of Physics has a nice website about Lawrence’s invention. The same story is told, perhaps more elegantly, in Richard Rhode’s masterpiece The Making of the Atomic Bomb (see Chapter 6, “Machines”). Lawrence played a major role in the Manhattan Project, using modified cyclotrons as massive mass spectrometers to separate the fissile uranium isotope ²³⁵U from the more abundant ²³⁸U.

Finally, I think it is appropriate that Intermediate Physics for Medicine and Biology should have a section about the cyclotron, because my coauthor Russ Hobbie (who was the sole author of the first three editions of the textbook) obtained his PhD while working at the Harvard Cyclotron. Thus, an unbroken path leads from Ernest Lawrence and the cyclotron to the publication of our book and the writing of this blog.

Feedback Loops

2011-12-02T06:26:00.003-05:00

Negative feedback is an important concept in physiology. Russ Hobbie and I discuss feedback loops in Chapter 10 of the 4th edition of Intermediate Physics for Medicine and Biology. In the text and homework problems, we discuss several examples of negative feedback, including the regulation of breathing rate by the concentration of carbon dioxide in the alveoli, the prevention of overheating of the body by sweating, and the control of blood glucose levels by insulin. You can never have enough of these examples. Therefore, here is another homework problem related to negative feedback: regulation of blood osmolarity by antidiuretic hormone. Warning: the model is greatly simplified. It should be correct qualitatively, but not accurate quantitatively.

Section 10.3

Problem 15 ½ The osmolarity of plasma (C, in mosmole) is regulated by the concentration of antidiuretic hormone (ADH, in pg/ml, also known as vasopressin). As antidiuretic hormone increases, the kidney reabsorbs more water and the plasma osmolarity decreases, C=700/ADH. When osmoreceptors in the hypothalamus detect an increase of plasma osmolarity, they stimulate the pituitary gland to produce more antidiuretic hormone, ADH = C-280 for C greater than 280, and zero otherwise.
(a) Draw a block diagram of the feedback loop, including accurate plots of the two relationships.
(b) Calculate the operating point and the open loop gain (you may need to use four to six significant figures to determine the operating point accurately).
(c) Suppose the behavior of the kidney changed so now C=750/ADH. First determine the new value of C if the regulation of ADH is not functioning (ADH is equal to that found in part b), and then determine the value of C taking regulation of ADH by the hypothalamus into account.

You should find that this feedback loop is very effective at holding the blood osmolarity constant. For more about osmotic effects, see Chapter 5 of Intermediate Physics for Medicine and Biology.

Here is how Guyton and Hall describe the physiological details of this feedback loop in their Textbook of Medical Physiology (11th edition):

“When osmolarity (plasma sodium concentration) increases above normal because of water deficit, for example, this feedback system operates as follows:

1. An increase in extracellular fluid osmolarity (which in practical terms means an increase in plasma sodium concentration) causes the special nerve cells called osmoreceptor cells, located in the anterior hypothalamus near the supraoptic nuclei, to shrink.

2. Shrinkage of the osmoreceptor cells casuse them to fire, sending nerve signals to additional nerve cells in the supraoptic nuclei, which then relay these signals down the stalk of the pituitary gland to the posterior pituitary.

3. These action potentials conducted to the posterior pituitary stimulate the release of ADH, which is stored in secretory granules (or vesicles) in the nerve endings.

4. ADH enters the blood stream and is transported to the kidneys, where it increases the water permeability of the late distal tubules, cortical collecting tubules, and the medullary collecting ducts.

5. The increased water permeability in the distal nephron segments causes increased water reabsorption and excretion of a small volume of concentrated urine.

Thus, water is conserved in the body while sodium and other solutes continue to be excreted in the urine. This causes dilution of the solutes in the extracellular fluid, thereby correcting the initial excessively concentrated extracellular fluid.”

Feedback loops are central to physiology. Guyton and Hall write in their first introductory chapter

“Thus, one can see how complex the feedback control systems of the body can be. A person’s life depends on all of them. Therefore, a major share of this text is devoted to discussing these life-giving mechanisms.”

The Second Law of Thermodynamics

2011-11-25T05:26:00.005-05:00

Russ Hobbie and I discuss thermodynamics in Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology. We take a statistical perspective (similar to that used so effectively by Frederick Reif in Statistical Physics, which is Volume 5 of the Berkeley Physics Course), and discuss many topics such as heat, temperature, entropy, the Boltzmann factor, Gibbs free energy, and the chemical potential. But only at the very end of the chapter do we mention the central concept of thermodynamics: The Second Law.

“In some cases, thermal energy can be converted into work. When gas in a cylinder is heated, it expands against a piston that does work. Energy can be supplied to an organism and it lives. To what extent can these processes, which apparently contradict the normal increase of entropy, be made to take place? The questions can be stated in a more basic form.

1. To what extent is it possible to convert internal energy distributed randomly over many molecules into energy that involves a change of a macroscopic parameter of the system? (How much work can be captured from the gas as it expands the piston?)

2. To what extent is it possible to convert a random mixture of simple molecules into complex and highly organized macromolecules?

Both these questions can be reformulated: under what conditions can the entropy of a system be made to decrease?

The answer is that the entropy of a system can be made to decrease if, and only if, it is in contact with one or more auxiliary systems that experience at least a compensating increase in entropy. Then the total entropy remains the same or increases. This is one form of the second law of thermodynamics. For a fascinating discussion of the second law, see Atkins (1994).”

The book by Peter Atkins, The Second Law, is published by the Scientific American Library, and is aimed at a general audience. It is a wonderful book, and provides the best non-mathematical description of thermodynamics I know of. Atkins’ preface begins

“No other part of science has contributed as much to the liberation of the human spirit as the Second Law of thermodynamics. Yet, at the same time, few other parts of science are held to be so recondite. Mention of the Second Law raises visions of lumbering steam engines, intricate mathematics, and infinitely incomprehensible entropy. Not many would pass C. P. Snow’s test of general literacy, in which not knowing the Second Law is equivalent to not having read a work of Shakespeare.

In this book I hope to go some way toward revealing the workings of the Law, and showing its span of application. I start with the steam engine, and the acute observations of the early scientists, and I end with a consideration of the processes of life. By looking under the classical formulation of the Law we see its mechanism. As soon as we do so, we realize how simple it is to comprehend, and how wide is its application. Indeed, the interpretation of the Second Law in terms of the behavior of molecules is not only straightforward (and in my opinion much easier to understand that the First Law, that of the conservation of energy), but also much more powerful. We shall see that the insight it provides lets us go well beyond the domain of classical thermodynamics, to understand all the processes that underlie the richness of the world.”

Atkins’s book is at the level of a Scientific American article, with many useful (and colorful) pictures and historical anecdotes. The writing is excellent. For instance, consider this excerpt:

“The Second Law recognizes that there is a fundamental dissymmetry in Nature…hot objects cool, but cool objects do not spontaneously become hot; a bouncing ball comes to rest, but a stationary ball does not spontaneously begin to bounce. Here is the feature of Nature that both Kelvin and Clausius disentangled from the conservation of energy: although the total quantity of energy must be conserved in any process…, the distribution of that energy changes in an irreversible manner….”

I particularly like Atkins’ analysis of the equivalence of two statements of the Second Law: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work (Kelvin statement); and No process is possible in which the sole result is the transfer of energy from a cooler to a hotter body (Clausius statement). Atkins writes

‘The Clausius statement, like the Kelvin statement, identifies a fundamental dissymmetry in Nature, but ostensibly a different dissymmetry. In the Kelvin statement the dissymmetry is that between work and heat; in the Clausius statement there is no overt mention of work. The Clausius statement implies a dissymmetry in the direction of natural change: energy may flow spontaneously down the slope of temperature, not up. The twin dissymmetries are the anvils on which we shall forge the description of all natural change.”

Peter Atkins has written several books, including another of my favorites: Peter Atkins’ Molecules. Here is a video of Atkins discussing his book the Four Laws that Drive the Universe. Not surprisingly, the four laws are the laws of thermodynamics.

Plessey Semiconductor Electric Potential Integrated Circuit

2011-11-18T06:36:00.006-05:00

The electrocardiogram, or ECG, is one of the most common and useful tools for diagnosing heart arrhythmias. Russ Hobbie and I discuss the ECG in Chapter 7 (The Exterior Potential and the Electrocardiogram) of the 4th edition of Intermediate Physics for Medicine and Biology. The November issue of the magazine IEEE Spectrum contains an article by Willie D. Jones about new instrumentation for measuring the ECG. Jones writes

“In October, Plessey Semiconductors of Roborough, England, began shipping samples of its Electric Potential Integrated Circuit (EPIC), which measures minute changes in electric fields. In videos demonstrating the technology, two sensors placed on a person’s chest delivered electrocardiogram (ECG) readings. No big deal, you say? The sensors were placed on top of the subject’s sweater, and in future iterations, the sensors could be integrated into clothes or hospital gurneys so that vital signs could be monitored continuously—without cords, awkward leads, hair-pulling sticky tape, or even the need to remove the patient’s clothes.”

Apparently the Plessey device is an ultra high input impedance voltmeter. The electrode is capacitively coupled to the body, so no electrical contact is necessary. You can learn more about it by watching this video. I don’t want to sound like an advertisement for Plessey Semiconductors, but I think this device is neat. (I have no relationship with Plessey, and I have no knowledge of the quality of their product, other than what I saw in the IEEE Spectrum article and the video that Plessey produced.)

According to the Plessey press release, “most places on earth have a vertical electric field of about 100 Volts per metre. The human body is mostly water and this interacts with the electric field. EPIC technology is so sensitive that it can detect these changes at a distance and even through a solid wall.”

I don’t have any inside information about this device, but let me guess how it can detect a person at a distance. The body would perturb a surrounding electric field because it is mostly saltwater, and therefore a conductor. In Section 9.10 of Intermediate Physics for Medicine and Biology, Russ and I explain how a conductor interacts with applied electric fields. For the case of a dc field, the conducting tissue completely shields the interior of the body from the field. To understand how a body could affect an electric field, try solving the following new homework problem

Section 9.10

Problem 34 ½ Consider how a spherical conductor, of radius a, perturbs an otherwise uniform electric field, E_o. The conductor is at a uniform potential, which we take as zero. As in Problem 34, assume that the electric potential V outside the conductor is V = A cosθ/r² – E_o r cosθ.
(a) Use the boundary condition that the potential is continuous at r=a to determine the constant A.
(b) In the direction θ=0, determine the upward component of the electric field, - dV/dr.
(c) The perturbation of the electric field by the conductor is the difference between the fields with and without the conductor present. Calculate this difference. How does it depend on r?
(d) Suppose you measure the voltage in two locations separated by 10 cm, and that your detector can reliably detect voltage differences of 1 mV. How far from the center of a 1 m radius conductor can you be (assuming θ=0) and still detect the perturbation caused by the conductor?

You may be wondering why there is a 100 V/m electric field at the earth’s surface. The Feynman Lectures (Volume 2, Chapter 9) has a nice discussion about electricity in the atmosphere. The reason that this electric field exists is complicated, and has to do with 1) charging of the earth by lightning, and 2) charge separation in falling raindrops.

The Making of the Pacemaker: Celebrating a Lifesaving Invention

2011-11-11T05:20:00.003-05:00

I’m still thinking about Wilson Greatbatch, one of the inventors of the implantable pacemaker, who died a few weeks ago (see my September 30 blog entry honoring him). Here is an interesting excerpt from his book The Making of the Pacemaker: Celebrating a Lifesaving Invention, about how he created the circuit in the first pacemaker.

“My marker oscillator used a 10k basebias resistor. I reached into my resistor box for one but misread the colors and got a brown-black-green (one megohm) instead of a brown-black-orange. The circuit started to ‘squeg’ with a 1.8 ms pulse, followed by a one second quiescent interval. During the interval, the transistor was cut off and drew practically no current. I stared at the thing in disbelief and then realized that this was exactly what was needed to drive a heart. I built a few more. For the next five years, most of the world’s pacemakers used a blocking oscillator with a UTC DOT-1 transformer, just because I grabbed the wrong resistor.”

Here is another story from the book about how he met William Chardack, his primary collaborator in developing the first pacemaker.

“In Buffalo we had the first local chapter in the world of the Institute of Radio Engineers, Professional Group in Medical Electronics (the IRE/PBME, now the Biomedical Engineering Society of the Institute of Electrical and Electronic Engineers [IEEE]). Every month twenty-five to seventy-five doctors and engineers met for a technical program. We strove to attract equal numbers of doctors and engineers. We had a standing offer to send an engineering team to assist any doctor who had an instrumentation problem. I went with one team to visit Dr. Chardack on a problem deadline with a blood oximeter. Imagine my surprise to find that his assistant was my old high school classmate, Dr. Andrew Gage. We couldn’t help Dr. Chardack much with his oximeter problem, but when I broached my pacemaker idea to him, he walked up and down the lab a couple times, looked at me strangely, and said, ‘If you can do that, you can save ten thousand lives a year.’ Three weeks later we had our first model implanted in a dog."

This excerpt is interesting:

“I had $2,000 in cash and enough set aside to feed my family for two years. I put it to the Lord in prayer and felt led to quit all my jobs and devote my time to the pacemaker. I gave the family money to my wife. I then took the $2,000 and went up into my wood-heated barn workshop. In two years I built fifty pacemakers, forty of which went into animals and ten into patients. We had no grant funding and asked for none. The program was successful. We got fifty pacemakers for $2,000. Today, you can’t buy one for that.”

This one may be my favorite. You gotta love Eleanor. They were married in 1945 and stayed together until her death in January of this year.

“Many of the early Medtronic programs were first worked out in Clarence, New York, and then taken to Minneapolis. I had two ovens set up in my bedroom. My wife did much of the testing. The shock test consisted of striking the transistor with a wooden pencil while measuring beta (current gain). We found that a metal pencil could wreck the transistor, but a wooden pencil could not. Many mornings I would awake to the cadence of my wife Eleanor tap, tap, tapping the transistors with her calibrated pencil. For some months every transistor that was used worldwide in Medtronic pacemakers got tapped in my bedroom.”

You can learn more about pacemakers and defibrillators in the 4th edition of Intermediate Physics for Medicine and Biology.

Countercurrent Heat Exchange

2011-11-04T06:32:00.006-04:00

Problem 17 in Chapter 5 of the 4th edition of Intermediate Physics for Medicine and Biology considers a countercurrent heat exchanger. Countercurrent transport in general is discussed in Section 5.8 in terms of the movement of particles. However, Russ Hobbie and I conclude the section by applying the concept to heat exchange.

“The principle [of countercurrent exchange] is also used to conserve heat in the extremities—such as a person’s arms and legs, whale flippers, or the leg of a duck. If a vein returning from an extremity runs closely parallel to the artery feeding the extremity, the blood in the artery will be cooled and the blood in the vein warmed. As a result, the temperature of the extremity will be lower and the heat loss to the surroundings will be reduced."

Problem 17 provides an example of this behavior, and cites Knut Schmidt-Nielsen’s book How Animals Work (1972, Cambridge University Press), which describes countercurrent exchange in more detail. (His comments below about the nose refer to an earlier section of the book, in which Schmidt-Nielsen discusses heat exchange in the nose of the kangaro o rat).

“The heat exchange in the nose has a great similarity to the well-known countercurrent heat exchange which takes place, for example, in the extremities of many aquatic animals, such as in the flippers of whales and the legs of wading birds. The body of a whale that swims in water near the freezing point is well insulated with blubber, but the thin streamlined flukes and flippers are uninsulated and highly vascularized and would have an excessive heat loss if it were not for the exchange of heat between arterial and venous blood in these structures. As the cold venous blood returns to the body from the flipper, the vessels run in close proximity to the arteries, in fact, they completely surround the artery, and heat from the arterial blood flows into the returning venous blood, which is thus reheated before it returns to the body (figure 3). Similarly, in the limbs of many animals both arteries and veins split up into a large number of parallel, intermingled vessels each with a diameter of about 1 mm or so, forming a discrete vascular bundle known as a rete…Whether the blood vessels form such a rete system, or in some other way run in close proximity, as in the flipper of the whale, is a question of design and does not alter the principle of the heat recovery mechanism. The blood flows in opposite directions in the arteries and veins, and heat exchange takes place between the two parallel sets of tubes; the system is therefore known as a countercurrent heat exchanger.”

Schmidt-Nielsen also wrote Scaling: Why is Animal Size So Important?, which Russ and I cite often in Chapter 2 and which I included in my top ten list of biological physics books. I have also read Schmidt-Nielsen's autobiography The Camel’s Nose: Memoirs of a Curious Scientist. (See the review of this book in the New England Journal of Medicine.) His Preface begins

“This is a personal story of a life spent in science. It tells about curiosity, about finding out and finding answers. The questions I have tried to answer have been very straightforward, perhaps even simple. Do marine birds drink sea water? How do camels in hot deserts manage for days without drinking when humans could not survive without water for more than a day? How can kangaroo rats live in the desert without any water to drink? How can snails find water and food in the most barren deserts? Can crab-eating frogs really survive in sea water?

These are important questions. The answers not only tell us how animals overcome seemingly insurmountable obstacles in hostile environments; they also give us insight into general principles of life and survival.”

Schmidt-Nielsen died in 2007, and Steven Vogel (who I quoted in last week’s blog entry) wrote an article about him for the Biographical Memoirs of Fellows of the Royal Society (volume 54, pages 319-331, 2008). See also his obituar y in the Journal of Experimental Biology. A statue of Schmidt-Nielsen with a camel (which he famously studied) graces the Duke University campus.

Murray’s Law

2011-10-28T06:57:00.004-04:00

Homework Problem 33 in Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology is about Murray’s law, a relationship describing the radii of branching vessels.

“A parent vessel of radius R_p branches into two daughter vessels of radii R_d1 and R_d2. Find a relationship between the radii such that the shear stress on the vessel wall is the same in each vessel. (Hint: Use conservation of the volume flow.) This relationship is called ‘Murray’s Law’. Organisms may use shear stress to determine the appropriate size of vessels for fluid transport [LaBarbera (1990)].”

The reference is to

LaBarbera, M. (1990). Principles of design of fluid transport systems in zoology. Science, 249:992-1000.

In his book Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems, Steven Vogel provides a clear and engaging discussion of Murray’s law.

“Our problem of figuring the cheapest arrangement of pipes turns out to involve nothing more nor less than calculating the relative dimensions of pipes so that the steepness of the speed gradient at all walls is the same. This calculation was done by Cecil D. Murray, of Bryn Mawr College, back in 1926, and is spoken of, when (uncommonly) it’s mentioned, as ‘Murray’s law’.

Murray’s law isn’t especially complicated, and anyone with a hand calculator can play around with it (but you can ignore the specifics without missing the present message). The rule is that the cube of the radius of the parental vessel equals the sum of the cubes of the radii of the daughter vessels. If a pipe with a radius of two units splits into a pair of pipes, each of the pair ought to have a radius of about 1.6 units. (To check, cube 1.6 and then double the result—you get about 2 cubed.) The daughters are smaller, but only a little (Figure 5.6). Still, if the parental one eventually divides into a hundred progeny, the progeny do come out substantially smaller, each about a fifth of the radius of the parent. (Their aggregate cross-section area is, of course, greater than the parental one—to be specific, four fold greater.)

The relationship predicts the relative sizes of both our arteries and our veins quite well. It only fails for the very smallest arterioles and capillaries….

It would be indefensibly anthropocentric to suppose that we’re the only creatures to follow Mr. Murray. My friend, Michael LaBarbera (who introduced me to the whole issue) has tested the law on several systems that are very unlike us structurally and functionally, and very distant from us evolutionarily…Murray’s law again proves applicable…

The mechanism … is becoming clear. Without getting into the details, it looks as if the cells lining the blood vessels can quite literally sense changes in the speed gradient next to them. An increase in the speed of flow through a vessel increases the speed gradient at its walls. An increase in gradient stimulates cell division, which would increase vessel diameter as appropriate to offset the faster flow. Neither change in blood pressure nor cutting the nerve supply makes any difference—this is apparently a direct effect of the gradient on synthesis of some chemical signal by the cells. Perhaps the neatest feature of the scheme is that a cell needn’t know anything about the size of the vessel of which it’s a part. As a consequence of Murray’s Law, it can be given the same specific instruction wherever it might be located, a command telling it to divide when the speed gradient exceeds a specific value.”

Vogel is a faculty member in the Biology Department at Duke University. He has published several fine books, including Vital Circuits quoted above and the delightful Life in Moving Fluids (Princeton University Press, 1994), both cited in Intermediate Physics for Medicine and Biology.

A Useful Website

2011-10-21T06:30:00.003-04:00

While I have many goals when writing this blog (with the top being to sell textbooks!), sometimes I simply like to point out useful websites relevant to readers of the 4th edition of Intermediate Physics for Medicine and Biology. One example is the website of Rob MacLeod, a professor of bioengineering at the University of Utah. MacLeod’s research, like mine, centers on the numerical simulation of cardiac electrophysiology, so we find many of the same topics interesting.

I particularly enjoy his list of “Background Links for Rob’s Courses”. You will find many books listed, some of which Russ Hobbie and I cite in Intermediate Physics for Medicine and Biology, and some that we don’t cite but should. For example, MacLeod speaks highly of the book Mathematical Physiology by Keener and Sneyd, but somehow Russ and I never reference it. I didn’t know Malmivuo and Plonsey’s book Bioelectromagnetism (which we do cite) is now available online and free of charge. The Welcome Trust Heart Atlas is beautiful, as is the Virtual Heart website. MacLeod’s list of books about “Cardiology and Medicine” look fascinating, with a heavy emphasis on the relevant history and biography. If I start running out of topics for these blog posts, I could probably find a year of material by exploring the sources listed on this page.

If you visit MacLeod's website (and I hope you do), make sure to click on the link “Information on Writing”. I am an admirer of good writing, especially in nonfiction, and am frustrated when presented with a poorly written scientific book or paper. (I review a lot of papers for journals, and often find myself venting and fuming.) My advice to a young scientist is: Learn To Write. Throughout your scientific career you will be judged primarily on your papers and your grant proposals, which are both written documents. Maybe your science is so good that it can overcome poor writing and still impress the reader, but I doubt it. Learn to write.

Bethesda

2011-10-14T06:44:00.003-04:00

A couple months ago I went to Bethesda, Maryland to review grant proposals for the National Institutes of Health. They swear us to secrecy, so I can’t divulge any details about the specific research. But I will share a few general observations.

Winston Churchill said that “Democracy is the worst form of government except all the others that have been tried.” That sums up my opinion of the NIH review process. There are all sorts of problems with the way we select the best research to fund, but I cannot think of a better way than that used by NIH. Each time I participate, I come away with a great respect for the process. Of course, from the outside the review process can resemble a casino, but I don’t see how you can eliminate some randomness while at the same time keeping the process fair, with wide input, and a focus on the significance and impact of the research.
If you are a young biomedical researcher, or hope to be one someday, then you should take advantage of any opportunity to review grant proposals. It is like going to grant writing school. No book, no website, no video, no workshop is more useful for learning how to prepare a proposal. It is a lot of work, but you will gain much, especially the first time or two you do it. However, if you simply are not able to participate in a review panel, then at least watch this video, which is a fairly accurate description of what goes on.
After reviewing grant proposals, I am optimistic about the future of the scientific enterprise in the United States, because of all the fascinating and important research being proposed. I am also pessimistic about my chances for winning additional funding, because the competition is so fierce. But, we must soldier on. To quote Churchill again, “Never give in, never give in, never, never, never, never.” So I’ll keep trying.
Research is becoming more and more interdisciplinary, and many proposals now come from multidisciplinary teams. Each individual researcher cannot know everything, but they must know enough to understand each other, and to talk to each other intelligently. I believe this is one of the virtues of the 4th edition of Intermediate Physics for Medicine and Biology. It helps bridge the gap between physicists and engineers on the one side, and biologists and medical doctors on the other. The book won’t turn a physicist into a biologist, but it may help a physicist talk to and better appreciate a biologist. This is crucial for performing modern collaborative research, and for obtaining funding to pay for that research. After reviewing all those proposals, I came away proud of our textbook.

We finished our review session a couple hours earlier than anticipated, so I used the time to visit the new Martin Luther King Memorial in Washington, DC. It is just across the tidal basin from the Jefferson Memorial, and the statues of King and Jefferson stare at each other across the water. If you happen to be going to DC soon, prepare yourself for a shock. The beautiful Reflecting Pool between the Washington Monument and the Lincoln Memorial is now a dried up, plowed up mud flat. Apparently they are renovating it. But the other attractions are as beautiful as ever, including the Vietnam Veterans Memorial, the Korean War Veterans Memorial, the National World War II Memorial, and the Franklin Delano Roosevelt Memorial. I even saw one I had somehow missed in previous visits: the George Mason Memorial, near the Jefferson Memorial. All this site seeing was a little bonus after reviewing all those grants (packed into two frantic hours between leaving the review session and reaching the airport).

The Mathematics of Diffusion

2011-10-07T06:53:00.005-04:00

Diffusion is one of those topics that is rarely covered in an introductory physics class, but is essential for understanding biology. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss diffusion and its biomedical applications. One of the books we cite is The Mathematics of Diffusion by John Crank. Hard-core mathematical physicists who are interested in biology and medicine will find Crank’s book to be a good fit. Physiologists who want to avoid as much mathematical analysis as possible may prefer to learn their diffusion from Random Walks in Biology, by Howard Berg.

Crank died five years ago this week. Like Wilson Greatbatch, who I discussed in my last blog entry, Crank was one of those scientists who came of age serving in the military during World War Two (Tom Brokaw would call them members the “Greatest Generation”). Crank’s 2006 obituary in the British newspaper The Telegraph states:

“John Crank was born on February 6 1916 at Hindley, Lancashire, the only son of a carpenter's pattern-maker. He studied at Manchester University, where he gained his BSc and MSc. At Manchester he was a student of the physicist Lawrence Bragg, the youngest-ever winner of a Nobel prize, and of Douglas Hartree, a leading numerical analyst.

Crank was seconded to war work during the Second World War, in his case to work on ballistics. This was followed by employment as a mathematical physicist at Courtaulds Fundamental Research Laboratory from 1945 to 1957. He was then, from 1957 to 1981, professor of mathematics at Brunel University (initially Brunel College in Acton).

Crank published only a few research papers, but they were seminal. Even more influential were his books. His work at Courtaulds led him to write The Mathematics of Diffusion, a much-cited text that is still an inspiration for researchers who strive to understand how heat and mass can be transferred in crystalline and polymeric material. He subsequently produced Free and Moving Boundary Problems, which encompassed the analysis and numerical solution of a class of mathematical models that are fundamental to industrial processes such as crystal growth and food refrigeration.”

Crank is best known for a numerical technique to solve equations like the diffusion equation, developed with Phyllis Nicolson and known as the Crank-Nicolson method. The algorithm has the advantage that it is numerically stable, which can be shown using von Neuman stability analysis. They published this method in a 1947 paper in the Proceedings of the Cambridge Philosophical Society:

Crank, J., and P. Nicolson (1947) A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc. 43:50–67.

Rather than describe the Crank-Nicolson method, I will let the reader explore it in a new homework problem.

Section 4.8

Problem 24 ½ The numerical approximation for the diffusion equation, derived as part of Problem 24, has a key limitation: it is unstable if the time step is too large. This problem can be avoided using the Crank-Nicolson method. Replace the first time derivative in the diffusion equation with a finite difference, as was done in Problem 24. Next, replace the second space derivative with the finite difference approximation from Problem 24, but instead of evaluating the second derivative at time t, use the average of the second derivative evaluated at times t and t+Δt.
(a) Write down this numerical approximation to the diffusion equation, analogous to Eq. 4 in Problem 24.
(b) Explain why this expression is more difficult to compute than the expression given in the first two lines of Eq. 4. Hint: consider how you determine C(t+Δt) once you know C(t).
The difficulty you discover in part (b) is offset by the advantage that the Crank-Nicolson method is stable for any time step. For more information about the Crank-Nicolson method, stability, and other numerical issues, see Press et al. (1992).

The citation is to my favorite book on computational methods: Numerical Recipes (of course, the link is to the FORTRAN 77 version, which is the edition that sits on my shelf).

Wilson Greatbatch (1919-2011)

2011-09-30T06:45:00.006-04:00

This week we lost a giant of engineering: Wilson Greatbatch, inventor of the implantable cardiac pacemaker.

The cardiac pacemaker represents one of the most important contributions of physics and engineering to medicine. In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the pacemaker:

“Cardiac pacemakers are a useful treatment for certain heart diseases [Jeffrey (2001); Moses et al. (2000); Barold (1985)]. The most frequent are an abnormally slow pulse rate (bradycardia) associated with symptoms such as dizziness, fainting (syncope), or heart failure. These may arise from a problem with the SA node (sick sinus syndrome) or with the conduction system (heart block)….

A pacemaker can be used temporarily or permanently. The pacing electrode can be threaded through a vein from the shoulder to the right ventricle (transvenous pacing, Fig. 7.31) or placed directly in the myocardium during heart surgery.”

Several years ago, I taught a class about Pacemakers and Defibrillators as part of Oakland University’s honors college. The class was designed to challenge our top undergraduates, but not necessarily those majoring in science. Among the readings for the class was a profile in the March 1995 issue of IEEE Spectrum about Wilson Greatbatch (Volume 32, Pages 56-61). The article tells the story of Greatbatch’s first implantable pacemaker:

“Greatbatch was on one team that had been summoned by William C. Chardack, chief of surgery at Buffalo's Veteran's Administration Hospital, to deal with a blood oximeter. The engineers could not help with that problem, but the meeting for the inventor was momentous: finally, after many previous attempts, he had met a surgeon who was enthusiastic about prospects for an implantable pacemaker. The surgeon estimated such a device might save 10000 lives a year.

Three weeks later, on May 7, 1958, the engineer brought what would become the worlds first implantable cardiac pacemaker to the animal lab at Chardack's hospital. There Chardack and another surgeon, Andrew Gage, exposed the heart of a dog, to which Greatbatch touched the two pacemaker wires. The heart proceeded to beat in synchrony with the device, made with two Texas Instruments 910 transistors. Chardack looked at the oscilloscope, looked back at the animal, and said, ‘Well, I'll be damned.’ "

Another source the honors college students studied from was Kirk Jeffrey’s excellent book “Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care.” Jeffrey tells the long history of how pacemakers and defibrillators were developed. In a chapter titled “Multiple Invention of Implantable Pacemakers” he describes Greatbatch’s contributions as well as others, including Elmqvist and Senning in Sweeden. Jeffrey writes

“If theirs [Chardack and Greatbatch] was not the only pacemaker of the 1950s, it appears to be the only one that survives today in the collective memory of the community of physicians, engineers, and businesspeople whose careers are tied to the pacemaker…The Chardack-Greatbatch pacamaker stood out from other prototype implantables of the late 1950s not because it was first or clearly a better design, but because it succeeded in the U.S. market as did no other device.”

Jeffrey also discusses at length Greatbatch’s contributions to developing the lithium battery.

“Because of his prestige in the pacing community and his effectiveness as a champion of technology be believed in, Greatbatch was able almost single-handedly to turn the industry to lithium; in fact by 1978, a survey of pacing practices indicated that only 5 percent of newly implanted pulse generators still used mercury-zinc batteries.”

Greatbatch was inducted into the National Inventor’s Hall of Fame in 1986. His citation says:

“Wilson Greatbatch invented the cardiac pacemaker, an innovation selected in 1983 by the National Society of Professional Engineers as one of the two major engineering contributions to society during the previous 50 years. Greatbatch has established a series of companies to manufacture or license his inventions, including Greatbatch Enterprises, which produces most of the world's pacemaker batteries.

Invention Impact

His original pacemaker patent resulted in the first implantable cardiac pacemaker, which has led to heart patient survival rates comparable to that of a healthy population of similar age.

Inventor Bio

Born in Buffalo, New York, Greatbatch received his preliminary education at public schools in West Seneca, New York. In 1936 he entered military service and served in the Atlantic and Pacific theaters during World War II. He was honorably discharged with the rating of aviation chief radioman in 1945. He attended Cornell University and graduated with a B.E.E. in electrical engineering in 1950. Greatbatch received a master's from the State University of New York at Buffalo in 1957 and was awarded honorary doctor's degrees from Houghton College in 1970 and State University of New York at Buffalo in 1984. Although trained as an electrical engineer, Greatbatch has primarily studied interdisciplinary areas combining engineering with medical electronics, agricultural genetics, the electrochemistry of pacemaker batteries, and the electrochemical polarization of physiological electrodes.”

Below are some links related to Wilson Greatbatch that you might find useful.

A video honoring Wilson Greatbatch, the 1996 Lemelson-MIT Lifetime Achievement Award Winner: http://www.youtube.com/watch?v=WLZBl118Ads&list=PLE9482E912FAB47BA&index=4

An article about Greatbatch published by the Lemelson Center for the Study of Invention and Innovation: http://invention.smithsonian.org/centerpieces/ilives/lecture09.html

A video about Greatbatch produced by the Vega Science Trust: http://www.vega.org.uk/video/programme/248

Biography of Wilson Greatbatch on the Heart Rhythm Society website:
http://www.hrsonline.org/News/ep-history/notable-figures/wilsongreatbatch.cfm

New York Times obituary: http://www.nytimes.com/2011/09/28/business/wilson-greatbatch-pacemaker-inventor-dies-at-92.html

BBC obituary: http://www.bbc.co.uk/news/world-us-canada-15085056

Optical Mapping

2011-09-23T06:33:00.007-04:00

In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I mention an optical technique that is used to measure the transmembrane potential in the heart.

“Experimental measurements of the transmembrane potential often rely on the use of a voltage sensitive dye whose fluorescence changes with the transmembrane potential [Knisley et al. (1994); Neunlist and Tung (1995); Rosenbaum and Jalife (2001)].”

This method, often called optical mapping, has revolutionized cardiac electrophysiology, because it allows you to use optical methods to make electrical measurements. If you want to learn more, take a look at the book Optical Mapping of Cardiac Excitation and Arrhythmias, by David Rosenbaum and Jose Jalife (2001). The chapters in this book were written by the stars of this field.

Optical Mapping: Background and Historical Perspective. Guy Salama.
Mechanisms and Principles of Voltage-Sensitive Fluorescence. Leslie M. Loew.
Optical Properties of Cardiac Tissue. William T. Baxter.
Optics and Detectors Used in Optical Mapping. Kenneth R. Laurita and Imad Libbus.
Optimization of Temporal Filtering for Optical Transmembrane Potential Signals. Francis X. Witkowski, Patricia A. Penkoske, and L. Joshua Leon.
Optical Mapping of Impulse Propagation within Cardiomyocytes. Herbert Windisch.
Optical Mapping of Impulse Propagation between Cardiomyocytes. Stephan Rohr and Jan P. Kucera.
Role of Cell-to-Cell Coupling, Structural Discontinuities, and Tissue Anisotropy in Propagation of the Electrical Impulse. André G. Kléber, Stephan Rohr, and Vladimir G. Fast.
Optical Mapping of Impulse Propagation in the Atrioventricular Node: 1. Todor N. Mazgalev and Igor R. Efimov.
Optical Mapping of Impulse Propagation in the Atrioventricular Node: 2. Guy Salama and Bum-Rak Choi.
Optical Mapping of Microscopic Propagation: Clinical Insights and Applications. Albert L. Waldo.
Mapping Arrhythmia Substrates Related to Repolarization: 1. Dispersion of Repolarization. Kenneth R. Laurita, Joseph M. Pastore, and David S. Rosenbaum.
Mapping Arrhythmia Substrates Related to Repolarization: 2. Cardiac Wavelength. Steven Girouard and David S. Rosenbaum.
Video Imaging of Cardiac Fibrillation. Richard A. Gray and José Jalife.
Video Mapping of Spiral Waves in the Heart. William T. Baxter and Jorge M. Davidenko.
Video Imaging of Wave Propagation in a Transgenic Mouse Model of Cardiomyopathy. Faramarz Samie, Gregory E. Morley, Dhjananjay Vaidya, Karen L. Vikstrom, and José Jalife.
Optical Mapping of Cardiac Arrhythmias: Clinical Insights and Applications. Douglas L. Packer.
Response of Cardiac Myocytes to Electrical Fields. Leslie Tung.
New Perspectives in Electrophysiology from The Cardiac Bidomain. Shien-Fong Lin and John P. Wikswo, Jr..
Mechanisms of Defibrillation: 1. Influence of Fiber Structure on Tissue Response to Electrical Stimulation. Stephen B. Knisley.
Mechanisms of Defibrillation: 2. Application of Laser Scanning Technology. Stephen M. Dillon.
Mechanisms of Defibrillation: 3. Virtual Electrode-Induced Wave Fronts and Phase Singularities; Mechanisms of Success and Failure of Internal Defibrillation. Igor R. Efimov and Yuanna Cheng.
Optical Mapping of Cardiac Defibrillation: Clinical Insights and Applications. Douglas P. Zipes.

For those who are tired of reading, two videos have recently been published in the Journal of Visualized Experiments that explain the technique step-by-step. One video is about studying the rabbit heart and the other is about the mouse heart. These excellent video clips were filmed in the laboratory of Igor Efimov, of Washington University in Saint Louis.

My former graduate student, Debbie Janks, is now a post doc in Efimov’s lab. Regular readers of this blog may recognize Janks’ name, as she provides many insightful comments following these blog entries. Janks studied optical mapping from a theoretical perspective when she was here at Oakland University. She published a nice paper that examined the question of averaging over depth during optical mapping. The optical method does not measure the transmembrane potential at the tissue surface. Rather, light penetrates some distance into the tissue, and the optical signal is a weighted average of the transmembrane potential over depth. Janks looked at the effect of this averaging during an electrical shock. Rather than explaining the whole story, I will present it as a new homework problem. That way, you can figure it out for yourself. Enjoy.

Section 7.10

Problem 47 1/2 The signal measured during optical mapping, V, is a weighted average of the transmembrane potential, V_m(z), as a function of depth, V=∫₀^∞V_m(z)w(z)dz, where w(z) is a normalized weighting function. Suppose the light decays with depth exponentially, with an optical length constant δ. Then w(z) = exp(-z/δ)/δ. Often a shock will cause V_m(z) to fall off exponentially with depth, V_m(z)=V_o exp(-z/λ), where V_o is the transmembrane potential at the tissue surface and λ is the electrical length constant (see Sec. 6.12).
(a) Perform the required integration to find an analytical expression for the optical signal, V, as a function of V_o, δ and λ.
(b) What is V in the case δ much less than λ? Explain this result physically.
(c) What is V in the case δ much greater than λ? Explain this result physically.
(d) For which limit do you obtain an accurate measurement of the transmembrane potential at the surface, V=V_o?

In cardiac tissue, δ is usually on the order of a millimeter, whereas λ is more like a quarter of a millimeter, so averaging over depth significantly distorts the measured signal. For a more detailed analysis of this problem, see Janks and Roth (2002).

Does cell biology need physicists?

2011-09-16T06:55:00.003-04:00

The American Physical Society has an online journal, Physics, with the goal of making recent research accessible to a wide audience. The journal website states:

“Physics highlights exceptional papers from the Physical Review journals. To accomplish this, Physics features expert commentaries written by active researchers who are asked to explain the results to physicists in other subfields. These commissioned articles are edited for clarity and readability across fields and are accompanied by explanatory illustrations.”

One recent paper that caught my eye was an essay written by Charles Wolgemuth, titled “Does Cell Biology Need Physicists?”. Wolgemuth asks key questions in the introduction to his essay:

“The past has shown that cell biologists are extremely capable of making great progress without much need for physicists (other than needing physicists and engineers to develop many of the technologies that they use). Do the new data and new technological capabilities require a physicist’s viewpoint to analyze the mechanisms of the cell? Is physics of use to cell biology?”

Later in the essay, Wolgemuth asks his central question in a more specific way:

“It is possible that the physics that cells must deal with is slave to the reactions; i.e., the protein levels and kinetics of the biochemical reactions determine the behavior of the system, and any physical processes that a cell must accomplish are purely consequences of the biochemistry. Or, could it be that cellular biology cannot be fully understood without physics?”

Readers of the 4th edition of Intermediate Physics for Medicine and Biology are likely to scream “Yes!” to these questions. I too enthusiastically answer yes, but I agree with Wolgemuth that it is proper to ask such basic questions occasionally.

I should add that Russ Hobbie and I tend to look primarily at macroscopic phenomena in Intermediate Physics for Medicine and Biology, such as the biomechanics of walking with a cane, the interpretation of an electrocardiogram, or the algorithm required to form an image of the brain using a CAT scan. We occasionally look at events on the atomic scale, but for the most part we ignore molecular biophysics. Yet, the cellular scale is an interesting intermediate level that is becoming a fertile field for the applications of physics to biology. Indeed, I examined this issue when discussing the textbook Physical Biology of the Cell last year in this blog. The discussion that Russ and I give to fluid dynamics, diffusion, and bioelectricity in Intermediate Physics for Medicine and Biology is relevant to cellular topics.

To answer his question, Wolgemuth provides five examples in which physics provides key insights into cellular biology: 1) Molecular motors, 2) Cellular movement, 3) How cells swim, 4) Cell growth and division, and 5) How cells interact with the environment. One of my favorite parts of the essay is the consideration of potential pitfalls for physicists in biology.

“Fifteen years ago, around the time that I began working in biophysics, there were very few collaborations between physicists and cell biologists, especially if the physicists were theorists. Theory was, and still is to a good degree, a word that should be avoided in the presence of biologists. Those of us who use math and computers to try to understand how cells work tend to call ourselves modelers instead of theorists. My suspicion is that many of the first physicists and mathematicians who tried to develop models for how biology works attempted to be too abstract or too general. As physicists we like to try to find universal laws, and though there are undoubtedly general principles at play in cell biology, it is likely that there are no real universal rules. Evolution need not find only one way to do something but more often probably finds many. Rather than search out generalities, we will serve biology better if we deal with specifics. As Aharon Katchalsky, who is largely credited with bringing nonequilibrium thermodynamics to biology, purportedly said, ‘It is easier to make a theory of everything than a theory of something.’

In recent years, physicists have done a much better job at addressing specific problems in biology. However, there still remains a divide between the two communities. Indeed, good physical biology that comes out of the physics community often goes unnoticed or is under appreciated. The burden falls on us to properly convey our work so as to be accessible to biologists. We need to make conscious efforts at communication and dissemination of our results. Two useful approaches toward this end are to publish in broader audience journals that reach both communities, and for papers that contain theoretical analyses to provide a qualitative description of the modeling in the main text, while leaving the more mathematical details for the appendices or supplemental material (for further discussion of this topic, see Ref. [55]). It is also of prime importance to maintain and to forge new connections between physicists and biologists.”

Wolgemuth comes closest to answering his own questions near the end of the essay, where he predicts

“To be truly successful, we must provide an understanding of biology that spans the gorge from biochemistry and genetics to cellular function, and do it in such a way that our models and experiments are not only informative about physics, but directly impact biology.

Cell biology is awaiting these descriptions. And it may be that physicists are the most able to draw these connections between the protein level description of cellular biology that currently dominates and a more intuitive, yet still quantitative, description of the behavior of cells and their responses to their environments.”

Radon Transform

2011-09-09T06:35:00.003-04:00

In Chapter 12 of the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the Radon transformation. It consists of finding the projections F(θ, x’) at different angles θ from a function f(x,y). But why is it called the “Radon” transformation, and does it have anything to do with the radioactive gas radon discussed in Chapter 16?

Well, it has nothing to do with the element radon. Instead, and predictably, the term honors Johann Radon, the Austrian mathematician who investigated this transformation. In “A Tribute to Johann Radon” in the IEEE Transactions on Medical Imaging (Volume 5, Page 169, 1986, reproduced long after his death to honor his memory) Hans Hornich wrote

“With the death in Vienna on 25 May 1956 of Dr. Johann Radon, Professor of the University of Vienna, not only the mathematical world and Austrian science but also the German Mathematical Union has suffered a severe loss, as have also many other scientific bodies of which the deceased was a prominent member, and who spent most of his teaching life in German universities.

Radon was born in the small town of Tetschen in Bohemia near the border of Saxony on December 16, 1887. He studied at Vienna University where, alongside Mertens and Wirtinger, Escherisch above all was the great influence on Radon's development: Escherisch had, as one of the first in Austria, imparted to his students the world of ideas of Weierstrass and his rigorous foundations of analysis. Through Escherich, Radon was led next to variational calculus….

A few years later appeared his "Habilitationsschrift" "Theory and application of absolute additive weighting functions" (S. Ber. math. naturw., Kl. K. Akad. Wiss. Wien II Abt., vol. 122, pp. 1295-1438, 1913), which played a leading role in the development of analysis; the Radon integral and the Radon theorem laid the foundations of functional analysis. As an application Radon somewhat later treated the first and second boundary value problem of the logarithmic potential in a very general way."

The Radon transformation has important applications in medical imaging, and plays a crucial role in computed tomography, positron emission tomography, and single photon emission tomography. I found a nice layman's description of the Radon Transform in an essay at the website http://www.ams.org/samplings/feature-column/fcarc-tomography, written by Bill Casselman.

“The original example of this sort of technology [involving a collaboration between medicine and mathematics], and the ancestor of many of these technologies, is what is now called computed tomography, for which Allan Cormack, a physicist whose research became more and more mathematical as time went on, laid down the theoretical foundations around 1960. He shared the 1979 Nobel prize in medicine for his work in this field.

In fact the basic idea of tomography had been discovered for purely theoretical reasons in 1917 by the Austrian mathematician Johann Radon, and it had been rediscovered several times since by others, but Cormack was not to know this until much later than his own independent discovery. The problem he solved is this: Suppose we know all the line integrals through a body of varying density. Can we reconsruct the body itself? The answer, perhaps surprisingly, is that we can, and furthermore we can do so constructively. In practical terms, we know that a single X-ray picture can give only limited information because certain things are obscured by other, heavier things. We might take more X-ray pictures in the hope that we can somehow see behind the obscuring objects, but it is not at all obvious that by taking a lot - really, a lot - of X-ray pictures we can in effect even see into objects, which is what Radon tells us, at least in principle. Making Radon's theorem into a practical tool was not a trivial matter.”

You can listen to a lecture on tomography and inverting the Radon transform here.

Fraunhofer Diffraction

2011-09-02T06:33:00.005-04:00

Last week’s blog entry discussed Fresnel diffraction, which Russ Hobbie and I analyzed in the 4th edition of Intermediate Physics for Medicine and Biology when we examined the ultrasonic pressure distribution produced near a circular piezoelectric transducer. This week, I will analyze diffraction far from the wave source, known as Fraunhofer diffraction, named for the German scientist Joseph von Fraunhofer.

The mathematics of Fraunhofer diffraction is a bit too complicated to derive here, but the gist of it can be found by inspecting the first equation in Section 13.7 (Medical Uses of Ultrasound), found at the bottom of the left column on page 351. The pressure is found by integrating 1/r times a cosine function over the transducer face. When you are far from the transducer, r is approximately a constant and can be taken out of the integral. In that case, you just integrate cosine over the transducer area. This becomes similar to the two-dimensional Fourier transform defined in Chapter 12 (Images). In fact, the far field pressure distribution produced by a circular transducer given in Eq. 13.40 is the same Bessel function result as derived in Problem 10 of Chapter 12.

The intensity distribution in Eq. 13.40 is known as the Airy pattern, after the English scientist and mathematician George Biddell Airy. As shown in Fig. 13.15, the pattern consists of a central peak, surrounded by weaker secondary maxima. The Airy pattern occurs during imaging using a circular aperture, such as when viewing stars through a telescope. Two adjacent stars appear as two Airy patterns. Distinguishing the two stars is difficult unless the separation between the images is greater than the separation between the peak of the Airy pattern and its first zero. This is called the Rayleigh criterion, after Lord Rayleigh. Rayleigh (1842-1919, born John William Strutt)--one of those 19th century English Victorian physicists I like so much--did fundamental work in acoustics, and published the classic textbook Theory of Sound.

Fresnel Diffraction

2011-08-26T06:32:00.003-04:00

In Section 13.7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the medical uses of ultrasound. One important problem we analyze is the pressure distribution produced by a piezoelectric transducer.

“There are some important features of the radiation pattern from a transducer which we review next. Consider a circular transducer, the surface of which is oscillating back and forth in a fluid….Each small element of the vibrating fluid creates a wave that travels radially outward, the points of constant phase being expanding hemispheres. The amplitude of each spherical wave decreases as 1/r, the intensity falling as 1/r². We want the pressure at a point z on the axis of the transducer. It is obtained by summing up the effect of all the spherical waves emanating from the face of the transducer….

The [average intensity] is plotted in Fig. 13.13 for a fairly typical but small transducer (a = 0.5 cm, f = 2 MHz)... Close to the transducer there are large oscillations in intensity along the axis: there are corresponding oscillations perpendicular to the axis, as shown in Fig. 13.14. The maxima and minima form circular rings. This is called the near field or Fresnel zone…The depth of the Fresnel zone is approximately a²/λ [where a is the radius of the transducer, λ is the wavelength, and f is the frequency].”

The calculated intensity along the axis, as shown in our Fig. 13.13, is interesting. In the Fresnel zone, the intensity has many points where it is zero. In Intermediate Physics for Medicine and Biology we calculate why this happens mathematically, but it is illuminating to describe what is happening physically. Basically, this is a result of wave interference. Our statement that “each small element of the vibrating fluid creates a wave that travels radially outward” is often called Huygens principle. Each point on the face of the transducer produces such a wavelet. To understand the pressure distribution, we must examine the phase relationship among these various wavelets. Very near the face of the transducer, the waves that contribute significantly to the pressure are in phase; they all interfere constructively and you get a maximum (evaluate Eq. 13.39 at z=0 and you get a nonzero constant). However, as you move away, more distant points on the transducer face contribute to the pressure on the axis, and these points may be out of phase with the pressure produced by the point at the center. For some value of z the in-phase and out-of-phase wavelets interfere destructively, resulting in zero intensity. Increase z a little more, and not only do the in-phase points at the center and the out-of-phase points just away from the center contribute to the pressure, but so do some in-phase points even farther from the center. When you add it all up, you get a net constructive interference and a non-zero intensity. And so it goes, as you move out farther and farther along the z axis.

The radial distribution of the intensity is surprisingly rich and complex, given the rather simple integral that underlies the behavior. If you want to explore the radial distribution in more detail, go to the excellent website http://wyant.optics.arizona.edu/fresnelZones/fresnelZones.htm, where you can perform these calculations yourself. You can adjust the parameters as you wish and create plots such as those in Fig. 13.14, and also produce grayscale images of the full intensity distribution that provide much insight. The website was produced with optics in mind, so you have to put in strange looking parameters to model ultrasound. To reproduce the middle panel of Fig 13.14, input 770,000 for the wavelength in nm, 10,000 for the aperture diameter in microns, and 15.75 for the observation distance in mm. To my eye, the agreement between the website’s calculation and Fig. 13.14 is impressive. At small values of z the plots get very complex and beautiful. For the same wavelength and aperture, I like the richness of z=5 mm, and for z=4 mm you get a fairly uniform brightness except for a dramatic dark spot right at the center. It reminds me of Poisson’s spot, which I discussed in the September 17, 2010 entry in this blog, about Augustin-Jean Fresnel. Indeed, the physics behind the calculations in Fig. 13.14 and Poisson’s spot in optics are nearly identical. The circular aperture is a classic problem in Fresnel diffraction. You can find a more detailed discussion of this topic in the textbook Optics (4th edition), by Eugene Hecht. (My bookshelf contains the first edition, by Hecht and Zajac, that I used in my undergraduate optics class at the University of Kansas).

If you want to be clever, you could make the ultrasound transducer vibrate only at those radii that result in constructive interference along the axis, and have it remain stationary at radii that cause destructive interference. (Of course, this would mean you would have to design your transducer face cleverly so concentric rings vibrate, separated by rings that do not, which might make constructing the transducer more difficult.) Using such a trick eliminates the dark spots along the z axis, increasing the intensity there. This method is commonly used to focus light waves, and is called a zone plate. It has been used occasionally with ultrasound.

The Nonlinear Poisson-Boltzmann Equation

2011-08-19T06:23:00.004-04:00

Last week’s blog entry was about the Gouy-Chapman model for a charged double layer at an electrode surface. The model is based on the Poisson-Boltzmann equation (Eq. 9.10 in the 4th edition of Intermediate Physics for Medicine and Biology). One interesting feature of the Poisson-Boltzmann equation is that it is nonlinear. In applications when the thermal energy of ions in solution is much greater than the energy of the ions in an electrical potential, the equation can be linearized (Eq. 9.13). That is not always the case.

Homework problem 9 in Chapter 9 of Intermediate Physics for Medicine and Biology was added in the 4th edition. It begins

“Problem 9 Analytical solutions to the nonlinear Poisson-Boltzmann equation are rare but not unknown. Consider the case when the potential varies in one dimension (x), the potential goes to zero at large x, and there exists equal concentrations of monovalent cations and anions. Chandler et al. (1965) showed that the solution to the 1-d Poisson-Boltzmann equation, d²ζ/dx²=sinh(ζ), is…”

You will need to get a copy of the book to see this lovely solution. It is a bit too complicated to write in this blog, but it involves the exponential function, the hyperbolic tangent function, and the inverse hyperbolic tangent function. I like this homework problem, because you can solve both the nonlinear and linear equations exactly, with the same boundary conditions, and compare them to get a good intuitive feel for the impact of the nonlinearity. I admit, the problem is a bit advanced for an intermediate-level book, but upper-level undergraduates or graduate students studying from our text should be up to the challenge.

The full citation to the paper by Knox Chandler, Alan Hodgkin, and Hans Meves mentioned in the problem is

Chandler, W. K., A. L. Hodgkin, and H. Meves (1965). The effect of changing the internal solution on sodium inactivation and related phenomena in giant axons. J. Physiol. 180: 821-836.

I always thought it odd that one finds a really elegant analytical solution to the nonlinear Poisson-Boltzmann equation in a paper about sodium channel inactivation in a squid nerve axon (with Nobel Prize-winning physiologist Alan Hodgkin as a coauthor). The solution is buried in the discussion (in a section set of in a smaller font than the rest of the paper). The reason for its appearance is that Chandler et al. found changes in membrane behavior with intracellular ion concentration, and postulated that the measured voltage drop between the inside and outside of the axon consisted of a voltage drop across the membrane itself (which affects the ion channel behavior) and a voltage drop within a double layer adjacent to the membrane. It is the double layer voltage that they model using the Poisson-Boltzmann equation.

Nowadays, the nonlinear Poisson-Boltzmann equation is typically solved using numerical methods. See, for example, the paper that Russ Hobbie and I cite in Intermediate Physics for Medicine and Biology, written by Barry Honig and Anthony Nicholls: “Classical Electrostatics in Biology and Chemistry,” Science, Volume 268, Pages 1144-1149, 1995 (it now has over 1500 citations in the literature). Their abstract states

“A major revival in the use of classical electrostatics as an approach to the study of charged and polar molecules in aqueous solution has been made possible through the development of fast numerical and computational methods to solve the Poisson-Boltzmann equation for solute molecules that have complex shapes and charge distributions. Graphical visualization of the calculated electrostatic potentials generated by proteins and nucleic acids has revealed insights into the role of electrostatic interactions in a wide range of biological phenomena. Classical electrostatics has also proved to be a successful quantitative tool yielding accurate descriptions of electrical potentials, diffusion limited processes, pH-dependent properties of proteins, ionic strength-dependent phenomena, and the solvation free energies of organic molecules.”

Such calculations continue to be an active area of research. See, for example, “The Role of DNA Shape in Protein-DNA Recognition” by Remo Rohs, Sean West, Alona Sosinsky, Peng Liu, Richard Mann and Barry Honig (Nature, Volume 461, Pages 1248-1253, 2009).

Gouy and Chapman

2011-08-12T05:41:00.004-04:00

Sometimes when I am studying physics, I run across a model or equation with names attached to it, and I wonder "just who are these people?" For example, in Chapter 9 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Gouy-Chapman model.

“In this section we study one model for how ions are distributed at the interface in Donnan equilibrium. The model was used independently by Gouy and by Chapman to study the interface between a metal electrode and an ionic solution. They investigated the potential changes along the x axis perpendicular to a large plane electrode. The same model is used to study the charge distribution in a semiconductor.”

So who are Gouy and Chapman? I can tell they lived a long time ago, because in the next section we write “In an ionic solution, ions of opposite charge attract one another. A model of this neutralization was developed by Debye and Huckel a few years after Guoy and Chapman developed [their] model.” I know Peter Debye worked in the early days of quantum mechanics, so Gouy and Chapman had to be a bit earlier.

As if often the case with scientific developments in the 19th century, a Frenchman and an Englishman share credit for the discovery. Louis Georges Gouy (1854-1926) was a French experimental physicist from the University of Lyon. He is best known as the inventor of the Gouy balance, a device for measuring magnetic susceptibility. Gouy had an interest in Brownian motion at a time when the atomic nature of matter was still an open question. Russ and I discuss Brownian motion in Section 4.3 of our book.

“[The] movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by the English botanist Robert Brown in 1827 and is called Brownian motion.”

Albert Einstein wrote a fundamental paper on Brownian motion in 1905, his miraculous year. In a subsequent paper on the same topic, Einstein began (Annalen der Physik, 1906, Volume 19, Pages 371-381)

“Soon after the appearance of my paper on the movements of particles suspended in liquids demanded by the molecular theory of heat, Siedentopf (of Jena) informed me that he and other physicists—in the first instance, Prof. Gouy (of Lyons)—had been convinced by direct observation that the so-called Brownian motion is caused by the irregular thermal movements of the molecules of the liquid.”

Gouy’s model sought to explain the electrical potential around small Brownian particles, or colloids. He described the double layer of charge that develops at the surface, with one charge layer bound to the surface of the particle, and a layer of counterions in the surrounding fluid.

David Leonard Chapman (1869-1958) was an English physical chemist at Oxford. He was interested in the theory of detonation in gasses, and developed the Chapman-Jouget condition describing their behavior. About three years after Gouy, Chapman derived a model describing the double layer at a charged surface. The Gouy-Chapman model is an application of what is now known as the Poisson-Boltzmann equation (Eq. 9.13 in our book).

In 1924, physicist Otto Stern extended the Gouy-Chapman model by noting that ions cannot be represented as point charges when they are within a few ion radii of the surface. This leads to the Stern layer of immobile counter-ions right next to the surface, and a diffuse layer of counter-ions whose concentration decays exponentially.

Fisher-Kolmogorov equation

2011-08-05T06:18:00.005-04:00

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss many of the important partial differential equations of physics, such as Laplace’s equation, the diffusion equation, and the wave equation. One lesser-known PDE that we don’t discuss is the Fisher-Kolmogorov equation. However, our book supplies most of what you need to understand this equation.

In Section 2.10, we examine the logistic equation, an ordinary differential equation governing population growth,

du/dt = b u (1-u) .

For u much less than one, the population grows exponentially with rate b. As u approaches one, the population levels off near a steady state value of u=1. Our Eq. 2.28 gives an analytical solution to this nonlinear equation.

In Section 4.8, we drive the diffusion equation, which for one dimension is

du/dt = D d²u/dx² .

This linear partial differential equation is one of the most famous in physics. It describes diffusion of particles, and also the flow of heat by conduction. D is the diffusion constant.

To get the Fisher-Kolmogorov equation, just put the logistic equation and the diffusion equation together:

du/dt = D d²u/dx² + b u (1-u) .

The Fisher-Kolmogorov equation is an example of a “reaction-diffusion equation.” Russ and I discuss a similar reaction-diffusion equation in Homework Problem 24 of Chapter 4, when modeling intracellular calcium waves. The only difference is that we use a slightly more complicated reaction term rather than the logistic equation.

In his book Mathematical Biology, James Murray discusses the Fisher-Kolmogorov equation in detail. He states

“The classic simplest case of a nonlinear reaction diffusion equation … is [The Fisher-Kolmogorov equation]… It was suggested by Fisher (1937) as a deterministic version of a stochastic model for the spatial spread of a favoured gene in a population. It is also the natural extension of the logistic growth population model discussed in Chapter 11 when the population disperses via linear diffusion. This equation and its travelling wave solutions have been widely studied, as has been the more general form with an appropriate class of functions f(u) replacing ku(1-u). The seminal and now classical paper is that by Kolmogoroff et al. (1937)…. We discuss this model equation in the following section in some detail, not because in itself it has such wide applicability but because it is the prototype equation which admits travelling wavefront solutions. It is also a convenient equation from which to develop many of the standard techniques for analyzing single-species models with diffusive dispersal.”

The Fisher-Kolmogorov equation was derived independently by Ronald Fisher (1890-1962), an English biologist, and Andrey Kolmogorov (1903-1987), a Russian mathematician. The key original papers are

Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics, 7:353-369.

Kolmogoroff, A., I. Petrovsky, and N. Piscounoff (1937) Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Moscow Univ, Bull. Math., 1:1-25.

The terahertz

2011-07-29T05:50:00.006-04:00

In the opening section of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I provide a table of common prefixes used in the metric system:

giga G 10⁹
mega M 10⁶
kilo k 10³
milli m 10⁻³
micro μ 10⁻⁶
nano n 10⁻⁹
pico p 10⁻¹²
femto f 10⁻¹⁵
atto a 10⁻¹⁸

We went all the way down to “atto” on the small side, but stopped at “giga” on the large side. I now wish we had skipped “atto” and instead included “tera,” or “T”, corresponding to 10¹². Why? Because the prefix tera can be very useful in optics when discussing the frequency of light.

To better appreciate why, take a look at the letter to the editor “Let’s Talk TeraHertz!” in the April 2011 issue of my favorite journal, the American Journal of Physics. There Roger Lewis--from the melodious University of Wollongong--argues that the terahertz is superior to the nanometer when discussing light. Lewis writes

“…the terahertz shares the desirable properties of the nanometer as a unit in teaching optics. … Like the nanometer, the terahertz conveniently represents visible light to three digits in numbers that fall in the midhundreds. … The terahertz has other desirable properties that the nanometer lacks. First, the frequency is a more fundamental property of light than the wavelength because the frequency does not change as light traverses different media, whereas the wavelength may. Second, the energy of a photon is directly proportional to its frequency. … The visible spectrum is often taken to span 400–700 nm, corresponding to 749–428 THz, falling in the octave 400–800 THz. …”

I suspect that the reason I have always preferred wavelength over frequency when discussing light is that the nanometer provides such a nice, easy-to-remember unit to work with. Had I realized from the start that terahertz offered an equally useful unit for discussing frequency, I might naturally think in terms of frequency rather than wavelength. Incidentally, Planck’s constant is 0.00414 (or about 1/240) in the units of eV/THz.

After reading Lewis’s letter, I checked Intermediate Physics for Medicine and Biology to see how Russ and I characterized the properties of visible light. On page 360 I found our Table 14.2, which lists the different colors of the electromagnetic spectrum in terms of wavelength (nm), energy (eV) and frequency. We didn’t explicitly mention the unit THz, but we did list the frequency in units of 10¹² Hz, so terahertz was there in every way but name. As a rule I don’t like to write in my books, but nevertheless I suggest that owners of Intermediate Physics for Medicine and Biology take a pencil and replace “(10¹² Hz)” in Table 14.2 with “(THz)”.

Russ and I discuss the terahertz explicitly in Chapter 14 about atoms and light.

14.6.4 Far Infrared or Terahertz Radiation

For many years, there were no good sources or sensitive detectors for radiation between microwaves and the near infrared (0.1–100 THz; 1 THz = 10¹² Hz). Developments in optoelectronics have solved both problems, and many investigators are exploring possible medical uses of THz radiation (“T rays”). Classical electromagnetic wave theory is needed to describe the interactions, and polarization (the orientation of the E vector of the propagating wave) is often important. The high attenuation of water in this frequency range means that studies are restricted to the skin or surface of organs such as the esophagus that can be examined endoscopically. Reviews are provided by Smye et al. (2001), Fitzgerald et al. (2002), and Zhang (2002).

The citations are to

Fitzgerald, A. J., E. Berry, N. N. Zinonev, G. C. Walker, M. A. Smith and J. M. Chamberlain (2002). An introduction to medical imaging with coherent terahertz frequency radiation. Phys. Med. Biol. 47: R67–R84.

Smye, S. W., J. M. Chamberlain, A. J. Fitzgerald and E. Berry (2001). The interaction between terahertz radiation and biological tissue. Phys. Med. Biol. 46: R101–R112.

Zhang, X-C. (2002). Terahertz wave imaging: horizons and hurdles. Phys. Med. Biol. 47: 3667–3677.

So not only is the terahertz useful when talking about visible light, but also it is useful if working in the far infrared, when the frequency is about 1 THz. Such “T rays” (I hate that term) are being used nowadays for imaging during airport security and as a tool to study cell biology and cancer.

Euler: The Master of Us All

2011-07-22T06:29:00.003-04:00

Swiss mathematician Leonard Euler (1707-1783) is a fascinating man. I discussed him once before in this blog, during an entry about the book e: The Story of a Number. Euler’s name never appears in the 4th edition of Intermediate Physics for Medicine and Biology, but his influence is there.

William Dunham describes Euler's life and work in his book Euler: The Master of Us All. In the Preface, Dunham writes

“This book is about one of the undisputed geniuses of mathematics, Leonhard Euler. His insight was breathtaking, his vision profound, his influence as significant as that of anyone in history. Euler contributed to long-established branches of mathematics like number theory, analysis, algebra, and geometry. He also ventured into the largely unexplored territory of analytic number theory, graph theory, and differential geometry. In addition, he was his century’s foremost applied mathematician, as his work in mechanics, optics, and acoustics amply demonstrates. There was hardly an aspect of the subject that escaped Euler’s penetrating gaze. As the twentieth-century mathematician Andre Weil put it, ‘All his life…he seems to have carried in his head the whole of the mathematics of his day, both pure and applied.’”

In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss one of Euler’s best known contributions, his relationship between the exponential function, trigonometric functions, and complex numbers.

“The numbers that we have been using are called real numbers. The number i = √−1 is called an imaginary number. A combination of a real and imaginary number is called a complex number. The remarkable property of imaginary numbers that make them useful in this context is that e^iθ = cosθ + i sinθ.”

Dunham wrote about this identity:

“ ‘From these equations,’ Euler noted with evident satisfaction, ‘we understand how complex exponentials can be expressed by real sines and cosines.’ His enthusiasm has been echoed by mathematicians ever since. Few would argue that Euler’s identity is among the most beautiful formulas of all.”

Euler didn’t invent complex numbers, but he did contribute significantly to their development, including a derivation of this gem (“which seems extraordinary to me,” wrote Euler)

iⁱ = e^-π/2.

Dunham’s book gives examples of Euler’s contributions to number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, and combinatorics. For instance, Dunham describes an discovery Euler made when in his 20s.

“One of his earliest triumphs was a solution of the so-called ‘Basel Problem’ that perplexed mathematicians for the better part of the previous century. The issue was to determine the exact value of the infinite series

1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/k² + … .

… The answer was not only a mathematical tour de force but a genuine surprise, for the series sums to π²/6. This highly non-intuitive result made the solution all the more spectacular and its solver all the more famous.”

As he grew older, Euler slowly became blind. His accomplishments despite his handicap remind me of Beethoven composing his majestic 9th symphony after going deaf. Dunham writes about Euler

“Although unable to see, he not only maintained but even increased his scientific output. In the year 1775, for instance, he wrote an average of one mathematical paper per week. Such productivity came in spite of the fact that he now had to have others read him the contents of scientific papers, and he in turn had to dictate his work to diligent scribes. During his descent into blindness, he wrote an influential textbook on algebra, a 775-page treatise on the motion of the moon, and a massive, three-volume development of integral calculus, the Institutiones calculi integralis. Never was his remarkable memory more useful than when he could see mathematics only in his mind’s eye.

That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler’s courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.”

Dunham concludes

“Euler left behind a legacy of epic proportions. So prolific was he that the journal of the St. Petersburg Academy was still publishing the backlog of his papers a full 48 years after his death. There is hardly a branch of mathematics—or for that matter of physics—in which he did not play a significant role.”

The leibniz

2011-07-15T06:16:00.003-04:00

In order to motivate the study of thermal physics, Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology begins with an examination of how many equations are required to simulate the motion of all the molecules in one cubic millimeter of blood. Russ Hobbie and I write

“It is possible to identify all the external forces acting on a simple system and use Newton’s second law (F = ma) to calculate how the system moves … In systems of many particles, such calculations become impossible. Consider, for example, how many particles there are in a cubic millimeter of blood. Table 3.1 shows some of the constituents of such a sample [including 3.3 × 10¹⁹ water molecules]. To calculate the translational motion in three dimensions, it would be necessary to write three equations for each particle using Newton’s second law. Suppose that at time t the force on a molecule is F. Between t and t + Δt, the velocity of the particle changes according to the three equations

v_i(t+Δt) = v_i(t) + F_iΔt/m, (i = x, y, z).

The three equations for the change of position of the particle are of the form x(t + Δt) = x(t) + v_x(t)Δt … Solving these equations requires at least six multiplications and six additions for each particle. For 10¹⁹ particles, this means about 10²⁰ arithmetic operations per time interval … It is impossible to trace the behavior of this many molecules on an individual basis.

Nor is it necessary. We do not care which water molecule is where. The properties of a system that are of interest are averages over many molecules: pressure, concentration, average speed, and so forth. These average macroscopic properties are studied in statistical or thermal physics or statistical mechanics.”

It is difficult to gain an intuitive feel for just how many differential equations are needed in such a calculation, just as it is difficult to imagine just how many molecules make up a macroscopic bit of matter. Chemists have solved the problem of dealing with large numbers of molecules by introducing the unit of a mole, corresponding to Avogadro’s number (6 × 10²³) of molecules. Other quantities involving Avogadro’s number are similarly defined. For instance, the Faraday corresponds to the magnitude of the charge of one mole of electrons (I admit, the Faraday is more of a constant than a unit); see page 60 and Eq. 3.32 of Intermediate Physics for Medicine and Biology. In Problem 2 of Chapter 14, Russ and I discuss the einstein, a unit corresponding to a mole of photons. When doing large-scale numerical simulations on a computer, it would be useful to have a similar unit to handle very large numbers of differential equations, such as are required to model a drop of blood.

Fortunately, such a unit exists, called the leibniz. Sui Huang and John Wikswo coined the term in their paper Dimensions of Systems Biology, published in the Reviews of Physiology, Biochemistry & Pharmacology (Volume 157, Pages 81-104, 2006). They write

“The electrical activity of the heart during ten seconds of fibrillation could easily require solving 10¹⁸ coupled differential equations (Cherry et al. 2000). (N.B., Avogadro’s number of differential equations may be defined as one Leibnitz, so 10 s of fibrillation corresponds to a micro-Leibnitz problem.) Multiprocessor supercomputers running for a month can execute a micromole of floating point operations, but in the cardiac case such computers may run several orders of magnitude slower than real time, such that modeling 10 s of fibrillation might require 1 exaFLOP/s×year.”

The leibniz appeared again in Wikswo et al.’s paper Engineering Challenges of BioNEMS: The Integration of Microfluidics, Micro- and Nanodevices, Models and External Control for Systems Biology in the IEE Proceedings Nanobiotechnology (Volume 153, Pages 81-101, 2006).

“What distinguishes the models of systems biology from those of many other disciplines is their multiscale richness in both space and time: these models may eventually have millions of dynamic variables with complex non-linear interactions. It is conceivable that the ultimate models for systems biology might require a mole of differential equations (called a Leibnitz) and computations that require a yottaFLOPs (floating point operations per second) computer.”

If we take the leibniz (Lz) as our unit of simulation complexity, the calculation Russ and I consider at the start of Chapter 3 requires solving approximately 6 × 10¹⁹ differential equations, or about 0.1 mLz. Note that we describe two first order differential equations for each molecule, but others might prefer to speak of a single second-order differential equation. This would make a difference of a factor of two in the number of equations. I propose that when using the leibniz we consider only first order ODEs. Moreover, when using a differential equation governing a vector, we count one equation per component.

For those not familiar with Gottfried Leibniz (1646 – 1716), he is a German mathematician and a co-inventor of the calculus, along with Isaac Newton. In fact, Leibniz and Newton got into one of the biggest priority disputes in the history of science about this landmark development. Newton has his unit, so its only fair that Leibniz has one too. Leibniz also made contributions to information theory and computational science, so the liebniz is a particularly appropriate way to honor this great mathematician.

John Wikswo, my PhD advisor when I was in graduate school at Vanderbilt University, notes that there are two alternative spellings of Leibniz's name: Leibnitz and Leibniz. I favor “Leibniz”, the spelling on Wikipedia, and so does Wikswo now, but he points out that there’s plenty of support for “Leibnitz” used in his earlier publications. I had high hopes of enjoying a bit of fun at my friend’s expense by adding an annoying “[sic]” after each appearance of “Leibnitz” in the above quotes, but then Wikswo pointed out that Richard Feynman used “Leibnitz” in the Feynman Lectures on Physics. What can I say; you can’t argue with Feynman.

Gasiorowicz

2011-07-08T06:29:00.005-04:00

One of the standard topics in any modern physics class is blackbody radiation. Indeed, it was the study of blackbody radiation that led to the development of quantum mechanics. In Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

“The spectrum of power per unit area emitted by a completely black surface in the wavelength interval between λ and λ+dλ is … a universal function called the blackbody radiation function. …The description of [this] function …by Planck is one of the foundations of quantum mechanics… We can find the total amount of power emitted per unit surface area by integrating¹⁰ Eq. 14.32 [Planck’s blackbody radiation function]…[The result] is the Stefan-Boltzmann law.”

As I was reading over this section recently, I was struck by the footnote number ten (present in earlier editions of our book, so I know it was originally written by Russ).

“¹⁰This is not a simple integration. See Gasiorowicz (1974, p. 6).”

This is embarrassing to admit, but although I am a coauthor on the 4th edition, there are still topics in our book that I am learning about. I always feel a little guilty about this, so recently I decided it is high time to take a look at the book by Stephen Gasiorowicz and see just how difficult this integral really is. The result was fascinating. The integral is not terribly complicated, but it involves a clever trick I would have never thought of. Because math is rather difficult to write in the html of this blog (at least for me), I will explain how to evaluate this integral through a homework problem. When revising our book for the 4th edition, I enjoyed finding “missing steps” in derivations and then creating homework problems to lead the reader through them. For instance, in Problem 24 of Chapter 14, Russ and I asked the reader to “integrate Eq. 14.32 over all wavelengths to obtain the Stephan-Boltzmann law, Eq. 14.33.” Then, we added “You will need the integral [integrated from zero to infinity]

∫ x

³/(e^x-1) dx = π⁴/15 ."

Below is a new homework problem related to footnote ten, in which the reader must evaluate the integral given at the end of Problem 24. I base this homework problem on the derivation I found in Gasiorowicz. In our book, we cite the 1974 edition

“Gasiorowicz, S. (1974). Quantum Physics. New York, Wiley.”

This is the edition in Kresge library at Oakland University, and is the one I used to create the homework problem. However, I found using amazon.com’s “look inside” feature that this derivation is also in the more recent 3rd edition (2003). In addition, I found the derivation repeated in another of Gasiorowicz’s books, The Structure of Matter.

Problem 24 ½ Evaluate the integral given in Problem 24.
(a) Factor out e^-x, and then use the geometric series 1 + z + z² + z³ + …=1/(1-z) to replace the denominator by an infinite sum.
(b) Make the substitution y = (n+1) x.
(c) Evaluate the resulting integral over y, either by looking it up or (better) by repeated integration by parts.
(d) Make the substitution m=n+1
(e) Use the fact that the sum of 1/m⁴ from 1 to infinity is equal to π⁴/90 to evaluate the integral.

Really, who would have thought to replace 1/(1-z) by an infinite series? Usually, I am desperately trying to do just the opposite: get rid of an infinite series, such as a geometric series, by replacing it with a simple function like 1/(1-z). The last thing I would have wanted to do is to introduce a dreaded infinite sum into the calculation. But it works. I must admit, this is a bit of a cheat. Even if in part (c) you don’t look up the integral, but instead laboriously integrate by parts several times, you still have to pull a rabbit out of the hat in step (e) when you sum up 1/m⁴. Purists will verify this infinite sum by calculating the Fourier series over the range 0 to 2π of the function

f(x) = π⁴/90 – π² x²/12 + π x³/12 – x⁴/48

and then evaluating it at x=0. (Of course, you know how to calculate a Fourier series, since you have read Chapter 11 of Intermediate Physics for Medicine and Biology.) When computing Fourier coefficients, you will need to do a bunch of integrals containing powers of x times cos(nx), but you can do those by--you guessed it--repeated integration by parts. Thus, even if lost on a deserted island without your math handbook or a table of integrals, you should still be able to complete the new homework problem using Gasiorowicz’s method. I’m assuming you know how to do some elementary calculus--integrate by parts and simple integrals of powers, exponentials, and trigonometric functions--without looking it up. (Full disclosure: I found the function f(x) given above by browsing through a table of Fourier series in my math handbook. On that lonely island, you would have to guess f(x), so let’s hope you at least remembered to bring along plenty of scrap paper.)

I checked out the website for Gasiorowicz’s textbook. There is a lot of interesting material there. The book covers many of the familiar topics of modern physics: blackbody radiation, the photoelectric effect, the Bohr model for hydrogen, the uncertainty principle, the Schrodinger equation and more, all the way up to the structure of atoms and molecules. I learned this material from Eisberg and Resnick’s Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (1985), cited several times in Intermediate Physics for Medicine and Biology, when I used their book in my undergraduate modern physics class at the University of Kansas. For an undergraduate quantum mechanics class, I like Griffith’s Introduction to Quantum Mechanics, in part because I have taught from that book. But Gasiorowicz’s book appears to be in the same class as these two. I noticed that Gasiorowicz is from the University of Minnesota, so perhaps Russ knows him.

P.S. Did any of you dear readers notice that Russ and I spelled the name “Stefan” of the “Stefan-Boltzmann law” differently in the text of Chapter 14 and in Problem 24? I asked Google, and it found sites using both spellings, but the all-knowing Wikipedia favors “Stefan”. I’m not 100% certain which is correct (it may have to do with the translation from Slovene to English), but we should at least have been consistent within our book.

Physiology is the link between the basic sciences and medicine

2011-07-01T06:24:00.004-04:00

When Russ Hobbie and I need to cite a general physiology reference in the 4th edition of Intermediate Physics for Medicine and Biology, we often choose the Textbook of Medical Physiology. The book was originally written by Arthur Guyton, but in the most recent editions the lead author is John Hall. We cite the book in several places: 1) In Chapter 1, we reproduce one of Guyton’s figures of the human circulatory system (we reference the 8th edition of the Textbook of Medical Physiology, 1991), 2) We cite Guyton in Chapter 5 when discussing osmotic pressure and when analyzing countercurrent transport, 3) When discussing nerve synapses and neurotransmitters we cite the 10th edition (2000) on which Hall is also first author, 4) In our chapter on feedback loops (Chapter 10) we reproduce a figure showing how alveolar ventilation responds to exercise from the 9th edition (1995, Guyton sole author), and 5) In Chapter 11 on the method of least squares we base Homework Problem 15 on data from Guyton’s textbook regarding the secretion of cortisol by the adrenal gland.

When I was a graduate student at Vanderbilt University, I decided to sit in on the physiology and biochemistry classes that the medical students took. The physiology class was based on Guyton’s book (likely the 6th or 7th edition). I took the class seriously, but since I had little formal coursework in biology (only one introductory class as an undergraduate at the University of Kansas, plus a high school course), and because I didn’t get as much out of the lectures as I should have, my main accomplishment was reading the Textbook of Medical Physiology, cover to cover. Unfortunately, my copy of the book has been lost (probably loaned out to someone who forgot to return it). It is a pity, because I have fond memories of that book, and all the physiology I learned while reading it.

The 12th edition of the Textbook of Medical Physiology (2010) was published after the 4th edition of Intermediate Physics for Medicine and Biology went to press. Hall and Guyton’s preface states

“The first edition of the Textbook of Medical Physiology was written by Arthur C. Guyton almost 55 years ago. Unlike most major medical textbooks, which often have 20 or more authors, the first eight editions of the Textbook of Medical Physiology were written entirely by Dr. Guyton, with each new edition arriving on schedule for nearly 40 years. The Textbook of Medical Physiology, first published in 1956, quickly became the best-selling medical physiology textbook in the world. Dr. Guyton had a gift for communicating complex ideas in a clear and interesting manner that made studying physiology fun. He wrote the book to help students learn physiology, not to impress his professional colleagues.

I worked closely with Dr. Guyton for almost 30 years and had the privilege of writing parts of the 9th and 10th editions. After Dr. Guyton's tragic death in an automobile accident in 2003, I assumed responsibility for completing the 11th edition.

For the 12th edition of the Textbook of Medical Physiology, I have the same goal as for previous editions—to explain, in language easily understood by students, how the different cells, tissues, and organs of the human body work together to maintain life.

This task has been challenging and fun because our rapidly increasing knowledge of physiology continues to unravel new mysteries of body functions. Advances in molecular and cellular physiology have made it possible to explain many physiology principles in the terminology of molecular and physical sciences rather than in merely a series of separate and unexplained biological phenomena.

The Textbook of Medical Physiology, however, is not a reference book that attempts to provide a compendium of the most recent advances in physiology. This is a book that continues the tradition of being written for students. It focuses on the basic principles of physiology needed to begin a career in the health care professions, such as medicine, dentistry and nursing, as well as graduate studies in the biological and health sciences. It should also be useful to physicians and health care professionals who wish to review the basic principles needed for understanding the pathophysiology of human disease.

I have attempted to maintain the same unified organization of the text that has been useful to students in the past and to ensure that the book is comprehensive enough that students will continue to use it during their professional careers.

My hope is that this textbook conveys the majesty of the human body and its many functions and that it stimulates students to study physiology throughout their careers. Physiology is the link between the basic sciences and medicine. The great beauty of physiology is that it integrates the individual functions of all the body's different cells, tissues, and organs into a functional whole, the human body. Indeed, the human body is much more than the sum of its parts, and life relies upon this total function, not just on the function of individual body parts in isolation from the others…”

If you are a physicist studying from Intermediate Physics for Medicine and Biology with little background in biology and medicine, you will need to find a good general source of information about physiology. The Guyton and Hall Textbook of Medical Physiology is a good choice. Another book Russ and I cite a lot is Textbook of Physiology by Patton, Fuchs, Hille, Scher and Steiner. However, I cannot find an edition more recent than 1989, so it would not be a good choice for getting up-to-date information.

Arthur Guyton (1919-2003) was a famous physiologist, known for his research on the circulatory system. An obituary published in the Physiologist says

“Arthur Guyton’s research contributions, which include more than 600 papers and 40 books, are legendary and place him among the greatest figures in the history of cardiovascular physiology. His research covered virtually all areas of cardiovascular regulation and led to many seminal concepts that are now an integral part of our understanding of cardiovascular disorders such as hypertension, heart failure, and edema. It is difficult to discuss cardiovascular regulation without including his concepts of cardiac output and venous return, negative interstitial fluid pressure and regulation of tissue fluid volume and edema, regulation of tissue blood flow and whole body blood flow autoregulation, renal-pressure natriuresis, and long-term blood pressure regulation.

Perhaps his most important scientific contribution, however, was a unique quantitative approach to cardiovascular regulation through the application of principles of engineering and systems analysis. He had an extremely analytical mind and an uncanny ability to integrate bits and pieces of information, not only from his own research but also from others, into a quantitative conceptual framework. He built analog computers and pioneered the application of large-scale systems analyses to modeling the cardiovascular system before digital computers were available. With the advent of digital computers, his cardiovascular models expanded dramatically in the 1960’s and 70’s to include the kidneys and body fluids, hormones, autonomic nervous system, as well as cardiac and circulatory functions. He provided the first comprehensive systems analysis of blood pressure regulation and used this same quantitative approach in all areas of his research, leading to new insights that are now part of the everyday vocabulary of cardiovascular researchers.

Many of his concepts were revolutionary and were initially met with skepticism, and even ridicule, when they were first presented. When he first presented his mathematical model of cardiovascular function at the Council for High Blood Pressure Research meeting in 1968, the responses of some of the hypertension experts, recorded at the end of the article, reflected a tone of disbelief and even sarcasm. Guyton’s systems analysis had predicted a dominant role for the renal pressure natriuresis mechanism in long-term blood pressure regulation, a concept that seemed heretical to most investigators at that time. One of the leading figures in hypertension research commented “I realize that it is an impertinence to question a computer and systems analysis, but the answers they have given to Guyton seem authoritarian and revolutionary.” Guyton’s concepts were authoritarian and revolutionary, but after 35 years of experimental studies by investigators around the world, they have also proved to be very powerful in explaining diverse physiological and clinical observations. His far-reaching concepts will continue to be the foundation for generations of cardiovascular physiologists."

If you are interested in the interface between physics and physiology, you will find the Guyton and Hall Textbook of Medical Physiology to be a valuable resource.

William Beaumont

2011-06-24T06:33:00.003-04:00

I spent last weekend at Mackinac Island in northern Lake Huron. It’s an interesting little place that you reach by ferry and that does not allow any vehicles (except for a few fire engines and ambulances). The ferry ride is dominated by a view of the Mackinac Bridge (the “Mighty Mac”) connecting the upper and lower peninsulas of Michigan. It is a gorgeous piece of engineering (read about its construction in Henry Petroski’s book Engineers of Dreams: Great Bridge Builders and the Spanning of American). On the island, people walk, bike, and ride in horse-drawn carriages. An old 18th century fort dominates the coastline on the south side of the island, and the nearby town has many shops and restaurants (a cynic might call the town a tourist trap). We visited the fort, observed the firing of a civil war-era cannon, had a carriage tour, stopped at “Arch Rock,” and saw the famous Grand Hotel. Last week happened to the their annual Lilac festival, which included an actual “Dog and Pony Show” (the theme this year was board games, and the little terriers carrying big Scrabble pieces on their backs won first prize).

Buying fudge is a Mackinac Island tradition. We stopped at one of the iconic fudge shops, Murdick’s Fudge, and bought a few slabs. One of the Murdick clan, Ryan Murdick, attended Oakland University, where I teach, and obtained a master’s degree in physics. He and I published several papers together, including one about the bidomain model of the electrical properties of cardiac tissue (see Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology), one about magnetocardiography (the magnetic field produced by the heart, Chapter 8), and one about how eddy currents induced in electroencephalogram electrodes can influence measurements of the magnetoencephalogram (Chapter 8; the effect on the MEG is very small). The papers are:

Murdick, R. and B. J. Roth, 2003, Magneto-encephalogram artifacts caused by electro-encephalogram electrodes. Med. & Biol. Eng. & Comput.,41:203-205.

Murdick, R. A. and B. J. Roth, 2004, A comparative model of two mechanisms from which a magnetic field arises in the heart. J. Appl. Phys., 95:5116-5122.

Roth, B. J., S. G. Patel, and R. A. Murdick, 2006, The effect of the cut surface during electrical stimulation of a cardiac wedge preparation. IEEE Trans. Biomed. Eng., 53:1187-1190.

I wasn’t expecting to find material for this blog on Mackinac Island, but I did. In 1822, Alexis St. Martin was accidently shot in the abdomen in a small trading post near Fort Mackinac. Dr. William Beaumont was summoned to treat St. Martin, and was able to save his life. However, the wound healed in an odd way, leaving an opening providing access to the inside of his stomach.

Readers of the 4th edition of Intermediate Physics for Medicine and Biology will appreciate what happened next. The resourceful Beaumont took advantage of the situation to conduct experiments on digestion. He tied different foods to a string, threaded them into St. Martin’s stomach, left them to digest for a while, and then pulled them out to see what had happened. These ground-breaking experiments were instrumental in establishing how digestion works. I toured a small museum dedicated to Beaumont, which describes these experiments in graphic (perhaps too graphic) detail.

I am interested in Beaumont not just because of his experiments studying digestion. Beaumont Hospital, in Royal Oak Michigan, is the clinical partner for a new medical school recently established at Oakland University. The first class of students at the Oakland University William Beaumont School of Medicine arrives this August. This will be a landmark event in OU’s history, and we are all excited about it.

I can’t help but be intrigued by the juxtaposition of these two stories: William Beaumont’s experiments on Alexis St. Martin, and the establishment of a new medical school bearing Beaumont’s name. St. Martin lived into his 80’s. I expect our new medical school will have a similarly long and productive life.

Opus 200

2011-06-17T06:20:00.004-04:00

In August 2007 I began posting entries to this blog, in order to highlight topics discussed in the 4th edition of Intermediate Physics for Medicine and Biology. Since then, I have posted an entry every Friday morning, without fail. This is my 200th (excluding two rare non-Friday posts).

Why do I keep this blog? First, I hope it sells books. Second, I want a way to keep the book up-to-date. Third, some topics Russ and I only mention in passing, and this blog lets me explore these issues in more detail. Fourth, in the blog I often feature past scientists who contributed to the intersection between physics and biology. Fifth and finally, I enjoy it. I like writing, and I find the topics fascinating.

I get some help. Russ Hobbie often sends me ideas and suggestions. My daughter Kathy posted some key entries when I was in Paris and had very limited computer access. I particularly like comments (thanks Debbie). Feel free to voice your opinion. (However, I’m glad the bozo who posted links to porno sites in the comments has stopped.) I hope the readers find this blog useful.

Remember, the book website contains many useful items, including an errata (listing all known errors in the book), a reprint of our 2009 Resource Letter that appeared in the American Journal of Physics, a link to an interview with Russ Hobbie that appeared in the December 2006 Newsletter of the Division of Biological Physics, which is part of the American Physical Society, and (my personal favorite) a link to Russ Hobbie’s MacDose video on YouTube.

Finally, if you use Facebook, you can join the group “Intermediate Physics for Medicine and Biology” and receive postings about the book there.

National Academies Press

2011-06-10T06:24:00.006-04:00

Getting correct and detailed information about the applications of physics to biology and medicine is important. The 4th edition of Intermediate Physics for Medicine and Biology is an excellent source of such information. Yet I know that you, dear reader, are probably saying “Yes, but I want a FREE source of information.” Well, for those cheapskates like me, there is some good news this week from the National Academies Press (forwarded to me via Russ Hobbie). First, what is the National Academies Press? Their website explains:

“The National Academies Press (NAP) was created by the National Academies to publish the reports issued by the National Academy of Sciences, the National Academy of Engineering, the Institute of Medicine, and the National Research Council, all operating under a charter granted by the Congress of the United States. The NAP publishes more than 200 books a year on a wide range of topics in science, engineering, and health, capturing the most authoritative views on important issues in science and health policy. The institutions represented by the NAP are unique in that they attract the nation’s leading experts in every field to serve on their award-wining panels and committees. The nation turns to the work of NAP for definitive information on everything from space science to animal nutrition.”

Now, what is the good news? An email from the NAP states

“As of June 2, 2011, all PDF versions of books published by the National Academies Press (NAP) will be downloadable free of charge to anyone. This includes our current catalog of more than 4,000 books plus future reports published by NAP.

Free access to our online content supports the mission of NAP--publisher for the National Academy of Sciences, National Academy of Engineering, Institute of Medicine, and National Research Council--to improve government decision making and public policy, increase public education and understanding, and promote the acquisition and dissemination of knowledge in matters involving science, engineering, technology, and health. In 1994, we began offering free content online. Before today's announcement, all PDFs were free to download in developing countries, and 65 percent of them were available for free to any user.

Like no other organization, the National Academies can enlist the nation's foremost scientists, engineers, health professionals, and other experts to address the scientific and technical aspects of society's most pressing problems through the authoritative and independent reports published by NAP. We invite you to sign up for MyNAP --a new way for us to deliver free downloads of this content to loyal subscribers like you, to offer you customized communications, and to reward you with exclusive offers and discounts on our printed books.”

Intermediate Physics for Medicine and Biology cites several NAP reports. For instance, in Section 9.10 about the possible effects of weak external electric and magnetic fields, Russ and I cite and quote from the NAP report Possible Health Effects of Exposure to Residential Electric and Magnetic Fields. I tested the website (free just seemed too good to be true), and was able to download a pdf version of the document with no charge (although I did have to give them my email address when I logged in). I got 379 pages of expert analysis about the biological effects of powerline fields. Russ and I quote the bottom line of this report in our book:

“There is no convincing evidence that exposure to 60-Hz electric and magnetic fields causes cancer in animals.... There is no evidence of any adverse effects on reproduction or development in animals, particularly mammals, from exposure to power-frequency 50- or 60-Hz electric or magnetic fields.”

In Chapter 16 on the medical use of X rays, we cite three of the Biological Effects of Ionizing Radiation (BEIR) reports: V, VI, and VII. These reports provide important background about the linear nonthreshold model of radiation exposure. Then in Chapter 17 on nuclear physics and nuclear medicine we cite BEIR reports IV and VI when discussing radiation exposure caused by radon gas. The full citations listed in our book are:

"BEIR IV (1988). Committee on the Biological Effects of Ionizing Radiations. Health Risks of Radon and Other Internally Deposited Alpha-Emitters. Washington, D.C., National Academy Press.

BEIR Report V (1990). Health Effects of Exposure to Low Levels of Ionizing Radiation. Washington, DC, National Academy Press. Committee on the Biological Effects of Ionizing Radiation.

BEIR VI (1999). Committee on Health Risks of Exposure to Radon. Health Effects of Exposure to Radon. Washington, D.C., National Academy Press.

BEIR Report VII (2005). Health Risks from Exposure to Low Levels of Ionizing Radiation. Washington, DC, National Academy Press. Committee to Assess Health Risks from Exposure to Low Levels of Ionizing Radiation."

Besides the reports cited in our book, there are many others you might like to read. In a previous blog entry, I discussed the report BIO2010: Transforming Undergraduate Education for Future Research Biologists, published by NAP. You can download a copy free. It discusses how we should teach physics to future life scientists. In another blog entry I discussed the book In the Beat of a Heart, which explores biological scaling. It is also published by the NAP.

Yet another report, published just last year, that will be of interest to readers of Intermediate Physics for Medicine and Biology is the NAP report Research at the Intersection of the Physical and Life Sciences. The report summary explains the goals of the report.

“Today, while it still is convenient to classify most research in the natural sciences as either biological or physical, more and more scientists are quite deliberately and consciously addressing problems lying at the intersection of these traditional areas. This report focuses on their efforts. As directed by the charges in the statement of task (see Appendix A), the goals of the committee in preparing this report are several fold. The first goal is to provide a conceptual framework for assessing work in this area—that is, a sense of coherence for those not engaged in this research about the big objectives of the field and why it is worthy of attention from fellow scientists and programmatic focus by funding agencies. The second goal is to assess current work using that framework and to point out some of the more promising opportunities for future efforts, such as research that could significantly benefit society. The third and final goal of the report is to set out strategies for realizing those benefits—ways to enable and enhance collaboration so that the United States can take full advantage of the opportunities at this intersection.”

An older report that covers much of the material that is in the last half of Intermediate Physics for Medicine and Biology is Mathematics and Physics of Emerging Biomedical Imaging (1996). Finally, yet another useful report is Advancing Nuclear Medicine Through Innovation (2007).

All this and more is now available free. Who says there is no such thing as a free lunch?

Jean Perrin and Avogadro’s Number

2011-06-03T06:19:00.006-04:00

Regular readers of this blog may recall that last summer I visited Paris for my 25th wedding anniversary, which was followed by a string of blog entries about famous French scientists. During this trip, my wife and I toured the Pantheon, where we saw the burial site of French scientist Jean Baptiste Perrin (1870-1942). Russ Hobbie and I mention Perrin in a footnote on page 85 of the 4th edition of Intermediate Physics for Medicine and Biology.

“The Boltzmann factor provided Jean Perrin with the first means to determine Avogadro’s number [N_A]. The density of particles in the atmosphere is proportional to exp(-mgy/k_BT), where mgy is the gravitational potential energy of the particles. Using particles for which m was known, Perrin was able to determine [Boltzmann’s constant] k_B for the first time. Since the gas constant R was already known, Avogadro’s number was determined from the relationship R=N_Ak_B.”

This brief footnote does not do justice to Perrin’s extensive accomplishments. He played a key role in establishing that matter is not a continuum, but rather is made out of atoms. He performed experiments not only on the exponential distribution of particles (described above, and also known as sedimentation equilibrium), but also on Brownian motion. Russ and I describe this phenomenon in Chapter 4:

“This movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by English botanist Robert Brown in 1827 and is called Brownian motion.”

One can learn more about Perrin in the book Molecular Reality: A Perspective on the Scientific Work of Jean Perrin, by Mary Jo Nye. I would not rank this book with the best histories of science I have read (my top three would be The Making of the Atomic Bomb, The Eighth Day of Creation, and The Maxwellians), or among the best scientific biographies (such as Subtle is the Lord: The Science and Life of Albert Einstein). However, it did provide some valuable insight into Perrin’s achievements. Ney states in her introduction that

“What has struck me in a perusal of the literature on these topics [discoveries in physics during the early 20th century] is the tendency to assume what so many of the physical scientists of this pivotal period did not for one minute assume—the discontinuity of the matter which underlies visible reality. In looking back upon the discoveries and theories of particles, one perhaps fails to realize that the focus was not simply upon the nature of the molecules, ions and atoms, but upon the very fact of their existence….

In analyzing the role of Jean Perrin in the eventual acceptance of this assumption among the outspoken majority of the scientific community, I have concentrated upon the period of experimental, theoretical, philosophical and popular science which climaxed with the Solvay conference of 1911 and with the publication of Perrin’s book Les Atomes [read an online English translation here] in 1913….

In conclusion, I have discussed the reception of Perrin’s scientific experimentation and propagandisation on the subject of molecular reality, especially his work on Brownian movement, which climaxed in 1913 with the completion of a number of national and international conferences and the publication of Les Atomes. Though Perrin himself did not view his task as completed at that time, the question was no longer central to the basic working assumptions of scientists, and polemics on this question were no longer an impediment or impetus to the progress of general scientific conceptualization. That Perrin’s role was historically essential to this denouement cannot, in my opinion, be doubted.”

Nye’s first chapter on 19th-century background contains a little too much philosophy of science for my taste. But her historical review does indicate that, despite what our footnote says, Perrin did not provide the first estimate for Avogadro’s number, but rather provided a definitive early measurement of that value. Her second chapter about Young Perrin: Initial Investigations was better, and the book really captured my attention in the third chapter on The Essential Debate.

“The exponential law which Perrin announced in 1908, describing the vertical distribution of a colloid at equilibrium, was the fruit of laborious experiments on Brownian movement after several years of apprenticeship in the study of colloids. Included in his first 1908 paper on Brownian movement was a successful application of the concepts of osmotic pressure and mean kinetic energy to the visible Brownian particles, as well as a convincing calculation of Avogadro’s number. These endevours were but the prelude to a five-year drama devoted to the erection of an unassailable edifice to house the dictum of molecular reality, a structure buttressed at its most vulnerable point of criticism by the observed laws of visible Brownian movement.”

I was particularly fascinated by how Perrin knew the mass of the particles he studied.

“In order to find m, Perrin utilized Stoke’s law [see Section 4.5 of Intermediate Physics for Medicine and Biology], applying it to a column of the emulsion in a vertical capillary tube, and observing the fact that when the emulsion is very far from equilibrium, the Brownian granules in the upper layers of the column fall as if they were droplets of a cloud. Using Stokes’ formula relating the velocity of a spherical droplet, its radius, and the viscosity of the medium, Perrin found the radius of the granules [on the order of a micron].”

Then from the known density, he could determine the mass. Perrin had to go to great lengths to obtain particles with a uniform distribution of radii, starting with 1200 grams of particles and, after repeated centrifugation, ending with less than a gram of uniform particles.

In 1926, Jean Perrin won the 1926 Nobel Prize in physics "for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium".

e, The Story of a Number

2011-05-27T06:06:00.008-04:00

On page 33 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the constant e.

“The number e is approximately equal to 2.71828… and is called the ‘base of the natural logarithms.’ Like π (3.14159…) e has a long history [Maor (1994)].”

The citation is to the delightful book e: The Story of a Number, by Eli Maor. In his preface, Maor explains why he wrote the book.

“My goal is to tell the story of e on a level accessible to readers with only a modest background in mathematics. I have minimized the use of mathematics in the text itself, delegating several proofs and derivations to the appendixes. Also, I have allowed myself to digress from the main subject on occasion to explore some side issues of historical interest. These include biographical sketches of the many figures who played a role in the history of e, some of whom are rarely mentioned in textbooks. Above all, I want to show the great variety of phenomena--from physics and biology to art and music--that are related to the exponential function e^x, making it a subject of interest in fields well beyond mathematics.”

Our Chapter 2, about exponential growth, centers on the exponential and logarithm functions, and our Appendix C lists many of the properties of these functions. Maor explores all sorts of interesting facts about e. For instance, 878/323 is a very good rational approximation to this irrational number. You can recall the first ten digits of e by remembering 2.7 (Andrew Jackson)² [Jackson was elected president in 1828]. In his Chapter 13, Maor presents some beautiful continued fractions for e that I will not attempt to reproduce here using html.

When developing the Fourier series in Chapter 11 of Intermediate Physics for Medicine and Biology, Russ and I note that “the remarkable property of imaginary numbers that make them useful in this context is that e^iθ=cosθ + i sinθ.” (Here, i is the square root of minus one.) Maor sets θ=π to obtain an equation studied by the Swiss mathematician Leonhard Euler

e^iπ = -1 ,

and claims

“it must surely rank among the most beautiful formulas in all of mathematics. Indeed, by rewriting it as e^πi + 1 = 0, we obtain a formula that connects the five most important constants of mathematics (and also the three most important mathematical operations--addition, multiplication, and exponentiation). These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 0 and 1; algebra, by i; geometry, by π; and analysis, by e.”

Taking a less aesthetic view, Russ and I downplay the use of complex exponentials in Intermediate Physics for Medicine and Biology.

“The Fourier transform is usually written in terms of complex exponentials. We have avoided using complex exponentials. They are not necessary for anything done in this book. The sole advantage of complex exponentials is to simplify the notation. The actual calculations must be done with real numbers.”

Another reason I often steer clear of complex exponentials is that I place great importance on being able to visualize physically what a mathematical expression is saying, and I find trigonometric functions far easier to envision than complex exponentials. So, while I concede the abstract beauty of the formula e^iπ=-1, I don’t find it so useful when thinking about physics.

While educating his readers about e, Maor also introduces them to many famous mathematicians, including Archimedes, Napier, Newton, Gauss, the Bernoulli's, and above all Euler, who is apparently one of Maor’s favorites.

“Leonhard Euler (1707-1783) is unquestionably the Mozart of mathematics, a man whose immense output--not yet published in full--is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion."

Maor discusses the uses of logarithms and exponentials in biology. He talks about the logarithmic spiral and its role in growth, for instance, of a nautilus shell. He also makes an interesting comparison between the ear and the eye.

“The remarkable sensitivity of the human ear to frequency changes is matched by its audibile range--from about 20 cycles per second to about 20,000 (the exact limits vary somewhat with age). In terms of pitch, this corresponds to about ten octaves (an orchestra rarely uses more then seven). By comparison, the eye is sensitive to a wavelength range from 4,000 to 7,000 angstroms (10^-8 cm)--a range of less than two 'octaves.' [Doesn't Maor mean: less than one 'octave'?]"

I’m particularly fond of Maor’s recreation of a meeting between Bach and one of the Bernoulli's

“Let us imagine a meeting between Johann Bernoulli (Johann I, that is) and Johann Sebastian Bach. The year is 1740. Each is at the peak of his fame. Bach, at the age of fifty-five, is organist, composer, and Kapellmeister (musical director) at St. Thomas's Church in Leipzig. Bernoulli, at seventy three, is the most distinguished professor of the University of Basel.”

The resulting imagined conversation is fascinating and amusing. Musicians interested in the "equal tempered scale" will enjoy this section.

I will close this blog entry the same way Maor ended his book, by letting e take a final bow. Here is e to one hundred decimal places:
2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274

Non-Newtonian Fluids and the Rheology of Blood

2011-05-20T06:32:00.005-04:00

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I explain the difference between a Newtonian fluid and a non-Newtonian fluid.

“A fluid can support a viscous shear stress if the shear strain is changing. One way to create such a situation is to immerse two parallel plates, each of area S, in the fluid, and to move one parallel to the other … The variation of velocity between the plates gives rise to a velocity gradient dv_x/dy …

In order to keep the top plate moving and the bottom plate stationary, it is necessary to exert a force of magnitude F on each plate: to the right on the upper plate and to the left on the lower plate. The resulting shear stress or force per unit area is in many cases proportional to the velocity gradient:

F/S = η dv_x/dy . (1.33)

The constant η is called the coefficient of viscosity … Fluids that are described by Eq. 1.33 are called Newtonian fluids. Many fluids are not Newtonian.”

At the end of the chapter, we give an example of a biologically important non-Newtonian fluid.

“Blood is not a Newtonian fluid. The viscosity depends strongly on the fraction of volume occupied by red cells (the hematocrit).”

An excellent review of blood's fluid behavior can be found in the article “Rheology of Blood” by Edward Merrill (Physiological Reviews, Volume 49, Pages 863-888, 1969). Rheology is the part of fluid mechanics that deals with non-Newtonian fluids. Merrill explains clearly the difference between a Newtonian fluid with a high viscosity and a Non-Newtonian fluid.

“A Newtonian liquid is one in which the viscosity, at fixed temperature and pressure, is independent of the shear stress. Thus, a non-Newtonian liquid is one in which the viscosity depends on shear stress. Water and honey are Newtonian, but many aqueous suspensions of fine particulate matter such as water-base paint, plaster, and oil emulsions are non-Newtonian. The distinction is qualitatively obvious if one imagines two spoons, one in a pot of honey (Newtonian) and the other in a pot of mayonnaise (non-Newtonian emulsion). The honey is harder to stir (has a higher viscosity) than the mayonnaise, but when the spoons are removed and held above the pots, the honey continues to drizzle off its spoon, whereas the mayonnaise coating the other spoon clings indefinitely to it without flow, thus exhibiting 'infinite' viscosity.”

An important concept when discussing the rheology of blood is yield stress. Merrill explains

“Blood … exhibits a 'yield stress.' This means that, if …one increases from zero the stress, but keeps it less than a critical value, the response will be elastic … On removal of the stress, the shape of the blood film will be unaltered, i.e., no flow will have occurred. However, if the yield stress is exceeded, irreversible deformation will occur.”

In other words, it acts like a solid at low stress, and a fluid at high stress. Merrill concludes by discussing the physiological significance of the non-Newtonian nature of blood.

“In summary, the relevance of blood rheology to physiological fluid mechanics is to make stopping of flows easier, starting of flows more difficult, and slow flows more energy consuming than would be expected if blood were a simple, cell-less, micromolecular fluid of equal viscosity--and these effects are increasingly emphasized with increase of hematocrit and fibrinogen concentration.”

Besides blood, another dramatic example of a non-Newtonian fluid is a mixture of corn starch and water. My Oakland University colleague Alberto Rojo (who’s office is next door to mine) has made a fun video demonstrating how you can “walk on water” by taking advantage of this mixture's non-Newtonian properties. The effect is fascinating.

Drawing Figures

2011-05-13T16:16:00.004-04:00

Two of my favorite figures in the 4th edition of Intermediate Physics for Medicine and Biology are Fig. 7.13 (the extracellular potential produced by an action potential along a nerve axon) and Fig. 8.14 (the magnetic field produced by the same axon). John Wikswo and I prepared these figures when I was in graduate school at Vanderbilt University.

Soon after entering graduate school in 1982, I took a class taught by John based on Russ Hobbie’s first edition of Intermediate Physics for Medicine and Biology. Clearly, the book had a significant influence on my subsequent career. (I remember the bright yellow cover of the first edition: my office is probably one of the few places outside of Minnesota where all four editions of the book sit proudly, side-by-side, on a bookshelf.) When preparing the second edition, Russ added a chapter on biomagnetism, and asked John to contribute a figure showing the magnetic field produced by an axon. Of course, this is just the sort of work graduate students are good for, and I was given the task of preparing the figure (actually two figures, as we decided to make a similar figure for the extracellular potential). This was not a big job, because I already had access to the computer code that my friend Jim Woosley had written for his master’s thesis, and which I used when preparing our paper “The Magnetic Field of a Single Axon: A Volume Conductor Model,” (Woosley, Roth, and Wikswo, 1985, Mathematical Biosciences, 76:1-36).

In the mid-1980s, three-dimensional graphics programs were not as common as they are now, but we had one and I was able to create the figure. What we didn’t have was a publication-quality printer or software to prepare and manipulate figures. Therefore, once I had the plots created, they went to the drafting room to be finished. John usually had one or more undergraduates hired for the sole task of preparing figures. I don’t remember exactly who worked on the two figures for the 2nd edition, but it may have been David Barach, son of Vanderbilt physics professor John Barach. The daftsman’s job was to retrace the figure, thereby providing a higher quality appearance than a dot-matrix printer could provide. As I recall, his job was also to remove hidden lines (I don’t think that our 3-d graphics program was “smart” enough to remove hidden lines on its own). He also labeled all the axes using some really neat rub-on letters that John was able to purchase in both Roman and Greek fonts (note the “μ” in μV in Fig. 7.13). I remember David working on figures at a large, slanted drafting table, using very high quality, vellum-like paper. He had rulers, triangles, and “French curves” of all types. First the drawing was done in pencil, and then traced with black ink. Once finished, additional copies were made using photography by a center in the Vanderbilt Medical School dedicated to such work. Before Photoshop, Powerpoint, and other such programs, that is the way figures were prepared. John had a policy that all graduate students had to get some experience at the drafting table, which I didn’t mind at all. At the risk of sounding like a Luddite who is nostalgic for the days of buggy whips, I think those figures have a little more personality and visual appeal than computer-generated figures drawn today.

The figures appeared in the second edition of Russ’s book, and have continued on through subsequent editions (including the 4th edition, on which I have the high honor of becoming a coauthor). Figures like that required much time and expense to prepare, and are difficult to edit. But my, it was more fun to really “draw” those figures than it is to churn out figures using graphics software.

Central Slice Theorem and Ronald Bracewell

2011-05-06T05:13:00.008-04:00

Chapter 12 of the 4th edition of Intermediate Physics for Medicine and Biology deals with images and tomography. One of the central ideas in tomography is the “central slice theorem.” Russ Hobbie and I write in Section 12.4 that

“The Fourier transform of the projection at angle θ is equal to the two-dimensional Fourier transform of the object, evaluated in the direction θ in Fourier transform space. This result is known as the projection theorem or the central slice theorem (Problem 17). The transforms of a set of projections at many different angles provide values of C and S [the cosine and sine parts of the 2-d Fourier transform] throughout the k_xk_y plane [frequency space] that can be used in Eq. 12.9a [the definition of the 2-d Fourier transform] to calculate f(x,y).”

I consider the central slice theorem to be one of the most important concepts in medical imaging. How was this fundamental idea first developed? The answer to that question provides a fascinating example of how physics and engineering can contribute to medicine.

Ronald Bracewell first developed the central slice theorem while working in the field of radio astronomy. His 2007 New York Times obituary states

“Ronald N. Bracewell, an astronomer and engineer who used radio telescopes to make early images of the Sun’s surface, in work that also led to advances in medical imaging, died on Aug. 12 at his home in Stanford, Calif. He was 86…

With his colleagues at Stanford University in the 1950s, Dr. Bracewell designed a specialized radio telescope, called a spectroheliograph, to receive and evaluate microwaves emitted by the Sun….

Later, in the 1970s, the techniques and a formula devised by Dr. Bracewell were applied by other scientists in developing X-ray imaging of tumors, called tomography, and other forms of medical imaging that scan electromagnetic and radio waves. Dr. Bracewell advised researchers at Stanford and other institutions, but did not conduct laboratory research in the field.”

Russ and I cite Bracewell’s 1990 paper “Numerical Transforms” (Science, Volume 248, Pages 697-704). The central slice theorem was published in 1956 in the Australian Journal of Physics (Volume, 9, Pages 198-217). Early in this career Bracewell published a lot in that journal, which is now defunct but maintains a website with free access to all the papers. Bracewell also wrote a marvelous book: The Fourier Transform and Its Applications (originally published in 1965, the revised 2nd edition is published by McGraw-Hill, New York, 1986). When writing this blog entry, I checked this book out of Kresge Library here at Oakland University. Once I opened it, I realized it is an old friend. I am sure I read this book in graduate school. It contains many pictures that allow the student to gain an intuition about the Fourier transform; an extraordinarily valuable skill to develop. The Introduction states

“The present work began as a pictorial guide to Fourier transforms to complement the standard lists of pairs of transforms expressed mathematically. It quickly became apparent that the commentary would far outweigh the pictorial list in value, but the pictorial dictionary of transforms is nevertheless important, for a study of the entries reinforces the intuition, and many valuable and common types of function are included which, because of their awkwardness when expressed algebraically, do not occur in other lists.”

The text also does a fine job describing convolutions.

“Convolution is used a lot here. Experience shows that it is a fairly tricky concept when it is presented bluntly under its integral definition, but it becomes easy if the concept of a functional is first understood.”

Many of the ideas that Russ and I present in Chapter 11 of Intermediate Physics for Medicine and Biology are examined in more detail in Bracewell’s book. I recommend it as a reference to keep at your side as your plow through the mathematics of Fourier analysis.

Finally, Bracewell’s view of homework problems, as stated in his Preface to the second edition, mirrors my own.

“A good problem assigned at the right stage can be extremely valuable for the student, but a good problem is hard to compose. Among the collection of supplementary problems now included at the end of the book are several that go beyond being mathematical exercises by inclusion of technical background or by asking for opinions.”

Bursting

2011-04-29T05:54:00.002-04:00

Last week in this blog I talked briefly about bursting in pancreatic beta cells. A bursting cell fires several action potential spikes consecutively, followed by an extended quiescent period, followed again by another burst of action potentials, and so on. One of the first and best-known models for bursting was developed by James Hindmarsh and Malcolm Rose (A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations, Proc. Roy. Society Lond, B, Volume 221, Pages 87-102, 1984). Their analysis was an extension of the FitzHugh-Nagumo model, with an additional variable governed by a very slow time constant. Their system of equations is

dx/dt = y – x³ + 3 x² – z + I

dy/dt = 1 – 5 x² – y

dz/dt = 0.001 [ 4(x + 1.6) – z]

where x is the membrane potential (appropriately made dimensionless), y is a recovery variable (like a sodium channel inactivation gate), z is the slow bursting variable, and I is an external stimulus current. For some values of I, this model predicts bursting behavior.

There is an entire book dedicated to this topic: Bursting--The Genesis of Rhythm in the Nervous System, by Stephen Coombes and Paul Bressloff (World Sci. Pub. Co., 2005). The first chapter, co-written by Hindmarsh, provides a little of the history behind the Hindmarsh-Rose model:

“The collaboration that led to the Hindmarsh-Rose model began in 1979 shortly after Malcolm Rose joined Cardiff University. The particular project was to model the synchronization of firing of two snail neurons in a relatively simple way that did not use the full Hodgkin-Huxley equations... A natural choice at the time was to use equations of the FitzHugh [type]….

A problem with this choice was that these equations do not provide a very realistic description of the rapid firing of the neuron compared to the relatively long interval between firing. Attempts were made to achieve a more realistic description by making the time constants … voltage dependent. In particular so the rates of change of x and y were much smaller in the subthreshold or recovery phase. These were not convincing and it was not until Malcolm raised the question about whether the FitzHugh equations could account for “tail current reversal” that progress was made.

The modification of the FitzHugh equations to account for tail current reversal was crucial for the development of the Hindmarsh-Rose model.”

For those not familiar with the FitzHugh-Nagumo model, see Problem 33 in Chapter 10 of the 4th edition of Intermediate Physics for Medicine and Biology, or see the Scholarpedia article by FitzHugh himself, written before he died in 2007. If you want to see some bursting patterns, check out this youtube video. It is not great, but you will get the drift of what the model predicts.

My Friend Artie Sherman also had a chapter in the bursting book, titled “Beyond Synchronization: Modulatory and Emergent Effects of Coupling in Square-Wave Bursting.” He has been working on bursting in pancreatic beta cells for years, as a member (and now chief) of the Laboratory of Biological Modeling in the Mathematical Research Branch, the National Institute of Diabetes, Digestive and Kidney Diseases, part of the National Institutes of Health. His work is the best I am aware of for modeling bursting.

Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations

2011-04-22T05:25:00.007-04:00

I enjoy taking a scientific paper and reducing it to a homework problem. For example, one of the new homework problems in the 4th edition of Intermediate Physics for Medicine and Biology is Problem 23 of Chapter 4 (Transport in an Infinite Medium), based on a paper by John Wagner and Joel Keizer.

Problem 23 Calcium ions diffuse inside cells. Their concentration is also controlled by a buffer:
Ca + B ⇐⇒ CaB.
The concentrations of free calcium, unbound buffer, and bound buffer ([Ca], [B], and [CaB]) are governed, assuming the buffer is immobile, by the differential equations
∂[Ca]/∂t= D∇²[Ca] − k⁺[Ca][B] + k⁻[CaB],
∂[B]/∂t= −k⁺[Ca][B] + k⁻[CaB],
∂[CaB]/∂t= k⁺[Ca][B] − k⁻[CaB].
(a) What are the dimensions (units) of k⁺ and k⁻ if the concentrations are measured in mole l⁻¹ and time in s?
(b) Derive differential equations governing the total calcium and buffer concentrations, [Ca]_T = [Ca]+[CaB] and [B]_T= [B] + [CaB] . Show that [B]_T is independent of time.
(c) Assume the calcium and buffer interact so rapidly that they are always in equilibrium:
[Ca][B]/[CaB]= K,
where K = k⁻/k⁺.Write [Ca]_T in terms of [Ca] , [B]_T , and K (eliminate [B] and [CaB]).
(d) Differentiate your expression in (c) with respect to time and use it in the differential equation for [Ca]_T found in (b). Show that [Ca] obeys a diffusion equation with an “effective” diffusion constant that depends on [Ca]:
D_eff = D/(1 + K [B]_T/(K+[Ca])²) .
(e) If [Ca] < < K and [B]_T = 100K (typical for the endoplasmic reticulum), determine D_eff/D.
For more about diffusion with buffers, see Wagner and Keizer (1994).

The reference and abstract of the paper is given below:

John Wagner and Joel Keizer, Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations. Biophysical Journal, Volume 67, Pages 447-456, 1994.

Based on realistic mechanisms of Ca²⁺ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca²⁺ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca²⁺ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca²⁺ that depends on local Ca²⁺ concentrations. Mobile buffers, such as fura^-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca²⁺ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca²⁺ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca²⁺ channel in the ER membrane to illustrate the way in which Ca²⁺ buffering can affect both the amplitude and existence of Ca²⁺ oscillations.

John Wagner is currently with the Functional Genomics and Systems Biology Group of the IBM T. J. Watson Research Center. In the mid 1990s he was a research assistant with Joel Keizer.

Joel Keizer was a long-time member of the University of California at Davis. A UC Davis website states

“Joel’s scientific legacy encompassed several fields. Joel originally trained as a chemist at the University of Oregon under Terrell Hill, where he received his doctorate in theoretical physical chemistry, and did postdoctoral work in chemical physics at the Battelle Institute in Columbus, Ohio. He began his career in 1971 at the University of California, Davis, as an assistant professor of chemistry. He pioneered an approach to the thermodynamics of non-equilibrium steady states, which culminated in the monograph, “Statistical Thermodynamics of Nonequilibrium Processes” in 1987. By this time, he had over 60 journal publications to his credit.

In the 1980s, Joel gradually shifted his research program and focused his powerful intellect on problems within the biological sciences, first on mathematical models of insulin secretion, and later on intracellular calcium oscillations and diffusion. He subsequently transferred his appointment to the Division of Biological Sciences, where both theoreticians and empiricists respected and admired Joel for his strong modeling work and his insightful collaborations with experimental biologists.”

I never met Joel Keizer, but I did know a couple of his collaborators, John Rinzel and Arthur Sherman, both at NIH when I was there in the early 1990s. They worked on bursting in pancreatic beta-cells, and published some influential papers with Keizer (for example, see: Sherman, Rinzel, and Keizer, Emergence of Organized Bursting in Clusters of Pencreatic Beta-Cells by Channel Sharing, Biophysical Journal, Volume 54, Pages 411-425, 1988).

Finally, the paper by Allbritton et al. cited in the Wagner and Keizer paper is:

Allbritton, Meyer, Stryer, Range of Messenger Action of Calcium-Ion and Inositol 1,4,5-Trisphosphate. Volume 258, Pages 1812-1815, 1992.

Superconductivity

2011-04-15T06:39:00.006-04:00

This month marks the hundredth anniversary of the discovery of superconductivity. An article in the magazine IEEE Spectrum states:

On April 8, 1911, physicist Heike Kamerlingh Onnes of Leiden University used an intricate glass cryostat to cool mercury down to just a few degrees above absolute zero. Then he scribbled down three words that ultimately marked the discovery of an entirely new physical phenomenon.

The phrase, jotted more than halfway down the page of a messy lab notebook, didn’t really match the occasion. What Kamerlingh Onnes wrote was 'Mercury practically zero', or, according to a more literal translation, 'Quick[silver] near-enough dull'. But what he saw was the first evidence of superconductivity, the ability of some substances to conduct electricity with no resistance at all."

You can learn more about this landmark event in a Physics Today September 2010 article “The Discovery of Superconductivity,” by Dirk van Delft and Peter Kes, and the article “Superconductivity’s Smorgasbord of Insights: A Movable Feast,” in the April 8, 2011 issue of Science by Adrian Cho. Also, see the biography of Onnes on Nobelprize.org.

One of my favorite books is The Quest for Absolute Zero, by Kurt Mendelssohn. He starts his tale in 1877 with the liquefaction of oxygen and then tells the subsequent history of low temperature physics, including the fascinating story of how Onnes liquefied helium and his early superconductivity studies. According to Mendelssohn, the reason mercury was used for the first experiment is because it could be purified:

“There was one other metal which might be obtained in an even purer state than gold, and that was mercury. Being a liquid at room temperatures, it can be distilled and re-distilled again and again until an extreme degree of purity is reached. The results were communicated to the Netherlands Royal Academy on the 28th April 1911, when Onnes reported that mercury, as well as a sample of very pure gold, had, at helium temperature, reached resistivities so low that his instruments had failed to detect them. He was particularly intrigued with the behavior of the mercury sample because it had still a fairly high resistance at liquid hydrogen temperatures and could also be recorded at the boiling point of liquid helium but then vanished at lower temperatures.”

Russ Hobbie and I discuss superconductivity in Section 8.9 (Detection of Weak Magnetic Fields) of the 4th edition of Intermediate Physics for Medicine and Biology.

“The [magnetic] signals from the body are weaker, and their measurement requires higher sensitivity and often special techniques to reduce noise. Hämäläinen et al. (1993) present a detailed discussion of the instrumentation problems. Sensitive detectors are constructed from superconducting materials. Some compounds, when cooled below a certain critical temperature, undergo a sudden transition and their electrical resistance falls to zero. A current in a loop of superconducting wire persists for as long as the wire is maintained in the superconducting state. The reason there is a superconducting state is a well-understood quantum-mechanical effect that we cannot go into here. It is due to the cooperative motion of many electrons in the superconductor [Eisberg and Resnick (1985), Sec. 14.1; Clarke (1994)].”

We then go on to discuss superconducting quantum interference device (SQUID) magnetometers, which are often used to measure the small magnetic fields produced by the brain or the heart. Although not discussed in our book, superconductivity is also used in many MRI machines to produce the strong static magnetic field without losses due to heating of a copper coil.

The citations in the quote from our book are to:

Clarke, J. (1994). SQUIDS. Sci. Am. Aug. 1994: 46–53.

Eisberg, R., and R. Resnick (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. New York, Wiley.

Hämäläinen, M., R. Harri, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa (1993). Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65(2): 413–497.

Point/Counterpoint Revisited

2011-04-08T05:56:00.005-04:00

In one of my first entries in this blog, I introduced readers to the Point/Counterpoint articles in the journal Medical Physics. I enjoy these articles immensely, and they provide valuable insight into important and controversial questions in the medical physics field. Each issue of Medical Physics contains one Point/Counterpoint, in which a proposition is stated and two prominent medical physicists debate it, one for and one against. They each have “opening statements” and then provide a “rebuttle” to their opponent’s claims. The articles make for more lively reading than a typical scientific paper full of jargon and technical content.

The September 2010 issue of Medical Physics contains a Point/Counterpoint article titled "Ultrasonography is Soon Likely to Become a Viable Alternative to X-Ray Mammography for Breast Cancer Screening". Arguing for the proposition is Carri Glide-Hurst, a Senior Associate Physicist at Henry Ford Hospital in Detroit. Her opponent is Andrew Maidment, Associate Professor of Radiology at the University of Pennsylvania in Philadelphia. I am in favor of the proposition, for the silly reason that I always root for the home team. (Oakland University, where I work, is about 20 miles north of Detroit, and our medical physics program has several adjunct faculty at Henry Ford Hospital.)

The 4th edition of Intermediate Physics for Medicine and Biology provides much of the scientific background needed to understand this debate. Russ Hobbie and I added a new chapter to the 4th edition that describes ultrasound and its applications to medical imaging (Chapter 13, Sound and Ultrasound). After introducing the wave equation, we describe the decibel intensity scale, attenuation, medical uses of ultrasound, and the Doppler effect. Two chapters are dedicated to understanding x rays and x-ray imaging. In Chapter 15 (Interaction of Photons and Charged Particles with Matter) we analyze the basic mechanisms by which an x-ray photon affects tissue, including the photoelectric effect, Compton scattering, and pair production. Chapter 16 (Medical Use of X Rays) focuses on applications, including a section dedicated entirely to mammography.

Magnetic Resonance Imaging (MRI) is another modality that produces no ionizing radiation. It is described in Chapter 18 of Intermediate Physics for Medicine and Biology. However, MRI is expensive, takes a long time, and cannot be used in some patients, such as those with surgical clips. Therefore the American Cancer Society recommends MRI only for a small group of patients.

Which side wins the debate? It’s always hard to say. I’m sure the moderator, Colin Orton, chooses only those questions that do not have obvious answers. Glide-Hurst concludes her opening statement by arguing

"Ultrasound poses a practical and affordable solution for screening younger women with dense breasts, pregnant females, and those who do not meet the risk level requirements of breast MRI screening. Overall, whole-breast ultrasound is advantageous because it is volumetric, noninvasive, and nonionizing, and the current literature supports the routine implementation for breast cancer screening, particularly for women with dense breasts."

Maidment ends his opening statement by stating

“Since ultrasound can distinguish solid tumors from fluidfilled cysts, it has a clear clinical role as a diagnostic tool in breast imaging. However, ultrasound does not appear useful for routine screening because of lower sensitivity and specificity compared to mammography, the suboptimal imaging of microcalcifications with ultrasonography, and the projected costs.”

All things considered, I do know who wins the debate. The winner is the reader, who witnesses two experts carefully weighing the evidence, analyzing the physics, and predicting future trends. I encourage any student reading Intermediate Physics for Medicine and Biology to also browse through recent issues of Medical Physics. If, like me, you are often short on time, skip the articles and just read Point/Counterpoint. You won’t regret it.

Fukushima Nuclear Reactors

2011-04-01T06:01:00.005-04:00

Because of the scary events at the Fukushima nuclear reactors in Japan, the health hazards of radiation is in the news a lot. One place I turn to for authoritative information is the Health Physics Society. Here is what their website says:

“As you are well aware, the Japanese experienced the worst earthquake in their history, followed by a devastating tsunami. These natural disasters have had a serious impact on several Japanese nuclear reactors, principally those at the Fukushima Daiichi site. The Health Physics Society is concerned about radiation exposures associated with these reactor problems and desires to keep our members and the concerned public advised on current events associated with the Japanese nuclear plants.

For information on the potential for radiation from the Japanese Nuclear Plants reaching the United States, see this Health Physics Society Ask the Experts FAQ. For information on radiation particle effects on food, read this Bloomberg FAQ.

Details of the status of the reactors at Fukushima are available in a document issued by the Japan Atomic Industrial Forum that is provided here. We will be updating this news item periodically to provide current information.”

The Health Physics Society links to an interesting youtube video: an interview with John Boice of Vanderbilt University. He says “the fear is out of proportion to the risk,” and claims this event is no where near the situation in the Chernobyl diasater. (Warning: The interview was on March 20, and events seem to change daily.) The website also links to the following statement:

"RADIATION RISKS TO HEALTH
A Joint Statement from the American Association of Clinical Endocrinologists, the American Thyroid Association, The Endocrine Society, and the Society of Nuclear Medicine
March 18, 2011

The recent nuclear reactor accident in Japan due to the earthquake and tsunami has raised fears of radiation exposure to populations in North America from the potential plume of radioactivity crossing the Pacific Ocean. The principal radiation source of concern is radioactive iodine including iodine-131, a radioactive isotope that presents a special risk to health because iodine is concentrated in the thyroid gland and exposure of the thyroid to high levels of radioactive iodine may lead to development of thyroid nodules and thyroid cancer years later. During the Chernobyl nuclear plant accident in 1986, people in the surrounding region were exposed to radioactive iodine principally from intake of food and milk from contaminated farmlands. As demonstrated by the Chernobyl experience, pregnant women, fetuses, infants and children are at the highest risk for developing thyroid cancer whereas adults over age 20 are at negligible risk.

Radioiodine uptake by the thyroid can be blocked by taking potassium iodide (KI) pills or solution, most importantly in these sensitive populations. However, KI should not be taken in the absence of a clear risk of exposure to a potentially dangerous level of radioactive iodine because potassium iodide can cause allergic reactions, skin rashes, salivary gland inflammation, hyperthyroidism or hypothyroidism in a small percentage of people. Since radioactive iodine decays rapidly, current estimates indicate there will not be a hazardous level of radiation reaching the United States from this accident. When an exposure does warrant KI to be taken, it should be taken as directed by physicians or public health authorities until the risk for significant exposure to radioactive iodine dissipates, but probably for no more than 1-2 weeks. With radiation accidents, the greatest risk is to populations close to the radiation source. While some radiation may be detected in the United States and its territories in the Pacific as a result of this accident, current estimates indicate that radiation amounts will be little above baseline atmospheric levels and will not be harmful to the thyroid gland or general health.

We discourage individuals needlessly purchasing or hoarding of KI in the United States. Moreover, since there is not a radiation emergency in the United States or its territories, we do not support the ingestion of KI prophylaxis at this time. Our professional societies will continue to monitor potential risks to health from this accident and will issue amended advisories as warranted."

News sources have been reporting that higher-than-normal radiation levels were detected in the United States. These observations say more about our ability to detect small amounts of radiation than about any risk to Americans. People living in the United States are at no risk of health hazards from radiation exposure caused by the Fukushima reactors.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of radiation in Section 13 of Chapter 16 (Medical Use of X Rays). We introduce the unit of the sievert (Sv), one of the most important units used when discussing radiation risk.

“Both the sievert and the gray are J kg^-1. Different names are used to emphasize the fact that they are quite different quantities. One is physical, and the other includes biological effects. An older unit … is the rem. 100 rem = 1 Sv.”

We then analyze the natural background dose, which is about 3 mSv per year, and which arises from several sources, including cosmic radiation, terrestrial rocks, and inhalation of radon gas.

If you prefer learning from a video, watch Understanding the Reactor Meltdown in Fukushima, Japan from a Physics Perspective on YouTube.

Time will tell if this event turns into a full-scale disaster. At the moment, it is a serious situation, but does not appear to be a serious health hazard, except perhaps for the workers trying to repair the power plants.

Maxwell Equation Sesquicentennial

2011-03-25T06:30:00.008-04:00

I am a big James Clerk Maxwell fan. In fact, I have made my living applying Maxwell’s equations to biology and medicine. Yes, I own one of those tee shirts with Maxwell’s equations written on it. I keep a copy of Maxwell’s A Treatise on Electricity and Magnetism in my office (although I have never read it in its entirety…Oh how I wish Maxwell had access to modern vector notation!). I have read The Maxwellians (outstanding) and The Man Who Changed Everything: The Life of James Clerk Maxwell (good). So, this month I am celebrating with gusto the sesquicentennial of the publication of Maxwell’s famous equations. The March 17 issue of the journal Nature has a special section containing four articles about Maxwell’s equation. In an editorial titled “A bold Unifying Leap” (Volume 471, Page 265) the editor writes

“In this issue we celebrate the first expression of those equations by Scottish physicist Maxwell in the Philosophical Magazine 150 years ago. There he drew together several strands of understanding about the behaviour of electricity, of magnetism, of light, and of the ways in which these fundamental aspects of nature behave in matter. As Albert Einstein remarked, ‘so bold was the leap’ of this work that it took decades for physicists to grasp its full significance. And although it was a wonderful expression of science at its purest, it was forged in the thoroughly practical culture of intellects at that time.”

Russ Hobbie and I mention Maxwell’s equations in the 4th edition of Intermediate Physics for Medicine and Biology. We added a new homework problem to the 4th edition in Chapter 8 (Biomagnetism):

“Problem 22 Write down in differential form (a) the Faraday induction law, (b) Ampere’s law including the displacement current term, (c) Gauss’s law, and (d) Eq. 8.7. … These four equations together constitute 'Maxwell’s equations.' Together with the Lorentz force law (Eq. 8.2), Maxwell’s equations summarize all of electricity and magnetism.”

All four of Maxwell’s equations are discussed in our book. Section 6.3 is dedicated to Gauss’s law, governing the electric field produced by a collection of charges, and we analyze the usual suspects: a line of charge and a charged sheet. Ampere’s law appears in Section 8.2 (The Magnetic Field of a Moving Charge or Current), and--in one of my favorite homework problems--we show in Problem 13 of Chapter 8 how "one can obtain a very different physical picture of the source of a magnetic field using the Biot Savart law than one gets using Ampere’s law, even though the field is the same." Faraday’s law is presented in Section 8.6 on Electromagnetic Induction, followed by a discussion of magnetic stimulation of the brain. Even Gauss’s law for a magnetic field (Eq. 8.7, stating that the magnetic field has no divergence) is introduced. Maxwell’s great insight was to add the displacement current term to Ampere’s law. We show how the charging of a capacitor implies the existence of this additional term on page 207, and explore its role in biomagnetism (slight).

Russ and I never analyze what may be the greatest prediction of Maxwell’s equations: the wave nature of light. We state in Section 14.1 that “the velocity of light traveling in a vacuum is given by electromagnetic theory as c = 1/√(ε₀ μ₀)”, but we never derive this result from Maxwell’s equations. Many of the applications of electromagnetic waves—such as wave guides, antennas, diffraction, radiation, and all of optics—are barely mentioned, if mentioned at all, in our text. For those who want to learn these topics (and all students of physics should want to learn these topics), I suggest Griffith’s Introduction to Electrodynamics (undergraduate) or Jackson’s Classical Electrodynamics (graduate). Richard Feynman introduces Maxwell’s equations in his celebrated book The Feynman Lectures on Physics. In Chapter 18 of Volume 2, he writes

“It was not customary in Maxwell’s time to think in terms of abstract fields. Maxwell discussed his ideas in terms of a model in which the vacuum was like an elastic solid. He also tried to explain the meaning of his new equation in terms of the mechanical model. There was much reluctance to accept his theory, first because of the model, and second because there was at first no experimental justification. Today, we understand better that what counts are the equations themselves and not the model used to get them. We may only question whether the equations are true or false. This is answered by doing experiments, and untold numbers of experiments have confirmed Maxwell’s equations. If we take away the scaffolding he used to build it, we find that Maxwell’s beautiful edifice stands on its own. He brought together all of the laws of electricity and magnetism and made one complete and beautiful theory.”

Anyone with a historical bent may want to read Maxwell’s original papers and accompanying commentary in Maxwell on the Electromagnetic Field: a Guided Study, by Thomas Simpson. The book contains a detailed analysis of Maxwell’s papers, including On the Physical Lines of Force, which is the publication we celebrate this month. Simpson’s book is the best place I know of to learn about the “scaffolding” Maxwell used to build his theory.

I will close with one of my favorite quotes, again from the Feynman Lectures. At the end of his first chapter introducing electromagnetism, Feynman writes

“From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.”

Murderous Microwaves

2011-03-18T06:31:00.007-04:00

I have written previously on the topic of cell phone electromagnetic radiation and cancer, but the issue remains a concern among the general public. Kenneth Foster reviewed three new books about the risks associated with cell phones in the March issue of IEEE Spectrum (disclaimer: I have not read any of these books):

Disconnect: The Truth About Cell Phone Radiation, What the Industry Has Done to Hide It, and How to Protect Your Family, by Devra Davis;

Zapped: Why Your Cell Phone Shouldn’t Be Your Alarm Clock and 1268 Ways to Outsmart the Hazards of Electronic Pollution, by Ann Louise Gittleman

Dirty Electricity: Electrification and the Diseases of Civilization, by Samuel Milham.

Foster writes

“Do you feel zapped, disconnected, electronically polluted by electromagnetic fields in your homes and workplace? Are you fearful of your electricity? These three books will feed your fears.

But are such fears justified? Public debates have been going on for more than a century about the possible health hazards of electromagnetic fields from power lines and radio-frequency energy from broadcast transmitters—and now cellphones. At the same time, health agencies have repeatedly reviewed the scientific literature and found no clear evidence of a problem. How can these totally different perspectives be reconciled?”

Foster ultimately concludes that these perspectives can’t be reconciled. He counters these alarmist books with exhaustive scientific studies, such as Exposure to High Frequency Electromagnetic Fields, Biological Effects and Health Consequences (100 kHz–300 GHz), Edited by Paolo Vecchia et al., International Commission on Non-Ionizing Radiation Protection, 2009; and Risk Analysis of Human Exposure to Electromagnetic Fields, by Zenon Sienkiewicz, Joachim Schüz, Aslak Harbo Poulsen, and Elisabeth Cardis, report of the European Health Risk Assessment Network on Electromagnetic Fields Exposure, 2010. The first report concludes that

“In the last few years the epidemiologic evidence on mobile phone use and risk of brain and other tumors of the head has grown considerably. In our opinion, overall the studies published to date do not demonstrate a raised risk within approximately ten years of use for any tumor of the brain or any other head tumor. However, some key methodologic problems remain - for example, selective non-response and exposure misclassification. Despite these methodologic shortcomings and the still limited data on long latency and long-term use, the available data do not suggest a causal association between mobile phone use and fast-growing tumors such as malignant glioma in adults, at least those tumors with short induction periods. For slow-growing tumors such as meningioma and acoustic neuroma, as well as for glioma among long-term users, the absence of associations reported thus far is less conclusive because the current observation period is still too short. Currently data are completely lacking on the potential carcinogenic effect of exposures in childhood and adolescence.”

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I examine this topic in Section 9.10, Possible Effects of Weak External Electric and Magnetic Fields. We focus on power line (60 Hz) fields (another story….), but many of the same conclusions apply to cell phone (1 GHz) fields. A key factor is the energy of a microwave photon.

“Radiated energy is in the form of discrete packets or photons, whose energy is related to the frequency of oscillation of the fields. The energy of each photon is E = hν, where h is Planck’s constant and ν the frequency. At room temperature, the energy of random thermal motion is k_BT = 4 × 10⁻²¹ J. At 60 Hz, the energy in each photon is much smaller: 4 × 10⁻³² J. At 100 MHz it is 7 × 10⁻²⁶ J.”

Therefore, cell phone frequencies correspond to photon energies that are nearly 10,000 times less than thermal energies. Moreover, the energy required to break chemical bonds is hundreds of times greater than thermal energies. If cancer is caused by the breaking of bonds in DNA by photons, then cell phone photons are one millions times too weak to cause cancer. If enough photons were present, the tissue temperature could rise, but no one has evidence that there is a significant heating of the brain by photons; the fields are not that strong. We are left with no plausible mechanism connecting microwaves and cancer.

With weak epidemiological evidence and no mechanism, I remain a hard-boiled skeptic. In fact, my only reservation with Foster’s review is that his criticisms may have been too tame. My views are closer to physicist Bob Park, who is a vocal (and often sarcastic) critic of those who insist that cell phones cause cancer. Nevertheless, even Foster’s mild criticisms triggered a heated debate in the comments section following his review. Interestingly, most of the comments make emotional arguments, not scientific ones, indicating the need for a better understanding by the pubic of the basic physics of how electromagnetic fields interact with tissue. (At this point, I again plug our book, Intermediate Physics for Medicine and Biology, as the best source to learn the physics …. although I admit on this one claim I may be slightly biased.)

So who should you believe in this debate? How about the National Cancer Institute? It is hard to think of a more unbiased or authoritative source of information. Their fact sheet provides a science-based analysis of the issue. But Ken Foster is a pretty reliable source of information too. He has spent nearly 40 years studying electricity and magnetism, with much of that analyzing the biological effects of E&M fields. His 1989 paper “Dielectric-Properties of Tissues and Biological Materials: A Critical Review,” (Crit. Rev. Biomed. Eng., Volume 17, Pages 25-104), written with Herman Schwan, is a highly-cited classic. Foster’s article “Risk Management: Science and the Precautionary Principle” (Science, Volume 288, Pages 979-981, 2000) provides useful insight into the role of scientific evidence in evaluating risk. Russ and I cite several of Foster’s papers in the 4th edition of Intermediate Physics for Medicine and Biology, including

Foster, K. R. (1996). Electromagnetic field effects and mechanisms: In search of an anchor. IEEE Eng. Med. Biol. 15(4): 50–56.

Foster, K. R., and H. P. Schwan (1996). Dielectric properties of tissues. In C. Polk and E. Postow, eds. Handbook of Biological Effects of Electromagnetic Fields. Boca Raton, FL, CRC Press, pp. 25–102.

Moulder, J. E., and K. R. Foster (1995). Biological effects of power-frequency fields as they relate to carcinogenesis. Proc. Soc. Expt. Biol. Med. 209: 309–323.

Moulder, J. E., and K. R. Foster (1999). Is there a link between power-frequency electric fields and cancer? IEEE Engineering in Medicine and Biology 18(2): 109–116.

(Note: when preparing this blog entry, I found that we have the title to the last paper incorrect in our book. It should be "Is there a link between exposure to power-frequency electric fields and cancer?" I will correct that in the erratum, found at the book website.)

In conclusion, I don’t believe the evidence supports the hypothesis that cell phones cause cancer. Give me some convincing new evidence or a plausible mechanism, and I’ll reconsider.

Retinal Injuries from a Handheld Laser Pointer

2011-03-11T06:43:00.005-05:00

Are laser pointers safe? Apparently, it depends on the laser pointer. A recent article by Christine Negroni in the New York Times (Feb. 28, 2011) states that

“Eye doctors around the world are warning that recent cases of teenagers who suffered eye damage while playing with high-power green laser pointers are likely to be just the first of many.”

Negroni cites a letter that appeared last September in the New England Journal of Medicine (Wyrsch, Baenninger, and Schmid, “Retinal Injuries from a Handheld Laser Pointer”, N. Engl. J. Med., 2010, Volume 363, Pages 1089-1091), which says

“In the past, laser pointers sold to the public had a maximal output of 5 mW, which is regarded as harmless because the human eye protects itself with blink reflexes. The measured output of the laser in [the case of a person who was injured] was 150 mW. The use of lasers that are threatening to the eye is normally restricted to occupational and military environments; laser accidents outside these fields are very rare. However, powerful laser devices, with a power of up to 700 mW, are now easily obtainable through the Internet, despite government restrictions. These high-power lasers are advertised as ‘laser pointers’ and look identical to low-power pointers. The much higher power of such devices may produce immediate, severe retinal injury. Despite their potential to cause blinding, such lasers are advertised as fun toys and seem to be popular with teenagers. In addition, Web sites now offer laser swords and other gadgets that use high-power lasers.”

I attended a talk just last week where the speaker waved his green laser pointer around like a light saber. I don’t know the power of his pointer, but I wonder if I was in danger.

One concern arises from the bozos who point lasers at airplanes. The U.S. Congress plans to toughen the laws on this sort of horseplay, making shining a laser at a plane a federal crime with up to five years imprisonment. I’m all for high school students learning science by hands-on activities, but do it right. Buy a 5 mW red helium-neon laser pointer and use it safely to do some optics experiments (I suggest observing Young’s double slit interference pattern). Don’t buy a 700 mW green laser pointer and start shining it up into the sky! Do you think I am being a schoolmarm out to ruin your fun? Consider this: the website laserpointersafety.com reports that

“A $5000 reward is being offered for information leading to the arrest of the person(s) who aimed a laser into the cockpit of a Southwest Airlines flight approaching Baltimore-Washington International Airport. The flight, which originated in Milwaukee, was 2000 feet over the town of Millersville, near Old Mill Road and Kenora Drive, when it was illuminated around 6:45 pm on Sunday, Feb. 20, 2011. Millersville is about 8 miles from BWI Airport.”

You better be careful; someone may be watching.

How do you tell the difference between a safe, educational experience and a potentially disastrous prank? You begin by learning about light and its biological impact. Russ Hobbie and I discuss light in Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology. We address topics related to light and safety, although we don’t analyze the particular concern of laser damage to the eye. For instance, we discuss how ultraviolet light damages the eye (Section 14.9.6) and how light can be used to heat tissue (Section 14.10), as well as a detailed discussion of radiometry (the measurement of radiant energy, Section 14.11) and the anatomy and optics of the eye (Section 14.12).

In another New York Times article, Negroni relates how high powered laser pointers can pose a risk to pilots. And on her blog, she explains why helicopters may be at a greater risk than airplanes.

“A helicopter cockpit has glass extending below the level of the pilots' eyes toward the ground exactly where the lasers are. Rotor craft fly at low altitudes over residential areas and busy highways. They are not flying autopilot and they may be piloted by a single person. They hover and may make inviting targets. That was the case on Tuesday when a Los Angeles television station sent its chopper to follow and report on the police activity and it was hit by a laser. ”

If you prefer video, watch and listen to Christine Negroni on MSNBC.

The interaction of laser light and vision is one more example of why a firm understanding of physics applied to medicine and biology is so important.

The Role of Magnetic Forces in Biology and Medicine

2011-03-04T05:55:00.005-05:00

The current issue of the journal Experimental Biology and Medicine contains a minireview about The Role of Magnetic Forces in Biology and Medicine by yours truly (Volume 236, Pages 132-137). It fits right in with Section 8.1 (The Magnetic Force on a Moving Charge) in the 4th Edition of Intermediate Physics for Medicine and Biology. The abstract states:

“The Lorentz force (the force acting on currents in a magnetic field) plays an increasingly larger role in techniques to image current and conductivity. This review will summarize several applications involving the Lorentz force, including (1) magneto-acoustic imaging of current; (2) ‘Hall effect’ imaging; (3) ultrasonically-induced Lorentz force imaging of conductivity; (4) magneto-acoustic tomography with magnetic induction; and (5) Lorentz force imaging of action currents using magnetic resonance imaging.”

The review was easy to write, because it consisted primarily of the background and significance section of a National Institutes of Health grant proposal I wrote several years ago (and which is now funded). The review describes ground-breaking work by many authors, but here I want to highlight studies by three talented undergraduate students who worked with me at Oakland University during several summers.

Kaytlin Brinker

Kayt studied a method to measure conductivity called Magneto-Acoustic Tomography with Magnetic Induction, or MAT-MI (Brinker and Roth, 2008, The Effect of Electrical Anisotropy During Magnetoacoustic Tomography with Magnetic Induction. IEEE Trans. Biomed. Eng., 55:1637–1639). This technique was developed by Bin He and his group at the University of Minnesota. You apply two magnetic fields, one static and one changing with time. The rapidly changing magnetic field induces eddy currents in the tissue, which then experience a Lorentz force from the static field, causing the material to move and initiating a sound wave. Measurement of the acoustic signal allows you to gain information about the conductivity distribution. Kayt’s task was to determine how anisotropy (the conductivity depends on direction) would influence MAT-MI. She “found that when imaging nerve or muscle, electrical anisotropy can have a significant effect on the acoustic signal and must be accounted for in order to obtain accurate images.”

Nancy Tseng

Nancy, who had just graduated from high school when she worked with me, analyzed a technique originally pioneered by Han Wen and then developed further by Amalric Montalibet. A sound wave is propagated through the tissue in the presence of a magnetic field. The Lorentz force causes charge separation, inducing an electrical potential and current. Measurement of the electrical signal provides information about the conductivity. Tseng looked at this effect in anisotropic tissue (Tseng and Roth, 2008, The Potential Induced in Anisotropic Tissue by the Ultrasonically-Induced Lorentz Force. Med. Biol. Eng. Comput., 46:195–197). She found “a novel feature of the ultrasonically-induced Lorentz force in anisotropic tissue: an oscillating electrical potential propagates along with the ultrasonic wave.” The effect has not yet been measured experimentally, but represents a fundamentally new mechanism for the induction of bioelectric signals.

Kevin Schalte

Kevin derived a tomographic method to determine tissue conductivity using the ultrasonically-induced Lorentz force (Roth and Schalte, 2009, Ultrasonically-Induced Lorentz Force Tomography, Med. Biol. Eng. Comput., 47:573-577). “The strength and timing of the electric dipole caused by the ultrasonically-induced Lorentz force determines the amplitude and phase of the Fourier transform of the conductivity image. Electrical measurements at a variety of [ultrasonic] wavelengths and directions are therefore equivalent to mapping the Fourier transform of the conductivity distribution in spatial frequency space. An image of the conductivity itself is then found by taking the inverse Fourier transform.” I would never have undertaken this project had I not been a coauthor on the 4th edition of Intermediate Physics for Medicine and Biology. Only by working on the textbook did I come to fully understand and appreciate the power of tomography (see Chapter 12 on Images and Section 16.9 about Computed Tomography).

I often read about how the United States is falling behind other nations in math and science, but working with outstanding undergraduates such as these three gives me confidence that we remain competitive.

Finally, let me reproduce the all-important acknowledgments section of the minireview:

“I thank Steffan Puwal and Katherine Roth for their comments on this manuscript. I also thank Bruce Towe, Han Wen, Amalric Montalibet and Xu Li for permission to reproduce their figures in this review. This work was supported by the National Institutes of Health grant R01EB008421.”

Round-Number Handbook of Physics for Medicine and Biology

2011-02-25T06:47:00.011-05:00

The 4th edition of Intermediate Physics for Medicine and Biology contains a list of fundamental constants in Appendix O. Russ Hobbie and I got the values of these constants from a 2002 study, but the National Institute of Science and Technology (NIST) website we cite no longer exists. A new NIST website, http://physics.nist.gov/cuu/Constants/index.html, gives the most up-to-date values for these constants, often including many significant figures.

For some applications, knowing the electron mass to, say, nine significant figures is important. But in biology and medicine, most quantities are not known with such precision. If a number is known to one percent, that is impressive. When I teach biological and medical physics, I would much rather see my students have an approximate feel for the size of important constants, without having them bother to memorize more than one or two significant figures. To know that the speed of light is 299,792,458 m/s is nice, but what I really want them to remember (forever!) is that the speed of light is about 3 × 10⁸ m/s. If they need more precision, they can look it up.

Edward Purcell, one of the great ones, published his “Round-Number Handbook of Physics” in the January 1983 issue of the American Journal of Physics. He presented a list of important physical constants, but only to one or two significant figures. It was meant not as a reference to look up precise values, but as a list of approximate values that every physicist should know without needing to consult a reference. Unfortunately, Purcell used cgs units, which are becoming more and more obsolete.

Below I present my version of a “Round-Number Handbook of Physics for Medicine and Biology”. I take the constants from Appendix O and approximate them as round numbers in (mostly) mks units. These are the numbers you should remember.

c	Speed of light	3 × 10⁸ m/s
e	Elementary charge	1.6 × 10^-19 C
F	Faraday constant	10⁵ C/mole
g	Acceleration of gravity	10 m/s²
h	Planck’s constant	2π × 10^-34 J s
ℏ	Planck’s constant (reduced)	10^-34 J s
k_B	Boltzmann’s constant	1.4 × 10^-23 J/K
m_e	Electron mass	9 × 10^-31 kg
m_ec²	Electron rest energy	0.5 MeV
m_p	Proton mass	1.7 × 10^-27 kg
m_pc²	Proton rest energy	1000 MeV
N_A	Avogadro’s number	6 × 10²³ 1/mole
r_e	Classical radius of the electron	3 × 10^-15 m
R	Gas constant	8 J/(mole K)
		2 cal/(mole °C)
ε₀	Electrical permittivity	9 × 10^-12 F/m
1/4πε₀	Coulomb’s law constant	9 × 10⁹ N m²/C²
σ_SB	Stefan Boltzmann constant	6 × 10^-8 W/(m² K⁴)
λ_C	Compton wavelength of the electron	2.4 pm
μ_B	Bohr magneton	10^-23 J/T
μ₀	Magnetic permeability	4π × 10^-7 H/m
μ_N	Nuclear magneton	5 × 10^-27 J/T

Tc-99m Production: Losing the Reactor

2011-02-18T06:32:00.004-05:00

Periodically in this blog I have discussed the growing technetium-99m shortage that faces medical physics (see, for instance, here, here, here, and here). Russ Hobbie and I discuss technetium in the 4th edition of Intermediate Physics for Medicine and Biology.

“The most widely used isotope is ^99mTc. As its name suggests, it does not occur naturally on earth, since it has no stable isotopes … The isotope is produced in the hospital from the decay of its parent, ⁹⁹Mo, which is a fission product of ²³⁵U and can be separated from about 75 other fission products. The ⁹⁹Mo decays to ^99mTc.”

Interestingly, the ^99mTc shortage here in the United States may be solved in part by our friends up north (or, for those of us living in the Detroit area, our friends down south; look at a map), the Canadians. You can learn more in an article on medicalphysicsweb.org (and I hope you are a regular reader of that very useful website).

"Technetium-99m (Tc-99m) is the most widely used medical imaging isotope, employed in more than 30 million procedures worldwide each year. The isotope is created via decay of molybdenum-99 (Mo-99), which itself is produced in nuclear reactors. And herein lies the problem.

The nuclear reactor is needed to generate neutrons that bombard uranium-235 targets, with the resulting fission reaction producing Mo-99 around 6% of the time. This Mo-99 then decays into Tc-99m. Unfortunately, over 90% of the world's Mo-99 is produced by just five ageing reactors, resulting in an extremely fragile supply chain - the vulnerability of which was highlighted recently when unexpected shutdowns and routine maintenance closures combined to create serious shortages.

But there are other ways to create Tc-99m, and ways that don't require nuclear reactors or a uranium target – itself a cause for concern as most facilities currently process highly-enriched (weapons-grade) uranium. Instead, researchers are investigating production methods based on cyclotrons and linear accelerators. Such processes exploit nuclear reactions within targets of Mo-100, bypassing the need for uranium completely.

In a bid to advance such technologies, the government of Canada has invested $35 million in four development programmes. The projects are headed up by: TRIUMF (Vancouver, BC); Canadian Light Source (Saskatoon, SK); Advanced Cyclotron Systems (Richmond, BC); and Prairie Isotope Production Enterprise (Winnipeg, MB) ….

In terms of practical implementation, the cyclotron-based method produces Tc-99m, which has a half-life of just six hours and must therefore be manufactured at or very near to clinical sites. This approach can, however, take advantage of a wide network of existing medical cyclotrons.

The electron accelerator approach creates Mo-99, which has a half-life of 66 hours and, as such, can be shipped. "One or two linacs could probably supply most of Canada," Barnard said. This method also benefits from being more similar to, and thus able to exploit, the existing Tc-99m supply chain based on shipping of Mo-99."

The article was written by medicalphysicsweb's editor, Tami Freeman, who has worked as a journalist for the Institute of Physics for the last dozen years, and who has a PhD in physics.

P.S. There is a nice article in the February issue of Physics Today about U.S. attempts to address the Tc-99m shortage (see the comments to this blog entry).

The Framingham Heart Study

2011-02-11T06:28:00.006-05:00

The Framingham Heart Study is one of the oldest and most widely cited research studies in the history of medicine. Russ Hobbie and I mention the study briefly In Section 2.4 of the 4th edition of Intermediate Physics for Medicine and Biology, when discussing exponential decay.

“Figure 2.8 shows the survival of patients with congestive heart failure for a period of nine years. The data are taken from the Framingham study [McKee et al. (1971)]; the death rate is constant during this period.”

The data in Fig. 2.8 is from a paper with over 1400 citations in the scientific and medical literature: P. A. McKee, W. P. Castelli, P. M. McNamara, and W. B. Kannel (1971). The natural history of congestive heart failure: The Framingham study. New Engl. J. Med. 285:1441-1446. The abstract to the paper states

“The natural history of congestive heart failure was studied over a 16-year period in 5192 persons initially free of the disease. Over this period, overt evidence of congestive heart failure developed in 142 persons. In almost every five-year age group, from 30 to 62 years, the incidence rate was greater for men than for women. Although the usual etiologic precursors were found, the dominant one was clearly hypertension, which preceded failure in 75 per cent of the cases. Coronary heart disease was noted at an earlier examination in 39 per cent, but in 29 per cent of the cases it was accompanied by hypertension. Precursive rheumatic heart disease, noted in 21 per cent of cases of congestive heart failure, was accompanied by hypertension in 11 per cent. Despite modern management, congestive heart failure proved to be extremely lethal. The probability of dying within five years from onset of congestive heart failure was 62 per cent for men and 42 per cent for women.”

In 2005, Daniel Levy and Susan Brink published A Change of Heart: How the Framingham Heart Study Helped Unravel the Mysteries of Cardiovascular Disease. The book is a fascinating history of this landmark study. Levy (the study’s current director) and Brink (formerly a writer for U.S. News & World Report) write

“A turning point in our evolving understanding of heart disease was the establishment of the Framingham Heart Study in 1948. It was a large and ambitious community-based research project unlike anything that had been conducted before. It came at a time of growing awareness that cardiovascular disease was sweeping the country, even slowing down what should have been a steady rise in life expectancy. It was also a time, three years after the end of World War II, when resources from the national treasury, no longer needed for military purposes, could be used for research into the nation’s leading killer….

In light of this ignorance [of how to treat coronary disease], the U.S. government in 1948 made a twenty-year commitment to uncovering the root causes of heart disease. That scientific resolve was sponsored by the U.S. Public Health Service with half a million dollars of start-up funding from Congress. A cadre of physicians, scientists, government officials, and academics—many of whom knew each other from having served together at military hospitals during the war—selected a New England town in which to carry out this national scientific experiment. The Framingham Heart Study turned out to be instrumental in changing the attitudes, if not the behavior, of virtually every American, and it put the otherwise ordinary town of Farmingham, Massachusetts, on the map….

They [the Heart Study researchers] needed the 5209 men and women from Framingham at first, followed by 5124 of their sons and daughters, and now 3500 of their grandchildren who have donated their medical histories to science. It is ironic, perhaps, that this most respected—even beloved—piece of epidemiology centers on the heart, the organ that symbolically aches, breaks, longs, and loves like no other. It took a commitment from thousands of volunteers to make the study a success.”

I found Chapter 5, “The People Who Changed America’s Heart: Voices from Framingham,” to be particularly inspiring. For instance, they quote Evelyn Langley—housewife, mother, and PTA president—who played an early role in promoting the study among potential participants, and was a participant herself.

“Langley’s heart still lies with the Study. ‘When they call me up and tell me it’s time to come in for an exam, I know I have that ritual to do,’ she says. She has made the trip to the clinic twenty-seven times so far. ‘I am trying to give back to the Heart Clinic [Study] what they have given me. I always feel as if I am part of something bigger than myself. It’s not just for the people who live in this town. Many lives have been saved because of the Heart Study.' ”

You can learn more about the Framingham Heart Study at the study’s website: http://www.framinghamheartstudy.org. Also, you can view a video about it from CBS’s Sunday Morning with Charles Osgood. The study is currently funded by the National Heart, Lung, and Blood Institute (part of the National Institutes of Health) and Boston University. Let me finish with a fitting quote from the acknowledgments of A Change in Heart:

“This book would not have been possible without the more than fifty years of dedication and commitment from three generations of Framingham Heart Study volunteers. We would like to thank them all for providing a gift to the world that has changed untold millions of lives.”

Britton Chance (1913-2010)

2011-02-04T06:33:00.007-05:00

Britton Chance died late last year. The website www.brittonchance.org states that

“Britton Chance, M.D., Ph.D., D.Sc., for more than 50 years one of the giants of biochemistry and biophysics and a world leader in transforming theoretical science into useful biomedical and clinical applications, died on November 16, 2010, at age 97 in Philadelphia, PA. Dr. Chance had the rare distinction of being the recipient of a National Medal of Science (1974), a Gold Medal in the Olympics (1952, Sailing, Men’s 5.5 Meter Class), and a Certificate of Merit for his sensitive work during World War II.”

His obituary in the New York Times describes his early work.

“Over a lifetime of research, Dr. Chance focused on the observation and measurement of chemical reactions within cells, tissue and the body. But unlike most researchers, he also had expertise in mechanics, electronics and optics, and a great facility in instrument-building. His innovations helped transform theoretical science into biochemical and biophysical principles, the stuff of textbooks, and useful biomedical and clinical applications.

Early in his career he invented a tool, known as a stopped-flow apparatus, for measuring chemical reactions involving enzymes; it led to the establishment of a fundamental principle of enzyme kinetics, known as the enzyme-substrate complex.”

Another obituary, in the December 17 issue of Science magazine, observed that

“in his mid-70s, Chance (then emeritus) launched a new field of optical diagnostics that rests on the physics of light diffusion through scattering materials such as living tissue. He showed that scattered near-infrared light pulses could not only measure the dynamics of oxy- and deoxyhemoglobin levels in performing muscles, but also reveal and locate tumors and cancerous tissue in muscles and breast as well as injury in the brain. Because changing patterns of oxy- and deoxyhemoglobin in the brain reflect cognitive activity, the applications of this diagnostic approach widened to include assessing neuronal connectivity in premature babies.”

Chance appears in the 4th edition of Intermediate Physics for Medicine and Biology because of his research on light diffusion. In Section 14.4 (Scattering and Absorption of Radiation), Russ Hobbie and I analyze the absorption and scattering coefficients of infrared light, and then give typical values that “are eyeballed from data from various tissues reported in the article by Yodh and Chance (1995),” with the reference being to “Yodh, A. and B. Chance (1995). Spectroscopy and imaging with diffusing light. Phys. Today. March: 34-40.”

Then in Sec. 14.5 (The Diffusion Approximation to Photon Transport), we analyze pulsed measurements of infrared light.

“A technique made possible by ultrashort light pulses from a laser is time-dependent diffusion. It allows determination of both [the scattering coefficient] and [the absorption coefficient]. A very short (150-ps) pulse of light strikes a small region on the surface of the tissue. A detector placed on the surface about 4 cm away records the multiply-scattered photons. A typical plot of the detected photon fluence rate is shown in Fig. 14.13.”

Figure 14.13 is a figure from "Patterson, M. S., B. Chance, and B. C. Wilson (1989). Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties. Appl. Opt. 28:2331-2336," which has been cited over 1000 times in the scientific literature.

Finally, in Sec. 14.6 (Biological Applications of Infrared Scattering), we reproduce a figure from the Physics Today article by Yodh and Chance, which shows the absorption coefficient for water, oxyhemoglobin and deoxyhemoglobin.

“The greater absorption of blue light in oxygenated hemoglobin makes oxygenated blood red…The wavelength 800 nm at which both forms of hemoglobin have the same absorption is called the isosbestic point. Measurements of oxygenation are made by comparing the absorption at two wavelengths on either side of this point.”

This property of infrared absorption of light is the basis for pulse oximeters that measure oxygenation. Not all measurements of blood oxygen use pulsed light. Russ and I cite one of Chance’s papers that uses a continuous source: "Liu, H., D. A. Boas, Y. Zhang, A. G. Yodh, and B. Chance (1995). Determination of optical properties and blood oxygenation in tissue using continuous NIR light. Phys. Med. Biol., 40:1983-1993." A fourth of Chance’s paper that we include in our references is "Sevick, E. M., B. Chance, J. Leigh, S. Nioka, and M. Maris (1991). Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation. Analyt. Biochem. 195:330-351."

In 1987, Chance won the Biological Physics Prize (now known as the Max Delbruck Prize in Biological Physics) from the American Physical Society

"for pioneering application of physical tools to the understanding of Biological phenomena. The early applications ranged from novel spectrometry that elucidated electron transfer processes in living systems to analog computation of nonlinear processes. Later contributions have been equally at the forefront."

The Quantum Ten

2011-01-28T06:46:00.003-05:00

Over the holiday break, I read The Quantum Ten: A Story of Passion, Tragedy, Ambition and Science, by Sheilla Jones. The book is about the development of quantum mechanics in the 1920’s.

“The seeds of the shift currently taking place in science were sown eighty years ago, from 1925 to 1927. That’s when a dramatic two-year revolution in physics reached a climax, the denouement set the course for what was to follow. It’s the story of a rush to formalize quantum physics, the work of just a handful of men fired by ambition, philosophical conflicts, and personal agendas….

Remarkably, this dramatic shift in science was primarily the work of ten men, and they were ten fallible men, some famous and some not so famous, although they also had a large supporting cast. The triumphs and tragedies, loves and betrayals, dreams realized and ambitions thwarted, shaped the competition over who would get to define truth and reality. There never was a consensus. By the time of the pivotal Fifth Solvay Conference in Brussels in 1927, there was so much ill will and disappointment among the creators of quantum physics over their various competing theories and over who deserved credit that most were barely on speaking terms.

The Brussels conference was the first time so many of them had come together: Albert Einstein, the lone wolf; Niels Bohr, the obsessive but gentlemanly father figure; Max Born, the anxious hypochondriac; Werner Heisenberg, the intensely ambitious one; Wolfgang Pauli, the sharp-tongued critic with a dark side; Paul Dirac, the quiet one; Erwin Schrodinger, the enthusiastic womanizer; Prince Louis de Broglie, the French aristocrat; and Paul Ehrenfest, who was witness to it all. Their coming together, however, lasted only for the duration of the conference.”

I enjoyed the book, but couldn’t help wishing that it would focus less on the personal problems of the scientists and more on their science. I prefer my scientific biographies to be a bit more rigorous with an emphasis on the science, like Pais’s Subtle is the Lord. Nevertheless, the story was fascinating in a gossipy sort of way. The book is full of tidbits like this:

“From time to time [Schrodinger] did consult on the mathematics with his Zurich colleague Hermann Weyl, who was at that point embroiled in a passionate love affair with Schrodinger’s wife, Anny. Wince the Weyls were part of the same sexually permissive crowd as the Schrodingers, the affair was no cause for tension between the two colleagues.”

I found myself oddly attracted to Paul Ehrenfest, “an intense physicist with a debilitating streak of self-doubt who could rarely see the valuable gift he offered to physics and a passionate friend to both Einstein and Bohr.” Then, near the end of the book, I discovered--to my horror--that not only did Ehrenfest take his own life (I had heard that before), but that just before he committed suicide he shot and killed his son. My admiration vanished.

There was no biological physics in The Quantum Ten, but I couldn’t help wonder how these great scientists fared in the 4th edition of Intermediate Physics for Medicine and Biology. A quick survey gave the following results:

Albert Einstein. I discussed Einstein’s presence in our textbook about a year ago in this blog, and concluded that “we rarely mention Einstein by name in our book, but his influence is present throughout, and most fundamentally when we discuss the idea of a photon.”
Niels Bohr. His model for the hydrogen atom is referred to, but not derived. His contributions to calculating the stopping power of a charged particle in tissue are discussed in Chapter 15 (Interaction of Photons and Charged Particles with Matter).
Paul Ehrenfest. His name never appears in our book.
Max Born. The Born charging energy is discussed in Chapter 6 (Impulses in Nerve and Muscle Cells).
Erwin Schrodinger. The Schrodinger equation is mentioned in Chapter 3 (Systems of Many Particles), but never written down.
Wolfgang Pauli. The Pauli exclusion principle is stated in Chapters 14 (Atoms and Light) and 15 (Interaction of Photons and Charged Particles with Matter).
Louis de Broglie. His name is not in the book, although I have mentioned him in this blog before.
Werner Heisenberg. He and his uncertainty principle are not in the book.
Paul Dirac. I discussed Dirac in the blog before. His delta function shows up in Chapter 11 (The Method of Least Squares and Signal Analysis).
Pascual Jordan. His name never appears in our book.

I am not overly concerned that the quantum ten don’t figure prominently in Intermediate Physics for Medicine and Biology. Russ Hobbie and I do not focus on microscopic phenomena, where quantum mechanics is essential. Probably the greatest contribution to biological physics from any of the quantum ten is Schrodinger’s book What is Life?, which had a major impact on the early development of molecular biology (see The Eighth Day of Creation).

P.S. We had a significant revision of the errata this week. It is available at the book’s website: http://www.oakland.edu/~roth/hobbie.htm. A big thank you to Gabriela Castellano for finding many mistakes and pointing them out to us. If you, dear reader, find additional mistakes, please let us know.

Gaussian integration

2011-01-21T06:36:00.003-05:00

Chapter 8 in the 4th edition of Intermediate Physics for Medicine and Biology covers Biomagnetism: the measurement of the magnetic field produced by electrical currents in nerve and muscle. One issue that arises during biomagnetic recordings is that the magnetic field is not measured at a point, but is averaged over a pickup coil. Therefore, when comparing theoretical calculations to experimental data, you need to integrate the calculated magnetic field over the coil.

One way to do this is Gaussian quadrature, which approximates the integral by a weighted sum. Homework problem 40 in Chapter 8 shows a three-point Gaussian quadrature formula for integrating over a circular coil. At the end of the problem Russ Hobbie and I write “Higher-order formulas for averaging the magnetic field can be found in Roth and Sato (1992).” The reference is to “Roth, B. J. and S. Sato (1992) Accurate and efficient formulas for averaging the magnetic field over a circular coil. In M. Hoke, S. N. Erne, T. C. Okada, and G. L. Romani, eds. Biomagnetism: Clinical Aspects. Amsterdam, Elsevier.” This book is actually the proceedings of the 8th International Conference on Biomagnetism, held in Munster, Germany on August 19-24, 1991. I didn’t attend that meeting, but my colleague Susumu Sato did. Sato is a senior scientist in the Epilepsy Research Branch of the National Institute of Neurological Disorders and Stroke, part of the National Institutes of Health in Bethesda, Maryland. When I worked with him he had an active research program in magnetoencephalography (MEG), including a large and expensive shielded room and a multi-channel SQUID magnetometer.

The introduction of our paper states

“The MEG is measured by detecting the magnetic flux through a pickup coil, usually circular, that is coupled to a SQUID magnetometer. Often the source of the MEG is modeled as a current dipole, whose position, orientation and strength are determined iteratively by fitting the MEG data to a dipolar magnetic field pattern. To obtain an accurate result, this dipole field must be integrated over the pickup coil area to obtain the magnetic flux. Since this integration is repeated for each dipole considered in the iteration, the numerical algorithm used to estimate this integral should be efficient. In this note, several integration formulas are presented that allow the magnetic field to be integrated over the coil area quickly with little error. These formulas are examples of a general technique of approximating multiple integrals described by Stroud [1].”

Reference [1] is to: Stroud AH (1971) Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, New Jersey, pp. 277-289.

I remember deriving several of these formulas independently before discovering Stroud's textbook (it is always deflating to find you’ve been scooped). The derivation requires solving a system of nonlinear equations (which I rather enjoyed). Each formula requires evaluating the magnetic field at N points, and the integral is accurate to m^th order. We presented a 1-point formula accurate to first order, a 3-point formula accurate to second order (this was the formula examined in the homework problem), a 4-point formula accurate to third order, a 6-point formula accurate to fourth order, a 7-point formula accurate to fifth order, and a 12-point formula accurate to seventh order.

The general formulation of Gaussian quadrature was developed by Carl Friedrich Gauss (1777-1855), one of the greatest mathematicians of all time. Gauss's name appears often in the 4th edition of Intermediate Physics for Medicine and Biology, including the Gaussian function (Chapter 4), Gauss's law (Chapter 6), the cgs unit for the magnetic field of a gauss (Chapter 8), the Fast Fourier transform (FFT, Chapter 11) about which we write "the grouping used in the FFT dates back to Gauss in the early nineteenth century," and the Gaussian Probability Distribution (Appendix I).

DNA animation by Drew Berry

2011-01-14T06:30:00.003-05:00

I know that the 4th edition of Intermediate Physics for Medicine and Biology doesn’t discuss much about the physics of life at the molecular level. In the preface, Russ Hobbie and I wrote that “molecular biophysics has been almost completely ignored.” Nevertheless, I recently ran across an animation of DNA that is so good I just have to tell you about it.

My story starts with the January-February issue of American Scientist, the science and technology magazine published by Sigma Xi, The Scientific Research Society. The cover of this issue shows DNA, packed “tightly in some chromosomal territories and loosely in others, forming sheer walls and intergenic fissures, as seen in the cover image from a 3D animation by renowned molecular animator Drew Barry.” When I read this, I asked myself: Who is Drew Berry, and where can I find his animations?

It turns out you can find Berry’s wonderful animation “Molecular Visualizations of DNA” on youtube: http://www.youtube.com/watch?v=7wpTJVWra7I. Trust me, you really want to watch this video. It explains DNA packing into chromosomes, transcription, and translation in a visual way that is unforgettable. Other Berry animations can be found at http://www.wehi.edu.au/education/wehitv.

In 2010, Berry was awarded a MacArthur Fellowship from the John D and Catherine T MacArthur Foundation, the so-called “genius award”. The MacArthur website says

“Drew Berry is a biomedical animator whose scientifically accurate and aesthetically rich visualizations are elucidating cellular and molecular processes for a wide range of audiences. Trained as a cell biologist as well as in light and electron microscopy, Berry brings a rigorous scientific approach to each project, immersing himself in the relevant research in structural biology, biochemistry, and genetics to ensure that the most current data are represented. In three- and four-dimensional renderings of such key biological concepts as cell death, tumor growth, and the packaging of DNA, Berry captures the details of molecular shape, scale, behavior, and spatio-temporal dynamics in striking form. His groundbreaking series of animations of the intricate biochemistry of DNA replication, translation, and transcription demonstrates these multifaceted processes in ways that enlighten both scientists and the scientifically curious. The sequence and pace of each molecular interaction are precisely coordinated, at the same time as the ceaseless motion of the molecules reveals the complex and seemingly random choreography of the molecular world. Committed to educating the public about critical topics in medical research, Berry created a two-part animation of the malaria life cycle that illustrates the pathogen’s development in the mosquito host and its invasion of and diffusion throughout human cells. In these and many other projects in progress, Berry synthesizes data across a variety of fields and presents them in engaging and lucid animations that both inspire a sense of wonder and enhance understanding of biological systems.

Drew Berry received B.Sc. (1993) and M.Sc. (1995) degrees from the University of Melbourne. Since 1995, he has been a biomedical animator at the Walter and Eliza Hall Institute of Medical Research. His animations have appeared in exhibitions and multimedia programs at such venues as the Museum of Modern Art, the Guggenheim Museum, the Royal Institute of Great Britain, and the University of Geneva.”

Convergence

2011-01-07T06:26:00.003-05:00

This week researchers at the Massachusetts Institute of Technology released a white paper titled “The Third Revolution: The Convergence of the Life Sciences, Physical Sciences, and Engineering”. It begins

“There are few challenges more daunting than the future of health care in this country. This paper introduces the dynamic and emerging field of convergence—which brings together engineering and the physical and life sciences—and explains how convergence provides a blueprint for addressing the health care challenges of the 21st century by producing a new knowledge base, as well as a new generation of diagnostics and therapeutics. We discuss how convergence enables the innovation necessary to meet the growing demand for accessible, personalized, affordable health care. We also address the role of government agencies in addressing this challenge and providing funding for innovative research. Finally, we recommend strategies for embedding convergence within agencies like the National Institutes of Health (NIH), which aims to optimize basic research, improve health technology, and foster important medical advances.”

If “convergence” is the melding of physics and engineering with the life sciences, then I suggest that a good place to start your search for convergence is the 4th edition of Intermediate Physics for Medicine and Biology. The MIT white paper is singing our song about the integration of physics with biology. But I am a Johnny-come-lately to convergence compared to my coauthor, Russ Hobbie, who pioneered this approach decades ago.

“Between 1971 and 1973 I audited all the courses medical students take in their first two years at the University of Minnesota. I was amazed at the amount of physics I found in these courses.”

You can find more about the white paper in an article in the Science Insider. The authors talk about three revolutions in biomedicine: the first was molecular and cellular biology, the second was genomics, and the third will be convergence. I must admit that I find the white paper a little self-serving; most of their examples feature MIT researchers (says the guy who writes a weekly blog about physics in medicine and biology with the goal of peddling textbooks!). But I agree with its premise. Indeed, the first sentence of their concluding paragraph sounds as if it could be a promotion for our book.

“The merger of the life, engineering, and physical sciences promises to fundamentally alter and speed our scientific trajectory. NIH and other affected agencies, if adequately funded and made ready, can be thought leaders in this next scientific revolution. The time is right for NIH and other agencies to take up convergence as the wave of the future, creating dramatic new opportunities in medicine for new therapies and diagnostics, economic opportunity, as well as promise in many other scientific fields, from energy to climate to agriculture.”

Brownian Motion

2010-12-31T05:18:00.005-05:00

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Brownian motion. We first address this topic in Chapter 3 when deriving the equipartition of energy: the average thermal kinetic energy of an object at temperature T is 3k_BT/2, where k_B is Boltzmann’s constant.

“This result is true for particles of any mass: atoms, molecules, pollen grains, and so forth. Heavier particles will have a smaller velocity but the same average kinetic energy. Even heavy particles are continually moving with this average kinetic energy. The random motion of pollen particles in water was first seen by a botanist, Robert Brown, in 1827. This Brownian motion is an important topic in the next chapter.”

We next address this topic in Chapter 4, as we motivate the reader for a discussion of diffusion.

“This movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by the English botanist Robert Brown in 1827 and is called Brownian motion. Solute particles are also subject to this random motion. If the concentration of particles is not uniform, there will be more particles wandering from a region of high concentration to one of low concentration than vice versa. This motion is called diffusion.”

As so often happens when you look deeply into a subject, the story is more complicated than can be described in an introductory (or even an intermediate) textbook. In the December 2010 issue of the American Journal of Physics, Philip Pearle and his colleagues published the fascinating article What Brown Saw and You Can Too (Volume 78, Pages 1278-1289).

“A discussion of Robert Brown’s original observations of particles ejected by pollen of the plant Clarkia pulchella undergoing what is now called Brownian motion is given. We consider the nature of those particles and how he misinterpreted the Airy disk of the smallest particles to be universal organic building blocks. Relevant qualitative and quantitative investigations with a modern microscope and with a 'homemade' single lens microscope similar to Brown’s are presented.”

One interesting conclusion of their study is that Brown did not actually see pollen grains move.

"We emphasize that Brown did not observe the pollen move. Instead, he observed the motion of much smaller objects that reside within the pollen.¹⁵ Nonetheless, statements that Brown saw the pollen move are common.¹⁶"

Fortunately, reference 16 does not cite Intermediate Physics for Medicine and Biology. But it raises the question: Did Russ and I get it wrong? Our discussion in Chapter 4 seems safe. The text in Chapter 3 depends on if you interpret “pollen particles” as the entire “pollen grain” or “particles arising from pollen”. Russ may have been aware of this distinction when he wrote the original text, but I confess I was not. I always thought Brown saw the entire pollen grain move.

Pearle et al. show electron microscope pictures of pollen grains, which are 50-100 microns in diameter. I summarize their analysis about what Brown actually saw move as a new homework problem for Chapter 4

Problem 5 1/2. This problem looks at the original observations of Robert Brown that established Brownian motion.
(a) Combine Eqs. 4.23 and 4.71 to determine an expression for the average distance a particle of radius a will diffuse through a fluid of viscosity η in time t.
(b) Assume you observe a pollen grain with a radius of 50 microns in water at room temperature, and that your visual perception is particularly sensitive to motions occurring over a time of about one second. What is the average distance you observe the grain to move?
(c) Now assume your eye cannot see movements that occur over angles of less than 1 minute of arc, or 3 × 10^-4 radians (in Chapter 14, we estimate 3 minutes of arc, but use 1 arc min to be conservative). Most eyes cannot focus on objects closer than 25 cm. Determine the smallest displacement you can observe with the naked eye.
(d) Robert Brown had a microscope that could magnify objects by a factor of about 370. What is the smallest displacement he could observe with his microscope? Is this larger or smaller than the displacement of a pollen grain in one second?
In fact, Brown did not observe the motion of entire pollen grains. He observed fat and starch particles about 2 microns in diameter that are released by pollen. For more on Brown’s original observations, see Pearle et al. (2010).

The authors also analyze the microscope that Brown used, and estimate the diffraction effects he had to contend with. Using an analysis similar to that presented in Section 13.7 (Medical Uses of Ultrasound) of Intermediate Physics for Medicine and Biology, they show that Brown probably could not resolve some of the smaller particles, but instead observed their diffraction pattern. As in Eq. 13.40 in our textbook, the diffraction pattern involves a Bessel function, and implies that the apparent size of an object is larger than the real size. The effect is minor for large objects but dominates for small objects.

I find the history and analysis of Brown’s original studies to be fascinating. For me, Pearle et al.’s paper reminds me that 1) the American Journal of Physics is still my favorite journal, and 2) physics has much to offer biology and medicine.

giga	G	10⁹
mega	M	10⁶
kilo	k	10³
milli	m	10⁻³
micro	μ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²
femto	f	10⁻¹⁵
atto	a	10⁻¹⁸