Wilhelm Roentgen

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.

Asimov's Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

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

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’s 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

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 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 Chapter 18 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss gadolinium.

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 Kishor Karki, 2009.

The Cyclotron

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 Rhodes’ 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’s 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

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.

Textbook of Medical Physiology,
by Guyton and Hall.

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.