Leon Glass wins Winfree Prize

From Clocks to Chaos,
by Glass and Mackey.

Leon Glass was honored recently by the Society for Mathematical Biology. Their website states

The Society for Mathematical Biology is pleased to announce that this year’s recipient of the Arthur T. Winfree prize is Prof. Leon Glass of McGill University. Awarded every two years to a scientist whose work has “led to significant new biological understanding affecting observation/experiments,” this prize commemorates the creativity, imagination and intellectual breadth of Arthur T. Winfree.

Beginning with simple and brilliantly chosen experiments, Leon launched the study of chaos in biology. Among the applications he and his many collaborators and students pursued was the novel idea of “dynamical disease” and the better understanding of pathologies like Parkinson’s disease and cardiac arrhythmias. His elegant work (with Michael Guevara and Alvin Shrier) on periodic stimulation of heart cells demonstrated and explained how the interaction of nonlinearities with oscillations create complex dynamics and chaos.

The book From Clocks to Chaos, which he co-authored with Michael Mackey, was an instant classic that illuminated this difficult subject for a whole generation of mathematical biologists. His combination of imagination, experimental and mathematical insight, and ability to communicate fundamental principles has launched new fields of research and inspired researchers ranging from applied mathematicians to medical researchers.

Leon Glass is the Isadore Rosenfeld Chair in Cardiology at McGill University. Russ Hobbie and I cite From Clocks to Chaos (discussed previously in this blog) in the 4th edition of Intermediate Physics for Medicine and Biology, especially in Chapter 10 when discussing nonlinear dynamics. According to Google Scholar, the book has been cited 1800 times. Even more highly cited (over 2600 times) is Mackey and Glass’s paper “Oscillation and Chaos in Physiological Control Systems” (Science, Volume 197, Pages 287–289, 1977), which Russ and I also cite.

Other books and papers mentioned in IPMB include

Bub, G., A. Shrier, and L. Glass (2002) “Spiral wavegeneration in heterogeneous excitable media,” Phys. Rev. Lett., Volume 88, Article Number 058101.

Glass, L., Y. Nagai, K. Hall, M. Talajic, and S. Nattel (2002) “Predicting the entrainment of reentrant cardiacwaves using phase resetting curves,” Phys. Rev. E, Volume 65, Article Number 021908.

Guevara, M. R., L. Glass, and A. Shrier (1981) “Phaselocking,period-doubling bifurcations and irregular dynamicsin periodically stimulated cardiac cells,” Science Volume 214, Pages 1350–1353.

Glass, L. (2001) “Synchronization and rhythmicprocesses in physiology,” Nature, Volume 410, Pages 277–284.

Kaplan, D., and L. Glass (1995) Understanding NonlinearDynamics. New York, Springer-Verlag.

You can listen to Glass talk about cardiac arrhythmias below.

Leon Glass talks about cardiac arrhythmias.

https://www.youtube.com/embed/ebZUhi5lDSc

1932: A Watershed Year in Nuclear Physics

I should have posted this article last year, to mark the 80th anniversary of the annus mirabilis of nuclear physics. Unfortunately, I didn’t think of it until I read “1932: A Watershed Year in Nuclear Physics” by Joseph Reader and Charles Clark, which appeared in the March 2013 issue of Physics Today. The article describes four major discoveries that changed nuclear physics forever.

 

Deuterium

The first landmark result was published by Harold Urey on January 1, 1932, in which he reported his discovery of deuterium, the isotope ²H. Russ Hobbie and I mention deuterium in Homework Problem 45 of Chapter 4 in the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 45 Using the definitions in Problem 44, write the diffusion constant in terms of λ and v_rms. By how much do you expect the diffusion constant for “heavy water” (water in which the two hydrogen atoms are deuterium, ²H) to differ from the diffusion constant for water? Assume the mean free path is independent of mass.

Unlike many elements, for which the various stable isotopes differ in mass be only a few percent, deuterium has twice the mass of normal hydrogen. Even when deuterium is incorporated into water, the H₂O molecule’s mass increases by a significant 11%. Heavy water has been used as a non-radioactive biological tracer.

The Neutron

A second advance of 1932, and in my opinion the most important, is the discovery of the neutron by James Chadwick. Of course, the idea of a neutron is central to nuclear physics, and you cannot make sense of isotopes without it (I wonder how Urey interpreted the deuterium before the neutron was discovered). Russ and I discuss neutrons throughout Chapter 17 on Nuclear Medicine, and in particular we describe boron neutron capture therapy in Chapter 16

Boron neutron capture therapy (BNCT) is based on a nuclear reaction which occurs when the stable isotope ¹⁰B is irradiated with neutrons, leading to the nuclear reaction (in the notation of Chapter 17)

¹⁰₅B + ¹₀n → ⁴₂α + ⁷₃Li

... Both the alpha particle and lithium are heavily ionizing and travel only about one cell diameter. BNCT has been tried since the 1950s; success requires boron-containing drugs that accumulate in the tumor.

The Positron

Discovery number three is the positron, the first example of antimatter. Carl Anderson found evidence of this positive electron in cosmic ray tracks in cloud chambers. Positrons appear in IPMB in two key places. In Chapter 15 (The Interaction of Photons and Charged Particles with Matter) positrons are key to pair production.

A photon with energy above 1.02 MeV can produce a particle–antiparticle pair: a negative electron and a positive electron or positron… Since the rest energy (m_ec²) of an electron or positron is 0.51 MeV, pair production is energetically impossible for photons below 2m_ec² = 1.02 MeV.

The positron appears again in our discussion of β⁺ decay in Chapter 17.

Two modes of decay allow a nucleus to approach the stable line. In beta (β⁻ or electron) decay, a neutron is converted into a proton. This keeps A [mass number] constant, lowering N [neutron number] by one and raising Z [atomic number] by one. In positron (β⁺) decay, a proton is converted into a neutron. Again A remains unchanged, Z decreases and N increases by 1. We find β⁺ decay for nuclei above the line of stability and β^- decay for nuclei below the line.

Isotopes that undergo β⁺ decay are used in positron emission tomography.

If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence….

PET can provide a functional image with information about metabolic activity. A very common positron agent is ¹⁸F fluorodeoxyglucose—glucose in which a hydroxyl group has been replaced with ¹⁸F. The PET signal is largest in those cells that have taken up the ¹⁸F because they are actively metabolizing glucose. PET has become particularly important in studies of brain function, where active neurons are detected by an increase in their metabolism, and in locating metastatic cancer.

Accelerators

The last of the four great developments of 1932 is the first use of accelerators to study nuclear reactions. John Cockcroft and Ernest Walton built an accelerator to produce high energy protons, which smashed into ⁷Li to produce two alpha particles. Their work was soon followed by the invention of the cyclotron by Ernest Lawrence, which is now the main tool for producing the unstable isotopes used in PET. Russ and I explain that

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.

The Making of the Atomic Bomb,
by Richard Rhodes.

Soon after the miraculous year of 1932 Hitler came to power in Germany, and nuclear physics became much more than a scientific curiosity. The story of how the discoveries of Urey, Chadwick, Anderson, Cockcroft and Walton led relentlessly to the Manhattan Project is told masterfully in Richard Rhodes’ book The Making of the Atomic Bomb.

I have a few personal connections to this watershed year. My father Ron Roth, now retired and living in Lenexa Kansas, was born in 1932, proving that we are not so far removed from that historic time. In addition, my academic genealogy goes back to James Chadwick and Ernest Rutherford (whose lab Cockcroft and Walton worked in). Finally, Carl Anderson worked under the supervision of Robert Millikan, who was born in Morrison, Illinois, the small town I grew up in.

Barouh Berkovits (1926-2012)

When my March 2013 issue of the journal Heart Rhythm arrived this week, I found in it an obituary for Barouh Berkovits, who died last year.

Barouh Vojtec Berkovits passed away on October 23, 2012, at the age of 86 years. Berkovits was a master of science and an electrical engineer. Born in 1926 in Lucenec, Czechoslovakia (today Czech Republic), he worked as a technician behind the enemy lines. He escaped the Holocaust, but his parents and sister Eva perished in Auschwitz, Poland. In 1949 he immigrated to Israel and in 1956 to the United States… Berkovits invented and patented the first demand pacemaker capable of sensing the R wave…For his contributions to the treatment of cardiac arrhythmias, Berkovits received the “Distinguished Scientist Award” in 1982 by the Heart Rhythm Society.

Machines in Our Hearts,
by Kirk Jeffrey.

The story of how Berkovits invented the demand pacemaker is told in Machines in Our Hearts, by Kirk Jeffrey.

Barouh V. Berkovits (b. 1924), an engineer at the American Optical Company, was already well known as the inventor of the DC defibrillator and the cardioverter, a device that interrupts a rapid heart rate (tachycardia) with low-energy shocks. He knew that when the cardioverter discharged randomly into the tachycardia, it would “occasionally not only not stop the tachyarrhythmia…but would produce ventricular fibrillation.” Cardioversion has to be synchronized to fall within the QRS complex and avoid the vulnerable period of the heartbeat. In 1963, Berkovits applied this principle to cardiac pacing. To solve the problem of competition [between the SA node and the artificial pacemaker], Berkovits in 1963 designed a sensing capability into the pacemaker. His invention behaved exactly like an asynchronous pacer until it detected a naturally occurring R wave, the indication of a ventricular contraction. This event would reset the timing circuit of the pacemaker, and the countdown to the next stimulus would begin anew. Thus the pacer stimulated the heart only when the ventricles failed to contract. It worked only “on demand.” As an added benefit, non-competitive pacing extended the life of the battery.

The 4th edition of Intermediate Physics for Medicine and Biology does not mention Berkovits by name, but Homework Problem 45 in Chapter 7 does analyze the demand pacemaker.

Problem 45 A patient with “intermittent heart block” has an AV node which functions normally most of the time with occasional episodes of block, lasting perhaps several hours. Design a pacemaker to treat the patient. Ideally, your design will not stimulate the heart when it is functioning normally. Describe

(a) whether you will stimulate the atria or ventricles

(b) which chambers you will monitor with a recording electrode

(c) what logic your pacemaker will use to determine when to stimulate. Your design may be similar to a “demand pacemaker” described in Jeffrey (2001), p. 132.

Of course, the reference is to Machines in Our Hearts. Berkovits’s phenomenal career is yet another example of how knowledge of engineering and physics can allow you to contribute to medicine and biology.

The Technology of Medicine

In Chapter 5 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the artificial kidney as an example of the use of physics and engineering to solve a medical problem. Rather than delving into the engineering details, today we consider the larger question of technology in medicine. Russ and I write

The reader should also be aware that this “high-technology” solution to the problem of chronic renal disease is not an entirely satisfactory one… The distinction between a high-technology treatment and a real conquest of a disease has been underscored by Thomas (1974, pp. 31–36).

The Lives of a Cell,
by Lewis Thomas.

The citation is to the book The Lives of a Cell by Lewis Thomas (physician, poet, essayist, administrator). To me, his essays—written 40 years ago—sound surprisingly modern. For instance, the introduction to his essay about “The Technology of Medicine” is relevant today as we struggle with the role of expensive technology in the ever-increasing cost of health care.

Technology assessment has become a routine exercise for the scientific enterprises on which the country is obliged to spend vast sums for its needs. Brainy committees are continually evaluating the effectiveness and cost of doing various things in space, defense, energy, transportation, and the like, to give advice about prudent investments for the future. Somehow medicine, for all the $80-odd billion that it is said to cost the nation [$2-something trillion in 2013], has not yet come in for much of this analytical treatment. It seems taken for granted that the technology of medicine simply exists, take it or leave it, and the only major technologic problem which policy-makers are interested in is how to deliver today's kind of health care, with equity, to all the people.

When, as is bound to happen sooner or later, the analysts get around to the technology of medicine itself, they will have to face the problem of measuring the relative cost and effectiveness of all the things that are done in the management of disease. They make their living at this kind of thing, and I wish them well, but I imagine they will have a bewildering time… 

The analysts have finally gotten around to it. As our nation spends 15% of our Gross Domestic Product on health care, the costs of technology in medicine are no longer taken for granted. The Affordable Care Act (a.k.a. Obamacare) focuses on research into the comparative effectiveness of treatments, often measured by the incremental cost-effectiveness ratio. And as Thomas predicted, the analysts are having a bewildering time dealing with it.

Thomas divides “technology” into three types. His first type is not really technology at all (“no-technology”). It is “supportive therapy”, that “tides patients over through diseases that are not, by and large, understood.” There is not much physics in this category, so we will move on.

The second level of technology, which he calls “halfway technology,” represents “the kinds of things that must be done after the fact, in efforts to compensate for the incapacitating effects of certain diseases whose course one is unable to do very much about. It is a technology designed to make up for disease, or to postpone death.” The artificial kidney, as well as kidney transplants, fall into this category, and “almost everything offered today for the treatment of heart disease is at this level of technology, with the transplanted and artificial hearts as ultimate examples.” There is lots of physics in this category. Yet, Thomas sees it as, at best, an intermediate—and expensive—type of medical solution. “In fact, this level of technology is, by its nature, at the same time highly sophisticated and profoundly primitive. It is the kind of thing that one must continue to do until there is a genuine understanding of the mechanisms involved in disease.”

The third type of technology is based on a complete understanding of the causes of disease. Thomas writes that it “is the kind that is so effective that it seems to attract the least public notice; it has come to be taken for granted. This is the genuinely decisive technology of modern medicine, exemplified best by modern methods for immunization against diphtheria, pertussis, and the childhood virus diseases, and the contemporary use of antibiotics and chemotherapy for bacterial infections.”

Is there physics in this third category? I think so. Biological mechanisms will be based, ultimately, on the constraints of physical laws, and one can’t hope to understand biology without physics (at least, this is what I believe). Perhaps we can say that physics and engineering are essential for the second type of technology, whereas physics and biology are essential for the third type.

Thomas clearly favors the third category. He concludes

The point to be made about this kind [the third type] of technology—the real high technology of medicine—is that it comes as the result of a genuine understanding of disease mechanisms, and when it becomes available, it is relatively inexpensive, and relatively easy to deliver.

Offhand, I cannot think of any important human disease for which medicine possesses the outright capacity to prevent or cure where the cost of the technology is itself a major problem. The price is never as high as the cost of managing the same diseases during the earlier stages of no-technology or halfway technology…

It is when physicians are bogged down by their incomplete technologies, by the innumerable things they are obliged to do in medicine when they lack a clear understanding of disease mechanisms, that the deficiencies of the health-care system are most conspicuous. If I were a policy-maker, interested in saving money for health care over the long haul, I would regard it as an act of high prudence to give high priority to a lot more basic research in biologic science. This is the only way to get the full mileage that biology owes to the science of medicine, even though it seems, as used to be said in the days when the phrase still had some meaning, like asking for the moon.

As we face the looming crisis of budget sequestration, with its devastating cutbacks in funding for the National Institutes of Health and the National Science Foundation, and as the calls for translational medical research increase, I urge our legislators to heed Thomas’s advice and “give high priority to a lot more basic research.”

Helium Shortage!

A recent article in the New York Times discusses the looming shortage of helium.

A global helium shortage has turned the second-most abundant element in the universe (after hydrogen) into a sought-after scarcity, disrupting its use in everything from party balloons and holiday parade floats to M.R.I. machines and scientific research….

Experts say the shortage has many causes. Because helium is a byproduct of natural gas extraction, a drop in natural gas prices has reduced the financial incentives for many overseas companies to produce helium. In addition, suppliers’ ability to meet the growing demand for helium has been strained by production problems around the world. Helium plants that are being built or are already operational in Qatar, Algeria, Wyoming and elsewhere have experienced a series of construction delays or maintenance troubles.

One medical use of helium is discussed in the 4th edition of Intermediate Physics for Medicine and Biology. In Chapter 8, Russ Hobbie and I write about the role of helium in magnetoencephalography—the biomagnetic measurement of electrical activity in the brain—using Superconducting Quantum Interference Device (SQUID) magnetometers.

The SQUID must be operated at temperatures where it is superconducting. It used to be necessary to keep a SQUID in a liquid-helium bath, which is expensive to operate because of the high evaporation rate of liquid helium. With the advent of high-temperature superconductors, SQUIDS have the potential to operate at liquid-nitrogen temperatures, where the cooling problems are much less severe [for additional information, see here].

A more wide-spread use of helium in medicine is during magnetic resonance imaging. Chapter 18 of our book discusses MRI, but it does not describe how the strong, static magnetic field required by MRI is created. In a clinical MRI system, a magnetic field (typically 2 to 4 T) must exist over a large volume. Producing such a magnetic field using permanent magnets would, if possible, require giant, massive, expensive structures. A more reasonable method to create this field is using coils carrying a large current. One way to minimize the resulting Joule heating losses in the coils is to make them out of superconducting wire, which must be cooled cryogenically. An article on the Time Magazine online newsfeed states

Liquid helium has an extremely low boiling point—minus 452.1 degrees Fahrenheit, close to absolute zero—which makes it a perfect substance for cooling the superconducting magnets found in MRI machines. Hospitals are generally the first in line for helium, so the shortage isn’t affecting them yet. But prices for hospital-grade helium may continue to go up, leading to higher health-care costs or, in the worst-case scenario, the need for a backup plan for cooling MRI machines.

More detail about the use of helium during MRI can be found in an online book titled The Basics of MRI by Joseph Hornak. Below I quote some of the text, but you will need to go the book website to see the pictures and animations.

The imaging magnet is the most expensive component of the magnetic resonance imaging system. Most magnets are of the superconducting type. This is a picture of a first generation 1.5 Tesla superconducting magnet from a magnetic resonance imager. A superconducting magnet is an electromagnet made of superconducting wire. Superconducting wire has a resistance approximately equal to zero when it is cooled to a temperature close to absolute zero (−273.15° C or 0 K) by immersing it in liquid helium. Once current is caused to flow in the coil it will continue to flow as long as the coil is kept at liquid helium temperatures. (Some losses do occur over time due to infinitely small resistance of the coil. These losses can be on the order of a ppm of the main magnetic field per year.)

The length of superconducting wire in the magnet is typically several miles. The coil of wire is kept at a temperature of 4.2 K by immersing it in liquid helium. The coil and liquid helium is kept in a large dewar. The typical volume of liquid Helium in an MRI magnet is 1700 liters. In early magnet designs, this dewar was typically surrounded by a liquid nitrogen (77.4 K) dewar which acts as a thermal buffer between the room temperature (293 K) and the liquid helium. See the animation window for a cross sectional view of a first generation superconducting imaging magnet.

In later magnet designs, the liquid nitrogen region was replaced by a dewar cooled by a cryocooler or refrigerator. There is a refrigerator outside the magnet with cooling lines going to a coldhead in the liquid helium. This design eliminates the need to add liquid nitrogen to the magnet, and increases the liquid helium hold time to 3 to 4 years. The animation window contains a cross sectional view of this type of magnet. Researchers are working on a magnet that requires no liquid helium.

With the discovery of high temperature superconductivity (HTS), MRI magnets cooled at higher temperatures, avoiding the need for liquid helium, are possible. The ideal solution to the helium shortage would be superconducting coils cooled with liquid nitrogen. Nitrogen makes up 80% of our atmosphere, so it is free and virtually limitless. However, a 2010 article by scientists at the MIT Francis Bitter Magnet Laboratory (FBML) suggests that a more practical solution might be the use of solid nitrogen to reach temperatures of 20 K, for which superconducting materials such as magnesium diboride (MgB₂) exist that have the properties required for magnet coils.

A tremendous progress achieved in the past decade and is continuing today has transformed selected HTS materials into “magnet-grade” conductors, i.e., meet rigorous magnet specifications and are readily available from commercial wire manufacturers [1]. We are now at the threshold of a new era in which HTS will play a key role in a number of applications— here MgB₂ (T_c=39 K) is classified as an HTS. The HTS offers opportunities and challenges to a number of applications for superconductivity. In this paper we briefly describe three NMR/MRI magnets programs currently being developed at FBML that would be impossible without HTS: 1) a 1.3 GHz NMR magnet; 2) a compact NMR magnet assembled from YBCO [yttrium barium copper oxide] annuli; and 3) a persistent-mode, fully-protected MgB₂ 0.5-T/800-mm whole-body MRI magnet.

Even if new MRI magnets using solid nitrogen or some other abundant substance as the coolant were developed, there are thousands of existing MRI devices that still would require liquid helium and would be very expensive to replace. Congress is currently considering legislation to address the helium shortage (see article here). We urgently need to preserve our helium supply to ensure its availability for important medical devices.

P.S. I saw this article just a few days ago. High temperature superconductors for MRI may be just around the corner!

Magnetoacoustic Tomography with Magnetic Induction

Magnetoacoustic tomography with magnetic induction is a new method to image the distribution of electrical conductivity in tissue. Bin He, the director of the Institute for Engineering in Medicine at the University of Minnesota, developed this technique with his student Yuan Xu in a 2005 publication (Physics in Medicine and Biology, Volume 50, Pages 5175–5187). They describe MAT-MI in their introduction.

We have developed a new approach called magnetoacoustic tomography with magnetic induction (MAT-MI) by combining ultrasound and magnetism. In this method, the object is in a static magnetic field and a time-varying (μs) magnetic field... The time-varying magnetic field induces an eddy current in the object. Consequently, the object will emit ultrasonic waves through the Lorentz force produced by the combination of the eddy current and the static magnetic field. The ultrasonic waves are then collected by the detectors located around the object for image reconstruction. MAT-MI combines the good contrast of EIT [electrical impedance tomography] with the good spatial resolution of sonography.

One nice feature of MAT-MI is that it fits so well into the 4th edition of Intermediate Physics for Medicine and Biology, in which Russ Hobbie and I analyze both eddy currents caused by Faraday induction (Chapter 8) and ultrasound imaging (Chapter 13). Another characteristic of MAT-MI is that the physics is simple enough that it can be summarized in a homework problem. So, dear reader, here is a new problem that will help you understand MAT-MI.

Section 8.6

Problem 25 ½  Assume a sheet of tissue having conductivity σ is placed perpendicular to a uniform, strong, static magnetic field B₀, and a weaker spatially uniform but temporally oscillating magnetic field B₁(t).
(a) Derive an expression for the electric field E induced by the oscillating magnetic field. It will depend on the distance r from the center of the sheet and the rate of change of the magnetic field.
(b) Determine an expression for the current density J by multiplying the electric field by the conductivity.
(c) The force per unit volume, F, is given by the Lorentz force, J×B₀ (ignore the weak B₁). Find an expression for F.
(d) The source of the ultrasonic pressure waves can be expressed as the divergence of the Lorentz Force. Derive an expression for ∇ · F.
(e) Draw a picture showing the directions of J, B₀, and F.

While this example is simple enough to serve as a homework problem, it does not illustrate imaging of conductivity; the conductivity is uniform so there is no variation to image. As He and Yuan explain, if the conductivity varies with position, this will also contribute to ∇ · F, and therefore influence the radiated ultrasonic wave. Thus, information about the conductivity distribution σ(x,y) is contained in the pressure. Subsequent papers by He and his colleagues explore methods for extracting σ(x,y) from the ultrasonic signal. Potential applications include using MAT-MI to image breast cancer tumors.

I’ve worked on MAT-MI a little bit. University of Michigan student Kayt Brinker and I published a paper describing MAT-MI in anisotropic tissue like skeletal muscle, where the conductivity is much higher parallel to the muscle fibers than perpendicular to them [Brinker, K. and B. J. Roth (2008) “The effect of electrical anisotropy during magneto-acoustic tomography with magnetic induction,” IEEE Transactions on Biomedical Engineering, Volume 55, Pages 1637–1639]. For some reason the figures published by the journal were not of high quality, so here I reproduce a better version of Figure 6, which shows the pressure wave produced during MAT-MI.

Fig. 6. Pressure at 20, 40, 60, and 80 μs in isotropic and anisotropic tissue.
Each panel represents a 400 mm by 400 mm area.

In isotropic tissue, the wave propagates outward, the same in all directions. In electrically anisotropic tissue, the pressure is greater in the direction perpendicular to the fiber axis (vertical) than parallel to it (horizontal). The main difference between our calculation and that in the new homework problem given above is that Kayt and I restricted the oscillating magnetic field B₁ to a small region (40 mm radius) at the center of the tissue sheet.

The Response of a Spherical Heart to a Uniform Electric Field

In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the bidomain model of cardiac tissue.

Myocardial cells are typically about 10 μm in diameter and 100 μm long. They have the added complication that they are connected to one another by gap junctions, as shown schematically in Fig. 7.27. This allows currents to flow directly from one cell to another without flowing in the extracellular medium. The bidomain (two-domain) model is often used to model this situation [Henriquez (1993)]. It considers a region, small compared to the size of the heart, that contains many cells and their surrounding extracellular fluid.

The citation is to the 20-year-old-but-still-useful review article by Craig Henriquez of Duke University.

Henriquez, C. S. (1993) “Simulating the electrical behavior of cardiac tissue using the bidomain model,” Crit. Rev. Biomed. Eng., Volume 21, Pages 1–77.

According to Google Scholar, this landmark paper has been cited over 450 times (including a citation on page 202 of IPMB).

During the early 1990s I collaborated with another researcher from Duke, Natalia Trayanova. Our goal was to apply the bidomain model to the study of defibrillation of the heart. In the same year that Craig’s review appeared, Trayanova, her student Lisa Malden, and I published an article in the IEEE Transactions on Biomedical Engineering titled “The Response of the Spherical Heart to a Uniform Electric Field: A Bidomain Analysis of Cardiac Stimulation” (Volume 40, Pages 899–908). I’m fond of this paper for several reasons:

• Like most physicists, I like simple models that highlight and clarify basic mechanisms. Our spherical heart model had that simplicity.
• The article was the first to show that fiber curvature provides a mechanism for polarization of cardiac tissue in response to an electrical shock. Since our paper, researchers have appreciated the importance of the fiber geometry in the heart when modeling electrical stimulation.
• The model emphasizes the role of unequal anisotropy ratios in the bidomain model. In cardiac tissue, both the intracellular and extracellular spaces are anisotropic (the electrical conductivity is different parallel to the myocardial fibers then perpendicular to them), but the intracellular space is more anisotropic than the extracellular space. Fiber curvature will only result in polarization deep in the heart wall if the tissue has unequal anisotropy ratios.
• The calculation has important clinical implications. Fibrillation of the heart is a leading cause of death in the United States, and the only way to treat a fibrillating heart is to apply a strong electric shock: defibrillation. I’ve performed a lot of numerical simulations in my career, but none have the potential impact for medicine as my work on defibrillation.
• The IEEE TBME publishes brief bios of the authors. Back in those days I published in this journal often, and my goal was to have my entire CV included, bit by bit, in these small bios. The one in this paper read “Bradley J Roth was raised in Morrison, Illinois. He received the BS degree in physics from the University of Kansas in 1982, and the PhD in physics from Vanderbilt University in 1987. His PhD dissertation research was performed in the Living State Physics Laboratory under the direction of Dr. J. WIkswo. He is now a Senior Staff Fellow with the Biomedical Engineering and Instrumentation Program, National Institutes of Health, Bethesda, MD. One of this research interests is the mathematical modeling of the electrical behavior of the heart. He is also interested in the production and interactions of magnetic fields with biological tissue, e.g. biomagnetism, magnetic stimulation, and magnetoacoustic imaging.”
• The acknowledgments state “the authors thank B. Bowman for his assistance in editing the manuscript.” Barry was a great help to me in improving my writing skills during my years at NIH, and I’m glad that we mentioned him.
• The paper cites several of my favorite books, including When Time Breaks Down by Art Winfree, Classical Electrodynamics by John David Jackson, and Handbook of Mathematical Functions with Formulas, Graphs, and and Mathematical Tables, by Abramowitz and Stegun.
• The paper has been fairly influential. It’s been cited 97 times, which is small potatoes compared to Henriquez’s review, but not too shabby nevertheless; an average of almost five citations a year for 20 years.
• It was a pleasure to collaborate with Natalia Trayanova, who I was to work with again seven years later on another study of cardiac electrical behavior (Lindblom, Roth, and Trayanova, Journal of Cardiovascular Electrophysiology, Volume 11, Pages 274–285, 2000).
• The paper led to subsequent simulations of defibrillation that are much more realistic and sophisticated than our simple spherical model of twenty years ago. Trayanova has led the way in this research, first at Duke, then at Tulane, and now at Johns Hopkins. You can listen to her discuss her research here. If you have a subscription to the Journal of Visualized Experiments you can hear more here. For a recent review, see Trayanova et al. (2012). Also, see this article recently put out by Johns Hopkins University. 

Listen to Natalia Trayanova discuss developing computer simulations to improve arrhythmia treatments.

https://www.youtube.com/watch?v=oW40gJUBSQ4

Cardiac Bioelectric Therapy:
Mechanisms and Practical Implications.

To learn more about how physics and engineering can help us understand defibrillation, consult the book Cardiac Bioelectric Therapy: Mechanisms and Practical Implications, which has chapters by Trayanova and many of the other leading researchers in the field (including yours truly).

The Joy of X

The Joy of X,
by Steven Strogatz.

Steven Strogatz’s latest book is The Joy of X: A Guided Tour of Math, From One to Infinity. I have discussed books by Strogatz in previous entries of this blog, here and here. The preface defines the purpose of The Joy of X.

The Joy of X is an introduction to math’s most compelling and far-reaching ideas. The chapters—some from the original Times series [a series of articles about math that Strogatz wrote for the New York Times]—are bite-size and largely independent, so feel free to snack wherever you like. If you want to wade deeper into anything, the notes at the end of the book provide additional details, and suggestions for further reading.

My favorite chapter in The Joy of X was “Twist and Shout” about Mobius strips. Strogatz’s discussion was fine, but what I really enjoyed was the lovely video he called my attention to: “Wind and Mr. Ug”. Go watch it right now; it’s less than 8 minutes long. It is the most endearing mathematical story since Flatland.

Wind and Mr. Ug.

https://www.youtube.com/watch?v=4mdEsouIXGM

Of course, I’m always on the lookout for medical and biological physics, and I found it in Strogatz’s chapter called “Analyze This!,” in which he describes the Gibbs phenomenon. I have written about the Gibbs phenomenon in this blog before, but not so eloquently. Russ Hobbie and I introduce the Gibbs phenomenon in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. When talking about the fit of a Fourier series to a square wave, we write

As the number of terms in the fit is increased, the value of Q [a measure of the goodness of the fit] decreases. However, spikes of constant height (about 18% of the amplitude of the square wave or 9% of the discontinuity in y) remain…These spikes appear whenever there is a discontinuity in y and are called the Gibbs phenomenon.

It turns out that the Gibbs phenomenon is related to the alternating harmonic series. Strogatz writes

Consider this series, known in the trade as the alternating harmonic series:

1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + … .

[…] The partial sums in this case are

S1 = 1

S2 = 1 – 1/2 = 0.500

S3 = 1 – 1/2 + 1/3 = 0.833 …

S4 = 1 – 1/2 + 1/3 – 1/4 = 0.583…

And if you go far enough, you’ll find that they home in on a number close to 0.69. The series can be proven to converge. Its limiting value is the natural logarithm of 2, denoted ln2 and approximately equal to 0.693147. […]

Let’s look at a particularly simple rearrangement whose sum is easy to calculate. Supposed we add two of the negative terms in the alternating harmonic series for every one of its positive terms, as follows:

[1 – 1/2 – 1/4] + [1/3 – 1/6 – 1/8] + [1/5 – 1/10 – 1/12] + …

Next, simplify each of the bracketed expressions by subtracting the second term from the first while leaving the third term untouched. Then the series reduces to

[1/2 – 1/4] + [1/6 – 1/8] + [1/10 – 1/12] + …

After factoring out ½ from all the fractions above and collecting terms, this becomes

½ [ 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + …].

Look who’s back: the beast inside the brackets is the alternating harmonic series itself. By rearranging it, we’ve somehow made it half as big as it was originally—even though it contains all the same terms!”

Strogatz then relates this to a Fourier series

f(x) = sinx – 1/2 sin 2x + 1/3 sin 3x – 1/4 sin 4x + …

This series approaches a sawtooth curve. But when he examines its behavior with different numbers of terms in the sum, he finds the Gibbs phenomenon.

Something goes wrong near the edges of the teeth. The sine waves overshot the mark there and produce a strange finger that isn’t in the sawtooth wave itself… The blame can be laid at the doorstep of the alternating harmonic series. Its pathologies discussed earlier now contaminate the associated Fourier series. They’re responsible for that annoying finger that just won’t go away.

In the notes about the Gibbs phenomenon at the end of the book, Strogatz points us to a fascinating paper on the history of this topic

Hewitt, E. and Hewitt, R. E. (1979) “The Gibbs-Wilbraham Phenomenon: An Episode in Fourier Analysis,” Archive for the History of Exact Sciences, Volume 21, Pages129–160.

He concludes his chapter

This effect, commonly called the Gibbs phenomenon, is more than a mathematical curiosity. Known since the mid-1800s, it now turns up in our digital photographs and on MRI scans. The unwanted oscillations caused by the Gibbs phenomenon can produce blurring, shimmering, and other artifacts at sharp edges in the image. In a medical context, these can be mistaken for damaged tissue, or they can obscure lesions that are actually present.

Photodynamic Therapy

I am currently teaching Medical Physics (PHY 326) at Oakland University, and for our textbook I am using (surprise!) the 4th edition of Intermediate Physics for Medicine and Biology. In class, we recently finished Chapter 14 on Atoms and Light, which “describes some of the biologically important properties of infrared, visible, and ultraviolet light.”

Once a week, class ends with a brief discussion of a recent Point/Counterpoint article from the journal Medical Physics (see here and here for my previous discussion of Point/Counterpoint articles). I find these articles to be useful for introducing students to cutting-edge questions in modern medical physics. The title of each article contains a proposition that two leading medical physicists debate, one for it and one against it. This week, we discussed an article about photodynamic therapy (PDT) by Timothy C. Zhu (University of Pennsylvania, for the proposition) and E. Ishmael Parsai (University of Toledo, against the proposition):

Zhu, T. C., and E. I Parsai (2011) “PDT is Better than Alternative Therapies Such as Brachytherapy, Electron Beams, or Low-Energy X Rays for the Treatment of Skin Cancers,” Medical Physics, Volume 38, Pages 1133–1135.

When reading through the article, I thought I would check how extensively we discuss of PDT in IPMB. I found that we say nothing about it! A search for the term “photodynamic” or “PDT” comes back empty. So, this week (with an eye toward the 5th edition) I am preparing a very short new section in Chapter 14 about PDT.

14.8 ½ Photodynamic Therapy

Photodynamic therapy (PDT) uses a drug called a photosensitizer that is activated by light [Zhu and Finlay (2008), Wilson and Patterson (2008)]. PDT can treat accessible solid tumors such as basal cell carcinoma, a type of skin cancer [see Sec. 14.9.4]. An example of PDT is the surface application of 5-aminolevulinic acid, which is absorbed by the tumor cells and is transformed metabolically into the photosensitizer protoporphyrin IX. When this molecule interacts with light in the 600-800 nm range (red and near infrared), often delivered with a diode laser, it converts molecular oxygen into a highly reactive singlet state that causes necrosis, apoptosis (programmed cell death), or damage to the vasculature that can make the tumor ischemic. Some internal tumors can be treated using light carried by optical fibers introduced through an endoscope.

The two citations are to the articles 

Wilson, B. C. and M. S. Patterson (2008) “The Physics, Biophysics and Technology of Photodynamic Therapy,” Physics in Medicine and Biology, Volume 53, Pages R61–R109.

Zhu, T. C. and J. C. Finlay (2008) “The Role of Photodynamic Therapy (PDT) Physics,” Medical Physics, Volume 35, Pages 3127–3136.

The first PhD dissertation from the Oakland University Medical Physics graduate program dealt with photodynamic therapy: In Vivo Experimental Investigation on the Interaction Between Photodynamic Therapy and Hyperthermia, by James Mattiello (1987).

You can learn more about photodynamic therapy here and here. Please don’t confuse PDT with the alternative medicine (bogus) treatment “Sono Photo Dynamic Therapy.”

The Page 99 Test

English editor Ford Madox Ford advised people who are debating if they should read a particular book to “open the book to page ninety-nine and read, and the quality of the whole will be revealed to you.” This approach is now called the Page 99 Test. Although arbitrary, it provides a way to decide quickly if a book will interest you. Let’s try the Page 99 Test with the 4th edition of Intermediate Physics for Medicine and Biology. Section 4.12 comparing drift and diffusion ends on Page 99, and Section 4.13 about the solution to the diffusion equation begins. The page contains five displayed equations (four of them numbered, Eqs. 4.70 to 4.73) and three figures (Figs. 4.17 to 4.19). An example of the text of page 99 is the opening paragraph at the start of Sec. 4.13.

If C(x, 0) is known for t = 0, it is possible to use the result of Sec. 4.8 to determine C(x,t) at any later time. The key to doing this is that if C(x,t) dx is the number of particles in the region between x and x+dx at time t, it may be be interpreted as the probability of finding a particle in the interval (x, dx) multiplied by the total number of particles. (Recall the discussion on p. 91 about the interpretation of C(x,t).) The spreading Gaussian then represents the spread of probability that a particle is between x and x + dx.

Page 99 appears in the Table of Contents:

4.13 A General Solution for the Particle Concentration as a Function of Time . . . . . 99

and the title of this section appears as the running title at the top of the page. Page 99 appears three times in the index, under 1) Diffusion equation, general solution, 2) Fick’s law (frankly, I'm not sure why page 99 is listed for Fick’s law, as I don't see it mentioned explicitly anywhere on that page), and 3) Gaussian distribution. According to the Symbol List at the end of Chapter 4, the first use of the Greek symbol xi for position was on page 99. Somewhat unusually, no references are cited on page 99 (there are citations on the page before and the page after). No corrections to page 99 appear in the errata, and no words are emphasized using italics.

Does Intermediate Physics for Medicine and Biology pass the page 99 test? I think so. The topic—diffusion—is a physical phenomenon that is crucial for understanding biology. The mix of equations and figures is similar to the remainder of the book. Calculus is used without apology. If you like page 99, I think you will enjoy the rest of the book. And if you like page 99, you are going to love page 100, which contains more equations and figures, plus error functions, Green’s functions, random walks, and citations to classic texts such as Benedek and Villars (2000), Carslaw and Jaeger (1959), and Crank (1975). And if you liked page 100, on page 101 you find......