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......