The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart

Readers of Intermediate Physics for Medicine and Biology may enjoy this story about some of my research as a graduate student, working for John Wikswo at Vanderbilt University. My goal was to determine if the biomagnetic field contains new information that cannot be obtained from the electrical potential.

In 1988, Wikswo, fellow grad student Wei-Qiang Guo, and I published an article in Mathematical Biosciences (Volume 88, Pages 191-221) about the magnetic field at the apex of the heart.

The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart.

B. J. Roth, W.-Q. Guo, and J. P. Wikswo, Jr. 

Living State Physics Group, Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235

This paper describes a volume-conductor model of the apex of the heart that accounts for the spiraling tissue geometry. Analytic expressions are derived for the potential and magnetic field produced by a cardiac action potential propagating outward from the apex. The model predicts the existence of new information in the magnetic field that is not present in the electrical potential.

The analysis was motivated by the unique fiber geometry in the heart, as shown in the figure below, from an article by Franklin Mall. It shows how the cardiac fibers spiral outward from a central spot: the apex (or to use Mall’s word, the vortex).

The apex of the heart.
From Mall, F. P. (1911) “On the Muscular Architecture of the Ventricles of the Human Heart.” American Journal of Anatomy, Volume 11, Pages 211-266.

Our model was an idealization of this complicated geometry. We modeled the fibers as making Archimedean spirals throughout a slab of tissue representing the heart wall, perfused by saline on the top and bottom.

The geometry of a slab of cardiac tissue. The thickness of the tissue is l, the conductivity of the saline bath is σ_e, and the conductivity tensors of the intracellular and interstitial volumes are σ̃_i and σ̃_o. The variables ρ, θ, and z are the cylindrical coordinates, and the red curves represent the fiber direction. Based on Fig. 2 of Roth et al. (1988).

Cardiac tissue is anisotropic; the electrical conductivity is higher parallel to the fibers than perpendicular to them. This is taken into account by using conductivity tensors. Because the fibers spiral and make a constant angle with the radial direction, the tensors have off-diagonal terms when expressed in cylindrical coordinates.

Consider a cardiac wavefront propagating outward, as if stimulated at the apex. Two behaviors occur. First, ignore the spiral geometry. A wavefront produces intracellular current propagating radially outward and extracellular current forming closed loops in the bath (blue). This current produces a magnetic field above and below the slab (green).

The current (blue) and magnetic field (green) created by an action potential propagating outward from the apex of the heart if no off-diagonal terms are present in the conductivity tensors. Based on Fig. 5a of Roth et al. (1988).

Second, ignore the bath but include the spiral fiber geometry. Although the wavefront propagates radially outward, the anisotropy and fiber geometry create an intracellular current that has a component in the θ direction (blue). This current produces its own magnetic field (green).

The azimuthal component of the current (blue) and the electrically silent components of the magnetic field (green) produced by off-diagonal terms in the conductivity tensor, with σ_e = 0. Based on Fig. 5b of Roth et al. (1988).

Of course, both of these mechanisms operate simultaneously, so the total magnetic field distribution looks something like that shown below.

The total magnetic field at the apex of the heart. This figure is only qualitatively correct; the field lines may not be quantitatively accurate. Based on Fig. 5e of Roth et al. (1988).

The original versions of these beautiful figures were prepared by a draftsman in Wikswo’s laboratory. I can’t remember who, but it might have been undergraduate David Barach, who prepared many of our illustrations by hand at the drafting desk. I added color for this blog post.

The main conclusion of this study is that there exists new information about the tissue in the magnetic field that is not present from measuring the electrical potential. The ρ and z components of the magnetic field are electrically silent; the spiraling fiber geometry has no influence on the electrical potential.

Is this mathematical model real, or just the musings of a crazy physics grad student? Two decades after we published our model, Krista McBride—another of Wikswo’s grad students, making her my academic sister—performed an experiment to test our prediction, and obtained results consistent with our calculations.

I’m always amazed when one of my predictions turns out to be correct.

Murray Eden

In 1992, when I was working at the National Institutes of Health, I wrote a review article about magnetic stimulation with my boss’s boss, Murray Eden. We submitted it to IEEE Potentials, a magazine aimed at engineering students. I liked our review, but somehow we never heard back from the journal. I pestered them a few times, and finally gave up and focused on other projects. I hate to waste anything, however, so I give the manuscript to you, dear readers (click here). It’s well written (thanks to our editor Barry Bowman, who improved many of my papers from that era) and describes the technique clearly. You can use it to augment the discussion in Section 8.7 (Magnetic Stimulation) in Intermediate Physics for Medicine and Biology. Unfortunately the article is out of date by almost thirty years.

I reproduce the title page and abstract below.

 

Eden was our fearless leader in the Biomedical Engineering and Instrumentation Program. He was an interesting character. You can learn more about him in an oral history available at the Engineering and Technology History Wiki. In our program, Eden was known for his contribution to barcodes. He was on the committee to design the ubiquitous barcode that you find on almost everything you buy nowadays. Just when the design was almost complete, Eden piped up and said they should include written numbers at the bottom of the barcode, just in case the barcode reader was down. There they have been, ever since (thank goodness!). I didn’t work too closely with Eden; I generally interacted with him through my boss, Seth Goldstein (inventor of the everting catheter). But Eden suggested we write the article, and I was a young nobody at NIH, so of course I said yes.

In Eden’s oral history interview, you can read about the unfortunate end of his tenure leading BEIP.

The world changed and I got a new director in the division, a woman who had been Director of Boston City Hospital’s Clinical Research Center. She and I battled a good deal and I just didn’t like it. By this time I was well over seventy and I said, “Okay, the hell with it. I’m going to retire.” I retired in the spring of ’94. It’s a very sad thing; I don’t like to talk about it very much. My program was essentially destroyed. A few years thereafter NIH administration took my program out of her control. They are currently trying to build the program up again, but most of the good people left.

I was one of the people who left. That woman who became the division director (I still can’t bring myself to utter her name) made it clear that all of us untenured people would not have our positions renewed, which is why I returned to acedemia after seven wonderful years at NIH. I shouldn’t complain. I’ve had an excellent time here at Oakland University and have no regrets, but 1994–1995 was a frustrating time for me.

After I left NIH, I stopped working on magnetic stimulation. I was incredibly lucky to be at NIH at a time when medical doctors were just starting to use the technique and needed a physicist to help. Even now, my most highly cited paper is from my time at NIH working on magnetic stimulation.

Announcement of Murray Eden's retirement in the NIH Record, March 15, 1994.

NMR Imaging of Action Currents

Vanderbilt Notebook 11,
Page 69, dated April 3, 1985

In graduate school, I kept detailed notes about my research. My PhD advisor, John Wikswo, insisted on it, and he provided me with sturdy, high-quality notebooks that are still in good shape today. I encourage my students to keep a notebook, but most prefer to record “virtual” notes on their computer, which is too newfangled for my taste.

My Vanderbilt Notebook 11 covers January 28 to April 25, 1985 (I was 24 years old). On page 69, in an entry dated April 3, I taped in a list of abstracts from the Sixth Annual Conference of the IEEE Engineering in Medicine and Biology Society, held September 15–17, 1984 in Los Angeles. A preview of the abstracts were published in the IEEE Transactions on Biomedical Engineering (Volume 31, Page 569, August, 1984). I marked one as particularly important:

NMR Imaging of Action Currents 

J. H. Nagel

The magnetic field that is generated by action currents is used as a gradient field in NMR imaging. Thus, the bioelectric sources turn out to be accessible inside the human body while using only externally fitted induction coils. Two- or three-dimensional pictures of the body’s state of excitation can be displayed.

That’s all I had: a three sentence abstract by an author with no contact information. I didn’t even know his first name. Along the margin I wrote (in blue ink):

I can’t find J H Nagel in Science Citation Index, except for 3 references to this abstract and 2 others at the same meeting (p. 575, 577 of same Journal [issue]). His address is not given in IEEE 1984 Author index. Goal: find out who he is and write him for a reprint.

How quaint; I wanted to send him a little postcard requesting reprints of any articles he had published on this topic (no pdfs back then, nor email attachments). I added in black ink:

3–25–88 checked biological abstracts 1984–March 1, 1988. None

Finally, in red ink was the mysterious note

See ROTH21 p. 1

In Notebook 21 (April 11, 1988 to December 1, 1989) I found a schedule of talks at the Sixth Annual Conference. I wrote “No Nagel in Session 14!” Apparently he didn’t attend the meeting.

Why tell you this story? Over the years I’ve wondered about using magnetic resonance imaging to detect action currents. I’ve published about it:

Wijesinghe, R. and B. J. Roth, 2009, Detection of peripheral nerve and skeletal muscle action currents using magnetic resonance imaging. Ann. Biomed. Eng., 37:2402-2406.

Jay, W. I., R. S. Wijesinghe, B. D. Dolasinski and B. J. Roth, 2012, Is it possible to detect dendrite currents using presently available magnetic resonance imaging techniques? Med. & Biol. Eng. & Comput., 50:651-657.

Xu, D. and B. J. Roth, 2017, The magnetic field produced by the heart and its influence on MRI. Mathematical Problems in Engineering, 2017:3035479.

I’ve written about it in this blog (click here and here). Russ Hobbie and I have speculated about it in Intermediate Physics for Medicine and Biology:

Much recent research has focused on using MRI to image neural activity directly, rather than through changes in blood flow (Bandettini et al. 2005). Two methods have been proposed to do this. In one, the biomagnetic field produced by neural activity (Chap. 8) acts as the contrast agent, perturbing the magnetic resonance signal. Images with and without the biomagnetic field present provide information about the distribution of neural action currents. In an alternative method, the Lorentz force (Eq. 8.2) acting on the action currents in the presence of a magnetic field causes the nerve to move slightly. If a magnetic field gradient is also present, the nerve may move into a region having a different Larmor frequency. Again, images taken with and without the action currents present provide information about neural activity. Unfortunately, both the biomagnetic field and the displacement caused by the Lorentz force are tiny, and neither of these methods has yet proved useful for neural imaging. However, if these methods could be developed, they would provide information about brain activity similar to that from the magnetoencephalogram, but without requiring the solution of an ill-posed inverse problem that makes the MEG so difficult to interpret.

Notebook 11.

Apparently all this activity began with my reading of Nagel’s abstract in 1985. Yet, I was never able to identify or contact him. Recent research indicates that the magnetic fields in the brain are tiny, and they produce effects that are barely measurable with modern technology. Could Nagel really have detected action currents with nuclear magnetic resonance three decades ago? I doubt it. But there is one thing I would like to know: who is J. H. Nagel? If you can answer this question, please tell me. I’ve been waiting 35 years!

Practice Problems in Bioelectricity and Biomagnetism

It’s funny how your memory can deceive you. I thought my interest in writing homework problems began with my work on Intermediate Physics for Medicine and Biology. Recently, however, I was rummaging through some old papers and discovered that I’ve been writing homework problems for a lot longer. This habit traces back to my graduate school days at Vanderbilt University, when I worked for John Wikswo. Among my old documents, I found a brittle yellowed copy of “The Magnetic Field of a Single Axon: Practice Problems.” It begins

These problems are presented to help someone to become familiar with the analytic volume conduction models of electric potentials and magnetic fields produced by nerve axons or bundles of nerve or muscle fibers developed between 1982 and 1988 in the Living State Physics Group. The problems vary in difficulty, with the very difficult ones marked by a *.

The problems are drawn from eight publications I helped write back in the day. If you need copies of these articles so you can solve the problems, just email me: roth@oakland.edu. (Technically the journal owns the copyright, so I won’t link to the pdfs in this blog. 😞)

J. K. Woosley, B. J. Roth, and J. P. Wikswo, Jr. (1985) “The Magnetic Field of a Single Axon: A Volume Conductor Model.” Mathematical Biosciences, Volume 75, Pages 1-36.

B. J. Roth and J. P. Wikswo, Jr. (1985)  “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment.” Biophysical Journal, Volume 48, Pages 93-109.

B. J. Roth and J. P. Wikswo, Jr. (1985) “The Electrical Potential and Magnetic Field of an Axon in a Nerve Bundle.” Mathematical Biosciences, Volume 76, Pages 37-57.

B. J. Roth and J. P. Wikswo, Jr. (1986) “A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue.” IEEE Transactions on Biomedical Engineering, Volume 33, Pages 467-469.

B. J. Roth and J. P. Wikswo, Jr. (1986) “Electrically Silent Magnetic Fields.” Biophysical Journal, Volume 50, Pages 739-745.

B. J. Roth and F. L. H. Gielen (1987) “A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle.” Annals of Biomedical Engineering, Volume 15, Pages 591-602.

J. P. Wikswo, Jr. and B. J. Roth (1988) “Magnetic Determination of the Spatial Extent of a Single Cortical Current Source: A Theoretical Analysis.” Electroencephalography and Clinical Neurophysiology, Volume 69, Pages 266-276.

B. J. Roth (1987) “Longitudinal Resistance in Strands of Cardiac Muscle.” Ph.D. Thesis, Vanderbilt University, Nashville, Tennessee.

Many of these problems require analyzing Bessel functions and Fourier transforms; I was enamored by those mathematical methods in the 80’s. You can get a hint of what these old homework problems are like by looking at Problem 16 in Chapter 8 of IPMB, where the reader must use these techniques to calculate the magnetic field of a nerve axon.

Let me give you another example. Problem 10 in this ancient collection is based on the  paper by Woosley et al.:

Prove that Eq. (36) and (45) are equal.

Equation (36) is the magnetic field of a nerve axon derived using the law of Biot and Savart, and Equation (45) is the magnetic field derived using Ampere’s law. I’ve discussed before in this blog how I could not prove these two equations were equivalent until I found a Wronskian relating Bessel functions.

I’ve also analyzed the IEEE TBME paper in a blog post from 2018.

These practice problems are not for the faint of heart. Nevertheless, most of what you need to solve them is in IPMB. If you want to learn some advanced methods in theoretical bioelectricity and biomagnetism, download the practice problems and give them a try. If nothing else, they will provide insight into what I used to work on as a graduate student. Besides, with the coronavirus pandemic holding you in quarantine, what else do you have to do?

Influence of a Perfusing Bath on the Foot of the Cardiac Action Potential

Roth BJ, “Influence of a Perfusing Bath on the
Foot of the Cardiac Action Potential,”
Circ. Res., 86:e19-e22, 2000.

Twenty years ago this week, I published a Research Commentary in Circulation Research about the “Influence of a Perfusing Bath on the Foot of the Cardiac Action Potential” (Volume 86, Pages e19-e22, 2000). I like this article for several reasons: it’s short and to the point; it’s a theoretical paper closely tied to data; it’s well written; and it challenges a widely-accepted interpretation of an experiment by a major figure in cardiac electrophysiology.

Back in my more pugnacious days, I wouldn’t hesitate to take on senior scientists when I disagreed with them. In this case, I critiqued the work of Madison Spach, a Professor at Duke University and a towering figure in the field. In 1981, Spach led an all-star team that measured cardiac action potentials propagating either parallel to or perpendicular to the myocardial fibers.

Spach MS, Miller WT III, Geselowitz DB, Barr RC, Kootsey JM, Johnson EA. “The Discontinuous Nature of Propagation in Normal Canine Cardiac Muscle: Evidence for Recurrent Discontinuities of Intracellular Resistance that Affect the Membrane Currents.” Circulation Research, Volume 48, Pages 39–45, 1981.

Spach et al., “The Discontinuous Nature of
Propagation in Normal Canine Cardiac Muscle:
Evidence for Recurrent Discontinuities of
Intracellular Resistance that Affect the
Membrane Currents,” Circ. Res., 48:39-45, 1981.

They found that the rate-of-rise of the action potential and the time constant of the action potential foot depend on the direction of propagation. Continuous cable theory predicts that the rate-of-rise and time constant should be the same, regardless of direction. Therefore, they concluded, cardiac tissue is not continuous. Instead, they claimed that their experiment revealed the tissue’s discrete structure.

To be sure, cardiac tissue is discrete in a sense. It’s made of individual cells, coupled by intercellular junctions to form a “syncytium.” Often, however, you can average over the cellular structure and treat the tissue as a continuum, just as you can often treat a material as a continuum even through it’s made from discrete atoms. For example, the bidomain model is a continuous description of the electrical properties of a microscopically heterogeneous tissue (See Section 7.9 of Intermediate Physics for Medicine and Biology for more about the bidomain model).

I’m skeptical of Spach’s interpretation of his data, and I’m not convinced that his observations imply the tissue’s discrete nature. I didn’t waste any time making this point in my article; I mention Spach by name in the first sentence of the Introduction. (In all quotes, I don’t include the references.)

In 1981, Spach et al observed a smaller maximum rate of rise of the action potential, V̇_max, and a larger time constant of the action potential foot, τ_foot, during propagation parallel to the myocardiac [sic] fibers (longitudinal) than during propagation perpendicular to the fibers (transverse). They attributed these differences to the discrete cellular structure of the myocardium. Their research has been cited widely and is often taken as evidence for discontinuous propagation in cardiac tissue.

Several researchers have suggested that the observations of Spach et al may be caused by the bath perfusing the tissue rather than the discrete nature of the tissue itself... The purpose of this commentary is to model the experiment of Spach et al using a numerical simulation and to show that the perfusing bath plays an important role in determining the time course of the action potential foot.

I performed a computer simulation of wave fronts propagating through a slab of cardiac tissue that is perfused by a tissue bath. The tissue is represented as a bidomain, so its discrete nature was not incorporated into the model. I found that the rate-of-rise of the action potential is slower when propagation is parallel to the fibers compared to perpendicular to the fibers, just as Spach et al. observed. However, when I eliminated the purfusing bath this effect disappeared and the rate-of-rise was the same in both directions.

My favorite part of the article is in the Discussion, where I summarize my conclusion using a syllogism.

The data of Spach et al are cited widely as evidence for discontinuous propagation in cardiac tissue. Their hypothesis of discontinuous propagation is supported by the following logic: (1) During 1-dimensional propagation in a tissue with continuous electrical properties, the time course of the action potential (including V̇_max and τ_foot) does not depend on the intracellular and interstitial conductivities; (2) experiments indicate that in cardiac tissue V̇_max and τ_foot differ with the direction of propagation and therefore with conductivity; and (3) therefore, the conductivity of cardiac tissue is not continuous. A flaw exists in this line of reasoning: when a conductive bath perfuses the tissue, the propagation is not 1-dimensional. The extracellular conductivity is higher for the tissue near the surface (adjacent to the bath) than it is for the tissue far from the surface (deep within the bulk). Therefore, gradients in V_m exist not only in the direction of propagation, but also in the direction perpendicular to the tissue surface. Reasoning based on the 1-dimensional cable model (such as used in the first premise of the syllogism above) is not applicable.

In biology and medicine, the main purpose of computer simulations is to suggest new experiments, so I proposed one.

One way to distinguish between the 2 mechanisms ([the discrete structure] versus perfusing bath) would be to repeat the experiments of Spach et al with and without a perfusing bath present. The tissue would have to be kept alive when the perfusing bath was absent, perhaps by arterial perfusion. The results … indicate that when the bath is eliminated, the action potential foot should become exponential, with no differences between longitudinal and transverse propagation. Furthermore, the maximum rate of rise of the action potential should increase and become independent of propagation direction. Although this experiment is easy to conceive, it would be susceptible to several sources of error. If V_m were measured optically, the data would represent an average over a depth of a few hundred microns. Because the model predicts that V_m changes dramatically over such distances, the data would be difficult to interpret. Microelectrode measurements, on the other hand, are sensitive to capacitative [sic] coupling to the perfusing bath, and the degree of such coupling depends on the bath depth. The rapid depolarization phase of the action potential is particularly sensitive to electrode capacitance. Although it is possible to correct the data for the influence of electrode capacitance, these corrections would be crucial when comparing data measured at different bath depths.

A later paper by Oleg Sharifov and Vladimir Fast (Heart Rhythm, Volume 3, Pages 1063-1073, 2006) suggests a better way to perform this experiment: use optical mapping but with the membrane dye introduced through the perfusing bath so it stains only the surface tissue. In this case, there is no capacitive coupling (no microelectrode) and little averaging over depth (the optical signal arises from only surface tissue). This would be an important experiment, but it hasn’t been performed yet. Until it is, we can’t resolve the debate over discrete versus continuous behavior. 

The last paragraph in the paper sums it all up. I particularly like the final sentence.

We cannot conclude from our study that [discrete structures] are not important during action potential propagation. Nor can we conclude that discontinuous propagation does not occur (particularly in diseased tissue). These factors may well play a role in propagation. We can conclude, however, that the influence of a perfusing bath must be taken into account when interpreting data showing differences in the shape of the action potential foot with propagation direction... Therefore, differences in action potential shape with direction cannot be taken as definitive evidence supporting discontinuous propagation... if a perfusing bath is present. Finally, without additional experiments, we cannot exclude the possibility that in healthy tissue the difference in the shape of the action potential upstroke with propagation direction is simply an artifact of the way the tissue was perfused.

Has my commentary had much impact? Nope. Compared to other papers I’ve written, this one is a citation dud. It has been cited only 27 times (22 if you remove self-citations); barely once a year. Spach’s 1981 paper has over 800 citations; over 20 per year. Even a response by Spach and Barr (Circ. Res., Volume 86, Pages e23-e28, 2000) to my commentary has almost twice as many citations as my original commentary. Does this difference in citation rate arise because I’m wrong and Spach’s right? Maybe. The only way to know is to do the experiment.

The Electric Field Induced During Magnetic Stimulation

Chapter 8 of Intermediate Physics for Medicine and Biology discusses electromagnetic induction and magnetic stimulation of nerves. It doesn't, however, explain how to calculate the electric field. You can learn how to do this from my article “The Electric Field Induced During Magnetic Stimulation” (Electroencephalography and Clinical Neurophysiology, Supplement 43, Pages 268-278, 1991). It begins:

“The Electric Field Induced
During Magnetic Stimulation.”

Magnetic stimulation has been studied widely since its use in 1982 for stimulation of peripheral nerves (Polson et al. 1982), and in 1985 for stimulation of the cortex (Barker et al. 1985). The technique consists of inducing current in the body by Faraday’s law of induction: a time-dependent magnetic field produces an electric field. The transient magnetic field is created by discharging a capacitor through a coil held near the target neuron. Magnetic stimulation has potential clinical applications for the diagnosis of central nervous system disorders such as multiple sclerosis, and for monitoring the corticospinal tract during spinal cord surgery (for review, see Hallett and Cohen 1989). When activating the cortex transcranially, magnetic stimulation is less painful than electrical stimulation.

Appendix 1.

Although there have been many clinical studies of magnetic stimulation, until recently there have been few attempts to measure or calculate the electric field distribution induced in tissue. However, knowledge of the electric field is important for determining where stimulation occurs, how localized the stimulated region is, and what the relative efficacy of different coil designs is. In this paper, the electric field induced in tissue during magnetic stimulation is calculated, and results are presented for stimulation of both the peripheral and central nervous systems.

In Appendix 1 of this article, I derived an expression for the electric field E at position r, starting from

where N is the number of turns in the coil, μ₀ is the permeability of free space (4π × 10^-7 H/m), I is the coil current, r' is the position along the coil, and the integral of dl' is over the coil path. For all but the simplest of coil shapes this integral can't be evaluated analytically, so I used a trick: approximate the coil as a polygon. A twelve-sided polygon looks a lot like a circular coil. You can make the approximation even better by using more sides.

A circular coil (black) approximated by
a 12-sided polygon (red).

With this method I needed to calculate the electric field only from line segments. The calculation for one line segment is summarized in Figure 6 of the paper.

Figure 6 from “The Electric Field
Induced During Magnetic Stimulation.”

I will present the calculation as a new homework problem for IPMB. (Warning: t has two meanings in this problem: it denotes time and is also a dimensionless parameter specifying location along the line segment.)

Section 8.7

Problem 32 ½. Calculate the integral

for a line segment extending from x₂ to x₁. Define δ = x₁ - x₂ and R = r - ½(x₁ + x₂).

(a) Interpret δ and R physically.

(b) Define t as a dimensionless parameter ranging from -½ to ½. Show that r' equals r – R – tδ.

(c) Show that the integral becomes

(d) Evaluate this integral. You may need a table of integrals.

(e) Express the integral in terms of δ, R, and φ (the angle between R and δ).

The resulting expression for the electric field is Equation 15 in the article

Equation (15) in “The Electric Field Induced During Magnetic Stimulation.”

The photograph below shows the preliminary result in my research notebook from when I worked at the National Institutes of Health. I didn't save the reams of scrap paper needed to derive this result.

The November 10, 1988 entry
in my research notebook.

To determine the ends of the line segments, I took an x-ray of a coil and digitized points on it. Below are coordinates for a figure-of-eight coil, often used during magnetic stimulation. The method was low-tech and imprecise, but it worked.

The November 17, 1988 entry
in my research notebook.

Ten comments:

• My coauthors were Leo Cohen and Mark Hallett, two neurologists at NIH. I recommend their four-page paper “Magnetism: A New Method for Stimulation of Nerve and Brain.”
• The calculation above gives the electric field in an unbounded, homogeneous tissue. The article also analyzes the effect of tissue boundaries on the electric field.
• The integral is dimensionless. “For distances from the coil that are similar to the coil size, this integral is approximately equal to one, so a rule of thumb for determining the order of magnitude of E is 0.1 N dI/dt, where dI/dt has units of A/μsec and E is in V/m.”
• The inverse hyperbolic sine can be expressed in terms of logarithms: sinh^-1z = ln[z + √(z² + 1)]. If you're uncomfortable with hyperbolic functions, perhaps logarithms are more to your taste. 
  
• This supplement to Electroencephalography and Clinical Neurophysiology contained papers from the International Motor Evoked Potential Symposium, held in Chicago in August 1989. This excellent meeting guided my subsequent research into magnetic stimulation. The supplement was published as a book: Magnetic Motor Stimulation: Principles and Clinical Experience, edited by Walter Levy, Roger Cracco, Tony Barker, and John Rothwell. 
• Leo Cohen was first author on a clinical paper published in the same supplement: Cohen, Bandinelli, Topka, Fuhr, Roth, and Hallett (1991) “Topographic Maps of Human Motor Cortex in Normal and Pathological Conditions: Mirror Movements, Amputations and Spinal Cord Injuries.”
• To be successful in science you must be in the right place at the right time. I was lucky to arrive at NIH as a young physicist in 1988—soon after magnetic stimulation was invented—and to have two neurologists using the new technique on their patients and looking for a collaborator to calculate electric fields.
• A week after deriving the expression for the electric field, I found a similar expression for the magnetic field. It was never published. Let me know if you need it.
• If you look up my article, please forgive the incorrect units for μ₀ given in the Appendix. They should be Henry/meter, not Farad/meter. In my defense, I had it correct in the body of the article. 
• Correspondence about the article was to be sent to “Bradley J. Roth, Building 13, Room 3W13, National Institutes of Health, Bethesda, MD 20892.” This was my office when I worked at the NIH intramural program between 1988 and 1995. I loved working at NIH as part of the Biomedical Engineering and Instrumentation Program, which consisted of physicists, mathematicians and engineers who collaborated with the medical doctors and biologists. Cohen and Hallett had their laboratory in the NIH Clinical Center (Building 10), and were part of the National Institute of Neurological Disorders and Stroke. Hallett once told me he began his undergraduate education as a physics major, but switched to medicine after one of his professors tried to explain how magnetic fields are related to electric fields in special relativity.

A map of the National Institutes of Health campus
in Bethesda, Maryland.

A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue

My research notebooks from graduate school.

In graduate school, I worked with John Wikswo measuring the magnetic field of a nerve axon. We isolated a crayfish axon and threaded it through a wire-wound ferrite-core toroid immersed in saline. As the action currents propagated by, they produced a changing magnetic field that induced a signal in the toroid by Faraday induction. Ampere’s law tells us that the signal is proportional to the net current through the toroid, which is the sum of the intracellular current and the fraction of the current in the saline that passes through the toroid, called the return current.

For my PhD dissertation, Wikswo had me make similar measurements on strands of cardiac tissue, such as a papillary muscle. The instrumentation was the same as for the nerve, but the interpretation was different. Now the signal had three sources: the intracellular current, the return current in the saline, and extracellular current passing through the interstitial space within the muscle called the “interstitial return current.” Initially neither Wikswo nor I knew how to calculate the interstitial return current, so we were not sure how to interpret our results. As I planned these experiments, I recorded my thoughts in my research notebook. The July 26, 1984 entry stressed that “Understanding this point [the role of interstitial return currents] will be central to my research and deserves much thought.”

Excerpt from the July 26, 1984 entry in my Notebook 8, page 64.

When working on nerves, I had studied articles by Robert Plonsey and John Clark, in which they calculated the extracellular potential in the saline from the measured voltage across the axon’s membrane: the transmembrane potential. I was impressed by this calculation, which involved Fourier transforms and Bessel functions (see Chapter 7, Problem 30, in Intermediate Physics for Medicine and Biology). I had used their result to calculate the magnetic field around an axon (see Chapter 8, Problem 16 in IPMB), so I set out to extend their analysis again to include interstitial return currents.

Page 1 of Notebook 9, from Sept 13, 1984.

The key was to use the then-new bidomain model, which accounts for currents in both the intracellular and interstitial spaces. The crucial advance came in September 1984 after I read a copy of Les Tung’s PhD dissertation that Wikswo had loaned me. After four days of intense work, I had solved the problem. My results looked a lot like those of Clark and Plonsey, except for a few strategically placed additional factors and extra terms.

Excerpt from Notebook 9, page 13, the Sept 16, 1984 entry.

First page of Roth and Wikswo (1986).

Wikswo and I published these results as a brief communication in the IEEE Transactions on Biomedical Engineering.

Roth, B. J. and J. P. Wikswo, Jr., 1986, A bidomain model for the extracellular potential and magnetic field of cardiac tissue. IEEE Trans. Biomed. Eng., 33:467-469.

Abstract—In this brief communication, a bidomain volume conductor model is developed to represent cardiac muscle strands, enabling the magnetic field and extracellular potential to be calculated from the cardiac transmembrane potential. The model accounts for all action currents, including the interstitial current between the cardiac cells, and thereby allows quantitative interpretation of magnetic measurements of cardiac muscle.

Rather than explain the calculation in all its gory detail, I will ask you to solve it in a new homework problem.

Section 7.9

Problem 31½. A cylindrical strand of cardiac tissue, of radius a, is immersed in a saline bath. Cardiac tissue is a bidomain with anisotropic intracellular and interstitial conductivities σ_ir, σ_iz, σ_or, and σ_oz, and saline is a monodomain volume conductor with isotropic conductivity σ_e. The intracellular and interstitial potentials are V_i and V_o, and the saline potential is V_e.

a) Write the bidomain equations, Eqs. 7.44a and 7.44b, for V_i and V_o in cylindrical coordinates (r,z). Add the two equations.

b) Assume V_i = σ_iz/(σ_iz+σ_oz) [A I₀(kλr) + (σ_oz/σ_iz) V_m] sin(kz) and V_o = σ_iz/(σ_iz+σ_oz) [A I₀(kλr) - V_m] sin(kz) where λ² = (σ_iz+σ_oz)/(σ_ir+σ_or). Verify that V_i - V_o equals the transmembrane potential V_m sin(kz). Show that V_i and V_o obey the equation derived in part a). I₀(x) is the modified Bessel function of the first kind of order zero, which obeys the modified Bessel equation d²y/dx² + (1/x) dy/dx = y. Assume V_m is independent of r.

c) Write Laplace’s equation (Eq. 7.38) in cylindrical coordinates. Assume V_e = B K₀(kr) sin(kz). Show that V_e satisfies Laplace’s equation. K₀(x) is the modified Bessel function of the second kind of order zero, which obeys the modified Bessel equation.

d) At r = a, the boundary conditions are V_o = V_e and σ_ir∂V_i/∂r + σ_or∂V_o/∂r = σ_e∂V_e/∂r. Determine A and B. You may need the Bessel function identities dI₀/dx = I₁ and dK₀/dx = - K₁, where I₁(x) and K₁(x) are modified Bessel functions of order one.

In the problem above, I assumed the potentials vary sinusoidally with z, but any waveform can be expressed as a superposition of sines and cosines (Fourier analysis) so this is not as restrictive as it seems.

In part b), I assumed the transmembrane potential was not a function of r. There is little data supporting this assumption, but I was stuck without it. Another assumption I could have used was equal anisotropy ratios, but I didn’t want to do that (and initially I didn’t realize it provided an alternative path to the solution).

The calculation of the magnetic field is not included in the new problem; it requires differentiating V_i, V_o, and V_e to find the current density, and then integrating the current to find the magnetic field via Ampere’s law. You can find the details in our IEEE TBME communication.

Some of you might be thinking “this is a nice homework problem, but how did you get those weird expressions for the intracellular and interstitial potentials used in part b)”? Our article gives some insight, and my notebook provides more. I started with Clark and Plonsey’s result, used ideas from Tung’s dissertation, and then played with the math (trial and error) until I had a solution that obeyed the bidomain equations. Some might call that a strange way to do science, but it worked for me.

I was very proud of this calculation (and still am). It played a role in the development of the bidomain model, which is now considered the state-of-the-art model for simulating the heart during defibrillation.

Wikswo and I carried out experiments on guinea pig papillary muscles to test the calculation, but the cardiac data was not as clean and definitive as our nerve data. We published it as a chapter in Cell Interactions and Gap Junctions (1989). The cardiac work made up most of my PhD dissertation. Vanderbilt let me include our three-page IEEE TBME communication as an appendix. It’s the most important three pages in the dissertation.


Coda: While browsing through my old research notebooks, I found this gem passed from Prof. Wikswo to his earnest but naive graduate student, who dutifully wrote it down in his research notebook for posterity.

Excerpt from Notebook 10, Page 62.

Frequency Locking of Meandering Spiral Waves in Cardiac Tissue

In Chapter 10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss spiral waves of electrical activity in the heart.

The study of spiral waves in the heart is currently an active field .... They can lead to ventricular tachycardia, they can meander, much as a tornado does, and their breakup into a pattern resembling turbulence is a possible mechanism for the development of ventricular fibrillation.

Twenty years ago, I published a paper about meandering in Physical Review E.

Roth, B. J., 1998, Frequency locking of meandering spiral waves in cardiac tissue. Phys. Rev. E, 57:R3735-3738.

The influence of anisotropy on spiral waves meandering in a sheet of cardiac tissue is studied numerically. The FitzHugh-Nagumo model represents the tissue excitability, and the bidomain model characterizes the passive electrical properties. The anisotropy ratios in the intracellular and extracellular spaces are unequal. This condition does not induce meandering or destabilize spiral waves; however, it imposes fourfold symmetry onto the meander path and causes frequency locking of the rotation and meander frequencies when the meander path has nearly fourfold symmetry.

A spiral wave meandering in a sheet of cardiac tissue.

Above is a picture of a meandering spiral wave. Color indicates the transmembrane potential: purple is resting tissue and yellow is depolarized. The thin red band indicates where the transmembrane potential is half way between rest and depolarized. The red region, however, can be in one of two states. The outer red band (next to the deeper purple) is where the transmembrane potential is increasing (depolarizing) during the action potential upstroke, and the inner red band (next to the royal blue) is where the transmembrane potential is decreasing (repolarizing) during the refractory period. The point where the two red bands meet near the center of the tissue is called the phase singularity. There, you can’t tell if the transmembrane potential is increasing or decreasing (to learn more about phase singularities, try Homework Problem 44 in Chapter 10 of IPMB). The spiral wave rotates about the phase singularity, in this case counterclockwise.

One interesting feature about a rotating spiral wave is that its phase singularity sometimes moves around: it meanders. In the above picture, the meander path is white. Often this path looks like it was drawn while playing Spirograph. The motion consists of two parts, each with its own frequency: one corresponds to the rotation of the spiral wave and another creates the petals of the flower-like meander. All this was known long before I entered the field (see, for instance, Art Winfree’s lovely paper: “Varieties of spiral wave behavior: An experimentalist's approach to the theory of excitable media,” Chaos, Volume 1, Pages 303–334, 1991).

What I found in my 1998 paper was that the bidomain nature of cardiac tissue can entrain the two frequencies (force them to be the same, or lock them in to some simple integer ratio). In the bidomain model the intracellular and extracellular spaces are both anisotropic (the electrical resistance depends on direction), but the amount of anisotropy is different in the two spaces. The intracellular space is highly anisotropic and the extracellular space is less so. This property of unequal anisotropy ratios causes the two frequencies to adjust so that the meander path has four-fold symmetry.

My 2004 paper “Art Winfree and the Bidomain Model of Cardiac Tissue” tells the rest of the story (I quote from my original submission, available on ResearchGate, and not the inferior version ultimately published in the Journal of Theoretical Biology).

Most of the mail I get each day is junk, but occasionally, something arrives that has a major impact on my research. One day in June, 2001, I opened my mail to find a letter and preprint from a Canadian mathematician I had never heard of, named Victor LeBlanc. To my astonishment, Victor’s preprint contained analytical proofs specifying what conditions result in locking of the meander pattern to the underlying symmetry of the tissue, and what conditions lead to drift [another type of spiral wave meander]. These conclusions, which I had painstakingly deduced after countless hours of computer simulations, he could prove with paper and pencil. Plus, his analysis predicted many other cases of locking and drifting that I had not examined. I am not enough of a mathematician to understand the proofs, but I could appreciate the results well enough. I contacted Victor, and we tested his predictions using my computer program. The analytical and computational results were consistent in every case we tested. Ironically, Victor predicted that the meander path should have a two-fold symmetry, not the four-fold symmetry that originally motivated my study, and he was correct.... My last email correspondence with Art [Winfree], just a few months before he died, was about a joint paper Victor and I published, describing these results.

I will close with a photo that appeared in the 1997 Annual Report of the Whitaker Foundation, which funded my work on spiral wave meandering. Enjoy!

A picture of a spiral wave and me from the 1997 Whitaker Foundation Annual Report.

Cover of the 1997 Whitaker Foundation Annual Report.

Sepulveda, Roth and Wikswo (1989): How to Write a Scientific Paper

In 1989, Nestor Sepulveda, John Wikswo and I published “Current Injection into a Two-Dimensional Anisotropic Bidomain” (Biophysical Journal, 55:987–999). Of my papers, this is one of my favorites.

When I teach my graduate Bioelectric Phenomena class here at Oakland University, we study the Sepulveda et al. (1989) article. The primary goal of the class is to introduce students to bioelectricity, but a secondary goal is to analyze how to write scientific papers. When we get to our paper, I let students learn the scientific content from the publication itself. Instead, I use class time to analyze scientific writing. The paper lends itself to this task: It is written well enough to serve as an example of technical writing, but it is written poorly enough to illustrate how writing can be improved. Critically tearing apart the writing of someone else’s paper in front of students would be rude, but because this writing is partly mine I don’t feel guilty.

Many readers of Intermediate Physics for Medicine and Biology will eventually write papers of their own, so in this post I share my analysis of scientific writing just as I present it in class. Students read “Current Injection into a Two-Dimensional Anisotropic Bidomain” in advance, and then during class we go through the writing page by page, and often line by line, using a powerpoint presentation that I have placed on the IPMB website. I use the “animation” feature of powerpoint so edits, revisions, and corrections can be considered one at a time. To see for yourself, download the powerpoint and click “slide show.” Then, start using the right arrow to analyze the paper.

A screen shot of the first page of the powerpoint. It looks a mess, but the animation feature lets you consider all these suggestions one by one. You can download it and use it to teach your students.

When using this powerpoint, keep these points in mind:

• One reason I use Sepulveda et al. (1989) as my example is that it has the classic format of a scientific paper: Introduction, Methods, Results, and Discussion. It also contains an Abstract, References, and other sections of a scientific publication. 
• Often I highlight a sentence or two of text and ask students to revise and improve it. If you are leading a class using this powerpoint, stop and let the students struggle with the revision. Then compare their revised text with mine. The class should be interactive.
• I have talked before in this blog about the importance of writing. In the powerpoint, I mention two publications that have helped me become a better writer. First is Strunk and White’s book Elements of Style. The powerpoint illustrates much of their advice—such as their famous admonition to “omit needless words”—with concrete examples. You can read Elements of Style online here. Second is N. David Mermin’s essay “What’s Wrong with These Equations” published in Physics Today (download it here). Mermin explains how to integrate math with prose, and introduces the “Good Samaritan Rule” (remind your reader what an equation is about when you refer to it, rather than just saying “Eq. 4”) and other concepts. 
• Some of the points raised in my powerpoint are trivial, such as the difference between “there,” “their,” and “they’re.” Others are more substantial, such as sentence construction and clarity. I find it takes most of a 90 minute class to finish the whole thing. 
• On the sixth page of the powerpoint I have a note reminding me to “Tell Story.” The story is one I wrote about in the original version of my paper “Art Winfree and the Bidomain Model of Cardiac Tissue.” “Nestor Sepulveda, a research assistant professor from Columbia who was working in John [Wikswo]'s lab, had written a finite element computer program that we modified to do bidomain calculations. One of the first simulations he performed was of the transmembrane potential induced in a two-dimensional sheet of cardiac tissue having 'unequal anisotropy ratios' (different degrees of anisotropy in the intracellular and extracellular spaces). Much to our surprise, Nestor found that when he stimulated the tissue through a small cathodal electrode, depolarization (a positive transmembrane potential) appeared under the electrode, but hyperpolarization (a negative transmembrane potential) appeared near the electrode along the fiber direction (Fig. 2). The depolarization was stronger in the direction perpendicular to the fibers, giving those voltage contour lines a shape that John named the ‘dogbone.’ Only Nestor understood the details of his finite element code, and I was a bit worried that his program might contain a bug that caused this weird result. So I quietly returned to my office and developed an entirely different numerical scheme, using Fourier transforms, to do the same calculation. Of course, I got the same result Nestor did (there was no bug). Although I didn’t realize it then, I would spend the next 15 years exploring the implications of Nestor’s result.” During class, I often take off on tangents telling old  “war stories” like this. I can’t help myself.
• John Wikswo, my coauthor and PhD dissertation advisor, is still active, and he and I continue to collaborate. I learned much about scientific writing from him, but our writing styles are different and he might not agree with all the suggestions in the powerpoint. Tragically, Nestor Sepulveda has passed away; a great loss for bioelectricity research. I miss him.
• If you are teaching and want to discuss how to write a scientific paper, feel free to use this powerpoint. I encourage you to download it and modify it to suit your needs. Students could even use it for self study, although they would not see some essential hand waving.

Although the powerpoint suggests many changes to the Sepulveda et al. (1989) paper, I nevertheless consider that article to be a success. According to Google Scholar, it has been cited 379 times. I believe it had an impact on the field of pacing and defibrillation of the heart. Overall, I am proud of the writing.

Let me close by emphasizing that writing is an art. Your style might not be the same as mine. Take my suggestions in the powerpoint as just that: suggestions. Yet, whether or not you agree with my suggestions, I believe your students will benefit by going through the process of revising a scientific paper. It’s the next best thing to assigning them to write their own paper. Enjoy!