Electric and Magnetic Fields From Two-Dimensional Anisotropic Bisyncytia

Page 223 of Intermediate Physics for Medicine and Biology.

Figure 8.18 on page 223 of Intermediate Physics for Medicine and Biology contains a plot of the magnetic field produced by action currents in a slice of cardiac tissue. The measured magnetic field contours have approximately a four-fold symmetry. The experiment by Staton et al. that produced this data was a tour de force, demonstrating the power of high-spatial-resolution biomagnetic techniques. 

 

Sepulveda and Wikswo (1987).

In this post, I discuss the theoretical prediction by Nestor Sepulveda and John Wikswo in the mid 1980s that preceded and motivated the experiment.

Sepulveda NG, Wikswo JP (1987) “Electric and Magnetic Fields From Two-Dimensional Anisotropic Bisyncytia,” Biophysical Journal, Volume 51, Pages 557-568.

Their abstract is presented below.

Cardiac tissue can be considered macroscopically as a bidomain, anisotropic conductor in which simple depolarization wavefronts produce complex current distributions. Since such distributions may be difficult to measure using electrical techniques, we have developed a mathematical model to determine the feasibility of magnetic localization of these currents. By applying the finite element method to an idealized two-dimensional bisyncytium [a synonym for bidomain] with anisotropic conductivities, we have calculated the intracellular and extracellular potentials, the current distributions, and the magnetic fields for a circular depolarization wavefront. The calculated magnetic field 1 mm from the tissue is well within the sensitivity of a SQUID magnetometer. Our results show that complex bisyncytial current patterns can be studied magnetically, and these studies should provide valuable insight regarding the electrical anisotropy of cardiac tissue.

Sepulveda and Wikswo assumed the tissue was excited by a brief stimulus through an electrode (purple dot in the illustration below), resulting in a circular wave front propagating outward. The transmembrane potential coutours at one instant are shown in red. The assumption of a circular wave front is odd, because cardiac tissue is anisotropic. A better assumption would have been an elliptical wave front with its long axis parallel to the fibers. Nevertheless, the circular wave front captures the essential features of the problem.

If the tissue were isotropic, the intracellular current density would point radially outward and the extracellular current density would point radially inward. The intracellular and extracellular currents would exactly cancel, so the net current (their sum) would be zero. Moreover, the net current would vanish if the tissue were anisotropic but had equal anisotropy ratios. That is, if the ratios of the electrical conducivities parallel and perpendicular to the fibers were the same in the intracellular and extracellular spaces. The only way to produce a net current (shown as the blue loops in the illustration below) is if the tissue has unequal anisotropy ratios. In that case, the loops are four-fold symmetric, rotating clockwise in two quadrants and counterclockwise in the other two.

Current loops produce magnetic fields. The right-hand-rule implies that the magnetic field points up out of the plane in the top-right and the bottom-left quadrants, and down into the plane in the other two. The contours of magnetic field are green in the illustration below, and the peak magnitude for a 1 mm thick sheet of is about one fourth of a nanotesla.

Jut for fun, I superimposed the transmembrane potential, net current density, and magnetic field plots in the picture below.

Notes:

1. The measurement of the magnetic field is a null detector of unequal anisotropy ratios. In other words, in tissue with equal anisotropy ratios the magnetic field vanishes, so the mere existence of a magnetic field implies the anisotropy ratios are unequal. The condition of unequal anisotropy ratios has many implications for cardiac tissue. One is discussed in Homework Problem 50 in Chapter 7 of IPMB. 
2. If the sheet of cardiac tissue is superfused by a saline bath, the magnetic field distribution changes.
   
3. Wikswo was a pioneer in the field of biomagnetism. In particular, he developed small scanning magnetometers that had sub-millimeter spatial resolution. He was in a unique position of being able to measure the magnetic fields that he and Sepulveda predicted, which led to the figure included in IPMB. 
4. I was a graduate student in Wikswo’s laboratory when Sepulveda and Wikswo wrote this article. Sepulveda, a delightful Columbian biomedical engineer and a good friend of mine, worked as a research scientist in Wikswo’s lab. He was an expert on the finite element method—the numerical technique used in his paper with Wikswo—and had written his own finite element code that no one else in the lab understood. He died a decade ago, and I miss him. 
5. Sepulveda and Wikswo were building on a calculation published in 1984 by Robert Plonsey and Roger Barr (“Current Flow Patterns in Two Dimensional Anisotropic Bisyncytia With Normal and Extreme Conductivities,” Biophys. J., 45:557-571). Wikswo heard either Plonsey or Barr give a talk about their results at a scientific meeting. He realized immediately that their predicted current loops implied a biomagnetic field. When Wikswo returned to the lab, he described Plonsey and Barr’s current loops at a group meeting. As I listened, I remember thinking “Wikswo’s gone mad,” but he was right.
   
6. Two years after their magnetic field article, Sepulveda and Wikswo (now with me included as a coauthor) calculated the transmembrane potential produced when cardiac tissue is stimulated by a point electrode. But that’s another story.
   

I’ll give Sepulveda and Wikswo the last word. Below is the concluding paragraph of their article, which looks forward to the experimental measurement of the magnetic field pattern that was shown in IPMB.

The bidomain model of cardiac tissue provides a tool that can be explored and used to study and explain features of cardiac conduction. However, it should be remembered that “a model is valid when it measures what it is intended to measure” (31). Thus, experimental data must be used to evaluate the validity of the bidomain model. This evaluation must involve comparison of the model's predictions not only with measured intracellular and extracellular potentials but also with the measured magnetic fields. When the applicability of the bidomain model to a particular cardiac preparation and the validity and reliability of our calculations have been determined experimentally, this mathematical approach should then provide a new technique for analyzing normal and pathological cardiac activation.

Members of Wikswo's lab at Vanderbilt University in the mid 1980s: John Wikswo is on the phone, Nestor Sepulveda is in the white shirt and gray pants, and I am on the far right. The other people are Frans Gielen (the tall guy with arms crossed on the left), Ranjith Wijesinghe (between Gielen and Sepulveda), Peng Zhang (between Wikswo and Sepulveda), and Pat Henry (knelling).