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 σe, and the conductivity tensors of the intracellular and interstitial volumes are σ̃i and σ̃o. 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 σe = 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.

Originally published at http://hobbieroth.blogspot.com.