Conduction in the Heart Wall: Helicoidal Fibers Minimize Diffusion Bias

Abstract

The mammalian heart must function as an efficient pump while simultaneously conducting electrical signals to drive the contraction process. In the ventricles, electrical activation begins at the insertion points of the Purkinje network in the endocardium. How does the diffusion component of the subsequent excitation wave propagate from the endocardium in a healthy heart wall without creating directional biases? We show that this is a consequence of the particular geometric organization of myocytes in the heart wall. Using a generalized helicoid to model fiber orientation, we treat the myocardium as a curved space via Riemannian geometry, and then use stochastic calculus to model local signal diffusion. Our analysis shows that the helicoidal arrangement of myocytes minimizes the directional biases that could lead to aberrant propagation, thereby explaining how electrophysiological principles are consistent with local measurements of cardiac fiber geometry. We discuss our results in the context of the need to balance electrical and mechanical requirements for heart function.

Figure 1

Left: a rat left ventricle, with a slice of fiber data from diffusion MRI (color denotes transmural position). Middle: displays the fiber field (from the white cube of the left inset), as well as a local GHM fit (shown as the larger, thicker curves). Right: shows the diffusion behavior in the planes of corresponding border color from the central inset (where columns represent times t = 0.02, 0.3, 1.0) using the GHM parameters from the local fit (see Fig. 5 for further details).

Figure 2

Three examples of Brownian motions (BMs) in the GHM manifold, under various speed settings, for t = 0 to 0.5 (with 1200 integration steps) with k_B = 0.9 and k_N = k_T = 0. Projections of each trajectory are shown on the axial planes in lighter shades. Simulation was computed via an order 1.0 Stochastic Runge-Kutta algorithm^,. Left: v_f = 3, v_t = 1. Right: v_f = 1, v_t = 1. The line segments on the left represent the local fiber directions along the z-axis. Notice that the anisotropic case (left) leads to spatially skewed diffusion, in comparison with the behavior of the isotropic case (right). This is particularly noticeable along the x axis, which coincides with the fiber orientation at the origin, where each trajectory begins.

Figure 3

An illustration of an exemplar GHM, including a depiction of the local transmural direction (i.e. the z-axis, parallel to the unit vector $\hat{k}$), as well as the meaning of the local θ value, determined by the GHM orientation function. In this example k_T = k_N = 0.

Figure 4

Illustration of GHM parameter values via streamline plots. Parameters (k_N, k_T, k_B) for each column are given left to right as (0.02, 0, 0.7), (0, −0.02, 0.7), and (0, 0, 0.7). Planes (green, red, and blue) are at fixed points on the z-axis (i.e. normal to the transmural direction z, in the heart). Notice the in-plane fanning and bending effects, respectively, of the in-plane curvatures k_N and k_T, and the transmural turning induced by k_B.

Figure 5

A visualization of the diffusion equation $\frac{\partial u}{\partial t}={\mathrm{\Delta }}_{g}u/2$ on the GHM manifold, with parameters k_N = 0.2, k_B = 0.9, k_T = 0, v_f = 3, v_t = 1. In this example, values of u are shown (both in color and in vertical height) for an excitation given by a delta function at (0, 0, 0), approximated by a Gaussian with σ = 0.01, placed at a site on the endocardium (bottom row, z = 0) for t = 0.1, 0.4, 0.8 (left to right), and for a nearby location just above it in the heart wall (top row, z = 1) for t = 0.2, 0.6, 1.0. Black lines are the local streamlines of the GHM fibers at that z-slice. The vertical height axis is from 0 to 2.5 for the bottom left 3D inset and from 0 to 0.5 for the rest. The horizontal domain in all cases are x, y ∈ [−4, 4] × [−4, 4]. Notice the elongating effect of the fibers on the behavior of the propagating signal, as well as the asymmetric spreading due to the fiber fanning due to non-zero k_N (e.g. in the lower-right inset).

Figure 6

Variance (solid lines) and means (dashed lines) over time for the x (red/left), y (blue/middle), and z (green/right) directions, respectively. Black lines are theoretical predictions. Top row: v_f = 3, v_t = 1; bottom row: v_f = 1, v_t = 1 (i.e. isotropy). Shaded regions are 99% confidence intervals using large-sample normal approximations to the sampling distributions of the sample mean and variance. Simulation parameters are the same as in Fig. 1 (bottom right), run 5000 times.