At the present time the technology for directly imaging cardiac muscle fibers (e.g., diffusion tensor MRI) is not sufficiently developed for our purposes. Instead, we approximate the muscle fibers in the left ventricle using a method motivated again by the observations of Streeter, et al. [6] and by an asymptotic analysis of Peskin [3]. From measurements on the hearts of macaques, and treating the left ventricle as a nest of ellipsoidal surfaces of revolution, Streeter observed that muscle fibers in the left ventricle follow trajectories that are approximately geodesic paths on those surfaces. Using an asymptotic approach, also treating the wall of the left ventricle as a nest of surfaces (but not necessarily ellipsoids) of revolution, Peskin was able to derive the relation between the angle made by the geodesics as a function of their distance from the midwall surface. The angle is measured relative to the latitude lines of an appropriate coordinate system constructed on the surface of revolution. Figure 3(R) (redrawn from [3]) shows the relation between angle and distance from the midwall surface.

We treat the midwall surface shown in Fig. 3(L) as if it were a surface of revolution, even though it is not. A coordinate system is constructed on this surface consisting of geodesic curves which radiate out from the apex and terminate at the upper cross-sections of the data set, above the mitral or aortic valve ring. These coordinates can be thought of as lines of longitude. Lines of latitude are constructed by joining points of equal arclength measured from the apex along the lines of longitude. It is an interesting theorem of differential geometry that the latitudes and longitudes so constructed form an orthogonal net, even when the surface on which the above construction is done is not a surface of revolution. Because the midwall surface is not in fact a surface of revolution, some of these longitude lines intersect. When a pair of longitude lines intersects, the longer of the longitudes is terminated at the intersection. This insures that when following a line of constant latitude there is a monotonic change in longitude.

Following Streeter’s observation, and the theoretical curve of Fig. 3(R), muscle fibers are parallel to the lines of latitude on the midwall surface. We construct families of model fibers by computing geodesic paths on the CT scan midwall surface, starting at locations equally spaced on, and initially parallel to, a line of constant latitude. Geodesics paths are computed in both directions (increasing longitude and decreasing longitude), terminating whenever a valve ring is encountered. When a geodesic crosses a longitude line, its angle with respect to the latitude line is calculated, and the intersection point is projected toward the epicardial or endocardial surface by the distance given by the theoretical curve of Fig. 3(R). In this way model muscle fibers fill the space between the epicardial and endocardial surfaces, and have an angular distribution through the wall which matches the observed distribution.

How are geodesics constructed? Recall that the surface is triangulated. Each triangle in such a structure “knows” which triangles are its neighbors. Each triangle shares each of its edges either with another triangle or with nothing (in the case of edges on the upper end of the CT scan). No triangle shares any two of its edges with the same other triangle. It is straightforward to construct a map that indicates the other triangles with which any triangle shares edges. A triangle is a plane figure, so on each triangle a local coordinate system may be constructed having one unit vector normal to the plane of the triangle and two unit tangent vectors in the plane of the triangle. If a line is drawn from a point within any triangle in a known direction in the plane of the triangle, its intersection with one of the edges of the triangle is easily calculated. From these considerations geodesic curves on the surface are constructed as follows: draw a straight line or ray emanating from a point on a latitude line in one direction along the latitude line. Call the triangle containing the starting point “the current triangle”. The drawn ray, straight and in the plane of the triangle, is a geodesic of the triangle by definition. Calculate the intersection point of the drawn ray with an edge of the current triangle (there can be only one such point). Determine which triangle shares that edge with the current triangle, and call that triangle “the next triangle”. The vector in the direction of the drawn ray intersecting the edge can be decomposed into two components, along the edge and normal to the edge within the current triangle, and then recomposed along the edge and normal to the edge within the next triangle. The component along the edge is unchanged since the edge is shared; the component which was normal to the edge within the current triangle retains its magnitude but changes its direction to be normal to the edge within the next triangle. The ray drawn in the next triangle line has a known starting location (on the edge) and a known direction. Rename the next triangle as the current triangle, and repeat the drawing process just described. The result is a piecewise linear geodesic path on a triangulated surface. Note that the path is not necessarily the globally shortest path between the endpoints, but is the shortest path given the particular starting direction, which is sufficient for a geodesic path.

Figure 4 (upper row) shows the lines of longitude on the midwall surface (left panel), and a family of midwall-surface geodesics starting on a latitude line near the apex (right panel). Notice that even though geodesics are initially perpendicular to longitude lines (near the apex), they become more longitudinal as they rise (toward the base). Figure 4 (middle row, left panel) shows several families of geodesics on the midwall surface, each family arising from one of a set of regularly spaced latitude lines.