Segmental calculation of left ventricular wall stresses

A procedure for calculating left ventricular wall stresses segmentally was devised. Rectangular coordinates of the wall surfaces as seen in longitudinal section were plotted with the long axis as the x-axis. For each cavity point, a third-order polynomial (cubic spline) was fitted to the point together with several adjacent points on either side of it; the cavity radius (normal to cavity surface) at the point was found algebraically from the spline's coefficients. Each cavity radius was matched with the most symmetrical one from the opposite cavity surface. The point of intersection of the cavity radius with the outer surface was found, and a midwall point was identified from logarithmic means of cavity and outer radial lengths. For each midwall point, a cubic spline was fitted to that point together with several adjacent points on either side of it, and the midwall radius at that point was determined algebraically from the spline's coefficients. Each midwall radius was matched to the most symmetrical one from the opposite midwall. The locus of points at equal radial distances from opposite midwalls forms the axis. The midwall radius of curvature (r theta) orthogonal to the meridian at each point was taken as the radial distance from the midwall to the axis. Midwall meridional radius of curvature (r phi) was calculated from the spline's coefficients. Thickness (h) was calculated from intersections between the midwall radius and the inner and outer surfaces. For each point, meridional tension (T phi) was calculated as T phi = Pr theta/2, and hoop tension (T theta) was calculated as T theta = (Pr theta/2)(2 - r theta/r phi) where P is transmural pressure. Stresses were calculated as tensions divided by thicknesses (sigma phi = T phi/h, sigma theta = T theta/h), or more directly as sigma phi = Pr theta/2h and sigma theta = (Pr theta/2h)(2 - r theta/r phi). This procedure was validated with simple chamber shapes, and it has been applied to left ventricles.