Myocardial stress equations: fiberstresses of the prolate spheroid

There are occasions in physiological research and medical practice where it is desirable to estimate the average fiberstress in a chamber wall, knowing only the pressure and dimensions. Because the contribution of a strained wall element to pressure depends on its location whereas its contribution to average stress is independent of location, an equation of this kind must involve an assumption about the stress distribution. When applied to a particular chamber, it will give an exact result only if the chamber's stress distribution is in some sense like that of the model for which the equation was derived. Since the fibers of biological chambers are continually being deposited and resorbed, they tend to exhibit similar stretches under the average conditions of the chamber. To the extent that this is so, P = (2/3) sigma v ln V0/Vc, is the best simple fiberstress equation for biological chambers. (P = transmural pressure, sigma v = volume-averaged fiberstress, V0 = volume enclosed by outside surface, Vc = cavity volume). It expresses the pressure-dimension-average-fiberstress relation of a chamber of any shape whose stresses exhibit the simplest possible distribution. One can add a term to the right side to account for the influence of stress profile complexities. That term takes the form of a moment whose value is zero at one state of distension. This "stress moment" expresses the unequal weighting of complexities on the two sides of the midwall isobar. Judging from the sarcomere length profile of the left ventricular wall, the stress moment is zero and the average fiberstress equation above is exact for average developed stress (without a second term) when cavity volume is somewhere near end-diastolic. Moreover, the departures from the relation (the effects of stress moment) are small so long as the inner and outer stresses do not differ by a factor greater than two.