Mathematical analysis of left ventricular elastance with respect to afterload change during ejection phase

Since the left ventricle (LV) has pressure (Plv) and volume (Vlv), we can define LV elastance from the ratio between Plv and Vlv, termed as “instantaneous elastance.” On the other hand, end-systolic elastance (Emax) is known to be a good index of LV contractility, which is measured by the slope of several end-systolic Plv—Vlv points obtained by using different loads. The word Emax originates from the assumption that LV elastance increases during the ejection phase and attains its maximum at the end-systole. From this concept, we can define another elastance determined by the slope of isochronous Plv—Vlv points, that is Plv—Vlv points at a certain time after the ejection onset time by using different loads. We refer to this elastance as “load-dependent elastance.”To reveal the relation between these two elastances, we used a hemodynamic model that included a detailed ventricular myocyte contraction model. From the simulation results, we found that the isochronous Plv—Vlv points lay in one line and that the line slope corresponding to the load-dependent elastance slightly decreased during the ejection phase, which is quite different from the instantaneous elastance.Subsequently, we analyzed the mechanism determining these elastances from the model equations. We found that instantaneous elastance is directly related to contraction force generated by the ventricular myocyte, but the load-dependent elastance is determined by two factors: one is the transient characteristics of the cardiac cell, i.e., the velocity–dependent force drops characteristics in instantaneous shortening. The other is the force–velocity relation of the cardiac cell. We also found that the linear isochronous pressure–volume relation is based on the approximately linear relation between the time derivative of the cellular contraction force and the cellular shortening velocity that results from the combined characteristics of LV and aortic compliances.Author summary: Characteristic that the end-systolic pressure volume relation (ESPVR) of left ventricle becomes linear is one of the well known property of the heart, and its slope is considered as a good index of the heart contractility. However, the reason of the linearity is still not clarified. Recent biophysical model can reproduce the linear ESPVR, and since the model is written in the mathematical equation, the basis of the ESPVR linearity can be analytically analyzed.