Cardiac timing modulates the contribution of compliance to effective arterial elastance

Effective arterial elastance (E_a) is a lumped descriptor of arterial load. The contributions of arterial characteristics to E_a, and their interactions with cardiac timing parameters, have not been fully elaborated. Previous studies have shown that E_a can be approximated by equations of the form E_a ≈ R/T + k/C, where R is either total or peripheral resistance, T is cardiac cycle time, and C is arterial compliance, but reported values for the coefficient k vary substantially, and some analyses suggest that compliance contributes little to E_a. Here, an analytical approximation is derived from Sunagawa's Windkessel expression for E_a by replacing the exponential term with a truncated Taylor expansion. The resulting approximation predicts that k is not constant but varies with cardiac timing, such that k ≈ 0.5 (t_d/T)², where t_d is diastolic time. Approximation accuracy was assessed across 693 combinations of resistance, compliance, and heart rate and in a virtual population of 4374 adults derived from a published one-dimensional vascular model. The linear approximation for E_a closely tracked the Windkessel expression across the full parameter space (R² > 0.9999, mean absolute percentage error 0.51%). In the virtual population, the approximation showed good agreement with the E surrogate P_ES/SV (end-systolic pressure/stroke volume), surpassing previous approximations. These findings show that the contribution of compliance to E_a is modulated by cardiac timing and provide a mechanistic explanation for previously reported values of k.