Elastohydrodynamic Scaling Law for Heart Rates

Animal hearts are soft shells that actively pump blood to oxygenate tissues. Here, we propose an allometric scaling law for the heart rate based on the idea of elastohydrodynamic resonance of a fluid-loaded soft active elastic shell that buckles and contracts axially when twisted periodically. We show that this picture is consistent with numerical simulations of soft cylindrical shells that twist-buckle while pumping a viscous fluid, yielding optimum ejection fractions of 35%–40% when driven resonantly. Our scaling law is consistent with experimental measurements of heart rates over 2 orders of magnitude, and provides a mechanistic basis for how metabolism scales with organism size. In addition to providing a physical rationale for the heart rate and metabolism of an organism, our results suggest a simple design principle for soft fluidic pumps.

Figure 1

(a) Structure of a four-chambered heart. LA, LV, RA, and RV denote the left atrium, left ventricle, right atrium, and right ventricle, respectively (drawing adapted from [14]). (b) Transverse section of the ventricles of a rat, sheep, and horse (schematics adapted from [13]). The sections have been enlarged to emphasize their close resemblance. (c) Schematic of the apical loop of the ventricular myocardial band. Adapted from [15]. Periodic twisting and untwisting of the ventricle driven actively by myocardial band contraction leads to fluid pumping. (d) Simplified ventricle geometry, reduced to an elastic shell of thickness $h$, radius $R$, density ${\rho }_{\mathrm{wall}}$, elastic modulus $E$, and containing a fluid (blood) of density ${\rho }_{\text{blood}}$. Passive end-twisting of the cylinder causes it to buckle and pump fluid.

Figure 2

(a) Snapshots from numerical simulations of a cylindrical shell buckling under twist. The bottom end is kept fixed while the top end is rotated by ${\pi /2}^c$. The rotation step between each picture is ${\pi /6}^c$. As the shell buckles into a low-order mode that has an internal volume that is smaller than that of the straight cylinder, it ejects fluid during the process. (b) Net ejection fraction versus the driving Reynolds number for different thickness ratios computed using direct numerical simulations of a deforming elastic shell coupled with a Navier-Stokes solver (see Supplemental Material [28] for details). Inset shows dependence of the frequency with the highest ejection fraction for each thickness ratio versus the thickness ratio which roughly follows a scaling $f\sim (R/h{)}^{-3/2}$ for fixed $L/R=3$, consistent with (1).

Figure 3

Comparison between experimentally measured animal heart rates $f_e$ (see Supplemental Material [28]) and the theoretical law for elastohydrodynamic resonance $f_t$; the straight line is the linear relation (1) $f_e=f_t\simeq c_{\text{shape}}/(2\pi )\sqrt{E/{\rho }_f}{h}^{3/2}/R^{5/2}$, with $c_{\text{shape}}\simeq 1/\sqrt{6}$ (for cylindrical shapes).