Modelling carotid artery adaptations to dynamic alterations in pressure and flow over the cardiac cycle

Abstract

Motivated by recent clinical and laboratory findings of important effects of pulsatile pressure and flow on arterial adaptations, we employ and extend an established constrained mixture framework of growth (change in mass) and remodelling (change in structure) to include such dynamical effects. New descriptors of cell and tissue behavior (constitutive relations) are postulated and refined based on new experimental data from a transverse aortic arch banding model in the mouse that increases pulsatile pressure and flow in one carotid artery. In particular, it is shown that there was a need to refine constitutive relations for the active stress generated by smooth muscle, to include both stress- and stress rate-mediated control of the turnover of cells and matrix and to account for a cyclic stress-mediated loss of elastic fibre integrity and decrease in collagen stiffness in order to capture the reported evolution, over 8 weeks, of luminal radius, wall thickness, axial force and in vivo axial stretch of the hypertensive mouse carotid artery. We submit, therefore, that complex aspects of adaptation by elastic arteries can be predicted by constrained mixture models wherein individual constituents are produced or removed at individual rates and to individual extents depending on changes in both stress and stress rate from normal values.

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Appendix A.

The numerical finding in Humphrey & Na (2002), that inertial forces are generally negligible in wall mechanics over the cardiac cycle, is supported here for the special case of the mouse common carotid artery via a simple analysis of orders of magnitude of the terms in (2.5). Using the radius a as a reference length, the cardiac period T as a reference time, the mean pressure P_m as a reference pressure and wall density ρ as a reference mass density, the radial equilibrium equation (2.5) can be written in the non-dimensional form as

$${\frac{1}{{\lambda }_z(s,t)}\frac{\partial {\omega }^{*}}{\partial \lambda \vartheta }|}_{s^{*},t^{*}}+h^{*}(s^{*},t^{*}){\sigma }_{\vartheta }^{\text{act}*}(s^{*},t^{*})=P^{*}(s^{*},t^{*})a^{*}(s^{*},t^{*})-\left[\frac{\rho a^2}{P_{\text{m}}{T}^2}\right]h^{*}{\left(s^{*},t^{*}\right)}^2{\ddot{a}}^{*}(s^{*},t^{*}),$$ (A.1)

where superscript ^* denotes non-dimensional quantities. Non-dimensionalization of G&R time s would require definition of another reference time S that does not appear explicitly in the equation of motion because it is written over the cardiac cycle timescale and there are no derivatives with respect to s. For a mouse carotid artery, assuming P_m ∼100 mmHg, a ∼10⁻¹ mm, T ∼10⁻¹ s and ρ ∼10³ kg/m³, the non-dimensional quantity in the square bracket, which represents the ratio between the inertial forces per unit surface ρa²/T² and the pressure P_m, is of order 10⁻⁷, suggesting that the inertial force is indeed negligible. Hence, we used the quasi-static relation (2.6) similar to the one employed previously on the G&R timescale.

Fig. A1.

Flow chart of the computational scheme for G&R simulations. N-R denotes the Newton–Raphson method.

Appendix B.

Here, we provide a brief description of the numerical scheme used to simulate dynamic arterial G&R (see Fig. A1). We adopt a marching technique in G&R time s, advancing in fixed steps Δs. Before starting the march, we check initial equilibrium and compute homeostatic values of stress and stress rate measures (given values of deposition stretch; Table 1) that will serve as targets through the computation. At each integration time step, we perform a nested loop to satisfy radial equilibrium (2.6) at the mean pressure P_m and to account for the updated turnover of constituents; the loop starts by estimating values of the mass production rates $m_{(i)}^k$ based on previous values and then solves the radial equilibrium (2.6) via a Newton–Raphson method, thus estimating a new value of the radius a_(i). It is then possible to compute new values for the stress measures σ^k and thus the mass production $m_{(i+1)}^k$ and removal q^k rates. The nested loop stops when mass production rates are estimated with sufficient resolution (i.e. when $\sum \left|m_{(i+1)}^k-m_{(i)}^k\right|$ is smaller than a given tolerance). The solution on the cardiac cycle is computed offline and associated rate measures, which influence the evolution of individual mass productions, removals, remodelling and damaging, are evaluated after the nested loop for computational efficiency.