Constrained Mixture Model of Arterial Growth and Remodeling

Whereas the constrained mixture framework is general, the constitutive assumptions are problem specific. Our choice of constitutive relations seeks to yield appropriate emergent solutions for evolving geometry and material properties with the simplest relations possible while building on previous successes (cf. Valentín et al 2009 ). Here, we briefly describe our method for parameter estimation based largely on experimental findings in Wu et al (2014) (link) and Bersi et al (2016) (link) for a particular mouse model of hypertension for which information is available on the time course of changes in blood pressure, wall composition, and material properties with known roles of inflammatory cells.
Bersi et al (2014) (link) confirmed that biaxial stress-stretch data can be accurately reproduced from knowledge of the geometry, constitutive relation with best-fit parameters, and loading conditions. Hence, using information from Tables S1 and S2 in Bersi et al (2016) (link), we recreated the biaxial data and determined best-fit values for the original (labeled Sham in Bersi et al (2016 (link))) and evolved (4wk Ang II ) biaxial stress-stretch data. Because of the observed maladaptive evolution in Bersi et al (2016) (link), we let the elastic material parameters for collagen and smooth muscle (which turnover continuously) potentially evolve, whereby we determined: material parameters in the neoHookean (constant) and Fung-type (original/evolved) relations for elastin, collagen, and circumferential smooth muscle as well as appropriate deposition streches for elastin (constant) as well as collagen and smooth muscle (original/evolved). Because the previous constrained mixture model yields a mass-based rule-of-mixture expression for stresses for transient elastic (biaxial) responses, we also recreated (original and evolved) histological fractions from Figure 3 in Bersi et al (2016) (link) consistent, again, with predicted relations for our constrained mixture model for G&R (cf. Latorre and Humphrey 2018 ). Finally, because collagen fibers can be oriented in circumferential (θ), axial (z), and symmetric diagonal (d, with angle α₀) directions, the fit to biaxial data incorporated layer- and orientation-specific fractions of collagen within each layer (cf. Bellini et al 2014 (link)):
${\beta }_M^z={\beta }_A^z$ , and
${\beta }_A^{\theta }$ , with
${\beta }_M^d=1-{\beta }_M^z$ and
${\beta }_A^d=1-{\beta }_A^{\theta }-{\beta }_A^z$ , and where
${\beta }_M^{\theta }=0$ because medial circumferential collagen and smooth muscle are combined into a single contribution, collectivelly referred to as medial circumferential smooth muscle m. Due to the many model parameters to be determined from experimental data, each having its own significance, a progressive determination procedure was designed. See Appendix 1 for additional details regarding parameter estimation.
Finally, we recently showed that given appropriate timescales s_G&R/s_ext ≪ 1, where s_G&R is a characteristic time of the G&R process and s_ext is a characteristic time of the external loading stimulus, one can derive rate-independent relations that approximate predictions of a full (hereditary integral based) constrained mixture formulation (Latorre and Humphrey 2018 ). Based on the data in Bersi et al (2016) (link), s_ext ~7 – 14 days (for both pressure elevation and inflammatory cell infiltration) whereas we estimated s_G&R ~7 – 14 days in induced hypertension (cf. Nissen et al 1978 ). Hence, s_G&R/s_ext ~ 1 and we used the full constrained mixture model. Yet, prior simulations suggested that this rate-independent approximation is better in cases of sustained changes in pressure over long periods than for cases of sustained changes in flow or especially changes in axial stretch. For evaluative purposes, therefore, we also performed some simulations using the conceptually simpler, and computationally more efficient, rate-independent formulation. Briefly, a quasi-equilibrium formulation of this type presumes that the arterial wall adapts almost instantaneously to the given external perturbations at each G&R time s, hence making possible a pre-integration of Eq. (14) at each s, which yields
Similarly, an approximate integration of Eq. (4) for smooth muscle and collagen, which yet turn over continuously during the actual evolution, yields
$${\mathit{\sigma }}_{\mathit{\Gamma }}^0(s)\approx {\varphi }_{\mathit{\Gamma }}^{\alpha }(s){\stackrel{\sim }{\mathit{\sigma }}}_{\mathit{\Gamma }}^{\alpha },\forall s$$ with
${\stackrel{\sim }{\mathit{\sigma }}}_{\mathit{\Gamma }}^{\alpha }={\mathbf{G}}_{\mathit{\Gamma }}^{\alpha }{\widehat{\mathbf{S}}}_{\mathit{\Gamma }}^{\alpha }({\mathbf{G}}_{\mathit{\Gamma }}^{\alpha 2}){\mathbf{G}}_{\mathit{\Gamma }}^{\alpha }$ , so all of the integral equations are simplified to algebraic equations and one need not store the past history of each cohort of the structurally-significant constituents within the arterial wall. Indeed, consideration of Eqs. (15) and (16), instead of the hereditary integral-type Eqs. (14) and (4), along with additional equilibrium relations evaluated at the current G&R time s, lead to an equivalent nonlinear system of algebraic equations that can be solved easily and efficiently at each time s (Latorre and Humphrey 2018 ), which we will use as explained below.