A constrained mixture model for arterial adaptations to a sustained step change in blood flow

Abstract

A sustained change in blood flow results in an arterial adaptation that can be thought to consist of two general steps: an immediate vasoactive response that seeks to return the wall shear stress to its homeostatic value, and a long-term growth and remodeling process that seeks to restore the intramural stresses and, if needed, the wall shear stress toward their homeostatic values. Few papers present mathematical models of arterial growth and remodeling in general, and fewer yet address flow-induced changes. Of these, most prior models build upon the concept of "kinematic growth" proposed by Skalak in the early 1980s (Skalak R (1981) In: Proceedings of the IUTAM Symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347-355). Such approaches address important consequences of growth and remodeling, but not the fundamental means by which such changes occur. In this paper, therefore, we present a new approach for mathematically modeling arterial adaptations and, in particular, flow-induced alterations. The model is motivated by observations reported in the literature and is based on a locally homogenized, constrained mixture theory. Specifically, we develop a 3-D constitutive relation for stress in terms of the responses of the three primary load-bearing constituents and their time-varying mass fractions, with the latter accounting for the kinetics of the turnover of cells and extracellular matrix in changing, stressed configurations. Of particular importance is the concept that the natural configurations of the individual constituents can evolve separately and that this leads to changes in the overall material properties and empirically inferred residual stress field of the vessel. Potential applications are discussed, but there is a pressing need for new, theoretically motivated data to allow the prescription of specific functional forms of the requisite constitutive relations and the values of the associated material parameters.