A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue

Most mathematical models of the growth and remodeling of load-bearing soft tissues are based on one of two major approaches: a kinematic theory that specifies an evolution equation for the stress-free configuration of the tissue as a whole or a constrained mixture theory that specifies rates of mass production and removal of individual constituents within stressed configurations. The former is popular because of its conceptual simplicity, but relies largely on heuristic definitions of growth; the latter is based on biologically motivated micromechanical models, but suffers from higher computational costs due to the need to track all past configurations. In this paper, we present a temporally homogenized constrained mixture model that combines advantages of both classical approaches, namely a biologically motivated micromechanical foundation, a simple computational implementation, and low computational cost. As illustrative examples, we show that this approach describes well both cell-mediated remodeling of tissue equivalents in vitro and the growth and remodeling of aneurysms in vivo. We also show that this homogenized constrained mixture model suggests an intimate relationship between models of growth and remodeling and viscoelasticity. That is, important aspects of tissue adaptation can be understood in terms of a simple mechanical analog model, a Maxwell fluid (i.e., spring and dashpot in series) in parallel with a "motor element" that represents cell-mediated mechanoregulation of extracellular matrix. This analogy allows a simple implementation of homogenized constrained mixture models within commercially available simulation codes by exploiting available models of viscoelasticity.