Constrained Mixture Models of Soft Tissue Growth and Remodeling – Twenty Years After

Abstract

Soft biological tissues compromise diverse cell types and extracellular matrix constituents, each of which can possess individual natural configurations, material properties, and rates of turnover. For this reason, mixture-based models of growth (changes in mass) and remodeling (change in microstructure) are well-suited for studying tissue adaptations, disease progression, and responses to injury or clinical intervention. Such approaches also can be used to design improved tissue engineered constructs to repair, replace, or regenerate tissues. Focusing on blood vessels as archetypes of soft tissues, this paper reviews a constrained mixture theory introduced twenty years ago and explores its usage since by contrasting simulations of diverse vascular conditions. The discussion is framed within the concept of mechanical homeostasis, with consideration of solid-fluid interactions, inflammation, and cell signaling highlighting both past accomplishments and future opportunities as we seek to understand better the evolving composition, geometry, and material behaviors of soft tissues under complex conditions.

Appendix

Table A.1 The full (heredity integral based) constrained mixture theory requires three classes of constitutive relations for each structurally significant constituent $\alpha =1, 2, 3,\dots ,N$ to capture the evolving composition and material properties of a mature tissue for all G&R times $\tau \in [0,s]$: constitutive-specific rates of mass density production $m^{\alpha } ( \tau ) >0$, mass survival functions $q^{\alpha } ( s-\tau ) \in [0,1]$, and stored energy functions $\hat{W}^{\alpha } ( \mathbf{F}_{n ( \tau )}^{\alpha } (s) ) >0$, the latter of which include information on the deposition stretch and orientation. Although the basic framework has persisted over the past 20 years, notation has evolved to increase clarity: for example, G&R time $t$ changed to time $s$, subscript or superscript $h$ for homeostatic changed to $o$ for original homeostatic (to accommodate possible adaptive homeostasis), constituent index $k$ changed to $\alpha$, the rate parameter $K_{q}^{\alpha }$ changed to $k^{\alpha }$, and dimensional rate-gain parameters $K_{g}^{\alpha }$ changed to non-dimensional gain parameters $K_{j}^{\alpha }$, which can be subscripted to denote $j = \text{stress}$ related or inflammation related. Of course, because these functions are constitutive, they necessarily differ for different problems, depending on 2D versus 3D formulations as well as, in some cases, on particular stresses of concern (constituent-specific versus tissue level), the presence of blood clots, damage to or healing of the functional cells, and the presence of inflammation. Listed here are a few of the key forms for the production and removal functions; the stored energy functions tend to follow standard forms, as, for example, neoHooken or Fung exponential. To facilitate comparisons here, some notations have been changed as appropriate for consistency with current conventions