A mechanobiologically equilibrated constrained mixture model for growth and remodeling of soft tissues

Abstract

Growth and remodeling of soft tissues is a dynamic process and several theoretical frameworks have been developed to analyze the time-dependent, mechanobiological and/or biomechanical responses of these tissues to changes in external loads. Importantly, general processes can often be conveniently separated into truly non-steady contributions and steady-state ones. Depending on characteristic times over which the external loads are applied, time-dependent models can sometimes be specialized to respective time-independent formulations that simplify the mathematical treatment without compromising the goodness of the particularized solutions. Very few studies have analyzed the long-term, steady-state responses of soft tissue growth and remodeling following a direct approach. Here, we derive a mechanobiologically equilibrated formulation that arises from a general constrained mixture model. We see that integral-type evolution equations that characterize these general models can be written in terms of an equivalent set of time-independent, nonlinear algebraic equations that can be solved efficiently to yield long-term outcomes of growth and remodeling processes in response to sustained external stimuli. We discuss the mathematical conditions, in terms of orders of magnitude, that yield the particularized equations and illustrate results numerically for general arterial mechano-adaptations.

Fast, Rate-Independent, Finite Element Implementation of a 3D Constrained Mixture Model of Soft Tissue Growth and Remodeling

“…where we used ρ ξ Rh = J h ρ ξ h and Eqs. (8), which are equivalent to those obtained in [3] between the two evolving constituents considered therein (smooth muscle cells "m" and collagen fibers "c", with a single η = η q η Υ ). Finally, the constraint φ α h = 1, with φ ζ h (J h ) from Eq.…”

“…Summation of mass (2) n α (Υ α − 1) =m (4) and ρv = div σ + ρb (5) where ρ R = ρ α R = Jρ, m = m α , n = n α , σ = σ α t , and ρb = ρ α b α . As in [3], and because n α > 0, we observe from Eqs. (2) and (4) that a necessary condition for a soft tissue to be in mechanobiological equilibrium (to preserve its mixture mass, composition, and properties) is…”

“…which generalizes rate equations forρ α R obtained in [3] or [4] from an integral-based approach based on a first-order kinetics decay n α = k α ρ α , with k α a rate parameter that defines the removal. On other hand, the spatial linear momentum balance relation for constituent α, with vanishing interactive forces among constituents due to their concurrent motion, may be written as…”

“…Since constituents ζ are deposited within κ o = κ(0), associated contributions at the mixture level (per unit reference volume of mixture) are weighted directly by respective original volume fractions Φ ζ o as W ζ R = Φ ζ oŴ ζ . Noting that Φ ζ o coincides with the mass fraction φ ζ o for mixtures of constituents with the same mass density [3], as assumed herein,…”

“…Herein, we exploit a recent concept of mechanobiologically equilibrated G&R [3] and show computational advantages for illustrative cases that allow direct comparison with full constrained mixture solutions. Towards this end, we also extend prior kinematics to account for general three-dimensional G&R with finite deformations and possible rotations.…”