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A mechanobiologically equilibrated constrained mixture model for growth and remodeling of soft tissues
Author(s) -
Latorre Marcos,
Humphrey Jay D.
Publication year - 2018
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201700302
Subject(s) - yield (engineering) , algebraic number , nonlinear system , algebraic equation , process (computing) , mathematics , steady state (chemistry) , term (time) , set (abstract data type) , biological system , computer science , mathematical optimization , mechanics , mathematical analysis , physics , chemistry , thermodynamics , quantum mechanics , biology , programming language , operating system
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.