On the theory of reactive mixtures for modeling biological growth

Abstract

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents is lacking in the biomechanics literature. This study provides the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents. The constituents are intrinsically incompressible and may carry an electrical charge. The interface jump condition on the mass flux of individual constituents is shown to define a surface growth equation, which predicts deposition or removal of material points from the solid matrix, complementing the description of volume growth described by the conservation of mass. A formu-lation is proposed for the reference configuration of a body whose material point set varies with time. State variables are defined which can account for solid matrix volume growth and remodeling. Constitutive constraints are provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochem cal potential of the fluids. Simplifications appropriate for biological tissues are also proposed, which help reduce the governing equations into a more practical format. It is shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework.

Simulation of the Progression of Intervertebral Disc Degeneration Due to Decreased Nutritional Supply

“…28–30 The disc is modeled as an inhomogeneous, porous, mixture consisting of a charged solid phase (with cells and PG fixed to the solid phase), a fluid (i.e., water) phase, and a solute phase [with different species of sodium ion Na + (+), chloride ion Cl − (−), glucose (g), oxygen (o), and lactate (l)]. In the model, the PG content is estimated by the chondroitin sulfated GAG content.…”

Temporal changes of mechanical signals and extracellular composition in human intervertebral disc during degenerative progression

“…In this study, the cell-activity coupled mechano-electrochemical theory (Zhu et al, 2012), which was developed based on the triphasic theory (Ateshian, 2007; Gu et al, 1998; Lai et al, 1991), was extended to include PG, a charged component in the solid matrix. The equation of mass balance for PG (estimated by GAG concentration) was described as follows: $$\frac{\partial \left(C^{\mathrm{italicGAG}}\right)}{\partial t}+\nabla ·false(C^{\mathit{GAG}}{\mathbf{v}}^{\mathbf{s}}false)=Q^{\mathit{GAG}},$$ where C GAG is the GAG concentration (mole per tissue volume), v s is the velocity of the solid phase, and Q GAG (mole per tissue volume per time) is the rate of GAG production (or consumption), which is assumed to be dependent on the cell density and GAG content by: $$Q^{\mathit{GAG}}={\lambda }_1{\rho }^{\mathit{cell}}-{\lambda }_2{C}^{\mathit{GAG}}.$$…”

Computational modeling of chemical reactions and interstitial growth and remodeling involving charged solutes and solid-bound molecules

“…Since ${\rho }_r^{\alpha }$ is the mass of α in the current configuration per volume of the mixture in the reference configuration (an invariant quantity), this parameter represents a direct measure of the evolving mass content of α in the mixture, which may thus be used as a state variable in a framework that accounts for chemical reactions (Ateshian 2007; Myers and Ateshian 2013). A distinction is now made between solid and solute species in the mixture, since they are often treated differently in an analysis.…”

“…In a biological mixture under isothermal conditions, the minimum set of state variables needed to describe reactive mixtures that include a solid matrix are: the (uniform) absolute temperature θ , the solid matrix deformation gradient F (or related strain measures), and the molar content c α of the various constituents (Ateshian 2007). This set differs from the classical treatment of chemical kinetics in fluid mixtures by the inclusion of F and the subset of constituents bound to the solid matrix.…”

“…Atheoretical framework for modeling reactive mixtures of neutral or charged soluble constituents (Ateshian 2007), formulated based on the classical theory of mixtures (Truesdell and Toupin 1960; Eringen and Ingram 1965; Bowen 1968, 1976), has provided the necessary foundation for the computational approach presented in this study. That theoretical framework has had a number of ramifications in relation to growth mechanics that were investigated in a recent series of reports.…”