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Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

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Abstract

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material’s anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.

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Notes

  1. For an infinitesimal strain analysis we may assume that the material and spatial configurations are nearly identical, \(x\approx X^{s}\).

  2. Chemistry textbooks typically adopt temperature and pressure as state variables, so that the Gibbs energy emerges as the natural scalar potential for examining the thermodynamics of reactions. In contrast, our treatment employs temperature and strain (thus, volume) as state variables, so that the Helmholtz free energy emerges as the natural choice of scalar potential.

  3. An explicit dependence of \(E\) on \(\rho_{r}^{\alpha }\) may also be adopted, e.g., assuming that each bundle has the same density \(\rho_{r}^{\alpha }=\rho_{0}/n\), where \(\rho_{0}\) is the mixture density; then, letting \(E ( \rho_{r}^{ \alpha } ) = ( \rho_{r}^{\alpha }/\rho_{0} ) E_{0}\) would reproduce the relation adopted in (3.14b).

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Acknowledgements

Research reported in this publication was supported by the National Institutes of Health under Award Number R01 GM083925. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.

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    Robert J. Nims & Gerard A. Ateshian

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Nims, R.J., Ateshian, G.A. Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues. J Elast 129, 69–105 (2017). https://doi.org/10.1007/s10659-017-9630-9

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