Computational Modeling of Growth and Remodeling in Biological Soft Tissues: Application to Arterial Mechanics

Abstract

Traditional approaches in continuum biomechanics have revealed tremendous insights into the behavior of biological soft tissues, including arteries. Nevertheless, such approaches cannot describe or predict perhaps the most important characteristic behavior, the ability of soft tissues to adapt to changes in their chemo-mechanical environment. In this chapter, we introduce and illustrate one possible approach to modeling commonly observed cases of biological growth and remodeling of arteries in response to altered mechanical stimuli. In particular, we introduce a constrained rule-of-mixtures approach to modeling that can account for individual mechanical properties, natural (stress-free) configurations, and rates and extents of turnover of the different structurally significant constituents that make-up an artery. We illustrate how this theoretical framework can be implemented in a nonlinear finite element model of an evolving intracranial fusiform aneurysm and conclude with guidance on future needs for continued research. There is a pressing need, for example, for additional mechanobiological data that can guide the formulation of appropriate constitutive relations for stress-mediated production and removal of structural constituents by the different types of cells that reside within the arterial wall.

Appendix

Increasing experience with our basic G&R framework continues to reveal new approaches that can be effective and conceptually straightforward in particular problems. For example, Humphrey and Rajagopal [17] suggested G&R relations based on Cauchy stress whereas Baek et al. [2, 4] and Valentin et al. [39] suggested that it was preferable to pose these relations in terms of constituent strain energies per unit reference area in 2D formulations. In other words, it is easier to identify or prescribe scalar constitutive functions than tensorial functions. Associated simulations have predicted salient aspects of arterial G&R in diverse cases, thus supporting this strain energy based approach. In anticipation of the need for 3-D computations, however, constituent stored energies defined per mass rather than per volume may be computationally more expedient. In particular, this may facilitate the defining of stored energy with respect to the evolving times of material deposition. This change from a per reference area/volume to a per mass (consistent with classical ideas for defining Helmholtz potentials) is accomplished easily via the determinant of the appropriate deformation gradient between the mixture reference configuration and the mixture configuration at the time of deposition. Hence, whereas Baek et al. [2, 4] and Valentin et al. [39] wrote (without explicit subscripts R for reference)

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$$\begin{array}{rcl}{ W}_{R}^{k}(s) = \frac{{\rho }_{R}^{k}(0)} {\rho (s)} {Q}^{k}(s){W}^{k}\left ({\mathbf{C}}_{ n(0)}^{k}(s)\right )+{\int \nolimits \nolimits }_{0}^{s}\frac{{m}_{R}^{k}(\tau )} {\rho (s)} {q}^{k}(s - \tau ){W}^{k}\left ({\mathbf{C}}_{ n(\tau )}^{k}(s)\right )d\tau & & \\ & &\end{array}$$

(1)

where W _R ^k(s), ρ _R ^k(0), and m _R ^k(τ) are the strain energy per reference volume (or area) at time s, mass per reference volume (or area) at time 0, and rate of mass production per reference volume (or area) at time τ, W ^k is a strain energy density of the constituent k, and ρ(s) is the density of mixture defined per unit current volume. We now suggest that it may be preferable to write

$${ W}_{R}^{k}(s) = {\rho }_{ R}^{k}(0){Q}^{k}(s){\psi }^{k}\left ({\mathbf{C}}_{ n(0)}^{k}(s)\right )+{\int \nolimits \nolimits }_{0}^{s}{m}_{ R}^{k}(\tau ){q}^{k}(s-\tau ){\psi }^{k}\left ({\mathbf{C}}_{ n(\tau )}^{k}(s)\right )d\tau $$

(2)

where ψ^k C _n(τ) ^k(s) is the stored energy of constituent k that has been produced at time τ per unit mass. This letter framework has been implemented by Figueroa et al. [8] in a fluid-solid-growth (FSG) simulation.