THE COUPLING OF DIFFUSION, REACTION AND MECHANICS IN BIOLOGICAL GROWTH: THEORY AND NUMERICAL MODELS

Growth in biological tissue depends upon cascades of complex biochemical reactions involving several species, as well as their transport through the extra cellular matrix and diffusion across cell membranes. In this work, a theoretical and numerical framework for the macroscopic treatment of growth is formulated within the context of open system continuum thermodynamics. Assumptions central to classical mechanics being too restrictive to capture such detail, this treatment involves the introduction of additional quantities (including mass sources/sinks, mass fluxes, terms for energy and momentum transfer between species) and deduces implications for balance laws. The framework, consistent with classical mixture theory, accounts for the multiple inter-converting and interacting species present in the tissue. Systematic adherence to fundamental physical principles, such as frame indifference of the response in the reference configuration and non-violation of the dissipation inequality, result in constitutive laws whose forms are clearly motivated and arise naturally from the treatment. Notably, the transport of the extra cellular fluid relative to the matrix is shown to be driven by the gradients of stress, concentration and chemical potential—a coupling of mass transport and mechanics that emerges directly. A finite element formulation employing a staggered scheme is implemented to solve the coupled partial differential equations that arise from the theory. Nonlinear projection methods are utilised to handle incompressibility, mixed methods for stress-gradient driven fluxes and energy-momentum conserving algorithms are used for dynamics. Our current tissue of interest being engineered in vitro tendon, the numerical examples are in this context. The examples serve to demonstrate aspects of the coupled phenomena as the tissue grows. The classes of initial and boundary conditions imposed, and model geometry match parallel experiments (which form another integral part of a larger project studying growth and remodelling of engineered tissues). Representatively, concentration or flux boundary conditions (tissue exposed to fluid in a bath, fluid injected in at the boundary) are imposed for the balance of mass governing the fluid phase. Conditions in an experiment being consistent with specification of quantities in the current configuration, noteworthy differences that arise when solving these equations in terms of quantities in this configuration, as opposed to the reference configuration, where the equations are initially developed, are accentuated. During the course of applying the theory to our system of interest, various meaningful modelling choices are made to tailor it to be representative of tendon constructs. While this tends to hint at a loss of generality, it is important to recognise that the fundamental theory and physical principles employed are still applicable to a larger class of problems, both from biology (injury mechanisms, wound healing, scarring and surgical repair, . . . ) as well as from other diverse fields (from porous soil mechanics, to diffusion of air through anisotropic rubber materials in automobile tyres). With further refinements to the theory and concurrent maturation of the resulting computational framework, it is of future interest to extend the application of the theory to other classes of problems.