A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Abstract

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and uxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a uid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to uxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a uniÿed fashion from previous formulations of the problem. The present work demonstrates these e ects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation. ?

On the theory of reactive mixtures for modeling biological growth

“…Conversely, a number of classical studies have explored growth and remodeling of tissues modeledas solids (Skalaketal, 1982;Cowin, 1983;Rodriguez et al, 1994), with no explicit reference to chemical reactions. More recently, studies have combined mixture analysis with chemical reactions (Humphrey and Rajagopal, 2002;Klisch et al, 2003;Garikipati et al, 2004), though they have been applied in narrower contexts than presented here.…”

A finite strain fibre-reinforced viscoelasto-viscoplastic model of plant cell wall growth

“…However, the complex nature of the cell wall growth and existence of competing modelling methodologies for the growth of soft tissue, e.g. single phase modelling [39] [46][47][48], mixture theory [49] and modelling based on the concept of natural configuration [50], imply that there is no single approach which is superior to others in all aspects of modelling. Therefore, the present study focuses on single phase (solid) modelling and tries to present a general framework for modelling cell wall growth based on solid mechanics.…”

Numerical pathologies in snap-through simulations

“…Many authors used conserving A to solve various types of problems [14]. In recent work, Garikipati et al [15] used this algorithm to model growth in biological tissue. Bathe [13] showed that the conserving B algorithm, which is incoporated in the commercial FEM software ADINA 8.7, is able to solve a specific type of problems where the Newmark algorithm is unstable and does not conserve energy and momentum.…”