A model for the volumetric growth of a soft tissue

A new model is developed for the volumetric growth of a soft biological tissue with finite strains. Unlike previous models, we distinguish the mere growth (material production) and forced deformation of a tissue caused by physical and chemical stimuli. The model accounts for an 'internal' inhomogeneity of a growing tissue, where any elementary (from the standpoint of the mechanics of continua) volume contains elements produced at different instants. We derive constitutive equations for an incompressible, growing, viscoelastic medium, subjected to aging, and apply them to two problems of interest in biomechanics. The first deals with growth of a viscoelastic bar driven by compressive forces. The model reflects functional adaptation in large femoral bones caused by trauma. We analyze numerically the effect of compressive load on the rate of cellular activity and demonstrate that a linear law of growth implies a finite material supply. The other problem is concerned with radial deformation of a growing viscoelastic cylinder under internal pressure. The model reflects mass production in large blood arteries and veins. It is shown, that the linear law of growth implies some conclusions regarding the rate of growth which contradict experimental data. A more sophisticated equation for the growth rate is suggested, which implies qualitatively adequate results.