Thermodynamics of Surface Growth with Application to Bone Remodeling

Abstract

In physics, surface growth classically refers to processes where material reorganize on the substrate onto which it is deposited (like epitaxial growth), but principally to phenomena associated to phase transition, whereby the evolution of the interface separating the phases produces a crystal (Kessler, 1990; Langer, 1980). From a biological perspective, surface growth refers to mechanisms tied to accretion and deposition occurring mostly in hard tissues, and is active in the formation of teeth, seashells, horns, nails, or bones (Thompson, 1992). A landmark in this field is Skalak (Skalak et al., 1982, 1997) who describe the growth or atrophy of part of a biological body by the accretion or resorption of biological tissue lying on the surface of the body. Surface growth of biological tissues is a widespread situation, with may be classified as either fixed growth surface (e.g. nails and horns) or moving growing surface (e.g. seashells, antlers). Models for the kinematics of surface growth have been developed in (Skalak et al., 1997), with a clear distinction between cases of fixed and moving growth surfaces, see (Ganghoffer et al., 2010a,b; Garikipati, 2009) for a recent exhaustive literature review. Following the pioneering mechanical treatments of elastic material surfaces and surface tension by (Gurtin and Murdoch, 1975; Mindlin, 1965), and considering that the boundary of a continuum displays a specific behavior (distinct from the bulk behavior), subsequent contributions in this direction have been developed in the literature (Gurtin and Struthers, 1990; Gurtin, 1995, Leo and Sekerka, 1989) for a thermodynamical approach of the surface stresses in crystals; configurational forces acting on interfaces have been considered e.g. in (Maugin, 1993; Maugin and Trimarco, 1995) – however not considering surface stress -, and (Gurtin, 1995; 2000) considering specific balance laws of configurational forces localized at interfaces. Biological evolution has entered into the realm of continuum mechanics in the 1990’s, with attempts to incorporate into a continuum description time-dependent phenomena, basically consisting of a variation of material properties, mass and shape of the solid body. One outstanding problem in developmental biology is indeed the understanding of the factors that may promote the generation of biological form, involving the processes of growth (change of mass), remodeling (change of properties), and morphogenesis (shape changes), a classification suggested by Taber (Taber, 1995). The main focus in this chapter is the setting up of a modeling platform relying on the thermodynamics of surfaces (Linford, 1973) and configurational mechanics (Maugin, 1993)