A Growth Mixture Theory for Cartilage

In this paper we present a model of growth for cartilaginous tissues in which there exists a saturated solid matrix composed of multiple constituents that may grow and remodel independently of each other. Klisch and Hoger recently developed a general theory of volumetric growth for a mixture of ν-1 growing elastic materials and an inviscid fluid, which included a treatment of two special types of internal constraints that are relevant to cartilage. Here, that theory is specialized to construct a cartilage growth model. This theory allows the constituents of the solid matrix to grow independently of each other, and can model the evolution of the constituent pre-stresses and the tissue’s mechanical properties during developmental growth and degeneration. A simple example is presented which illustrates these features of the theory. INTRODUCTION Cartilaginous tissues are composed primarily of three constituents: water, collagen, and proteoglycans. The results of numerous studies suggest that mechanical stimuli cause growth abnormalities in conditions such as hip dysplasia (Pauwels, 1976), traumatic injury near the growth plate (Ogden, 1988), and scoliosis of the spine (Brickley-Parsons and Glimcher, 1984). There also exists experimental evidence from in vivo (Kiviranta et al., 1994) and in vitro (Sah et al., 1989) experiments that growth-related parameters such as biochemical composition, mechanical properties, and cartilage thickness vary due to changes in mechanical stimuli such as stress and strain. In addition, these studies suggest that the proteoglycan and collagen constituents of the solid matrix may grow and remodel independently of each other. Continuum mixture theory has been used to model the finite deformation and flow-dependent mechanical properties of both articular cartilage (Ateshian et al., 1997) and the intervertebral disc (Klisch and Lotz, 2000). In those theories, the mixture is modeled as being composed of a solid constituent representing the collagen-proteoglycan matrix and a fluid constituent representing the water. As a first step towards developing a model of growth and remodeling for cartilaginous tissues, a general theory of volumetric growth for compressible elastic materials was presented by Klisch et al. (2000). That paper