An Idealised Biphasic Poroelastic Finite Element Model of a Tibial Fracture

The outcome of a bone fracture partly depends upon the mechanical environment experienced by the fracture callus (reparative tissue) during the healing. Therefore biomechanics of bone fracture healing has been examined in many clinical or biological, mathematical or finite element studies (Cheal et al. 1991, DiGioia et al. 1986, Claes et al. 1999, Doblaré et al. 2004 and Oh et al. 2010). Most of the studies model the components of bone fractures as monophasic, homogenous materials, which may not be appropriate considering the large inter fragmentary displacements and high porosity of the reparative tissue. Therefore, this study describes an idealised mathematical model of a healing bone fracture with biphasic approach when the callus bone is modelled as mixture of solids and fluids. Markel et al. (1990) reported that the porosity of the callus in a healing canine osteotomy decreased from 99.6% at 2 weeks to 38% at 12 weeks. Therefore, the biphasic, poroelastic model for fracture callus and bone has been suggested in the literature (Carter et al. 1998, Simon et al. 1992, Prendergast et al1997, Spilker et al. 1990). Biphasic poroelastic models for soft tissues (Mow et al. 1980, Simon et al. 1985, Van Driel et al. 1998, Prendergast et al. 1997, Spilker et al. 1990) have been developed and applied to model cartilage (Mow et al. 1980) and intervertebral discs (Simon et al. 1985). Van Driel et al. (1998) and Prendergast et al. (1997) modelled tissue adjacent to prostheses using poroelastic material properties to investigate tissue differentiation. In the field of fracture healing however, only monophasic material properties of callus have been simulated (Carter 1988, Carter 1998, Blenman 1989, Cheal 1991, DiGioia 1986, Claes 1999, Gardner 1998 and 2000). This is probably because of the paucity of data in the literature on the values of parameters required to define the biphasic material properties of fracture callus. Simulation of a biphasic, compressible, anisotropic, linear poroelastic material model requires forty material constants (Simon 1992). Even the very simplified simulation of an isotropic material requires a minimum of five material constants. However, the number of material constants required to simulate a biphasic, poroelastic medium can be further reduced to three if the solid and fluid media are assumed to be incompressible (Simon 1992). These three independent material parameters are Lame's material stiffness parameters (λ and μ) and hydraulic permeability (k). Alternatively, Zienkiewicz and Taylor (1994b) suggested a method to model poroelastic behaviour under `undrained' condition using the modulus of elasticity, Poisson's ratio, the