A Triple Porosity Model of Stress Induced Fluid Flow in Cortical Bone

A theoretical model to characterize the stress induced fluid flow in cortical bone has been developed using triple porosity theory. The construction of the model starts with assumption involving three interacting fluid compartments: microporosity of the bone matrix, the lacunar-canalicular porosity and the vascular porosity associated with the Haversian Canals. On the basis of deductive considerations and using available experimental and anatomical data, the model is consistently specified. The model is able to account for the observations of macroscopic streaming potentials at cyclic and step loading. The predicted results are: independent of specimen thickness as experimentally measured by Otter et al., (1994); and are characterized by two relaxation times that are present in the data of Pienkowski (1982), and Gross and Williams (1982). INTRODUCTION In order to describe the macroscopic observations of streaming potentials, Salzstein et al., [1] developed two phase model of stress induced fluid flow in bone. The formulation was successful in modeling the frequency dependence of the streaming potential amplitude and phase relative to load. This model however is in conflict with later experimental studies of Otter et al., [2] and MacGinitie et al., [3]. They experimentally observed that decreasing the thickness of the specimen does not alter the effective time constant for the flow. Otter explains the observation with the assumption that in multi-osteonal specimens, the principle fluid relaxation occurs to Haversian Canals rather than to the bone surface. The aim of the present study is to offer macroscopic model of the stress induced fluid flow in bone, congruent with all existing macroscopic type experimental data at the present time. Our strategy is to start from triple porosity continuum model description of cortical bone and on the basis of existing information about the structure of bone, and applicable experimental data, to simplify and specify the model. TRIPLE POROSITY MATHEMATICAL MODEL Considering bone as a triple porosity deformed body, we model the bone utilizing multiple porosity consolidation theory (Wilson and Aifantis, [4], Cowin, [5]). Previously (Petrov et al., [6]), applying single porosity consolidation theory for an osteon, we assumed material isotropy and that the fluid pressure influence on the deformation state of the matrix is small. Using the same assumptions we have the following mathematical formulation for the problem: