Gailani, Gaffar

Russian Doll Poroelasticity

[PhD thesis]. New York, NY: The City University of New York

Cowin, Stephen (supervisor)

2009

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- Abstract
In this thesis I solve three interconnected problems related to the understanding of cortical bone poroelasticity. The first and most basic problem addressed is the unconfined compression of a cylindrical annular disk of fluid-filled, porous, elastic solid between smooth, impermeable plates. The theory of anisotropic poroelastic materials is used to develop an analytical solution for this axisymmetric problem in which the plane of transverse isotropy is perpendicular to the cylindrical axis. A general solution is obtained from which the stress relaxation response is obtained. The solution to this boundary value problem is determined for three different boundary conditions on the pore pressure. These boundary conditions are (i) free fluid passage across the outer and inner cylindrical boundaries of the disk; (ii) free fluid passage across the outer cylindrical boundary and no flow across the inner boundary (i.e. zero pressure gradient); (iii) free fluid passage across the inner cylindrical boundary and no flow across the outer boundary. These solutions appeared in Gailani and Cowin (2008).

The second problem is to detail a design and execute an experimental method to measure the permeability of the lacunar-canalicular porosity (PLC) of bone tissue. The analytical solution of problem #1 is used as a theoretical basis for outlining a step-by-step procedure to experimentally determine the permeability of the disk. In addition, procedures will be described for measuring the permeability of the inner walls (Haversian canal) of an osteon. This is achieved by experimentally employing any of the three different solutions obtained for the basic problem described above, particularly the one that is described as (iii) above. An iterative optimization technique, the least squares method, is used to obtain the material properties including the permeability which yield the best curve fit of the analytical solution to the experimental data.

The two problems described are a prelude to constructing a more detailed model of the porous structure of osteonal cortical bone. This new porous structure model is said to be “nested” in the sense that smaller pore size porosity, the PLC, only interchanges interstitial bone fluid with the larger pore size porosity, the vascular porosity (PV). In the construction of the nested porosity model, called Russian doll poroelasticity, the analytical solution of problem #1 can be used twice, once at the PLC level (the osteon) and again at the PV level (the whole bone cross-section). The significance of the result is basic to the understanding of interstitial flow in bone tissue which, in turn, is basic to understanding nutrient transport from the vasculature to the bone cells buried in the bone tissue and to mechanotransduction in bone tissue. In addition it gives an insight into the elastic response of soft tissues versus hard tissues and saturated swampy soils versus saturated rocks and as an analytical tool to analyze unconfined compression experiments performed on these materials.

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