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Image-Based Multiscale Modeling of Poroelastic Biological Materials with Application to Bones


Biot's theory has been widely used to construct the poroelasticity models for describing the mechanical behavior of biological materials. This phenomenological framework, however, does not take the explicit microstructural configuration and the corresponding solid-fluid coupling into consideration. This work investigates how the microstructural configuration and material properties of porous materials constitute the macroscopic poroelastic material behavior described by the classical Biot's theory. We introduced an asymptotic based homogenization method to correlate the macro- and micro-mechanical behaviors of poroelastic materials, where an elastic solid and Newtonian fluid of low viscosity are considered. Through this homogenization process, the generalized Darcy's law, homogenized macroscopic continuity equation, and homogenized macroscopic equilibrium equation were obtained, where the homogenized macroscopic continuity and equilibrium equations reassemble the governing equations in Biot's theory.

For an effective modeling of microstructures, a numerical solution for PDEs based on a strong form collocation that employs image pixels as the discretization points is proposed. To achieve this objective, a gradient reproducing kernel collocation method (G-RKCM) formulated based on the partition of nullity and gradient reproducing conditions was developed. This approach reduces the order of differentiation to the first order when solving second order PDEs with strong form collocation. We showed that the same number of collocation points and source points can be used in G-RKCM for optimal convergence, unlike other strong form collocation methods. In addition, same order of convergence rate in the solution and its first order derivative are achieved, owing to the imposition of gradient reproducing conditions. The computational complexity of G-RKCM is also shown to be an enhancement over other strong form collocation methods, such as the reproducing kernel collocation method (RKCM).

In this work, we introduced the active contour model based on variational level set formulation for interface identification and boundary segmentation for the discretization of microstructures based on medical images. Using pixel point discretization, we introduced the RKCM and G-RKCM to solve the level set equation. In particular, the G-RKCM has been shown be effective since the second derivatives of the level set function involved in the regularization term are approximated by the first order differentiations of the gradient RK shape functions. We further showed that a B-spline kernel function with lower continuity can be preferably used to avoid the oscillation of level set functions in the two-color images.

The image based G-RKCM was applied to model trabecular bone microstructures with complex geometry for both solid and fluid phases. The corresponding numerical issues such as interface discretization and kernel function support size selection have been addressed. The investigation on the proper choice of unit cell dimension and image resolution has been performed, which provides guidance in the image-based trabecular bone modeling. The validation of the proposed image based multiscale modeling framework has been carried out by comparing the numerical prediction of effective material properties with experimental data of trabecular bone in the literature and solving a macroscopic trabecular bone problem using the homogenized material constants.

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