FLUID-SOLID HOMOGENIZATION OF BONE TISSUE AT THE CANALICULAR CHANNEL AND CELLULAR LEVELS

Participants: B.R. McCreadie, S.A. Goldstein, S.J. Hollister

Keywords: bone, osteocytes, computational model

Introduction

Many computational models have been created which attempt to analyze either the fluid flow through bone or the deformation of the solid components of bone tissue. Generally these models are only able incorporate fluid flow or solid deformation, but not both. The purpose of this study was to develop and evaluate models at two hierarchical levels to begin to determine how fluid and solid components of the bone tissue interact during bone loading.

Materials and Methods

Confocal microscopy images at 0.2 micron intervals were obtained from a human trabecular bone specimen stained with basic fuchsin. The images were analyzed using a program written in PV-Wave to determine coordinates of a single canalicular channel. These coordinates were transferred to Patran to create finite element models of a single canalicular channel in a rectangular block of bone tissue. The cross-sectional geometry of the canalicular channels and the cross-sectional dimensions of the matrix block were varied according to literature values to investigate how these parameters affected the results. Other models were shortened by reducing only the coordinate in the direction approximately parallel to the channel to half the original value, resulting in an increased tortuosity. The models were analyzed using a fluid-solid homogenization program written by Terada, et al.(Computer Methods in Applied Mechanics and Engineering 153:223) to determine the homogenized material properties (stiffnesses, Poisson's ratios, permeability) of the bone tissue, including canaliculi.

At the cell level, a model was created which included the osteocyte, surrounding matrix, and a "gap" region between the two. An eighth of the geometry of the cell, matrix, and gap was used, taking advantage of symmetry in three directions. The material properties for the matrix were taken from the canalicular level homogenization analysis. The properties of the cell were set so the cell had the bulk modulus of water. The cell was modeled as both permeable and impermeable in separate model runs. Since the material(s) that are between the cell and matrix are largely unknown, this region was also given the bulk modulus of water. It was assumed this region was permeable with a large fluid fraction. Most of the model material properties and assumptions were varied in parametric studies to elucidate which were most critical in determining the results.

Cell level boundary conditions were applied according to the symmetry assumption. A deformation was applied parallel to the long axis of the lacuna over a period of 0.01 s, and resulted in an overall strain of approximately 800 microstrain. The "soils" procedure in ABAQUS was used for the consolidation analysis. The results shown here are at the time of maximum displacement (0.01 s).

Results

The permeability of the canalicular channel level model ranged from 0.85 to 5.1 x 10^-2 mm/s. An increased tortuosity resulted in a slightly reduced permeability. The stiffnesses and Poisson's ratios varied little among the models.

Results for the cell level model that included properties from the homogenization analysis and the initial assumptions described above are shown below. Decreasing the matrix permeability increased the pore pressure, but left other results essentially unchanged. Reducing the gap modulus resulted in higher principal strains in the gap and lower strains in the cell, as well as a change in the distribution of strains around the cell. Reducing the cell modulus reduced the pore pressure immediately around the cell and lowered the maximum principal strains. When both the cell and gap moduli were reduced equally, the results were nearly identical to the original model. Replacing the gap with bone matrix so the cell was rigidly attached to the matrix resulted in uniform strains throughout the model, with fluid velocity similar to the original model.

These results suggest that the bone matrix determines the deformation of the cell/gap, while the relative properties of the cell and gap determine how each of these components deforms. Although the permeability of the cell membrane does not appear to be an important parameter, the relative properties of the cell and gap, which are not currently known, are important areas for future study.

Progress

A manuscript is expected to be submitted by spring 2001.

Axis of canalicular channel Canalicular channel level model

Fluid velocity Pore pressure

Maximum principal strain Von Mises strain