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Research Papers

A Mixed Boundary Representation to Simulate the Displacement of a Biofluid by a Biomaterial in Porous Media

[+] Author and Article Information
René P. Widmer1

Institute for Surgical Technology and Biomechanics, University of Bern, Stauffacherstrasse 78, 3014 Bern, Switzerlandrene.widmer@istb.unibe.ch

Stephen J. Ferguson

Institute for Surgical Technology and Biomechanics, University of Bern, Stauffacherstrasse 78, 3014 Bern, Switzerlandstephen.ferguson@istb.unibe.ch

1

Corresponding author.

J Biomech Eng 133(5), 051007 (Apr 28, 2011) (12 pages) doi:10.1115/1.4003735 History: Received July 14, 2010; Revised February 08, 2011; Posted March 02, 2011; Published April 28, 2011; Online April 28, 2011

Characterization of the biomaterial flow through porous bone is crucial for the success of the bone augmentation process in vertebroplasty. The biofluid, biomaterial, and local morphological bone characteristics determine the final shape of the filling, which is important both for the post-treatment mechanical loading and the risk of intraoperative extraosseous leakage. We have developed a computational model that describes the flow of biomaterials in porous bone structures by considering the material porosity, the region-dependent intrinsic permeability of the porous structure, the rheological properties of the biomaterial, and the boundary conditions of the filling process. To simulate the process of the substitution of a biofluid (bone marrow) by a biomaterial (bone cement), we developed a hybrid formulation to describe the evolution of the fluid boundary and properties and coupled it to a modified version of Darcy’s law. The apparent rheological properties are derived from a fluid-fluid interface tracking algorithm and a mixed boundary representation. The region- specific intrinsic permeability of the bone is governed by an empirical relationship resulting from a fitting process of experimental data. In a first step, we verified the model by studying the displacement process in spherical domains, where the spreading pattern is known in advance. The mixed boundary model demonstrated, as expected, that the determinants of the spreading pattern are the local intrinsic permeability of the porous matrix and the ratio of the viscosity of the fluids that are contributing to the displacement process. The simulations also illustrate the sensitivity of the mixed boundary representation to anisotropic permeability, which is related to the directional dependent microstructural properties of the porous medium. Furthermore, we compared the nonlinear finite element model to different published experimental studies and found a moderate to good agreement (R2=0.9895 for a one-dimensional bone core infiltration test and a 10.94–16.92% relative error for a three-dimensional spreading pattern study, respectively) between computational and experimental results.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Fluid interface model in terms of a fixed mesh and cells. One may approximate the abrupt interface (a) of the two fluids using a piecewise linear, noncontinuous function (black straight lines) or a mixed boundary representation, indicated by the faded elements (b). The numbers represent the filling factor function δe, i.e., the elementwise volume fractions of the displacing fluid.

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Figure 2

Homogenized mixed boundary representation of a fluid interface on an element Ωe. (a) depicts the one-dimensional, arbitrarily oriented column model and (b) indicates the schematic representation.

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Figure 3

The spherical flow domains and the associated boundary conditions used to validate the mixed boundary representation model. Subdomains indicated by the square markers represent fluid sources that are fixed to the inlet pressure pin. The domains indicated by the circle markers or dotted lines represent fluid sinks that are fixed to the outlet pressure pout. The domain configuration shown in (a) is used to demonstrate the anisotropic permeability and validate the fluid interface evolution coincident and oblique to the principal coordinate system axes. The computational model shown in (a) is used to illustrate the dependence of the spreading pattern circularity on the biofluid and biomaterial viscosity.

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Figure 4

The experimental setup of Baroud , adapted from Ref. 41

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Figure 5

All dimensions are given in millimeters—the sketch of the artificial vertebra model (left figure). The area drawn in light gray represents the area, where the PMMA is injected through a cannula. Notice that the open porous aluminum foam is sealed in an acrylic enclosure, mimicking the cortical shell. Hence fluids can leave the model only by flowing through the circular areas with 4 mm diameter each, drawn in dark gray. All drill holes were placed at the face centers. The right image shows the experimental setup of Löffel (18), consisting of the sealed aluminum foam, the injection device, and the image intensifier for the monitoring of the injection process.

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Figure 6

μ1and μ2 are given in Pa s—displacement of a fluid with viscosity μ1 by a fluid with viscosity μ2 in a spherical domain. (d)–(f) illustrate the fluid interface development given an orthotropic permeability tensor (thus, the off-diagonal tensor components are zero and the axes of the ellipsoid are aligned to the Cartesian coordinate axes); (g)–(i) illustrate the interface development related to an anisotropic permeability tensor. For this type of domain and boundary condition assignment, the interface shape is theoretically insensitive to the viscosity ratio μ2/μ1. The simulations confirm this expectation.

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Figure 7

μ1and μ2 are given in Pa s and ks=5×10−8 m2—the circularity of the spreading pattern depends, in a flow domain made of point sources and sinks, on the viscosity ratio μ2/μ1. A high ratio (c) results in an almost perfect and a small ratio (g) gives a ragged spreading pattern. The matrix implies that the spreading pattern does not depend on the absolute viscosity level ((d)–(h), (a)-(e)-(i), and (b)–(f)).

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Figure 8

Plunger injection pressure versus time after the infiltration start. Comparison of the experimental data and simulation results: The increase in the pressure with respect to time after complete infiltration (t≈6.3 s) occurs because of the ongoing hardening of the PMMA during the testing.

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Figure 9

All dimensions are given in millimeters—spreading patterns after the injection of 4 ml PMMA into an open porous foam rated at 20 PPI. Figures from the left column depict the spreading patterns found by Löffel (18) whereas the figures on the right result from the simulations.

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