This example describes a parameter optimization analysis for extracting material properties of cartilage from an unconfined compression stress-relaxation experiment. It uses experimental data from a stress-relaxation test performed on a disk of immature bovine articular cartilage (thanks to Brian Jones).
In this analysis cartilage is modeled as a biphasic material whose porous matrix is described by a solid mixture of a "neo-Hookean" ground matrix (e.g., to model the contribution of proteoglycans to the compressive modulus) and a "spherical fiber distribution" to model the collagen fibrillar matrix (fibrils can only sustain tension). The hydraulic permeability is assumed to be constant.
The model has a total of five material constants: Young's modulus (E) and Poisson's ratio (v) for the neo-Hookean solid, the fibril modulus ksi and the power-law exponent beta for the spherical fiber distribution, and the constant hydraulic permeability k. It is assumed a priori that v=0 and beta=2 so that the parameter optimization need only be performed on E, ksi and k.
The material description in the model file (biphasic_ucsrlx_model.feb) is as follows:
Notice that explicit names are provided for each <solid> component of the solid mixture, so that these names may be referenced in the optimization file (biphasic_ucsrlx_opt.feb) as follows:
This analysis is based on actual experimental data where the compressive displacement of the loading platen was recorded simultaneously with the platen load response. Though the desired platen displacement should have a linear ramp-and-hold profile, the actual displacement does not follow that profile exactly due to compliance in the loading apparatus. Therefore, the model file includes a load curve for the prescribed platen displacement which is based on the experimentally measured displacement.
The parameter optimization analysis can be executed with the command
The length units in this analysis are millimeters, force is in Newtons, time is in seconds. The analysis should converge to E=0.325 MPa, ksi=1.75 MPa and k=5.67E-04 mm^4/N.s. The quality of the fit is shown in the attached figure biphasic_ucsrlx_results.jpg.
Gerard
In this analysis cartilage is modeled as a biphasic material whose porous matrix is described by a solid mixture of a "neo-Hookean" ground matrix (e.g., to model the contribution of proteoglycans to the compressive modulus) and a "spherical fiber distribution" to model the collagen fibrillar matrix (fibrils can only sustain tension). The hydraulic permeability is assumed to be constant.
The model has a total of five material constants: Young's modulus (E) and Poisson's ratio (v) for the neo-Hookean solid, the fibril modulus ksi and the power-law exponent beta for the spherical fiber distribution, and the constant hydraulic permeability k. It is assumed a priori that v=0 and beta=2 so that the parameter optimization need only be performed on E, ksi and k.
The material description in the model file (biphasic_ucsrlx_model.feb) is as follows:
Code:
<material id="1" name="Cartilage" type="biphasic"> <phi0>0.2</phi0> <solid type="solid mixture"> <solid name="fibers" type="spherical fiber distribution"> <beta>2</beta> <ksi>4</ksi> </solid> <solid name="ground" type="neo-Hookean"> <density>1</density> <E>0.33</E> <v>0</v> </solid> </solid> <permeability type="perm-const-iso"> <perm>0.001</perm> </permeability> </material>
Code:
<Parameters> <param name="Cartilage.permeability.perm">0.001, 0.0001, 0.002, 0.005</param> <param name="Cartilage.solid.ground.E">0.33, 0.1, 1, 1</param> <param name="Cartilage.solid.fibers.ksi">0.9, 0.2, 3, 1</param> </Parameters>
The parameter optimization analysis can be executed with the command
Code:
febio -s biphasic_ucsrlx_opt.feb
Gerard
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