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Thread: Parameter Optimization for Biphasic Unconfined Compression Stress-Relaxation

  1. #1
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    Default Parameter Optimization for Biphasic Unconfined Compression Stress-Relaxation

    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:
    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>
    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:
    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>
    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
    Code:
    febio -s biphasic_ucsrlx_opt.feb
    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
    Last edited by ateshian; 12-04-2013 at 09:31 AM.

  2. #2
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    To run this example in FEBio2, use biphasic_ucsrlx_model2.feb for the model file and biphasic_ucsrlx_opt2.feb for the optimization file. The command is
    Code:
    febio2 -i biphasic_ucsrlx_model2.feb -s biphasic_ucsrlx_opt2.feb
    Gerard

  3. #3

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    Hi Prof. Ateshian,

    I want to estimate the biomechanical properties of cartilage constructs from unconfined compression stress relaxation experiments. First I build the model and use biphasic theory (E, v and k). But no reaction force in z-direction are calculated. What is wrong? I have attached my model files. Please let me know what you think.

    ucc_rl_dtk1.prv

    Best regards,

    Thomas

  4. #4
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    Hi Thomas,

    Would you please post the .feb file? I am unable to open the .prv file.

    Thanks,

    Gerard

  5. #5

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    Hi Prof. Ateshian,

    sorry for the late answer. Here is the .feb file. Please let me know what you think.

    ucc_rl_dtk1.feb

    Best regards,

    Thomas

  6. #6

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    Hi Prof. Ateshian,

    I cannot find the problem in my model. The reaction force in z-direction are not calculated. What is wrong! Please let me know what you think. I have attached my model files.

    ucc_rl_dtk1.feb

    Best,

    Thomas

  7. #7
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    Hi Thomas,

    It appears that the loading platen has its outward normal facing up (away from the cartilage surface). You need to flip the platen 180 degrees so that the normal faces down toward the cartilage surface.

    Also, your symmetry plane has a width that matches the radius of the cartilage wedge exactly. This means that with compression, as the cartilage expands laterally, it will lose contact with the symmetry plane. I recommend that you increase the width of that plane.

    Best,

    Gerard

  8. #8

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    Hi Prof. Ateshian,

    thanks a lot. Now it works. The compression plot shows a little bit strange! Because the last part of the cartilage surface has no contact to the plate during the compression phase. Please, look at the following picture.

    compression.jpg

    Although the model is attached.

    ucc_rl_dtk1.feb

    Please, let me know what you thing. Maybe I have to change the biphasic contact?

    Best,

    Thomas

  9. #9
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    Dec 2007
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    Hi Thomas,

    From the picture it seems that you anchored the bottom surface to prevent it from moving in the radial direction. If it was your intention to model an adhesive interface, this result is not surprising, but you would have to add more elements along the axial direction to accurately reflect the response using this boundary condition. Otherwise, if your intent is to have a frictionless interface at the bottom, you should release the fixed conditions that constrain the radial expansion. In that case you will not observe a loss of contact at the radial edge.

    Best,

    Gerard

  10. #10

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    Hi Prof. Ateshian,

    thanks a lot! Now, the model works well.

    Best,

    Thomas

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