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Thread: "Donnan equilibrium" versus "triphasic" steady-state swelling

  1. #1
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    Default "Donnan equilibrium" versus "triphasic" steady-state swelling

    A triphasic material swells as a result of the Donnan osmotic pressure arising in its interstitial fluid. This osmotic pressure is produced by the imbalance between the interstitial and environmental osmolarities. This imbalance is caused by the presence of electric charges fixed to the solid matrix (the fixed charge density), and the requirement that the combined electric charge of all species (ions and charged solid) should be zero everywhere (the electroneutrality condition).

    When the fixed charge density is distributed inhomogeneously throughout a triphasic material, swelling is also inhomogeneous and may produce shape alterations. For example, consider a ring excised from a rat aorta, which has been slit axially at one location on its circumference for the purpose of examining the resulting opening angle. The intima and media (inner) layers of the aorta contain a relatively large amount of negatively charged proteoglycans (versican), whereas the adventitia (outer layer) contains a negligible amount. If the osmolarity of the environmental fluid bath is decreased from a hypertonic value (e.g., 4000 mM) to a hypotonic value (e.g., 2 mM), the media and intima will swell due to the resulting increase in the Donnan osmotic pressure of their interstitial fluid. Since the adventitia is not charged, it experiences no such Donnan swelling, and the resulting disparity in the responses of these layers causes the slit aorta to open up.

    This response may be modeled in FEBio using a triphasic material in a steady-state "Multiphasic/Solutes" analysis (see attached CutRatAortaTR.feb). This full-fledged triphasic analysis produces results for all the dependent variables in the analysis (including the solute concentrations, the fixed charge density, and the Donnan osmotic pressure and electric potential). This capability has been implemented starting with FEBio 1.5.
    http://bio7.mech.columbia.edu/MRLFor...RatAortaTR.mov

    Equivalently, it is possible to reproduce the same swelling response using the "Donnan equilibrium" material previously introduced in FEBio (see attached CutRatAortaDE.feb). This material behavior may be solved using a "Structural Mechanics" analysis, which has the benefit of using only solid displacement degrees of freedom. Therefore, a "Donnan equilibrium" model is computationally more efficient than a "triphasic" model. It also requires much less effort in setting up initial and boundary conditions, since solute concentrations and fluid pressure are implicitly incorporated in the analysis. The downside of a "Donnan equilibrium" model is that variables such as solute concentrations, fixed charge density, osmotic pressure and electric potential are not explicitly provided in the output file.
    Attached Files Attached Files

  2. #2

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    Hello. Mr. Ateshian:
    I am a beginner in using this software. Can you please provide the preview version of the uploaded files? So I know how to do it

    Sincerely

  3. #3
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    Dec 2007
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    Default

    Hi Meng Ting,

    You can import these files into PreView from the File->Import... menu.

    Best,

    Gerard

  4. #4

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    Hello, Herard:
    Thanks for the reply. I tried to reproduce the 1 D swelling model in the journal. The mesh was axially compressed by 15% in the z direction. After full relaxation, the external sal concentration was decreased to 0.0001M. After equilibrium was reached the salt concentration was increased to 0.05M. After equilibrium the salt concentration was increased with 0.05M. This process was repeated until an external salt concentration of 2M was reached. I am not sure how to do boundary conditions for the time steps for those concentration. I attached the febio file for what I have done. It kept showing NAN error. I don't know why. Please give me some help. Greatly Appreciated


    Sincerely,
    Meng-Ting
    Attached Files Attached Files

  5. #5
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    Hi Meng-Ting,

    Don't forget to prescribe initial conditions for ion concentrations and fluid pressure. If you don't, the default value is zero, which is incompatible with the non-zero fixed charge density you have prescribed. That's probably the reason that you are getting NaN.

    I also noticed that your permeability is set to 0, this will cause problems as well.

    Best,

    Gerard

  6. #6

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    Hello, Gerard:
    Thank you very much. I did prescribe the initial conditions of Na and Cl to effective concentration of 150M and fluid pressure of -0.730836. I just added the permeability to 0.000914 but it still shows NAN. Please give me some idea what the problem might be.


    Sincerely,
    Meng-Ting Hsieh
    Attached Files Attached Files

  7. #7
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    Dec 2007
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    Hi Meng-Ting,

    Are you modeling confined compression with a porous indenter? If so, you should not use a FixedFluidPressure boundary condition on the top face of the cylinder, since that sets the fluid pressure to zero, when in fact you want it to be -0.730836. I suspect that changing this BC will fix the problem.

    Other things you may want to consider: If the lateral surface of the cylinder is confined in a rigid impermeable chamber, there is no need to prescribe the ion effective concentrations and effective fluid pressure on that boundary. The natural boundary conditions (i.e., not prescribing anything) will correctly enforce zero ion flux and fluid flux across that boundary. Also remember that it is always better to use a biased mesh for boundaries on which the fluid pressure is prescribed, since a boundary layer is likely to be produced at that interface.

    Best,

    Gerard

  8. #8

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    Hello, Gerard:
    I actually was doing a swelling model that goes through several external concentration change. I defined a rigid indenter to capture the reaction force in z direction for parameter optimization later on. It showed me negative Jacobian and NAN error. I attached the swelling model titled "fitswellingmodel" in the mail please take a look and give me some comments.
    Attached Files Attached Files

  9. #9

    Default Triphasic model subjected to compression and several concentration change

    Hello, Dr. Ateshian:
    I redid the triphasic model. It now has rigid indentor that compressed the model on the top and porous filter at the bottom. The edge was confined in radial directions. The model was also subjected to multiple external concentration change so I prescribed the bottom nodes with effective concentration of Na and Cl ions and effective pressure. It run with normal termination up to time step 95 on the transient step. If I increase the time step, it will result in error termination. I am positive that I have the right boundary condition this time. Please give me any possible correction. Deeply Appreciated. Thank you very much for taking a look!


    Sincerely,
    Meng-Ting Hsieh
    Attached Files Attached Files

  10. #10
    Join Date
    Dec 2007
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    Hi Meng-Ting,

    Here are a few observations I can make regarding your model:

    1) You use a two-step analysis where the first step (t=0 to 2.5) is a steady-state response and the second step (t=2.5 to 97.5) is a transient response. However, you haven't set must points, so the time point of 2.5 is skipped over in the analysis. As a general rule, I recommend using must point to make sure that the analysis passes through the start and end of each step.

    2) You prescribe a rigid body displacement on the rigid indenter which starts at t=0 and ends at t=10. This range spans over the transition from steady-state to transient response (t=2.5), which is probably not what you want to do.

    3) You start decreasing the ion concentrations on the bottom surface from 150 at t=50 to 0 at t=100. Since your fixed-charge density is not equal to zero (you have used a value equal to 1), you should not allow the ion concentrations to go to zero because you would violate electroneutrality (there will be no ions to balance the fixed charge on the solid). That's probably the reason you get NaN when you try to run the analysis to t=100, because the code attempts to produce an infinite electric potential.

    Best,

    Gerard

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