Variable Solid Volume Fraction

[pcundergrad]

Hi,

I am trying to simulate diffusion of a solute from a biodegradable porous matrix into an external bath. One of the requirements of my model is that the material degrades with respect to time i.e pore size increases as a result of the material degrading. I am aware of the fact that the solid volume fraction (SVF) has no effect on a biphasic simulation that uses constant permeability, so am I right in saying that I need to introduce the varying function to the permeability property of the model in order to achieve variable SVF? (Essentially, I want to make the permeability dependent on the SVF and make the SVF dependent on time)

As far as I can tell, the permeability materials offered in FE Bio are all strain-dependent materials via Holmes-Mow relationship using the Jacobian, J. So I am not sure how I can model degradation of the material that experiences increasing pore size and insignificant deformation. I would really appreciate any advice on this!

Thanks,
Peter

[ateshian]

Hi Peter,

The multiphasic material domain allows you to specify solid-bound molecules whose referential mass density (mass-based concentration, per volume in the reference configuration) may evolve with time (e.g., representing the scaffold material as it degrades). By picking the true density of the solid-bound molecule material using a realistic value, the evolution of this solid-bound molecule referential density will alter the porosity of the multiphasic material. Then, any material whose response is dependent on the porosity (such as Holmes-Mow permeability in Section 4.7.2.2, and referentially isotropic diffusivity in Section 4.8.3.3 of the FEBio 2.9 User's Manual) will automatically change according to this dependence (regardless of changes in J, although changes in J will compound this effect).

If you want the porosity to depend exclusively on the concentration of the solid-bound molecule, you should set the referential solid volume fraction "phi0" to zero in the "multiphasic" material. Otherwise, the referential solid volume fraction (i.e., in the absence of deformation) is calculated as the sum of phi0 + the volumetric contribution of the solid-bound molecule (see Section 4.10.1 of the User's Manual, specifically equation 9 in the online version).

A solid-bound molecule is declared in the <Globals> section such as

<SolidBoundMolecules>
<solid_bound id="1" name="bound">
<charge_number>0</charge_number>
<molar_mass>1</molar_mass>
<density>1</density>
</solid_bound>
</SolidBoundMolecules>

Here, <density> represents the true density.

The actual referential mass density of a solid-bound molecule in a <multiphasic> material is declared as

<solid_bound sbm="1">
<rho0>0.1</rho0>
<rhomin>0.001</rhomin>
<rhomax>1</rhomax>
</solid_bound>

Here, <rho0> is the referential mass density in this multiphasic mixture. If the true density is 1 (as shown above), the referential solid volume fraction contributed by this solid-bound molecule is 0.1/1 = 0.1 (i.e., 90% referential porosity if phi=0 in the <multiphasic> material). The typical value for <rhomin> is zero, but the option here is to set it to some floor value in case a value of zero is problematic (e.g., in some materials, Young's modulus may depend on this value and one may not want to have zero Young's modulus). The typical value for <rhomax> is the true density, but again this could be problematic for some materials since it would set the porosity to zero (e.g., permeability and diffusivity materials that depend on porosity would produce zero values which will prevent the analysis from proceeding).

The evolution of the solid-bound molecule concentration can be controlled in two ways: (1) Simplistically, one can associate a loadcurve with the solid-bound molecule referential density <rho0>, e.g., <rho0 lc="1">0.1</rho0>, where the loadcurve decreases from 1 to 0 over the desired time. (2) More generally one can associate a chemical reaction that controls the evolution of <rho0>, for example see cr01.feb in the test suite.

Let me know if you require more information on this.

Best,

Gerard

[pcundergrad]

Hi Gerard,

Thank you for your detailed response to my query. This was very useful and I have since been working to introduce this type of analysis to my model.

Initially, my model was based on the Fickian Diffusion model presented in "Gerard A. Ateshian Michael B. Albro Steve Maas Jeffrey A. Weiss Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation".

The next step I wanted to carry out was to convert the porous matrix into one of a biodegradable nature i.e. increasing porosity. Therefore, following your advice, I added a solid-bound molecule with an evolving referential density over time via a loadcurve. After that, I set phi0 equal to zero and I converted my material from perm-const-iso to perm-Holmes-Mow and the diffusivity of my solute from diff-const-iso to diff-ref-iso. (At this point, I assumed these properties would evolve over time as a result of the solid-bound molecule, and therefore, affect the results in terms of time-profiles).

I then re-run my simulation, however, it doesn't appear as though very much has changed; the concentration profiles are identical across the two models and the only change that I can see is a slight scaling factor in the solute flux and effective fluid pressure. But overall, there doesn't seem to be any difference over time between the two models and I am struggling to understand why this might be. One question I have is whether or not the permeability in perm-Holmes-Mow will actually change as a result of a change in the referential solid volume fraction as, from 4.7.2.2 in the User Manual, the permeability is dependent on phi0 rather than the referential solid volume fraction?

Another question I have is, why would one not simply associate a loadcurve with the solid volume fraction, phi0, to simulate increasing porosity rather than using the method of solid-bound molecules that you recommended?

Sorry for the lengthy reply. I appreciate your help.

Regards,

Peter

[ateshian]

Hi Peter,

If you are running a diffusion analysis with very little convection, changes to the permeability will have virtually no effect on the solute concentration profile, so I am not surprised that you don't see much of an effect.

Regarding diffusivity, remember that the diffusivity d in the tissue is affected by phi0, but not the free diffusivity d0. For the changes in phi0 that you are investigating, how much does d change?

Finally, yes you can associate a loadcurve with the solid volume fraction phi0 to alter the diffusivity and permeability. There is no need to create a solid-bound molecule. The only reason I mentioned that option is because degradation is typically governed by some reactive process and the only way to control phi0 that way is to define a sbm. If you don't want a reactive process for degradation, there is no need to use an sbm.

Best,

Gerard