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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.
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.
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.
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.
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