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Thread: Well-Stirred Bath with Finite Solute Content

  1. #1
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    Dec 2007
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    Default Well-Stirred Bath with Finite Solute Content

    In some applications of solute transport, the user may want to model solute transport from a well-stirred bath into a tissue. One way to solve this problem is to assume that the bath has an infinite solute content, in which case the bath solute concentration may be prescribed as a boundary condition on the tissue. In that case, the bath does not need to be modeled explicitly.

    However, there are times where the user may want to account for the fact that the bath has a finite amount of solute that may get depleted as solute transports into the tissue. In that case, the bath solution must be modeled explicitly and the solute content in the bath should be prescribed as an initial condition for the bath (not a boundary condition on the tissue). If the bath is well stirred (e.g., if there is a stir bar in the bath, or if the tissue and bath are set on an orbital shaker), we expect that the solute concentration in the bath should stay uniform.

    To simulate this problem of a well-stirred bath with uniform (but time-varying) solute content, it suffices to set the solute diffusivity in the bath much higher than the diffusivity in the tissue (e.g., 1000x higher), and then analyze the problem on a time-scale relevant to the solute transport in the tissue.

    In the attached example (WellStirredBath.feb), a tissue sample (8 mm x 3 mm) is placed in a slightly larger bath (12 mm x 5 mm). Due to symmetry, only one-half of the actual geometry is modeled. The initial solute concentration in the bath is set to 1 mM, whereas the solute concentration in the tissue is initially zero. The solute diffusivity in the tissue is D=0.5e-3 mm^2/s and the free diffusivity is D0=1.0e-3 mm^2/s (this accounts for hindrance by the porous solid matrix). The solute diffusivity in the bath is set to D=D0=1 mm^2/s to simulate well-stirred condition.

    The attached movie shows the solute concentration contours as a function of time. Note the following:

    1) The bath concentration remains essentially uniform over the entire temporal response.
    2) The bath concentration decreases over time as solute transports into the tissue, thereby getting partially depleted from the bath.
    3) The solute concentration in the tissue diffuses much more slowly as expected, initially exhibiting a narrow boundary layer near the tissue periphery.

    This example illustrates how easy it is to simulate a well-stirred bath containing a finite amount of solute by simply picking a much higher diffusivity than in the tissue. The same principle may be applied for related analyses, such as solute desorption from the tissue into a well-stirred bath (same analysis as above, just invert the initial conditions by setting the tissue initial concentration to 1 mM and the bath initial concentration to 0 mM). If the bath is not stirred, all that needs to be done is set D=D0=1e-3 mm^2/s in the bath (i.e., the true free diffusivity of the solute).

  2. #2
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    Oct 2010
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    Hi Gerard,

    How can I extend this example to the case where the bath is not well-stirred and that the bath contains charged solutes (like Hexabrix)? The tissue is of course having a fixed charge density too.

    Regards,
    Amir

  3. #3
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    Hi Amir,

    If the bath is not well-stirred, you can use the true diffusivity of the solutes in the bath (instead of 1000x higher). This means that there will be transient boundary layers in the bath as well as the tissue and you should mesh accordingly (if you want to capture the transient gradients in concentration in the boundary layers).

    If the bath solute is charged (e.g., Hexabrix), you have to make sure that the bath also contains solutes of the opposite charge such that electroneutrality is satisfied in the bath. I don't know what are the constituents of Hexabrix, but I can give you a similar example: Suppose that you want to have a bath that contains the sulfate ion SO4(2-) at a concentration of [SO4] = 2 mM. The bath also needs a positively charged ion, e.g., Na+. Since the charge number of sulfate is zSO4 = -2 and that of Na is zNa = +1, electroneutrality is satisfied if zNa [Na] + zSO4 [SO4] = 0, which means that the sodium concentration should be [Na] = 4 mM.

    The bath could also contain additional dissociated salts if you want, e.g., NaCl at 150 mM, which means that you would have additional concentrations of [Na] and [Cl] to take into account.

    Best,

    Gerard

  4. #4
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    Hi Gerard,

    Thanks for the reply.

    Regarding the positively charged ions, is this requirement of having the electro-neutrality satisfied a physical requirement or something that is required due to the formulation of the model and may not hold in reality? The problem of studying the diffusion of anionic contrast agents into cartilage often involves baths with high concentrations of the anionic contrast agents. Can these experiments be modeled using the multi-phasic theory?

    Kind regards,

    Amir

  5. #5
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    Dec 2007
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    Hi Amir,

    The electroneutrality condition is an assumption that idealizes the real behavior of charged mixtures, akin to the assumption of incompressibility for fluids or solids: It is a constraint placed on the behavior of the mixture which helps simplify the analysis in some way. In the case of electroneutrality, the implied simplification is that the mixture cannot act as a capacitor (there can be no charge accumulation anywhere); the mixture only behaves as a resistor. For biological tissues, this is generally a good assumption as long as the mixture is not subjected to high frequency electrical potentials and currents.

    The response of a biological tissue to the varying osmolarity of an electrolytic bath is well described with multiphasic theory. You should be able to use that framework for this type of modeling.

    Best,

    Gerard

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