Consider an elemental volume (δV) (see Fig. 1A) of tissue, in a cylindrical coordinate system (r,θ,z) with a linear microdialysis probe running along the z axis (i.e. the probe is centred at r= 0) of length L. The probe carries a solution (flow rate qp) at an initial solute concentration (at z= 0) of Cin and a final solute concentration (at z=L) of Cout.
Figure 1. Model of combined probe and muscle system A, an elemental volume of muscle in a cylindrical coordinate system (r,θ,z) is shown with a linear microdialysis probe running along the z-axis, centred at r= 0. Steady-state mass fluxes are shown into and out of the elemental volume. B, elemental slice of the probe and muscle system showing mass flow out of the probe in the radial (r) direction and mass flow out of the muscle due to blood flow. C, mass flow through an elemental slice of the probe along the length (z) direction.
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From the axial symmetry of the situation it can be assumed that there is no concentration gradient in the θ direction so that a mass balance of the solute for this elemental volume only involves radial and axial gradients. For completeness we develop the formalism for the general case with radial and axial solute diffusion. For the present experimental system the various spatial scales involved – see Appendix 1 for a discussion – enable us to make a convenient simplification in our modelling, leading to a closed analytic expression for the O/I ratio, as discussed below. For the present experimental conditions we show that one can also neglect the diffusional flow associated with the z-direction concentration gradient with minimal error and further simplification. The more general model, presented in Appendix 2, provides a framework to verify the applicability of these simplifications for the present experiments.
Considering the more general situation initially for completeness, conservation of mass in the elemental volume (steady state conditions) leads to
where the terms on the left-hand side represent the total mass of solute entering the element via a mass flux with radial component and axial component . The terms on the right-hand side represent the total mass of solute leaving the element, due to mass fluxes resulting from a concentration gradient (first and second terms) and removal by a specific blood flow (Q b in m3 of blood per second per m3 of tissue, of which γ is the nutritive fraction of the total specific blood flow) in the element. This model of solute removal by blood flow assumes that the blood initially contains no solute, and that the effective diffusion coefficient into the blood is sufficiently large for the blood to reach the same solute concentration (C) as the surrounding tissue. As such, it should be noted that the nutritive fraction incorporates a blood–tissue partitioning coefficient at equilibrium.
Eqn (1) can be simplified to give:
and expanding eqn (2) and neglecting second-order terms (i.e. terms in δr2) yields
Substituting for the mass fluxes, in terms of a two dimensional version of Fick's law:
into eqn (3) and making the assumption that the effective tissue diffusion coefficient (Dt) is independent of position, results in:
Eqn (4) is a second-order, homogeneous linear partial-differential equation which (because r > 0 and Dt≠ 0) can be rewritten in the form:
where contains the quantity of interest: the nutritive fraction γ.
Differentiating eqn (6) with respect to r and using the relations I′0=I1 and K′0=−K1 from eqn (9.6.27) of Abramowitz & Stegun (1964) gives:
Applying the boundary in eqn (7b) to eqn (8) gives:
Now (eqn (9.7.1) of Abramowitz & Stegun, 1964) while (eqn (9.7.2) of Abramowitz & Stegun, 1964) so that eqn (9a) reduces to:
Substituting eqn (9b) into eqn (6) and applying the boundary condition in eqn (7a) yields:
Eqn (10) gives the solute concentration at any position in the tissue, dependent on the solute concentration at the outer surface of the microdialysis probe (β (z)). Note that we have not considered any boundary conditions associated with the value of the concentration or the corresponding flux in the z-direction – for example at the ends of the cylindrical region defined by the extent of the probe. Rather we have postulated the exponential decline in concentration in the tissue at the probe wall with distance along the flow. The discussion in Appendix 2 details the complexity involved in treating those boundaries exactly.
The next step is to relate β (z) to the solute concentration in the probe. Consider an elemental slice (of thickness δz) of the combined probe and tissue system as shown in Fig. 1B. The mass lost from an elemental section of the probe, by diffusion through its wall , is related to Cp(z), the solute concentration in the probe (assumed to depend only on the coordinate along the probe axis, z) by a mass balance. For the elementally thin slice (thickness δz) of the probe, as shown in Fig. 1C, steady state conservation of mass has:
or in terms of solute concentration in the probe (Cp):
Where Cp(z) is the solute concentration in the probe, qp is the flow rate in the probe and Pp is the effective probe permeability (a constant incorporating the probe geometry and mass transport effects between the dialysate flow within the dialysis solution and the membrane). We expect Pp to be dominated by diffusion through the membrane, but it is readily determined experimentally (see section below and Discussion).
Rearranging and simplifying gives:
for an arbitrary external surface concentration β (z) (assumed independent of θ) where ψ= 2πPprp, with general solution:
This general solution is used in Appendix 2 in deriving the series solution. For our postulated form for the concentration in the muscle (eqn (7a) and eqn (10) above), eqn (12) becomes:
and the solution (from eqn (13)) is:
This expression generally contains two axial length scales, 1/λ originating from the form of β (z), and the length scale for the homogeneous part of eqn (12), qp/ψ, which controls the decrease of concentration along the probe if there is no external solute concentration (β0= 0). The latter distance is much shorter than the typically observed scale in the present perfusion studies, as discussed in detail in Appendix 1. Accordingly we will make the further assumption that β (z) has a z-dependence that is directly proportional to Cp(z): β (z) =ηCp(z) so that eqn (12) takes the form:
with the solution:
which is compatible with our choice for β (z), and identifies λ, relating it to the proportionality factor η and the homogeneous scale factor qp/ψ:
As physically loss from the probe by diffusion through the wall requires a concentration gradient (Cp(z) > β (z)) we expect η < 1, so that the length scale associated with the axial direction (1/λ) is increased over qp/ψ by the presence of solute in the surrounding muscle as expected (see eqn (18)).
The value of λ is related to the major experimentally determined quantity in the microdialysis process, the O/I ratio because:
The total mass flow of solute through the probe membrane is simply the solute lost from the probe:
while the total mass of solute removed by the blood flow in the tissue is:
Substituting the expression for muscle concentration from eqn (10):
The radial integral in eqn (21) is a standard form:
(using eqn (6.561.8) and eqn (6.561.16) of Gradshteyn & Rhyzhik (1980)). Substituting back into eqn (21), and substituting for β0 yields:
Now for steady state we require that Mp=Mb so combining eqn (19) and eqn (22) gives:
which simplifies to:
which indicates that, for a given O/I ratio in the probe, as λ is known (as is η) via eqn (18): , a can be found by solving the eqn (23), and accordingly ao and hence γ can be determined.
In summary, the ratio of the outflowing solute concentration to the inflowing solute concentration, for a probe of length L is simply given by:
If one can also neglect the contribution of diffusion in the tissue in the axial direction, arguing (see Appendix 1) that a0≫λ2 this expression simplifies to:
which has the advantage that it is slightly more straightforward to search for the optimal value of a0 and hence for the nutritive fraction γ.