Axially symmetric semi-infinite domain models of microdialysis and their application to the determination of nutritive flow in rat muscle
1 Tasmanian Partnership for Advanced Computing, University of Tasmania, Private Bag 37, Hobart 7001, Tasmania, Australia
2 Biochemistry, Medical School, University of Tasmania, Private Bag 58, Hobart 7001, Tasmania, Australia
3 Department of the Environment and Heritage, Australian Antarctic Division, Private Bag 80, Hobart 7001, Tasmania, Australia
Abstract
Theoretical models for the description of microdialysis outflow:inflow (O/I) ratio for 3H2O and [14C]ethanol were developed, taking into account the nutritive fraction of total blood flow in muscle. The models yielded an approximately exponential decay expression for the O/I ratio, dependent on the physical dimensions of a linear probe (length and radius), the flow rate through the probe, muscle blood flow (including the nutritive fraction) and the diffusion coefficients for the tracer in the probe and muscle. The models compared favourably with experimental data from the constant-flow perfused rat hindlimb. Estimates of the nutritive fraction of total blood flow from experimental data were determined by minimizing the error between model and experimental data. The nutritive fraction was found to be 0.22 ± 0.04 under basal perfusion conditions. When 70 nM noradrenaline (norepinephrine) was included in the perfusion medium, the nutritive fraction was 0.91 ± 0.06 (P < 0.05). The inclusion of 300 nM serotonin, decreased the nutritive fraction to 0.05 ± 0.01 (P < 0.05). This model can be applied to the determination of nutritive fraction of skeletal muscle blood flow in physiologically relevant microvascular conditions such as during exercise and in disease states.
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Introduction
Recent work has provided evidence for the concept of two vascular routes in muscle (see Clark et al. 1995, 2000) consistent with observations by previous researchers (Renkin, 1955; Hyman et al. 1959; Barlow et al. 1961; Grant & Wright, 1970). In the constant-flow perfused rat hindlimb, these two routes are controlled by two types of vasoconstrictors, type A which increase general muscle metabolism and type B which decrease metabolism (Clark et al. 1995), neither of which affects total flow to individual muscles as determined by 15-μm microsphere markers (Clark et al. 2000). The metabolic effects of vasoconstrictors are dependent on the vasculature as the same vasoconstrictors do not affect metabolism of incubated muscle preparations (Colquhoun et al. 1990) and vasodilators abolish both the constrictive and metabolic effects of both types of vasoconstriction during hindlimb perfusion (Rattigan et al. 1993, 1995). In addition, the type A vasoconstrictors have been shown to increase blood flow to vessels supplying the muscle cells (nutritive; Newman et al. 1996; Clark et al. 2000), while type B vasoconstrictors direct flow away from muscle cells to vessels with limited capacity for exchange (non-nutritive; Newman et al. 1996; Clark et al. 2000) in part associated with septa of muscle (Newman et al. 1997).
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The use of the microdialysis technique in studies of muscle metabolism has increased in recent years. The two main uses for the technique are determining changes in the interstitial concentration of biologically important compounds (Chaurasia, 1999; Henriksson, 1999) and monitoring changes in blood flow (Arner, 1999). This second use involves the inclusion of ethanol (either unlabelled or labelled with 14C; Hickner et al. 1995) or in some cases 3H2O to the microdialysis inflow solution (Stallknecht et al. 1999). As the blood flow rate changes, the removal of tracer from the interstitial fluid is altered, as is the amount of tracer that can subsequently diffuse across the microdialysis membrane. The ratio of the concentration of ethanol (or 3H2O) emerging from the probe (outflow) to the inflowing concentration is monitored. This outflow:inflow concentration ratio (O/I ratio) has been shown to vary inversely with the total blood flow rate (Hickner et al. 1995; Stallknecht et al. 1999). The O/I ratio has been used as an indicator of changes in total flow to the muscle during exercise (Hickner et al. 1994) and during hyperinsulinaemia (Rosdahl et al. 1998).
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A number of mathematical models have been developed to describe the changes observed in microdialysis studies (Bungay et al. 1990; Morrison et al. 1991; Stahle, 2000). Most of the earlier models were focused on microdialysis in brain and were mostly concerned with modelling the recovery of endogenous compounds across the microdialysis membrane (Kehr, 1993). Although the recovery of an exogenous compound such as ethanol can be shown to be equivalent to 1–O/I, these early models were not developed to model the ethanol technique. It was not until 1995 that a model describing the O/I ratio of ethanol in muscle was developed (Wallgren et al. 1995). Indeed, this model has been used to calculate absolute values of blood flow in humans during rest and exercise (Hickner et al. 1997). However, this model may only be valid when the proportion of nutritive flow is unaltered, as it has been shown that the redistribution of blood flow within muscle in the absence of changes in total flow affects the microdialysis O/I ratio (Newman et al. 2001). The addition of the type A vasoconstrictors, such as noradrenaline (NA), decreases the O/I ratio, while the type B vasoconstrictors, such as serotonin (5-HT), increases the ratio (Newman et al. 2001). This paper therefore develops models of microdialysis O/I ratio that allows the determination of the nutritive fraction of blood flow in muscle.
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Methods
Perfusion
Experimental data from a previously published study on the O/I ratio in the perfused rat hindlimb (Newman et al. 2001) were re-analysed in this study. The experimental procedures are described in detail in that study (Newman et al. 2001) and were approved by the Committee on the Ethical Aspects of Research Involving Animals of the University of Tasmania. Briefly, rats (180–200 g) were anaesthetized with 6 mg (100 g body weight)–1 of pentobarbital sodium (I.P.). Surgery was performed as previously described (Newman et al. 2001) and the animal was killed by an overdose of the anaesthetic (12 mg) into the heart. One hindlimb was perfused under constant flow with a red blood cell containing medium. Linear microdialysis probes were constructed as detailed by Newman et al. (2001). Briefly, the point of a 23G Terumo needle was blunted by filing and the syringe adaptor removed. A 25-mm length of microdialysis tubing (Bioanalytic Systems, West Lafayette, IN, USA; molecular weight cutoff, 30 kDa; outer diameter, 320 μm) was inserted into the blunted end of the needle to a depth of 5 mm. Two probes were inserted into the calf muscles of the perfused leg, passing through the gastrocnemius red, gastrocnemius white, tibialis and extensor digitorum longus such that the total exposed surface for exchange was approximately 10 mm in length. These probes contained a 0.9% NaCl solution supplemented with 10 mM [14C]ethanol (100 nCi ml–1) and 3H2O (500 nCi ml–1). In each experiment, one probe flow rate was set to 1 μl min–1, while the other was set to 2 μl min–1. The hindlimb perfusion blood flow rate was varied between 1 and 9 ml min–1 for a period of 30 min at each flow rate and the nutritive fraction was varied by the addition of 70 nM NA or 300 nM 5-HT to the perfusion medium. The addition of a saline vehicle (0.9% NaCl) to the perfusion medium was used as a control. The O/I ratios for [14C]ethanol and 3H2O were determined at each perfusion flow rate over the last 10 min of experimentation at each perfusion flow rate. One 10-min sampling period was used for the 1 μl min–1 probe and two sequential 5-min sampling periods for the 2 μl min–1 probe. The two samples from the 2 μl min–1 probe were found to not significantly differ and thus a mean value was taken. The model was used to determine the nutritive fraction at each perfusion flow rate and for each probe flow rate and tracer.
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Model
Consider an elemental volume ( V) (see Fig. 1A) of tissue, in a cylindrical coordinate system (r,,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.
A, an elemental volume of muscle in a cylindrical coordinate system (r,,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 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.
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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 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.
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Eqn (1) can be simplified to give:
and expanding eqn (2) and neglecting second-order terms (i.e. terms in r2) 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:
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where contains the quantity of interest: the nutritive fraction .
In Appendix 2 a detailed series expansion solution to eqn (5) is presented, together with a complete treatment of the appropriate boundary conditions. This was developed to check the validity of our simpler models, and may be of interest for cases where the experimental parameters differ significantly from the present study, as well as providing a starting point for the development of solutions for other experimental geometries. Presently, as the concentration should decrease along the direction of the probe axis (in the direction of flow of the microdialysis solution) we explore simpler models by postulating that a simple, separable solution to eqn (5) is appropriate to our problem:
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where I0 is the modified Bessel function of the first kind, K0 the modified Bessel function of the third kind (also known as the Macdonald function), C1 and C2 are integration constants associated with the radial direction, a = a0 – 2 and the positive parameter is (at this stage) undetermined. The integration constants can be found from suitable boundary conditions: the concentration value at the probe wall (r = rp), and the vanishing of the radial flux at a great distance from the probe axis:
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(7a)
(7b)
where (z) is the solute concentration at the outer edge of the probe and is the radial derivative of the solute concentration.
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:
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or
(9a)
Now (eqn (9.7.1) of Abramowitz & Stegun, 1964) while (eqn (9.7.2) of Abramowitz & Stegun, 1964) so that eqn (9a) reduces to:
(9b)
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 ( (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.
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The next step is to relate (z) to the solute concentration in the probe. Consider an elemental slice (of thickness z) 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 z) of the probe, as shown in Fig. 1C, steady state conservation of mass has:
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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).
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Rearranging and simplifying gives:
(12)
for an arbitrary external surface concentration (z) (assumed independent of ) where = 2 Pprp, with general solution:
(13)
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:
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(14)
and the solution (from eqn (13)) is:
(15)
This expression generally contains two axial length scales, 1/ originating from the form of (z), and the length scale for the homogeneous part of eqn (12), qp/, which controls the decrease of concentration along the probe if there is no external solute concentration (0 = 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 (z) has a z-dependence that is directly proportional to Cp(z): (z) = Cp(z) so that eqn (12) takes the form:
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(16)
with the solution:
(17)
which is compatible with our choice for (z), and identifies , relating it to the proportionality factor and the homogeneous scale factor qp/:
(18)
As physically loss from the probe by diffusion through the wall requires a concentration gradient (Cp(z) > (z)) we expect < 1, so that the length scale associated with the axial direction (1/) is increased over qp/ by the presence of solute in the surrounding muscle as expected (see eqn (18)).
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The value of is related to the major experimentally determined quantity in the microdialysis process, the O/I ratio because:
or
The total mass flow of solute through the probe membrane is simply the solute lost from the probe:
(19)
while the total mass of solute removed by the blood flow in the tissue is:
(20)
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Substituting the expression for muscle concentration from eqn (10):
(21)
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 0 yields:
(22)
Now for steady state we require that Mp = Mb so combining eqn (19) and eqn (22) gives:
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which simplifies to:
(23)
which indicates that, for a given O/I ratio in the probe, as is known (as is ) via eqn (18): , a can be found by solving the eqn (23), and accordingly ao and hence 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:
where:
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(24)
or equivalently:
(24a)
and:
If one can also neglect the contribution of diffusion in the tissue in the axial direction, arguing (see Appendix 1) that a0 >> 2 this expression simplifies to:
(25)
which has the advantage that it is slightly more straightforward to search for the optimal value of a0 and hence for the nutritive fraction .
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In vitro
The effective permeability constant of the microdialysis probe (Pp) was determined in vitro. A probe of known exchange length was constructed by gluing a second blunted 23 G needle to the outflow end of the microdialysis probe (in this particular case the exposed probe length was 11 mm). This was then immersed in a well-stirred saline (0.9% NaCl) solution maintained at 37°C to match the perfusion temperature. In this situation, eqn (24) or eqn (25) can be reduced (because 0) to
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(26)
as in such a situation, the bulk movement of the stirred medium means that the term Q b input in eqn (24) or eqn (25) is effectively infinite (or at least many orders of magnitude greater than those for muscle blood flow). Thus the term is much greater than 1, and hence the ratio of the two Bessel functions in the expressions for tends to 1. Furthermore, the term would be expected to be much greater than Pp (which can then be verified upon calculation of Pp). Eqn (26) can be rewritten in the form Cout/Cin = e–PpS/qp where S is the probe surface area. This is analogous to the equation pioneered by Renkin (1955) to describe the extraction fraction (EF) of substances across a vascular bed (EF = 1 – e–PS/Q where PS is the permeability–surface area product and Q is the blood flow rate).
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In a similar fashion to the perfusion experiments, the tracers 3H2O and [14C]ethanol were added to the inflow solution (0.9% NaCl) and the O/I ratio for each tracer was determined at probe flow rates between 2 and 10 μl min–1 in order to calculate Pp. At least 10 min was allowed between each flow rate change for the probe to equilibrate. Two sequential 5-min samples were taken in order to ensure the probe had equilibrated.
Statistical analysis
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One way analysis of variance (ANOVA) was performed using SigmaStat (SPSS Science; Chicago, IL, USA) with comparisons made between treatments using the Student–Newman–Kuels post hoc test. Significance differences were accepted at the P < 0.05 level. Data are presented as means ± S.E.M.
Results
Figure 2 shows the O/I ratio and the calculated Pp for 3H2O and [14C]ethanol for a probe in a stirred saline solution across a number of probe flow rates based on a probe length and radius of 0.011 m and 1.6 x 10–4 m, respectively. The data for the lowest flow rate (2 μl min–1) were excluded because the total amount of radioactivity in these samples were only twice the background level. Therefore the other data were used to determine the average Pp to be 1.59 ± 0.01 x 10–5 m s–1 and 0.88 ± 0.01 x 10–5 m s–1 for 3H2O and [14C]ethanol, respectively.
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A known length (0.011 m) of microdialysis membrane was immersed in a well-stirred saline solution and pumped through at varying rates with a saline solution containing 3H2O ()and [14C]ethanol (). The O/I ratio was determined and the effective permeability of the probe (Pp) was calculated in duplicate according to eqn (26).
The model can now be used to determine the ratio of nutritive blood flow to total muscle blood flow () from previously published experimental data (Newman et al. 2001). The values of which minimized the root mean square (RMS) error between the approximate model of eqn (25) and experiment, for the different drug treatment regimens and microdialysis flow rates, are given below. Minimum overall RMS errors were calculated for experiments involving the same drug treatment regimens. Typical RMS error as a function of is shown in Fig. 3 (in this case 3H2O in vehicle perfusions with qp = 2 μl min–1). Calculations were based on parameters as shown in Table 1. These were done for each treatment (vehicle, NA or 5-HT), tracer and qp, the results of which are summarized in Table 2. The performance of the approximate model is shown in Fig. 4 (3H2O, qp = 1 μl min–1), Fig. 5 ([14C]ethanol, qp = 1 μl min–1), Fig. 6 (3H2O, qp = 2 μl min–1) and Fig. 7 ([14C]ethanol, qp = 2 μl min–1).
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For each drug treatment regimen, the RMS error between the approximate model and experimental data (Newman et al. 2001) was determined over a range of 0–1. The best estimate for was taken to be that which minimized the error. Data are shown for vehicle perfusions using a probe flow rate of 2 μl min–1 and 3H2O as tracer. Other parameters are L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s –1.
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Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 0.88 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
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Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 0.88 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Figure 8 shows a contour plot of the percentage difference between the approximate (eqn (25)) and full model (Appendix 2) for the range of and Q b values appropriate for the present study.
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The percentage difference between the predicted O/I ratios from the simplified model (eqn (25)) and the series expansion model of Appendix 2, is presented as a function of blood flow rate Q b and nutritive factor (for ranges appropriate to the present study) by contours ((approximate–full)/full x (100). In this case, L = 0.01 m, qp = 1 μl min–1, Dt = 1 x 10–9 m2 s–1 and Pp = 1 x 10–5 m s–1 (as representative of either tracer). Numbers indicate the percentage difference value for each contour.
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In addition, as discussed further in Appendix 2, Fig. 9 shows the variation of the concentration in the probe along the probe axis based on the full and approximate models. Figure 10 shows the corresponding concentrations at the outer surface of the probe, while concentration profiles in the radial direction midway along the probe are shown in Fig. 11.
The concentration Cp(z), normalized by Cp(0) = Cin, is presented as function of distance (z) along the probe axis. Results are shown for the probe concentration derived from taking one, three and 11 term approximations in the series solution model of Appendix 2 for Cp(z) (eqn (A3)), and for the simplified exponential model for Cout/Cin, i.e. eqn (17) for Cp(z), with the scale factor in the exponential generated by eqn (25).
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The concentration of solute in the muscle is presented as a function of distance (z) along the probe, just outside the probe wall, C(rp, z), using the one, three and 11 term approximations in the series solution model for C(r, z) (eqn (A1)), at r = rp, and also for the simplified model where the z dependence of C(r, z), (z) of eqn (10), is taken as proportional to Cp(z), with proportionality factor controlled by eqn (25) through eqn (18).
The concentration of solute in the muscle is presented as a function of radial distance from the probe (r > rp) at the midpoint of the probe length (z = L/2), C(r, L/2), using the one, three and 11 term approximations in the series solution model for C(r, z), and also for the simplified model where the r dependence of C(r, z) is given by the Bessel function expression eqn (10), and involves the dependent scale factor a, which is related to Cout/Cin by eqn (25).
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Discussion
This paper details the derivation of theoretical expressions for the microdialysis O/I ratio in tissue. From eqn (24) or eqn (25) it is evident that this expression is an exponentially decaying function dependent on the physical dimensions of the probe (length, L and radius, rp), the flow rate through the probe (qp) and blood flow rate (Q b), the effective permeability of the substance of interest in the probe (Pp) and the tissue diffusion coefficient (Dt). There is also a term involving the nutritive fraction () of the total blood flow. Recognition of this dependence leads to the main experimental results of this paper – the estimates of the values of that minimize the error from experimental data using the perfused rat hindlimb preparation (Newman et al. 2001) for 3H2O and [14C]ethanol. The experimental data and model performance are shown in Figs 4–7 with the minimized errors for in Table 2. It is clear that the microdialysis techniques and the model provide sufficient discrimination to observe the influence of the two vasoconstrictors on the nutritive fraction.
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The simplest derived model as shown in eqn (25) is consistent with the model of Bungay et al. (1990) except for the use of different clearance rates. In the present case Qb is the main clearance parameter while Bungay et al. (1990) used first-order rate constants including removal by the vasculature and intra- and extracellular metabolism). In addition, eqn (25) is consistent with the active tissue model developed by Stahle (2000) although this latter model does not specify the contribution of blood flow and metabolism to the clearance rates. The model pioneered by Bungay et al. (1990) incorporates resistances in series, that of dialysate resistance (Rd), membrane resistance (Rm) and extracellular resistance (Re) and they show that Re >> Rm >> Rd. This relative influence of resistances in the model means that it is advantageous to determine the effective permeability of the entire probe (in the current model Pp) for each species experimentally, incorporating all probe influences (including the probe's cross-sectional geometry) into this parameter. The present work also indicates the complications involved in modelling diffusion in the tissue parallel to the probe axis.
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The experimental difficulties involved in the use of this model focus on an accurate assessment of the parameters. It is a simple enough matter to measure the physical dimensions of the probe and to set the flow rate through the probe. In this study the effective permeability (Pp) of 3H2O and [14C]ethanol through the probe were determined in vitro (see Fig. 2). In the current experiments to determine Pp there is necessarily a balance between the length of the probe and flow rate through the probe to ensure that background radioactivity does not interfere with the measurement of the probe outflow solution. Eqn (26) suggests that a proportional change in probe length (L) has the same effect as a reciprocal change in probe flow rate (qp). It was considered prudent to keep L the same as the perfusion experiments and vary qp, as there is a lower fractional error in the measurement of the probe length at L = 10 mm compared to L = 2–3 mm if qp was set to 1 or 2 μl min–1 as in the perfusion experiments.
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The other important parameter is that of the tissue diffusion coefficient (Dt). Values for the diffusion coefficient of water in muscle can be obtained from the literature and range between 0.68 and 1.82 x 10–9 m2 s–1 (Hansen, 1971; Chang et al. 1972; Cleveland et al. 1976; Schultz & Armstrong, 1978; Morvan & Leroy-Willig, 1995). In the current study, a compromise position was taken so that a value for Dt of 1.0 x 10–9 m2 s–1 was used. It was not possible to find values for the diffusion coefficient of ethanol in muscle. The diffusion coefficient of ethanol has been measured in the bronchial mucosa of dogs and was found to be 0.563 x 10–9 m2 s–1 (George et al. 1996). Gonzales et al. (1998) found the diffusion coefficient of ethanol in brain was 1.2 x 10–9 m2 s–1. These values for ethanol diffusion coefficient are similar to the diffusion coefficient of water in muscle. As a consequence it was considered reasonable to take the diffusion coefficient of ethanol in muscle to be the same as that for water. As Wallgren et al. (1995) found, the strong influence of this parameter on the results was disappointing. It does suggest, however, that the use of 3H2O as opposed to ethanol is preferable as diffusion coefficients for water are better known as well as the fact that evaporative errors are less likely with water.
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The accurate determination of the total blood flow to the muscle or muscles through which the probe passes (Q b) is also required. Post-mortem analysis of the perfused hindlimb showed that the probe passed through the gastrocnemius red, gastrocnemius white, tibialis and extensor digitorum longus in proportions of 0.2, 0.2, 0.5 and 0.1, respectively. There are previously published data on the blood flow to these muscles as assessed by 15-μm microsphere entrapment in the red blood cell-perfused rat hindlimb under basal conditions (Youd et al. 1999). It has also been shown that these blood flow rates are not affected by the addition of vasoconstrictors such as NA or 5-HT (Clark et al. 2000). For this study, therefore it was possible to estimate the average blood flow to the muscles through which the probe passed. Data from Youd et al. (1999) show that the weighted average proportion of total perfusion blood flow going to the muscles the probe passes through was 0.6%. This corresponds to a flow rate surrounding the probe of 1.73 x 10–2 ml min–1 (g muscle)–1 for a typical 180- to 200-g rat at a total perfusion flow rate of 1 ml min–1. The conversion factor from Table 1 then gives a Q b of 3.2 x 10–4 m3 s–1 (m3 muscle)–1.
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The range of blood flows over which the model is tested is approximately 3–30 ml min–1 (100 g muscle)–1. While this easily encompasses the flow rates typical for resting muscle, it is a limitation of the current testing of the model. However, it was necessary to restrict the flow rates to this range in order that the perfusion pressure generated with the vasoconstrictors would not become excessive. As reported in the study from which the experimental data were obtained, the perfusion pressure with NA and 5-HT reached approximately 200 mmHg at the highest flow rate (Newman et al. 2001). Higher flow rates with the vasoconstrictors led to uncontrolled oedema. Determination of the nutritive fraction of blood flow at higher flows (particularly in vivo during exercise) will be a particularly useful extension of the current model.
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The model described in this paper assumes the system is at steady state; however, some of the experimental data may not satisfy this condition. While the model does not allow the calculation of equilibration times, Morrison et al. (1991) and Bungay et al. (2001) suggest that it is possible to estimate the e-folding time in the approach to steady state as the reciprocal of the clearance rate constant (in this case Q b). These calculations imply that during vehicle infusion (Q b = 0.0013 s–1 and = 0.22), the e-folding time would be approximately 1 h. However, it is not obvious that Qb is the only determinant of equilibration time. Data from Newman et al. (2001) show a set of experiments where the hindlimb was allowed to equilibrate for 40 min at a pump flow rate of 4 ml min–1 (corresponding to Q b = 0.0013 s–1). Subsequently vehicle, NA, 5-HT or the vasodilator nitroprusside was added to the blood and left for a further 40 min. These data showed that the O/I ratio measured during the last 10 min of vehicle infusion were 98.5 ± 2.5% and 101.8 ± 1.2% of the basal ratio for 3H2O and [14C]ethanol, respectively. This suggests that 40 min is sufficient time to reach steady state under these conditions which is considerably shorter than the e-folding time predicted by 1/Qb.
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During 5-HT perfusion, the largest rate constant (Q b = 0.0028 s–1 and = 0.05) implies an e-folding time of 2 h. From the previous example, it is likely that this is a considerable overestimate and as the experiment had been in progress for nearly 3 h by this time, it is likely that these data are at steady state. In addition, all steady-state approach times during NA perfusion were less than 1 h. Hence while some of the low Q b data may be under non-steady-state conditions, the highest Q b data would appear to be at equilibrium. If only these data are used then, for example from Fig. 6, the value of for vehicle is 0.32 ± 0.08 (n = 10). Corresponding values for 5-HT and NA are 0.13 ± 0.05 (n = 8, P < 0.05) and 0.93 ± 0.05 (n = 10, P < 0.05), respectively. These order of magnitude calculations for equilibrium approach times could be used in the design of future experiments even if a value of unity is assumed for . For example, a typical muscle blood flow rate during rest of 1.5 ml min–1 (100 g)–1 would suggest that the approximate e-folding time for the approach to steady state was 1 h. While this is likely to be a significant overestimate, it does provide a time frame in which to experimentally determine the time required to reach equilibrium.
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This model could also be used with compounds other than 3H2O or [14C]ethanol as well as to model the microdialysis recovery of compounds. As microdialysis recovery (R) of an exogenous compound is related to the O/I ratio by the equation R = 1–O/I, then eqn (25) can be modified accordingly. The only condition would be that the compound was not metabolized in addition to being removed by the blood. In addition, a volume fraction for extracellular space may need to be incorporated if the compound is excluded from cells. Depending on the compound used, the direct evaluation of the ratio of two Bessel functions may be problematic (see Appendix 3 for details).
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This paper makes the assumption that the non-nutritive flow route is unable to exchange ethanol and water, which may be an oversimplification. However, at present there is no way of determining the relative exchange capacity of each pathway; the use of would seem to be a valid start. Another assumption in this model is that the [14C]ethanol and 3H2O are not taken up by tissue and metabolized. As such it varies from other models of microdialysis such as that by Morrison et al. (1991) which took into account the effects of metabolism on the model. Skeletal muscle does not appreciably metabolize ethanol as it contains no alcohol dehydrogenase. In addition as the results for [14C]ethanol and 3H2O are similar, it was not considered necessary to include a term for metabolism in the model.
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In conclusion, this study demonstrates that the microdialysis O/I ratio can be modelled mathematically while taking into account the nutritive fraction of blood flow. In particular, the relative influences of vasoconstrictors on nutritive fractions can be observed. This model can be applied to the determination of nutritive fraction of skeletal muscle blood flow in physiologically relevant microvascular conditions such as during exercise and in disease states.
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The analysis presented in the text leading to eqn (24) and eqn (25) is adequate for the parameters of the experiments of the present study, although this is not quite obvious without some consideration of the various spatial scales involved. It is also relatively straightforward, compared to the complete solution of eqn (5) and the full set of boundary conditions.
The relevant spatial scales in the microdialysis problem are as follows.
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The probe radius (1.6 x 10–4 m).
The length of the probe (10–2 m).
The length scale associated with diffusion in the absorbing muscle which for the value of Dt = 1 x 10–9 m2 s–1 and the likely range of Qb (, 0.1 – 1; Qb, 2.5 x 10–4–3.5 x 10–3 s–1) for the present experiments, gives a range of 6.3 x 10–3–5.3 x 10–4 m, all lying between the probe radius and the probe length.
Two candidate length scales for the variation in concentration in the axial direction present themselves. The first is the decay length for concentration in the probe qp/ = qp/ 2 rp Pp in the absence of appreciable external solute concentration (which appears in eqn (12)), which for the present experiments ranges from 1.0 x 10–3 to 3.8 x 10–3 m. These values fall between the probe length and the probe radius, but overlap the diffusion scales above (for the higher probe flow rate compared with low values of Qb reflecting some combination of low nutritive fractions and muscle blood flow rates), but this is not particularly relevant for the actual microdialysis experiments.
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The more relevant axial length scale arises where there is muscle present, and an appreciable solute concentration at the outer wall of the probe. If we assume, as in our modelling, that the probe concentration has an exponential z-dependence, then as the O/I ratios observed in the present experiments are typically in the range 0.1–0.8, then for the present probe length of 10–2 m, the effective length scales for the axial dependence ought to range between 4.34 x 10–3 and 4.48 x 10–2 m.
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The lower extent of the expected range of axial length scales approaches the upper limits of muscle diffusion scales ((iii) above) which apply for cases where Qb is small, but such low Qb experiments tend to have a smaller reduction in probe concentration, and consequently (in the present experiments) correspond to O/I values nearer unity, and longer implied axial length scales. Accordingly, the concept that the radial length scale is dominant over the axial length scale, and that the diffusion along the z-direction should be relatively small, remains sound for the present experimental situation.
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We have not carried out an exhaustive search over the wide range of possible experimental geometries, flow rates, probe materials and solute properties to see how frequently the conditions prevail that make eqn (25) an excellent approximation to the more complicated model of Appendix 2. The calculated radial concentrations presented in Fig. 4 of Bungay et al. (1990) do suggest that one solute species (sucrose) has a radial scale comparable with its relatively short probe length. It is suggested that consideration should be given to the various length scales in any proposed study – preferably at the experimental design stage. While the complication of using the model of Appendix 2 in data fitting for estimating (for example) optimal values of nutritive fraction might be daunting, it would be prudent to scan the ranges of values likely to be encountered to check (as was done here) that the full and approximate models give comparable results.
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Appendix 2
In this appendix a more complete solution to the diffusion problems of probe wall and tissue is presented. This necessitates some additional computational machinery. While it is not essential, the analysis was made simpler by the use of the symbolic and numerical mathematics package Mathematica (Wolfram Research Inc., Mathematica Version 5, Champaign, IL, USA).
The treatment in the main text ignores the boundary conditions that should be satisfied by the diffusion along the z-direction in the muscle (parallel to the perfusion probe). The geometrical arrangement of the probe and the muscle in the present studies is such that there should be no flow in the z-direction across the planes at z = 0 and z = L. This is not satisfied by the simple, separable solution (eqn (6)) to the diffusion eqn (5), although for the present experimental conditions the effect should be small, because as discussed in Appendix 1 the axial length scales are long compared to the length scales associated with radial diffusion.
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To solve the diffusion eqn (5) with the boundary conditions of vanishing fluxes across the planes at z = 0 and z = L, and at a large radial distance (r ), requires that C(r, z) takes the form:
(A1)
where
The remaining boundary condition is the mass balance at the probe wall – the mass flux through the membrane at each point along the z-direction must equal the diffusive mass flux in the adjacent muscle.
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(A2)
However as we have already noted (eqn (13)), Cp(z) itself depends on the external concentration, and (defining for convenience the probe scale parameter = /qp) eqn (13) yields:
(A3)
Substituting for Cp(z) in eqn (A2), the boundary conditions imply that:
(A4)
An infinite set of relationships between the (infinite number) of series coefficients bn can be determined by multiplying eqn (A4) by for each non-negative integer m, and integrating over z from 0 to L. The general form of these integrations can be performed analytically, and by truncating the series sum in eqn (A1) to a finite number of terms, a system of linear equations for the coefficients (bn), can be generated and solved numerically, providing solutions of any required accuracy for C(r, z) and Cp(z), as functions of a0, and hence of Qb. The form of the general system is:
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(A5)
and
(A6)
Truncating the sums in eqn (A5) and eqn (A6) produces a set of linear equations that can be solved numerically for any set of given experimental parameters and a value for , and the coefficients can then be substituted into eqn (A1) and eqn (A3) to enable the muscle concentration fields or the O/I ratios to be calculated.
We demonstrate the convergence of the series solution, by solving eqn (A5) and eqn (A6), for the coefficients bn, for a truncation of the series expansion (eqn (A1)), to one, three and 11 terms. We use the resulting concentration functions, C(r, z) and Cp(z) (eqn (A1) and eqn (A3)) to demonstrate the close agreement with the simplified model presented in the main text, for the range of experimental parameters in the present study. The variation of the concentration in the microdialysis probe Cp(z) along the probe axis is presented in Fig. 9, together with the prediction of the simple model based on eqn (25). Clearly even with three terms the results of the two models for Cout/Cin, i.e. the value at z = L, agree closely. In Fig. 10, the corresponding results for the z-dependence of the concentration in the muscle just outside the probe wall are presented. The small discrepancy between the 11-term series solution and the simplified model is clearest at the ends of the probe, z = 0 and z = L, where the series solution has vanishing z-derivative, to match the no flux boundary conditions there. Finally, Fig. 11 compares the radial dependence of the concentration in the muscle, in a plane normal to the probe and located midway along it. The agreement of the simple Bessel function solution (from eqn (10)) and the series solutions is achieved for even three-series terms. These computations indicate that we can apply the simplified model of the main text, whose core is the relation between Cout/Cin and the nutritive fraction through eqn (25), to the data from the present study with confidence.
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Appendix 3
Direct evaluation of the ratio of two Bessel functions (i.e. ) may be problematic as the individual Bessel functions rapidly tend to zero, leading to a loss in precision for large arguments. A much more accurate and easier to implement calculation for the ratio of the two Bessel functions can be found using the relationship leading to and substituting the polynomial approximation from (section 9.8.6 of Abramowitz & Stegun, 1964) of
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This yields an approximation where
and
For x > 2.5, || < 1.5 x 10–7.
In the present study, direct evaluation is quite adequate as the argument of the Bessel functions is always less than 0.3 (rp = 1.6 x 10–4 m, = 1, Q b = 3.5 x 10–3 m s–1 and Dt = 1 x 10–9 m2 s–1). For the direct evaluation of the ratio of the two Bessel functions to become problematic a combination of high probe radius and high blood flow (possibly during exercise) and a compound with a very low diffusion coefficient in muscle (a protein or other high molecular weight molecule) would be required.
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2 Biochemistry, Medical School, University of Tasmania, Private Bag 58, Hobart 7001, Tasmania, Australia
3 Department of the Environment and Heritage, Australian Antarctic Division, Private Bag 80, Hobart 7001, Tasmania, Australia
Abstract
Theoretical models for the description of microdialysis outflow:inflow (O/I) ratio for 3H2O and [14C]ethanol were developed, taking into account the nutritive fraction of total blood flow in muscle. The models yielded an approximately exponential decay expression for the O/I ratio, dependent on the physical dimensions of a linear probe (length and radius), the flow rate through the probe, muscle blood flow (including the nutritive fraction) and the diffusion coefficients for the tracer in the probe and muscle. The models compared favourably with experimental data from the constant-flow perfused rat hindlimb. Estimates of the nutritive fraction of total blood flow from experimental data were determined by minimizing the error between model and experimental data. The nutritive fraction was found to be 0.22 ± 0.04 under basal perfusion conditions. When 70 nM noradrenaline (norepinephrine) was included in the perfusion medium, the nutritive fraction was 0.91 ± 0.06 (P < 0.05). The inclusion of 300 nM serotonin, decreased the nutritive fraction to 0.05 ± 0.01 (P < 0.05). This model can be applied to the determination of nutritive fraction of skeletal muscle blood flow in physiologically relevant microvascular conditions such as during exercise and in disease states.
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Introduction
Recent work has provided evidence for the concept of two vascular routes in muscle (see Clark et al. 1995, 2000) consistent with observations by previous researchers (Renkin, 1955; Hyman et al. 1959; Barlow et al. 1961; Grant & Wright, 1970). In the constant-flow perfused rat hindlimb, these two routes are controlled by two types of vasoconstrictors, type A which increase general muscle metabolism and type B which decrease metabolism (Clark et al. 1995), neither of which affects total flow to individual muscles as determined by 15-μm microsphere markers (Clark et al. 2000). The metabolic effects of vasoconstrictors are dependent on the vasculature as the same vasoconstrictors do not affect metabolism of incubated muscle preparations (Colquhoun et al. 1990) and vasodilators abolish both the constrictive and metabolic effects of both types of vasoconstriction during hindlimb perfusion (Rattigan et al. 1993, 1995). In addition, the type A vasoconstrictors have been shown to increase blood flow to vessels supplying the muscle cells (nutritive; Newman et al. 1996; Clark et al. 2000), while type B vasoconstrictors direct flow away from muscle cells to vessels with limited capacity for exchange (non-nutritive; Newman et al. 1996; Clark et al. 2000) in part associated with septa of muscle (Newman et al. 1997).
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The use of the microdialysis technique in studies of muscle metabolism has increased in recent years. The two main uses for the technique are determining changes in the interstitial concentration of biologically important compounds (Chaurasia, 1999; Henriksson, 1999) and monitoring changes in blood flow (Arner, 1999). This second use involves the inclusion of ethanol (either unlabelled or labelled with 14C; Hickner et al. 1995) or in some cases 3H2O to the microdialysis inflow solution (Stallknecht et al. 1999). As the blood flow rate changes, the removal of tracer from the interstitial fluid is altered, as is the amount of tracer that can subsequently diffuse across the microdialysis membrane. The ratio of the concentration of ethanol (or 3H2O) emerging from the probe (outflow) to the inflowing concentration is monitored. This outflow:inflow concentration ratio (O/I ratio) has been shown to vary inversely with the total blood flow rate (Hickner et al. 1995; Stallknecht et al. 1999). The O/I ratio has been used as an indicator of changes in total flow to the muscle during exercise (Hickner et al. 1994) and during hyperinsulinaemia (Rosdahl et al. 1998).
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A number of mathematical models have been developed to describe the changes observed in microdialysis studies (Bungay et al. 1990; Morrison et al. 1991; Stahle, 2000). Most of the earlier models were focused on microdialysis in brain and were mostly concerned with modelling the recovery of endogenous compounds across the microdialysis membrane (Kehr, 1993). Although the recovery of an exogenous compound such as ethanol can be shown to be equivalent to 1–O/I, these early models were not developed to model the ethanol technique. It was not until 1995 that a model describing the O/I ratio of ethanol in muscle was developed (Wallgren et al. 1995). Indeed, this model has been used to calculate absolute values of blood flow in humans during rest and exercise (Hickner et al. 1997). However, this model may only be valid when the proportion of nutritive flow is unaltered, as it has been shown that the redistribution of blood flow within muscle in the absence of changes in total flow affects the microdialysis O/I ratio (Newman et al. 2001). The addition of the type A vasoconstrictors, such as noradrenaline (NA), decreases the O/I ratio, while the type B vasoconstrictors, such as serotonin (5-HT), increases the ratio (Newman et al. 2001). This paper therefore develops models of microdialysis O/I ratio that allows the determination of the nutritive fraction of blood flow in muscle.
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Methods
Perfusion
Experimental data from a previously published study on the O/I ratio in the perfused rat hindlimb (Newman et al. 2001) were re-analysed in this study. The experimental procedures are described in detail in that study (Newman et al. 2001) and were approved by the Committee on the Ethical Aspects of Research Involving Animals of the University of Tasmania. Briefly, rats (180–200 g) were anaesthetized with 6 mg (100 g body weight)–1 of pentobarbital sodium (I.P.). Surgery was performed as previously described (Newman et al. 2001) and the animal was killed by an overdose of the anaesthetic (12 mg) into the heart. One hindlimb was perfused under constant flow with a red blood cell containing medium. Linear microdialysis probes were constructed as detailed by Newman et al. (2001). Briefly, the point of a 23G Terumo needle was blunted by filing and the syringe adaptor removed. A 25-mm length of microdialysis tubing (Bioanalytic Systems, West Lafayette, IN, USA; molecular weight cutoff, 30 kDa; outer diameter, 320 μm) was inserted into the blunted end of the needle to a depth of 5 mm. Two probes were inserted into the calf muscles of the perfused leg, passing through the gastrocnemius red, gastrocnemius white, tibialis and extensor digitorum longus such that the total exposed surface for exchange was approximately 10 mm in length. These probes contained a 0.9% NaCl solution supplemented with 10 mM [14C]ethanol (100 nCi ml–1) and 3H2O (500 nCi ml–1). In each experiment, one probe flow rate was set to 1 μl min–1, while the other was set to 2 μl min–1. The hindlimb perfusion blood flow rate was varied between 1 and 9 ml min–1 for a period of 30 min at each flow rate and the nutritive fraction was varied by the addition of 70 nM NA or 300 nM 5-HT to the perfusion medium. The addition of a saline vehicle (0.9% NaCl) to the perfusion medium was used as a control. The O/I ratios for [14C]ethanol and 3H2O were determined at each perfusion flow rate over the last 10 min of experimentation at each perfusion flow rate. One 10-min sampling period was used for the 1 μl min–1 probe and two sequential 5-min sampling periods for the 2 μl min–1 probe. The two samples from the 2 μl min–1 probe were found to not significantly differ and thus a mean value was taken. The model was used to determine the nutritive fraction at each perfusion flow rate and for each probe flow rate and tracer.
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Model
Consider an elemental volume ( V) (see Fig. 1A) of tissue, in a cylindrical coordinate system (r,,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.
A, an elemental volume of muscle in a cylindrical coordinate system (r,,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 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.
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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 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.
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Eqn (1) can be simplified to give:
and expanding eqn (2) and neglecting second-order terms (i.e. terms in r2) 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:
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where contains the quantity of interest: the nutritive fraction .
In Appendix 2 a detailed series expansion solution to eqn (5) is presented, together with a complete treatment of the appropriate boundary conditions. This was developed to check the validity of our simpler models, and may be of interest for cases where the experimental parameters differ significantly from the present study, as well as providing a starting point for the development of solutions for other experimental geometries. Presently, as the concentration should decrease along the direction of the probe axis (in the direction of flow of the microdialysis solution) we explore simpler models by postulating that a simple, separable solution to eqn (5) is appropriate to our problem:
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where I0 is the modified Bessel function of the first kind, K0 the modified Bessel function of the third kind (also known as the Macdonald function), C1 and C2 are integration constants associated with the radial direction, a = a0 – 2 and the positive parameter is (at this stage) undetermined. The integration constants can be found from suitable boundary conditions: the concentration value at the probe wall (r = rp), and the vanishing of the radial flux at a great distance from the probe axis:
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(7a)
(7b)
where (z) is the solute concentration at the outer edge of the probe and is the radial derivative of the solute concentration.
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:
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or
(9a)
Now (eqn (9.7.1) of Abramowitz & Stegun, 1964) while (eqn (9.7.2) of Abramowitz & Stegun, 1964) so that eqn (9a) reduces to:
(9b)
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 ( (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.
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The next step is to relate (z) to the solute concentration in the probe. Consider an elemental slice (of thickness z) 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 z) of the probe, as shown in Fig. 1C, steady state conservation of mass has:
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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).
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Rearranging and simplifying gives:
(12)
for an arbitrary external surface concentration (z) (assumed independent of ) where = 2 Pprp, with general solution:
(13)
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:
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(14)
and the solution (from eqn (13)) is:
(15)
This expression generally contains two axial length scales, 1/ originating from the form of (z), and the length scale for the homogeneous part of eqn (12), qp/, which controls the decrease of concentration along the probe if there is no external solute concentration (0 = 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 (z) has a z-dependence that is directly proportional to Cp(z): (z) = Cp(z) so that eqn (12) takes the form:
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(16)
with the solution:
(17)
which is compatible with our choice for (z), and identifies , relating it to the proportionality factor and the homogeneous scale factor qp/:
(18)
As physically loss from the probe by diffusion through the wall requires a concentration gradient (Cp(z) > (z)) we expect < 1, so that the length scale associated with the axial direction (1/) is increased over qp/ by the presence of solute in the surrounding muscle as expected (see eqn (18)).
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The value of is related to the major experimentally determined quantity in the microdialysis process, the O/I ratio because:
or
The total mass flow of solute through the probe membrane is simply the solute lost from the probe:
(19)
while the total mass of solute removed by the blood flow in the tissue is:
(20)
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Substituting the expression for muscle concentration from eqn (10):
(21)
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 0 yields:
(22)
Now for steady state we require that Mp = Mb so combining eqn (19) and eqn (22) gives:
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which simplifies to:
(23)
which indicates that, for a given O/I ratio in the probe, as is known (as is ) via eqn (18): , a can be found by solving the eqn (23), and accordingly ao and hence 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:
where:
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(24)
or equivalently:
(24a)
and:
If one can also neglect the contribution of diffusion in the tissue in the axial direction, arguing (see Appendix 1) that a0 >> 2 this expression simplifies to:
(25)
which has the advantage that it is slightly more straightforward to search for the optimal value of a0 and hence for the nutritive fraction .
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In vitro
The effective permeability constant of the microdialysis probe (Pp) was determined in vitro. A probe of known exchange length was constructed by gluing a second blunted 23 G needle to the outflow end of the microdialysis probe (in this particular case the exposed probe length was 11 mm). This was then immersed in a well-stirred saline (0.9% NaCl) solution maintained at 37°C to match the perfusion temperature. In this situation, eqn (24) or eqn (25) can be reduced (because 0) to
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(26)
as in such a situation, the bulk movement of the stirred medium means that the term Q b input in eqn (24) or eqn (25) is effectively infinite (or at least many orders of magnitude greater than those for muscle blood flow). Thus the term is much greater than 1, and hence the ratio of the two Bessel functions in the expressions for tends to 1. Furthermore, the term would be expected to be much greater than Pp (which can then be verified upon calculation of Pp). Eqn (26) can be rewritten in the form Cout/Cin = e–PpS/qp where S is the probe surface area. This is analogous to the equation pioneered by Renkin (1955) to describe the extraction fraction (EF) of substances across a vascular bed (EF = 1 – e–PS/Q where PS is the permeability–surface area product and Q is the blood flow rate).
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In a similar fashion to the perfusion experiments, the tracers 3H2O and [14C]ethanol were added to the inflow solution (0.9% NaCl) and the O/I ratio for each tracer was determined at probe flow rates between 2 and 10 μl min–1 in order to calculate Pp. At least 10 min was allowed between each flow rate change for the probe to equilibrate. Two sequential 5-min samples were taken in order to ensure the probe had equilibrated.
Statistical analysis
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One way analysis of variance (ANOVA) was performed using SigmaStat (SPSS Science; Chicago, IL, USA) with comparisons made between treatments using the Student–Newman–Kuels post hoc test. Significance differences were accepted at the P < 0.05 level. Data are presented as means ± S.E.M.
Results
Figure 2 shows the O/I ratio and the calculated Pp for 3H2O and [14C]ethanol for a probe in a stirred saline solution across a number of probe flow rates based on a probe length and radius of 0.011 m and 1.6 x 10–4 m, respectively. The data for the lowest flow rate (2 μl min–1) were excluded because the total amount of radioactivity in these samples were only twice the background level. Therefore the other data were used to determine the average Pp to be 1.59 ± 0.01 x 10–5 m s–1 and 0.88 ± 0.01 x 10–5 m s–1 for 3H2O and [14C]ethanol, respectively.
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A known length (0.011 m) of microdialysis membrane was immersed in a well-stirred saline solution and pumped through at varying rates with a saline solution containing 3H2O ()and [14C]ethanol (). The O/I ratio was determined and the effective permeability of the probe (Pp) was calculated in duplicate according to eqn (26).
The model can now be used to determine the ratio of nutritive blood flow to total muscle blood flow () from previously published experimental data (Newman et al. 2001). The values of which minimized the root mean square (RMS) error between the approximate model of eqn (25) and experiment, for the different drug treatment regimens and microdialysis flow rates, are given below. Minimum overall RMS errors were calculated for experiments involving the same drug treatment regimens. Typical RMS error as a function of is shown in Fig. 3 (in this case 3H2O in vehicle perfusions with qp = 2 μl min–1). Calculations were based on parameters as shown in Table 1. These were done for each treatment (vehicle, NA or 5-HT), tracer and qp, the results of which are summarized in Table 2. The performance of the approximate model is shown in Fig. 4 (3H2O, qp = 1 μl min–1), Fig. 5 ([14C]ethanol, qp = 1 μl min–1), Fig. 6 (3H2O, qp = 2 μl min–1) and Fig. 7 ([14C]ethanol, qp = 2 μl min–1).
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For each drug treatment regimen, the RMS error between the approximate model and experimental data (Newman et al. 2001) was determined over a range of 0–1. The best estimate for was taken to be that which minimized the error. Data are shown for vehicle perfusions using a probe flow rate of 2 μl min–1 and 3H2O as tracer. Other parameters are L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s –1.
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Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 0.88 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 1.59 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
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Individual experimental data points are shown for vehicle (), 300 nM 5-HT () and 70 nM NA () from Newman et al. (2001). The predicted response of the approximate model using minimized RMS error values for from Table 2 are also shown for vehicle (continuous line), 5-HT (dotted line) and NA (dashed line). Parameters used were L = 0.01 m, rp = 0.16 mm, Pp = 0.88 x 10–5 m s–1 and Dt = 1 x 10–9 m2 s–1.
Figure 8 shows a contour plot of the percentage difference between the approximate (eqn (25)) and full model (Appendix 2) for the range of and Q b values appropriate for the present study.
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The percentage difference between the predicted O/I ratios from the simplified model (eqn (25)) and the series expansion model of Appendix 2, is presented as a function of blood flow rate Q b and nutritive factor (for ranges appropriate to the present study) by contours ((approximate–full)/full x (100). In this case, L = 0.01 m, qp = 1 μl min–1, Dt = 1 x 10–9 m2 s–1 and Pp = 1 x 10–5 m s–1 (as representative of either tracer). Numbers indicate the percentage difference value for each contour.
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In addition, as discussed further in Appendix 2, Fig. 9 shows the variation of the concentration in the probe along the probe axis based on the full and approximate models. Figure 10 shows the corresponding concentrations at the outer surface of the probe, while concentration profiles in the radial direction midway along the probe are shown in Fig. 11.
The concentration Cp(z), normalized by Cp(0) = Cin, is presented as function of distance (z) along the probe axis. Results are shown for the probe concentration derived from taking one, three and 11 term approximations in the series solution model of Appendix 2 for Cp(z) (eqn (A3)), and for the simplified exponential model for Cout/Cin, i.e. eqn (17) for Cp(z), with the scale factor in the exponential generated by eqn (25).
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The concentration of solute in the muscle is presented as a function of distance (z) along the probe, just outside the probe wall, C(rp, z), using the one, three and 11 term approximations in the series solution model for C(r, z) (eqn (A1)), at r = rp, and also for the simplified model where the z dependence of C(r, z), (z) of eqn (10), is taken as proportional to Cp(z), with proportionality factor controlled by eqn (25) through eqn (18).
The concentration of solute in the muscle is presented as a function of radial distance from the probe (r > rp) at the midpoint of the probe length (z = L/2), C(r, L/2), using the one, three and 11 term approximations in the series solution model for C(r, z), and also for the simplified model where the r dependence of C(r, z) is given by the Bessel function expression eqn (10), and involves the dependent scale factor a, which is related to Cout/Cin by eqn (25).
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Discussion
This paper details the derivation of theoretical expressions for the microdialysis O/I ratio in tissue. From eqn (24) or eqn (25) it is evident that this expression is an exponentially decaying function dependent on the physical dimensions of the probe (length, L and radius, rp), the flow rate through the probe (qp) and blood flow rate (Q b), the effective permeability of the substance of interest in the probe (Pp) and the tissue diffusion coefficient (Dt). There is also a term involving the nutritive fraction () of the total blood flow. Recognition of this dependence leads to the main experimental results of this paper – the estimates of the values of that minimize the error from experimental data using the perfused rat hindlimb preparation (Newman et al. 2001) for 3H2O and [14C]ethanol. The experimental data and model performance are shown in Figs 4–7 with the minimized errors for in Table 2. It is clear that the microdialysis techniques and the model provide sufficient discrimination to observe the influence of the two vasoconstrictors on the nutritive fraction.
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The simplest derived model as shown in eqn (25) is consistent with the model of Bungay et al. (1990) except for the use of different clearance rates. In the present case Qb is the main clearance parameter while Bungay et al. (1990) used first-order rate constants including removal by the vasculature and intra- and extracellular metabolism). In addition, eqn (25) is consistent with the active tissue model developed by Stahle (2000) although this latter model does not specify the contribution of blood flow and metabolism to the clearance rates. The model pioneered by Bungay et al. (1990) incorporates resistances in series, that of dialysate resistance (Rd), membrane resistance (Rm) and extracellular resistance (Re) and they show that Re >> Rm >> Rd. This relative influence of resistances in the model means that it is advantageous to determine the effective permeability of the entire probe (in the current model Pp) for each species experimentally, incorporating all probe influences (including the probe's cross-sectional geometry) into this parameter. The present work also indicates the complications involved in modelling diffusion in the tissue parallel to the probe axis.
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The experimental difficulties involved in the use of this model focus on an accurate assessment of the parameters. It is a simple enough matter to measure the physical dimensions of the probe and to set the flow rate through the probe. In this study the effective permeability (Pp) of 3H2O and [14C]ethanol through the probe were determined in vitro (see Fig. 2). In the current experiments to determine Pp there is necessarily a balance between the length of the probe and flow rate through the probe to ensure that background radioactivity does not interfere with the measurement of the probe outflow solution. Eqn (26) suggests that a proportional change in probe length (L) has the same effect as a reciprocal change in probe flow rate (qp). It was considered prudent to keep L the same as the perfusion experiments and vary qp, as there is a lower fractional error in the measurement of the probe length at L = 10 mm compared to L = 2–3 mm if qp was set to 1 or 2 μl min–1 as in the perfusion experiments.
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The other important parameter is that of the tissue diffusion coefficient (Dt). Values for the diffusion coefficient of water in muscle can be obtained from the literature and range between 0.68 and 1.82 x 10–9 m2 s–1 (Hansen, 1971; Chang et al. 1972; Cleveland et al. 1976; Schultz & Armstrong, 1978; Morvan & Leroy-Willig, 1995). In the current study, a compromise position was taken so that a value for Dt of 1.0 x 10–9 m2 s–1 was used. It was not possible to find values for the diffusion coefficient of ethanol in muscle. The diffusion coefficient of ethanol has been measured in the bronchial mucosa of dogs and was found to be 0.563 x 10–9 m2 s–1 (George et al. 1996). Gonzales et al. (1998) found the diffusion coefficient of ethanol in brain was 1.2 x 10–9 m2 s–1. These values for ethanol diffusion coefficient are similar to the diffusion coefficient of water in muscle. As a consequence it was considered reasonable to take the diffusion coefficient of ethanol in muscle to be the same as that for water. As Wallgren et al. (1995) found, the strong influence of this parameter on the results was disappointing. It does suggest, however, that the use of 3H2O as opposed to ethanol is preferable as diffusion coefficients for water are better known as well as the fact that evaporative errors are less likely with water.
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The accurate determination of the total blood flow to the muscle or muscles through which the probe passes (Q b) is also required. Post-mortem analysis of the perfused hindlimb showed that the probe passed through the gastrocnemius red, gastrocnemius white, tibialis and extensor digitorum longus in proportions of 0.2, 0.2, 0.5 and 0.1, respectively. There are previously published data on the blood flow to these muscles as assessed by 15-μm microsphere entrapment in the red blood cell-perfused rat hindlimb under basal conditions (Youd et al. 1999). It has also been shown that these blood flow rates are not affected by the addition of vasoconstrictors such as NA or 5-HT (Clark et al. 2000). For this study, therefore it was possible to estimate the average blood flow to the muscles through which the probe passed. Data from Youd et al. (1999) show that the weighted average proportion of total perfusion blood flow going to the muscles the probe passes through was 0.6%. This corresponds to a flow rate surrounding the probe of 1.73 x 10–2 ml min–1 (g muscle)–1 for a typical 180- to 200-g rat at a total perfusion flow rate of 1 ml min–1. The conversion factor from Table 1 then gives a Q b of 3.2 x 10–4 m3 s–1 (m3 muscle)–1.
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The range of blood flows over which the model is tested is approximately 3–30 ml min–1 (100 g muscle)–1. While this easily encompasses the flow rates typical for resting muscle, it is a limitation of the current testing of the model. However, it was necessary to restrict the flow rates to this range in order that the perfusion pressure generated with the vasoconstrictors would not become excessive. As reported in the study from which the experimental data were obtained, the perfusion pressure with NA and 5-HT reached approximately 200 mmHg at the highest flow rate (Newman et al. 2001). Higher flow rates with the vasoconstrictors led to uncontrolled oedema. Determination of the nutritive fraction of blood flow at higher flows (particularly in vivo during exercise) will be a particularly useful extension of the current model.
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The model described in this paper assumes the system is at steady state; however, some of the experimental data may not satisfy this condition. While the model does not allow the calculation of equilibration times, Morrison et al. (1991) and Bungay et al. (2001) suggest that it is possible to estimate the e-folding time in the approach to steady state as the reciprocal of the clearance rate constant (in this case Q b). These calculations imply that during vehicle infusion (Q b = 0.0013 s–1 and = 0.22), the e-folding time would be approximately 1 h. However, it is not obvious that Qb is the only determinant of equilibration time. Data from Newman et al. (2001) show a set of experiments where the hindlimb was allowed to equilibrate for 40 min at a pump flow rate of 4 ml min–1 (corresponding to Q b = 0.0013 s–1). Subsequently vehicle, NA, 5-HT or the vasodilator nitroprusside was added to the blood and left for a further 40 min. These data showed that the O/I ratio measured during the last 10 min of vehicle infusion were 98.5 ± 2.5% and 101.8 ± 1.2% of the basal ratio for 3H2O and [14C]ethanol, respectively. This suggests that 40 min is sufficient time to reach steady state under these conditions which is considerably shorter than the e-folding time predicted by 1/Qb.
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During 5-HT perfusion, the largest rate constant (Q b = 0.0028 s–1 and = 0.05) implies an e-folding time of 2 h. From the previous example, it is likely that this is a considerable overestimate and as the experiment had been in progress for nearly 3 h by this time, it is likely that these data are at steady state. In addition, all steady-state approach times during NA perfusion were less than 1 h. Hence while some of the low Q b data may be under non-steady-state conditions, the highest Q b data would appear to be at equilibrium. If only these data are used then, for example from Fig. 6, the value of for vehicle is 0.32 ± 0.08 (n = 10). Corresponding values for 5-HT and NA are 0.13 ± 0.05 (n = 8, P < 0.05) and 0.93 ± 0.05 (n = 10, P < 0.05), respectively. These order of magnitude calculations for equilibrium approach times could be used in the design of future experiments even if a value of unity is assumed for . For example, a typical muscle blood flow rate during rest of 1.5 ml min–1 (100 g)–1 would suggest that the approximate e-folding time for the approach to steady state was 1 h. While this is likely to be a significant overestimate, it does provide a time frame in which to experimentally determine the time required to reach equilibrium.
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This model could also be used with compounds other than 3H2O or [14C]ethanol as well as to model the microdialysis recovery of compounds. As microdialysis recovery (R) of an exogenous compound is related to the O/I ratio by the equation R = 1–O/I, then eqn (25) can be modified accordingly. The only condition would be that the compound was not metabolized in addition to being removed by the blood. In addition, a volume fraction for extracellular space may need to be incorporated if the compound is excluded from cells. Depending on the compound used, the direct evaluation of the ratio of two Bessel functions may be problematic (see Appendix 3 for details).
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This paper makes the assumption that the non-nutritive flow route is unable to exchange ethanol and water, which may be an oversimplification. However, at present there is no way of determining the relative exchange capacity of each pathway; the use of would seem to be a valid start. Another assumption in this model is that the [14C]ethanol and 3H2O are not taken up by tissue and metabolized. As such it varies from other models of microdialysis such as that by Morrison et al. (1991) which took into account the effects of metabolism on the model. Skeletal muscle does not appreciably metabolize ethanol as it contains no alcohol dehydrogenase. In addition as the results for [14C]ethanol and 3H2O are similar, it was not considered necessary to include a term for metabolism in the model.
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In conclusion, this study demonstrates that the microdialysis O/I ratio can be modelled mathematically while taking into account the nutritive fraction of blood flow. In particular, the relative influences of vasoconstrictors on nutritive fractions can be observed. This model can be applied to the determination of nutritive fraction of skeletal muscle blood flow in physiologically relevant microvascular conditions such as during exercise and in disease states.
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The analysis presented in the text leading to eqn (24) and eqn (25) is adequate for the parameters of the experiments of the present study, although this is not quite obvious without some consideration of the various spatial scales involved. It is also relatively straightforward, compared to the complete solution of eqn (5) and the full set of boundary conditions.
The relevant spatial scales in the microdialysis problem are as follows.
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The probe radius (1.6 x 10–4 m).
The length of the probe (10–2 m).
The length scale associated with diffusion in the absorbing muscle which for the value of Dt = 1 x 10–9 m2 s–1 and the likely range of Qb (, 0.1 – 1; Qb, 2.5 x 10–4–3.5 x 10–3 s–1) for the present experiments, gives a range of 6.3 x 10–3–5.3 x 10–4 m, all lying between the probe radius and the probe length.
Two candidate length scales for the variation in concentration in the axial direction present themselves. The first is the decay length for concentration in the probe qp/ = qp/ 2 rp Pp in the absence of appreciable external solute concentration (which appears in eqn (12)), which for the present experiments ranges from 1.0 x 10–3 to 3.8 x 10–3 m. These values fall between the probe length and the probe radius, but overlap the diffusion scales above (for the higher probe flow rate compared with low values of Qb reflecting some combination of low nutritive fractions and muscle blood flow rates), but this is not particularly relevant for the actual microdialysis experiments.
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The more relevant axial length scale arises where there is muscle present, and an appreciable solute concentration at the outer wall of the probe. If we assume, as in our modelling, that the probe concentration has an exponential z-dependence, then as the O/I ratios observed in the present experiments are typically in the range 0.1–0.8, then for the present probe length of 10–2 m, the effective length scales for the axial dependence ought to range between 4.34 x 10–3 and 4.48 x 10–2 m.
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The lower extent of the expected range of axial length scales approaches the upper limits of muscle diffusion scales ((iii) above) which apply for cases where Qb is small, but such low Qb experiments tend to have a smaller reduction in probe concentration, and consequently (in the present experiments) correspond to O/I values nearer unity, and longer implied axial length scales. Accordingly, the concept that the radial length scale is dominant over the axial length scale, and that the diffusion along the z-direction should be relatively small, remains sound for the present experimental situation.
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We have not carried out an exhaustive search over the wide range of possible experimental geometries, flow rates, probe materials and solute properties to see how frequently the conditions prevail that make eqn (25) an excellent approximation to the more complicated model of Appendix 2. The calculated radial concentrations presented in Fig. 4 of Bungay et al. (1990) do suggest that one solute species (sucrose) has a radial scale comparable with its relatively short probe length. It is suggested that consideration should be given to the various length scales in any proposed study – preferably at the experimental design stage. While the complication of using the model of Appendix 2 in data fitting for estimating (for example) optimal values of nutritive fraction might be daunting, it would be prudent to scan the ranges of values likely to be encountered to check (as was done here) that the full and approximate models give comparable results.
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Appendix 2
In this appendix a more complete solution to the diffusion problems of probe wall and tissue is presented. This necessitates some additional computational machinery. While it is not essential, the analysis was made simpler by the use of the symbolic and numerical mathematics package Mathematica (Wolfram Research Inc., Mathematica Version 5, Champaign, IL, USA).
The treatment in the main text ignores the boundary conditions that should be satisfied by the diffusion along the z-direction in the muscle (parallel to the perfusion probe). The geometrical arrangement of the probe and the muscle in the present studies is such that there should be no flow in the z-direction across the planes at z = 0 and z = L. This is not satisfied by the simple, separable solution (eqn (6)) to the diffusion eqn (5), although for the present experimental conditions the effect should be small, because as discussed in Appendix 1 the axial length scales are long compared to the length scales associated with radial diffusion.
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To solve the diffusion eqn (5) with the boundary conditions of vanishing fluxes across the planes at z = 0 and z = L, and at a large radial distance (r ), requires that C(r, z) takes the form:
(A1)
where
The remaining boundary condition is the mass balance at the probe wall – the mass flux through the membrane at each point along the z-direction must equal the diffusive mass flux in the adjacent muscle.
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(A2)
However as we have already noted (eqn (13)), Cp(z) itself depends on the external concentration, and (defining for convenience the probe scale parameter = /qp) eqn (13) yields:
(A3)
Substituting for Cp(z) in eqn (A2), the boundary conditions imply that:
(A4)
An infinite set of relationships between the (infinite number) of series coefficients bn can be determined by multiplying eqn (A4) by for each non-negative integer m, and integrating over z from 0 to L. The general form of these integrations can be performed analytically, and by truncating the series sum in eqn (A1) to a finite number of terms, a system of linear equations for the coefficients (bn), can be generated and solved numerically, providing solutions of any required accuracy for C(r, z) and Cp(z), as functions of a0, and hence of Qb. The form of the general system is:
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(A5)
and
(A6)
Truncating the sums in eqn (A5) and eqn (A6) produces a set of linear equations that can be solved numerically for any set of given experimental parameters and a value for , and the coefficients can then be substituted into eqn (A1) and eqn (A3) to enable the muscle concentration fields or the O/I ratios to be calculated.
We demonstrate the convergence of the series solution, by solving eqn (A5) and eqn (A6), for the coefficients bn, for a truncation of the series expansion (eqn (A1)), to one, three and 11 terms. We use the resulting concentration functions, C(r, z) and Cp(z) (eqn (A1) and eqn (A3)) to demonstrate the close agreement with the simplified model presented in the main text, for the range of experimental parameters in the present study. The variation of the concentration in the microdialysis probe Cp(z) along the probe axis is presented in Fig. 9, together with the prediction of the simple model based on eqn (25). Clearly even with three terms the results of the two models for Cout/Cin, i.e. the value at z = L, agree closely. In Fig. 10, the corresponding results for the z-dependence of the concentration in the muscle just outside the probe wall are presented. The small discrepancy between the 11-term series solution and the simplified model is clearest at the ends of the probe, z = 0 and z = L, where the series solution has vanishing z-derivative, to match the no flux boundary conditions there. Finally, Fig. 11 compares the radial dependence of the concentration in the muscle, in a plane normal to the probe and located midway along it. The agreement of the simple Bessel function solution (from eqn (10)) and the series solutions is achieved for even three-series terms. These computations indicate that we can apply the simplified model of the main text, whose core is the relation between Cout/Cin and the nutritive fraction through eqn (25), to the data from the present study with confidence.
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Appendix 3
Direct evaluation of the ratio of two Bessel functions (i.e. ) may be problematic as the individual Bessel functions rapidly tend to zero, leading to a loss in precision for large arguments. A much more accurate and easier to implement calculation for the ratio of the two Bessel functions can be found using the relationship leading to and substituting the polynomial approximation from (section 9.8.6 of Abramowitz & Stegun, 1964) of
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This yields an approximation where
and
For x > 2.5, || < 1.5 x 10–7.
In the present study, direct evaluation is quite adequate as the argument of the Bessel functions is always less than 0.3 (rp = 1.6 x 10–4 m, = 1, Q b = 3.5 x 10–3 m s–1 and Dt = 1 x 10–9 m2 s–1). For the direct evaluation of the ratio of the two Bessel functions to become problematic a combination of high probe radius and high blood flow (possibly during exercise) and a compound with a very low diffusion coefficient in muscle (a protein or other high molecular weight molecule) would be required.
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