A Theoretical Description of Microdialysis with Mass Transport

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Anal. Chem. 2000, 72, 2042-2049

A Theoretical Description of Microdialysis with Mass Transport Coupled to Chemical Events Hua Yang, Jennifer L. Peters, Cassandra Allen, Shyh-Shi Chern, Rob D. Coalson, and Adrian C. Michael*

Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

A random-walk simulation of microdialysis is used to examine how a reaction that consumes analyte in the medium external to the probe affects the extraction and recovery processes. The simulations suggest that such a reaction can promote the extraction process while simultaneously inhibiting the recovery process, which appears to be consistent with recent experimental evidence of asymmetry in the extraction and recovery of the neurotransmitter, dopamine, during brain microdialysis. This suggests that quantitative microdialysis strategies that rely on the extraction fraction as a measure of the probe recovery value, such as the no-net-flux method, will produce an underestimate of the analyte concentration in the external medium when that analyte is consumed by a reaction in the external medium. Furthermore, if experimental conditions arise under which the kinetics of the reaction are changed, then changes in the extraction and recovery processes are likely to occur as well. The implications of these theoretical findings for the quantitative interpretation of in vivo microdialysis results obtained for the neurotransmitter dopamine are examined. Microdialysis sampling is widely used for monitoring neurotransmitter levels in the extracellular space of the brain.1 Dopamine has been the target of numerous in vivo microdialysis studies because of its central role in several CNS disorders, such as Parkinson’s disease,2,3 schizophrenia,4 and substance abuse.5 Some microdialysis-based estimates of the extracellular concentration of dopamine in the rat striatum, a region of the brain that is densely innervated by dopamine-containing nerve fibers,6 fall in the single-digit nanomolar range.7 This low-nanomolar concentra* Corresponding author. Phone:(412) 624-8560 (voice), (412) 624-8611 (FAX), Email:[email protected]. (1) Robinson, T. E.; Justice J. B., Jr. In Techniques in the Behavioral and Neural Sciences, Vol. 7: Microdialysis in the Neurosciences; Robinson, T. E., Justice J. B., Jr., Eds.; Elsevier: Amsterdam, The Netherlands; 1991. (2) Zigmond, M. J.; Stricker, E. D. Int. Rev. Neurobiol. 1989, 31, 1-79. (3) Starr, M. S. Synapse 1995, 19, 264-293. (4) Grace, A. A. Neuroscience 1991, 41, 1-24. (5) Koob, G. F.; Bloom, F. E. Science (Washington, D.C.) 1988, 242, 715723. (6) Kuhar, M. J.; Couceyro, P. R.; Lambert, P. D. In Basic Neurochemistry: Molecular, Cellular and Medical Aspects, 6th ed.; Siegel, G. J.; Agranoff, B. W.; Albers, R. W.; Fisher, S. K.; Uhler, M. D., Eds.; Lippincott-Raven: Philadelphia, PA, 1999; pp 243-261. (7) (a) Olson, R. J.; Justice, J. B., Jr. Anal. Chem. 1993, 65, 1017-1022. (b) Smith, A. D.; Justice, J. B., Jr. J. Neurosci. Methods 1994, 54, 75-82. (c) Sam, P. M.; Justice, J. B., Jr. Anal. Chem. 1996, 68, 724-728.

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tion, however, is approximately 3 orders of magnitude below the reported affinity of the D1 receptor,8 the most abundant of the five known dopamine receptors in the rat striatum. In the presence of a low-nanomolar dopamine concentration, the D1 receptor should have little opportunity to participate in dopaminergic neurotransmission. Numerous studies suggest, however, that D1 receptors do participate in dopaminergic neurotransmission.9 This has prompted us to consider the possibility that microdialysis has underestimated the extracellular concentration of dopamine.10 THEORY To state the issue to be addressed in this paper, we first provide a brief overview of the quantitative aspects of microdialysis. There can be two contributions to the concentration of analyte in a microdialysate sample. One is from analyte added to the perfusion fluid that enters via the probe inlet, while the other is from analyte present in the medium external to the probe. The relationship between the concentration at the probe outlet, Cout, that at the probe inlet, Cin, and that in the external medium, Cext, can be expressed as follows10

Cout ) (1 - E)Cin + RCext,∞

(1)

where E is the extraction fraction, R is the relative recovery, and Cext,∞ refers to the analyte concentration in the external medium beyond any concentration gradients created by the probe.11 Microdialysis results are often presented in the concentration differences format, which is a plot of the difference between Cin and Cout as a function of Cin.12 The algebraic form of such a plot is obtained by rearranging eq 2:

Cin - Cout ) ECin - RCext,∞

(2)

The linear form of eq 2 is consistent with concentration differences (8) Missale, C.; Nash, S. R.; Robinson, S. W.; Jaber, M.; Caron, M. C. Physiol. Rev. 1998, 78, 189-225. (9) See for example: (a) Keys, A. S.; Mark, G. P. Neuroscience 1998, 86, 521531. (b) Darbin, O.; Risso, J. J.; Rostain, J. C. Neurosci. Lett. 1999, 267, 149-152. (10) (a) Lu, Y.; Peters, J. L.; Michael, A. C. J. Neurochem. 1998, 70, 584-593. (b) Peters, J. L.; Michael, A. C. J. Neurochem. 1998, 70, 594-603. (c) Yang, H.; Peters, J. L.; Michael, A. C. J. Neurochem. 1998, 71, 684-692. (d) Yang, H.; Peters, J. L.; Michael, A. C. In Monitoring Molecules in Neuroscience; Rollema, H.; Abercrombie, E.; Sulzer, D., Zackheim, J., Eds.; The State University of New Jersey: Newark, 1999; pp 81-82. (11) Benveniste, H.; Hansen, A. J.; Ottosen, N. S. J. Neurochem. 1989, 52, 17411750. 10.1021/ac991186r CCC: $19.00

© 2000 American Chemical Society Published on Web 03/29/2000

plots obtained under a variety of both in vitro and in vivo conditions.7,13 The x intercept of the concentration differences plot has become known as the point of no-net-flux, since at this point Cin and Cout are equal to each other. Equation 3 is obtained by rearranging eq 2

ECnnf ) RCext,∞

(3)

where Cnnf is the no-net-flux concentration, i.e., the common value of Cin and Cout at the point of no-net-flux. Equation 3 raises the issue addressed in this paper. The low-nanomolar estimates of striatal extracellular dopamine were based on the measured no-net-flux concentration.7 According to eq 3, however, the no-net-flux concentration can be a direct measure of the external concentration only if the extraction and recovery values are equal. However, the existence of this equality during the in vivo microdialysis of dopamine has not been demonstrated. According to eq 2, the in vivo value of E can be obtained from the slope of a concentration differences plot. On the other hand, eq 2 also implies that R cannot be determined from microdialysis results unless Cext,∞ is independently known, which is generally not the case during the in vivo experiments. Hence, confirmation that E and R for dopamine are equal in vivo has not been obtained. In fact, recent evidence suggests that E and R for dopamine are not equal during in vivo microdialysis. For example, drugs that inhibit the dopamine uptake mechanism, such as nomifensine and cocaine, cause dopamine extraction, as measured by the slope of a concentration differences plot, to decrease.7 On the other hand, studies in which voltammetry and microdialysis results were compared have led to the conclusion that such drugs cause dopamine recovery to increase.10 Taken together, these observations suggest that uptake, a process that consumes dopamine in the medium external to the microdialysis probe, creates an asymmetry in the extraction and recovery of dopamine during in vivo microdialysis. This supports our suspicion that, in the case of dopamine, Cnnf underestimates Cext,∞. The mathematical description of microdialysis developed by Bungay and co-workers14 appears to provide theoretical justification for the no-net-flux method. That model, however, was presented in terms of a dialysis efficiency parameter, ED:

ED )

Cin - Cout Cin - Cext,∞

(4)

The dialysis efficiency parameter is obtained from eq 1 by invoking the equality that ED ) E ) R. Consequently, it appears that the ED parameter should not be applied to situations where E and R may not be equal. Furthermore, as with R, ED cannot be (12) (a) Lonnroth, P.; Jansson, P. A.; Smith, U. Am. J. Physiol. 1987, 253, E228E231. (b) Parsons, L. H.; Justice, J. B., Jr. Crit. Rev. Neurobiol. 1994, 8, 189-220. (13) (a) Zhao, Y.; Liang, X.; Lunte, C. E. Anal. Chim. Acta 1995, 316, 403-410. (b) Khramov, A. N.; Stenken, J. A. Anal. Chem. 1999, 71, 1257-1264. (14) Bungay, P. M.; Morrison, P. F.; Dedrick, R. L. Life Sci. 1990, 46, 105-119. (b) Morrison, P. F.; Bungay, P. M.; Hsiao, J. K.; Mefford, I. N.; Dykstra, K. H.; Dedrick, R. L. In Techniques in the Behavioral and Neural Sciences; Vol. 7: Microdialysis in The Neurosciences; Robinson, T. E.; Justice, J. B., Jr., Eds.; Elsevier: Amsterdam, The Netherlands, 1991; p 47.

Figure 1. Schematic of the (A) axial and (B) radial geometry used for random-walk simulations of microdialysis (a is the impermeable inner cannula of the microdialysis probe; b is the flow channel of the microdialysis probe; c is the passive zone; d is the external medium; e is the probe outlet). The cubes in the external medium represent release and uptake sites (traps). The dashed lines denote the boundaries between the flow channel, the passive zone, and the external medium. The solid lines with hatch marks are impermeable boundaries: particles colliding with these boundaries were reflected back into the neighboring solution.

determined unless Cext,∞ is independently known, which is generally not the case during in vivo experiments. Hence, the goal of this work is to present a mathematical description of microdialysis that does not invoke the a priori assumption that E is equal to R. Specifically, simulations are used to consider the possibility that an asymmetry in the extraction and recovery processes can be created by a chemical reaction that consumes analyte in the medium external to the probe. A fully quantitative mathematical description of in vivo microdialysis is beyond the scope of this work: what follows is a qualitative model. Nevertheless, we have attempted to construct a model that includes the main features of the in vivo microdialysis of dopamine. In the model, for example, kinetic events in the external medium take place at an array of reactive volume elements, since dopamine is released and taken up only at specific sites in brain tissue, i.e., at neuronal terminals. Whenever possible, nominal experimental values10 are used for parameters in the model and approximations or simplifications are explicitly noted. SIMULATION METHODS Simulation Geometry. Figure 1 explains the geometry used for these simulations. The central axis of a concentric-style microdialysis probe was positioned along the line at x ) y ) 0 in a three-dimensional Cartesian coordinate system. The simulation space included (a) the cylindrical inner cannula of the probe, (b) the annular flow channel, (c) an annular passive zone that represents the dialysis membrane and any passive tissue adjacent to the probe, (d) the external medium, and (e) the outlet of the flow channel. The hatched boundaries in Figure 1 represent impermeable boundaries. The radius of the inner cannula was 75 µm, the width of the flow channel was 5 µm, and the width of the passive zone was varied between 5 and 30 µm. The external medium contained an array of small cubic elements, which qualitatively represent nerve terminals, referred to henceforth as “traps”. The edge length of the traps was 150 nm, and their centerAnalytical Chemistry, Vol. 72, No. 9, May 1, 2000

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to-center spacing was 2 µm, which approximates the size and spacing of dopamine terminals in the rat brain.15 Although Figure 1 only shows a small segment of the simulation space near the probe, the actual simulation space extended to infinity in all directions. Extraction and Recovery. In this work, diffusion was simulated by means of a random-walk algorithm, which is described below. To simulate recovery, diffusing particles were initially released from one of the traps in the external medium. To simulate extraction, diffusing particles were released from the flow channel of the probe. Any particle that diffused into a trap was subject to a first-order unimolecular decay reaction, while particles that diffused into the flow channel were subject to convective flow toward the probe outlet. A count was maintained of the number of particles consumed by a trap and the number recovered by the microdialysis probe (a particle was counted as recovered if it crossed the boundary at the probe outlet, “e” in Figure 1). Simulation of Diffusion. A random-walk algorithm was used to simulate diffusion of an ensemble of infinitely small, noninteracting particles.16 Initially, the ensembles contained 5000 particles and the simulations were carried out until 95% of those particles were either consumed or recovered. Simulations were run in triplicate, and the average results are presented below. All the particles in an ensemble were assigned initially to the same starting location. Then, the x, y, and z coordinates of each particle were adjusted during a series of time iterations by the quantity Ωx6D∆t, where Ω is a random number uniformly distributed between +1 and -1, D is the diffusion coefficient, and ∆t is the duration of the iterations. The root mean square of Ω is x3-1, so the root-mean-square diffusion step was x2D∆t, in keeping with the definition of the diffusion coefficient.17 A different random number was used to generate the step sizes in each direction. Diffusion was assumed to be isotropic, and D was set to the value measured for dopamine in the extracellular space of the rat striatum, 2.4 × 10-6 cm2/s.18 Since the diffusion coefficient measured in the brain is only slightly smaller than the solution value, the same diffusion coefficient was used in the flow channel, the passive zone, and the external medium. Since all the zones are predominantly aqueous, partition coefficients at the zone boundaries were taken to be unity. Simulation of Unimolecular Decay. Rate constants for the unimolecular reaction, kr, were selected to match the reported range of pseudo-first-order rate constants for dopamine uptake in the rat striatum before and after the administration of uptake inhibitors (∼0.25-10 s-1).19 For a reaction that occurs everywhere in solution, the fraction, φr, of particles that undergo the firstorder decay during each iteration would be φr ) 1 - e-kr∆t. Here, however, particles only undergo the reaction if they happen to be inside a trap. Hence, the fraction of particles available to react (15) Doucet, G.; Descarries, L.; Garcia, S. Neuroscience 1986, 19, 427-445. (16) (a) Bard, A. J.; Faulkner, L. R. In Electrochemical Methods: Fundamentals and Applications; Wiley: New York, NY, 1980; p 127. (b) Bartol, T. M., Jr.; Land, B. R.; Salpeter, E. E.; Salpeter, M. M. Biophys. J. 1991, 59, 12901307. (c) Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, NJ, 1993. (d) Agmon, N.; Edeistein, A. L. Biophys. J. 1995, 68, 815-825. (e) Garris, P. A. Soc. Neurosci. Abstr. 1997, Abstract no. 274.14. (17) Levine, I. N. Physical Chemistry; McGraw-Hill: New York, NY, 1978; p 442. (18) Nicholson, C.; Rice, M. E. In Volume Transmission in The Brain; Fuxe, K., Agnati, L. F., Eds.; Raven Press: New York, NY, 1991; p 279. (19) Kawagoe, K. T.; Garris, P. A.; Wiedemann, D. J.; Wightman, R. M. Neuroscience 1992, 51, 55-64.

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can be approximated by the volume fraction of the external medium occupied by the traps, φv (∼4 × 10-4 for the geometry in Figure 1). So, the following expression was used to relate kr to ∆t:

∆t ) -

ln(1 - φr φv ) kr

(5)

It should be noted, however, that eq 5 only specifies the kinetic control of the reaction, which may also be rate-limited by diffusion to the traps (see below). If a particle was determined to be inside a trap, then a random number between 0 and 1 was used to decide if the particle was consumed by the reaction. If the random number was smaller than φr, then the particle was counted as trapped and was removed from the diffusing ensemble. The minimum value of φr was set so that the rms diffusion distance was at least one-half the edge length of the traps, which allowed the particles to diffuse throughout the traps during a single iteration. The value of φr was varied during tests of the stability of the random-walk algorithm, but was otherwise set to 1 to give the quickest possible simulations. Simulation of Convection. Particles in the flow channel were subject to convective flow toward the probe outlet. The z-position of particles in the flow channel was adjusted in the direction of flow by the quantity υ•∆t, where υ is the average linear flow velocity. Here, as before,10,14 parabolic velocity profiles were replaced with an average linear velocity because the flow channel is thin and the flow velocity is small. In most cases, the flow velocity was 0.05 cm/s, which is approximately the value used in this laboratory.10 For purposes of comparison, however, some simulations were conducted with infinitely fast flow so that particles were recovered instantly when they entered the flow channel. We assumed that all flow was confined to the flow channel. SIMULATION RESULTS Accuracy and Stability of the Random-Walk Algorithm. The accuracy of the random-walk algorithm was evaluated by comparing simulated results with available analytical solutions (Figures 2 and 3). Simulations were conducted with several values of kr and φr and with the ensemble initially positioned sufficiently far from the probe such that the particles never encountered the probe or the passive zone. Figure 2 compares the simulated time evolution of the number of diffusing particles with the analytical solution for a first-order reaction, Nt ) N0e-kr∆t, where Nt is the number of particles remaining at time t and N0 is the initial number of particles in the ensemble. At small values of kr the time evolution of the number of particles matches well with the analytical solution, but as kr increases the simulated time course begins to lag behind the analytical solution. The lag is due to the onset of a diffusion limitation of the overall reaction, which is a two-step process involving diffusion to a reaction site (i.e, a trap) followed by the reaction itself. Since the steps occur in series, the overall rate constant, k, is20

k)

krkd kr + kd

(6)

Figure 2. Comparison of the rate of decay of the number of diffusing particles in random-walk simulations (symbols) with the exponential decay expected for homogeneous first-order kinetics (lines). The simulation was carried out with active tissue occupying all space around the release site. The symbols show the average value from simulations performed in triplicate. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; φv, 4.22 × 10-4; φr, 0.25 (circles); φr, 0.5 (up triangles); φr, 1 (down triangles).

where kd is the rate constant for the diffusion step in the reaction. A value for kd can be estimated from D/δ2, where D is the diffusion coefficient and δ is a diffusion distance.21 Combining the diffusion coefficient used here, 2.4 × 10-6 cm2/s,18 with the half-distance between the traps, ∼1 µm, gives a kd of ∼240 s-1. This value is sufficiently large compared to the kr values of interest here that diffusion limitation of the reaction rate is expected to play only a slight role, which is consistent with Figure 2. Figure 2 confirms the appropriateness of eq 5 for the simulation of the first-order trapping reaction. Figure 3 compares the three-dimensional distribution of particles, again in the absence of the probe, with the analytical solution to Fick’s law of diffusion. For this comparison the x, y, and z coordinates were divided into small bins and the number of particles in each bin were counted during each iteration of the simulation. The analytical distribution was calculated with the following expression:22

Nd,t )

{

}

N0 ∆+d ∆-d + erf erf exp{- krt} 2b x 2 Dt 2xDt

(7)

where Nd,t is the number of particles at time t in the bin centered at distance d from the point of origin, N0 is the initial number of particles in the simulation, “erf” is the error function, ∆ is the quantity x6D∆t, and b is the number of bins per distance ∆, which was 2 in this case. The comparisons in Figure 3 were generated with kr ) 1 and φr ) 1; similar agreement was obtained for all other values of kr and φr examined. The random-walk results are scattered about the analytical solution, which is as expected since the analytical solution applies in the limit of an infinitely (20) Levine, I. N. Physical Chemistry; McGraw-Hill: New York, NY, 1978; p 492. (21) Albery, W. J. Electrode Kinetics; Clarendon Press: Oxford, UK, 1975; p 128. (22) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, UK, 1975; p 15.

Figure 3. Comparison between the simulated (symbols) and analytical (lines) particle distribution in the x (circles), y (squares) and z (triangles) directions after 100 (top panel), 300 (middle panel) and 500 (bottom panel) ms of diffusion. The simulation was carried out with active tissue occupying all space around the release site. The symbols show the average value from simulations performed in triplicate. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; kr, 1 s-1; φv, 4.22 × 10-4; φr, 1.

large population of diffusing particles. Figure 3 confirms that the random-walk algorithm used here produces results consistent with Fick’s laws of diffusion. Simulations were conducted in which an ensemble of particles diffused out of a trap located in the vicinity of a microdialysis probe. Figure 4 shows an example of the time evolution of the number of freely diffusing particles, the number of particles recovered by the microdialysis probe, and the number of trapped particles. The simulation was repeated with different values of φr to examine the stability of the results. Although analytical solutions are not available in this case, Figure 4 demonstrates the stability of the simulation algorithm as reflected by the reproducibility of the simulation results when the ∆t and φr were varied. Here, ∆t was varied over almost 2 orders of magnitude (∼4 × 10-5 to ∼2 × 10-3 s). The stability of the results evident in Figure 4 provides a clear indication of the reliability of the random-walk algorithm for simulations of microdialysis. Particle Distributions near the Surface of a Microdialysis Probe. The random-walk algorithm provides insight into complex mass transport processes by permitting animation of the motion of the individual particles. Figure 5 shows snapshots from one such animation. Figure 5 reports the axial (panels A and C) and radial (panels B and D) distributions of the particles still remaining 200 ms after an ensemble began to diffuse from a trap near the Analytical Chemistry, Vol. 72, No. 9, May 1, 2000

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Figure 4. A random-walk simulation of microdialysis showing the time evolution of the number of free, recovered, and consumed particles. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; kr, 1 s-1; φv, 4.22 × 10-4; φr, 0.25 (circles); φr, 0.5 (up triangles); φr, 1 (down triangles); release site positioned at the boundary between the active and passive zones; inner cannula radius, 75 µm; flow channel thickness, 5 µm; passive zone thickness, 5 µm; probe outlet was 10 µm downstream from the release site; perfusion velocity, 0.05 cm/s.

Figure 6. Examples of the stochastic trajectories followed by individual particles during their lifetimes. The top panels show the trajectories of two particles that were recovered by the microdialysis probe, i.e., they eventually crossed the boundary (labeled “e”) that defines the probe outlet. The lower panels show the trajectories of two particles that eventually underwent the spatially nonuniform uptake reaction in the active tissue zone. The particles shown here spent part of their lifetime in the active tissue zone, part in the passive zone, and part in the flow channel. The radial position of each particle was calculated from the x and y position using the Pythagorean theorem. The labeling of regions with lower case letters corresponds to the labeling scheme in Figure 1. Simulation variables were the same as those given in Figure 5, except that the perfusion velocity was 0.05 cm/s in all cases.

Figure 5. The distribution of free particles 200 ms after the start of diffusion for the cases of rapid (panels A and B) and slow (panels C and D) perfusion. The distribution in the axial (panels A and C) and radial (panels B and D) directions are reported. In panels A and C, the radial position of each particle was calculated from the x and y position using the Pythagorean theorem. The labeling of regions with lower case letters corresponds to the labeling scheme in Figure 1. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; kr, 1 s-1; φv, 4.22 × 10-4; φr, 1; release site positioned at the boundary between the active and passive zones at the point (0,0) in the (y,z) plane; inner cannula radius, 75 µm; flow channel thickness, 5 µm; passive zone thickness, 5 µm; probe outlet was 50 µm downstream from the release site; perfusion velocity in C and D, 0.05 cm/s.

microdialysis probe. Panels A and B report the distribution when the perfusion rate was sufficiently rapid that particles were recovered instantly upon entering the flow channel. Panels C and 2046 Analytical Chemistry, Vol. 72, No. 9, May 1, 2000

D report the distribution obtained when the perfusion velocity was 0.05 cm/s. Comparison of Figure 5A with 5C shows that, if the time required for particles to reach the probe outlet is long in comparison to the time needed to diffuse the width of the flow channel, then an obvious distortion of the axial particle distribution occurs. While en route to the probe outlet, particles have an opportunity to diffuse back out of the flow channel and maybe out of the passive zone as well. This process, which will henceforth be called back-extraction, is significant in that it affords particles additional encounters with traps in the external medium. Figure parts B and D show that the axial perfusion process, as expected, does not affect the radial particle distribution. Figure 6 shows examples of the stochastic trajectories followed by four individual particles released from a trap positioned at the boundary between the passive zone and the external medium, i.e., as close as possible to the probe. The top panels are the trajectories of two particles that were eventually recovered by the probe, while the bottom panels are the trajectories of particles that were eventually trapped in the external medium. These trajectories clearly show that individual particles spend time

Figure 7. The total number of recovered particles when the simulations were carried out until 95% of the particles released were either recovered or consumed. The source location is the distance in the x direction from the outer boundary of the passive zone. In panels A and C, each line is the result for source locations of 0 to 20 µm (top to bottom) at 2-µm increments. In panels B and D, each line is the result for a different value of kr: 0 (filled circles), 0.25 (open circles), 1 (filled triangles), 3 (open triangles), 5 (filled squares), and 10 s-1 (open squares). The symbols show the average value from simulations performed in triplicate. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; φv, 4.22 × 10-4; φr, 1; release sites positioned at the point (0,0) in the (y,z) plane; inner cannula radius, 75 µm; flow-channel thickness, 5 µm; passive-zone thickness, 5 µm (panels A and B) and 30 µm (panels C and D); perfusion velocity was sufficiently rapid that particles were recovered immediately upon entering the flow channel.

diffusing both in the probe and in the external medium, regardless of whether they are eventually recovered or eventually trapped. Hence, Figures 5 and 6 show that the recovery process is affected by a reaction that consumes analyte in the external medium even though the reaction is not functional in the passive zone. Recovery and Extraction as Functions of the Diffusion Source Location, Clearance Kinetics, and Perfusion Velocity. Figure 7 reports the number of particles recovered with a passive zone thickness of 5 µm (Figure 7A and 7B) and 30 µm (Figure 7C and 7D), as a function of the reaction rate constant, kr (Figure 7A and 7C) and as a function of the location of the diffusion source (Figure 7B and 7D). The location of the diffusion source is expressed as the distance in the x direction from the outer boundary of the passive zone. The perfusion velocity was sufficiently rapid that particles were recovered immediately upon diffusing into the flow channel. The number of recovered particles decreased as the reaction rate constant increased, as the distance between the diffusion source and the probe increased, and as the width of the passive zone increased. Overall, Figure 7 shows that the number of recovered particles decreased when the time required for diffusion to the probe increased relative to the time scale of the clearance reaction. Hence, the clearance reaction acts to prevent particles from reaching the probe. This exacerbates the tendency of the back-extraction event to prevent the particles from reaching the probe outlet even if they do reach the probe (Figures 5 and 6). Figure 8 compares how changes in the reaction rate constant and the width of the passive zone affect the recovery (top panel)

and extraction (bottom panel) processes. Increasing the width of the passive zone leads to a decrease in both the number of particles recovered from the external medium and the number of particles extracted from the probe. From this it appears that masstransport phenomena have symmetric effects on the extraction and recovery processes. On the other hand, decreasing the rate constant of the trapping reaction leads to an increase in the number of recovered particles but a decrease in the number of extracted particles, which appears to confirm that kinetic phenomena can create asymmetry in the extraction and recovery processes. Hence, Figure 8 theoretically confirms that the extraction and recovery values may not be equal under conditions, such as those simulated here, where analyte mass transport is coupled to chemical events. Figure 9 shows how the number of recovered particles depends on the axial and radial position of the diffusion source relative to the probe outlet. This simulation was performed with a perfusion velocity of 0.05 cm/s. Figure 9, therefore, includes the combined effects of back extraction (Figure 5) and the tendency of the clearance process to prevent diffusion of particles to the probe (Figure 7). As the distance between the diffusion source and the probe outlet increases, in either the axial or radial direction, the fraction of particles recovered decreases. Under the conditions simulated here, only sources located short (micrometer) distances from the probe outlet contribute significantly to the number of recovered particles. Particles released too far from the probe outlet are not recovered but rather are trapped either while on their way to the probe or after back-extraction from the probe. Analytical Chemistry, Vol. 72, No. 9, May 1, 2000

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recovery value estimated in that work, 0.005, is 2-3 orders of magnitude less than previous estimates based on the slope of concentration differences plots, i.e., the extraction fraction. Figure 9 suggests that, as a consequence of the actions of the dopamineuptake mechanism, dopamine is only recovered from a very small annular volume of brain tissue located just at the very tip of the microdialysis probe. Figures 5-8 predict that the volume of tissue from which dopamine is recovered will increase upon uptake inhibition, which provides a theoretical explanation for the increase in dopamine recovery that we observed after giving animals drugs that inhibit dopamine uptake.10

Figure 8. The effect of the dimension of the passive zone and the rate constant of the chemical reaction on the number of particles recovered by (top panel) and extracted from (bottom panel) the microdialysis probe. In the top panel, the diffusion source was placed 10 µm from the outer boundary of the passive zone, while in the bottom panel the diffusion source was at the center of the flow channel. The sources were 50 µm in the axial direction from the probe outlet. The symbols show the average value from simulations performed in triplicate. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; φv, 4.22 × 10-4; φr, 1; inner cannula radius, 75 µm; flow channel thickness, 5 µm; perfusion velocity, 0.05 cm/s.

Figure 9. The influence of the axial and radial location of the diffusion source on the number of particles recovered under slow perfusion conditions. The symbols show the average value from simulations performed in triplicate. Simulation variables: N0, 5000; D, 2.4 × 10-6 cm2/s; kr, 5 s-1; φv, 4.22 × 10-4; φr, 1; release sites positioned at the point (0,0) in the (y,z) plane; inner cannula radius, 75 µm; flow channel thickness, 5 µm; passive zone thickness, 5 µm; perfusion velocity, 0.05 cm/s.

Figure 9 provides a theoretical explanation for the very small dopamine recovery values estimated on the basis of simultaneous in vivo voltammetry and microdialysis measurements.10d The 2048 Analytical Chemistry, Vol. 72, No. 9, May 1, 2000

DISCUSSION The model presented above provides theoretical insight into several aspects of the in vivo microdialysis of dopamine recently brought to light by the use of voltammetric electrodes placed in brain tissue nearby microdialysis probes.10 That work produced unexpectedly small estimates of the in vivo dopamine recovery value and suggested that the recovery value increases after animals are given drugs that inhibit the dopamine uptake mechanism. The model was therefore formulated to focus attention on the question of whether a reaction that consumes analyte in the medium external to the probe can create asymmetry in the extraction and recovery processes. Consequently, the model is a simplified rendition of an actual in vivo microdialysis experiment. It is necessary to mention, therefore, the main simplifications that have been invoked. In the model, there is a discrete boundary between the passive zone and the external medium. Such a discrete boundary might exist if the passive zone contained only the dialysis membrane and the tissue adjacent to the membrane were “normal”. Recent histological analysis has reported, however, that the tissue surrounding a microdialysis probe is traumatized in such a way as to produce a gradient of biological activity extending several hundred micrometers from the probe.23 Since the details of this gradient in terms of the kinetics of dopamine release and uptake have not yet been elucidated, it is not yet possible to include such details in models of in vivo microdialysis. The model does not consider the possibility that analyte transport by blood flow might be coupled to the extraction and recovery processes. Dopamine, however, being charged at physiological pH and being subject to rapid clearance from the extracellular fluid, is unlikely to cross the blood-brain barrier to any substantial extent. Mass transport to and from the blood compartment is relevant, however, to the microdialysis of pharmaceutical agents.24 In the present model, a first-order reaction consumes analyte in the external medium, whereas dopamine uptake follows Michaelis Menten kinetics. This simplification allows the diffusing particles to be treated as noninteracting random walkers. This is not possible with Michaelis Menten kinetics because the uptake rate would be concentration-dependent. Thus, the rate of dopamine uptake at any given location would be dependent on the contributions to the local dopamine concentration from all other dopamine release sites. To meaningfully include this feature into a model requires highly detailed information about the spatio(23) Clapp-Lilly, K. L.; Roberts, R. C.; Duffy, L. K.; Irons, K. P.; Hu, Y.; Drew, K. L. J. Neurosci. Methods 1999, 90, 129-142. (24) Song, Y.; Lunte, C. E. Anal. Chim. Acta 1999, 379, 251-262.

temporal features of ongoing dopamine release and uptake in the brain; such information is not currently available. The use of firstorder rather than Michaelis Menten kinetics should not alter the main conclusion of this work, i.e., that consumption of analyte in the external medium can create asymmetry between the extraction and recovery processes. CONCLUSION This work has focused on the impact of a clearance reaction in the external medium on the microdialysis recovery process. Understanding the recovery process is central to the neurochemical interpretation of analytical results obtained by in vivo microdialysis. Until recently,10 direct information about in vivo recovery values was not available, so the quantitative interpretation of in vivo microdialysis results has relied heavily on the measured value of the extraction fraction. This strategy requires, however, that the extraction fraction and relative recovery be equal under the experimental conditions at hand. Recent evidence has emerged, however, to suggest that in the case of dopamine the in vivo relative recovery of dopamine is 2-3 orders of magnitude less

than the extraction fraction. This, in turn, suggests that the resting extracellular concentration of dopamine is 2-3 orders of magnitude higher than current nanomolar estimates. Hence, we conclude that extracellular dopamine levels in the striatum are actually in the micromolar range, which is in accordance with the affinity of the D1 receptors found in this part of the rat brain.8 Recent evidence has also emerged to suggest that drugs that inhibit the dopamine uptake mechanism can change the relative recovery and, hence, the analytical performance of the probes.10 The simulations reported here also provide insight into the mechanism by which dopamine uptake inhibitors may change the dopamine recovery value. ACKNOWLEDGMENT This work was supported by NIH (NS 31442).

Received for review October 14, 1999. Accepted February 7, 2000. AC991186R

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