Journal of Neurochemistry, 2003, 86, 932–946 doi:10.1046/j.1471-4159.2003.01904.x
Microdialysis of dopamine interpreted with quantitative model incorporating probe implantation trauma
Peter M. Bungay,* Paige Newton-Vinson, Wanda Isele,* Paul A. Garris and Joseph B. Justice, Jr
*Division of Bioengineering & Physical Science, National Institutes of Health, DHHS, Bethesda, Maryland, USA Department of Chemistry, Emory University, Atlanta, Georgia, USA Department of Biological Science, Illinois State University, Normal, Illinois, USA
Abstract identified by histology and proposed to distort measurements Although microdialysis is widely used to sample endogenous of extracellular DA levels by the no-net-flux method. To and exogenous substances in vivo, interpretation of the address this issue, an existing quantitative mathematical results obtained by this technique remains controversial. The model for microdialysis was modified to incorporate a trau- goal of the present study was to examine recent criticism of matized tissue layer interposed between the probe and sur- microdialysis in the specific case of dopamine (DA) meas- rounding normal tissue. The tissue layers are hypothesized to urements in the brain extracellular microenvironment. The differ in their rates of neurotransmitter release and uptake. A apparent steady-state basal extracellular concentration and post-implantation traumatized layer with reduced uptake and extraction fraction of DA were determined in anesthetized rat no release can reconcile the discrepancy between DA uptake striatum by the concentration difference (no-net-flux) micro- measured by microdialysis and voltammetry. The model pre- dialysis technique. A rate constant for extracellular clearance dicts that this trauma layer would cause the DA extraction of DA calculated from the extraction fraction was smaller than fraction obtained from microdialysis in vivo calibration tech- the previously determined estimate by fast-scan cyclic vol- niques, such as no-net-flux, to differ from the DA relative tammetry for cellular uptake of DA. Because the relatively recovery and lead to an underestimation of the DA extracel- small size of the voltammetric microsensor produces little lular concentration in the surrounding normal tissue. tissue damage, the discrepancy between the uptake rate Keywords: dopamine, implantation trauma, in vivo microdi- constants may be a consequence of trauma from microdialy- alysis, in vivo voltammetry, striatum. sis probe implantation. The trauma layer has previously been J. Neurochem. (2003) 86, 932–946.
Microdialysis is widely used in neuroscience, pharmacology In particular, the manner and the degree to which the trauma and medicine for in vivo monitoring of endogenous and might compromise interpretation of microdialysis measure- exogenous substances. The technique is an invasive procedure ments are generally unclear. requiring the implantation into target tissue of probes that are Regulation of brain dopamine (DA), a neurotransmitter relatively large by comparison to dimensions of individual involved in the functions of cognition, motivation and motor cells and intercellular space. Trauma induced in the brain by control (Le Moal and Simon 1991; Schultz 1998), has been probe implantation is characterized by both short- and long- extensively studied by microdialysis (Robinson and Justice term histological, physiological and biochemical changes in 1991). Despite relative uniformity in the practice of the nearby neural tissue (Benveniste and Diemer 1987; Benveniste et al. 1987; Westerink and De Vries 1988; Ruggeriet al. 1990; Shuaib et al. 1990; Robinson and Camp 1991; Allen et al. Received January 15, 2003; revised manuscript received May 2, 2003; 1992; Camp and Robinson 1992; Georgieva et al. 1993; accepted May 6, 2003. Fumero et al. 1994; de Lange et al. 1995; Westergren et al. Address correspondence and reprint requests to Peter M. Bungay, NIH/DBEPS, Building 13/3N17 MSC 5766, Bethesda, MD 20892-5766, 1995; Morgan et al. 1996; Grabb et al. 1998; Groothuis et al. USA. E-mail: [email protected] 1998; Clapp-Lilly et al. 1999). The consequences of Abbreviations used: aCSF, artificial cerebrospinal fluid; DA, dopam- probe-associated trauma, however, are not well understood. ine; ECS, extracellular space; FSCV, fast scan cyclic voltammetry.
932 2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 Trauma model in dopamine microdialysis 933
microdialysis technique, various theoretical models used to drop in DA concentration of 485 nM (Kulagina et al. 2001). interpret results have led to predicted steady-state levels of Although the electrochemical technique employed, fast scan extracellular DA differing by three orders of magnitude cyclic voltammetry (FSCV), is not capable of determining an (Lindefors et al. 1989; Benveniste and Huttemeier 1990; absolute basal level of DA, it is well suited for monitoring Justice 1993). Failure to account properly for probe-induced concentration differences occurring over short sampling times trauma may also contribute to the wide range of concentration (Garris and Wightman 1995). Thus, evidence obtained by an estimates. Indeed, microdialysis sampling of DA would be alternative analytical method to microdialysis suggests that the particularly sensitive to the trauma layer adjacent to the probe extracellular DA concentration in the striatum greatly exceeds due to the high affinity neuronal transporter clearing released the low nanomolar estimates of the no-net-flux method. neurotransmitter from extracellular space (ECS) (Horn 1990; The present study extends the mathematical model previ- Giros et al. 1996). In general, the amount of tissue that ously developed to provide a quantitative relationship donates analyte to the dialysate depends inversely upon the between dialysate and extracellular concentration in steady- avidity of extracellular clearance mechanisms such as cellular state microdialysis (Bungay et al. 1990). The revised model uptake, interstitial catabolism or efflux to the blood. For DA as applies to general situations in which the mechanisms for well as any analyte rapidly cleared from interstitium, the analyte supply and removal are abnormal in a thin layer of condition of the tissue in close proximity to the probe is tissue adjacent to the probe. By distinguishing between these therefore of primary concern. Dopamine neurotransmission functional characteristics in the abnormal layer and the appears compromised in the trauma layer, as evidenced by the surrounding tissue, the model clarifies the difference between incomplete decrease in dialysate DA levels after perfusing the extraction fraction (determinable from dialysate measure- probe with a medium containing tetrodotoxin or depleted in ments) and relative recovery (potentially inaccessible by calcium (Westerink and De Vries 1988). Delaying the dialysate measurements alone). collection of dialysate for assay to allow constancy of In the present case, the modified theory uses the extraction indicators such as neurotransmitter concentrations, further- fraction from no-net-flux measurements to calculate a rate more, is not suitable as conditions may not represent normalcy constant for DA clearance. This clearance rate constant is found but rather a different quasi-steady state or a compensatory to be smaller than the one determined from FSCV for cellular adaptation to injury (Robinson and Camp 1991). uptake (Garris et al. 1994a). Because the small voltammetric The dominant approach in the last decade for estimating microsensors produce minimal tissue damage (Allen et al. steady-state levels of extracellular DA in the brain is the so 2001), discrepancies between the two clearance measurements called concentration-difference or no-net-flux measurement might arise from various factors, such as tissue damage related (Justice 1993). In this method, the analyte concentration to probe implantation, accumulation of fluid ultrafiltered perfused into the probe (inflow) is varied, and the difference through the probe membrane or formation of an abnormal between the inflow and outflow analyte concentration is tissue layer as a reaction of the tissue to the probe. Given that plotted versus the inflow concentration. Theory predicts that measurements to be reported are restricted to acute experiments the inflow concentration resulting in the zero concentration- in anesthetized rats, the most probable cause of local abnor- difference point indicates what the analyte extracellular mality is trauma. Consistency with the rate constant measured concentration would have been in the absence of the probe by voltammetry is achieved by invoking the hypothesis that, as (Bungay et al. 1990). Another prediction is that the extraction a consequence of implantation trauma, DA release is abolished fraction, which describes the exchange of DA between the and uptake is reduced in the traumatized tissue layer. probe and the tissue, equals the relative recovery of DA from the tissue under both steady-state (Bungay et al. 1990) and transient conditions (Morrison et al. 1991; Chen et al. 2002). Methods The low nanomolar (5–10 nM) concentrations in the striatum yielded by the no-net-flux approach is thought to reflect the Probe construction high efficiency and density of the DA transporter (Justice Two fused silica tubes (100 lm o.d.; 40 lm i.d., Polymicro 1993). However, in an elegant series of studies, Michael and Technologies, Inc., Phoenix, AZ, USA) were inserted into a plastic coworkers recently proposed that a trauma layer devoid of DA connector fitted with a guide cannula (Plastics One, Roanoke, VA, USA). While viewing under a microscope, the ends of the silica release and uptake capacity causes a difference between the tubes were placed such that one (outflow tube) was flush with the extraction fraction and relative recovery such that the extra- end of the guide cannula and the other (inflow tube) extended either cellular DA level is underestimated (Lu et al. 1998; Peters and 1, 2, 3, or 4 mm beyond the end of the guide cannula. The tubes Michael 1998; Yang et al. 1998). Strong evidence was were secured to the connector with Superglue. The guide cannula provided for compromised release, but uptake was not was removed and a cellulosic membrane with a 13-kDa molecular evaluated. The same research group has additionally demon- weight cutoff (220–240 lm o.d.; Spectrum Medical, Houston, TX, strated that focal application of a glutamate antagonist reduced USA) plugged at one end with polyimide resin (Alltech, Deerfield, a real-time voltammetric microsensor signal equivalent to a IL, USA) was placed over the two silica tubes and sealed to the
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 934 P. M. Bungay et al.
^ silica tubes with the polyimide resin. The membrane, as well, was À PÁAo=Qd Evivo ¼ 1 À e ; ð2Þ coated with resin down to the end of the outflow tube such that the area available for diffusion was limited to the region between the in which P^ is an overall permeability of the probe and tissue to the end of the outflow tube and the resin plug at the probe tip. The DA, Ao is the outer surface area of the membrane available for DA functional length of the membrane corresponded approximately to exchange and Qd is the perfusate volumetric flow rate. The overall the displacement between the ends of the silica tubes plus the gap permeability is composed of contributions from the probe and the between the inflow tube and the plug. At least 24 h was allowed for tissue that are interrelated in the usual sum of permeability curing of the polyimide resin before probe use. reciprocals for transport layers in series, In vivo microdialysis 1 1 À1 P^ ¼ þ : ð3Þ All in vivo experiments were performed using microdialysis probes Pp Pt with a 4-mm nominal active length. Male Wistar rats (275–350 g;
Harlan Sprague-Dawley Inc., Indianapolis, IN, USA) were anesthe- In the above, Pp is the probe permeability whose in vitro evaluation tized with 400 mg/kg of chloral hydrate administered intraperiton- is described in the next section. Pt is the permeability of the tissue. eally and placed in a stereotaxic apparatus. The guide cannula was The permeabilities are related to the mass transport resistances, R, implanted in the striatum using coordinates AP + 2.7 mm, ML employed previously (Bungay et al. 1990) by the general equation ) 2.7 mm from bregma and DV ) 2.7 mm from dura; incisor bar at
+5.0 mm (Pellegrino et al. 1979) and secured with cranioplastic P^ Á Ao ¼ 1=R: ð4Þ cement. The probe was inserted and perfusion with artificial cerebrospinal fluid (aCSF, 145 mM NaCl, 2.8 mM KCl, 1.2 mM One advantage of formulating the model in terms of permeabil-
CaCl2, 1.2 mM MgCl2, 0.25 mM ascorbic acid, 5.4 mMD-glucose, ities instead of resistances is that the former are much less dependent pH 7.2–7.4) was begun at 0.2 lL/min using a 500-lL gas-tight upon the size and other geometric aspects of the probe than the syringe (Hamilton, Reno, NV, USA) and either a Harvard Model latter. Other than the conversion from resistances to permeabilities, 2274 (Harvard Apparatus, Holliston, MA, USA) or KD Scientific equations 1–3 are of the same form as those previously obtained for Model 100 syringe pump (KD Scientific, New Hope, PA, USA). steady-state microdialysis in the absence of implantation trauma After 1 h the perfusion rate was gradually stepped up to 0.6 lL/min. (Bungay et al. 1990). An important difference from the previous After 1 h at this flow rate, sampling at intervals of 10 min was begun. expressions is in the appearance of the apparent extracellular app The perfusate concentration was then varied randomly between 100, concentration, Ce , in place of the normal spatially averaged 1 200, and 400 nM DA in aCSF. At least three samples at each extracellular concentration far from the probe, Ce . concentration were collected. Samples were stored on dry ice until The magnitude of the tissue permeability to an analyte, such as analysis. Rats were given 0.1 mL of a 400-mg/mL chloral hydrate DA, is determined by three quantities: the tissue extracellular volume solution as determined by response to tail-pinch. After the experi- fraction (/e), the diffusion coefficient for DA in the ECS (De), and ment, rats were given a lethal dose of chloral hydrate and perfused the rate constant for clearance of DA from the ECS (ke). Each of the intracardially with saline followed by 10% formalin. The brain was three quantities could vary spatially as a result of implantation harvested and sliced in 50-lm sections using a freezing microtome. trauma. Dykstra et al. (1992) estimated that the interstitial volume Probe placement in the striatum was confirmed from these slices. fraction in rat striatum is enlarged to about 0.35 for radial distances at All experiments were carried out according to the National least 1.5 mm from the probe during at least the first few hours Institute of Health Guide for the Care and Use of Laboratory following probe implantation. Since the exchange of DA with the Animals, using procedures approved by the Emory University probe is likely to occur over tissue distances more than an order of Institutional Animal Care and Use Committee. Efforts were made to magnitude shorter, the volume fraction is assumed to be uniformly minimize animal suffering, and reduce the number of animals used. enlarged. A uniform DA interstitial diffusion coefficient is likewise The steady-state DA concentrations measured in the outflow assumed. However, the model proposes that the avidity of DA out clearance will differ between the traumatized and normal tissue dialysate samples, Cd and the measured inflow perfusate concen- in in regions. The combined contribution of the two regions is represented trations, Cd (including the initial Cd ¼ 0), were plotted as in out in in the model by an apparent clearance rate constant, kapp. With these Cd À Cd against Cd . The mathematical model derived in the e Appendix for the effects of implantation trauma on microdialysis assumptions, the model suggests that the relationship between these in parameters and tissue permeability can be approximated by predicts that, for sufficiently small Cd , the plot should be described by the linear equation p app Pt ¼ /e ðDe Á ke Þ: ð5Þ Cin À Cout ¼ðCin À CappÞÁE : ð1Þ d d d e vivo This expression implicitly presumes that the extracellular concen-
in trations, Ce, are much less than the Michaelis–Menten constant for Whereas Cd is the supply concentration for exogenous DA in the app uptake, Km. perfusate, Ce is an apparent extracellular concentration providing the driving force for diffusion of endogenous DA towards the probe. In vitro probe characterization When these two concentrations are equal, there should be no net A flow cell used for the probe characterization is indicated exchange of DA between the perfusate and the tissue. This ‘point of schematically in Fig. 1. The cell held the probe concentrically in a no-net-flux’ x-axis intercept from the plot is then Capp, sometimes e 1/8 inch i.d. stainless steel flow channel through which aCSF was previously denoted by C . According to the model, the slope, nnf recirculated by a variable-speed Waters HPLC pump (Waters Corp., designated as the DA extraction fraction in vivo, is given by
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 Trauma model in dopamine microdialysis 935
Milford, MA, USA) from a 500-mL reservoir maintained at 37 C. A monobasic, 25 mM citric acid, 13% methanol (v/v), 0.1 mM screw-drive syringe pump (Harvard) delivered aCSF containing DA disodium ethylenediamine tetraacetate (Na2EDTA) and 1 mM octyl to the probe at the same flow rate of 0.6 lL/min employed to perfuse sodium sulfate, and was delivered with an ISCO LC-5000 syringe the probe internally in the in vivo experiments. The steady-state DA pump (ISCO, Lincoln, NE, USA). Dopamine was detected with out concentration in the outflow dialysate, Cd , was determined at flow- either an EG&G Princeton Applied Research (Model 400, Oak channel mean velocities, vext, of 0.1, 0.2, 0.4 and 4 cm/s produced by Ridge, TN, USA) or Bioanalytical System (Model LC-4C) pumping the external medium at volumetric flow rates of 0.42, 0.85, amperometric detector using a working and a reference electrode 1.7, and 17 mL/min. Extraction fractions for DA loss from the (Model RE1, West Lafayette, IN, USA) from Bioanalytical Systems perfusate were calculated as Inc. The applied potential was + 0.7 V versus Ag/AgCl. Dialysate samples were manually injected using a 10-lL Hamilton syringe. E Cout Cin vitro ¼ 1 Àð d = d Þ; ð6Þ Dopamine concentration was calculated using an external standard
in curve composed of the peak heights resulting from injection of for fixed inflow concentration of Cd ¼ 100 nM DA. A linear aliquots from the perfusate solutions of 0, 100, 200, and 400 nM regression of the values as a function of 1/vext was extrapolated to ws DA. the well-stirred limit, Evitro for 1/vext ¼ 0. Probes were used that had displacements between the inflow and In vitro outflow tube openings of 1, 2, 3 and 4 mm. The 4-mm probe batch Conditions were similar to those above except a CMA 200 was different than that employed in the in vivo experiments. The microsampler (CMA/Microdialysis, N. Chelmsford, MA, USA) predicted logarithmic dependence of Ews upon membrane length vitro maintained at 5 C with a 2-lL injection loop was used. Chroma- can be written as (Bungay et al. 1990) tograms were collected and quantitated using a Shimadzu Class VP
ws Pp Á Ao chromatography data acquisition program. À ln½1 À Evitro¼ : ð7Þ Qd
In the above, the outer surface area of the membrane is Results Ao ¼2pÆroÆlm, in which ro is the membrane outer radius and lm ¼ ln + la is the effective membrane length available for In vitro probe characterization exchange. The nominal length, ln, is based on the offset between the ends of the side-by-side inner tubes. The functional membrane Analysis of the in vivo measurements requires knowledge of length can exceed the nominal length by an additional length, la. the transport characteristics of the probe in the form of the permeability, Pp, evaluated with the aid of the in vitro system in Fig. 1. First, DA extraction fractions in vitro were Chromatography measured and found to be sensitive to the mean velocity of In vivo the external fluid flowing past the probe (Fig. 2). The Dialysate samples were analyzed for DA concentration using ws extraction fraction values (Evitro) at the well-stirred limit microbore HPLC with electrochemical detection using a 10-cm · 0.5-mm column packed with a 5-lm C18 stationary phase (Alltech) and a sample injection loop of either 0.5 or 1.0 lL. The pH 3.7 mobile phase consisted of 28.5 mM sodium phosphate
perfusate reservoir in Qd Cd flow cell
vext
out Qd Cd probe dialysate collection vial external medium recirculation loop
constant temperature Fig. 2 Determination of well-stirred in vitro dopamine extraction frac- ws tion, Evitro, by extrapolation. The points indicate extraction fraction
Fig. 1 Schematic diagram for the characterization of microdialysis values, Evitro, calculated according to the definition, equation 6, from probes in vitro. The dopamine concentrations in the perfusate and measured values of the inflow and steady-state outflow concentra- in out in out dialysate are indicated by Cd and Cd and the volumetric flow rate of tions, Cd and Cd , for various external solution mean velocities, vext. ws these solutions by Qd. The probe is held concentrically in a stream of The intercept at 1/vext ¼ 0 obtained by linear regression is Evitro.The ws j external medium pumped axially along the probe membrane at a mean nominal membrane lengths and Evitro (mean ± SE) values are: , velocity of mext. The probe and external medium are maintained at 37 C 1 mm, 0.244 ± 0.002, n ¼ 4; d, 2 mm, 0.405 ± 0.009, n ¼ 4; m, by immersion in a temperature controlled water bath. 3 mm, 0.529 ± 0.017, n ¼ 4; ., 4 mm, 0.604 ± 0.007, n ¼ 4.
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 936 P. M. Bungay et al.
Fig. 4 In vivo dopamine extraction fraction, Evivo, and apparent Fig. 3 Determination of the length of membrane available for dop- extracellular concentration, Capp, obtained by the concentration dif- amine exchange by regression of the well-stirred extraction fraction e ference (no-net-flux) technique. Probes implanted in anesthetized rat intercepts, Ews , from Fig. 2 plotted as a function of the nominal vitro striatum were alternately and randomly perfused with artificial cere- membrane length, ln. brospinal fluid containing 0, 100, 200 or 400 nM dopamine. According
to equation A20, Evivo is the coefficient of proportionality between the in ) out were obtained as the y-axis intercepts corresponding to perfusate concentration difference for inflow and outflow, Cd Cd , and the inflow value, Cin. Each point is a mean ± SE from five rats and extrapolation of the external solution velocity, vext,to d in out the slope and intercept at Cd ¼ Cd by linear regression are infinity. The intercept values, in turn, were plotted as app E 0.52 ± 0.01 and C 13 ± 13 nM, respectively, and ) ws vivo ¼ e ¼ –ln [1 E ] against the nominal membrane lengths, ln app vitro E ¼ 0.54 ± 0.10 and C ¼ 16 ± 7 nM if the 400 nM data are (Fig. 3), yielding a straight line in accordance with equa- vivo e excluded. tion 7. The x-axis intercept of ) 0.35 mm indicates that the length of the membrane accessible for DA exchange was where /e ¼ 0.35 is taken from the measurement for actually la ¼ 0.35 mm longer than the nominal value for anesthetized rat striatum surrounding an acutely implan- each probe that was derived from the difference in positions 2 ted probe (Dykstra et al. 1992), and De ¼ Dd/k ¼3.2 · of the ends of the inflow and outflow tubing. The probe )6 2 10 cm /s is calculated from the free solution value of Dd ¼ permeability (mean ± SE) calculated by applying equation )6 2 ws 7.6 · 10 cm /s and a tortuosity of k ¼ 1.54 (Rice 2000). 7 to the four Evitro values is Pp ¼ (2.92 ± 0.05) · 10)4 cm/s.
In vivo dopamine clearance Discussion The concentration differences plot for Cin concentrations from d Clearance rate constant 0 to 400 nM (Fig. 4) yielded a slope (mean ± SE) of Microdialysis measurements represent spatial averages over Evivo ¼ 0.52 ± 0.04 and an intercept at the point of no net app a tissue volume determined by the accessible length of the flux of Ce ¼ 13 ± 13 nM. The corresponding values for the in membrane and the distance over which the analyte has to narrower range of Cd ¼ 0–200 nM are Evivo ¼ 0.54 ± 0.10 app diffuse through the tissue to participate in exchange with the and C ¼ 16 ± 7 nM. From equation 2, the overall per- e perfusate. For this study, the value calculated for the apparent meability value for Evivo ¼ 0.54, outer radius, ro ¼ 0.12 mm, rate constant for DA clearance from rat striatum ECS, and effective in vivo membrane length, lm ¼ 4.35 mm, is ) ) app 1 4 ke ¼ 5s , is an average over axial distances (> 4 mm) P^ ¼ ) (Qd/Ao)Æln[1 ) Evivo] ¼ 2.4 · 10 cm/s. From equation 3, the tissue permeability is that are of the order of the transverse dimension of this tissue. This value differs from estimates obtained by FSCV for DA À1 clearance following electrically stimulated release. Small 1 1 À3 Pt ¼ À ¼ 1:4  10 cm/s; ð8Þ FSCV microdisk electrodes (glass-insulated carbon fibers P^ Pp with a 10-lm diameter) detect variation in uptake rates with electrode position indicative of considerable spatial hetero- and from equation 5, the DA apparent clearance rate constant geneity. Larger microcylinder electrodes of the order of is 200 lm in length yield signals that are much less sensitive to position (Lu et al. 1998), suggesting spatial averaging app 2 À1 ke ¼ðPt=/eÞ =De ¼ 4:9s ; ð9Þ similar to that for microdialysis probes. To obtain a spatial
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 Trauma model in dopamine microdialysis 937 mean over a distance comparable to the length of the microdialysis probe, we averaged values from 95 FSCV DA clearance rate measurements taken at different dorsoventral positions in the rat striatum (Garris et al. 1994a). This yielded a mean ± SE of Vm ¼ 3.0 ± 0.2 lM/s (range, 0.3– 8 lM/s). A rate constant can be obtained by dividing Vm by the Michaelis–Menten Km for the DA transporter. Reported values for Km in the range of 0.13–0.4 lM (Nicholson 1995) would correspond to spatial-average rate constants of )1 ke ¼ 7.5–23 s . The most commonly cited value of )1 Km ¼ 0.2 lM gives a value of ke ¼ 15 s . Correcting the FSCV DA clearance rates for diffusional distortion or peak Fig. 5 Geometry for proposed model of microdialysis with a trau- stimulated extracellular DA concentrations would likely matized layer of tissue immediately adjacent to the probe membrane. increase this ke estimate. We hypothesize that the discrepancy The radial distance from the probe axis is denoted by r and the axial between the microdialysis apparent rate constant for DA distance from the inlet end of the membrane is z. The inner and outer clearance and the average FSCV value primarily results from radii of the membrane (m) are ri and ro, respectively, and the thickness a reduction in the local DA clearance rate in the tissue that is of the trauma layer (tr) is d. The surrounding tissue layer (n) is of traumatized by the microdialysis probe implantation. effectively semi-infinite extent. A number of factors could contribute to the magnitude of the discrepancy between the microdialysis and voltammetry DA clearance values. FSCV may overestimate the basal rate Although albumin does not extravasate locally if adminis- constant in normal striatum because the elevated extracellular tered intravenously more than 30 min after probe insertion DA levels elicited by medial forebrain bundle stimulation (Dykstra et al. 1992), increased barrier permeability to small during FSCV could augment uptake rates through autore- molecules and 70 kDa dextran is reported to persist up to ceptor stimulation (Meiergerd et al. 1993; Dickinson et al. 4 weeks after probe insertion in the brain (Groothuis et al. 1999; Schmitz et al. 2002). The microdialysis estimate 1998). Dialysate levels of DA (Westerink and De Vries depends on several parameters not measured in the current 1988), and probably other brain constituents, are transiently experiments. An important parameter is the extracellular elevated following probe implantation. Initial changes in volume fraction, /e, as can be inferred from the inverse ECS DA following implantation could result from mechan- app square dependence of ke upon /e in equation 9. The value ical disruption and loss of intraneuronal DA to the ECS. for /e was estimated to be 0.35–0.4 for the tissue extending Residual intact neurons might rapidly clear DA that appears at least 1.5 mm from microdialysis probes acutely implanted in the ECS by direct damage. However, alterations in the in rat striatum (Dykstra et al. 1992). This is considerably chemical composition of the ECS could, as well, impair the higher than normal striatal values of about /e ¼ 0.2 function of intact neurons with respect to their avidity of DA estimated from other techniques (Patlak and Fenstermacher release and uptake. The mathematical model presented in the 1975; Rice and Nicholson 1991) indicating edema formation Appendix was formulated to simulate the effect of impaired as one major consequence of probe implantation. The analyte supply and removal processes confined to a thin alteration of the extracellular volume has a direct influence traumatized layer of tissue adjacent to the probe (region ‘tr’ app on the microdialysis calculation of ke and also indirect in Fig. 5). The parameters describing the rate of analyte effects. A direct influence is through the appearance of /e in supply and removal in the trauma layer and the surrounding app equation 9 that arises, in part, from defining ke on the basis tissue (region ‘n’) are independent variables that allow of extracellular volume. The second source of the appearance various possibilities to be explored. The possibilities are of /e in equation 9 is the restriction of DA diffusion through examined in the present context in terms of reduced DA tissue to the interstitial space. The variation of the DA release and uptake in the trauma layer. Impairment of interstitial diffusion coefficient, De, with /e has been different supply and removal mechanisms in the trauma layer neglected in the calculations. Tortuosity measurements in could occur for other analytes. rat cortical slices subjected to osmotic stress indicate that De for the low-molecular-weight cation, tetramethylammonium, Trauma layer thickness increased only slightly as /e increased from 0.24 in normal Interpretation of the degree of reduction in DA uptake in the medium to 0.42 in a hyperosmolar bathing solution (Nichol- trauma layer is a function of the unknown trauma layer son et al. 2000). thickness, d. In determining the influence of d, we begin by In addition to the induction of edema, other traumatic setting the uptake rate constant for the surrounding normal effects have been associated with probe implantation. The tissue equal to the FSCV value for Vm ¼ 3.0 ± 0.2 lM/s and blood–brain barrier is disrupted in the vicinity of the probe. Km ¼ 0.2 lM,
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 938 P. M. Bungay et al.
tissues. However, it has been suggested that quantitative microdialysis strategies relying upon the measurement of a DA extraction fraction underestimate the extracellular DA levels in normal tissue (Lu et al. 1998; Peters and Michael 1998; Yang et al. 1998, 2000) for reasons related to spatial non-uniformity in DA release and uptake. Implantation trauma was proposed as one possible cause for the non- uniformity. The present analysis supports the contention that 1 an underestimation of Ce could be one of the consequences of trauma. Our model predicts that, in general, the apparent extracellular DA concentration from probe calibration tech- app 1 niques, Ce , differs from Ce . This inference applies not only to the no-net-flux technique, but also to other in vivo Fig. 6 Because of the dependence on the unknown microdialysis probe calibration techniques that rely upon dialysate con- trauma layer thickness, d, the rate constant for dopamine clearance in centration measurements. This includes techniques of retro- tr app the trauma layer, ke , can be any value between zero and ke ,the dialysis (Wang et al. 1993) and variation of perfusate flow apparent clearance rate constant obtained from the no-net-flux ana- rate (Sam and Justice 1996; Tang and Gonzales 2001). tr lysis. For d greater than 13 lm, ke is within 5% of the apparent According to the model, all of these techniques provide clearance rate constant, kapp. app 1 e estimates of Ce rather than the desired Ce . The difference app 1 app between Ce and Ce arises because Ce represents a weighted average of the extracellular concentrations in both n À1 ke ¼ Vm=Km ¼ 15 s ; ð10Þ the trauma layer and the surrounding normal tissue. The weighting is expressed explicitly in equation A18 of the which yields a value for the tissue permeability in this region Appendix, )3 2 from equation A12 of Pn ¼ 2.5 · 10 cm /s. The trauma app à 1 layer permeability, Ptr, can then be obtained by quadratic Ce ¼ð1 À wÞÁCe þ w Á Ce ; ð11Þ solution of equation A15, together with the tissue permeab- ws ility, Pt, in equation 8 evaluated from the Evitro and Evivo in which w is a weighting factor defined in equation A19 and measurements. Iteration was employed to determine the is constrained to be 0 < w £ 1. The first term on the right dependence of the uptake rate constant in the trauma layer, side of equation 11, representing the contribution from the tr à tr ke ,ond and the results are displayed in Fig. 6. Any trauma trauma layer, contains the virtual concentration, Ce ¼ Str=ke , layer thickness greater than about 4 lm satisfies the interrelating the release rate, Str, and the clearance rate tr 1 n constraints imposed by the measurements and assumed constant, ke , in the trauma layer. Similarly, Ce ¼ Sn=ke is parameter values. For trauma layer thickness in the approxi- formed from the balance of the normal tissue release rate, Sn, n mate range, 4 lm
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 Trauma model in dopamine microdialysis 939 abnormal tissue and the probe from the surrounding normal tissue. Repetition of the electrical stimulation produced strong responses in the electrodes at all three locations in the presence of an inhibitor of the DA transport, nomifensine. This indicates that diffusion of DA through the abnormal tissue was possible. The lack of response to the first stimulation could have been the result of DA clearance in the abnormal tissue between the probe and viable DA release sites in the surrounding tissue, contrary to the presumption of Yang et al. Thus, in place of a passive layer lacking both release and uptake as suggested by Peters and Michael (1998) and Yang et al. (2000), our trauma model retains the possibility of clearance in the traumatized tissue, albeit with reduced avidity. Fig. 7 The discrepancy between the apparent extracellular dopamine Since neurons are responsible for both DA release and app concentration from the no-net-flux intercept, Ce , and the true value uptake, it might be anticipated that loss of neurons in the 1 far from the probe, Ce , increases as the trauma layer thickness trauma layer would eliminate both processes. The absence of increases. The same dependence on d applies to the discrepancy release in the probe vicinity suggests that impulse flow is between the true relative recovery, R, and the in vivo extraction frac- interrupted or ineffective in evoking DA exocytosis. Some tion, Evivo. For this illustrative calculation from equation 12, release is à neuronal damage mechanisms might lead to abolition of assumed to be abolished in the trauma layer (Ce ¼ 0). release without destruction of the neuron. For example, propagation of action potentials can be blocked by mechan- the model, the degree of underestimation is a strong function ical strain of axons. On the other hand, some level of DA of the unknown thickness of the trauma layer as shown by clearance could be conserved in the DA nerve terminals that the curve in Fig. 7 generated from equation 12, together with remain in the trauma layer. The process of DA uptake is Appendix equations A15–A17 and A19. For example, if the simpler and likely to be more robust, since it can be observed trauma layer thickness were 20 lm, then Fig. 7 predicts that 1 in synaptosomal preparations and in non-neuronal cells Ce would be about 150 nM for the value of app engineered to express the transporter. Enzymatic degradation Ce ¼ Cnnf ¼ 16 nM. Greater trauma layer thicknesses on and loss to blood across microvessels should still exist and the order of 30 lm might be able to reconcile the large could even be enhanced by trauma. DA uptake without discrepancy between reported DA Cnnf values and the basal 1 release would have implications for the maintenance of ECS value of Ce ¼ 485 nM indirectly estimated for DA in intracellular DA degradation. the rat striatum (Kulagina et al. 2001). However, the model The model simulates the observations of Yang et al.by in the present form is restricted by linearity assumptions to à setting Ce ¼ 0 in equation 11 to reflect the abolition of Ce values much less than Km ¼ 0.2 lM. The linearity of the in release in the trauma layer. The clearance rate constant in the data in Fig. 4 up to Cd of 400 nM suggests that non-linearity app trauma layer could be any value between zero and ke , in the combined DA clearance mechanisms is weak, even depending on the trauma layer thickness, as indicated in above the assumed Km value. Except for the computational Fig. 6. As noted above, for our measurements and assumed complexity incurred by introducing non-linear concentration parameter values, the model predicts the trauma layer dependencies, the range of validity of the model could be 1 thickness to be at least 4 lm. For no release in the trauma extended to higher Ce values, if the high Ce values are app 1 layer, the relationship predicted between Ce and Ce , verified. equation 11 reduces to Dissociation of extraction fraction from relative recovery app 1 Ce ¼ Cnnf ¼ w Á Ce : ð12Þ The above results can be viewed alternatively in terms of a difference between two measures of probe performance: The limiting value of w ¼ 1 corresponds to a trauma layer extraction fraction and relative recovery. The latter term will lacking both release and clearance. However, as noted above, be used in the restricted context of sampling endogenous DA, the observations of Yang et al. suggest some DA clearance out 1 in exists in the trauma layer requiring that w < 1. Clearance in R ¼ Cd =Ce for Cd ¼ 0: ð13Þ the trauma layer creates a barrier for diffusion to the probe of endogenous DA released in the surrounding normal tissue. This is the calibration factor that would be desirable for As a consequence, the intercept in the no-net-flux studies calculating the DA concentration in normal tissue ECS from 1 would underestimate the magnitude of Ce as previously the measurable levels of DA in the sampling mode probe suggested (Peters and Michael 1998; Yang et al. 1998). From effluent. The analysis predicts instead that all of the usual
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à tr 1 n in vivo ‘calibration’ procedures yield Evivo rather than R.Ifan layer (Ce ¼ Str/ke ¼ 0) is different than Ce ¼ Sn/ke in the abnormal layer lacking release is the traumatic consequence normal tissue. If the regulation of DA concentration were of probe implantation, then these two calibration factors are preserved in the trauma layer by a proportional change in the tr n discordant in the same proportion as the concentration ratio release rate and clearance rate constant, i.e. Str/ke ¼ Sn/ke , app 1 in Fig. 7. According to equation 12 and Appendix then Ce ¼ Ce and Evivo would not differ from R. equation A24, One limiting case in which equality of Evivo and R would hold is if both release and uptake were abolished, i.e. the app 1 R=Evivo ¼ Ce =Ce ¼ w: ð14Þ trauma layer was passive. This is illustrated by the point on tr the curves in Figs 6 and 7 at which Str ¼ ke ¼ 0 and w ¼ 1 Arguments have been previously presented that such a corresponding to a 4-lm thin passive layer. Thus, a passive distinction should be maintained between extraction fraction layer will not lead to a difference between Evivo and R under and relative recovery (Lu et al. 1998; Peters and Michael 1998; steady-state conditions. 1 Yang et al. 1998), as well as between Ce and the no-net-flux app intercept concentration, Cnnf ¼ Ce . Although there is con- Uptake inhibition decreases extraction fraction while siderable similarity in the underlying reasoning, there are increasing relative recovery important differences between this report and these prior Evivo and R will not be the same whenever the balance between analyses. First, the model presented in the Appendix is not analyte supply and clearance in the trauma layer differs from specific to DA or neurotransmitters. Second, the model is that in the surrounding tissue. For the DA analysis in which formulated in a cylindrical geometry appropriate to microdi- release is assumed absent in the trauma layer, R < Evivo. alysis probes, so the expressions developed from it have Michael and colleagues (Lu et al. 1998; Peters and Michael quantitative relevance to the interpretation of microdialysis 1998; Yang et al. 1998) have provided evidence that uptake measurements. Third, no artificial distinction is made between inhibition should cause an increase in R, while producing a endogenous and exogenous analyte, so that consistent bound- decrease in Evivo. Simulations from the trauma model support ary conditions are imposed on both the sampling of analyte this proposition. Values of Evivo and R have been calculated from the tissue by the perfusate and the delivery of analyte to for proportionately decreased values of the trauma and normal the tissue from the perfusate. As a result, the expressions tr n tissue uptake rate constants, ke and ke , to simulate the effect of obtained for the extraction fraction apply to both sampling and varying the degree of DA uptake inhibition. The results, delivery modes. There is no asymmetry between these plotted as a function of the fractional reduction in the rate operations, despite the spatial inhomogeneity in tissue prop- constants, are displayed in Fig. 8. The curves exhibit the erties that is introduced through subdividing the tissue into two predicted opposing trends in Evivo and R, for most of their distinct layers. Thus, the symmetry between sampling and range. The exception is that the trend in R reverses at levels of delivery extraction arises from the assumptions of linearity in nearly complete uptake inhibition. Another significant aspect analyte concentration dependence in the governing equations of the simulations that support experimental observations and boundary conditions, not from an assumption of uniform- (Parsons et al. 1991; Olson and Justice 1993; Smith and ity in tissue properties. No-net-flux measurements in a linear Justice 1994) is that uptake has to be nearly completely Cin Cout system produce straight lines when d À d is plotted versus inhibited to reduce Evivo by 50% from its value in the absence in Cd . The absence of a change in slope about the no-net-flux of inhibition (right-hand ordinate intercept). Cin Cout Capp intercept at d ¼ d ¼ e illustrates the sampling and The Evivo and R curves would converge to the same delivery extraction symmetry predicted by the model. This minimum, pure diffusion value in the limit of complete underlying symmetry of microdialysis as a process is not inhibition of clearance from ECS. The simulation employed inconsistent with a discrepancy between different measures of for Fig. 8 assumes diffusion occurs only in the radial probe performance, such as Evivo and R. direction. For purely radial diffusion, the values of Evivo By applying a consistent treatment of sampling and and R approach zero in the limit of no clearance. In actuality, delivery, the model provides a cogent explanation for this the common limiting value Evivo and R and would be non- distinction between extraction fraction and relative recovery, zero because axial diffusion through the ECS becomes namely that they are based on potentially different extracel- significant as the analyte penetration depth increases. Addi- app lular concentrations: Evivo relates to Ce from equation 1, tionally, other clearance mechanisms, such as chemical conversion and efflux to blood, have been neglected in the in out Cd À Cd simulations for Fig. 8. Evivo ¼ in app ; ð15Þ Cd À Ce The strong influence of trauma layer thickness on relative recovery is further illuminated by examining the associated 1 whereas R is defined in equation 13 in terms of Ce . In the influence on the axially averaged (Ææ) extracellular DA present analysis Evivo differs from R because the ratio of the concentration profiles shown in Fig. 9. For the trauma layer DA release rate, S, to uptake rate constant, ke, in the trauma thickness of d ¼ 10 lm, about one-quarter of the overall
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1 concentration drop from hCe i to ÆCdæ occurs across the probe membrane and three-quarters within the tissue ECS. How- ever, for d ¼ 30 lm, the concentration drop across the probe membrane would be only a small fraction of the total, while the majority of the drop appears across the tissue layers.
Some assumptions An implicit assumption in the analysis is that the tissue remains in intimate contact with the probe. It is possible that an additional resistance layer might be interposed, such as a coagulum formed from the disruption of blood vessels or dialysate fluid ultrafiltered through the membrane. As noted above, a passive 4-lm diffusion layer could account for the Fig. 8 If dopamine release is abolished in the layer of traumatized difference between the apparent clearance rate constant from tissue while uptake continues to occur throughout both this layer and microdialysis and the value from voltammetry. Since no the adjacent normal tissue, the extraction fraction, Evivo, exhibits a supply or removal processes exist in a passive layer, there is monotonic reduction as uptake is progressively inhibited. However, the no concentration gradient in the absence of a flux (Peters and true relative recovery, R, based on the dopamine extracellular con- Michael 1998). Consequently, passive layers would not centration in the normal tissue increases as the degree of uptake interfere with the determination of the extracellular concen- inhibition increases (except in range of nearly complete inhibition). tration in the surrounding tissue by a steady-state no-net-flux Uptake inhibition is simulated by proportionate reduction in both the procedure. However, such a thin layer could not explain the normal and trauma layer uptake rate constants, kn and ktr, respect- e e absence of evoked release in the vicinity of the microdialysis ively. The abscissa is the fraction reduction in these rate constants. In probe (Yang et al. 1998). the absence of trauma, the curve for R would become the same as that The assumption that the trauma is confined to a discrete for Evivo and likewise exhibit a monotonic decrease with uptake inhi- bition. For these illustrative curves, an arbitrary trauma layer thickness layer of abnormal tissue is probably an unrealistic idealiza- of d ¼ 20 lm was chosen. tion that has been employed to simplify the analysis. The deficit in DA release and uptake may well be spatially variable, which would imply that the distance over which the trauma extends is greater than that represented by the trauma layer thickness parameter. It is unclear whether our predic- tion that these functional deficits are confined to a thin region for acutely implanted probes is discordant with evidence of ultrastructural abnormalities at considerable distances from the probe observed more than 40 h post-implantation (Clapp- Lilly et al. 1999). The trauma model treats each tissue region as a medium of uniform properties by invoking spatial averaging of the microscale variations. For DA the length scale for the spatial averaging would be of the order of the average separation distance between release sites. The density of dopaminergic terminals in the striatum leads to an estimate for the average spacing between terminals of the order of 2–4 lm (Doucet et al. 1986; Garris et al. 1994b). This spacing between Fig. 9 Predicted radial profiles in endogenous dopamine extracellular release sites in normal striatum is similar to a characteristic concentration produced by microdialysis sampling (Cin 0) are d ¼ distance for diffusion of DA from the release sites of Ö(De/ strongly dependent upon the unknown thickness of the trauma layer. ke) 5 lm. This calculation assumes uptake is uniformly Profiles in the dialysate, probe membrane fluid and ECS of striatal distributed and employs the approximate average DA uptake tissue have been calculated from equations A27–A31 for d ¼ 10, 20 app rate constant estimated from voltammetry measurements, and 30 lm and the values, Ce ¼ 16 nM and Evivo ¼ 0.54. Concen- )1 ke ¼ 15 s , and the estimate for the DA diffusion coeffi- trations are normalized with respect to the distant undisturbed extra- )6 2 1 cient in the ECS of De ¼ 3.2 · 10 cm /s. This measure of cellular concentration , Ce . Axial-averaging over the membrane length is indicated by the angle brackets. The extracellular concentration at the DA diffusion distance in striatum is similar in magnitude the probe-tissue and trauma layer–normal tissue interfaces are to other conservative estimates: 9 lm (Rice 2000) and 7 lm o d denoted by Ce and Ce , respectively, and the radially averaged con- (Gonon et al. 2000). These measures provide support for the centration in the dialysate by Cd. continuum assumptions invoked by the model and the use of
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1 Ce as a characteristic spatially averaged concentration in the would not provide conclusive indication of the presence or distant normal tissue. Thus, the discrete nature of release and absence of trauma. uptake sites does not play a role in the model. Conclusions Importance of probe characterization in vitro Regardless of the degree of trauma, quantitative interpretation We have modified the theoretical framework for microdial- of the microdialysis measurements requires careful charac- ysis to allow for the existence of a tissue layer adjacent to the terization of the probe in vitro. The present simulations have probe in which the rates of analyte supply and clearance been based on the relatively small difference in extraction processes are abnormal. We have utilized the model to ws fraction measurements between Evitro ¼ 0.61 ± 0.01 in vitro explore the possibility that probe implantation traumatically and Evivo ¼ 0.54 ± 0.10 in vivo. In this regard, it is instruct- alters the local rates of DA release and uptake. The apparent ive to note that the model predicts that in the absence of a consequences for DA microdialysis would be a lack of trauma layer with the uptake rate constant set to the FSCV equality between extraction fraction and relative recovery )1 value of ke ¼ 15 s , the in vivo extraction would be and an underestimation of the normal tissue DA extracellular Evivo ¼ 0.57. It would be difficult experimentally to discrim- concentration. Besides the implications for the effects of inate between this in vivo value and the in vitro measurement. implantation trauma on the release and uptake of other Tang and Gonzales (Tang and Gonzales 2001; Tang et al. neurotransmitters, the model has additional potential uses. 2003) performed DA no-net-flux measurements in the One extension is underway to explore other implantation nucleus accumbens of freely behaving rats the day after trauma effects, such as alterations in the local rates of analyte probe implantation and found no statistically significant exchange between blood and tissue as suggested by reports difference between in vivo and stirred in vitro values. The of persistent modification in blood–brain barrier permeability reduction in the DA apparent clearance rate constant estima- (Morgan et al. 1996; Groothuis et al. 1998). ted in the present study may be indicative of a greater degree of trauma with acutely implanted probes. 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microdialysis probes in the rat hippocampus with silver alterations with a continuous and diffuse spatial variation, for degeneration staining. Neurosci. Lett. 112, 149–154. simplicity the trauma layer is assumed to be thin with Smith A. D. and Justice J. B. Jr (1994) The effect of inhibition of uniform properties that are distinct from those of the synthesis, release, metabolism and uptake on the microdialysis extraction fraction of dopamine. J. Neurosci. Methods 54, 75–82. surrounding tissue. Tang A. M. and Gonzales R. A. (2001) Extraction fraction for dopamine The model is formulated in cylindrical coordinates with r in vivo and in vitro: implications for detecting changes in dop- representing radial distance from the probe axis and z amine uptake, in 9th International Conference on in Vivo Methods representing axial distance from the inlet end of the (O’Connor W. J., Lowry J. P., O’Connor J. J. and O’Neill R. D., membrane. As indicated in Fig. 5, the inner and outer eds), pp. 109–110. University College Dublin, Dublin, Ireland. Tang A., Bungay P. M. and Gonzales R. A. (2003) Characterization of surfaces of the membrane are located at radial positions, ri and probe and tissue factors that influence interpretation of quantitative ro, respectively, and the thickness of the trauma layer is d. The microdialysis experiments for dopamine. J. Neurosci. Methods length of membrane accessible for diffusional exchange 126, 1–11. between perfusate and tissue is lm. The diffusional permeab- Wang Y., Wong S. L. and Sawchuk R. J. (1993) Microdialysis calibration ility of the perfusate and membrane will be combined into a using retrodialysis and zero-net flux: application to a study of the distribution of zidovudine to rabbit cerebrospinal fluid and thal- probe permeability, Pp, defined as the proportionality coef- amus. Pharm. Res. 10, 1411–1419. ficient between the flux of analyte into the probe and the Westergren I., Nystrom B., Hamberger A. and Johansson B. B. (1995) concentration difference driving the diffusion, Intracerebral dialysis and the blood–brain barrier. J. Neurochem. o 64, 229–234. Jp ¼ Pp ÁðCe À CdÞ: ðA1Þ Westerink B. H. and De Vries J. B. (1988) Characterization of in vivo dopamine release as determined by brain microdialysis after acute The diffusive flux, Jp, is the mass flow rate of analyte into the and subchronic implantations: methodological aspects. J. Neuro- probe per unit area of membrane outer surface. The driving chem. 51, 683–687. Yang H., Peters J. L. and Michael A. C. (1998) Coupled effects of mass force is given by the difference between the extracellular o transfer and uptake kinetics on in vivo microdialysis of dopamine. analyte concentration at the probe outer surface, Ce , and the J. Neurochem. 71, 684–692. dialysate concentration, Cd. The flux and the concentrations Yang H., Peters J. L., Allen C., Chern S. S., Coalson R. D. and Michael vary with the axial location, z, but probe permeability is A. C. (2000) A theoretical description of microdialysis with mass assumed to be uniform. The flux into the probe increases the transport coupled to chemical events. Anal. Chem. 72, 2042–2049. dialysate concentration along a differential length, dz, according to the balance, Appendix dC Q d ¼ 2p Á r Á J ; ðA2Þ d dz o p Mathematical model of steady-state microdialysis with probe implantation trauma in which Qd is the perfusate flow rate. Diffusion in the axial The principal effects of probe implantation trauma are direction is neglected. assumed to occur in a concentric layer of abnormal tissue The flux, Jp, is determined by a combination of the probe interposed between the probe and surrounding normal tissue. and tissue permeabilities. Solving mass balances for the two The mathematical model to be developed is an extension of tissue layers will yield expressions for the tissue permeabil- one previously proposed to describe microdialysis in a single ities. For simplicity, the spatially averaged rate of analyte tissue region of uniform properties (Bungay et al. 1990) on supply from sources other than the probe will be assumed the spatial scale relevant to current microdialysis probes uniform, but different, in each layer and independent of the whose membrane diameters and lengths are greater than local analyte concentration. The non-diffusional local rate of 0.2 mm and 1 mm, respectively. In these models the local analyte removal per unit volume of extracellular space (ECS) variations in properties and concentrations associated with will be described by the product, keÆCe, in which ke is the rate the individual cells and discrete sites of analyte supply and constant for clearance by all processes other than diffusion, removal, such as blood vessels and synapses, are spatially and Ce is the local analyte concentration in the ECS. Thus, Ce averaged, since associated length scales separating these sites is assumed to be sufficiently below the Michaelis–Menten are on the order of 0.05 mm or smaller. A number of Km value for any saturable clearance mechanisms, such as simplifications have been employed, such as the linearity in cellular uptake or microvascular efflux transporters. The analyte concentration dependence to describe analyte clear- value of the clearance rate constant is assumed to be uniform, ance processes. These have been described in the earlier but different, within the two layers. As yet the model neglects presentations (Bungay et al. 1990; Morrison et al. 1991). mechanisms for regulating analyte supply and removal rates. Few additional assumptions are invoked in adding the Since the trauma layer is assumed to be thin, the curvature of traumatized tissue layer. The types of trauma to be modeled the layer will be neglected. With these simplifications, the are alterations in the rates of analyte supply and removal steady-state ECS mass balances for the analyte in the ECS processes in this layer. Although it is possible to model these are of the form,
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 Trauma model in dopamine microdialysis 945
2 tr tr trp tr tr o C Ptr ¼ D Á / =Ctr ¼ / ðD Á k Þ; ðA11Þ trauma layer : 0 ¼ Dtr e À ktr Á C þ S ; ðA3Þ e e e e e e or2 e e tr o o and n 1 Ce n surrounding tissue : 0 ¼ De r À ke Á Ce þ Sn; r or or np n n Pn ¼ /e ðDe Á ke ÞÁK1½ro=Cn=K0½ro=Cn: ðA12Þ ðA4Þ
The modified Bessel functions of the second kind, K0 and K1, with De denoting the analyte diffusion coefficient in the each with dimensionless argument, ro/Gn, appear in equa- ECS and S denoting the supply rate per unit volume of ECS. tion A12 as a consequence of the cylindrical geometry. The The balances are to be solved subject to a number of ratio, K1/K0, approaches unity for large values of the constraints. Far from the probe the diffusional term in argument. equation A4 vanishes as the concentration approaches a The flux, Jp, can be equivalently expressed as the product uniform level of C1 determined by the balance of supply and e of an overall permeability, P^, and the tissue-to-perfusate removal rates, concentration difference driving analyte diffusion into the probe. The apparent DA extracellular concentration for the C1 ¼ S =kn: ðA5Þ e n e trauma layer and normal tissue composite will be denoted by Capp. The desired expression analogous to equation A1 is Analogously in the trauma layer, the supply and removal e processes are associated with a potential steady-state con- ^ app à Jp ¼P ÁðCe À CdÞ: ðA13Þ centration Ce , defined by
à tr The overall permeability for diffusion through the probe and C ¼ Str=k : ðA6Þ e e composite tissue in series is given by the inverse of the sum d of reciprocals of the probe and tissue permeabilities, At the interface between the two tissue layers, Ce ¼ Ce in both layers and the analyte flux across the interface is 1 1 À1 P^ ¼ þ : ðA14Þ oC oC Pp Pt J /tr Dtr e /n Dn e ; d ¼ e Á e Á o ¼ e Á e Á o r r ¼ r þ d r x ¼ r þ d o À o þ In the above, the tissue permeability, P , is a composite of the ðA7Þ t permeability properties of the two tissue layers, in which d– and d+ indicate evaluations in the trauma and À1 Ptr Á sh Pn Á sh surrounding tissue layers, respectively, at the interface, Pt ¼ Pn Á 1 þ 1 þ : ðA15Þ Pn Á ch Ptr Á ch r ¼ ro + d. At the membrane–tissue interface the flux leaving the tissue is the same as that entering the probe, Equation A15 was obtained by solving the tissue balances oC (equation A3 and equation A4) together with their boundary J ¼ /tr Á Dtr Á e : ðA8Þ p e e or conditions. For conciseness the hyperbolic functions have r ¼ ro been abbreviated by In the perfusate at the inlet and outlet ends of the membrane, sh ¼ sinh½d=Ctr and ch ¼ cosh½d=Ctr: ðA16Þ the analyte concentration is indicated by By analogy to equations A11 and A12, if the interstitial in out Cd ¼ Cd at z ¼ 0; and Cd ¼ Cd at z ¼ lm: volume fractions and diffusion coefficients are the same in tr n tr n ðA9Þ the two tissue layers (/e ¼ /e ¼ /e and De ¼ De ¼ De ), then Pt can be expressed in terms of an apparent clearance app Microdialysis induces spatial variations in extracellular rate constant, ke , concentration. The length scales over which the concentra- p app tion varies are characterized by ‘penetration depths’, Pt ¼ /e ðDe Á ke Þ: ðA17Þ
p tr tr p n n Ctr ¼ ðDe =ke Þ and Cn ¼ ðDe =ke Þ: ðA10Þ The apparent extracellular concentration of the analyte is given by The ease of permeation of each layer is given by a app à 1 permeability, Ce ¼ð1 À wÞÁCe þ w Á Ce : ðA18Þ
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946 946 P. M. Bungay et al.
In equation A18, w is a dimensionless weighting factor, application of the model is neither restricted to dopamine or other neurotransmitters, nor to situations in which analyte 1 w ¼ : ðA19Þ supply is lacking in the trauma layer. ch Á 1 þ PtrÁsh The axial variation in perfusate concentration is obtained PnÁch by integrating the perfusate balance (A2) combined with equation A13 from z ¼ 0toz, Since ch ‡ 1, equation A19 requires that 0 < w £ 1. Then, it follows from equation A18 that Capp must be intermediate e app in app À2pÁroÁP^Áz=Qd 1 à Cd½z¼Ce þðCd À Ce Þe : ðA26Þ between Ce and Ce . The factor, w, serves to weight the contributions of the two tissue layers to the apparent extracellular concentration. An appropriate mean perfusate concentration value is the Substituting equation A13 into the perfusate balance A2 axial-average, and integrating from z ¼ 0toz ¼ lm yields equation 1 of the Zlm Methods section, 1 app in app Qd Á Evivo hCdi¼ Cd½zdz ¼ Ce þðCd À Ce Þ : lm P^ Á Ao in out in app 0 Cd À Cd ¼ðCd À Ce ÞÁEvivo; ðA20Þ ðA27Þ in which Evivo is the extraction fraction in vivo that depends The concentrations in the tissue vary in both the r and z exponentially on P^, Qd and the membrane outer surface area, directions. A representative r-direction profile is, likewise, Ao, according to obtained by axial-averaging. This mean concentration profile
Evivo ¼ 1 À exp½ÀP^ Á Ao=Qd: ðA21Þ in the trauma layer, ro £ r £ ro + d,is
à o à In the context of no-net-flux experiments with a linearly hCei½r¼Ce þðhCe iÀCe ÞÁcosh½ðr À roÞ=Ctr d à o à behaving analyte, the concentration difference measurements þ½hCe iÀCe À ch ÁðhCe iÀCe Þ in ) out in ðA28Þ (Cd Cd ) are plotted as a function of Cd . Equation A20 sinh½ðr À r Þ=C Á o tr ; indicates that the resulting line has a slope of Evivo and a sh app point of no-net-flux intercept of Ce ¼ Cnnf. The above expressions hold for any value of Cin. For the d and in the surrounding tissue, r ‡ ro + d,is pure sampling mode, equation A20 simplifies to 1 d 1 hCei½r¼Ce þðhCe iÀCe iÞ Á K0½ðrÞ=C2=K0½ðro þ dÞ=C2: Cout ¼ Capp Á E for Cin ¼ 0: ðA22Þ d e vivo d ðA29Þ The true relative recovery for sampling, R, is defined as Equating the axial-averaged expressions for the flux into the probe, equations A1 and A13, gives a relationship for R ¼ Cout=C1 for Cin ¼ 0: ðA23Þ d e d calculating the probe interface concentration,
Combining equations A22 and A23 leads to a relationship of o 1 à à hJpi¼Pp ÁðhC iÀhCdiÞ ¼P^Á½w ÁðC À C ÞÀðhCdiÀC Þ; the form proposed by Yang et al. (1998) in connection with e e e e dopamine microdialysis, ðA30Þ
app 1 and the axial-average extracellular concentration at the R ¼ Evivo ÁðCe =Ce Þ: ðA24Þ interface between the two tissue layers is
Substituting equation A18 into the above yields the explicit o à 1 à d à Ptr ÁðhCe iÀCe ÞþPn Á sh ÁðCe À Ce Þ expression hCe i¼Ce þ : Ptr Á ch þ Pn Á sh à 1 R ¼ Evivo Á½w þð1 À wÞÁðCe =Ce Þ: ðA25Þ ðA31Þ
app Since 0 < w £ 1, R can be greater or less than Evivo and Ce 1 Ã can be greater or less than Ce depending upon whether Ce / 1 Ce is greater or less than unity. Expressions A18–A25 are the principal results of the model. The model is valid for any analyte satisfying the modeling assumptions. In particular,
2003 International Society for Neurochemistry, J. Neurochem. (2003) 86, 932–946