Computer-controlled Epidural Infusion to Targeted Cerebrospinal Fluid Concentrations in Humans: Clonidine

The study was approved by the Clinical Research Practices Committee; written informed consent was obtained; and seven volunteers reported to the General Clinical Research Center, Wake Forest University Medical Center, at 7:00 AM, having had nothing to eat or drink since midnight. A peripheral intravenous catheter was inserted for infusion of lactated Ringer's solution at 50-100 ml/h and a second intravenous catheter inserted and capped for sampling of venous blood. A 20-G single-distal-port epidural catheter was inserted 3-5 cm in the epidural space at the L2-L3 or L3-L4 interspace via an 18-G Tuohy needle and tested with injection of 5 ml 1.5% lidocaine. At the first interspace caudad to the epidural catheter, a 23-G single-distal-port catheter was inserted 3-5 cm in the intrathecal space through a ramped 19.5-G Sprotte tipped spinal needle. Volunteers were studied only after demonstration of bilateral sensory neural blockade consistent with an epidural injection of lidocaine and after complete resolution of blockade (minimum of 90 min from injection).

The following parameters were measured: blood pressure; heart rate; oxyhemoglobin saturation by pulse oximetry (SpO²); sedation and analgesia scored by visual analog scale (VAS); skin blood flow; and CSF sampling for clonidine, catecholamine, and ACh concentrations. After baseline measurements were obtained, epidural clonidine was infused to targeted concentrations in CSF of 25 ng/ml, then 50 ng/ml, then 75 ng/ml, and finally 150 ng/ml. Each infusion lasted 75 min. Immediately after the 60-min CSF sampling for each infusion period, measurements were obtained in the following order: SpO²; blood pressure; heart rate; VAS scores of sedation and analgesia; and skin blood flow. These measurements usually were completed within 10 min of the 60-min sampling of CSF.

CSF was sampled for clonidine and ACh analysis at 5, 15, 30, and 60 min after initiation of the infusion and after each change in the targeted concentration of the infusion. In each case the initial 0.5 ml CSF, representing 1.8 times the catheter dead space, was discarded, and 1.0 ml was obtained for clonidine and ACh analysis. At each 60-min point an additional 0.5 ml CSF was obtained for measurement of catecholamine concentrations. Blood was sampled before infusion and at each 60-min point during infusion for clonidine, catecholamine, and ACh analysis.

Blood pressure and heart rate were measured by a noninvasive oscillometric device at the same times as CSF sampling and immediately before each change in the targeted concentration. SpO²was measured before epidural infusion was begun and at the end of each targeted concentration infusion period as described above.

Volunteers were trained to perform an inspiratory gasp by using techniques previously described. [13]After a verbal command, volunteers inspired as quickly and deeply as they could and held the gasp for 4 s. They were instructed not to perform Valsalva's maneuver.

Digital skin perfusion was assessed by attaching a laser Doppler probe to the pulp on the long finger of one hand and the second toe of one foot and monitoring perfusion with a laser Doppler flowmeter (Dual Channel Laser Doppler flowmeter ALF 21D, Advance Electronics, Tokyo, Japan). The probe had a lightweight plastic head (type L probe) that was held on the finger and toe by double-sided adhesive disks. This flowmeter's laser probe transmitted coherent light at 780 nm, and the output of the flowmeter was sampled at a 0.1-s time constraint. The laser Doppler signal from each channel was recorded on a two-channel strip-chart recorder.

We made laser Doppler fluxmetric measurements before the epidural clonidine infusion, at the end of the 50-ng/ml infusion, and at the end of the 150-ng/ml infusion. During testing, each volunteer performed at least five inspiratory gasps. The changes in skin perfusion accompanying each inspiratory gasp were averaged. [14]The vascular responsiveness to the inspiratory gasp was calculated as a percentage change in cutaneous perfusion from baseline as previously described. [12]The skin was heated to 37 degrees Celsius with a regulated heating pad before Doppler flow measurements and the temperature confirmed by measurement with thermistors (YSI series 400, Yellow Springs Instruments, Yellow Springs, OH) taped to the finger and toe. Warming of the skin at the measurement site before reflex testing reduces variability in laser Doppler measurements within and between subjects. [15].

To assess analgesia, pain intensity was assessed by having the volunteers mark a 10-cm VAS after immersion of a foot, and 5 min later, a hand, in stirred ice water. A 60-s cutoff time was used, although volunteers were allowed to remove their hand or foot before this time if they experienced unbearable pain. The foot and hand contralateral to those used for skin blood flow measurements were used for analgesia testing. The volunteer's level of sedation was measured with a 10-cm VAS, anchored at "not drowsy at all" and "as drowsy as possible."

We modeled previously published data showing the CSF clonidine concentrations after epidural administration of a 700-micro gram clonidine bolus. [10]The model used was Equation 1where dose = 700 micro gram; C¹, C², and -(C¹+ C²) = the weighted coefficients; and lambda¹, lambda², and lambda³= the positive exponents of a triexponential relation relating a unit dose (i.e., a dose = 1 unit) at time 0 to the intrathecal concentration over time. The third exponential term has the coefficient -(C¹+ C²) instead of C³because at time 0 the concentration in the intrathecal space after administration of an epidural bolus is 0. This can happen only if each coefficient equals the negative of the sum of the other two coefficients.

The model was estimated by using the "solver" function in Excel version 4.0 spreadsheet software (Microsoft, Redmond, WA) and a spreadsheet incorporating the iteratively reweighted least-squares objective function: Equation 2. A pooled data approach was used in which n = the total number of samples in the study.

The observations from this study were used to estimate a second set of pharmacokinetic parameters to relate the dose given in this study to the measured concentrations. This second analysis was performed with the program MKMODEL, modified to incorporate the complex computer-controlled drug input function used, as previously described. [6,16].

Clonidine was infused epidurally in a concentration of 10 micro gram/ml with a pump (552275, Harvard, Quincy, MA) connected to a computer running the STANPUMP program on the MS-DOS (Microsoft) platform. STANPUMP incorporates a control strategy to achieve rapidly and then maintain a desired concentration at the site of observation when the site of observations is linked to the site of injection by a first-order transfer function. The control policy incorporated into STANPUMP is that described by Shafer and Gregg [17]and was implemented as suggested by Jacobs and Williams. [18].

STANPUMP was programmed with the pharmacokinetic constants estimated as reported above, transformed into V¹, k¹⁰, k¹², k²¹, and k^e0as described in the appendix. The algebraic manipulations described in the appendix were performed with the software program Mathematica version 2.2 enhanced (Wolfram Research, Champaign, IL). The transformation was performed because the STANPUMP program assumes that when a disequilibrium exists between the site of injection and the site of observation, there is a compartmental model for the plasma pharmacokinetics (described by V¹, k¹⁰, k¹², and k²¹) and a first-order transfer of drug from the plasma (the site of injection) to the effect site (described by k^e0). To take advantage of STANPUMP's ability to administer drug when disequilibrium exists between the site of drug injection and the site of observation, it was necessary to convert the disposition function of the intrathecal space, relative to epidural drug administration, estimated as described above, into a compartmental representation. STANPUMP, in turn, then computed the disposition function of the intrathecal space from the parameters V¹, k¹⁰, k¹², k²¹, and k^e0.

The measures of performance were based on the performance error (PE), defined for each observation as Equation 3. The performance error was based on the prediction rather than the target because in some settings the target and prediction may differ, as might occur if the pump is briefly interrupted while a target concentration is maintained.

Two summary measures were used to measure the aggregate trend of the weighted residuals: the median weighted performance error (MDPE), calculated as Equation 4which estimated the overall bias in the results, and the median absolute performance error (MDAPE), calculated as Equation 5which estimated the overall inaccuracy of the performance. These measures of performance are adapted from the recommendations of Varvel et al. [19]and have been previously used in prospective studies of computer-controlled drug delivery. [5-7].

The new pharmacokinetic parameter set derived from these data were measured for their ability to predict these observations as well. Because this second pharmacokinetic parameter set was derived from the same observations, errors in prediction are the "residual errors" from the pharmacokinetic analysis. Thus, we calculated the weighted residual precisely as we calculated the performance error, described above, except that the prediction was based on the new pharmacokinetic parameter set. We then calculated the median weighted residual and the median absolute weighted residual based on the weighted residual error, in a way exactly analogous to the calculations of the median weighted performance error and the median absolute performance error described above.

Clonidine was measured by Harris Laboratories (Lincoln, NE) with a radioimmunoassay with limit of sensitivity of 0.13 ng/ml. [10]Plasma clonidine concentrations below this limit were assigned a value of 0. Concentrations of catecholamines and ACh were measured by high-performance liquid chromatography with electrochemical detection as previously described. [20,21].

Clonidine hydrochloride, 100 micro gram/ml in preservative-free saline, was obtained from Fujisawa Pharmaceutical Co. (Deerfield, IL) under Investigational New Drug approval from the United States Food and Drug Administration and was diluted with preservative-free normal saline before epidural injection.

Unless otherwise indicated, data are presented as means plus/minus SEM. The effect of infusion on continuous variables, including VAS measurements, was determined by one-way analysis of variance for repeated measures followed by Dunnett's test. The relation between plasma or CSF clonidine concentrations and VAS pain scores on foot and hand testing was assessed by using Pearson's product-moment correlation.

All volunteers were women, 39 plus/minus 2 yr old, height 167 plus/minus 2 cm, and weight 78 plus/minus 7 kg. In all cases the lidocaine test dose resulted in bilateral sensory changes consistent with an epidural injection. All volunteers completed the study, and they received a total of 442 plus/minus 1 micro gram epidural clonidine. Two volunteers experienced a postdural puncture headache, although no volunteer required an epidural blood patch for treatment of the headache.

Subject 1 had a continuous decrease in concentration after 210 min, despite an increase in target concentration and continuous administration of clonidine from 210 to 300 min. This decrease may have been caused by intravascular migration of the epidural catheter, which would result in no delivery of clonidine to the epidural space after 210 min. We therefore excluded the data from subject 1 from subsequent analysis after 210 min. A single data point in subject 7 also was excluded from the analysis: at 210 min the observed clonidine concentration was a full order of magnitude less than the observed concentration at 180 and 230 min, despite continuous clonidine administration. This situation is biologically impossible. The observation at 210 min was probably an artifact of unclear cause (such as mislabeling of a sample or improper dilution in the assay).

With one exception, clonidine concentration in the CSF increased approximately 5 min after initiation of or changing of the targeted concentration. The pharmacokinetic analysis predicted a delay of approximately 15 min between a change in the target concentration and attainment of a new plateau in the CSF. Observation of the observed concentrations suggests that there was an initial delay of approximately 5-15 min between a change in target concentration and each plateau in the concentration in CSF (Figure 1). During the first plateau (target = 25 ng/ml), the average concentration was 50 plus/minus 13 ng/ml; during the second plateau (target = 50 ng/ml) the average concentration was 92 plus/minus 15 ng/ml; during the third plateau (target = 75 ng/ml) the average concentration was 140 plus/minus 19 ng/ml; and during the final plateau (target = 150 ng/ml) the average plateau was 215 plus/minus 37 ng/ml. Clonidine concentration in the plasma increased from 0.10 plus/minus 0.04 ng/ml after the first infusion to 0.30 plus/minus 0.10 ng/ml after the second, 0.70 plus/minus 0.08 ng/ml after the third, and to 1.1 plus/minus 0.12 ng/ml after the last infusion. One hour after discontinuation of epidural clonidine infusion, plasma clonidine concentration was 1.2 plus/minus 0.18 ng/ml.

(Table 1) shows the parameters of the triexponential disposition function relating intrathecal concentration to a unit epidural bolus dose and the transformation of those parameters into a compartmental model for use with the STANPUMP program, as described in the appendix. Each of the coefficients is the negative of the sum of the other coefficients, which gives the resulting curve a convex shape after bolus injection, with an intrathecal concentration of 0 at time 0, a subsequent peak, and then a return to an intrathecal concentration of 0 at time *symbol*. Table 1also shows the parameters of the new pharmacokinetic model estimated from these data.

The disposition functions of the intrathecal site predicted by these two models is shown in Figure 2. The original pharmacokinetic parameter set predicted a peak intrathecal concentration 22 min after epidural bolus administration. The revised pharmacokinetic parameter set predicts a much more rapid peak at 5 min after epidural bolus injection. The data (Figure 1) suggest that the peak drug concentration in the CSF after epidural bolus injection occurs closer to the 5 min predicted by the revised model than the 22 min predicted by the original model.

In general, the observed concentrations exceeded the target concentrations during the plateau phase. The overshoot ranged from nearly 100% at the lowest target to about 40% at the highest target. The median weighted performance error was +55%, and the median absolute performance error was 60% with the initial pharmacokinetic parameter set.

The computer-controlled infusion pump did reasonably well at achieving pseudo-steady-state concentrations and providing proportional increases in the pseudo-steady-state concentrations. However, there was considerable variability in the individual responses, as shown in Figure 1. The variability is demonstrated in the top graph of Figure 3. Each line has a fairly constant offset from the y = 1 line, indicating that the proportional error was fairly steady at each target.

The weighted residuals with the new pharmacokinetics are shown in the bottom graph of Figure 3. In this figure, the systematic overshoot is eliminated, in that the errors are now evenly divided into overshoot and undershoot of the predicted concentration. This comes at the expense of slightly more misspecification, in that the observed concentrations at 5 min after each change are, in general, slightly less than the model prediction, resulting in the sawtooth pattern seen in the residuals. The addition of a fourth exponential term to the pharmacokinetic model did not eliminate this pattern, suggesting that additional data about the early increase in concentration would be necessary to model properly the expected concentrations in CSF within the 1st 5 min of epidural bolus administration. The median weighted residual for the revised model is -8%, and the median absolute weighted residual is 37%.

Clonidine decreased mean arterial blood pressure and heart rate, and this decrease was related to the duration of infusion (Figure 4) and CSF clonidine concentration (data not shown). In contrast, clonidine had no effect on SpO²(Figure 4). Clonidine also caused dose-related sedation (Table 2).

The laser Doppler flow in the foot increased from 13.2 plus/minus 5.8 mV (arbitrary values) before epidural clonidine to 27.0 plus/minus 3.4 mV after targeted infusion to CSF clonidine concentration of 150 ng/ml (P < 0.05). The percentage change in skin perfusion in the foot at the time of maximum change after a deep gasp with 50 ng/ml clonidine was -63.1 plus/minus 6.4% and with 150 ng/ml clonidine was -54.8 plus/minus 8.7%. These changes did not differ from baseline. The time required to attain the minimum value of laser Doppler flow after initiation of the gasp reflex was approximately 7.5 s and did not vary with clonidine concentration. Similar results were observed in the hand tests.

Clonidine caused similar dose-related analgesia in both the foot and the hand (Figure 5). Because this finding was in contrast to the result of a previous study [10]that was of similar design but that used a single bolus of epidural clonidine rather than prolonged infusion, we examined the relation between lumbar CSF clonidine concentration and analgesia in the hand and the foot in both studies. This analysis revealed no difference between bolus and infusion administration in the linear regression of CSF clonidine and VAS pain score in the foot but a significant difference in this relation in the hand (Figure 6). The VAS pain score in the foot to ice-water immersion was more tightly correlated to lumbar CSF clonidine concentration (r = -0.70; P = 0.0000064) than to plasma clonidine concentration (r = -0.53; P = 0.0014) in the current study and in the study of bolus administration. [10].

Clonidine decreased the norepinephrine (NE) concentration but did not affect the epinephrine or dopamine concentration in plasma or CSF (Table 3). ACh concentration increased in CSF after initiation of clonidine infusion and was maintained throughout the clonidine infusion (Figure 7).

These data provide information regarding the feasibility of infusing drugs into the epidural space to targeted concentrations in CSF and regarding the mechanism of action of epidural clonidine analgesia.

Several assumptions implicit in this paradigm require explanation. We assumed that the presence of the intrathecal catheter would not affect distribution of clonidine in CSF after epidural infusion. Clearly a hole in the dura in close proximity to the epidural infusion can enhance the transfer of drug from epidural to intrathecal spaces, [22]and it is possible that the hole from the 19.5-G needle used to insert the catheter may have enhanced such a transfer of clonidine. Because the flux of drug across the dura is determined by the size of hole in the dura, [22]it is likely that the presence of the catheter in the hole decreased such flux. It is possible that the pharmacokinetic parameters calculated to predict more accurately the CSF clonidine concentrations in the current study differed from those obtained in the bolus study [10]in part for this reason, because the dural puncture was one interspace removed from the epidural catheter insertion site in the current study, compared with two interspaces removed in the former study. CSF was sampled several times during the clonidine infusion, and this procedure may have altered CSF circulation or movement of drug, although the volume of CSF sampled was small. Confirmation of the accuracy of these kinetic parameters in the presence of an intact dura awaits future studies with sampling of CSF by insertion of a small-gauge spinal needle for single time points.

A more basic assumption is that the time course of clonidine concentration in CSF closely reflects the time course of analgesic effect. Clearly, clonidine is rapidly absorbed into the systemic circulation after epidural injection, [10]and clonidine can produce analgesia after systemic administration [23]by actions at peripheral [24]and central [25]sites as well as spinal sites. Nonetheless, the major site of analgesia from epidural clonidine is most likely in the spinal cord, because it is reversed by epidural injection of a specific alpha²-adrenergic antagonist in a dose that has no effect when administered systemically. [26]Two other observations support a near identity of the time course of clonidine concentration in CSF after epidural administration and the time course of the analgesic effect. First is the minimal hysteresis observed in CSF clonidine versus analgesia over time in volunteers who received epidural clonidine by a single bolus. [10]This characteristic suggests rapid equilibration between clonidine in CSF and at its active site for analgesia. Second is the nearly identical relation between CSF clonidine and analgesia in the foot observed in that study, [10]with constantly changing CSF clonidine concentrations after a bolus, and that observed in the current study after 60 min of nearly steady CSF clonidine concentrations (Figure 6).

Clonidine administration by continuous lumbar epidural infusion resulted in analgesia in the upper extremities of volunteers in the current study, whereas analgesia in the upper extremities was not observed after a single lumbar epidural bolus in our previous study. [10]This result may reflect continuous addition of clonidine to lumbar CSF, allowing cephalad spread of drug in CSF as has been observed previously in humans receiving opioids of lipophilicity similar to that of clonidine. [27,28].

Although the computer-controlled infusion pump was fairly successful at providing pseudo-steady-state concentrations and making proportional changes in those concentrations, its absolute accuracy was poor because of a large and consistent overshoot. The large and fairly consistent bias in the results suggests that the intrathecal pharmacokinetics of epidurally administered clonidine could be further refined from our initial estimates after bolus drug administration. [10]Reports suggest that refining pharmacokinetic parameters intended for use in a computer-controlled infusion pump are best obtained from studies using computer-controlled infusions. [16]The improvement in the median weighted residual and the median absolute weighted residual with the revised models suggests that the technique is very promising for controlling intrathecal concentrations with rational epidural administration.

As discussed above, the close correlation between CSF clonidine concentration and analgesia in the current study argues strongly for a spinal site of action for analgesia after epidural clonidine administration. On the other hand, we observed nearly identical analgesia in both the hand and the foot after epidural clonidine infusion in this study. It has been suggested that clonidine produces equivalent (and poor) analgesia after intravenous or epidural administration. [29]This line of reasoning would suggest that equivalent analgesia in the hand and the foot after lumbar injection in the current study supports a systemic, nonspinal mechanism of action. However, we did observe analgesia restricted to the foot after epidural bolus administration of clonidine, [10]and "spread" of analgesia to the hand in the current study. This spread may reflect rapid movement of clonidine cephalad in CSF and measurement of analgesia 60 min after near-steady concentrations in CSF in the lumbar intrathecal space.

Although many factors affect the concentration of neurotransmitters in CSF, the effect of drugs on CSF neurotransmitter concentrations can provide clues concerning the actions of such drugs on synapses that release these neurotransmitters. The lack of effect of epidural clonidine on epinephrine or dopamine is consistent with the lack of demonstrated interaction between alpha²-adrenergic agonists and these neurotransmitters in the spinal cord. Increasing concentrations of clonidine accompanied decreasing concentrations of NE in CSF, which may reflect decreased synaptic release of NE in the superficial dorsal horn and intermediolateral cell column, the two major spinal sites of noradrenergic terminals. [30]This effect is most likely a result of classic presynaptic inhibition of NE release by stimulation of presynaptic alpha²-adrenoceptors, but it would not be a likely cause of clonidine analgesia, because NE is an analgesic neurotransmitter in the spinal cord. [31].

Increased CSF ACh concentration after epidural clonidine administration supports previous observations of cholinergic activation from spinal alpha²-adrenergic activation. Spinal clonidine-induced antinociception in animals is enhanced by spinal neostigmine and inhibited by spinal atropine. [32]Intrathecal clonidine administration increases ACh concentration in the CSF of sheep by an alpha²-adrenergic mechanism that is not attributable to nonspecific vasoconstriction and altered clearance of ACh. [21]We have previously observed an increase in CSF ACh in humans 1 h after a bolus epidural injection of clonidine, [21]and the current data demonstrate rapid increases in CSF ACh after epidural clonidine administration.

The cutaneous circulation is predominantly under sympathetic nervous system control. Under basal conditions, the overall "tone" or firing rate of the sympathetic nerves innervating the vasculature determines the overall cutaneous perfusion. The sympathetic nervous system also controls reflex responses of the cutaneous circulation. Reflex changes in cutaneous perfusion may be invoked by an inspiratory gasp. Cutaneous perfusion transiently decreases with this maneuver, and the magnitude of the decrease in perfusion accompanying the reflex reflects the degree of sympathetic activity evoked. It has been suggested, therefore, that this reflex may be sensitive to sympatholytic effects.

In the current study the cutaneous perfusion of the extremities was evaluated with laser Doppler fluxmetry, a technique sensitive to the average velocity and total number of cellular components in the blood. Both tonic and reflex sympathetic influences on the cutaneous perfusion were evaluated, and it was found that the epidural administration of clonidine produced increases in the basal cutaneous perfusion, a finding consistent with a reduction in sympathetic nervous system tone as reported by Kirno et al. [11]The overall magnitude of the vascular response elicited by a gasp was not significantly altered, however, and was close to the percentage change in perfusion evoked by this reflex as noted by others. [33]The pattern of sympathetically evoked responses found in the current study is similar to the work of Kirno et al., [11]who reported that a sympathetically mediated reflex of normal magnitude could be evoked after epidural clonidine but that the basal level of sympathetic nerve impulses was reduced. There are also cases in which reflex cutaneous responses to an inspiratory gasp have been evaluated under conditions in which basal levels of sympathetic tone are considered increased. [33]The latter study [33]found that the magnitude of the reflex change in cutaneous perfusion evoked by an inspiratory gasp was not different from that obtained under normal conditions. These patterns of responses of the cutaneous perfusion to an inspiratory gasp suggest that the magnitude of sympathetically mediated reflex changes may be less informative than the basal value of skin perfusion in predicting the overall status of resting sympathetic nerve activity.

In summary, lumbar epidural clonidine administration by computer-controlled infusion increases ACh concentration in CSF, and it produces analgesia in the foot and the hand that is tightly correlated to CSF clonidine concentration. These data support a spinal cholinergic mechanism of analgesia from epidural clonidine and demonstrate the feasibility of infusing drugs epidurally to targeted concentrations in CSF for research and clinical applications. The errors are only moderately larger than those observed with intravenous drug administration by computer-controlled infusion, and this approach may be particularly important as a research tool, just as intravenous computer-controlled administration has become a recognized tool for investigation of intravenous drugs.

In this study clonidine was administered into the epidural space to achieve a given intrathecal concentration profile. There was a disequilibrium between clonidine administration into the epidural space and the appearance of clonidine in the intrathecal space. There is an obvious parallel between this model and the conventional "effect-site" model that relates the time course of drug concentration in the plasma with the time course of concentration at the hypothetical site of drug effect.

The question of how to calculate the time course of epidural dosing to achieve a desired intrathecal concentration is similar to the question of how to calculate the time course of plasma dosing to achieve a target concentration at the effect site. Because we cannot inject drug directly into the effect site, it is impossible to obtain instantly a target effect-site concentration. There are many ways to give drug to achieve a specific target in the effect site, depending on how the advantages of a slow increase to the desired target are balanced with the advantages of a rapid increase to the target but with some overshoot. If we restrict ourselves to solutions in which the effect-site concentration increases as rapidly as possible to the desired target concentration without overshooting the target, and assume that the parameters are known a priori, then the desired dose is that given by the control policy developed by Shafer and Gregg. [17]This control strategy has been implemented in the computer program STANPUMP, using the computational approach suggested by Jacobs and Williams. [18].

To control the concentration of drug at the effect site, STANPUMP requires the micro-rate constants of a one-, two-, or three-compartment mamillary model plus the rate constant for drug elimination from the effect site (k^e0). In the current study we estimated a disposition function for the intrathecal space of the form; Equation 6where C^1T(t) = the intrathecal concentration over time; dose = the dose of clonidine administered (700 micro gram) [10]; C¹and C²and - (C¹+ C²) = the coefficients of a triexponential disposition function; and lambda¹, lambda²and lambda³are the exponents of the disposition function. Each coefficient is the negative of the sum of the other coefficients, which gives the disposition function the necessary shape, which is 0 at time 0, increases to a peak, and then decreases to 0 again at t = infinity. To administer clonidine by using this disposition function it is necessary to convert Equation 6to a series of rate constants representing a two-compartment mamillary model with an effect site, which is the model used internally by STANPUMP. This transformation is performed as follows.

The standard two-compartment pharmacokinetic model, [17]is represented by the following differential equations: Equation 7. We will transform this set of differential equations into an equation that describes the concentration over time after injection of a single unit of drug into the central compartment (V¹). Such an equation is called a "unit disposition function" and has the biexponential form (for this exercise); Equation 8where C^p(t) = the plasma concentration at time t that would be expected after an injection of 1 unit of drug, and A, B, alpha, and beta = pharmacokinetic parameters directly estimated or mathematically derived from the parameters estimated (e.g., the volumes and clearances of a two-compartment model or the rate constants shown in Equation 6).

The transformation between Equation 7and Equation 8is outlined in many pharmacokinetic textbooks but is reviewed here because several solutions developed from the transformation are required to convert the intrathecal disposition function (Equation 6) into a compartmental representation used by STANPUMP. Although symbolic manipulation programs such as Mathematica can directly perform the transformation between Equation 7and Equation 8in the time domain, the process generates huge intermediate equations. Therefore, to simplify the algebra, it is common first to transform Equation 7into the Laplace domain. Applying the Laplace transformation to Equation 7yields; Equation 9where s = the independent variable in the Laplace domain; X¹(s) and X²(s) = the Laplace transform x¹(t) and x²(t); and x¹(0) and x²(0) = the initial conditions of compartments 1 and 2, respectively. Because we are interested in the response of this system to the instantaneous injection of 1 unit of drug at time 0, the initial condition of x¹(0) is 1 (i.e., the initial amount in the central compartment is 1 unit). We assume, of course, that initially there is no drug anywhere else in the body, and so x²(0) is 0. Substituting 1 and 0 for x¹(0) and x²(0), respectively, into Equation 9and rearranging generates the following simultaneous equations: Equation 10.

If we solve Equation 10for X¹(s), the Laplace transform of the amount of drug in the central compartment as a function of s, we obtain; Equation 11.

Returning to Equation 8, and recognizing that the concentration in the plasma (C^p) is the amount of drug in the central compartment (x¹) divided by the volume of the central compartment (V sub 1), we can rewrite Equation 8in terms of x¹(t): Equation 12. The Laplace transformation of Equation 12is Equation 13.

If we are to transform Equation 7, the differential equations, into Equation 8, the unit disposition function, we must find definitions of A, B, alpha, and beta such that Equation 13can be transformed into Equation 11. Therefore our puzzle is to define A, B, alpha, and beta so that they satisfy the following identity, derived from Equation 11and Equation 13: Equation 14. To do si, we first recognize that the denominator of the left hand expression of identity 9 is a quadratic expression, Equation 15which can be factored by using the quadratic equation. Two solutions are generated: Equation 16. If we define alpha and beta as the negative of the roots of s shown in Equation 16, that is, Equation 17we can simplify identity 9 to the following identity: Equation 18. With some algebraic manipulation of the left side of identity 13, we can rearrange it to Equation 19. It follows directly from identity 14 that the following solutions exist for A and B: Equation 20.

Thus, when we define A and B as in Equation 20and alpha and beta as in Equation 17, then Equation 7(the differential equations for a two-compartment model) and Equation 8(a biexponential decay) become identical representations of the same system after a unit dose at time 0. Similar derivations to that above can be found in many pharmacokinetic texts.

To calculate the intravenous dose of drug required to produce a given effect-site concentration, STANPUMP must calculate the response of the effect site to an intravenous bolus of 1 unit. The response of the effect site to a bolus of 1 unit, the unit disposition function of the effect site relative to an intravenous bolus, can be computed as the convolution of the plasma concentrations over time after a bolus of 1 unit (e.g., Equation 8above) with the disposition function of the effect site itself, Equation 21where C^E(t) = the effect-site concentration over time. Convolution of two functions is easily performed as multiplication of the functions in the Laplace domain. The Laplace transformation of the right side of Equation 8is Equation 22and the Laplace transformation of the right side of Equation 21is Equation 23and the product of these is Equation 24. If we take the inverse Laplace transformation of Equation 24, we obtain the disposition function of the effect site for a unit intravenous bolus: Equation 25.

(Equation 6) can be rewritten to express the response of the intrathecal space to an epidurally administered unit dose as Equation 26. These are obviously parallel equations. Comparison of Equation 25and Equation 26shows that STANPUMP can administer drug to a targeted intrathecal concentration provided the following solutions that make Equation 25and Equation 26identical are observed: Equation 27. If we combine Equation 27above with Equation 17for the exponents alpha and beta and Equation 20for the coefficients A and B, we have a series of equations that can be solved simultaneously for the parameters required by STANPUMP to administer drug to the effect site, V¹, k¹⁰, k¹², k²¹, and k^e0in terms of the parameters estimated for the disposition function of the intrathecal space, C¹, C², lambda¹, lambda², and lambda³. The resulting definitions of V¹, k¹⁰, k¹², k²¹, and k^e0are as follows: Equation 28.

Using the above definitions, we entered into STANPUMP the micro-rate constants and k^e0for epidurally administered clonidine, enabling the program to target the intrathecal space as the effect site based on the control policy proposed by Shafer and Gregg. [17].

The mathematically inclined reader may attempt this derivation. If so, it should be noted that there are numerous approaches, only one of which is described here, and that solutions superficially different than those shown in Equation 28may be derived. Close inspection should reveal that the alternative solutions are mathematically equivalent to those shown above. The above derivation is available as a Mathematica notebook by FTP to pkpd.icon@palo_alto.med.va.gov in file /clonidine.dir/clonccip.ma.