A System Approach to Pharmacodynamics. Input-Effect Control System Analysis of Central Nervous System Effect of Alfentanil

A System Approach to Pharmacodynamics. Input-Effect Control System Analysis of Central Nervous System Effect of Alfentanil PETER VENG-PEDERSEN*XAND

NISHITB. MODI**

Received August 8, 1991, from the *University of lowa, CoNege of Pharmacy, lowa City, IA. Accepted for publication July 10, 1992. *Present address: Genentech Inc., 460 Point San Bruno Blvd., South San Francisco, CA 94080.

Abstract 0 Virtually all biological variables, including those affected by drugs, are subject to adaptive self regulation. In the description of the pharmacodynamics (PD) of drugs, it may be necessary to consider the endogenous control system (ECS) as an integral part of the PD. A PDECS model based on system analysis principles is presented and tested on PD data for alfentanil considering the central nervous system activity quantified by a power spectrum analysis of the electroencephalogram. The model was tested in terms of a proposed relativeprediction performance criterion that measures the accuracy of future predictions relativeto how well the model describes (fits) the past effect data. A mean value of 80% (standard deviation, 28) for relative prediction performance indicates that the model performs well when challenged by the complex multiple infusion scheme used in the test. The overshoot phenomenon observed in the data is considered by the PDECS model as a ECS-based tolerance phenomenon. The proposed development of tolerance is modeled as a variable gain in the ECS processing that influences the effect. Although the development and loss of tolerance is determined by a single rate constant in the tolerance model, the rates of increase and decrease of tolerance may be substantially different. Contrary to other PD tolerance models, the proposed PDECS approach models the tolerance in terms of an effect deviation from an ECS set point. The intrinsic (no tolerance) effect of the drug is isolated in terms of an open loop (no feedback) effect.

Only l i m i t e d attention has been given t o t h e pharmacodynamic (PD) role of biological self regulation, and most published work in t h i s area has been of more theoretical than practical significance.'-3 The purposes o f t h i s publication are (1)t o define a PD modeling objective aimed at drug effect prediction, (2) t o propose a system analysis-based PD endogenous control system (ECS) model (PDECS model) that addresses t h e stated modeling objective, and (3) to analyze t h e central nervous system (CNS) effect o f alfentanil with t h e sole purpose of demonstrating the suggested PD analyses procedures and test t h e proposcd PDECS model in terms of i t s predictive capability.

Experimental Section Seven New Zealand white rabbits of mixed sex weighing between 1.5 and 3.5 k g were used for the experiments. Two computercontrolled infusion pumps (Pump 22, Harvard Apparatus Inc., South Natick, MA) interfaced through a RS232 serial lines to an XT IBM computer were used to infuse the drug alfentanil (Janssen,lot 800/1). A slow infusion of normal saline (0.8 mL/h) was maintained throughout the study to keep the intravenous (iv) line patent. The iv line was established with a 24G 19-mm iv catheter (Jelco, Critikon, Tampa, IL)in the marginal ear vein. The experiments were highly computerized, simultaneously making use of five PC computers (four IBM XTs and one 20 MHZ 386 AT computer) for drug infusion control and real time digital data acquisition. The experiments were preprogrammed for automatic execution and required no intervention except for a few repositionings of a n IR solid-state video camera used for noninvasive pupillary measurement to adjust for slight head movements of the restrained rabbits. Six different PD responses were continuously and automatically acquired by the computers: (1) electroencephalogram (EEG), (2) electrocardiogram (ECG), (3)respi266 I Journal of Pharmaceutical Sciences Vol. 82, No. 3, March 1993

ratory dynamics, (4) respiratory gas analysis (end tidal COJ, (5) noninvasive IR measurements of pupillary response, and (6) rectal temperature. The EEG was obtained from measurements with specially designed bipolar miniature gold-plated electrodes (length, 4 mm; diameter, 1mm) that were surgically implanted equibilaterally 7 mm from the midsagittal cranial plane and 7 mm anterior to the basal uriclae. The electrodes were electrically isolated from all tissue except dura mater encephali and sinus dura matris. The electrodes contained a microminiature connector that enabled simple electrical disconnection between the experiments. The placement of the electrodes was done with ketamine * HC1 (40 mgkg, intramuscularly) general anesthesia and bupivacaine .HCl(5 mg) as a local anesthetic agent. Cranial penetration was done with minimal surgical insult with an electric cautery, leavinga 1-2-mm diameter hole through the skin, followed by a high-speed 1-mm drilling through the cranial bone. Use of adhesives or cement was not necessary because the double-tapered electrodes were placed by a press-fitting procedure that left them hermetically sealed and firmly placed. The animals were allowed to recover from surgery for 2 days before the experiments were done. The animals were restrained in a custom-designed adjustable restraining device that allowed easy access to the animals and posed minimal discomfort to the animal during the experiment. The experiments were conducted at the same time of the day with the animals in a sound- and light-attenuated chamber. The animals were adapted to the experimental chamber for 4 h before the start of the first drug infusion that was preceded by 30 min of predrug measurements. The pre-programmed computercontrolled infusion scheme of alfentanil .HCl was a repeated infusion of the following bracketed scheme [1.6 U for 15 min, 6.4 U next 15 min, nothing next 60 min, where U = pg/kg/min], followed by 6.4 U for 15min, 1.6 U for 15 min, and finally a 60-min washout period. The EEG signal was acquired at a rate of 200 Hz with a n EEG amplifier (model 7P511J, Grass Instruments Company, Quincy, MA) as a front end for a n AT computer-based data acquisition system (AT-CODAS, Dataq Instruments Inc., Akron, OH).The low-pass filterwas set as 0.3 Hz and the high-pass filter was set at 100 Hz with a sensitivity of 7.5. The EEG was analyzed by power spectrum analysis with the fast Fourier transform subroutine FTFREQ (IMSL, Houston, TX) as the basis for the computation of the 95% spectral edge frequency.4.5 The heart rate was acquired with subdermal electrodes (Grass E2, Grass Instruments) and a Hewlett-Packard model 78901A medical monitoring system with a 78203C module as a front end interfaced to an analogue to digital converter (DT2811, Data Translation Inc., Marlboro, MA) in a n XT IBM computer. The ECG was processed by a digital signal processing program developed by us that determines and stores in real time the beat-to-beat heart rate. At the completion of the experiment, the data from the five PC computers were transferred via a local area network directly to a micro VAX-I1 computer for subsequent PD analysis.

Theoretical Section M o d e l i n g Objective-The modeling objective o f this work is t o predict the effect [E(t)lo f a drug o n the basis of a controllable The r(t)value will be called the predictor infusion rate [r(t)]. variable. Thus, this work deals with input-effect systems in contrast t o the more traditional PD approaches that employ the concentration of t h e drug in plasma as t h e predictor variable for the effect. (A predictor variable is in this context defined as an observable variable that aEects the drug effect outcome.)

0022-3549/93/0300-0266502.50/0 0 1993,American PharmaceuticalAssociation

The present PD modeling objective is directly related to the ultimate aim of this research, which is to develop optimal, real-time control procedures for controlling the effect of drugs to optimize the drug treatment. It is well recognized in control theory that the better the behavior of a system can be predicted, the better it can be controlled. This work is aimed at the paramount first step in the control process; namely, prediction of future drug response. The input-effect control problem is aimed primarily at vascular drug administrations, that is, situations in which it is critically important to control the drug effect but in which this is not practical with drug level monitoring techniques because of insufficient drug sampling or because the pharmacokinetic (PK) and PD variables are changing rapidly relative to the time it takes to assay the drug. The terminology “effect” [E(t)lwill be used throughout to denote a measured nonpharmacokinetic variable that is observed directly (e.g., temperature) or obtained by signal processing of more complex biological signals (e.g., EEG power spectrum analysis). The effect measured is not assumed to be solely due to the drug. Instead E(t)may be considered to consist of a background/baseline/predrugcomponent [EO(t)l and a component due to the drug [Ed(t)l:

’0 )

(3)

It is proposed that the gain is changing according to the following very simple gain modification model:

G’(t)= K J E , - E(t))

(4)

The E , term in eq 4 is defined as the endogenous set point. The rate by which the system corrects an aberration in E is determined by the rate constant Ka (eq 4) that is appropriately called the gain change rate constant. Equations 3 and 4 define the predrug ECS model. Drug-ECS Interaction-It is proposed that the drug is acting by changing the gain block. Mechanistically this may be thought of as an occupancy of drug receptors, resulting in an attenuated processing of the neurological stimulation, leading to a reduction in the gain function. Thus, in this context, the drug effect is essentially a gain function effect, EG(t),so that the output of the gain block becomes KJCG(t)G(t) and eq 3 becomes:

E(t)= Er + K+?3~(t)G(t) Ec(O)=l, G ( 0 ) ~ l (5)

(2)

In most practical situations, it is simpler to work with E(t) than I(t). This is because Eo(t) is difficult to identify. It is valuable first to consider a modeling of the control of Eo(t)by the ECS in the predrug stage. Predrug ECS-Consider the case in which the effect is of neurological nature (e.g.,EEG). Figure 1illustrates a simple control system for the regulation of the predrug baseline effect. The parameter KO represents a general endogenous neurological stimulation (biological sensing and neurological communication) that is subject to neurological processing. The processing results in an output, KoG(t),that is determined by a variable gain, G(t),represented by the gain block. The gain is defined as the amplificationlattenuation of the endogenous neurological stimulation KO. The regulation of E(t)is achieved by a modification of the neurological processing [i.e., by a modification of G(t)l.This may, for example, be due to a change in the number and/or sensitivity of receptors responsible for the processing of the endogenous neurological stimulation. The output from the gain block is superimposed on a residual effect, E , 2 0 (Figure 1).The residual effect, E,, is included to consider a measured effect that may be heterogeneous due to an inadequate signal processing that does not remove the components in a signal that are extraneous and not affected by the particular control system and drug. This is typically the case in EEG signal processing. The ECS model (Figure 1)is described by eq 3: gain

(KO

It is assumed that the gain effect, E&), is determined by the drugs biophase concentration, cb(t),and a transduction function, N

The &(t) is commonly denoted the “intensity” [I@)]:

I ( t ) E ( t ) - Eo(t)

E ( t ) = Er + KoG(t)

Er

G(t) modification

Figure l-Basic ECS models for the regulation of the predrug baseline effect E0(t).

Therefore, eq 5 becomes:

E(t) = Er + (E(O)- Er)N(cb(t))G(t)

(7)

To delineate the gain effect, it is valuable to introduce a unit gain effect [E,(t)],defined as the open loop effect (i.e., the behavior of the system if the gain remained constant), that is equal to its initial value of one. The unit gain effect corresponds to a no feedback (open loop) situation. The unit gain effect (open loop effect) is accordingly given by:

(8) Corresponding to E,(t), there is a unit gain transduction function, N,:

In eq 9, N,(c,(O)) = E(0). For many drugs, it is likely true with good approximation that the PK is linear in the sense that there exists a linear (operational) relationship between the biophase drug level profile [ C b ( t ) ] and the predictor variable [r(t)].This relationship may be expressed in a structureless form by a linear operational response mapping operation, cb(t) = RMO{r(t)}.The above can now be summarized collectively by the following key equation defining the proposed PDECS model:

E(t) = Er + G(t)lE,(t)- Er1

(1OA)

El(t) = Nl(Cb(t))

(10B)

Cb(t) = RMO{r(t)}

(100

G’(t) = K,(E, - E(t))

G(0)=1

(10D)

Journal of Pharmaceutical Sciences 1 267 Vol. 82,No. 3, March 1993

In eq 10, the intrinsic effect of the drug is conveniently isolated in the unit gain effect, El(t). This enables easy evaluation of the change in the effect caused by the ECS, which is E(t) - El(t). The PDECS model (eq 10) is illustrated in control system block diagram form in Figure 2 in which the model is differentiated into its three basic components: (1) the PK component (RMO),(2) the PD transduction component (N1), and (3)the endogenous control system interaction component. Control System Stability-The PDECS model is stable if the gain function is stable. The gain function is, according to eq 10, given by the following simple linear differential equation of order 1:

Specific Procedure-For a drug with linear drug disposition, it is well recognized from linear system theory738 that the systemic drug level [c(t),arterial or veneousl is related to the rate of drug input Wt)]by the following convolution equation:

where G(O)E 1 and

The predictor variable is the rate of drug input [r(t) = fit)]. Thus, from eqs 13 and 14 it follows that:

It is well known from stability theory6 that eq 11 is stable under the condition that a(t)2 0 for t 2 0. Condition Cy(t)> 0 corresponds to development of tolerance. Condition a(t)= 0 for t 2 0 corresponds to a system without feedback (K, = 0). Condition a(t)< 0 is an unstable situation that corresponds to sensitization (i.e., the development of increased sensitivity to the drug). If the drug shows tolerance and has a suppressing effect, then K, should be positive and E, should be below the lowest effect experienced with the drug. Alternatively, if the drug shows tolerance and has a stimulating effect, K , should be negative and E, should be above the highest effect experienced with the drug. These conditions can readily be controlled by enforcing (if necessary) simple lowerlupper parameter bounds on the K , and E, parameters in the curve fitting of eq 10 to effect data. General Computational Procedures-G(t) is obtained by numerical integration of eq 11. Subsequently, E(t) is calculated according to eq 1OA. This procedure establishes the effect as a function of time accordingto the PDECS model that then can be fitted to effect versus time data by ordinary curve-fitting techniques. Alternatively, eq 10 may be solved analytically. However, the analytic solution requires a numerical integration (in the present case) that is more involved and numerically more problematic than the above numerical procedure that was used in the calculations.

PK mapping

~~

PD transduction

~

PD-ECS interaction

G(t) modification

I

Figure 2-Proposed PD model that interacts with the ECS (Figure 1). The PDECS model consists of three basic components: (1) a PK mapping of the predictor variable r (an observed PK variable or infusion scheme),(2) the PD transduction, and (3)the PD-ECS interaction. 268 I Journal of Pharmaceutical Sciences Vol. 82, No. 3, March 1993

In eq 13, c&t) is the unit impulse response. For a drug with linear distribution kinetics, it may, according to RMO principles,, be justified to propose that the biophase level is linearly related to the systemic drug level by a convolution operation involving a conduction function [+(t)110J1:

In eq 15, &(t)is an extended conduction function:

The function is extended in the sense that the biophase prediction extends all the way back to the input as the predictor variable. The function +Jt) was modeled as a biexponential function:

The common Em, and sigmoid Em, models (see Glossary for definitions) were considered as models for the transduction from the biophase level, cb, to the unit gain effect,El (eq 10B):

Both a regular Em, model (n = 1) and a sigmoid Em, model (n z 1)were considered in the analysis. The results reported correspond to n = 1 because the sigmoid Em, model did not result in any significant improvement as judged by the Akaike information criteria. In the same way, a twoexponential +Jt) (eq 17) was preferred over a threeexponential expression. The proposed PDECS model contains the following nine parameters that may be classified according to the structural components of the model (Figure 2): PK mapping parameters (gl, g2, yl, y2), PD transduction parameters [E,, EC,, (n = 111, and ECS parameters (K,, E,,EJ The predictor function (the infusion scheme) is a (up/down) staircase function represented by a straight-line spline function. The biophase level [Cb(t), eq 151 is, accordingly, calculated according to an analytical formula for the convolution between a straight line spline function [r(t)]and a biexponential[+Jt), eq 171. The biophase equilibration times t,, and t,, were calculated as previously described11 from the conduction function +,(t),as the solution to the following singlevariable nonlinear equation:

I,”

&(u) du = d l 0 0

(x = 50,95)

(19)

In eq 19, t, (e.g., t,,,t9,) is the time it takes to reach x percent of the predicted biophase steady state when the predictor variable is kept constant. Equation 19 assumes that conduction function is normalized [i.e., has a total area (t = 0 to m) of one].1OJ1

Results and Discussion The proposed model (eq 10, Figure 2) was tested for its suitability for predicting the future drug effect based on information of the past drug effect within a multiple-drug infusion scheme (Figure 3) lasting 5 h. The first 180-mineffect data (top panel, Figure 3) were used for the model calibration stage where the parameters of the proposed model (Tables I and 11)were estimated by regular, least-squares, curve-fitting procedures with the FUNFIT program.12 [The calibration stage took -10 min with an old 1mips computer (microVax I1 with DEC’s FORTRAN compiler). It should be possible to reduce the calibration time by a factor of 50-200 with a modern RISC-based workstation with optimized software, thereby bringing the procedure into the realm of real time PD control.] The calibrated model is subsequently used to predict the effect resulting from a future drug treatment (t > 180 min) as illustrated in the bottom panel of Figure 3. The future administration scheme (t > 180 min) includes washouts and a two-stage infusion that is applied in reverse order (high, low) to that of previous infusions. The predictive performance of the model was evaluated according to the following relative predictive performance (RPP) criteria:

RPP =

prediction accuracy

fitting accuracy

=J

1/%i (prediction) 1/ SS (fitting)

100

*

*

100

=

100

Jz

N (prediction) ss (prediction) * N (fitting) ss (fitting)

*

*

(20)

In eq 20, is mean squared residues and N is the number of data points. The above RPP criterion was preferred over the usual predicted sum of squared residual (PRESS) criterion. PRESS is an absolute measure that does not appear very informative because it is confounded by the degree of error in the effect data that typically is quite high. A poor (high)value of PRESS can be due to large errors in the data. It can be misleading because it may not necessarily reflect a poor prediction. This problem can be overcome by using instead a relative measure in which the prediction accuracy is measured relative to the fitting accuracy (e.g., eq 20). The measure is then less confounded by the degree of error in the effect data. For example, a RPP of 84.3%(subject R0324, Table 11) essentially means that the agreement between the predicted effect and observed effect (t > 180 min) is 84.3% as good as the agreement between the fitted effect and the observed effect (t c 180 rnin). The disadvantage of this relative RPP measure is that a high value does not necessarily mean the prediction is good if the model does not fit the data well in the calibration stage. However, the goodness of fit in the calibration stage is readily inspected, thereby making a valuable reference for the assessment of the RPP prediction measure. The present example (Figure 3, top panel) shows that the calibration fit is very satisfactory. This is particularly true considering the intrinsic erratic behavior of CNS activity data based on EEG measurements (Figure 3, top panel). A mean RPP value of 80.2% for the seven cases shows that the proposed model has a good predictive performance (Table 11). The poor prediction performance for subject R0104 (RPP = 28%,Table 11) was due to large errors in the predictor stage EEG data (t > 180 min) that were predominantly caused by motion artifacts. (EEG recordings are very sensitive to movements of the wires that carry the low voltage level from the brain electrodes to the EEG amplifier.)

0 SO 46

40

Figure 3-Demonstration and test of the proposed PDECS model (eq 10, Figure 2) using CNS activity data of alfentanil quantified by digital signal processing of EEG measurements. The top figure is the fit of the model to effect data for the first 180 min. The bottom figure shows the predicted (nonfitted)effect beyond the first 180 min. The predictor variable r (eq lOC, Figure 2), which is the infusion rate of the drug, is plotted as the irregular staircase function. Journal of Pharmaceutical Sciences I 269 Vol. 82,No. 3, March 1993

Table 1-Parameters of the Conduction Function (eq 16) Determined by Fitting the Proposed PDECS Model (eq 10, Figure 2) to CNS Activity Data of Alfentanll (Figure 3)'

Conduction Function Parameters

Subject 91.103

y,

176 64.1 28.1 177 1530 7.87 0.004

R0319 R0324 R0515 R0104 R0117 R0120 R0131

Mean

lo2, min-'

g**loz

15.6 9.56 4.82 6.85 2.70 1.45 5.24

- 17.6

6.60 4.78

283 554

SDb

Biophase Equilibration Times, min y2 lo2, min-'

tso

1

$6

11.1 13.9 - 13.5 9.74 8.29 6.23

139 33.8 33.5 8.48 22.5 18.2 6.23

5.20 4.62 5.83 14.3 9.09 12.5 11.1

20.0 27.2 51.O 53.3 90.0 164 48.0

13.3 33.1

37.4 46.1

8.95 3.84

64.8 49.1

'The biophase equilibration time parameters Go and G5 are calculated from the extended conduction function (eq 18). Standard deviation. Table Il-Parameters of the Transductlon Function (eq 21, n = 1) and the ECS (eq 10) Determlned by Fittlng the Proposed PDECS Model (eq 10, Figure 2) to CNS Actlvlty Data of Alfentanll (Figure 3). ECS Parameters

Transduction Parameters Subject R0319 R0324 R0515 R0104 R0117 R0120 R0131

Mean SD* a

RPP, 9/ob

q .lo4, Hz-'

P, Hz

ESTHz

-12.1 - 17.3 -38.9 -81.4 - 16.9

0.994 0.896 0.414 2.8 1.92 8.93 0.884

3.1 5 3.89 6.19 14.3 5.06 2.57 21.6

5.66 10.5 10.1 0.996 8.61 14.4 0.05

19.7 25.7 25.9 14.6 34.7 30.3 23.2

-27.6 25.5

2.42 2.9

12.2 10.9

7.19 5.25

24.9 6.62

P", Hz

EC50

- 10.6 - 16.0

min

84.8 84.3 68.5 28.0 109 74.0 113 80.2 28.4

Determined with eq 22 and a measure of the quality of the PDECS model as an effect predictor. Standard deviation.

The gain function in Figure 4 corresponding to the example in Figure 3 shows the progress of the development and partial loss of what appears to be acute tolerance. The gain increases relatively rapidly in the three infusion periods (30-60 min, 120-150 min, and 210-240 min; Figure 3).The gain continues to increase after the termination of the drug infusions in the early phases of the washout periods. Subsequently, there is a decline in the acquired gain in the later phases of the washout periods, corresponding to a loss of some of the acquired tolerance. This delay in the loss of tolerance is because the RRBBIT 0117

2 1

1.8

1

-

EEG

I

0

50

100

200 ~EE S 250

3dQ

MI Flgure M h a n g e in the gain function [G(f),eq 10D, Figure 2) resulting from multiple alfentanil infusions (Figure3) consideringthe CNS effect of alfentanil. 270 I Journal of Pharmaceutical Sciences Vol. 82, No. 3, March 1993

effect, E(t),in the washout period needs to return to the set point, E,, and shoot above the set point before there will be a loss in the gain (eq 10D).In the present example, E, is 34.7 Hz (Table 11).This value is reached at -100,175,and 250 min, corresponding to the three peaks of the gain function, respectively (Figure 4). Just before these peaks, E ( t ) is below E,, which, according to eq 10D,gives G' (t)> 0 [i.e., an increase in G(t)].After the overshoot [E(t) > E,], the opposite is the case, resulting in a decrease in the gain [G'W < 01. This rebound or overshoot phenomena is characteristic of control systems. The rebound is, for example, observed in the often pronounced abstinence or withdrawal effect following a discontinuation of prolonged andlor intense exposure to an opioid agonist. The model reveals the following important properties. The degree of overshoot increases with increasing gain (i.e., with increasing development of tolerance). This is seen in Figure 3 in which the broken horizontal line indicates the predrug baseline effect, E,. The model shows an increasing degree of overshoot in the later part of the three washout phases (Figure 3).This is consistent with the data that show that the fraction of effect data above the predrug baseline in these three phases increased in tune with the increase in the gain function (Figures 3 and 4). This property of the overshoot kinetics is consistent with the well-established fact that withdrawal symptoms of opiate agonists are more pronounced when more tolerance is developed. The PDECS model predicts that the degree of overshoot depends on the PK of the drug. The faster the drug is eliminated from the biophase [i.e., the faster c,(t) declines in the elimination phase], the greater the overshoot will be. This is because the ECS will have less time to compensate the gain.

a t

- 120

-

r

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46

26

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120

6

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-

4

-

3

-

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-

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0

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€30

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ido

160

ledo

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200

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Figure 5-Demonstration and test of the proposed PDECS model (eq 10, Figure 2) using alfentanil heart rate data. The top figure is the fit of the model to effect data for the first 180 min. The bottom figure shows the predicted (nonfitted)effect beyond the first 180 min. The predictor variable r (eq 1OC. Figure 2), which is the infusion rate of the drug, is plotted as the irregular staircase function.

From the above analysis it follows that the degree of overshoot depends on basically three factors: (1)the degree of gain (tolerance) acquired; (2) the rate of elimination of drug from the biophase, and (3) the rate by which the ECS can change the gain (i.e., K,, eq 10D). These properties of the proposed PDECS model appear in agreement with general observations made for opiate agonists13 and what logically can be expected from a general control system point of view. It may be argued that the observed overshoot phenomena is not related to tolerance but simply is a baseline shift. The logical way to resolve this issue is to observe the baseline in placebo treatments. However, that was not a practical approach because it was not possible, in the absence of the sedating analgesic and anesthetic effects of alfentanil, to keep the subjects sufficiently motionless for 5 h. The subjects quickly become restless and very forcefullytry to escape from the restraining device, leading to useless EEG recordings with excessive motion artifacts. The overshoot is believed to be real for the following reasons: (1) it agrees with the withdrawal effects observed for opiates, (2) it was consistently observed, (3)it agrees with what is logically expected from a control system point of view, and (4) it appears to be well predicted on the basis of control system principles. (Abaseline “drift” not kinetically determined would likely be of a less consistent, more stochastic behavior.) The quite constant maximum effect values (trough values) in Figure 3 do not contradict the tolerance hypothesis. Instead, it is consistent with a tolerance model in which the tolerance is not expressed in terms of a lowering of the Em, parameter (eq 20) but simply requires increasingly larger biophase levels to reach approximately the Em, value. Contrary to the EEG effect (Figure 31, the heart rate effect (Figure 5 ) does show a slight decrease in the magnitude of the maximum effect [through E(t)value1with repeated administrations. The proposed model accommodates such differences through the E, parameter. The heart rate ECS dynamics also appears to more rapidly adapt itself to the drug disturbance as indicated by a comparison of the gain plots (Figures 4 and 6).

Comparison of Proposed Model to Other Models-The proposed PDECS model appears different from other PD models dealing with acute tolerance in a number of different ways. The PDECS model considers the ECS as an integral part of the PD. Although control system theory has been considered in the discussion of the PD of drugs, work in this area has been mainly of a purely theoretical nature.l-3 There exists an extensive literature in which tolerance has been demonstrated experimentally,14 but very little has been published about the quantitative PD aspects of tolerance.15-21 Apparently, quantitative PD modeling of tolerance has been considered only in a noncontrol system context. Porchet et a l . 1 7 proposed a PD model for tolerance applied to nicotine with the heart rate as the effect variable. Some valid arguments were presented by these authors about the deficiencyof tolerance models presented by others relative to their model. The model of Porchet et al.considers the tolerance explained in terms of the creation and elimination of a hypothetical 1.4-

1.35 1.3-

t 0

1.25 1.2-

1.15-

1 4 0

’

7

50

160

M I N ~ ~ Y E260 S

250

300

Figure M h a n g e in the gain function [G(f);eq 10D; Figure 2) resulting from multiple alfentanil infusions (Figure 5) considering the effect of alfentanil on heart rate. Journal of Pharmaceutical Sciences 1271 Vol. 82, No. 3, March 1993

noncompetitive antagonist, although there is no evidence for the existence of such an antagonist. The PDECS tolerance model differs fundamentally from the tolerance model of Porchet et al. by basing the tolerance on the “effect exposure” not the drug exposure. The driving force for the gain (and loss) in tolerance is the deviation of the effect from the biological set point of the endogeneous control system (eq 10D). The development of tolerance occurs nonlinearly in the PDECS model [note that E ( t ) in eq 10D is nonlinearly related to r(t) when N, is nonlinear]. The linear, convolution type tolerance development in the model of Porchet et al. is based on a hypothetical antagonistic metabolite that, in the case of nicotine, has not been identified.22 Ekblad and Lico21developeda PD model that like the model of Porchet et al. considers the drug acting on the formation of a “substance”, denoted y by Ekblad and Licko, that is responsible for the effect. They considered y formed from a precursor x in a pseudo-first-order fashion with a rate ‘‘constant”that depends on the drug concentration. They assumed x is formed at a constant rate and y is eliminated in a first-order fashion. The Ekblad and Lico model is an attractive model because of its flexibility in describing even quite unusual behaviors involving tolerance and transient phenomena.21Holford23 discussed an interesting variation of the Ekblad and Licko model in the context of the tolerancedependent PD of cocaine. The model predicts an overshoot (explaining “hangover”) under certain kinetic conditions. Contrary to this, the PDECS model always predicts an overshoot. Similar to other proposed tolerance models, usage of the PDECS model requires some close attention to a number of confounding factors, such as nonlinear PK and timedependent PK changes24525 (e.g., enzyme induction), that if expressed in a significant way invalidates the model. The PDECS model, in its present implementation, also has the disadvantage that a numerical integration is involved, which is, for example, not required in the model of Porchet et al.17 Although this work has dealt only with a drug with an inhibitory effect and tolerance, it should be realized that the proposed PDECS model considers both inhibitory effects and excitory effects, tolerance as well as sensitization, and overshoot and undershoot. The PDECS model distinguishes itself from other proposed models in mainly two respects: (1) tolerance is considered in a control system context, and (2) modeling of the tolerance dynamics is done in terms of a deviation from an effect set point (eq 4). Integration of endogenous control system theory in PD modeling produces some new challenges in kinetic analysis. A better understanding of the PD-ECS interaction should result in more reliable, kinetically based drug treatment strategies.

Glossary Central neivous system Drug concentration (veneous or arterial) Unit impulse response Dose Effect Electroencephalogram Predrug effect Gain effect Electrocardiogram Unit gain effect (open loop gain effect) Residual effect Parameter of unit gain transduction function (eq 18)

272 1 Journal of Pharmaceutical Sciences Vol. 82, No. 3, March 1993

Set point of ECS Parameter of unit gain transduction function Endogeneous control system

e“ Gain function dGldt Parameter of conduction function (eq 17) Parameter of conduction function (eq 17) Biophase drug level Intensity of response Endogenous neural stimulation Rate constant determining rate of endogenous control system gain change (eq 4) Unit gain transduction function Parameter of unit gain transduction function Pharmacokinetics Pharrnacodynamics Predictor variable (PK variable or infusion scheme) dMO{ }, Response mapping operator t, Time ts0&, Biophase equilibration times 4(tL Conduction function 4,(t), Extended conduction function

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