Journal of Pharmacokinetics and Biopharmaceutics, Vol. 23, No. 6, 1995

A Pharmacokinetic-Pharmacodynamic Model of Tolerance to Morphine Analgesia During Infusion in Rats Dani~le M.-C. Ouellet I'2 and Gary M. Pollack 1'3 Received August 31, 1993--Final September 21, 1995 A pharmacokinetic-pharmacodynamic (PK-PD) model was constructed to describe the kinetics of tolerance development to morphine-induced antinoeiception. Tail-flick latencies in response to hot water (50~ were assessed in male Sprague-Dawley rats exposed to a 12-hr iv infusion of either morphine (1.4 to 3.0 mg/kg per hO or saline. Morphine-induced antinociception, expressed as the percentage of maximum possible response (% MPR), peaked after 120 rain of infitsion and decreased thereafter despite sustained systemic morphine concentrations. Both the rate and extent of tolerance development increased with increasing concentrations; an overall residual effect of approximately 24% MPR was observed at the end of the infusion regardless of the steady-state morphine concentration. The kinetics of tolerance offset were examined in a separate experiment by assessing tail-flick latency 15 min after morphine iv bolus (2 mg/kg) in tolerant and control rats. Recovery of response neared completion 18.5 days after a 12-hr exposure to morphine (2.0 mg/kg per hr). .4 PK-PD model was constructed to account for the delay in onset of antinociceptive effect and tolerance development relative to the blood concentration-time profile. According to this model, both the extent and the rate of tolerance development were modulated by the kinetics of the drug in the central compartment. Accumulation of a hypothetical "'inhibitor" acting either as a reverse agonist, a competitive or noncompetitive antagonist, or a partial agonist couM potentially account for the loss of pharmacologic effect in the presence of an agonist. The rate of tolerance development predicted fi'om the PK-PD model varied widely (28-fold) depending on the type of pharmacologic interaction selected to account for the loss of effect. Using the rate of tolerance offset to discriminate between the different models (tl/2 offset 5.4 days), onset and offset of tolerance was described accurately by postulating that the inhibitor behaves as a partial agonist with low intrinsic activity (5.5% MPR) and high binding a~nity for the receptor (ICso 15.0 ng/ml). KEY WORDS: pharmacokinetic-pharmacodynamic model; tolerance; morphine; antinociception; analgesia; opiates. Presented in part at the Seventh Annual Meeting of the American Association of Pharmaceutical Scientists, San Antonio, Texas, November 15-19, 1992. ~Division of Pharmaceutics, School of Pharmacy, The University of North Carolina at Chapel Hill, North Carolina 27599-7360. :Present address: Division of Drug Metabolism, Abbott Laboratories, Abbott Park, Illinois 60044-3500. 3To whom correspondence should be addressed. 531 0090.466x/95/1200.0531507.50/09 1995PlenumPublishingCorporation

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INTRODUCTION Tolerance is defined as a decrease in pharmacologic response after prolonged or repeated administration of a drug, and may be secondary to either pharmacokinetic (e.g., increase in systemic clearance) or pharmacodynamic (e.g., down-regulation of receptors) alterations during prolonged treatment. The phenomenon of tolerance has been observed for many different classes of therapeutic agents, particularly drugs with central nervous system activity (e.g., barbiturates, benzodiazepines, opiates) (1,2). Although pharmacologic treatment with morphine is considered the mainstay in management of severe pain associated with neoplastic disease, adequate control of chronic pain with morphine is difficult (3). Pain survey data indicate that a significant number of cancer patients receiving narcotic analgesics experience pain despite their pharmacologic treatment (4). A significant factor contributing to the failure of analgesic therapy with morphine is the development of tolerance, manifested clinically as an increase in the daily dose required to maintain a constant degree of analgesia (5-8). Pain management with morphine is complicated further by several factors. Substantial variability exists in morphine disposition, particularly oral bioavailability and systemic clearance, leading to a wide range of systemic concentrations after administration of a given dose (9,10). Morphine metabolites may possess significant pharmacologic activity; while morphine6-glucuronide (M6G) appears to contribute to morphine analgesia (11-14), morphine-3-glucuronide (M3G) may antagonize morphine antinociception (15-17). Finally, the progression of the underlying disease may lead to increased analgesic requirements over time. Although existing data (18-24) indicate that the mode (e.g., intermittent vs. continuous administration) and the route (e.g., iv vs. intrathecal) of opioid administration can modulate tolerance, the linkage between the kinet9ics of opioid administration and the rate and extent of tolerance development has yet to be explored in a quantitative manner. Pharmacokinetie-pharmacodynamic (PK-PD) models often are used to relate the time course of pharmacologic effect to systemic drug concentrations. Such models are capable, in theory, of predicting the intensity and duration of pharmacologic effect after administration of the drug under different dosing regimens. Only a few models have been developed to describe the time-dependent changes in pharmacologic response due to tolerance development (25-31). The availability of a PK-PD model of tolerance may be used to assess how tolerance development is modulated by the kinetics of drug administration, thus allowing the design of dosage regimens that minimize tolerance and maximize efficacy. Although PK-PD models are empirical in nature, quantitative aspects of the rate and extent of tolerance development might suggest directions for future mechanistic investigations.

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533

The objectives of the experiments reported herein were to (i) evaluate morphine-induced antinociceptive response during continuous infusion in the rat; (ii) characterize the kinetics of tolerance development and tolerance offset; and (iii) construct a PK-PD model to define the linkage between systemic disposition and the time course of morphine antinociception. The rat was selected for this project due to the availability of a quantitative model of analgesia, similarity to humans in terms of morphine disposition, and the absence of formation of M6G (32-34) which would complicate the development of the PK-PD model. EXPERIMENTAL

Determination of Antinociception Silicone rubber cannulas were implanted in the right jugular and right femoral veins of ether-anesthetized male Sprague-Dawley rats (250-400 g; n = 30) the day prior to the experiment. On the day of the study, either morphine (Research Biochemicals Inc., Natick, MA), 1.4 to 3.0 mg/kg per hr dissolved in normal saline or saline alone (1 ml/hr) was infused for 12 hr through the femoral vein cannula. Antinociceptive effect was evaluated prior to, and at timed intervals during, the infusion with a hot water-induced tailflick test according to a modification of the procedure described in Fennessy and Lee (35). Rats were placed in Plexiglas restraining cages 30 rain prior to tail-flick testing. The distal 5 cm of the tail was immersed in 50~ water, and the time to withdrawal of the tail was measured in duplicate. A cutoff time of 10 sec was used to minimize tissue damage. Antinociception was expressed as percentage of maximum possible response ( % M P R ) calculated as O/oMP R = Test L a t e n c y - Baseline Latency • 100 Cutoff Time - Baseline Latency

(1)

Animals not responding within 5 sec for both baseline measurements were excluded from further testing. Blood samples (0.3 ml, n = 11 per rat) were withdrawn from the jugular vein cannula immediately after pharmacological assessment. Tolerance offset was evaluated in a separate experiment. Rats (275325 g) were prepared according to the surgical procedure described above the day before a 12-hr infusion of morphine (2.0 mg/kg per hr; n=22) or saline (n=4). Baseline tail-flick response was determined prior to the 12-hr exposure, 0.5-day postinfusion, and immediately before the iv bolus test dose. Tolerance offset was evaluated at timed intervals following the 12hr exposure, ranging from 0.5 to 18.5 days postinfusion, by assessing the

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antinociceptive response 15 min after a 2-mg/kg iv bolus of morphine. The time for pharmacodynamic assessment was selected as the time to peak effect, determined in a separate experiment (data not shown) where morphine was administered as a 2-mg/kg iv bolus to male rats (n = 3) and the dynamic response was assessed at timed intervals after the dose (0, 7.5, 15, 30, 45, 60, 120, 180, and 240 min). The effect during the offset experiment was expressed as percentage of response in tolerant rats compared to salineexposed animals. Blood (0.3 ml) was sampled at the end of the infusion and immediately after tail-flick assessment. Morphine Analysis Blood morphine was quantitated by a sensitive and specific high performance liquid chromatographic (HPLC) method with fluorescence detection modified from procedures described by Turner and Murphy (36) and Venn and Michalkiewicz (37). Whole blood samples (0.3 ml added to 0.3 ml heparin, 100U/ml) were alkalinized with 0.6ml sodium bicarbonate (pH 9.0) after addition of 60 pl aqueous nalorphine (10 pg/ml) as an internal standard, and extracted with 3.5 ml chloroform. Following collection of the organic phase, the analyte molecules were back-extracted into 150 pl of 0.1 M phosphoric acid. The aqueous layer was subjected to analysis by HPLC. The analytical system consisted of a Beckman 112 solvent delivery module (Beckman Instruments Inc., Berkeley, CA), a Shimadzu SIL-9A autoinjector (Shimadzu Corp., Kyoto, Japan), and a Kratos FS970 LC Fluorometer (Kratos Analytical Instruments, Ramsey, N J). A Shimadzu (CR601) integrator was used to record detector output. Chromatographic separation was achieved on a 150 mm x 4.6 mm C6 (5 pm) column (Spherisorb, Phase Separations Inc, Norwalk, CT), with a mobile phase of 10% acetonitrile in an aqueous solution of 0.1% trifluoroacetic acid delivered at 1.0 ml/min. The fluorescence of column eluent was monitored continuously with excitation and emission wavelengths of 214 nm and 370 nm, respectively. Under these conditions, the retention times of morphine and the internal standard were 4 and 7 min, respectively. Standard curves constructed in blank rat blood were linear (r 2 > .999) up to 5000 ng/ml, with a detection limit of 25 ng/ml. Pharmacokinetic-Pharmacodynamic Modeling

Peak Effect Linear, Em~x,and sigmoidai Ema~models were used to fit the relationship between mean peak effect and morphine concentration, not accounting for any delays in the onset of action and assuming that the development of

A PK-PD Model of Tolerance to Morphine in Rats

535

tolerance was minimal at the peak response. Each of the equations listed below were fit to the peak effect vs. concentration data:

(2)

Linear Model: Peak Effect = Sex C

where So is the slope of the relationship of effect vs. morphine concentration (C). A ceiling effect was incorporated to adjust any predicted response equal to or greater than 100% to be the maximum measurable effect (i.e., 100%

MPR). Emax Model: Peak Effect- Emax x C r ECro + C r

(3)

where Emax is the maximum possible response (100% MPR), ECso is the morphine concentration producing 50% MPR, and ~, is the sigmoidicity factor describing the shape of the curve. For the simple Emax model, ), was set equal to 1. Tolerance Onset

The scheme depicting the PK-PD model of tolerance employed in the present study is shown in Fig. 1. This scheme was developed in part to

PK Model

~

PD Model

0

Effect K1 e

Ke0 . . . . .

zq

%,

Ce, Ve

Central 9

Net Effect

C, Vc

"Inhibitor" K KIO

li

ci, vi

K

i0

Fig. 1. Scheme describing the PK-PD model of tolerance. This scheme represents a single PK compartment with hypothetical effect and inhibitor compartments to describe the PD data. See text for equations of net effect and explanations of symbols.

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describe morphine concentrations during a constant iv infusion, and therefore assumes a single pharmacokinetic compartment with volume Vc, zeroorder input (k0), and first-order elimination (kl0). The model for pharmacodynamic response includes hypothetical effect (38) and "inhibitor" compartments linked to the central compartment by first-order rate constants (kl~ and kli, respectively). Separate effect and inhibitor sites are necessary to account for the delay observed in the onset of antinociceptive effect and tolerance development relative to the blood concentration-time profile. As originally proposed by Porchet et al. (28), the development of tolerance results from the accumulation of the hypothetical inhibitor in its compartment; tolerance is minimal following initial exposure to the drug and becomes significant with increasing concentrations of inhibitor. Thus, both the extent of tolerance and the rate of tolerance development depends on the kinetics of the drug in the central compartment. Disappearance from the effect and inhibitor compartments is characterized by the first-order elimination rate constants ke0 and k~0, respectively. The differential equations for concentrations in the central (C), effect (C~), and inhibitor (Ci) compartments are expressed as follows: (dC/dt)• V c = k o - k j o x V~x C

(4)

(dCe/dt) x V~=k,e x Vr x C-k~o x Vex C~

(5)

(dCJdt) x Vi=ku x gc x C - k i o x gi x Ci

(6)

This model is based on three assumptions: (i) first-order processes govern the onset (k~, k~i) and offset (ke0, ki0) of pharmacologic effect and tolerance development; (ii) the amount of drug distributing to the effect compartment is negligible relative to the total body load of the drug; and (iii) the clearances into and out of the effect and inhibitor compartments are equal. Net pharmacologic effect results from the interaction between the agonist in the effect compartment and the inhibitor. Increasing concentrations of a reverse agonist (negative intrinsic activity), a noncompetitive or competitive antagonist (no intrinsic activity), or a partial agonist (low intrinsic activity) will result in loss of pharmacologic effect in the presence of constant agonist concentrations, characterizing tolerance development. In their original paper, Porchet et al. (28) assumed that tolerance development to the cardioaccelerating effect of nicotine could be modeled by the presence of a hypothetical noncompetitive antagonist. However, the rate of tolerance development estimated by PK-PD modeling depends on the type of interaction selected. In the present study, different types of pharmacologic interactions between the agonist and the hypothetical inhibitor were considered in order to describe the net antinociceptive effect. The equations for net pharmacologic effect were derived from receptor theory (39-42) and are

A PK-PD Modal of Tolerance to Morphine in Rats

537

written as shown below for the four models. It should be noted that, in order to incorporate both an agonist and an inhibitor in the modeling scheme, it is necessary to utilize a model structure based on receptor-ligand interactions (i.e., the sigmoidal Em,x model) as opposed to more empirical mathematical relationships (e.g., a linear or log-linear relationship between effect and concentration of the agonist). Model 1. Agonist-Reverse Agonist: Emax • Cr~

/max • Ci~i

Net Effect = ECho + Cr~ I C ~ + C~ i

(7)

Model 2. Agonist-Noncompetitive Antagonist: Net Effect =

Emax• IC~'~ x C~ r ri ~ ri ECso • IC5o + Ce • IC5o + C7,' x ECho + C~ x C~ i

(8)

Model 3. Agonist-Competitive Antagonist: N e t Effect =

Em,x• C~ C~ + (EC~5o/IC~o) x C r, + ECho

(9)

Model 4. Agonist-Partial Agonist: Net Effect =

Emax • C~e • IC~o +/max • C~" • ECho

EC'~o x IC~o + ECho x C~ + IC~o • C~

(1 O)

The net effect results from the interaction of two variables, Ce and Ci, representing concentrations in the effect and inhibitor compartments, respectively. The equations are defined by the following parameters: Emax, the maximum possible response obtained in the presence of the agonist alone; ECso, the agonist concentration in the effect compartment producing 50% of E,,,x (reflecting the affinity of the agonist for its receptor); /max, the maximum response produced by the reverse agonist (negative response, Model 1) or partial agonist (positive response, Model 4); ICso, reverse or partial agonist concentration producing 50% Oflm~x (Models 1 and 4, respectively) or representing the concentration of noncompetitive or competitive antagonist corresponding to 50% of maximum occupancy (Models 2 and 3, respectively) ; and 7 (or ~q), the Hill exponent describing the shape of the concentration-response curve. Separate exponents were used only when two different receptors were postulated for the agonist and the inhibitor (Models 1 and 2). The addition of a separate sigmoidicity factor did not improve significantly the fit of Model 4 (partial agonist) and was not relevant in the case of a competitive antagonist (Model 3). It should be noted that the

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maximum effect reached is determined by the use of the cutoff time in the assessment of analgesia, rather than the effect of morphine itself. Therefore, deviations from the prediction of the model are expected at high concentrations since the maximum effect is artificially defined. For purposes of comparison to a previously published model of tolerance to morphine antinociception (30,31), the data were analyzed according to the following empirical equation that postulates that net pharmacologic effect results from the difference between an agonistic and a reverse effect, both of which are related linearly to concentrations in the effect and inhibitor compartments, respectively. Model 5. Linear Effect: Net Effect = S c x Cc - S i x Ci

(11)

where Se is the slope of the relationship between the agonist effect and concentration of the agonist in the effect compartment and S~ is the slope of the relationship between the reverse effect and concentration in the inhibitor compartment. This model structure is only relevant to the case in which tolerance development is driven by a reverse agonist. Tolerance Offiset

Since the objective of this study was to develop a model that could describe both onset and offset of tolerance, simulations were performed based on the parameters obtained from the five different P K - P D models evaluated to predict the rate of tolerance offset after a 12-hr morphine exposure. Predicted concentrations in the effect and inhibitor compartments were estimated at various times after a 12-hr infusion and at 15 min after an iv bolus dose of morphine. These concentrations were used to calculate the net pharmacologic effect 15 min after an iv bolus morphine challenge in morphine- and saline-exposed rats. To select the P K - P D model that best described the kinetics of pharmacologic response, the simulated profiles of recovery of pharmacologic response for the different models were compared to the experimental data. The effect data in the recovery experiment were expressed as percentage of control values, measured in saline-exposed rats receiving an iv bolus morphine challenge. For purposes of comparison across models, the degree of tolerance (initial Ci value) was adjusted in each model such that the predicted effect at 0.5-day postinfusion corresponded to the experimental value of 25% of control. The tolerance offset data were fitted with a hyperbolic function to generate an estimate of the apparent first-order rate constant associated with the dissipation of tolerance. Since the offset experiment was conducted over a longer period (18.5 days) than the infusion experiment (12hr), a better

A PK-PD Model of Tolerance to Morphine in Rats

539

estimate of k~owas obtained with the offset data. This value then was fixed, and the parameters of the final PK-PD model for the onset of tolerance were generated. D a t a Analysis

Due to the large interanimal variability in morphine clearance, rats were classified according to steady-state morphine blood concentration, rather than the infusion rate, into 6 different groups: 0 (saline control), 200-299, 300-399, 400-499, 500-599, and greater than 600 ng/ml (n =4-6 per group). Mean concentration- and effect-time profiles were used to generate the relevant pharmacokinetic and pharmacodynamic parameters. Clearance was estimated from the concentration-time profiles for each group. However, due to the limited information available to calculate distributional volume, this parameter (2.49 L/kg) was estimated in the pilot experiment described previously after administration of a 2-mg/kg iv bolus of morphine. Using the morphine clearance estimated from each group, the PK-PD model was fit simultaneously to all groups by nonlinear least-squares regression analysis (PCNONLIN, Lexington, KY). Assessment of the goodness of fit of the model to the observed data was based on Akaike's Information Criteria (AIC), residual plots, coefficients of determination, and standard error of the parameter estimates. RESULTS Mean blood morphine concentration vs. time profiles during continuous infusion are presented for each group in Fig. 2. Morphine clearance was extremely variable, with estimates of 55.6, 79.1, 89.1, 82.7, and 89.7 ml/min per kg for the >600, 500-599, 400-499, 300-399, and 200-299 ng/ml groups, respectively. No relationship was evident between clearance and infusion rate; the differences observed between groups were secondary to classification of the data based on steady-state concentrations, but are reflective of the degree of interanimal variability in morphine clearance. Tail-flick latencies remained constant during a 12-hr saline infusion in control rats. During morphine infusion, the antinociceptive effect peaked at approximately 120 min and decreased thereafter despite sustained systemic concentrations (Fig. 3). Tolerance development appeared to plateau at the end of the infusion. Averaging the results of all morphine-exposed animals, the residual antinociceptive response at 12 hr was approximately 24% MPR, with group means ranging from 12.5 to 31.9% MPR. Residual antinociceptire effect did not differ statistically among the experimental groups. The rate of tolerance development appeared concentration-dependent, with a

540

Ouellet and Pollack 2000 S" E 9.-.

1000

i C .m ,.C;

o

V

:f

[]

u

"o o o

lOO 0

200

400 Time

600

800

(min)

Fig. 2. Mean (+ SE) concentration-time profiles for the 200-299 (D), 300-399 (Vi--I), 400-499 (0), 500-599 (O), and >600 (A)ng/ml morphine groups. Solid lines represent the fit of the PK model to the data. m o r e rapid decline in the antinociceptive response at higher concentrations. The extent o f tolerance development was largest at the highest concentrations, with a loss o f p h a r m a c o l o g i c effect o f 62% M P R in the > 6 0 0 n g / m l g r o u p c o m p a r e d to only 17% M P R in the 200-299 n g / m l group. The results

120

A

ir fit.

'~

o~

q) i.u

-30 0

200

400

600

800

T i m e (min)

Fig. 3. Time course of morphine-induced antinociception at varying morphine steady-state concentrations. Symbols represent .~+SE (see Fig. 2); 9 represent the observed tail-flick response in saline-treated controls. For the purpose of clarity, only the fit of Model 4 (partial agonist) to the data is presented (solid lines).

A PK-PD Model of Tolerance to Morphine in Rats

541

100 ,-

A

rr fit.

/.'"

,/ T i O •

80 60

///~

./,"

40 LU

20 0 0

200

400

600

800

Blood Morphine (ng/mL) Fig. 4. Relationship between peak antinociceptive effect and morphine concentration. Symbols represent observed data (.~4-SE); lines indicate the fit of the linear ( . . . . ), the Emax (- - -), and the sigmoidal Emax (--) models.

of the fit of the linear, the E, . . . . and the sigmoidal E m a x models (Fig. 4) to the relationship between peak antinociceptive effect (assuming minimal tolerance development) and morphine concentration are listed in Table I. The goodness of fit was equivalent between the linear a n d sigmoidal Emax models, although the linear model was associated with a slight advantage based on the calculated AIC. A summary of the pharmacodynamic parameters generated by fitting mean data for the five PK-PD models is presented in Table II. The antinociceptive effect prior to significant development of tolerance could be related to morphine in the effect compartment with similar parameter estimates (ECso, y, keo) between the four models that were based on ligand-receptor Table L Parameter Estimates (Mean 4- SE) of the Fit of a Linear, Emax, and Sigrnoidal Emax Model to Peak Effect Versus Blood Morphine Concentrations ~ Parameter

S, (%MPR/ng per ml) E,,~ (%MPR)b ECso (ng/ml) 7 R2

AIC

Linear 0.139• n.a. n.a. n.a. 0.967 39.5

gmax

Sigmoidal Emax

n,a. 100 265 • 69 1 0.900 42.7

n.a. 100 323 • 36 2.11 + 0.64 0.961 40.3

aThe standard errors of the estimates reported were taken from the P C N O N L I N fit. n.a. = not applicable to the models. bE,~ax was fixed at the theoretical maximum, and was not determined as a parameter in analyzing the peak effect data.

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Ouellet and Pollack

Table !I. P a r a m e t e r E s t i m a t e s ( M e a n 5: SE) of the Fit o f E a c h P K - P D M o d e l to the T a i l - F l i c k D a t a D u r i n g M o r p h i n e Infusion"

Parameter

E,,,ax (%MPR)b ECso (ng/ml) r k~0 (hr -~)

Im,x (%MPR) IC5o (ng/ml) ~q k~ (day -~ ) Sr (%/ng per ml) Si (%/ng per ml)

AIC

Model 1 Reverse agonist 100 3974- 15 2.57:1:0.24 4.75:t: 1.27 48.3 4- 9.4 27.1 4-219 3.51 ~- 1.34 0.1714-1.43 n.a. n.a. 456.8

Model 2 Noncompetitive antagonist 100 4034-20 2.784-0.30 5.484- 1.75 n.a. 10.04- 601 1.844-0.81 0.0397:t:2.41 n.a. n.a. 469. I

Model 3 Competitive antagonist 100 3884- 14 2.684-0.24 4.584-1.10 n.a. 1194-120 n.a. 1.144-1.45 n.a. n.a. 449.2

Model 4 Partial agonist 100 389=1=15 2.644-0.24 4.334-1.07 7.1 + 17.5 13.8 4-230 n.a. 0.121+2.05 n.a. n.a. 452.0

Model 5 Linear (reverse agonist)

100 n.a. n.a. 1.824-0.46 n.a. n.a. n.a. 4.094-2.31 0.1754-0.029 0.1474-0.016 445.3

~ standard errors of the estimates reported were taken from the PCNONLIN fit. n.a. ~ not applicable to the model. bE,,~, was fixed at the theoretical maximum, and was not determined as a parameter in analyzing the data.

interactions. The resulting half-life for the delay in onset of antinociceptive effect relative to blood concentrations of morphine was approximately 9 rain. The fit of each of the models to the effect-time curves was comparable, resulting in similar AIC values. Based on the onset of tolerance data, the four models appeared equally good in describing the time course of morphine antinociceptive effect during infusion. However, depending on the model selected, a 28-fold variation in ki0 was observed, with values ranging from 0.0397 to 1.14 day -~. Standard errors of the estimates for kio and IC5o were rather large, with a high correlation between these two parameters. Thus, additional data (i.e., tolerance offset) were required for model selection. The kinetics of tolerance offset are presented in Fig. 5. Steady-state blood morphine ranged from 250 to 550 ng/ml (~ = 387 ng/ml) at the end of the infusion, contributing to the variability observed in the offset data. The offset of tolerance appeared to follow first-order kinetics, with the effect approaching 100% of control (i.e., complete resolution of tolerance) at 18.5 days postinfusion. Based on the simulated profiles, Model 4 (partial agonist) best described the offset of tolerance. The fit of the offset data generated an estimate for kio of 0.1284-0.027 day -1, close to the value obtained with Model 4. The parameter estimates of the final P K - P D model of tolerance are presented in Table III. No significant changes in baseline tail-flick latencies were observed in rats exposed to morphine between the preexposure and the 0.5-day postinfusion responses with mean values of 3.45 4-0.19 and 3.79 4- 0.15 sec, respectively. In agreement with these observations, Models 2 and 3 predicted that tail-flick latencies returned to baseline value at 0.5day postinfusion, while Model 4 predicted a slight positive increase (+5.1% MPR) most likely undetectable experimentally considering the variability observed in tail-flick latencies. A significant decrease in tail-flick response

A PK-PD Model of Tolerance to Morphine in Rats

120

543

Model 5

m

___!_

100 m

o L,. tO

O

80 60 40 20

~A_/ ~ - ~ - _-J------

\ .Model .... 2

-1 0

I

I

I

I

4

8

12

16

Time Post-Infusion

20

(day)

Fig. 5. Antinociceptive effect 15-min post-iv bolus morphine challenge (2mg/kg) in morphine-tolerant rats. Data are expressed as 24-SE relative to the effect in morphine-naive rats. Lines represent simulated data for each of the five models.

(-24% MPR, corresponding to a 0.7-sec tail-flick latency) was predicted by the reverse agonist model (Model 1); such hyperalgesia was not observed in the present study. To describe both onset and offset data, the rate of tolerance development to morphine antinociception is characterized most accurately by assuming an interaction of the agonist with a hypothetical partial agonist. Table i l l Parameter Estimates (Mean + SE) of the Fit of the Final PK-PD Model Selected to Describe Both Onset and Offset of Tolerance" Parameter Emax (%MPR) b ECso (ng/ml) ? k,0 (hr-')

lm~x (%MPR) 1C5o (ng/ml) ki0 (day-])c

Final model (partial agonist)

! O0 390 4- 13 2.61 4-0.23 4.424- 1.10 5.45 4- 12.56 15.0 + 2.7 0.128 4- 0.027

~ standard errors of the estimates reported were taken from the PCNONLIN fit. bE,a,, was fixed at the theoretical maximum, and was not determined as a parameter in analyzing the peak effect data. ~k~owas generated using the offset data and was fixed in modeling the onset of tolerance data.

544

Ouellet and Pollack

Tolerance then may be described by the following parameters: the estimated value of ki0, the first-order rate constant for tolerance onset and offset, was 0.128 day -j , indicating that the half-life of tolerance development was 5.4 days; an estimated IC5o of 15.0 ng/ml, indicating a relatively high affinity between the putative partial agonist and the receptor as compared to the agonist-receptor interaction (ECso 390 ng/ml) ; and finally, the/max value of 5.45% MPR, indicating the maximum loss of response at steady-state inhibitor concentrations would be 95%. Tolerance development and offset were associated with a long half-life despite the fact that tolerance appeared to plateau by the end of the 12-hr infusion. This apparent discrepancy can be explained by the fact that inhibitor concentrations exceeded the IC5o rapidly; further increases in inhibitor concentration during infusion no longer decreased significantly the net effect. DISCUSSION Previous studies investigating the pharmacodynamics of morphine have failed to show a direct relationship between the kinetics in plasma or brain and the time course of pharmacologic effect (43-45). Paalzow (46), through fitting an effect compartment model to the data published by Dahlstr6m et al. (45), described the kinetics of antinociceptive effect with an ECso of 442 ng/ml and a half-life for equilibration with the effect compartment of 10 rain. Antinociceptive effect was measured with vocalization responses to electrical stimulation after a single administration of morphine as an iv bolus (1.7, 2.5, 3.8, and 5.6 mg/kg). These parameters are in close agreement with the parameters generated by the model described herein prior to significant development of tolerance. Using a linear model to describe the relationship between antinociception and morphine concentration in the effect compartment, half-lives for onset of effect ranging from 13 to 34 min have been reported following administration of morphine as an infusion of different durations or as an iv bolus (30,31). Even though the fit of the relationship between peak effect and morphine concentration was slightly better with a linear model, the sigmoidal effect equation was preferred to describe the action of the agonist in the overall P K - P D model. The linear effect model is limited in the present context by its empirical nature and the inability to model agonist-inhibitor interactions with this approach. The primary objective of tbe present experiment was the construction of a P K - P D model to describe the kinetics of morphine-induced antinociception, including time-dependent changes due to the development of tolerance. The P K - P D model, which is similar to a previously published model (28) for tolerance development, included effect and inhibitor compartments linked to the central compartment with first-order rate constants as depicted in Fig.

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1. One of the assumptions required for PK-PD modeling is that clearances into and out of a hypothetical compartment are equal so that onset and offset of the dynamic response are described by a single first-order rate constant (e.g., keo, ki0). In the present study, the kinetics of the offset of tolerance to morphine-induced antinociception were determined independently in order to select the most accurate PK-PD model of tolerance. The rate of tolerance offset determined experimentally was compared to the time course predicted for the different models. Despite the acceptable goodness of fit for each of the PK-PD models evaluated to the pharmacodynamic data during the infusion, and the similarity between the parameter estimates for the onset of antinociception prior to significant development of tolerance, only the model involving the action of a partial agonist could predict accurately the rate of tolerance offset. Thus, the development of tolerance to morphine-induced antinociception can be described by postulating that increasing concentrations of a partial agonist displace the agonist from receptor binding sites, resulting in loss of pharmacologic effect. It should be emphasized that this model is empirical from the standpoint that it is based on the presence of a hypothetical metabolite (partial agonist), and that the actual existence of a partial agonist to explain the development of tolerance is not required. However, the data are consistent with the existence of partial agonists of either exogenous (e.g., morphine metabolites) or endogenous origin. Model 1 was based on the assumption that the action of a reverse agonist could account for the development of tolerance, with the net pharmacologic response resulting from the difference between two sigmoidal equations. A similar model based on the difference between two linear relationships (Model 5) has been used to describe the time course of pharmacologic effect during and after a 94-hr morphine infusion (30), as well as short-duration infusions of high-dose morphine in rats (31). A characteristic of this model is that, in the presence of significant inhibitor concentrations, the net pharmacologic effect will be negative with tail-flick latencies being shorter than the baseline value. Ekblom et al. (30) suggested that, in addition to modeling tolerance development, this approach is potentially useful to model withdrawal (essentially as a rebound effect). However, no rebound effect was observed in the present study. Due to the very short half-life for tolerance development predicted by this model (4 hr), the linear pharmacodynamic model was unable to predict the kinetics of tolerance offset. Investigations of the molecular mechanisms of opioid tolerance have generated conflicting results, and the phenomenon of tolerance is still poorly understood (47,48). Blfisig et al. (49) examined the mechanism responsible for tolerance development by constructing a series of dose-response curves in rats with different degrees of tolerance. Tolerance was induced with constant

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exposure to morphine by implanting different numbers of pellets subcutaneously (1 to 24 75-mg pellets) for different periods of time (4 to 10 days). The dose-response curves shifted to the right with increasing degrees of tolerance, with a reduction in the observed maximum effect. No further increase in the dose administered to tolerant rats could produce the maximum effect measured in naive animals. These results suggest that tolerance is secondary to noncompetitive changes rather than alterations in the binding affinity, and are consistent with the present PK-PD model of tolerance involving the interaction of a partial agonist with a considerably lower (more than 25-fold) IC5o compared to the ECso of the agonist. The nature of the interaction between a partial agonist and an agonist is by definition competitive; however, considering the difference in binding affinity (IC50 vs. ECso), it is unlikely that significant displacement of the partial agonist could be achieved even with high concentrations of the agonist. At steady state, the binding sites would be occupied preferentially by the partial agonist, which by itself could achieve a maximum antinociception of only 5.45% MPR. The rate of tolerance development estimated based on a noncompetitive interaction (Model 2) between the agonist and the inhibitor considerably underestimated the rate of offset of tolerance. This model has been used successfully to describe the rapid development and disappearance of tolerance to nicotine- (28) and caffeine-induced (50) pharmacologic effects. Not surprisingly, therefore, kinetic models of tolerance development to different classes of agents are consistent with different apparent underlying mechanisms of tolerance. The development of tolerance to morphine-induced antinociception has been studied extensively in animal models with a variety of methods for inducing tolerance. Although limited information is available on the quantitative aspects of tolerance development, the rate of tolerance development appears to be dependent on the mode of morphine administration: complete development of tolerance has been reported within 8 hr (21,22), 5-7 days (18,19), or 7-9 days (23,24) following administration of morphine as a continuous iv infusion, continuous intrathecal infusion, or intermittent boluses (subcutaneous or intraperitoneal), respectively. The kinetics of drug administration therefore appear to modulate tolerance development. This is consistent with the PK-PD model in which the kinetics of the partial agonist responsible for the development of tolerance are linked to systemic agonist concentrations. The predictability of the PK-PD model with alternative modes of morphine administration will be evaluated in prospective experiments. Several studies have suggested that the accumulation of the major metabolite of morphine, M3G, may contribute to the development of tolerance (15-17). Ekblom et al. (17) investigated the influence of M3G on

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morphine-induced antinociception measured as the vocalization threshold after electrical stimulation of the tail. The pharmacodynamic response to iv bolus morphine (10 m g / k g ) was decreased slightly in rats exposed to a 4day iv infusion of high-dose M 3 G (4 m g / k g per hr) compared to salineexposed animals. However, because of the high exposure to M 3 G and the presence o f morphine in the systemic circulation during M 3 G infusion (presumably due to hydrolysis of M 3 G to morphine), the exact role of this metabolite in tolerance development remains to be defined. However, this preliminary report suggests that it is unlikely that M 3 G contributes significantly to the decrease in morphine-induced pharmacologic response during prolonged morphine administration. Investigations of potential mechanisms of tolerance development have focused mainly on alterations at the receptor level, such as changes in binding affinity or capacity. Following the discovery of opioid receptors in the brain, peptides with morphine-like properties have been identified as endogenous ligands for these receptors; their role in endogenous pain control remains to be elucidated. It has been suggested that morphine administration m a y trigger the release of antiopioid peptides that would attenuate the pharmacologic response to morphine (51). The effect o f morphine administration on endogenous peptides, and in particular the role of these peptides in morphine tolerance, warrants further investigation. REFERENCES

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