Revisiting the effect compartment through timing errors in drug administration

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Revisiting the effect compartment through timing errors in drug administration Patrice Nony, Michel Cucherat and Jean-Pierre Boissel The variations in the pharmacological effects induced by timing errors in drug intake are compared for two drugs, one acting by way of an effect compartment and the other directly from the central compartment. A simulation was performed for two drugs having the same concentration– effect relationship at the receptor site, the same mean effect at equilibrium and identical concentrations in the central compartment. In this article, Patrice Nony, Michel Cucherat and Jean-Pierre Boissel discuss how, for the same variability of concentrations in the central compartment, the variations in mean effects are different. When there is large variability in the interval separating two consecutive doses, the model that includes an effect compartment dampens the pharmacokinetic variability present in the central compartment. Such an approach may be useful for the prescription recommendations of drugs, especially those with narrow therapeutic indices. In therapeutics, compliance is defined as the degree of coincidence between a person’s behaviour and the prescription instructions given by his or her physician1. If consideration is restricted to strict drug compliance, several patterns of non-compliance can be defined: (1) delay in the beginning and/or the termination of treatment; (2) non-prescribed drug intake; (3) omission of one or several doses; (4) errors in the size of the dose to be taken; and (5) inappropriate and irregular timing in administration. This article is restricted to considering the influence of irregular timing on the pharmacological effect and its variability. In order to investigate the influence of an effect compartment on the pharmacodynamic consequences of timing errors in drug intake, a method is proposed whereby the results of irregular intervals between two consecutive doses on the variability in the pharmacological effect can be predicted. It involves a computer simulation starting from the mathematical equations used to describe the kinetics of an in vivo pharmacological response2,3.

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Characteristics of the selected pharmacokinetic–pharmacodynamic models Two pharmacokinetic–pharmacodynamic (PK–PD) models are considered: (1) the direct link model, in which the PD effect is a function of the drug concentration in the central compartment; and (2) the indirect link model, in which there is a delay in equilibration between the drug concentration in the central compartment and the concentration of active substance at the effect site (an approach involving an effect compartment as developed by Segre4 and more recently by Holford and Sheiner)5,6. The effect compartment model has been used to describe the effects of various drugs such as d-tubocurarine for muscle paralysis, cimetidine for inhibition of gastric acid secretion, digoxin for left ventricular ejection time shortening, quinidine and disopyramide for QT prolongation5, cibenzoline for premature ventricular contraction reduction7, and propranolol for myocardial contractility inhibition8. Even for the same class of agents (dihydropyridine Ca2+ channel antagonists), both direct and indirect link models have been used respectively, for nifedipine and amlodipine to characterize their concentration–antihypertensive– effect relationships9,10. Here, the example of two drugs having the same concentration–effect relationship at the receptor site and the same concentration in the central compartment at equilibrium, will be studied. However, in one case the pharmacokinetic–pharmacodynamic relationship will be based on an indirect link model as mentioned above, and in the other case the pharmacological effect will be directly related to the concentration of the central compartment. Thus, the problem is to compare, for these two drugs administered in repeated doses, the pharmacological variability of the effect, as a result of changes in the time period separating two consecutive doses.

Models used Two systems (A and B) are considered, respectively, with and without an effect compartment (Fig. 1). Both systems include an absorption compartment (compartment 1) and a central compartment (compartment 2), with system A having an effect compartment. The elimination process for compartment 2 and the effect compartment is associated with the rate constants k20 and ke0, respectively. However, this last constant (ke0) could be considered, independently from a purely mathematical point of view, as a truly composite rate constant reflecting elimination, access to the biophase receptor from plasma, or on/off rates at the receptor, i.e. the time dimension of the equilibration between the kinetics of the concentration in the central compartment and the kinetics of the effect. For higher values of ke0 and after one dose, the rates of increase and decrease of the effect will be greater, whereas the peak time of the effect will be shortened; after repeated administrations, the

Copyright © 1998, Elsevier Science Ltd. All rights reserved. 0165 – 6147/98/$19.00 PII: S0165-6147(97)01159-0

TiPS – February 1998 (Vol. 19)

P. Nony, Cardiologist and Pharmacologist, M. Cucherat, Biostatistician and J-P. Boissel, Professor of Clinical Pharmacology, EA643, Division of Pharmacology, Cardiovascular Hospital, 69003 Lyon, France.

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Two models or relations are considered: E(t) = C(t), where the effect observed at time t is equal to the concentration measured at time t at the receptor site. This model assumes a slope a of 1 and an intercept b of 0 for the linear direct effect model [E(t) = a . C(t) + b], in order to reflect directly pharmacokinetics (i.e. concentrations in the effect compartment for system A and in the central compartment for system B).

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equilibrium of the effect will be reached faster, with a greater amplitude of the effect variation (maximum – minimum) in the interval of time separating two consecutive doses.

corresponding to a hyperbolic relation (Emax model) between the concentration measured at time t at the receptor site and the observed effect. For this model, two cases are then identified with C(t) ≈ C50 or C(t) >> C50.

Effect summary Over an interval of time (t2–t1), between two consecutive doses, different parameters can be proposed to summarize the pharmacodynamic effect E(t). Here, only three effect—summaries are considered: (1) E mean or E , the mean pharmacodynamic effect over a period of time (t2–t1), with, t2

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Pharmacokinetic model If yi (t) represents the quantity of drug in compartment i at time (t), and kij the microconstant of transfer from compartment i to compartment j, q0i the quantity of drug present in compartment i at t = 0, and vi the volume of compartment i, the pharmacokinetic model can be written as11: (1) for system A, dy1 (t) / dt = –k12 y1 (t), dy2 (t) / dt = k12 y1 (t) –k20 y2 (t) –k2e y2 (t), and dye (t) / dt = k2e y2 (t) –ke0 ye (t), where the index e indicates the effect compartment, and supposing k2e << k20; and (2) for system B; dy1 (t) / dt = –k12 y1 (t), and dy2 (t) / dt = k12 y1 (t) –k20 y2 (t). The initial conditions of these two systems are: q01 = F.D, where F is the bioavailability parameter, D the dose of drug administered, and q02 = q0e = 0. The volume of the effect compartment, ve, for system A was calculated under the equilibrium condition and for a fixed period between two consecutive doses in order to obtain equal mean concentrations at equilibrium — (Ci) between compartment 2 and the effect compartment — — (C2 = Ce). The value of ve [= ye(t) / Ce(t)] was then incorporated into the model for the numerical simulations. At equilibrium, if one compares one system — with the other, — C2 of system A is approximately equal to C2 of system B, because ve << v2. 50

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Emax × C(t) C50 + C(t)

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(2) Emax, the maximum pharmacodynamic effect at time t during the period of time (t2–t1); and (3) Emin, the minimum pharmacodynamic effect at time t during the period of time (t2–t1). Change as a function of time of the pharmacological effect E(t) is given in Fig. 2a for both systems A and B, supposing constant intervals of time separating two consecutive doses.

Poor adherence simulation Assumptions Poor adherence corresponds to the failure to take the drug at the scheduled times. Timing errors or erratic dosing time in drug administration only are considered here, and it is assumed that all doses are similar and taken by the patient12. As various factors may affect the outcome of the simulations, the following processes are assumed to be constant: absorption, distribution (including protein binding), biotransformation and drug elimination. It is also assumed that the earlier doses of the drug do not alter the basic pharmacokinetic parameters and the pharmacokinetic–pharmacodynamic relationships (superposition principle).

Imposed variability of the intervals of time (␶i) separating two consecutive doses

The values of the ␶i intervals will be normally distributed as follows: ␶i = ␮ + ⑀i with ␮ corresponding to a

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CV (␶i) = 0.5 (random variation of the normally distributed ␶i intervals starting immediately with the first dose, and having been applied similarly to both systems A and B).

theoretical fixed value and where {⑀i} are independently and identically normally distributed with mean 0 and standard deviation ␴. For our simulations we varied the coefficient of variation (CV) of ␶i (corresponding to the ratio between the standard deviation and the mean of its distribution). This variation is discrete and made according to an arithmetic progression from 0 to 0.5, in increments of 0.05. For CV(␶i) = 0.5, more than 95% of the simulated values of ␶i were positive whereas the remaining negative values were ignored. Figures 2b and 2c give examples of the changes as a function of time of the pharmacological effect E(t) for both systems A and B, supposing a random variation in intervals of time separating two consecutive doses with CV(␶i) = 0.5.

Numerical simulations For each system and for a given value of CV(␶i), the coefficients of variation of the pharmacological effect [i.e. — CV(E), CV(Emax) and CV(Emin)] were calculated for 60 consecutive doses of the drug, this calculation being repeated 30 times. The 60 consecutive doses allow us to obtain one CV of the pharmacological effect, and the 30 repetitions a mean CV(effect) with its confidence interval (2 s.e.m.). Two types of simulation were carried out: (1) a random variation in the ␶i intervals which only starts after equilibrium state concentrations have been obtained, for the two systems A and B (situation I); and (2) a random variation in the ␶i intervals starting immediately with the first dose (situation II). The parameters have been taken from the modified model, presented in the work of Gabrielson and Weiner13. Thus, for a molecule with analgesic properties: k12 = 1.385; k20 = 0.389; k2e = 0.001; ke0 = 0.13249; v1 = 1; v2 = 5; ve = 0.0377387; F = 0.65; D = 100; ␮ = 4.

For the hyperbolic model: Emax =—100; C50 = 8.333 (when — C2 ≈ C50) or C50 = 8.333/5 (when C2 >> C50). The simulation was performed using the program Mathematica, version 2.2.3 (Ref. 14).

Consequences of poor adherence on the variability of the pharmacological effect The results are presented in graphical form (Fig. 3) for each system: the variation in ␶i (i.e. the variability corresponding to erratic dosing time in drug administration) versus the variability of the pharmacological effect. A polynomial meta-model joins the different points, to better identify the behaviour of each of the systems A and B. According to the three concentration–effect relations chosen, Figs 3a, 3b and 3c show the relation between the CV(␶i) and the mean CV(effect), where the — effect is given respectively by E, Emax and Emin, and this for both situations I and II. For example, in situation I, for CV(␶i) = 0.5 (Fig. 3a, left — — column, E graph), CV(E) is equal to 0.2 for system— A and 0.32 for system B. As the theoretical mean effect E in the linear model corresponds to a target value — of 8.33 (see Fig. 2a at steady state), our results mean that E may vary from 4.6 to —16 in system B, whereas in system A the variation of E will be limited between 5.2 and 11. The two considered drugs have the same concentration–effect relation at the receptor site, the same — average effect (E) at equilibrium for ␶i constant, and the same concentration at each time point in the central compartment. Our results show for these two drugs only by the existence of an effect compartment for one, that for the same variability in their concentrations in the central compartment [for a known positive value of CV(␶i)], the variability of their effect will be different. In situation I, whatever the amplitude of the variability in ␶i, the system involving an effect compartment TiPS – February 1998 (Vol. 19)

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dampens the pharmacokinetic variability present in the central compartment. In situation II, it is only when there is a large variability in ␶i that such a damping occurs. The result is similar for the variability in Emax—and Emin, which exhibits the same behaviour as that of E when the variability of ␶i increases. The results also look very similar for the linear—model [i.e. E(t) = C(t)] and the hyperbolic model when C2 ≈C50, because the hyperbolic model for these concentrations is mathematically similar to the linear model. However, the variability of the pharmacological effect is much 52

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lower for the hyperbolic model when C2 >> C50 (i.e. when the plateau of the hyperbolic function is approached or reached), as compared with the two other concentration–effect relations.

Limitations and practical implications Drug non-compliance has many facets, including those of the pharmacokinetics and pharmacodynamics15. The present approach highlights the importance of distinguishing between substrates that act by way of an effect compartment and those that act directly.

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with C (t) >> C50 (scale on y axis from 0 to 0.25). Filled diamonds, system A (effect compartment); filled squares, system B (direct effect); error bar, 1 s.e.m.; solid line, polynomial fitting (system A); dotted line, polynomial fitting (system B).

Knowledge only of the concentration–effect relation and of the concentrations in the central compartment (and of their variability) is insufficient to predict the variability of the effect. Such a methodology may be helpful to better identify ‘forgiving drugs’16,17, i.e. in the present example, the drug acting through an effect compartment. However, in situation II (variation in the ␶i intervals starting immediately with the first dose) for a null or low variability in timing errors, the results are somewhat different and explained by the fact that the system with an effect compartment comes to equilibrium more slowly.

The impact of our results is probably limited to those drugs with very narrow therapeutic indices (such as antiarrhythmic agents): this is where the effect of variations in dose timing would be most relevant. Assuming situation II to be more clinically relevant, in the most extremely variable ␶ situation (50% CV), CV in Emean and Emax is less than 20% for the hyperbolic pharmacodynamic model and less than 30% for the more ‘sensitive’ direct effect model (which reflects exactly pharmacokinetic variability). Unless the drug has a very narrow therapeutic index and does not have an effect compartment,

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Acknowledgements The authors are grateful to Dr Pascal Girard for helpful suggestions, and wish to acknowledge the editorial assistance of Dr Alison Foote.

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this amount of variability is not likely to be of much clinical consequence. However, not all drugs require a sustained pharmacological effect to be maintained throughout the dosing interval to achieve a desired therapeutic effect, so that the results can possibly be extended to the instances where Emin is one of the critical determinants of therapeutic effect, because of the greater sensitivity to variability in ␶ (e.g. analgesic agents prescribed in chronic pain, or ␤-adrenoceptor antagonists used to decrease heart rate in patients with stable angina). The present approach belongs to the general framework of ‘prescription help’ and may provide insight as to whether a maximum variability in the time interval between two consecutive administrations should be recommended. However, recommended dose regimens are chosen for all drugs from extensive clinical experience. Therefore, deliberately or inadvertently, safety and therapeutic implications of non-compliance have already been explored during clinical trials and are reflected by dosage recommendations. In addition, the therapeutic effect of a drug is not limited only to the variability of the pharmacodynamic and pharmacokinetic profiles. For example, in the case of anti-hypertensive therapy18, although great efforts are taken to obtain a low variation in arterial pressure over time, it would be illusory to attribute the observed decrease in the mortality under diuretic or ␤-adrenoceptor antagonist treatments only to this factor. Thus, in some therapeutic situations, correlations between the history of drug administration and clinical response are likely to be complex19, and one should not assume that all drugs will demonstrate an important relation between compliance and clinical outcome. We explored only one pattern of poor compliance20,21, which does not reflect the real-life compliance behaviour of patients. Compliance is a spectrum of variable administration with more undercompliance than overcompliance, timing errors occurring irrespective of the severity of the disease. In addition to the ‘holiday’ pattern (i.e. sequential days without drug administration), ‘whitecoat’ compliance has also been identified, i.e. the improvement in compliance in the days prior to a scheduled medical examination. Errors in the doses taken have also not been considered in our simulations. We have assumed in situation I that the timing errors in drug administration happened after a steady state had been obtained. The reason for this was to better explain the results of our simulation; in medical practice, however, situation I is not clinically relevant, but situation II seems much more likely, i.e. timing errors occurring right from the beginning of drug intake. Using the same approach, other points could have been considered: other pharmacodynamic parameters describing the effect, such as the peak/trough ratio23; other concentration–effect relations, such as the sigmoidal (Hill) model or other indirect link models23. Lastly, our approach was limited to the individual level, but further simulations may be performed at a population level24.

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S Concluding remarks From this simulation study, it can be concluded that the consequences of irregular timings in drug intake, in terms of pharmacodynamic effect, are likely to be different for a drug with an effect compartment as compared to a drug with a direct concentration–effect relationship. In addition to therapeutic drug monitoring and pharmacokinetic–pharmacodynamic modelling, simulation studies of compliance will probably become routine in the near future for certain difficult therapeutic fields. However, the integration of these simulation methods, like any other new technique, will have to be carefully evaluated. Selected references 1 Haynes, R. B. (1979) in Compliance in Health Care (Haynes, R. B., Taylor, D. W. and Sackett, D. L., eds), Johns Hopkins University Press 2 Levy, G. (1966) Clin. Pharmacol. Ther. 7, 362–372 3 Holford, N. H. G. (1982) Pharmacol. Ther. 16, 143–166 4 Segre, G. (1968) Il Farmaco 23, 907–918 5 Holford, N. H. G. and Sheiner, L. B. (1981) Clin. Pharmacokinet. 6, 429–453 6 Sheiner, L. B. (1987) J. Pharmacokin. Biopharm. 15, 553–555 7 Holazo, A. A., Brazzell, R. K. and Colburn, W. A. (1986) J. Clin. Pharmacol. 26,336–345 8 Corbo, M., Wang, P. R., Li, J. K. J. and Chien, Y. W. (1989) J. Pharmacokin. Biopharm. 17, 551–570 9 Kleinbloesem, C. H., Van Brummelen, P., Van Harten, J., Danhof, M. and Breimer, D. D. (1985) Clin. Pharmacol. Ther. 37, 563–574 10 Donnelly, R., Meredith, P. A., Miller, S. H. K., Howie, C. A. and Elliot, H. L. (1993) Pharmacol. Ther. 54, 303–310 11 Gibaldi, M. and Perrier, D. (1982) in Pharmacokinetics (2nd edn), Marcel Dekker 12 Wang, W., Husan, J. F. and Chow, S. C. (1996) Stat. Med. 15, 659–669 13 Gabrielsson, J. and Weiner, D. (1994) in Pharmacokinetic and Pharmacodynamic Data Analysis. Concepts and Applications, pp. 523–530, Swedish Pharmaceutical Press 14 Wolfram, S. (1991) in Mathematica: a System for Doing Mathematics by Computer (2nd edn), Addison-Wesley Publishing 15 Urquhart, J. (1994) Clin. Pharmacokin. 27(3), 202–215 16 Urquhart, J. (1991) in Patient Compliance in Medical Practice and Clinical Trials (Cramer, J. A. and Spilker, B., eds), pp. 301–322, Raven Press 17 Levy, G. (1993) Clin. Pharmacol. Ther. 54, 242–244 18 Dahlof, B. et al. (1991) Lancet 338, 1281–1285 19 Efron, B. and Feldman, D. (1991) J. Am. Stat. Assoc. 86 (413), 7–17 20 Steward, R. B. and Cluff, L. E. (1972) Clin. Pharmacol. Ther. 13, 463–468 21 Cramer, J. A., Mattson, R. H., Prevey, M. L., Scheyer, R. D. and Ouellette, V. L. (1989) J. Am. Med. Assoc. 261, 3273–3277 22 Meredith, P. A. (1994) Drugs 48, 661–666 23 Dayneka, N. L., Garg, V. and Jusko, W. (1993) J. Pharmacokin. Biopharm. 21, 457–478 24 Sheiner, L. B., Rosenberg, B. and Marathe, N. (1977) J. Pharmacokin. Biopharm. 5, 445–479

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