Stochastic Control of Pharmacodynamic Processes with Application to Terbutaline

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System., Galveaco, Texas, USA,I994

Stochastic Control of Pharmacodynamic Processes with Application to Terbutaline David Z. D'Araenio and Kyungsoo Park Department of Biomedical Engineering, University of Southern California Los Angeles, California 90089-1451 INTRODUCTION Stochastic control methods have been applied by a number of investigators to the problem of dose regimen design for controlling pharmacokinetic processes in both the population and individuals. We have recently reported the application of stochastic control methods to the problem of pharmacodynamic process control [1). The work reported hereu. develops an open-loop feedback, stochastic control framework for controlling FEV I (forced expiratory volume in I second), using the anti asthmatic agent terbutaline following subcutaneous injection of drug. Central to implementing stochastic control methods is the ability to calculate the density of the model response as well as model parameters, in both prior and feedback stages. In the work to be presented, the a priori dose regimen design uses direct sampling from the given population distribution, while importance sampling is used in the feedback control phase for density calculations, as reported in [1]. In addition, a sample censoring procedure is used in the feedback stage calculations to improve the computational efficiency of the overall control algorithm. METHODS

CODtrol Problem The control problem consists of an a priori control stage: that involves the caiculation of the initial dose hI * tru.t optimizes the dynamic response at 8 Or (e l ), given tht population uncertainty in the kinetic/dynamic model, and feedback control stages that calculate doses delivered at 24 hr (b 2·) and 48 hr (b)·) which optimize the responses at 32 hr (e~ and 56 hr (eJ, respectively, given the FEVl measurements obtained at selected times during the therapy . Optimal doses are calculated based on a target interval control objective, similar to that applied in [3] and [4]. The initial optimal dose b l • is defmed as

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-4ZII'FfLp(e1)clel} U

(2)

where the target interval, [L,U], was chosen to be [2,2.5] L. The FEVl measurements following administration of bl * are modeled as

:J.t)=e. j=1.2.3

Model Structure for Terbutaline

(3)

where measurement error V(I) is assumed to be distributed as v(I.,}-N(O ,oj) with OJ = O.1e(I), and Ij = [1 ,8.23] hrs. The frrst feedback optimal dose b2• is then defmed as

A two compartment linear model is used to describe the plasma concentration of terbutaline following subcutaneous administration, and includes the following parameters: absorption rate constant lea (hr-I), intercompartmental rate constants kJ 2 (hr-I) and k21 (hr-I), elilnination rate constant kJO (hr-I), and the volume of central compartment Vc (L). An effect compartment model is used to describe the pharmacodynamic response of subcutaneous terbutaline (FEVI), e(t) , defmed as [2] e(t) EE()+ (Emtu-EO)Ce(t) EC50+Ce(t)

bo (hr-I). The model parameters are assumed to be distributed lognormally with population data as reported in [2], and denoted by the vector a. a = [lea kJ 2 k2 J kJO Vc bo EC50 Emax EO] .

(4)

where k = 2 and I = 1. Three additional measurements are assumed to be available following b 2• at (25,32 ,47] hrs, and are used together with the frrst three measurements, to calculate b)· dermed by Eq. (4) with k = 3 and I = 1.2. Brent's method was used to solve the optimization problems dermed in Eqs. (2) and (4).

(1)

where the three dynamic parameters, Emax (L), EO (L) and EC50 (nglml), denote the maximum drug effect, the baseline

Computational Methods

effect, and the concentration producing 50% of Emax, respectively. Ce(t) represents the hypothetical effect site concentration, which is a function of the five kinetic parameters, and one dynamic parameter, the rate constant,

A direct sampling scheme is used to calculate the required integration given in Eq. (2) as Jp(e l)de =: t even; )/N, where even; is defined to be 1 if e l E (L,U] for

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and 0 otherwise. Here, a/, i = 1,.. ,N, arc the samples The following algorithm describes the from pea). importance re-sampling scheme with sample censoring used to approximate the integral givCD in Eq. (4):

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Step 1. Take N' samples i = 1,.. ,N', from pea). Step 2. Forj -= I, m if Iz(t)~(t)1 > 30p then Pr(alr) = 0, and go to step 5 Step 3. Pr(alr) = l(alr) I :E,. /" l(alr) Step 4. event' = I if et e [L,U] = 0 else Step 5. Jp(eJde t It: :E,. IN' (Pr(alr) x event)

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where m = 3 (first feedback) and 6 (second feedback), and 1(.) denotes the likelihood function. RESULTS Fig. } (dotted) shows the distribution of e I for population given bl· = 0.5 mg, computed with 20,000 sample points. A target interval probability of 0.33 is obtained at the optimal dose. The distributions of e 2 (dashed) and e) (solid) given b· for a selected individual are also plotted in Fig. 1. The resulting target interval probabilities given b2• (3.8 mg) and b)· (4.0 mg) are 0.82 and 0.97, respectively, with N' = 500,000. The actual model responses for this individual are marked as • (e l ), ,. (eJ and ... (e~, showing that relatively low dynamic response at 8 hr was successfully moved into the target interval at 32 and 56 hrs. Fig. 2 shows the distributions of e l (dotted), e2 (dashed) and e) (solid) . obtamed from simulation of 300 individuals. The percent of simulated subjects for which FEV} fell within the target interval improved from 36% to 77% to 93%, for the prior, fust feedback and second feedback stages, respectively.

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DISCUSSION This work illustrates the application of stochastic control methods to the control of pharmacodynamic processes using an important class of simultaneous pharmacokineticl pharmacodynamic models. To calculate the required posterior densities, an importance re-sampling procedure that includes sample censoring was implemented, but other computational approaches, including rejection sampling method [5], can also ~ used. ACKNOWLEDGEMENT This work was supported m part by Nlli grant P41 RR01861. REFERENCES

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1. K.Park and D.Z.D'Argenio. Stochastic control of pharmacodynamic processes. Proceedings of the 15th IEEE EMBS Conference, San Diego, pp. 973-974, 1993. 2. B.Oosterhuis et a!. Pharmacokinetic-pharmacodynamic mode ling of terbutaline bronchodilation in asthma. Clin. Pharmacol. Ther. 40:469-75, 1986. 3. D.Z.D'Argenio and D.Katz. Application of stochastic ~ontrol methods to the problem of individualising mtravenous theophylline therapy. Biomed. Meas. Infor, Contr. 2:115-122, 1988 4. D.Z.D'Argenio and J.H.Rodman. Targeting the systemic exposure of teniposide in the population and the individual using a stochastic therapeutic objective. J . Pharmacokin. Biopharm. 21 :223-251, 1993 5. A.F.M.Smith and A.E.Gelfand. Bayesian Statistics Without Tears: A Sampling-Resampling Perspective. The American Statistician. 46:84-88, }992

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Figure 1. Dillribution of c. ( .. ) for population together with thc IUtributiODS of ". (-) IUId ". (-) for a aelcctcd individual.

260