Jawień, Wojciech

Krypel, Łukasz

Piekoszewski, Wojciech

Acta Poloniae Pharmaceutica - Drug Research

English

Jagiellonian Univeristy Repository

Acta Poloniae Pharmaceutica ñ Drug Research, Vol. 65 No. 1 pp. 129ñ134, 2008 ISSN 0001-6837Polish Pharmaceutical SocietyPopulation pharmacokinetics (PPK) tries tofind a model, which describes a drug course in thepatientís body and investigates statistical distribu-tion of parameters of that model in the populationunder study. The purpose of this analysis may bepurely utilitarian (e.g. in therapeutic drug monitor-ing, TDM) or cognitive (as one may wish to do withthe data gathered during clinical observation).The statistical theory makes parameter estimationpossible also in the case of sparse data, i.e. when thenumber of measured drug concentrations from agiven subject is less than or not much greater thanthe number of pharmacokinetic (PK) model param-eters. This is of great advantage, because it makesroutine clinical data usable for population analysis.However, an estimation of the model for sparse datais a very difficult statistical and computational task.A number of approaches to this problem have beendeveloped. Table 1 enumerates the most importantof them along with the appropriate computer pro-grams.Each of these methods constitutes significantlydistinct approach to the PPK problem, both in theaspect of the assumptions as well as numeric algo-rithms applied. The first three approaches have theirorigins in the maximum likelihood method, the veryfundamental statistical principle. The NONMEM program searches for a likeli-hood function maximum by means of numericalalgorithms. Next two methods search for likelihoodmaximizing parameters without necessity of explic-it calculation of likelihood function. These methodsdiffer in the way they express the parameter distri-bution. In the parametric EM, the parametric familyof distributions (like normal or log-normal) isassumed. On the other hand, the non-parametric EMreplaces continuous statistical distributions by theirdiscrete approximations (3, 5). This allows for agreater flexibility in a description of parameter dis-tributions. The principle of the MCMC approach is quitedifferent. It makes use of statistical experiment inorder to determine empirical distribution of eachmeasurable population characteristic under investi-gation (6, 7). It was of interest to investigate, for theapproaches considered:● to which degree estimated parameters agreewith their true values, ● to evaluate the efficacy of theA COMPARISON OF COMPUTATIONAL APPROACHES TO THE POPULATION PHARMACOKINETICS. AN EXAMPLE OF TOXICOLOGICAL DATAWOJCIECH JAWIE—1, , £UKASZ KRYPEL1 and WOJCIECH PIEKOSZEWSKI21 Deptartment of Pharmacokinetics and Physical Pharmacy, Faculty of Pharmacy, Jagiellonian University,Collegium Medicum, 9, Medyczna Str., PL 30-688 KrakÛw, Poland.2 The Prof. Dr. Jan Sehn Institute of Forensic Research, 9, Westerplatte Str., PL 31-033 KrakÛw, PolandAbstract: Four different approaches to population pharmacokinetic analysis were applied to routine clinicaldata on carbamazepine intoxications in epileptic and alcoholic patients. The computational issues were noticedfor maximum likelihood based methods, while Markov Chain Monte Carlo approach revealed that the estimatesof absorption and lag parameters might be unreliable. Non-identifiability of these parameters may be the sourceof computational problems with other methods. Simultaneous use of different approaches to population phar-macokinetics is therefore advised, since it allows for verification of the obtained results.Keywords: population pharmacokinetics, statistical estimation, maximum likelihood, expectation-maximiza-tion, Markov chain, Monte Carlo, carbamazepine, epilepsy, alcoholismAbbreviations: CBZ ñ carbamazepine, EM ñ expectation-maximization, MCMC ñ Markov Chain MonteCarlo, PK ñ pharmacokinetic, PPK ñ population pharmacokinetics, TDM ñ therapeutic drug monitoring129* Corresponding author: email: wojciech.jawien@uj.edu.pl130 WOJCIECH JAWIE— et al.approaches/programs and the agreement of theresults between them.The first point was the subject of previous sim-ulation studies (8, 9), in which the true parametervalues where known and it was possible to control afulfillment of the assumptions.In the present study, a focus is on the real clin-ical data. Not only PK parameters true values remainunknown, but also one does not know to whichdegree assumptions of the respective methods arecorrect.MATERIAL AND METHODPatients and samplesPatient records with evidence of CBZ intoxica-tion from the Clinic of Toxicology, JagiellonianUniversity, Collegium Medicum were used. Therewere selected patients belonging to the followinggroups:■ epileptic patients (E),■ patients with alcoholism with alcohol detect-ed in their blood at the time of admission to the hos-pital (A),■ patients with alcoholism with no alcohol inblood at the time of admission (N).Each patient had CBZ concentration determined atthe time of admission, and the additional 0-4 con-centrations measured in the course of therapy(Figure 1). Thus, the data are sparse. It is unknownwhat CBZ form was taken. In addition, the supposi-tions on the time and dose of CBZ administrationare only rough approximations.Pharmacokinetic and statistical modelThe clinical observations made in the course ofroutine medical care suggest the CBZ pharmacoki-netics for therapeutic doses may be described by theone-compartment model with the first order elimi-nation (10). Due to the lack of data on the drug formadministered, both zero and first order absorptionhad to be considered. In the case of the zero orderabsorption ñ which may be adequate for the pro-longed release drug forms ñ the time of absorptionwas chosen as a pharmacokinetic parameter. For thefirst order absorption, the related PK parameter isthe first order absorption constant, . There is also aparameter  in both models, which combines the lagtime with the correction for the time of administra-tion.The volume of distribution could not be select-ed as the model parameter because of the uncertain-ty of the drug form and its dose. Therefore, theparameter is a mixture of the volume of distribution,the fraction absorbed and the correction of dose toits true value. The clearance is the fourth parameterof the model. Unfortunately, important patient data,such as body weight or height, were missed. It madeinvestigation of regression models for clearance andparameter unfeasible.The Gaussian distribution was assumed forparameter in the studied populations. For the otherparameters log-normal distribution was assumed.The intraindividual variability of the measured con-centrations was assumed to be normally distributedwith a heteroschedastic variance model (the constantcoefficient of variance model).RESULTS AND DISCUSSIONComputational issuesThe computations with the NONMEM programwere not fully successful for any group and model.Despite the estimates of the PK parameters and theirvariances were obtained, the estimation of the resultsaccuracy failed and the software notified the numer-ical issues. For a certain initial data, the programunexpectedly exited without any message. Thechoice of proper initial parameter values was themain difficulty in the computations with NONMEM.Thermo Kinetica was less sensitive on initialparameters, but the stop criterion for iterationsappeared to be too liberal: using the final parametervalues as initial values for the next run sometimesled to significantly different final results and eachrun the results were different.The simulation studies yielded promisingresults for the NPAG program (8). Unfortunately, itTable 1. Methods and software for population pharmacokinetic analysis of sparse data.Method Program Authors/originatorsNonlinear mixed-effect modelling NONMEM (1) Sheiner L., Beal S. (UCSF)Expectation maximization (EM) Thermo Kinetica (2) MentrÈ F., Gomeni R.Nonparametric EM NPAG (3) Jelliffe R., Schumitzky A., Leary R. (USC)Markov Chain Monte Carlo (MCMC) WinBugs/PKBugs (4) Lunn D.J., St Maryís Hospital, London A comparison of computational approaches to the population pharmacokinetics... 131performed worse in the present study. Only the testversion (beta) was available to the authors. It is notknown, if the program is still under development.No significant dependency of WinBugs/PKBugs results on the (reasonably chosen) initialparameter guesses was found. The observed statisti-cal, post-hoc distributions of PK parameters providea certain insight into the reasons of computationalissues encountered with other programs. Figure 2shows probability density plots for the zero-orderabsorption model parameters. The distributions forclearance and  parameter are in the agreement withthe assumed log-normal distribution. On the otherhand, the shape of other parameters distributionappears irregular and it changes with the number ofiterations. This indicates the estimates of absorptiontime and of  parameter are subject to a greater uncer-tainty. Nevertheless, the plot suggests short absorp-tion times, not exceeding 2 hours. This is in agree-ment with the patientsí profiles (Figure 1), where,for a majority of subjects, the absorption phase ispoorly seen. It may thus be presumed that the pro-longed release drug forms were rarely used.Consequently, the use of the first order absorptionmodel as a population model is better motivated.The results of such modelling with a PKBugs pro-gram are shown in Figure 3. The clearance andparameter distributions are again regular and similarto those from the previous model. Also remainingtwo parameters behavior is similar to that previous-ly observed. Therefore, the first order absorptionconstant and  parameter cannot be estimated with asatisfactory accuracy.In the study described, the comparison of theabsorption and elimination parameters is of specialinterest. Table 2 compiles estimates of these param-eters obtained for the subsequent programs and bothabsorption models. Despite significant differencesfor the same data sets were obtained, their mutualrelations between groups are, in general, consistent.Besides one exception, for each method and absorp-tion model the lowest clearance values wereobtained for the A group and the highest values ofthat parameter for the N group. Contrary, the firstorder absorption constant seems to reach the highestvalues in the A group.The above suppositions require a thorough sta-tistical reasoning before they could be expressed asTable 2. Clearance and absorption parameter estimates for studied programs and both absorption models. First-order absorption Patient group1Program E (n = 76)2 N (n = 35) A (n = 46)NONMEM 5.75 ± 8.523 8.45 ± 18.44 2.17 ± 2.14Clearance Thermo Kinetica 4.06 ± 0.70 4.64 ± 0.74 2.17 ± 1.14[L h-1] NPAG 4.76 ± 2.81 5.54 ± 3.10 4.41 ± 2.60PKBugs 5.28 ± 4.71 6.21 ± 6.04 3.70 ± 3.72NONMEM 0.70 ± 0.007 8.42 ± 0.08 7.70 ± 0.07Absorption Thermo Kinetica 3.58 ± 0.70 4.60 ± 1.55 14.56 ± 0.94constant ka [h-1] NPAG 3.74 ± 3.93 4.61 ± 2.73 5.60 ± 8.26PKBugs 11.06 ± 4.20 8.90 ± 5.96 28.21 ± 20.29Zero-order absorptionNONMEM 6.16 ± 3.90 2.22 ± 0.84 1.83 ± 0.88Clearance Thermo Kinetica 4.43 ± 1.02 4.92 ± 0.94 2.31 ± 1.27[L h-1] NPAG 3.33 ± 2.34 5.46 ± 2.92 5.11 ± 3.32PKBugs 5.29 ± 4.72 5.77 ± 5.56 3.61 ± 3.51NONMEM 0.30 ± 0.19 0.17 ± 0.15 0.18 ± 0.15Absorption time Thermo Kinetica 2.68 ± 0.97 1.93 ± 0.57 1.94 ± 0.49[h] NPAG 5.22 ± 2.99 5.98 ± 4.06 9.93 ± 3.68PKBugs 0.72 ± 0.24 0.53 ± 0.78 0.28 ± 0.13  1 E ñ epilepsy; N ñ alcoholism, no alcohol in blood; A ñ alcoholism, alcohol detected in blood.2 Number of subjects in group.3 Population parameter value ± its standard deviation in population.132 WOJCIECH JAWIE— et al.Figure 1. Spaghetti plot of CBZ concentrations in three patient groups. E ñ epilepsy; N ñ alcoholism, no alcohol in blood; A ñ alcoholism,alcohol detected in blood.A comparison of computational approaches to the population pharmacokinetics... 133Figure 2. Empirical probability density of PK parameters for the one compartment model with zero order absorption: a ñ log of clearance,b ñ log of V parameter, c ñ log of absorption time, d ñ t0 parameter, e and f ñ same as c and d, respectively, but obtained for the randomsample twice as large as for a-d (600000 vs 300000).Figure 3. Empirical probability density of PK parameters for the one compartment model with first order absorption: a ñ log of clearance,b ñ log of V parameter, c ñ log of absorption rate constant, d ñ t0 parameter.134 WOJCIECH JAWIE— et al.credible conclusions. This might be a subject of aseparate study.CONCLUSIONSThe computational issues noticed are not neces-sarily a proof of malfunctioning of the given approachor bugs in a software. Rather, they indicate the assump-tions of these approaches do not hold. It is possible, notall the model parameters are identifiable (11). For themajority of profiles the absorption phase is poorlynoticeable, therefore one could not expect reliable esti-mates for parameters related to this process.Simultaneous use of different approaches topopulation pharmacokinetics is recommended,because it provides better inspection into possibleimperfections of population pharmacokinetic model.REFERENCES1. Beal S.L., Sheiner S.L.: Nonmem userís guide.NONMEM Project Group, University ofCalifornia, San Francisco, 1992.2. Thermo Electron Corporation: Thermo Kineticañ Help, 2005.3. Jelliffe R.W., Schumitzky A., van Guilder M.:User Manual for the Non-Parametric EMProgram for Population PharmacokineticModelling, USC School of Medicine, LosAngeles 1993.4. Lunn D. et al.: PKBugs User Guide, www.win-bugs-development.org.uk, 2007.5. Leary R.: http://www.lapk.org/pubsinfo/newad-vances.php (2004).6. Gilks W.R., Richardson S., Spiegelhalter D.J.(Eds.): Markov chain Monte Carlo in practice.Chapman & Hall/CRC, Boca Raton 2000.7. Wakefield J.: J. Am. Stat. Assoc. 91, 62 (1996).8. JawieÒ W.: [Not only NONMEM ñ on statisti-cal experiments in population pharmacokinet-ics.] [in Polish]. The 9th Meeting of theAssociation of Monitored Therapy, £Ûdü, 1-3.12. 2006.9. Krypel £.: [An analysis and evaluation of thecomputer programs for population pharmacoki-netics.] [in Polish]. Master thesis, JagiellonianUniversity, Faculty of Pharmacy, KrakÛw 2007.10. Jiao Z. et al.: Ther. Drug Monit. 25, 279 (2003).11. Godfrey K.R., Chapman M.J.: J. Pharmacokin.Biopharm. 17, 229 (1989).

A comparison of computational approaches to the population pharmacokinetics : an example of toxicological data

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A comparison of computational approaches to the population pharmacokinetics : an example of toxicological data

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