PHARMACOKINETICS

J Clln Pharmacol 1988;28:1059-1063 1059

Population Pharmacokinetics

T. M. Ludden, PhD

D uring the process of drug development and eval-uation, the biopharmaceutics and pharmacoki-

netics of therapeutic agents and their specific for-mulations are generally tested in young, healthy vol-unteers. While safety and efficacy studies areperformed in patients, there is seldom monitoring ofbiopharmaceutic and pharmacokinetic variables.Upon marketing the drug product, the clinician isoften left without clear guidelines for the design ofinitial, individualized, optimum dosage regimens orfor subsequent adjustments. Safety and efficiency ofdrug use could be improved if, at the time of market-ing, there were sufficient data available to describethe pharmacokinetic behavior of the drug in the pa-tient populations that were most likely to receive it.

DEFINITION AND RATIONALE

It has been appreciated for many years that drugdosage requirements in patients vary widely forpharmacokinetic and pharmacodynamic reasons.The influence of patient characteristics such as bodysize, gender, age, physical and pathophysiologicstates, genetics, environment and concurrent ther-apy on various pharmacokinetic parameters has re-ceived extensive attention.1 There are tabulations orequations for the calculation of the expected valuesfor pharmacokinetic parameters in various patientpopulations. Much of this information was obtainedafter the drugs were marketed. On the other hand,considerably less is known about pharmacodynamicparameters.2’3 There is a strong shift in research em-phasis toward obtaining a better understanding ofhow patient characteristics, environmental factorsand concurrent therapy quantitatively influence theexpected pharmacodynamics.

POPULATION PHARMACOKINETICS

This has become the general descriptor for both

pharmacokinetics and pharmacodynamics in groups

From the College of Pharmacy, The University of Texas-Austin and TheDepartment of Pharmacology, The University of Texas Health Science

Center, San Antonio, TX 78284. Address all correspondence to:Thomas M. Ludden, PhD The Department of Pharmacology The Uni-versity of Texas Health Science Center San Antonio, Texas 78284-7764.

of patients having similar characteristics. The pur-pose of population pharmacokinetics is to providequantitative or at least semi-quantitative guidelinesfor drug dosage individualization. Ideally, once a pa-tient has been adequately characterized by history,physical examination and clinical laboratory tests,the clinician could select an appropriate dose anddose interval. Of course, essentially all predictivetechniques are imperfect and therefore a dosage reg-imen predicted for an individual may differ to somedegree from the optimum regimen.

Sources of variation that contribute to differencesbetween expectation and outcome are usually cate-gorized as interindividual and intraindividual in ori-gin. The presence of interindividual variation sug-gests that even though expected parameter valuescan be calculated for an individual patient based onprevious research and experience, the particular pa-tient at hand may have parameter values that differfrom the expected values.

Intraindividual variation includes measurementerrors involved in quantitating drug concentrationor response and random changes in a patient’s pa-rameter values over time. Also model misspecifica-tion errors which arise because all mathematicalcalculations for predicting parameter values anddosage regimens are oversimplifications of realitypotentially influence intrasubject variation.

The probability of finding a particular value in asample of known size from a normally distributedpopulation can be calculated if the mean and vari-ance are known. Combined or joint probability oftwo particular parameter values yields a three di-mensional frequency profile as shown in the Figure.Joint probabilities of more than two parameters can-not be displayed geometrically but can be dealt withmathematically.

Knowledge about the quantitative aspects of in-terindividual variation can provide information toassess how well an individual value can be pre-dicted from patient characteristics and other factorsbut it does not help to make that prediction moreaccurate or precise. The ability to design appropriateinitial regimens rests completely on good prior esti-mates of the mean parameter values for the particu-lar patient population. However, if a feedback tech-nique is employed wherein measurements of drug

FCs

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1060 #{149}J Clln Pharmacol 1988;28:1059-1063

Figure. Example of a frequency curve for the combined distribu-

tion of two parameters, VBW, the volume of distribution relative to

body weight and Cl, the systemic clearance, (Reproduced from

Katz (4) with permission)

concentration or response are obtained during aninitial trial regimen, then this patient-specific infor-mation can be combined with prior informationabout the mean values their variances and the vari-

ance of the intrasubject errors. Prior population in-formation and current patient-specific data can becombined to predict the most probable parametervalues for an individual patient from the observedconcentrations or responses. The mathematical der-ivation of this procedure is based upon Bayestheorem for conditional probabilities.5’6’7

DETERMINATION OF PARAMETERS FORPATIENT POPULATIONS

Traditionally, mean pharmacokinetic parametervalues have been obtained by performing detailedstudies in a limited number of individuals. Numer-ous measurements of circulating drug concentrationwere obtained for each individual after a single drugdose or after achievement of steady-state. Parametervalues were estimated using unweighted orweighted least squares nonlinear regression analysisand an appropriate compartmental or noncompart-mental pharmacokinetic model. The estimated(multivariate) parameter value obtained for each in-dividual was taken to represent the true parametervalue for that individual. Individual values are then

used to calculate the mean and variance of the(multivariate) parameter. There are potential prob-lems with this two-stage approach. The individualparameter estimates may be imprecise estimates ofthe true individual parameter value due to intra-subject variation. If an identical study was repeatedin the same subject (whose true parameter valuewas not varying) on multiple occasions, somewhatdifferent parameter values would likely be obtainedon each occasion. Intrasubject variation in parame-ter estimates is small if there are a large number ofobservations per subject, if these observations aremade at times that provide information about theparameter, and if the parameter is actually time in-variant. However, for many reasons it is difficult toobtain large numbers of blood samples from patientsand, in all likelihood, the parameter value does varysomewhat from day-to-day even in a generallystable clinical situation. In spite of these problems,traditional studies appear to have provided reason-ably good estimates of parameter means. However,the resulting estimates of the variances, particularlyfrom studies with too few patients, tend to be inac-curate and inflated.8

NONMEM has evolved from the description bySheiner and co-workers9-’#{176} of a strategy for extract-ing population parameters, means and variances,from sparse data collected during routine patientcare. In general, there are no restraints on samplingtimes and data can be collected times after routinedoses over a period of several days. The method canextract whatever information is in the data regard-ing the parameters. However, it is likely that atten-tion to sampling times can increase the quality of theinformation obtained.11 Samples obtained at thetime of peak serum concentrations usually containthe most information about the volume of distribu-tion whereas samples obtained at later times are in-formative about clearance. Steady-state peak (postabsorption) and trough (predose) are useful whenboth Vd and Cl are to be estimated. Data obtainedprior to the achievement of steady-state provideeven more useful information about Vd and Cl.

The approach developed by Sheiner and asso-ciates uses all data as one set to separate intraindivi-dual from interindividual sources of variation. Dur-ing the analysis the samples from a given patientremain identified with that patient and yet the en-tire data set is computationally available. This per-mits the estimation of not only the mean parametersof the model but also the variances. A relatively newmethod (in pharmacokinetics) for nonlinear regres-sion analysis called extended least squares12’13 mustbe used to obtain estimates of the variances. (Under

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the assumption of normality this is a maximum like-lihood method.) Parameter means and variances aswell as intrasubject variance obtained by applica-tion of this method are ideally suited for develop-ment of a Bayesian regression algorithm for optimi-zation of therapy.

The digital computer software currently availablefor this analysis is NONMEM. The name NONMEMstands for nonlinear mixed effects models. Thephrase mixed effects describes the fact that themathematical model used in parameter estimationincludes both fixed and random factors that describethe data. The fixed factors or effects are assumed tohave no error and thus, for the purposes of the popu-lation model, are exact. Sampling times, dose sizes,patient age, weight, serum creatinine, etc. are fixedeffects. Although all of these data may contain someerror, it is usually small as compared to the othersources of variation. It should be kept in mindthough, that errors in fixed effects will influence theapparent interindividual and intraindividual vari-ance estimates. Random effects are of two types:those yielding interindividual differences in param-eter values not accounted for by the fixed effects andthose resulting in the intraindividual variation. TheNONMEM user must provide the general form of therelationships between fixed effects and the pharma-cokinetic parameters such as Cl, Vd, and F.

The output from the NONMEM program containsmany important components. These include the ob-jective function value, the parameter estimates (in-cluding the variance estimates), the standard errors,the covariance matrix, inverse covariance matrixand correlation matrix of the estimates and user de-fined tables and scatterplots. In addition, the objec-tive function value can be used to test statisticalhypotheses concerning alternative models, includ-ing the statistical models for intersubject and intra-subject variation. It can also be determined if thereare significant covariances among selected parame-ters.

The standard errors, and the covariance, inversecovariance and correlation matrices of the estimatesprovide information about the parameter estimationprocedure itself. It would be apparent if one or moreparameters were poorly estimated. This would mostlikely be seen as large relative standard errors of theestimate or as a high correlation between parameterestimates. Difficulty in parameter estimation mayarise because the data set contains inadequate in-formation about the parameters or because informa-tion is confounded. For example, if only elderlysubjects had decreased renal function and all youngsubjects had normal renal function, it would be dif-

ficult to separate out an effect of age on drug clear-ance in addition to the effect anticipated by the wellcharacterized effect of age on renal function. How-ever, the effect might be detected if the data setincluded patients with a wide range of creatinineclearances in both young and elderly subjects.

User defined scatterplots from NONMEM are ex-tremely useful for initial examination of the data setfor entry errors and for obvious correlations. Deter-mination of goodness of fit for a particular modelmust rely not only on the objective function valueand the precision of the parameter estimates butmust include inspection of plots of the weighted re-siduals versus predicted concentration, time andother fixed effects. A nonrandom distribution ofweighted residuals indicates that the model used isnot entirely appropriate. The model must then berevised, usually by adding parameters and/or fixedeffects. It may be possible to identify one or morefactors associated with the “outliers” that was notpreviously considered. Generally speaking, the ki-netic model to be used should be well defined priorto a population analysis. Identification of importantfixed effects and the appropriate models for randomeffects will provide adequate challenge to the inves-tigator.

Other approaches to population data analysis havebeen suggested and additional software packages arelikely to become available in the near future. A re-cently published book chapter discusses in muchgreater detail many of the topics touched upon inthis brief presentation and includes descriptions ofdifferent methods for population analysis.14

PHARMACOKINETIC SCREEN

Considerable debate has evolved since the first for-mal suggestion that a pharmacokinetic screen be in-corporated into the drug development process. Theability to model drug disposition and response using“real” patients has a great deal of intuitive appeal.However, there are certain obvious problems withthe opportunistic approach to data collection. If dataobtained only during the routine evaluation of pa-tient therapy is used then biased estimates of popu-lation parameters may result. The more difficult thepatient is to treat the more observations may bemade and, therefore, his data may have a greaterinfluence on the parameter estimates. One argu-ment is that this is exactly the type of patient whoneeds to be emphasized in the clinician’s “database” since this type of patient will be most likely torequire adjustment of his therapeutic regimen.

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However, biased estimates of parameter means cancreate problems for the initialization of drug regi-mens before patient-specific data are available forfeedback and Bayesian analysis. Unbiased popula-tion parameters may be obtained using prospectivephase II, III and IV studies designed to avoid unbal-anced, biased sampling or from pharmacokineticstudies by salvaging from the clinical laboratorysurplus blood, plasma or serum samples obtained forother clinical test and assaying these samples fordrug content. When “salvaged” samples are used,care must be taken to assure that the time of bloodsampling is known with appropriate accuracy andthat the biological sample has been stored to assuredrug stability. Benet and Sheiner15 have provided anexcellent summary of various approaches that canbe used to conduct a pharmacokinetic screen andthey discuss the costs, benefits and problems sur-rounding its implementation. If the predictions thatarise from the screen are subsequently validated byclinical experience then this approach may prove tobe more efficient, less costly and more relevant thanlimited studies in special populations. For example,the comparative study of pharmacokinetics inhealthy young and healthy elderly subjects mayhelp to answer the question: “Does age alone influ-ence the kinetics of this drug?” but does not addressthe problem of dosage individualization in an un-healthy elderly patient. Only by collecting and ana-lyzing data from such “real” patients can one hopeto answer the clinician’s questions: “What doseshould I initially prescribe for this patient and howdo I efficiently adjust his dose if the initial dose isinappropriate?” For this reason, it is probable thatthe population approach to data collection and anal-ysis will increase in popularity and acceptance. Al-ready several publications have appeared concern-ing the population pharmacokinetics of digoxin,1#{176}phenytoin,16 procainamide,” mexiletine,18 lido-caine,19 warfarin,2#{176} phenobarbital21 and alprazo-lam.22 The methodology has also been applied to thepharmacodynamics of theophylline23 and cimeti-dine. A review of the application of population anal-ysis to pharmacokinetics has been recently pub-lished.24

SUMMARY

The major strength of the population analysis ap-proach is that useful information can be extractedfrom sparse data using blood samples and pharmaco-logic monitoring during routine safety and efficacystudies conducted during the development of a drugproduct.15 The results of these analyses may lead to

integrated pharmacokinetic-pharmacodynamicmodels that can aid the clinician during the initia-tion and adjustment of therapeutic regimens. Insome cases it may be possible to develop closed-loopcontrol systems that monitor a drug concentration ora response and automatically adjust the drug admin-istration rate. Overall, an increase in the safety andefficiency of drug use can be anticipated.

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