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A model can be fit to joint PK/PD data (concentration and effect) either simultaneously or sequentially. The results of a companion paper suggested that when the data-analytic and true models agree, a particular sequential approach is computationally faster than the simultaneous one, yet produces hardly less precise PD parameter estimates, and for suitable designs, about as accurate PD standard error estimates. In this paper, we compare the performance of various methods for the case that the data-analytic model is misspecified. We illustrate these methods by applying them to a set of real data. Using NONMEM, population PK/PD observations were simulated under various study designs according to a one- or two-compartment PK model and direct Emax or sigmoid Emax model. A one-compartment PK model and Emax PD model were fit to the simulated observations by simultaneous and sequential methods. Predictive performance (interpolation and extrapolation) of PD and the type-I error rate of a likelihood ratio test are compared. The real data set consists of PK and (more frequent) PD observations after administration of the muscle relaxant vecuronium. When only the PK data-analytic model is misspecified, the simultaneous method has greater precision than the sequential methods. However a sequential method that uses a non-parametric PK model performs better than both other methods when PK model misspecification is severe. When the PD data-analytic model is misspecified, sequential and simultaneous methods perform similarly. The analysis of the real data shows that the PK fitted with the simultaneous method can be quite sensitive to PD model misspecification, yielding a possible diagnostic for this type of misspecification.

Dose [-concentration]-effect relationships can be obtained by fitting a predictive pharmacokinetic (PK)-pharmacodynamic (PD) model to both concentration and effect observations. Either a model can be fit simultaneously to all the data (“simultaneous” method), or first a model can be fit to the PK data and then a model can be fit to the PD data, conditioning in some way on the PK data or on the estimates of the PK parameters (“sequential” method). Using simulated data, we compare the performance of the simultaneous method with that of three sequential method variants with respect to computation time, estimation precision, and inference. Using NONMEM, under various study designs, observations of one type of PK and one type of PD response from different numbers of individuals were simulated according to a one-compartment PK model and direct Emax PD model, with parameters drawn from an appropriate population distribution. The same PK and PD models were fit to these observations using simultaneous and sequential methods. Performance measures include computation time, fraction of cases for which estimates are successfully obtained, precision of PD parameter estimates, precision of PD parameter standard error estimates, and type-I error rates of a likelihood ratio test. With the sequential method, computation time is less, and estimates are more likely to be obtained. Using the First Order Conditional Estimation (FOCE) method, a sequential approach that conditions on both population PK parameter estimates and PK data, estimates PD parameters and their standard errors about as well as the “gold standard” simultaneous method, and saves about 40% computation time. Type-I error rates of likelihood ratio test for both simultaneous and sequential approaches are close to the nominal rates.

When modeling new data with a complex population pharmacokinetic/pharmacodynamic model, there may not be sufficient information to obtain estimates of all parameters. In this case information from previous studies can also be used to help stabilize estimation. Using simulated data, we explored three different ways to do this. (i) Some parameter values were fixed to estimates obtained from earlier data. (ii) The earlier data were combined with the current data. (iii) The objective function based on the current data was augmented by a penalty function expressing summary information obtained from the earlier data. This last method is similar to the use of a Bayesian prior. It may be particularly useful when either the combined data set of method (ii) is very large and leads to large computation times or when the early data are not readily available. With this method, two different types of penalty functions were used. With our examples, the three methods all resulted in stabilized estimation. Methods (ii) and (iii) gave similar results for parameter and standard error estimation, especially with respect to fixed effects parameters. For hypothesis testing, results obtained with method (i) are very problematic. There are also problems with the results obtained with method (iii), but they are much less severe, and when the design for the earlier data is known, they can be corrected by using a computer-intensive simulation test procedure.

Over the past 25 years sophisticated data analytic techniques have been developed which can lead to improved analyses, but at additional computational cost. In particular, this applies to the approach where interindividual random effects are included in a data analytic model for population pharmacokinetic data, which can often lead to substantially improved estimates of fixed-effect parameters. However, there are also commonly occurring situations, notably with some types of pharmacodynamic data, where such improvement is not realized. This study simulates some simple population dichotomous data, and secondarily, some related continuous data. These data are analyzed using both mixed-effect (ME) models that include interindividual random effects and naive (NA) models that do not include interindividual random effects, and it is seen that use of an ME model does not inevitably lead to gains over use of an NA model. In fact, using maximum likelihood estimation with both types of models, the root mean square estimation errors for fixed effect parameters can actually be larger with an ME model than with the corresponding NA model. Using a form of restricted maximum likelihood estimation with the ME model, the two types of models yield root mean square errors which are comparable, but which still do not suggest that there is always marked advantage in using the ME model.

This paper presents and illustrates methodology for specifying, estimating, and evaluating a predictive model for repeated measures time-to-event responses. The illustrative example specifies a model of the antiemetic effect vs. concentration relationship for the 5-HT3 antagonist ondansetron in the human ipecac model for emesis. A key part of this model is a time-dependent log hazard function for emesis that is increased by ipecac administration and decreased by ondansetron concentration. The model is fit using an approximate maximum likelihood method. The data consist of the time free of emeses and, for those individuals with emetic episodes, the time(s) of the episode(s). Model evaluation is accomplished using residual plots adapted to time-to-event data and a "posterior predictive check" wherein observed data statistics are compared to those obtained from data simulated from the fitted model. The ondansetron concentration required to obtain a 50% reduction in the hazard of emesis is estimated to be 1.4 +/- 0.2 ng/ml, and the rate constant for elimination of ipecac-induced hazard is 1.5 +/- 0.2 hr-1.

Objective Our objective was to develop a population 1-compartment pharmacokinetic (PK) method of analysis to deal with suspect or missing prior dosage history. Methods Population PK data from a 1-compartment model with first-order elimination and absorption, described by PK parameters clearance, volume of distribution, and absorption rate constant, are simulated. A PK sample is drawn just before a test dose (Dt), followed by a (varying) number of additional samples over 1 interdose interval (τ). For 60% of the subjects, the true history of the scheduled dose (Ds) preceding Dt differs from that prescribed, whereas doses taken before Ds do not. Two settings are evaluated: considerable accumulation of drug in the body (typical drug half-life t½ approximately equal to τ) and very little such accumulation (t½ approximately equal to τ/5). Precision and bias of several PK analysis methods—Missing Dose Method (MDM), Missing Dose Mixture Method (MDMM) and Extrapolation-Subtraction Method (ESM), all of which essentially do not use prior dose history—are compared with those of the Prescribed Dose Method (PDM), which assumes nominal dosage, and an Ideal Method (IDM), which uses true (but unknown) pre-test dose history. Results At t½ approximately equal to τ, MDM and MDMM are the most precise methods. The accuracy of ESM and PDM is poor. At t½ approximately equal to τ/5, no significant differences, in terms of precision or bias, are observed between methods. Misspecification of the structural or statistical model seems not to influence these results. The results of analysis of a real (caffeine) data set are compatible with the findings from the simulations. Conclusion When a test dose is given and a predose baseline observation is taken as part of an “intensive” PK study during outpatient therapy of a 1-compartment drug, an analysis that assumes that the nominal dose history is correct is not robust to past dosage history misspecification, whereas methods that do not do this are robust and reliable. Clinical Pharmacology & Therapeutics (2004) 76, 441–451; doi: 10.1016/j.clpt.2004.07.010

When modeling new data with a complex population pharmacokinetic/pharmacodynamic model, there may not be sufficient information to obtain estimates of all parameters. In this case information from previous studies can also be used to help stabilize estimation. Using simulated data, we explored three different ways to do this. (i) Some parameter values were fixed to estimates obtained from earlier data. (ii) The earlier data were combined with the current data. (iii) The objective function based on the current data was augmented by a penalty function expressing summary information obtained from the earlier data. This last method is similar to the use of a Bayesian prior. It may be particularly useful when either the combined data set of method (ii) is very large and leads to large computation times or when the early data are not readily available. With this method, two different types of penalty functions were used. With our examples, the three methods all resulted in stabilized estimation. Methods (ii) and (iii) gave similar results for parameter and standard error estimation, especially with respect to fixed effects parameters. For hypothesis testing, results obtained with method (i) are very problematic. There are also problems with the results obtained with method (iii), but they are much less severe, and when the design for the earlier data is known, they can be corrected by using a computer-intensive simulation test procedure.

SUMMARY This bibliography lists all published methodological works (statistical methodology, implementation methodology, review papers, descriptions of software) in the area of population pharmacokinetics/ pharmacodynamics up to 1993 and all published works describing applications of population pharmacokinetics/pharmacodynamics up to 1992.

The influence of the choice of pharmacokinetic model on subsequent determination of covariate relationships in population pharmacokinetic analysis was studied using both simulated and real data sets. Simulations and data analysis were both performed with the program NONMEM. Data were simulated using a two-compartment model, but at late sample times, so that preferential selection of the two-compartment model should have been impossible. A simple categorical covariate acting on clearance was included. Initially, on the basis of a difference in the objective function values, the two-compartment model was selected over the one-compartment model. Only when the complexity of the one-compartment model was increased in terms of the covariate and statistical models was the difference in objective function values of the two structural models negligible. For two real data sets, with which the two-compartment model was not selected preferentially, more complex covariate relationships were supported with the one-compartment model than with the two-compartment model. Thus, the choice of structural model can be affected as much by the covariate model as can the choice of covariate model be affected by the structural model; the two choices are interestingly intertwined. A suggestion on how to proceed when building population pharmacokinetic models is given.

Only with knowledge of the (prior) distribution of dose-response parameters in a population, can one determine both the initial dose of a drug for chronic administration to an individual (such as the dose producing a fixed degree of response in a fixed proportion of the population) and an appropriate subsequent (adjusted) dose (such as the dose yielding a desirable response according to the posterior parameter distribution, given an observed response to an initial dose). The currently FDA-sanctioned design for a dose-ranging study, the parallel-dose design, assigns just one of several doses to each patient. It does not provide good information on the distribution of individual dose-response parameters. A cross-over design assigns several dose levels to each patient. It therefore can provide better information, but does not resemble clinical practice. Consequently, study participants must be restricted to patients who can tolerate such non-therapeutic drug exposure, posing problems in extrapolation of study results to other types of patients. A titration or dose-escalation design begins all patients on placebo and, except for those patients assigned to a placebo-only group, escalates the dose for a patient at preset intervals only when clinical response at lower doses is inadequate. It both exposes patients to several dose levels and resembles good clinical practice, allowing study of a representative patient sample. We report here the simulation results of parameter estimation for the three designs when the data arise from complex and realistic dose-response models and/or with certain complications in study execution. The dose-escalation design clearly performs better overall than the parallel-dose design for the models considered here, and generally, just a little worse than the cross-over design. These results support the conclusion that for dose ranging, depending on the demands of the clinical situation, one should use either the cross-over or the dose-escalation design.

Simultaneous dual immunofluorescence and flow cytometry was used to study sixteen lymphocyte phenotypes in 209 men including: healthy homosexuals, lymphadenopathy patients (LAN), and AIDS patients. Significant differences between the distribution of lymphocytes in healthy homosexuals and healthy heterosexuals were decreased percentages of helper/inducer T cells (Leu 3), increased cytotoxic/suppressor T cells (Leu 2), and consequently a decreased Leu 3/Leu 2 ratio. The increased Leu 2 cells were identified as functionally cytotoxic subset Leu 2+ 15- phenotype rather than suppressor cells which are Leu 15+. Leu 2 and Leu 3 bearing cells exhibited an excess of membrane-bound immunoglobulins which were easily elutable at 37 degrees C. An increased percentage of an HLA-DR framework determinant bearing T cells were also detected. Within the NK cell family, Leu 7 cells were moderately increased and the functionally unidentified Leu 2+ 7+ population was strikingly elevated. LAN or AIDS patients were compared to healthy homosexual controls. Lower percentages of Leu 3 cells and higher percentages of Leu 2 cells were evident in LAN patients. These subsets were similar in LAN and AIDS patients. The increase in Leu 2+ cells was due to the Leu 2+ 15- cytotoxic subset. Fewer T cells had immunoglobulin in LAN and AIDS. A definite increase in Leu 2+ DR+ cells but not Leu 3+ DR+ cells occurred in AIDS compared to LAN or healthy controls. NK cell changes already present in healthy homosexuals persisted in LAN and AIDS patients. No differences in the distribution of B cells was detected in any intergroup comparisons. Changes in monocytes or pan-T cells were relatively insensitive measures of immunologic alterations among any of the groups. These results indicate many of the changes in lymphocyte subsets seen in AIDS and LAN subjects are already present in a carefully screened population of healthy homosexuals in San Francisco. Many of the changes in Leu 2 and NK family of cells suggest a possible adaptive response to viral or neoplastic challenge. Whether these interesting phenotypic alterations relate to functional changes in response to such challenge of the identified subsets waits further investigation.

Individual pharmacokinetic parameters quantify the pharmacokinetics of an individual, while population pharmacokinetic parameters quantify population mean kinetics, interindividual kinetic variability, and residual variability, including intraindividual variability and measurement error. Individual pharmacokinetics are estimated by fitting a pharmacokinetic model to individual data. Population pharmacokinetic parameters have traditionally been estimated by doing this separately for each individual, and then combining the individual parameter estimates, the Standard Two Stage (STS) approach. Another approach, NONMEM, appropriately pools data across individuals and is therefore less dependent on individual parameter estimates. This study provides further evidence of NONMEM's validity and usefulness by comparing both approaches on simulated routine-type pharmacokinetic data arising from a monoexponential model. The estimates of population parameters (notably those describing interindividual variability) provided by the STS method are poorer than those provided by NONMEM, especially when there is considerable residual error. Further, NONMEM's estimates of population parameters do not require that the data be restricted to special types of routine data such as those obtained only at steady state, or only at peak or trough, nor do the estimates improve with such data. NONMEM's estimates do improve, however, when a data set is enhanced by the addition of single-observation-per-individual type data. Thus, population parameters can be estimated efficiently from data that simulate real clinical pharmacokinetic conditions.

One may attempt to individualize drug dosage by estimating an individual's pharmacokinetic parameters. Information useful for this purpose consists of certain population pharmacokinetic parameters (notably those describing the typical relationship between dosage and drug concentrations) and also measured drug concentrations from the individual of concern. Both types of information should be used. A (Bayesian) method that does so has been described in the pharmacokinetic literature. In this report an implementation of the Bayesian method that is readily adapted to a microcomputer is presented. Using simulated data it is compared with two other methods proposed by others, for estimating individual theophylline clearances. Both previously suggested methods are shown to be less precise than the Bayesian method: their typical error magnitudes are 20-70% larger.

Consider a finite population, called here the target population, and consider two conditions, each of which we may imagine applying to all the units of this population. Suppose that under each condition, each population unit will exhibit some binary (0-1) response. The problem is to "compare" the set of responses of all the units under one condition with the set of responses of all the units under the second condition. We call these two sets the two population responses. In order to make the comparison, an experiment, in which some responses are actually obtained, must be performed. The population may be large enough to make it impractical to obtain an entire population response, and consequently, part of the experiment may entail determining a sample of units whose response under one or both conditions will be obtained. Also, in a real experiment it may be impossible in general to actually apply both conditions to a unit. (The actual application of a condition to a particular unit may result in altering the unit so that the other condition cannot be applied to it later). Consequently part of the experiment may entail choosing the conditions which shall be applied to particular experimental units. A common experimental design in these situations involves 1) obtaining a random sample from the target population, 2) randomly allocating the sample units to two experimental groups, and 3) applying one condition to the units of one group, while applying the other condition to the units of the other group. Even when both conditions may actually be applied to all units, this design may still be helpful since both groups may undergo experimentation during the same time period, and therefore, total experiment time may be reduced. Fisher's well known hypergeometric test (2,3,8) or a test which is approximate to Fisher's test, such as the chi-squared test with Yates correction (1,2,3,8), is usually used to analyze the responses obtained from an experiment with the above design. When the sample size, N, is sufficiently small compared to the size, M, of the target population, one may regard the responses of the first (second) group as being approximately a sequence of Bernoulli 0-1 outcomes, with probability m1!M (m2!M) for outcome 1, where mi = the number of I's in the ith population response. Thus Fisher's test may be regarded to be approximately testing the null hypothesis m1!M = m2!M (or ml = M2). In this article we explain that the appropriateness of using Fisher's test to test m1 = m2 may be examined without recourse to the Bernoulli approximation. Such examination is especially important when NIM is so large that it is not really meaningful to talk about Bernoulli outcomes. An important requirement of the inferential procedure is that the sample be random. The usual assumption that the sample is indeed random is often rather shaky in application. This is often the case in medical experiments where an accessible group of patients is used for a sample, rather than a group of patients chosen from either a well-defined target population or sampling frame, and where the inference is about some population of interest which is characterized after the sample is determined and which should be identified here with the target population. For a proper interpretation of the experimental result, one may need to redefine the target population in such a way that the randomness assumption becomes more tenable (or, as it is said, in such a way that the sample becomes more representative of the population). In so doing, one may well add restrictions to the original definition of the target population, thereby decreasing the population size. In fact, it may be necessary to define the target population to be, in effect, the set of units comprising the sample; the sample is then undoubtedly random. Of course, when the population is made less encompassing, it may lose some interest value with respect to the conclusion to be drawn about it. However, this loss is often necessary for a valid inference to result. While constricting the population, the ratio of sample size to population size may increase to the point where the inferential argument should not depend on the use of the Bernoulli approximation. These considerations motivated this author's inquiry into the role of Fisher's test in the inference problem to compare the population responses.

A statistical model is set forth that can be taken as a very general description of most data sets that arise from population pharmacokinetic studies. A (nontraditional) method for estimating the parameters of the model—called the NONMEM method in previously published papers—is described. This method involves a linearization of the model. An investigation of the effect of this linearization is reported. This investigation is an empirical one, based on the simulation of data from special cases of the model and the application of a few different estimation methods to these data. From the limited evidence in this investigation it appears that the linearization per se does not significantly adversely affect the estimates.

Premarketing dose-ranging studies of a drug are done to establish a reasonable initial dose. According to the current procedure sanctioned by the Food and Drug Administration, each patient is given one of several possible doses, including placebo, after an initial placebo run-in period. Data analysis is based on a model in which the mean response at each dose is independent of the magnitude of the dose. The initial dose is the lowest dose tested that has a response that is statistically significantly greater than the response after placebo administration. We suggest that the present conceptual approach to, and standard study design and analysis for, dose-ranging studies be changed. We believe one must begin with a parametric model for patient-specific dose-response curves. Knowledge of the distribution of these curves in a population provides a basis for choice of an initial dose (e.g., the dose that achieves a given response in a given fraction of patients) and, after observation of response to an initial dose, for choice of an incremental dose for a specific patient (by use of Bayes rule). The current parallel-dose design can provide only poor information about the distribution of dose-response curves, biased estimates of the typical curve, and little information on interpatient variability. Crossover studies provide better information. In studies in which a parametric patient-specific dose-response model is used, a dose-escalation design provides no less information than a crossover design, and it has ethical advantages that allow a more representative patient group and clinical setting to be studied.

Individual pharmacokinetic par parameters quantify the pharmacokinetics of an individual, while population pharmacokinetic parameters quantify population mean kinetics, interindividual variability, and residual intraindividual variability plus measurement error. Individual pharmacokinetics are estimated by fitting individual data to a pharmacokinetic model. Population pharmacokinetic parameters are estimated either by fitting all individual's data together as though there was no individual kinetic differences (the naive pooled data approach), or by fitting each individual's data separately, and then combining the individual parameter estimates (the two-stage approach). A third approach, NONMEM, takes a middle course between these, and avoids shortcomings of each of them. A data set consisting of 124 steady-state phenytoin concentration-dosage pairs from 49 patients, obtained in the routine course of their therapy, was analyzed by each method. The resulting population parameter estimates differ considerably (population mean Km, for example, is estimated as 1.57, 5.36, and 4.44 micrograms/ml by the naive pooled data, two-stage, and NONMEN approaches, respectively). Simulations of the data were analyzed to investigate these differences. The simulations indicate that the pooled data approach fails to estimate variabilities and produces imprecise estimates of mean kinetics. The two-stage approach produces good estimates of mean kinetics, but biased and imprecise estimates of interindividual variability. NONMEN produces accurate and precise estimates of all parameters, and also reasonable confidence intervals for them. This performance is exactly what is expected from theoretical considerations and provides empirical support for the use of NONMEM when estimating population pharmacokinetics from routine type patient data.