Alan Schumitzky

A program package for simulation and parameter estimation in pharmacokinetic systems

A set of programs is presented which has been developed for parameter estimation and simulation of models arising from pharmacokinetic applications. The programs can accommodate linear and nonlinear models with multiple inputs and multiple outputs. When the model is defined by differential equations, non-uniform repetitive dosage regimens can be handled. The model may also be entered in integrated form when single dose studies or uniform multiple dose studies are being considered. The programs employ a variable-step, variable-order integration routine to solve the model differential equations, and the Nelder-Mead simplex procedure to determine the parameter values which minimize a weighted least squares criterion. The programs have been written for an interactive time-sharing environment with the experimental data and model equations stored in files for future use.

SAAM II: Simulation, analysis, and modeling software for tracer and pharmacokinetic studies

Kinetic analysis and integrated systems modeling have contributed substantially to our understanding of the physiology and pathophysiology of metabolic systems and the distribution and clearance of drugs in humans and animals. In recent years, many researchers have become aware of the usefulness of these techniques in the experimental design. With this has come the recognition that the discipline of kinetic analysis requires its own expertise. The expertise can impact experimental design in many ways, from the collaborative and service activities in which individuals interact in formal ways to the development of software tools to aid in kinetic analysis. The purpose of this report is to describe one such software tool, Simulation, Analysis, and Modeling Software II (SAAM II). In the first part, we describe in general how the user can take advantage of the capabilities of the software system, and in the second part, we give three specific examples using multicompartmental models found in lipoprotein (apolipoprotein B [apoB] kinetics) and diabetes (glucose minimal model) research.

Accurate detection of outliers and subpopulations with Pmetrics, a nonparametric and parametric pharmacometric modeling and simulation package for R.

Introduction: Nonparametric population modeling algorithms have a theoretical superiority over parametric methods to detect pharmacokinetic and pharmacodynamic subgroups and outliers within a study population. Methods: The authors created “Pmetrics,” a new Windows and Unix R software package that updates the older MM-USCPACK software for nonparametric and parametric population modeling and simulation of pharmacokinetic and pharmacodynamic systems. The parametric iterative 2-stage Bayesian and the nonparametric adaptive grid (NPAG) approaches in Pmetrics were used to fit a simulated population with bimodal elimination (Kel) and unimodal volume of distribution (Vd), plus an extreme outlier, for a 1-compartment model of an intravenous drug. Results: The true means (SD) for Kel and Vd in the population sample were 0.19 (0.17) and 102 (22.3), respectively. Those found by NPAG were 0.19 (0.16) and 104 (22.6). The iterative 2-stage Bayesian estimated them to be 0.18 (0.16) and 104 (24.4). However, given the bimodality of Kel, no subject had a value near the mean for the population. Only NPAG was able to accurately detect the bimodal distribution for Kel and to find the outlier in both the population model and in the Bayesian posterior parameter estimates. Conclusions: Built on over 3 decades of work, Pmetrics adopts a robust, reliable, and mature nonparametric approach to population modeling, which was better than the parametric method at discovering true pharmacokinetic subgroups and an outlier.

Individualizing drug dosage regimens: roles of population pharmacokinetic and dynamic models, Bayesian fitting, and adaptive control.

The role of population pharmacokinetic modeling is to store experience with drug behavior. The behavior of the model is then correlated with the clinical behavior of the patients studied, permitting selection of a specific serum level therapeutic goal that is based on each individual patients need for the drug and on the risk of adverse reactions, both of which must be considered. A dosage regimen is then computed to achieve that goal with maximum precision. The patient should not run a greater risk of toxicity than is justified, and should obtain the maximum possible benefit within the acceptable risk. The regimen is given and the patient monitored.

Model-based, goal-oriented, individualised drug therapy : Linkage of population modelling, new 'multiple model' dosage design, Bayesian feedback and individualised target goals

SummaryThis article examines the use of population pharmacokinetic models to store experiences about drugs in patients and to apply that experience to the care of new patients. Population models are the Bayesian prior. For truly individualised therapy, it is necessary first to select a specific target goal, such as a desired serum or peripheral compartment concentration, and then to develop the dosage regimen individualised to best hit that target in that patient.One must monitor the behaviour of the drug by measuring serum concentrations or other responses, hopefully obtained at optimally chosen times, not only to see the raw results, but to also make an individualised (Bayesian posterior) model of how the drug is behaving in that patient. Only then can one see the relationship between the dose and the absorption, distribution, effect and elimination of the drug, and the patient’s clinical sensitivity to it; one must always look at the patient. Only by looking at both the patient and the model can it be judged whether the target goal was correct or needs to be changed. The adjusted dosage regimen is again developed to hit that target most precisely starting with the very next dose, not just for some future steady state.Nonparametric population models have discrete, not continuous, parameter distributions. These lead naturally into the multiple model method of dosage design, specifically to hit a desired target with the greatest possible precision for whatever past experience and present data are available on that drug — a new feature for this goal-oriented, model-based, individualised drug therapy. As clinical versions of this new approach become available from several centres, it should lead to further improvements in patient care, especially for bacterial and viral infections, cardiovascular therapy, and cancer and transplant situations.

Parametric and nonparametric population methods: their comparative performance in analysing a clinical dataset and two Monte Carlo simulation studies.

Background and objectivesThis study examined parametric and nonparametric population modelling methods in three different analyses. The first analysis was of a real, although small, clinical dataset from 17 patients receiving intramuscular amikacin. The second analysis was of a Monte Carlo simulation study in which the populations ranged from 25 to 800 subjects, the model parameter distributions were Gaussian and all the simulated parameter values of the subjects were exactly known prior to the analysis. The third analysis was again of a Monte Carlo study in which the exactly known population sample consisted of a unimodal Gaussian distribution for the apparent volume of distribution (Vd), but a bimodal distribution for the elimination rate constant (ke), simulating rapid and slow eliminators of a drug.MethodsFor the clinical dataset, the parametric iterative two-stage Bayesian (IT2B) approach, with the first-order conditional estimation (FOCE) approximation calculation of the conditional likelihoods, was used together with the nonparametric expectation-maximisation (NPEM) and nonparametric adaptive grid (NPAG) approaches, both of which use exact computations of the likelihood.For the first Monte Carlo simulation study, these programs were also used. A one-compartment model with unimodal Gaussian parameters Vd and ke was employed, with a simulated intravenous bolus dose and two simulated serum concentrations per subject. In addition, a newer parametric expectation-maximisation (PEM) program with a Faure low discrepancy computation of the conditional likelihoods, as well as nonlinear mixed-effects modelling software (NONMEM), both the first-order (FO) and the FOCE versions, were used.For the second Monte Carlo study, a one-compartment model with an intravenous bolus dose was again used, with five simulated serum samples obtained from early to late after dosing. A unimodal distribution for Vd and a bimodal distribution for ke were chosen to simulate two subpopulations of ‘fast’ and ‘slow’ metabolisers of a drug. NPEM results were compared with that of a unimodal parametric joint density having the true population parameter means and covariance.ResultsFor the clinical dataset, the interindividual parameter percent coefficients of variation (CV%) were smallest with IT2B, suggesting less diversity in the population parameter distributions. However, the exact likelihood of the results was also smaller with IT2B, and was 14 logs greater with NPEM and NPAG, both of which found a greater and more likely diversity in the population studied.For the first Monte Carlo dataset, NPAG and PEM, both using accurate likelihood computations, showed statistical consistency. Consistency means that the more subjects studied, the closer the estimated parameter values approach the true values. NONMEM FOCE and NONMEM FO, as well as the IT2B FOCE methods, do not have this guarantee. Results obtained by IT2B FOCE, for example, often strayed visibly away from the true values as more subjects were studied.Furthermore, with respect to statistical efficiency (precision of parameter estimates), NPAG and PEM had good efficiency and precise parameter estimates, while precision suffered with NONMEM FOCE and IT2B FOCE, and severely so with NONMEM FO.For the second Monte Carlo dataset, NPEM closely approximated the true bimodal population joint density, while an exact parametric representation of an assumed joint unimodal density having the true population means, standard deviations and correlation gave a totally different picture.ConclusionsThe smaller population interindividual CV% estimates with IT2B on the clinical dataset are probably the result of assuming Gaussian parameter distributions and/or of using the FOCE approximation. NPEM and NPAG, having no constraints on the shape of the population parameter distributions, and which compute the likelihood exactly and estimate parameter values with greater precision, detected the more likely greater diversity in the parameter values in the population studied.In the first Monte Carlo study, NPAG and PEM had more precise parameter estimates than either IT2B FOCE or NONMEM FOCE, as well as much more precise estimates than NONMEM FO. In the second Monte Carlo study, NPEM easily detected the bimodal parameter distribution at this initial step without requiring any further information.Population modelling methods using exact or accurate computations have more precise parameter estimation, better stochastic convergence properties and are, very importantly, statistically consistent. Nonparametric methods are better than parametric methods at analysing populations having unanticipated non-Gaussian or multimodal parameter distributions.

On the equivalence between matrix riccati equations and Fredholm resolvents

It is shown that the solution to every matrix Riccati equation can be generated by the resolvent of a certain Fredholm integral operator and, conversely, this resolvent can be determined from the corresponding Riccati solution. This result leads to a computational scheme, based on initial-value methods, for solving a large class of Fredholm integral equations. A connection between this theory and the factorization of integral operators is also described.

Achieving target goals most precisely using nonparametric compartmental models and "multiple model" design of dosage regimens.

Multiple model (MM) design and stochastic control of dosage regimens permit essentially full use of all the information contained in either a Bayesian prior nonparametric EM (NPEM) population pharmacokinetic model or in an MM Bayesian posterior updated parameter set, to achieve and maintain selected therapeutic goals with optimal precision (least predicted weighted squared error). The regimens are visibly more precise in the achievement of desired target goals than are current methods using mean or median population parameter values. Bayesian feedback has now also been incorporated into the MM software. An evaluation of MM dosage design using an NPEM population model versus dosage design based on conventional mean population parameter values is presented, using a population model of vancomycin. Further feedback control was also evaluated, incorporating realistic simulated uncertainties in the clinical environment such as those in the preparation and administration of doses.

Two general methods for population pharmacokinetic modeling: non-parametric adaptive grid and non-parametric Bayesian

Population pharmacokinetic (PK) modeling methods can be statistically classified as either parametric or nonparametric (NP). Each classification can be divided into maximum likelihood (ML) or Bayesian (B) approaches. In this paper we discuss the nonparametric case using both maximum likelihood and Bayesian approaches. We present two nonparametric methods for estimating the unknown joint population distribution of model parameter values in a pharmacokinetic/pharmacodynamic (PK/PD) dataset. The first method is the NP Adaptive Grid (NPAG). The second is the NP Bayesian (NPB) algorithm with a stick-breaking process to construct a Dirichlet prior. Our objective is to compare the performance of these two methods using a simulated PK/PD dataset. Our results showed excellent performance of NPAG and NPB in a realistically simulated PK study. This simulation allowed us to have benchmarks in the form of the true population parameters to compare with the estimates produced by the two methods, while incorporating challenges like unbalanced sample times and sample numbers as well as the ability to include the covariate of patient weight. We conclude that both NPML and NPB can be used in realistic PK/PD population analysis problems. The advantages of one versus the other are discussed in the paper. NPAG and NPB are implemented in R and freely available for download within the Pmetrics package from www.lapk.org.

Population pharmacokinetics/pharmacodynamics modeling: parametric and nonparametric methods.

As clinicians acquire experience with the clinical and pharmacokinetic behavior of a drug, it is usually optimal to record this experience in the form of a population pharmacokinetic model, and then to relate the behavior of the model to the clinical effects of the drug or to a linked pharmacodynamic model. The role of population modeling is thus to describe and record clinical experience with the behavior of a drug in a certain group or population of patients or subjects.