To Estimate or to Forecast? Lessons From a Comparative Analysis of Four Bayesian Fitting Methods Based on Nonparametric Models.

ABSTRACT: Using pharmacokinetic (PK) models and Bayesian methods in dosing software facilitates the analysis of individual PK data and precision dosing. Several Bayesian methods are available for computing Bayesian posterior distributions using nonparametric population models. The objective of this study was to compare the performance of the maximum a posteriori (MAP) model, multiple model (MM), interacting MM (IMM), and novel hybrid MM(HMM) in estimating past concentrations and predicting future concentrations during therapy. Amikacin and vancomycin PK data were analyzed in older hospitalized patients using 2 strategies. First, the entire data set of each patient was fitted using each of the 4 methods implemented in BestDose software. Then, the 4 methods were used in each therapeutic drug monitoring occasion to estimate the past concentrations available at this time and to predict the subsequent concentrations to be observed on the next occasion. The bias and precision of the model predictions were compared among the methods. A total of 406 amikacin concentrations from 96 patients and 718 vancomycin concentrations from 133 patients were available for analysis. Overall, significant differences were observed in the predictive performance of the 4 Bayesian methods. The IMM method showed the best fit to past concentration data of amikacin and vancomycin, whereas the MM method was the least precise. However, MM best predicted the future concentrations of amikacin. The MAP and HMM methods showed a similar predictive performance and seemed to be more appropriate for the prediction of future vancomycin concentrations than the other models were. The richness of the prior distribution may explain the discrepancies between the results of the 2 drugs. Although further research with other drugs and models is necessary to confirm our findings, these results challenge the widely accepted assumption in PK modeling that a better data fit indicates better forecasting of future observations.

Prospective Trial on the Use of Trough Concentration versus Area under the Curve To Determine Therapeutic Vancomycin Dosing.

We hypothesized that dosing vancomycin to achieve trough concentrations of >15 mg/liter overdoses many adults compared to area under the concentration-time curve (AUC)-guided dosing. We conducted a 3-year, prospective study of vancomycin dosing, plasma concentrations, and outcomes. In year 1, nonstudy clinicians targeted trough concentrations of 10 to 20 mg/liter (infection dependent) and controlled dosing. In years 2 and 3, the study team controlled vancomycin dosing with BestDose Bayesian software to achieve a daily, steady-state AUC/MIC ratio of ≥400, with a maximum AUC value of 800 mg · h/liter, regardless of trough concentration. For Bayesian estimation of AUCs, we used trough samples in years 1 and 2 and optimally timed samples in year 3. We enrolled 252 adults who were ≥18 years old with ≥1 available vancomycin concentration. Only 19% of all trough concentrations were therapeutic versus 70% of AUCs (P < 0.0001). After enrollment, median trough concentrations by year were 14.4, 9.7, and 10.9 mg/liter (P = 0.005), with 36%, 7%, and 6% over 15 mg/liter (P < 0.0001). Bayesian AUC-guided dosing in years 2 and 3 was associated with fewer additional blood samples per subject (3.6, 2.0, and 2.4; P = 0.003), shorter therapy durations (8.2, 5.4, and 4.7 days; P = 0.03), and reduced nephrotoxicity (8%, 0%, and 2%; P = 0.01). The median inpatient stay was 20 days among nephrotoxic patients versus 6 days (P = 0.002). There was no difference in efficacy by year, with 42% of patients having microbiologically proven infections. Compared to trough concentration targets, AUC-guided, Bayesian estimation-assisted vancomycin dosing was associated with decreased nephrotoxicity, reduced per-patient blood sampling, and shorter length of therapy, without compromising efficacy. These benefits have the potential for substantial cost savings. (This study has been registered at ClinicalTrials.gov under registration no. NCT01932034.).

Accurately Achieving Target Busulfan Exposure in Children and Adolescents With Very Limited Sampling and the BestDose Software.

BACKGROUND: Busulfan dose adjustment is routinely guided by plasma concentration monitoring using 4-9 blood samples per dose adjustment, but a pharmacometric Bayesian approach could reduce this sample burden. METHODS: The authors developed a nonparametric population model with Pmetrics. They used it to simulate optimal initial busulfan dosages, and in a blinded manner, they compared dosage adjustments using the model in the BestDose software to dosage adjustments calculated by noncompartmental estimation of area under the time-concentration curve at a national reference laboratory in a cohort of patients not included in model building. RESULTS: Mean (range) age of the 53 model-building subjects was 7.8 years (0.2-19.0 years) and weight was 26.5 kg (5.6-78.0 kg), similar to nearly 120 validation subjects. There were 16.7 samples (6-26 samples) per subject to build the model. The BestDose cohort was also diverse: 10.2 years (0.25-18 years) and 46.4 kg (5.2-110.9 kg). Mean bias and imprecision of the 1-compartment model-predicted busulfan concentrations were 0.42% and 9.2%, and were similar in the validation cohorts. Initial dosages to achieve average concentrations of 600-900 ng/mL were 1.1 mg/kg (≤12 kg, 67% in the target range) and 1.0 mg/kg (>12 kg, 76% in the target range). Using all 9 concentrations after dose 1 in the Bayesian estimation of dose requirements, the mean (95% confidence interval) bias of BestDose calculations for the third dose was 0.2% (-2.4% to 2.9%, P = 0.85), compared with the standard noncompartmental method based on 9 concentrations. With 1 optimally timed concentration 15 minutes after the infusion (calculated with the authors' novel MMopt algorithm) bias was -9.2% (-16.7% to -1.5%, P = 0.02). With 2 concentrations at 15 minutes and 4 hours bias was only 1.9% (-0.3% to 4.2%, P = 0.08). CONCLUSIONS: BestDose accurately calculates busulfan intravenous dosage requirements to achieve target plasma exposures in children up to 18 years of age and 110 kg using only 2 blood samples per adjustment compared with 6-9 samples for standard noncompartmental dose calculations.

Achieving target voriconazole concentrations more accurately in children and adolescents.

Despite the documented benefit of voriconazole therapeutic drug monitoring, nonlinear pharmacokinetics make the timing of steady-state trough sampling and appropriate dose adjustments unpredictable by conventional methods. We developed a nonparametric population model with data from 141 previously richly sampled children and adults. We then used it in our multiple-model Bayesian adaptive control algorithm to predict measured concentrations and doses in a separate cohort of 33 pediatric patients aged 8 months to 17 years who were receiving voriconazole and enrolled in a pharmacokinetic study. Using all available samples to estimate the individual Bayesian posterior parameter values, the median percent prediction bias relative to a measured target trough concentration in the patients was 1.1% (interquartile range, -17.1 to 10%). Compared to the actual dose that resulted in the target concentration, the percent bias of the predicted dose was -0.7% (interquartile range, -7 to 20%). Using only trough concentrations to generate the Bayesian posterior parameter values, the target bias was 6.4% (interquartile range, -1.4 to 14.7%; P = 0.16 versus the full posterior parameter value) and the dose bias was -6.7% (interquartile range, -18.7 to 2.4%; P = 0.15). Use of a sample collected at an optimal time of 4 h after a dose, in addition to the trough concentration, resulted in a nonsignificantly improved target bias of 3.8% (interquartile range, -13.1 to 18%; P = 0.32) and a dose bias of -3.5% (interquartile range, -18 to 14%; P = 0.33). With the nonparametric population model and trough concentrations, our control algorithm can accurately manage voriconazole therapy in children independently of steady-state conditions, and it is generalizable to any drug with a nonparametric pharmacokinetic model. (This study has been registered at ClinicalTrials.gov under registration no. NCT01976078.).

A two-compartment population pharmacokinetic-pharmacodynamic model of digoxin in adults, with implications for dosage.

A population pharmacokinetic/pharmacodynamic model of digoxin in adult subjects was originally developed by Reuning et al in 1973. They clearly described the 2-compartment behavior of digoxin, the lack of correlation of effect with serum concentrations, and the close correlation of the observed inotropic effect of digoxin with the calculated amount of drug present in the peripheral nonserum compartment. Their model seemed most attractive for clinical use. However, to make it more applicable for maximally precise dosage, its model parameter values (means and SD's) were converted into discrete model parameter distributions using a computer program developed especially for this purpose using the method of maximum entropy. In this way, the parameter distributions became discrete rather than continuous, suitable for use in developing maximally precise digoxin dosage regimens, individualized to an adult patient's age, gender, body weight, and renal function, to achieve desired specific target goals either in the central (serum) compartment or in the peripheral (effect) compartment using the method of multiple model dosage design. Some illustrative clinical applications of this model are presented and discussed. This model with a peripheral compartment reflecting clinical effect has contributed significantly to an improved understanding of the clinical behavior of digoxin in patients than is possible with models having only a single compartment, and to the improved management of digoxin therapy for more than 20 years.

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.

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.

Population modeling and Monte Carlo simulation study of the pharmacokinetics and antituberculosis pharmacodynamics of rifampin in lungs.

Little information exists on the pulmonary pharmacology of antituberculosis drugs. We used population pharmacokinetic modeling and Monte Carlo simulation to describe and explore the pulmonary pharmacokinetics and pharmacodynamics of rifampin (RIF; rifampicin). A population pharmacokinetic model that adequately described the plasma, epithelial lining fluid (ELF), and alveolar cell (AC) concentrations of RIF in a population of 34 human volunteers was made by use of the nonparametric adaptive grid (NPAG) algorithm. The estimated concentrations correlated well with the measured concentrations, and there was little bias and good precision. The results obtained with the NPAG algorithm were then imported into Matlab software to perform a 10,000-subject Monte Carlo simulation. The ability of RIF to suppress the development of drug resistance and to induce a sufficient bactericidal effect against Mycobacterium tuberculosis was evaluated by calculating the proportion of subjects achieving specific target values for the maximum concentration of drug (C(max))/MIC ratio and the area under the concentration-time curve from time zero to 24 h (AUC(0-24))/MIC ratio, respectively. At the lowest MIC (0.01 mg/liter), after the administration of one 600-mg oral dose, the rates of target attainment for C(max)/MIC (> or =175) were 95% in ACs, 48.8% in plasma, and 35.9% in ELF. Under the same conditions, the target attainment results for the killing effect were 100% in plasma (AUC(0-24)/MIC > or = 271) but only 54.5% in ELF (AUC(0-24)/MIC > or = 665). The use of a 1,200-mg RIF dose was associated with better results for target attainment. The overall results suggest that the pulmonary concentrations obtained with the standard RIF dose are too low in most subjects. This work supports the need to evaluate higher doses of RIF for the treatment of patients with tuberculosis.

Recombinant human erythropoietin for the treatment of renal anaemia in children: no justification for bodyweight-adjusted dosage.

BACKGROUND: Drug doses for children are usually calculated by reducing adult doses in proportion to bodyweight. The clinically effective dose of recombinant human erythropoietin (epoetin) in children, however, seems to be higher than predicted by this calculation. OBJECTIVE: To determine the quantitative relationship between epoetin dose, bodyweight and response in children with end-stage renal disease. PATIENTS AND METHODS: The time-course of haemoglobin in 52 children during long-term treatment with epoetin beta was analysed by population pharmacodynamic modelling. Patients were 5-20 years old and weighed 16-53kg at the beginning of treatment. Epoetin beta was given intravenously three times per week after haemodialysis. Doses ranged from 110 to 7500IU (3-205 IU/kg). Haemoglobin versus time was described by assuming that the haemoglobin level rises after each dose due to the formation of new red blood cells, which then survive according to a logistic function. The initial rise after each dose was modelled in terms of absolute dose (not dose/kg). A parametric analysis was done with NONMEM, followed by a nonparametric analysis with NPAG. RESULTS: Dose-response was best described by a sigmoid maximum-effect (E(max)) model with median E(max) = 0.29 g/dL, median 50% effective dose (ED(50)) = 2400IU and shape parameter gamma = 2. The estimated median survival time of the epoetin-induced red blood cells, tau, was 76 days. Neither of the dose-response parameters E(max) and ED(50) showed dependence on bodyweight. The median haemoglobin response to a standard dose, 0.042 g/dL for 1000IU, was similar to that reported for adults with intravenous administration. CONCLUSIONS: Doses for children in this age range should be specified as absolute amounts rather than amounts per unit bodyweight. Initial doses can be calculated individually, based on haemoglobin level before treatment, the desired haemoglobin at steady state and the median population parameters E(max), ED(50) and tau.

Analysis of combination drug therapy to develop regimens with shortened duration of treatment for tuberculosis.

RATIONALE: Tuberculosis remains a worldwide problem, particularly with the advent of multi-drug resistance. Shortening therapy duration for Mycobacterium tuberculosis is a major goal, requiring generation of optimal kill rate and resistance-suppression. Combination therapy is required to attain the goal of shorter therapy. OBJECTIVES: Our objective was to identify a method for identifying optimal combination chemotherapy. We developed a mathematical model for attaining this end. This is accomplished by identifying drug effect interaction (synergy, additivity, antagonism) for susceptible organisms and subpopulations resistant to each drug in the combination. METHODS: We studied the combination of linezolid plus rifampin in our hollow fiber infection model. We generated a fully parametric drug effect interaction mathematical model. The results were subjected to Monte Carlo simulation to extend the findings to a population of patients by accounting for between-patient variability in drug pharmacokinetics. RESULTS: All monotherapy allowed emergence of resistance over the first two weeks of the experiment. In combination, the interaction was additive for each population (susceptible and resistant). For a 600 mg/600 mg daily regimen of linezolid plus rifampin, we demonstrated that >50% of simulated subjects had eradicated the susceptible population by day 27 with the remaining organisms resistant to one or the other drug. Only 4% of patients had complete organism eradication by experiment end. DISCUSSION: These data strongly suggest that in order to achieve the goal of shortening therapy, the original regimen may need to be changed at one month to a regimen of two completely new agents with resistance mechanisms independent of the initial regimen. This hypothesis which arose from the analysis is immediately testable in a clinical trial.