Stochastic Differential Equations in NONMEM®: Implementation,Application, and Comparison with Ordinary Differential Equations

Purpose The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling.Methods The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM^® for parameter estimation in SDE models.Results Various applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix.Conclusions The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.     

The importance of modeling interoccasion variability in population pharmacokinetic analyses

Individual pharmacokinetic parameters may change randomly between study occasions. Analysis of simulated data with NONMEM shows that ignoring such interoccasion variability (IOV) may result in biased population parameter estimates. Particular parameters affected and the extent to which they are biased depend on study design and the magnitude of IOV and interindividual variability. Neglecting IOV also results in a high incidence of statistically significant spurious period effects. Perhaps most important, ignoring IOV can lead to a falsely optimistic impression of the potential value of therapeutic drug monitoring. A model incorporating IOV was developed and its performance in the presence and absence of IOV was evaluated. The IOV model performs well with respect to both model selection and population parameter estimation in all circumstances studied. Analysis of two real data examples using this model reveals significant IOV in all parameters for both drugs and supports the simulation findings for the case that IOV is ignored: predictable biases occur in parameter estimates and previously nonexistent period effects are found.This work was supported by U.S. Department of Health and Human Services grants OM26691 and GM26676.     

Importance of characterizing determinants of variability in exposure: application to dasatinib in subjects with chronic myeloid leukemia

Characterizing the key determinants of variability in the exposure of orally administered drugs may be important in understanding the implications of exposure variability on clinical responses. In particular, partitioning overall variability into interoccasion variability (IOV) and interindividual variability (IIV) allows a better assessment of the clinical importance of exposure variability. The IOV characterizes the dose-to-dose variability in exposure within a subject and is likely to be less clinically relevant than IIV for chronically administered drugs as the effect of IOV averages out over repeated dosing. The main aims of this model-based analysis were (1) to characterize the IOV and IIV of dasatinib, a novel, orally administered, multitargeted kinase inhibitor of BCR-ABL and SRC family kinases that is indicated for the treatment of chronic myeloid leukemia and Philadelphia-positive acute lymphoblastic leukemia and (2) to demonstrate using simulated data that it is possible to estimate IIV and IOV in relative bioavailability (F(R)) of an orally administered drug, given an adequate sampling scheme. Variability in dasatinib exposure was estimated to be mainly due to IOV in F(R) (44% coefficient of variation [CV]) and, to a lesser extent, due to IIV in F(R) and IIV in clearance (32% and 25% CV, respectively). The IIV is expected to be more clinically relevant than IOV for chronically administered oral drugs such as dasatinib, as the overall variability in cumulative exposure will be mainly due to IIV. The analysis of simulated data demonstrated that models ignoring either IIV or IOV in F(R) resulted in upwardly biased estimates of interindividual or residual variability. Thus, it may be important to account for both IIV and IOV in F(R), particularly for orally administered agents that exhibit absorption-related variability in exposure.     

Mixed Effects Modeling Using Stochastic Differential Equations: Illustrated by Pharmacokinetic Data of Nicotinic Acid in Obese Zucker Rats

Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.KEY WORDS: extended Kalman filter, model uncertainty, nonlinear kinetics, parameter estimation, state prediction     

Comparison of different methods to evaluate population dose-response and relative potency: importance of interoccasion variability.

Different mixed-effects models were compared to evaluate the population dose-response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (Emax) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED50 and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose-response and relative potency, there are important differences in the assumptions made by each method.     

Stochastic nonlinear mixed effects: a metformin case study

In nonlinear mixed effect (NLME) modeling, the intra-individual variability is a collection of errors due to assay sensitivity, dosing, sampling, as well as model misspecification. Utilizing stochastic differential equations (SDE) within the NLME framework allows the decoupling of the measurement errors from the model misspecification. This leads the SDE approach to be a novel tool for model refinement. Using Metformin clinical pharmacokinetic (PK) data, the process of model development through the use of SDEs in population PK modeling was done to study the dynamics of absorption rate. A base model was constructed and then refined by using the system noise terms of the SDEs to track model parameters and model misspecification. This provides the unique advantage of making no underlying assumptions about the structural model for the absorption process while quantifying insufficiencies in the current model. This article focuses on implementing the extended Kalman filter and unscented Kalman filter in an NLME framework for parameter estimation and model development, comparing the methodologies, and illustrating their challenges and utility. The Kalman filter algorithms were successfully implemented in NLME models using MATLAB with run time differences between the ODE and SDE methods comparable to the differences found by Kakhi [10] for their stochastic deconvolution.     

Grey-box Modelling of Pharmacokinetic /Pharmacodynamic Systems

Grey-box pharmacokinetic/pharmacodynamic (PK/PD) modelling is presented as a promising way of modelling PK/PD systems. The concept behind grey-box modelling is based on combining physiological knowledge along with information from data in the estimation of model parameters. Grey-box modelling consists of using stochastic differential equations (SDEs) where the stochastic term in the differential equations represents unknown or incorrectly modelled dynamics of the system. The methodology behind the grey-box PK/PD modelling framework for systematic model improvement is illustrated using simulated data and furthermore applied to Bergmans minimal model of glucose kinetics using clinical data from an intravenous glucose tolerance test (IVGTT). The grey-box estimates of the stochastic system noise parameters indicate that the glucose minimal model is too simple and should preferably be revised in order to describe the complicated in vivo system of insulin and glucose following an IVGTT.     

Assessment of the Regulatory Methods for the Comparison of Highly Variable Dissolution Profiles

The objective is to compare the performance of dissolution-profile comparison methods when f ₂ is inadequate due to high variability. The 90% confidence region of the Mahalanobis distance and the 90% bootstrap confidence interval (CI) of the f ₂ similarity factor (f ₂-bootstrap) were explored. A modification of the Mahalanobis distance (new D-Mahalanobis) in which those points >85% were not taken into account for calculation was also used. A population kinetic approach in NONMEM was used to simulate dissolution profiles with the first-order or Weibull kinetic models. The scenarios were designed to have clearly similar, clearly non-similar or borderline situations. Four different conditions of variability were established: high (CV?=?20%) and low variability (CV?=?5%) for inter-tablet (IIV) and inter-batch variability (IBV) associated to the dissolution parameters (k _d or MDT) using an exponential model. Forty-four (44) scenarios were simulated, considering different combinations of IIV, IBV and typical dissolution parameters. The dissolution profiles simulated using a first-order model modified the profile slope. The Weibull model allows profiles with different shapes and asymptotes and crossing each other. The results show that the f ₂-bootstrap is the most adequate method in cases of high variability. The method based on the 90% confidence region of the Mahalanobis distance (D-Mahalanobis) is not able to detect large differences that can be detected simply with f ₂ (i.e. low specificity and positive predictive value due to false positives). The new D-Mahalanobis exhibits superior sensitivity to detect differences (i.e. specificity as a diagnostic test), but it is not as good as the f ₂-bootstrap method.     

Influence of <Emphasis Type="Italic">CYP2B6</Emphasis> 516G>T polymorphism and interoccasion variability (IOV) on the population pharmacokinetics of efavirenz in HIV-infected South African children

Objective

To investigate the influence of CYP2B6 516G>T polymorphism, as a covariate, and of interoccasion variability (IOV) on the oral clearance (CL/F) of efavirenz (EFV) in treatment-naïve black South African children over a period of 24 months post-antiretroviral therapy (ART) initiation.

Methods

HIV-infected black children (n?=?60, aged 3–16 years), with no prior exposure to ART, eligible to commence ART and attending an outpatient clinic were enrolled into this study. Blood samples were taken at mid-dose interval at 1, 3, 6, 12, 18 and 24 months post-ART initiation. EFV plasma samples were determined with an adapted and validated LC/MS/MS method. Genotyping of the CYP2B6 G516T single nucleotide polymorphism (SNP) was performed using polymerase chain reaction–restriction fragment length polymorphism (PCR-RFLP). NONMEM was used for the population pharmacokinetic modelling.

Results

EFV concentrations below 1 μg/mL accounted for 18% (116/649), EFV concentrations >4 μg/mL accounted for 29.5% (192/649) and concentrations within the therapeutic range (1–4 μg/mL) represented 52.5% (341/649) of all the samples determined. The covariates age, weight and CYP2B6 G516Tgenotype were included in the final model with population estimates for CL/F determined as 2.46, 4.60 and 7.33 L/h for the T/T, G/T and G/G genotype groups respectively.

Conclusions

The inclusion of both age and weight to predict accurate EFV CL values for the respective genotype groups within this paediatric population was required, whereas the addition of gender and body surface area did not improve the predictions. The importance of introducing IOV in a PK model for a longitudinal study with sparsely collected data was again highlighted by this investigation.

     

A matlab framework for estimation of NLME models using stochastic differential equations

The non-linear mixed-effects model based on stochastic differential equations (SDEs) provides an attractive residual error model, that is able to handle serially correlated residuals typically arising from structural mis-specification of the true underlying model. The use of SDEs also opens up for new tools for model development and easily allows for tracking of unknown inputs and parameters over time. An algorithm for maximum likelihood estimation of the model has earlier been proposed, and the present paper presents the first general implementation of this algorithm. The implementation is done in Matlab and also demonstrates the use of parallel computing for improved estimation times. The use of the implementation is illustrated by two examples of application which focus on the ability of the model to estimate unknown inputs facilitated by the extension to SDEs. The first application is a deconvolution-type estimation of the insulin secretion rate based on a linear two-compartment model for C-peptide measurements. In the second application the model is extended to also give an estimate of the time varying liver extraction based on both C-peptide and insulin measurements.     

Effects of <Emphasis Type="Italic">CYP2C19</Emphasis> and <Emphasis Type="Italic">CYP2C9</Emphasis> genotypes on pharmacokinetic variability of valproic acid in Chinese epileptic patients: nonlinear mixed-effect modeling

Purpose

To evaluate the effects of CYP2C19 and CYP2C9 genotypes on the pharmacokinetic variability of valproic acid (VPA) in epileptic patients using a population pharmacokinetic (PPK) approach.

Methods

VPA concentrations were measured in 287 epileptic patients, who were genotyped for CYP2C19*2/*3 and CYP2C9*3. Patients who were on monotherapy with VPA were divided into two groups, a PPK-model group (n?=?177) and a PPK-valid group (n?=?110). The PPK parameter values for VPA were calculated in the PPK-model group by using the NONMEM software. Ultimately, a biological model and a final model were established. Each model was then used to independently predict the concentrations of the PPK-valid group to validate the two models.

Results

There was a significant effect of the CYP2C19 and CYP2C9 genotypes on the pharmacokinetic (PK) variability (P?<?0.01) in the final PPK model of CL/F. The interindividual CL was calculated according to the final model: CL/F?=?0.0951?×?(1?+?e^{0.0267?×?(3???genotype)})?+?0.0071?×?age (L/h). The coefficient of variation (CV) (omega CL/F) of the final model was 29.3%, while that of the biological model was 31.7%. Based on the genotype, the individual PK parameters can be calculated more accurately than before.

Conclusion

The CYP2C19 and CYP2C9 genotypes significantly influenced the PK variability of VPA, as quantified by NONMEM software.

     

Comparison of Different Methods to Evaluate Population Dose–Response and Relative Potency: Importance of Interoccasion Variability

Different mixed-effects models were compared to evaluate the population dose–response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (E_max ) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED₅₀ and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose–response and relative potency, there are important differences in the assumptions made by each method.     

Exact Gradients Improve Parameter Estimation in Nonlinear Mixed Effects Models with Stochastic Dynamics

Nonlinear mixed effects (NLME) modeling based on stochastic differential equations (SDEs) have evolved into a promising approach for analysis of PK/PD data. SDE-NLME models go beyond the realm of standard population modeling as they consider stochastic dynamics, thereby introducing a probabilistic perspective on the state variables. This article presents a summary of the main contributions to SDE-NLME models found in the literature. The aims of this work were to develop an exact gradient version of the first-order conditional estimation (FOCE) method for SDE-NLME models and to investigate whether it enabled faster estimation and better gradient precision/accuracy compared to the use of gradients approximated by finite differences. A simulation-estimation study was set up whereby finite difference approximations of the gradients of each level were interchanged with the exact gradients. Following previous work, the uncertainty of the state variables was accounted for using the extended Kalman filter (EKF). The exact gradient FOCE method was implemented in Mathematica 11 and evaluated on SDE versions of three common PK/PD models. When finite difference gradients were replaced by exact gradients at both FOCE levels, relative runtimes improved between 6- and 32-fold, depending on model complexity. Additionally, gradient precision/accuracy was significantly better in the exact gradient case. We conclude that parameter estimation using FOCE with exact gradients can successfully be applied to SDE-NLME models.     

Estimating the effects of 17α-ethinylestradiol on stochastic population growth rate of fathead minnows: a population synthesis of empirically derived vital rates

Urban freshwater streams in arid climates are wastewater effluent dominated ecosystems particularly impacted by bioactive chemicals including steroid estrogens that disrupt vertebrate reproduction. However, more understanding of the population and ecological consequences of exposure to wastewater effluent is needed. We used empirically derived vital rate estimates from a mesocosm study to develop a stochastic stage-structured population model and evaluated the effect of 17α-ethinylestradiol (EE2), the estrogen in human contraceptive pills, on fathead minnow Pimephales promelas stochastic population growth rate. Tested EE2 concentrations ranged from 3.2 to 10.9 ng L^?1 and produced stochastic population growth rates (λ _S ) below 1 at the lowest concentration, indicating potential for population decline. Declines in λ _S compared to controls were evident in treatments that were lethal to adult males despite statistically insignificant effects on egg production and juvenile recruitment. In fact, results indicated that λ _S was most sensitive to the survival of juveniles and female egg production. More broadly, our results document that population model results may differ even when empirically derived estimates of vital rates are similar among experimental treatments, and demonstrate how population models integrate and project the effects of stressors throughout the life cycle. Thus, stochastic population models can more effectively evaluate the ecological consequences of experimentally derived vital rates.     

Population pharmacokinetics of mefloquine in military personnel for prophylaxis against malaria infection during field deployment

Objective The purpose of this study was to determine the population pharmacokinetics of mefloquine in healthy military personnel during prophylaxis for malaria infections. Methods The subjects were 1,111 Australian soldiers participating in two studies: a randomised double-blinded study (group A, 161 subjects) and an open-label study (group B, 950 subjects). Following a loading dose (250 mg mefloquine base daily, 3 days), subjects received an oral weekly maintenance dose of 250 mg over 6 months. Blood was collected after the last split loading dose then at weeks 4, 8 and 16 for group A, and at weeks 13 and 26 for group B. Plasma mefloquine concentrations were measured by high-performance liquid chromatography (HPLC). Pharmacokinetic modelling was performed using NONMEM. Results A two-compartment model with inter-occasion variability (IOV) for clearance satisfactorily described the pharmacokinetics. Typical values were clearance (CL/F, 2.09 l/h), central volume of distribution (V1/F, 528 l), absorption rate constant (KA, 0.24 h⁻¹⁾, inter-compartmental clearance (Q/F, 12.5 l/h), peripheral volume of distribution (V2/F, 483 l) and elimination half-life (t _1/2, 14.0 days). Weight had a positive influence on central volume but was insufficient to warrant dosage adjustments. The inter-individual variability (coefficient of variation, CV%) for CL/F and V1/F was 24.4% and 29.6%, respectively. The IOV for CL/F was 17.8%. The proportional residual error (CV%) for groups A and B was 11.5% and 19.5%, respectively, and the additive error standard deviation (SD) was 57 ng/ml and 149 ng/ml, respectively. Conclusion The typical parameter values were comparable with those estimated in much smaller cohorts of healthy subjects and in malaria patients treated with single-dose mefloquine. The lower unexplained variability in the blinded study suggested these subjects may have been more compliant in taking their medication than soldiers in the open-label study.     

Using a semi‐mechanistic model to identify the main sources of variability of metformin pharmacokinetics

Metformin pharmacokinetics (PK) is highly variable, and researchers have for years tried to shed light on determinants of inter‐individual (IIV) and inter‐occasion variability (IOV) of metformin PK. We set out to identify the main sources of PK variability using a semi‐mechanistic model. We assessed the influence of subject characteristics, including seven genetic variants. Data from three studies of healthy individuals with PK measurements of plasma and urine after single dose or at steady‐state were used in this study. In total, 87 subjects were included (16 crossover subjects). Single nucleotide polymorphisms in ATM, OCT1, OCT2, MATE1 and MATE2‐K were investigated as dominant, recessive or additive. A three‐compartment model with transit absorption and renal elimination with a proportional error was fitted to the data using NONMEM 7.3. Oral parameters were separated from disposition parameters as dose‐dependent absolute bioavailability was determined with support from urine data. Clearance was expressed as net renal secretion and filtration, assuming full fraction unbound and fraction excreted. Mean transit time and peripheral volume of distribution were identified as the main sources of variability according to estimates, with 94% IOV and 95% IIV, respectively. Clearance contributed only with 16% IIV. Glomerular filtration rate and body‐weight were the only covariates found to affect metformin net secretion, reducing IIV to 14%. None of the genetic variants were found to affect metformin PK. Based on our analysis, finding covariates explaining absorption of metformin is much more valuable in understanding variability and avoiding toxicity than elimination.     

Application of Pharmacometric Analysis in the Design of Clinical Pharmacology Studies for Biosimilar Development

This article provides an overview of four case studies to demonstrate the utility of pharmacometric analysis in biosimilar development to help design sensitive clinical pharmacology studies for the demonstration of biosimilarity. The two major factors that determine the sensitivity of a clinical pharmacokinetic/pharmacodynamic (PK/PD) study to demonstrate biosimilarity are the size of the potential difference to be detected (signal) and the inter-subject variability (noise), both of which can be characterized and predicted using pharmacometric approaches. To maximize the chance to detect any potential difference between the proposed biosimilar and the reference drug, the dose selected for the clinical pharmacology study should fall on the steep part of the dose-response curve. Pharmacometric analysis can be used to characterize the dose-response relationship using PD- or PK/PD-linked models. The understanding of the PD endpoints in terms of dynamic range of the response and the location of the studied dose on the dose-response curve can provide strategic advantage in the trial design. To reduce the inter-subject variability (noise), pharmacometric analysis can help avoid high variability associated with low doses, and decrease variability by controlling certain covariates in the inclusion/exclusion criteria. Pharmacometric analysis also can help select or justify margins for the equivalence test of PD endpoints. Pharmacometric analysis will assume an ever-increasing role in the clinical development of biosimilar drugs, as it helps to ensure that sufficient sensitivity is built into the study design to detect potential PK and PD differences.