Optimising covariate allocation at design stage using fisher information matrix for non-linear mixed effects models in pharmacometrics

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

A Simple and Flexible Computational Framework for Inferring Sources of Heterogeneity from Single-Cell Dynamics.

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

D-optimal designs for discrete-type responses have been derived using generalized linear mixed models, simulation based methods and analytical approximations for computing the fisher information matrix (FIM) of non-linear mixed effect models with homogeneous probabilities over time. In this work, D-optimal designs using an analytical approximation of the FIM for a dichotomous, non-homogeneous, Markov-chain phase advanced sleep non-linear mixed effect model was investigated. The non-linear mixed effect model consisted of transition probabilities of dichotomous sleep data estimated as logistic functions using piecewise linear functions. Theoretical linear and nonlinear dose effects were added to the transition probabilities to modify the probability of being in either sleep stage. D-optimal designs were computed by determining an analytical approximation the FIM for each Markov component (one where the previous state was awake and another where the previous state was asleep). Each Markov component FIM was weighted either equally or by the average probability of response being awake or asleep over the night and summed to derive the total FIM (FIM(total)). The reference designs were placebo, 0.1, 1-, 6-, 10- and 20-mg dosing for a 2- to 6-way crossover study in six dosing groups. Optimized design variables were dose and number of subjects in each dose group. The designs were validated using stochastic simulation/re-estimation (SSE). Contrary to expectations, the predicted parameter uncertainty obtained via FIM(total) was larger than the uncertainty in parameter estimates computed by SSE. Nevertheless, the D-optimal designs decreased the uncertainty of parameter estimates relative to the reference designs. Additionally, the improvement for the D-optimal designs were more pronounced using SSE than predicted via FIM(total). Through the use of an approximate analytic solution and weighting schemes, the FIM(total) for a non-homogeneous, dichotomous Markov-chain phase advanced sleep model was computed and provided more efficient trial designs and increased nonlinear mixed-effects modeling parameter precision.

We investigated the use of non-linear mixed effects modeling in two preclinical studies of the glycogen phosphorylase inhibitor 1,4-dideoxy-1,4-imino-D-arabinitol (DAB). In a 28-day repeated-dose toxicity study rats were dosed once daily p.o. with 0, 20, 45, 100, or 470 mg/kg of DAB in aqueous solutions by oral gavage. Three blood samples were obtained from each animal using a staggered sampling scheme. During the cause of model development, data were included from a safety pharmacological cardiovascular study, in which rats were dosed once orally with 0, 4, 40, or 400 mg/kg of DAB thereby enabling an extension of the dose range of the model. DAB was assayed in plasma using a validated LC/MS/MS method. Non-linear mixed effects modeling was performed using the software NONMEM. The covariate analysis comprised dose, sex and time. Exposure results (Cmax, AUC) obtained by mixed effects modeling were compared to results from noncompartmental analysis using naïve pooling of data. The final model was a one-compartment model with first order absorption and a saturation-like dose dependent increase of the (oral) clearance (CL/f) and volume of distribution (V/f). Furthermore, V/f increased (by 55%) from Day 1 to Day 28. The dose dependencies of CL/f and V/f were most likely due to dose dependent decreases of the fraction systemically absorbed (f). The mechanism behind the dose dependencies may be saturation of a (putative) carrier mediated transport or modulation of tight junctions causing a reduced paracellular transport across the intestinal epithelium. Exposure results obtained from the model compared well with results obtained using noncompartmental analysis. An analysis of the data requirements for non-linear mixed effects modeling showed that at least three concentration values per animal were required for model development. We conclude that non-linear mixed effects modeling is feasible even with dose dependent pharmacokinetics in preclinical studies, such as 28-day toxicity studies in rodents. Supplementing data from additional preclinical studies may be required in order to extend the dose range. Non-linear mixed effects models may prove to be valuable tools in early PK and PK-PD modeling during drug development.

Analysis of repeated measurement data using the nonlinear mixed effects model

Much of the literature on optimal design of experiments has focussed on experiments wherenthe behaviour of the system is approximated by a linear model, such as a low-order polynomial.nIn many areas such as pharmacology and chemistry, such approximations are notnappropriate, as the underlying mechanisms produce highly nonlinear or categorical responses.nThis thesis addresses some issues with the optimal design of experiments in these situations.n Commonly used criteria for the 'optimal' design of experiments relate to optimality innterms of efficient estimation of model parameters. However, quite often another importantnobjective of an experiment is to select the model structure which best describes the underlyingnbehaviour of the system. We examine existing criteria for model discrimination fornboth nonlinear and generalised linear models, and combine them with criteria for parameternestimation in order to create designs which address both objectives. We show that thesendesigns can be quite efficient in terms of each of the criteria.n Further complications in the design process arise with the use of mixed effects models,nthat is when some model parameters are allowed to vary randomly between blocks or clustersnof units. The Fisher information matrix is involved in the calculation of many optimalityncriteria, and this matrix cannot be written down in closed form for many nonlinear andngeneralised linear mixed effects models. We instead rely on approximations to the informationnmatrix to generate optimal designs. This thesis gives details of the use of an existingnapproximation to the matrix in the optimal design of a complex pharmacokinetic experimentninvolving nonlinear mixed effects models. We also investigate several alternative approximationsnto the matrix for logistic regression with random coefficients, with an application innpharmacodynamics: the design of a cross-over trial with a binary response Regardless of the criterion used to select a particular design, we require a method tonsearch the design space for points which maximise or minimise the criterion. The modelsnconsidered in this thesis are assumed to have predictor variables taken over a continuousnrange, so combinatorial optimisation techniques such as the tabu algorithm are not appropriate.nInstead we make use of the simulated annealing algorithm (modified for continuousnvariables) and a relatively new algorithm known as the cross-entropy method. Both algorithmsnare implemented in programs written for the MATLAB package.n

Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis–Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.

Non-linear mixed effect models (NLMEMs) are widely used for the analysis of longitudinal data. To design these studies, optimal design based on the expected Fisher information matrix (FIM) can be used instead of performing time-consuming clinical trial simulations. In recent years, estimation algorithms for NLMEMs have transitioned from linearization toward more exact higher-order methods. Optimal design, on the other hand, has mainly relied on first-order (FO) linearization to calculate the FIM. Although efficient in general, FO cannot be applied to complex non-linear models and with difficulty in studies with discrete data. We propose an approach to evaluate the expected FIM in NLMEMs for both discrete and continuous outcomes. We used Markov Chain Monte Carlo (MCMC) to integrate the derivatives of the log-likelihood over the random effects, and Monte Carlo to evaluate its expectation w.r.t. the observations. Our method was implemented in R using Stan, which efficiently draws MCMC samples and calculates partial derivatives of the log-likelihood. Evaluated on several examples, our approach showed good performance with relative standard errors (RSEs) close to those obtained by simulations. We studied the influence of the number of MC and MCMC samples and computed the uncertainty of the FIM evaluation. We also compared our approach to Adaptive Gaussian Quadrature, Laplace approximation, and FO. Our method is available in R-package MIXFIM and can be used to evaluate the FIM, its determinant with confidence intervals (CIs), and RSEs with CIs.

<p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d618785e60"> <b>Objectives:</b> Clinical trial simulation tools rely on mathematical models for the natural progression of disease, typically fit to historical data. Traditionally, nonlinear mixed effects models have been employed for this purpose, but generative AI holds promise for modernizing such simulation tasks. While AI-based methods can discover complex nonlinear relationships between multimodal covariates and longitudinal outcomes, the credibility of such methods must be assessed with respect to the specific context of use. Our goal is to build a framework to assess credibility for such models, using Alzheimer’s Disease as a case study. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d618785e65"> <b>Methods:</b> Historical trial data were curated from the Critical Path for Alzheimer’s Disease (CPAD) database (14 studies, N = 4428). A generative AI method was built based on the Conditional Variational Autoencoder (CVAE) framework [ <a class="xref-link" href="#r1">1</a>], with baseline patient covariates used to inform the conditional prior model, and Long Short-Term Memory (LSTM) models employed for the encoder and decoder layers. An existing regulatory-grade Nonlinear Mixed Effects (NLME) model [ <a class="xref-link" href="#r2">2</a>] was implemented in Stan for benchmarking. For both models, cross-validation splits were used to evaluate simulation performance, with Visual Predictive Checks (VPCs) and statistical metrics between real and simulated data used to assess performance. A comparison between the modeling approaches was then developed to assist in evaluating credibility with regards to model interpretability. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d618785e76"> <b>Results:</b> Both the NLME and CVAE-based methods produce realistic simulation output using baseline covariates to predict longitudinal outcomes. As assessed by VPCs, the CVAE method moderately improves upon the NLME model’s simulation performance, particularly in later times where the NLME model tends to overestimate severity; technical performance metrics comparing real and simulated distributions at fixed time points confirm this. These technical evaluations in combination with comparisons between the interpretability of the CVAE and NLME frameworks show that it is possible to establish credibility for a generative AI-based disease progression simulation model. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d618785e81"> <b>Conclusions:</b> Generative AI shows promise as a tool for simulating disease progression, but incorporating this new paradigm into existing simulation workflows requires careful consideration. Our goal is to demonstrate that the credibility of generative AI methods can be assessed in a way that highlights their strengths and provides decision-makers with the information necessary to determine their best usage. Future applications of this framework to additional disease areas and simulation targets are planned.

The aim of this study was to investigate the use of nonlinear mixed effects (NLME) models in a real bioequivalence study and compare it to noncompartmental analysis (NCA) which is proposed by regulatory agencies. NCA requires few hypotheses but a large number of samples per subject. On the other hand, NLME approach is more complex than NCA but it has some advantages such as it requires few samples per subject. A real data application was provided for the study, which was get from Ege University Drug Development and Pharmacokinetics Research Center. For NLME, we used the stochastic approach to expectation-maximization (SAEM) algorithm, whereas linear trapezoidal rule was used for NCA. We estimate pharmacokinetic parameters, area under the curve (AUC0-∞) and maximum concentration (Cmax), and perform a bioequivalence tests using NCA and NLME. According to real data analysis, NLME approach has smaller within subject error, narrower confidence intervals than non-compartmental analysis. However, NLME models have some limitations because of increasing type I error. Therefore, caution is needed for small sample size and data with high variability.

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.

Summary We examine the problem of variance component testing in general mixed effects models using the likelihood ratio test. We account for the presence of nuisance parameters, ie, the fact that some untested variances might also be equal to zero. Two main issues arise in this context, leading to a nonregular setting. First, under the null hypothesis, the true parameter value lies on the boundary of the parameter space. Moreover, due to the presence of nuisance parameters, the exact locations of these boundary points are not known, which prevents the use of classical asymptotic theory of maximum likelihood estimation. Then, in the specific context of nonlinear mixed effects models, the Fisher information matrix is singular at the true parameter value. We address these two points by proposing a shrunk parametric bootstrap procedure, which is straightforward to apply even for nonlinear models. We show that the procedure is consistent, solving both the boundary and the singularity issues, and we provide a verifiable criterion for the applicability of our theoretical results. We show through a simulation study that, compared to the asymptotic approach, our procedure has a better small sample performance and is more robust to the presence of nuisance parameters. A real data application on bird growth rates is also provided.

Non-linear mixed effects (NLME) models represent a powerful tool to simultaneously analyse data from several individuals. In this study, a compartmental model of leucine kinetics is examined and extended with a stochastic differential equation to model non-steady-state concentrations of free leucine in the plasma. Data obtained from tracer/tracee experiments for a group of healthy control individuals and a group of individuals suffering from diabetes mellitus type 2 are analysed. We find that the interindividual variation of the model parameters is much smaller for the NLME models, compared to traditional estimates obtained from each individual separately. Using the mixed effects approach, the population parameters are estimated well also when only half of the data are used for each individual. For a typical individual, the amount of free leucine is predicted to vary with a standard deviation of 8.9% around a mean value during the experiment. Moreover, leucine degradation and protein uptake of leucine is smaller, proteolysis larger and the amount of free leucine in the body is much larger for the diabetic individuals than the control individuals. In conclusion, NLME models offers improved estimates for model parameters in complex models based on tracer/tracee data and may be a suitable tool to reduce data sampling in clinical studies.

Tree height and diameter at breast height are two important forest factors. The best model from 23 height-diameter equations was selected as the basic model to fit the height-diameter relationships of Chinese fir with one level (sites or plots effects) and nested two levels (nested effects of sites and plots) Nonlinear Mixed Effects (NLME) models. The best model was chosen by smaller Bias, RMSE and larger R²_adj. Then the best random-effects combinations for the NLME models were determined by AIC, BIC and -2LL. The results showed that the basic model with three random effects parameters φ₁, φ₂ and φ₃ was considered the best mixed model. The nested two levels NLME model considering heteroscedasticity structure (power function) possessed with higher predictable accuracy and significantly improved model performance (LRT = 469.43, p<0.0001). The NLME model would be allowed for estimating accuracy the height-diameter relationships of Chinese fir and provided better height predictions than the models using only fixed-effects parameters.