Use of a linearization approximation facilitating stochastic model building
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Abstract
The objective of this work was to facilitate the development of nonlinear mixed effects models by establishing a diagnostic method for evaluation of stochastic model components. The random effects investigated were between subject, between occasion and residual variability. The method was based on a firstorder conditional estimates linear approximation and evaluated on three real datasets with previously developed population pharmacokinetic models. The results were assessed based on the agreement in difference in objective function value between a basic model and extended models for the standard nonlinear and linearized approach respectively. The linearization was found to accurately identify significant extensions of the model’s stochastic components with notably decreased runtimes as compared to the standard nonlinear analysis. The observed gain in runtimes varied between four to more than 50fold and the largest gains were seen for models with originally long runtimes. This method may be especially useful as a screening tool to detect correlations between random effects since it substantially quickens the estimation of large variance–covariance blocks. To expedite the application of this diagnostic tool, the linearization procedure has been automated and implemented in the software package PsN.
Keywords
Linearization Random effects Nonlinear mixed effects models Pharmacometrics DiagnosticsIntroduction
Population pharmacokinetics (PK) and pharmacodynamics (PD) models, i.e. pharmacometrics, play an increasingly important role in pharmaceutical sciences and drug development [1]. Nonlinear mixed effects (NLME) models are frequently used to describe population PK/PD and are composed of a structural component (fixed effects) and a stochastic component (random effects). Fixed effects describe the typical parameter values in a population and the effects of covariates. Random effects handle unexplained variability and can be further divided into variability assigned to parameters and residual variability (RV) assigned to the observations. Parameters can vary on multiple levels: between subjects (BSV), between occasions (BOV), between studies, etc.
For NLME, especially when used in simulations, the stochastic components of the model are crucial but the development procedure can be laborious. Despite a vast increase in available computational power during the past decade, the time for parameter estimation can still be a limiting factor since the complexity of the used models and the amount of fitted data typically are increasing likewise. For highly nonlinear models numerical instability of parameter estimation with common gradient based, local methods may cause further problems. Individual parameter estimates, the empirical Bayes estimates (EBEs), can be used as a diagnostic tool to aid the development procedure. However, this practice has notable shortcomings when data are uninformative on the individual level and shrinkage towards the population mean occurs. Shrinkage can cause diagnostics based on EBEs to mask, falsely induce or distort the shape of random effects relationships [2, 3]. The objective of this work was to develop and assess a fast and stable method which is not sensitive to shrinkage for diagnostics of BSV, BOV and RV model components. The here proposed method, henceforth called ‘linearization’, is based on a firstorder conditional estimation (FOCE) linear approximation which has previously successfully been applied for testing of covariates [4]. The linearization was evaluated in the modeling software NONMEM with a variety of different models and datasets. The results are presented as comparisons between the linearization and the corresponding nonlinear model, both in terms of estimation performance and runtimes.
Methods
Population models and linearization
The residual error model is commonly modeled as a function of \( \vec{\varepsilon }_{ij} \) which is assumed to be normally distributed with mean 0 and variance–covariance matrix Σ. Common forms of h _{ ij } are additive error (h _{ ij } = ɛ _{ ij }), proportional error \( (h_{ij} = f\left( {\vec{p}_{i} ,\vec{x}_{ij} } \right) \times \varepsilon_{ij} ) \) or a combined error with both an additive and a proportional element \( (h_{ij} = f\left( {\vec{p}_{i} ,\vec{x}_{ij} } \right) \times \varepsilon_{1ij} + \varepsilon_{2ij} ) \). Examples of extensions and more flexible forms have been described in the following references [5, 6, 7, 8].
Software
NONMEM versions 7.2 and 7.3 beta (ICON Development Solutions, Hanover, MD, USA) [10] with the estimation method FOCEI were used for the analysis. To overcome sensitivity to local minima when estimating individual parameters (ηvalues) the new NONMEM option MCETA was utilized. The default initial value for all η is zero. MCETA allows the user to define a number of vectors containing random samples of initial ηvalues that should be tested in addition to zero, whichever supplies the lowest OFV will be used as initial value in the estimation. The numbers in each vector of initial ηvalues are randomly drawn from the normal distribution with mean zero and variance–covariance matrix Ω. With a large enough number of initial ηvalues tested, the probability of the estimation to end up in a local minimum can be decreased.
Models were executed through PsN, using Piranha for documentation and creation of run records [11, 12, 13].
Evaluated model extensions

BSV of the residual error [14]

A power relation with the individual model predictions (F), implemented as \( h_{ij} = h_{ij(base)} \times F^{\theta } \)

Autocorrelation, i.e. serial correlation between the error of observations consecutive in time [5]

Time dependent residual error, implemented as a step function [5]
The NONMEM code of the RV extensions is provided in the supplementary material. For extensions of the BOV and/or the correlation structure, a base model already including some BSV terms was used while for extensions of the BSV structure the base model only included RV. To this model additional parameterspecific variability were added and/or the structure of the variance–covariance block changed. To enable estimation of the linearized model with additional variability parameters, the partial derivatives of the model with respect to the new parameters must be known. The partial derivatives can be obtained in NONMEM by including the code for the additional variability parameters already in the nonlinear base model but fixed to an arbitrary small value (fixing it to zero would result in that no derivatives are calculated).
All extended models were estimated with FOCEI both in the linearized form and as standard nonlinear models. The results were evaluated by comparing the difference in OFV (ΔOFV) between the extended model and the base model for the two approaches respectively. The runtimes for the estimation step were extracted from the result files. The runtime comparisons should be interpreted in terms of magnitude and not as exact figures, since all estimations were carried out at a cluster with many nodes, which have somewhat different capacities and are randomly assigned.
Datasets
 1.
Moxonidine [14]: The data were obtained from a phase II multicenter study of oral moxonidine in patients with congestive heart failure and contained 1,022 observations from 74 patients. The structural model describing the data was a onecompartment model with firstorder absorption and an additive residual error model.
 2.
Pefloxacin [15, 16]: The data were obtained from critically ill patients receiving 1 h intravenous infusions of pefloxacin and contained 337 observations from 74 patients. The structural model was a onecompartment model and a proportional residual error model.
 3.
Ethambutol [17]: The data were obtained from two prospective studies in tuberculosis patients receiving a standard treatment regimen including ethambutol. A total of 1869 observations from 189 patients were included in the dataset. The structural model was a onecompartment model with firstorder absorption through one transit compartment and a combined residual error model.
Results
Comparison of total runtimes for nonlinear and linearized models using the three example datasets
Data  Total runtime nonlinear (s)  Total runtime linearized (s)  Fraction time required linearized (%) 

Moxonidine  735.1  106.5  14.5 
Pefloxacin  47.78  6.96  14.6 
Ethambutol  103082  25983^{a}  25.2 
Discussion
A novel diagnostic method for evaluation of random effects was successfully developed and evaluated. The linearization was found to accurately identify significant extensions of models’ stochastic components with notably decreased runtimes as compared to standard nonlinear analysis. The three examples used had relatively short runtimes also in their nonlinear form. When the linearization was applied to more complex models, an even more substantial decrease in runtimes (>50×) was observed. For a PDmodel describing the effect of docetaxel on neutrophil counts (extension of [18] ) utilizing the full random effects approach (FREM [19] ) the runtime was decreased from 3 h and 50 to 5 min (2.2 %). For a PK model of bedaquiline plus two metabolites including a large covariance block (extension of [20] ) the runtime was reduced from 15 h and 52 to 6 min (0.6 %). For a PKmodel of rifabutin plus metabolite using a dataset combining 14 clinical studies (unpublished) for which the runtime of the base nonlinear model was several day, the estimation of 6 additional BSV parameters and extending the variance–covariance matrix from a diagonal to a full 12 × 12 block structure took only 18 and 89 min, respectively.
When applied to models with a proportional or a combined residual error structure the linearization was found to be sensitive to local minima when assigning the individual ηvalues. This happens due to the potential shape distortion of the individual EBE likelihood profiles that can be caused by the interaction term (including the partial derivative with respect to both ɛ and η of the residual error model) in the linearized model. The interaction term will always be zero for additive residual error models which explains why the problem was never observed in this case. Care must be taken to ensure that the OFV of the linearized base model agrees with the corresponding nonlinear model. If a deviation is detected, simple workarounds can preclude potential deviations. Transformations can render the error model additive in the transformed space, for example logtransformation for model with proportional error structure. Another solution is the MCETA option, available in NONMEM from version 7.3.0 [21], which decreases the risk of issues with local minima by supplying multiple initial estimates for the individual ηvalues. With MCETA values of between 10 and 1,000 all evaluated linearized models were able to obtain the same OFV value as the corresponding nonlinear model. However, run times were somewhat increase by use of the MCETA option. Yet another potential solution could be to input the EBE’s from the nonlinear base model as initial estimates for individual ηvalues in the linearized model. This is possible with the ETAS option, also available in NONMEM from version 7.3.0. The ETAS option should be faster than the MCETA option since it only tests one set of initial estimates instead of several, but it may be less reliable. In cases when the shape of the likelihood profiles contain multiple local minima, as have been observed for linearized models with proportional or combined error structures, the risk of terminating in local minima is substantial even with initial estimates close to the optimal values and therefore the use of the MCETA option could be safer.
To simplify the use of this novel methodology, an option called ‘linearize’ has been implemented in PsN (available from version 3.7.0). The procedure requires the provision of a nonlinear model, optimizes the parameters and outputs predictions and derivatives in a new dataset. This dataset constitutes the input for an automatically created linearized version of the same model. The linearized model can serve as starting point for evaluation of manually coded extensions of the random effects. Model building could potentially be made even more efficient by automated testing of a library of RV models, comparable to the stepwise covariate search method (the SCM) already implemented in PsN. Since the values of fixed effects are not estimated in the linearized model the method should be viewed as a diagnostic tool. It is recommended to reestimate with standard nonlinear methods once the best extended model is identified.
In conclusion, the successful use of a linear approximation method for fast diagnosis of a broad range of extended random effects models was demonstrated. The method may be especially valuable as a screening tool to detect correlations between random effects since estimating large variance–covariance blocks often is a computationally demanding and timeconsuming process but can be carried out magnitudes faster with the linearization.
Notes
Acknowledgments
We thank Joakim Nyberg for valuable help with understanding of the impact of the linearization on the likelihood function, Kajsa Harling for implementation in PsN and Ronald Niebecker and Hwiyeol Yun for examples of the use of the linearization with more complex models. The research leading to these results has received support from the Innovative Medicines Initiative Joint Undertaking under Grant Agreement No. 115156, resources of which are composed of financial contributions from the European Union’s Seventh Framework Programme (FP7/20072013) and EFPIA companies’ in kind contribution. The DDMoRe project is also supported by financial contribution from Academic and SME partners.
Supplementary material
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