Introduction: In NONMEM VI beta, an option exists for estimation of a nonparametric parameter (eta) distribution. In this, the parameter distributions are approximated by discrete probability density functions at a number of parameter values (support points) equal to or less than the number of individuals in the data set. It is thus similar to other nonparametric methods.[1] However, the support points are obtained from the empirical Bayes estimates (EBE) from a preceding parametric estimation step, which could be run with any of the standard methods available (e.g. FO or FOCE). The nonparametric objective function value (NPOFV) is calculated in the same manner as the OFV for mixture models with number of mixtures equal to number of support points. [2]

Aims: The present study aims at exploring the performance of this nonparametric estimation method through a Monte Carlo simulation study with special emphasis on the analysis of data with non-normal distribution of random effects.

Methods: Pharmacokinetic data sets (100 sets for each condition) were simulated from a one compartment iv bolus model in which CL and V parameter distributions were (i) log-normal, (ii) multimodal or (iii) heavy-tailed. Simulation settings were altered with respect to the number of individuals (200 vs 50 subjects) and informativeness of the data (assessed by varying the residual error magnitude). Each simulated data set was analyzed by models assuming (i) true (true model) (ii) log-normal (parametric model) and (iii) nonparametric (NP model) distributions. In each case both FO and FOCE was used and the nonparametric estimation was always preceded by an estimation assuming a log-normal parameter distribution, thus in most cases representing a model misspecification. The effect of shrinkage of EBE towards the population mean, in particular for sparse/uninformative data, was assessed by obtaining the EBE used as support points using (i) the estimated parametric variance and (ii) an inflated variance up to 5 times of the estimated standard deviation value. To compare the estimated and true distributions of CL and V, the relative estimation error and the absolute value of the relative bias (ARB) in parameter estimates were evaluated at the 10th, 25t, 50th, 75th and 90th percentile of the CL and V distributions.

Results: (i) Data with underlying log-normal true distribution With FO, the ARB in estimated distribution with parametric model (true model) ranged from 6.3 – 12.9 % at evaluated percentiles, while using the NP model, subsequent to these FO analyses, this bias became negligible (< 0.7 %). With FOCE, ARB with both true and NP models was minor (< 0.6 %).

(ii) Data with underlying multimodal true distribution With FO, the parametric model behaved poorly with an ARB ranging from 23 – 72 %. The NP model, although based on the preceding parametric step which incorrectly assumed a log-normal parameter distribution, reduced this ARB down to < 1.4 % and performed even better than the true mixture model, which resulted in an ARB ranging from 3.9 – 13.6 %. With FOCE, ARB seen with the true model and the NP model were similar, < 2.7 % and < 1.1 % respectively, while ARB ranged from 10 - 16 % for the parametric model.

(iii) Data with underlying heavy-tailed true distribution With FO, the parametric model performed poorly with ARB ranging from 8 to 98 %. The subsequent NP model reduced the ARB down to < 1.7 %. Again, the true model was less precise than the NP model (ARB ranging from 3.8 – 6.3 %). With FOCE, the true model and the NP model performed similarly with ARB < 2.3 % and < 1.5 %, respectively. The parametric model was imprecise with FOCE (ARB ranging from 9.4 - 14.4 %). In the case of uninformative data, with resulting EBE shrunk towards the population mean, results from all methods were less precise. However, the NP method resulted in a lower imprecision by use of support points (EBE) with an inflated variance, which was accompanied by a decrease in the NPOFV. This approach was an improvement also in comparison with the true model.

Discussion: The wide application of nonparametric methods has been limited for various reasons such as (i) methods are time–consuming, (ii) measures of precision are lacking, (iii) lack of investigation into model misspecification sensitivity, (iv) software allow limited flexibility in models and data format. The nonparametric method available in NONMEM VIβ may address some of these issues. In the present study, we have evaluated the method with respect to the parameter distribution estimation, which performed well in all studied cases (ARB < 2%), meaning that the estimated parameter distribution matched the true distribution, despite the method (FO or FOCE) used to assess the points of support and despite the preceding parametric model being misspecified (log-normal instead of multimodal or heavy-tailed). We have evaluated the NP method dependence on choice of support points, which in the case of EBE shrinkage may show increased bias at lower distribution percentiles. However, this bias disappears using the inflated variance for EBE computation. The appropriate inflation magnitude may be assessed using the NPOFV. The search for points of support is different in this method and NP methods available in other software. Having the EBE as a support points, the NP method available in NONMEM VI beta is likely to address the time aspect issue. NONMEM is recognized as a flexible tool for population analysis. As nonparametric distribution estimation is a follow-up on the parametric modeling, all flexible NONMEM features can be utilized, (e.g. complex residual error modeling). Furthermore, in general nonparametric methods condition on fixed residual error models. Within NONMEM, it is possible to use interindividual variation in residual error magnitude and estimate the residual error distribution using the NP method, representing an additional advantage. The potential uses of this method are seen as (i) an aid in building of parametric models, especially when the EBE distribution may be misleading or non-informative, (ii) replacement of the final parametric model, especially when the FO method is used and/or (iii) bridging the NONMEM with other NP software. However, issues such as lack of precision measures and handling of covariate effects remain to be addressed.

Conclusion: The nonparametric estimation method in NONMEM can identify non-normal parameter distributions and it is able to correct bias in parameter distribution estimates seen with FO. Overall, the method has shown promising properties when analyzing different types of PK data with both FO and FOCE methods as preceding steps.

Acknowledgements:
Comments and discussions on the topic with Prof. Stuart Beal are deeply and gratefully acknowledged.

References:
1. Mallet, A., et al., Nonparametric maximum likelihood estimation for population pharmacokinetics, with application to cyclosporine. J Pharmacokinet Biopharm, 1988. 16(3): 311-27.
2. Sheiner, L.B., Beal S.L., NONMEM Users Guide. 1992: NONMEM Project Group, University of California, San Francisco.