The reference corrected VPC – A more intuitive model diagnostic
Moustafa M. A. Ibrahim (1), Niclas Jonsson (1), Martin Bergstrand (1)
(1) Pharmetheus, Uppsala, Sweden.
Background:
The prediction corrected visual predictive check (pcVPC) is an informative model diagnostic that offer advantages over regular VPCs, in particular for cases with heterogenous study designs and adaptive dosing [1]. However, a drawback with these plots is that the prediction correction often results in y-axis values and trends that are nonintuitive and do not translate to any meaningful aspect of the study design, therefore challenging to communicate, especially to a wider audience.
Objectives:
To introduce the reference corrected VPC (rcVPC), that leverages a user defined set of independent variables, for a more intuitive model diagnostic that can be used for improved communication of modeling results.
Methods:
The rcVPC methodology is based on the definition of a reference dataset with user defined independent variables (e.g. covariates, dose, inter dose interval) but otherwise with the same dimensions as the analysis dataset. N number of simulations are conducted with this reference dataset (ref) as well as with the original analysis dataset. The observed and simulated dependent variable based on the analysis dataset is subject to the same type of reference correction (eq. 1 - 2). Where rcYij is the reference corrected dependent variable, Yij the dependent variable and PREDij the population typical prediction, each for the ith individual and jth observation. Yij,ref and PREDij,ref are the corresponding variable for the reference dataset simulations.
rcYij = exp(ln(PREDij,ref)+(ln(rcY*ij)-ln(PREDij,ref)) x sd(ln(Yij,ref,1 … Yij,ref,N)) / sd(ln(rcY*ij,ref,1 … rcY*ij,ref,N))) (eq. 1)
Where rcY*ij,ref is given by:
rcY*ij = Yij x PREDij,ref / PREDij (eq. 2)
The rcVPC approach was compared to pcVPCs and traditional VPCs for a range of simulated and real PK and PKPD examples. NONMEM version 7.5 in combination PsN was used for simulation and re-estimation purposes and all data post processing and graphical representation was done using R. A version of PsN with an option rcVPCs option is expected to be released in 2024.
Results:
The rcVPCs was demonstrated to show the model’s predictive performance with a more intuitive interpretation of the y-axis scale e.g. plasma concentrations normalized to an individual of 70 kg with weekly dosing of 400 mg for 16 weeks (original data with variable body weight and dosing regimens). The rcVPC approach was also found to indicated the true model misspecifications more clearly, including (1) the lack of an appropriate clearance maturation function for a pediatric PK example and (2) a misspecified exposure-response relationship (linear vs. Emax) in an example featuring simultaneous PKPD model for weight reduction. For the weight reduction example, the improved approach was used to normalize the y-axis variable to depict a 16 week end-of-treatment response versus trough plasma concentrations. This way the did not only make for a useful model diagnostic but also an efficient tool for communicating the established underlying exposure response relationship. Compared to pcVPCs the new method requires twice as many simulations (simulations both with the reference dataset and the original analysis dataset) but the correction method is independent of choice of binning and/or stratification.
Conclusions:
The proposed rcVPC methodology was demonstrated to offer a more intuitive interpretation and a more effective guidance to model development compared to the more traditional pcVPCs.
References:
[1] Bergstrand M, Hooker AC, Wallin JE, Karlsson MO. Prediction-corrected visual predictive checks for diagnosing nonlinear mixed-effects models. AAPS J. 2011 Jun;13(2):143-51.
