Cluster Newton Method for Sampling Multiple Solutions of Underdetermined Inverse Problems: Application to a Parameter Identification Problem in Pharmacokinetics

Abstract

A new algorithm is proposed for simultaneously finding multiple solutions of an underdetermined inverse problem. The algorithm was developed for an ODE parameter identification problem in pharmacokinetics for which multiple solutions are of interest. The algorithm proceeds by computing a cluster of solutions simultaneously, and is more efficient than algorithms that compute multiple solutions one-by-one because it fits the Jacobian in a collective way using a least squares approach. It is demonstrated numerically that the algorithm finds accurate solutions that are suitably distributed, guided by a priori information on which part of the solution set is of interest, and that it does so much more efficiently than a baseline Levenberg-Marquardt method that computes solutions one-by-one. It is also demonstrated that the algorithm benefits from improved robustness due to an inherent smoothing provided by the least-squares fitting. Introduction.Since the information we can obtain clinically from a live patient going through treatment is often much less extensive than the complexity of the internal activity in a patient's body, underdetermined inverse problems naturally appear in the field of mathematical medicine. Our interest in underdetermined inverse problems of this kind was initiated by the parameter identification problem of a pharmacokinetics model for the anticancer drug CPT-11 (also known as Irinotecan) [2]. This pharmacokinetics model is an ODE-based mathematical model for the transportation, metabolization, and excretion of the drug in a human body. In this problem, a large set of parameters needs to be estimated from a very small number of measurements that correspond to integrated (accumulated) quantities (area-underthe-curve) at a single final time Konagaya has proposed a framework called "virtual patient population convergence" [12] (see [13] for the English translation), whose essential idea is to estimate the parameters of a whole body pharmacokinetics model from the clinically observed patient data. The essential difference of this framework with other approaches is that instead of finding a single set of parameters that is suitable for the pharmacokinetics model to reproduce the clinically observed data, its aim is to find multiple sets of such parameters, because the ranges and extremal values of

Application of PBPK Modeling and Virtual Clinical Study Approaches to Predict the Outcomes of CYP2D6 Genotype‐Guided Dosing of Tamoxifen

“…Despite extensive in vitro investigations of tamoxifen and its metabolites, the information is rather limited to describe their PK profiles in humans . Previously, the CNM‐based parameter optimization was successfully implemented for cases in which multiple sets of parameters were simultaneously optimized with wide initial ranges . In those cases, the CNM‐based process first identified a large number of parameter sets that can adequately describe the observed data.…”

“…In the current study, we constructed comprehensive PBPK models of tamoxifen and its metabolites using a Cluster Newton method (CNM) for parameter optimization. 12 Via these efforts, we successfully captured interindividual variability in endoxifen levels in virtual Japanese patients with breast cancer with differing CYP2D6 genotypes and calculated the expected outcomes of the TARGET-1 study prior to its completion (our approach is outlined in Figure 1b). Our current efforts demonstrate the promising utility of the PBPK modeling and VCS approaches in designing effective clinical trials.…”

Virtual Clinical Studies to Examine the Probability Distribution of the AUC at Target Tissues Using Physiologically-Based Pharmacokinetic Modeling: Application to Analyses of the Effect of Genetic Polymorphism of Enzymes and Transporters on Irinotecan Induced Side Effects

“…This PBPK-based optimization was obtained using a CNM (18). Although the same optimization using a Levenberg–Marquardt method, which is a conventional fitting approach, was performed on MATLAB, parameters were not converged because of too many unknown parameters.…”

“…To provide the diversity of the problem solutions (optimized sets of parameters), AUC for each compound was used as the objective function for optimization. Only stage 1 of the CNM optimization (18) was performed to avoid overfitting and save computational time.Procedures 1 and 2 described above were repeated 100 times with 10,000 different initial sets of parameters.The blood concentration–time profile of irinotecan and its metabolites was simulated using 1,000,000 optimized sets of parameters. The weighted sum of squares (WSS) between the simulated and observed blood concentration–time profile described in Eq.…”

Prediction Uncertainty Estimation Despite Unidentifiability: An Overview of Recent Developments

“…Similarly, the problem of unidentifiability, and the need to consider multiple parameter sets, has been described also within the pharmacokinetics community, and methods have been proposed to characterize such sets. One such method is cluster Newton [27], where a cluster of parameter sets are considered together, used to approximate a surface, and where this surface then is used in the optimization. However, although cluster Newton has shown some strengths, e.g.…”