Global Sensitivity Analysis for Effects of Correlated Covariates Using Shapley Value
Takayuki Katsube
Shionogi & Co., Ltd., Japan
Introduction: Global sensitivity analysis (GSA) is a simulation-based approach to assess impacts of each factor on interested outcomes [1]. We reported that GSA was applicable to pharmacometric models to evaluate contribution of each covariate effect on the outcome [2]. In the variance-based GSA, Sobol index is used as a sensitivity index based on variance ratios [3]. The Sobol index is applicable to evaluate effects of independent factors, but not effects of correlated factors. Shapley values are indices proposed in cooperative games to allocate a mutual contribution of correlation and interaction of dependent factors [4].
Objectives: The objective of this study was to apply Shapley values to assess effects of correlated covariates in GSA for pharmacometric models.
Methods: The Shapley value is given in the reference [4]. Monte-Carlo simulations were conducted to simulate factors composed of covariates and random variables for inter-individual variability (IIV), which are defined as ETAs in NONMEM, and to calculate an interested outcome and Shapley values for each factor. As a tested model, a linear model of total clearance (CL) with bi-exponential covariates correlated (correlation coefficient [R] = 0.8) and IIV for CL was used to assess the Shapley values for the contribution of each of two covariate effects to CL. As an evaluation for a complex model, a quantitative systems pharmacology (QSP) model of lusutrombopag [5], a thrombopoietin-receptor agonist, was used to assess the Shapley values of each covariate for peak platelet counts. In the QSP model, hypothetical covariates to explain 50% of IIV were assumed for each of key parameters. The covariates on platelet baseline (PLT0) and concentration to achieve 50% of maximal effect (EC50) were set to correlate with R from -0.9 to 0.9. A physiologically-based pharmacokinetic (PBPK) model of midazolam [6], a cytochrome P450 (CYP) 3A substrate, was used to assess the Shapley values for the contribution of each covariate (sex, height, body mass index, microsomal protein per gram of liver [MPPGL], and abundances of CYP3A4 and 3A5) to area under the concentration-time curve (AUC) of midazolam in plasma. The abundances of CYP3A4 and CYP3A5 correlated (R = 0.52). The simulations were performed using R [7] for the linear CL model and the QSP model of lusutrombopag and MATLAB [8] for the PBPK model of midazolam. The Sharpley values were calculated using sobolshapley_knn in R library ”sensitivity” [9].
Results: For the linear CL model, the Shapley values for each of two covariate and a IIV were 0.39, 0.39, and 0.22, respectively. The sum of Shapley values was close to 1, which was consistent with the theoretical value. For the QSP model of lusutrombopag, the Sharpley values were 0.17 and 0.16 for each of covariates on PLT0 and EC50, respectively, for peak platelet counts without correlation (R = 0) between the covariates. The Shapley values of covariates on PLT0 and EC50 increased with increasing the magnitude of negative correlation between the covariates. These increases in the Shapley values were consistent with the model characteristics that PLT0 was positively related with the drug effect while EC50 was negatively related with the drug effect. For the PBPK model of midazolam, the Shapley values of MPPGL and abundances of CYP3A4 and CYP3A5 were > 0.2 for the AUC of midazolam. This result was consistent with that midazolam is primarily metabolized by CYP3A4 and CYP3A5.
Conclusions: The Shapley values would be applicable as indices for GSA to evaluate the contribution of correlated covariates to the outcomes in pharmacometric models.
References:
[1] Saltelli A et al. Comput Phys Commun. 2010. 18:259-70.
[2] Katsube T et al. J Pharmacokin Pharmacodyn. 2021. 48:851-60.
[3] Sobol IM. Math Model Comp Exp. 1993. 1:407–414.
[4] Idrissi M et al. Environ Model Software. 2021. 143: 105115.
[5] Shimizu R et al. CPT Pharmacomet Syst Pharmacol. 2021. 10:489–499.
[6] Melillo N et al. J Pharmacokin Pharmacodyn. 2021. 48:671-86.
[7] R Core Team.
[8] The MathWorks, inc.
[9] Iooss B et al. 2021. R package ‘sensitivity’ version 1.24.0. https://CRAN.R-project.org/package=sensitivity