A Two-Stages Global Sensitivity Analysis in presence of correlated inputs: Application on a Tumor-in-host-growth Inhibition model based on the Dynamic Energy Budget theory
Alessandro De Carlo (1), Elena Maria Tosca (1), Nicola Melillo (1,2), Paolo Magni (1)
(1) Laboratory of Bioinformatics, Mathematical Modelling and Synthetic Biology (BMS lab), Department of Electrical, Computer and Biomedical Engineering, University of Pavia, Pavia, Italy; (2) Systems Forecasting UK Ltd, Lancaster, UK
Objectives: Global sensitivity analysis (GSA) aims to evaluate the impact of variability and/or uncertainty of the model input parameters on given model outputs. GSA is a crucial step for Pharmacometrics models as their parameters can be affected by high uncertainty estimation due to the sparsity of data, especially in the preclinical settings. Unlike the commonly used one-at-a-time approaches, GSA provides an efficient exploration of the whole input space by sampling from the parameter joint probability density function (pdf) [1]. GSA methods commonly rely on the assumption of independence between model parameters [1]. However, ignoring known correlations between parameters may alter model predictions and then GSA results [2]. Variance-based GSA extensions were proposed to deal with dependent inputs [3,4,5], but how to appropriately interpret such results is still debated [6].
Here, we present a two-stages GSA workflow based on the moment independent δ index, that is well-defined in presence of correlated inputs [7], to analyse a tumor-in-host-growth inhibition model based on the Dynamic Energy Budget theory (DEB-TGI model) [8,9,10]. The aim was to evaluate the impact of the model parameter estimate uncertainty (including the correlation) on some metrics of interest: the tumor volume Doubling Time (DT) and the minimum threshold concentration necessary for tumor eradication, Ct.
Methods: Unlike the variance-based approach, the δ sensitivity index exploits the entire distribution of the model output rather than relying on a single moment. Thus, given a scalar model output, Y, the impact of a single parameter, Xi, on the entire output pdf, fY(Y), is computed by considering the expected value of the difference between fY(Y) and fY(Y|Xi). δ always ranges in [0,1].
To deal with correlated inputs, GSA was performed following a two-stages approach.
1) GSA under the hypothesis of uncorrelation between parameters was performed to establish which inputs have a direct (causal) effect on fY(Y), to avoid a misinterpretation of step 2 δ>0 related to those parameters that have only an indirect effect on the metric of interest (i.e. they do not appear in the metric equation)
2) GSA considering statistical dependencies was done to consider the ‘correct’ joint inputs pdf and to investigate the ‘indirect’ contribution of each Xi on fY(Y) due to the correlations with other parameters.
A parameter with δ>0 in the step 1 exerts a direct (causal) effect on fY(Y). Conversely, a parameter with δ=0 in step 1 and δ>0 in step 2 only impacts through its statistical dependencies (indirect effect). Parameters with δ>0 in both step 1 and 2 have both direct and indirect effects.
A multivariate Log-Normal distribution, based on previous model fittings on xenograft studies [8,9,11,12], was used to describe the joint pdf of the model inputs. Scenarios with accurate (small CV, CV1) and less accurate (large CV, CV2) estimates were considered.
The R package mvlognCorrEst (developed by University of Pavia BMS Lab) and its Matlab implementation were used to generate the log-normally distributed parameter sets, to deal with unknown correlations and non-positive definite covariance matrices [13].
Results: The flexibility of δ index with uncorrelated/correlated inputs allowed to successfully perform the two GSA steps on the DEB-TGI model in both CV1 and CV2 scenarios. Thus, parameters were ranked according to their impact on the output distribution, discerning the uncertainties to be necessarily reduced.
Considering known correlations between model parameters is pivotal to obtain reliable results with GSA. This workflow showed that their presence can mitigate the causal effects of some parameters: μu and gu had the highest direct effect on DT (δ≈0.30 in step 1 for CV2), however, step 2 showed that their contributes were minimal (δ<0.10 with CV2), because of a sort of compensation. Conversely, some parameters can become important due to the correlations. It is the case, for example, of the IC50 parameter when the Ct metric was considered (step 2 δ≈0.20 in both CV1 and CV2) as it is correlated to k2 (δ>0.5 in both steps and uncertainty cases).
Conclusions: This work introduced a novel two-stages approach exploiting the δ sensitivity index to perform GSA in presence of correlations. The proposed workflow was used to characterize the impact of the estimation uncertainty on output metrics of the DEB-TGI model.
References:
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