A Tutorial on Tackling High Dimensionality in QSP
Johannes Tillil1,2, Prof. Dr. Wilhelm Huisinga2, Jane Knöchel2,3
1PharMetrX Graduate Research Training Program: Pharmacometrics & Computational Disease Modelling, 2Institute of Mathematics, University of Potsdam, 3InSilicoTrials Technologies S.p.A.
Objectives Over the last years, QSP models have been increasingly employed to support important drug development questions, such as predicting first-in-human doses from animal and in-vitro data and extrapolating adult to pediatric doses [1]. In part due to recent advancements in combining machine learning (ML) and QSP modeling methodology, the models are better able to integrate diverse multi-omics data sources [2, 3]. These advancements, however, lead to and do not solve the large size (high dimensionality) of the models, which hinders their wider adoption in the drug development pipeline by making them computationally expensive and difficult to interpret. Model reduction techniques can simplify large QSP models while retaining their essential dynamics and mechanistic nature, thus facilitating their application in drug development [4]. While many model reduction techniques exist to simplify large QSP models, there is little to no guidance on which methods best to apply in what context. Additional, no single model reduction technique can reduce every model to a size sufficient for the application, and thus, a combination of methods might be needed [5, 6]. However, selecting and combining appropriate model reduction techniques remains a challenging problem. This tutorial presents a workflow to choose effective model reduction techniques for QSP models using the novel index analysis approach [7, 8]. The goal is to provide a practical framework for reducing model complexity while preserving the mechanistic meaning of the reduced model. Methods Index analysis provides a time-resolved view of the dynamic importance of model states for a specific input-output relationship (i.e., the downstream effect of a drug on a biomarker) via the input-response (ir) indices [7]. Index analysis is built on a set of indices, each designed to address a specific aspect of model reduction. The input-response (ir) indices provide an a-priori assessment of the importance of model states. States with a high ir-index at some point in time play an important role in propagating the input-output dynamics. The state-classification indices provide a time-resolved measure of the impact of different model reduction techniques on the approximation quality of that input-output relationship [8]. The term model reduction refers to a set of methods reducing the complexity of QSP models by simplifying the ODEs or by outright removing species and parameters that are uninfluential for a given input-output relationship [4]. Here, we focus on the common approaches of dimensionality reduction [9, 10], timescale separation [11], and lumping [12]. Different states may be more amenable to different reduction approaches. The state-classification indices provide a state-by-state assessment of the suitability of different reduction techniques. Results / Description of presentation We introduce the concepts of index analysis and model reduction and guide the audience through the workflow to effectively reduce a large QSP model. We first show how to calculate and interpret the indices based on a small example model. Second, we illustrate the application of index-guided model reduction to a QSP model of blood coagulation with respect to the input-output relationship between brown snake venom and fibrinogen suppression [13]. The model consists of 62 states and 171 parameters. We calculate the ir-indices and identify 18 states with a major contribution to the output dynamics (maximum normalized ir-index larger than 5%). For the remaining 44 less important states, we further calculate the state-classification indices and, based on the scheme introduced in [8], identify reduction approaches that can be applied to 38 of these states with little added approximation error. The reduced system consists of 24 states modeled via an ODE and 77 parameters. It approximates the dynamics of fibrinogen suppression with a relative error of 0.27%. With additional optimization, the model can be further reduced to 8 states and 32 parameters with a relative error of 4.6%. Furthermore, we compare the index-guided reduction to an automated proper lumping approach [12]. Setting a relative error bound of 5% yields a lumped model with 17 states, 7 of which remain unlumped and 10 are lumped states. The lumped system approximates the full model dynamics with a relative error of 2.95%. However, due to the nature of lumping, no unimportant states are removed from the model. Thus, it still depends on all 171 parameters. Conclusions / Take home message Index analysis provides time-resolved measures of the dynamic importance and nature of model states. In this tutorial, we illustrate how the insights gained from the indices can guide model reduction. In contrast to the model reduction techniques proper lumping and balanced truncation, the state-by-state analysis and reduction via index analysis retains the mechanistic meaning of the states and parameters. Importantly for clinical applications, it is possible to extend the combined framework of index analysis and model reduction to models including inter-individual variability (IIV) on the states and parameters [14, 15]. By combining index analysis with model reduction, we aim to make QSP models more accessible and computationally efficient, facilitating their broader application in drug discovery and development.