Stochastic Norton–Simon–Massagué Tumor Growth Modeling: Controlled and Mixed-Effect Uncontrolled Analysis

Abstract

Tumorigenesis is a complex process that is heterogeneous and affected by numerous sources of variability. This article proposes a stochastic extension of a biologically grounded tumor growth model, referred to as the Norton-Simon-Massagué (NSM) model. First, we study the uncontrolled version of the model where the effect of the chemotherapeutic drug agent is absent. Conditions on the model's parameters are derived to guarantee the positivity of the solution of the proposed stochastic NSM model and hence its validity to describe the dynamics of tumor volume. The proof of positivity makes use of a Lyapunov-type method and the classical Feller's test for explosion. To calibrate the proposed model, we utilize a population mixed-effect modeling formulation and a maximum likelihood-based estimation algorithm. The identification algorithm is tested by fitting previously published tumor volume mice data. Second, we study the controlled version of the model, which includes the effect of chemotherapy treatment. Analysis of the influence of adding the control drug agent into the model and how sensitive it is to the stochastic parameters is performed both in open- and closed-loop viewpoints. The designed closed-loop control strategy that solves an optimal cancer therapy scheduling problem relies on the model predictive control (MPC) combined with extended Kalman filter approaches. The simulation results and concluding guiding principles are provided for both the open-and closed-loop control cases.