Robust domain of attraction estimation for a tumor growth model

Abstract

This paper deals with the estimation of regions of attraction (RoAs) for a cancer dynamical model. The estimation of this type of sets is very interesting in the field of control for cancer dynamics, since it provides the set of possible initial health indicators, for which there exists a treatment protocol allowing to heal the patient. Therefore, we propose a methodology to estimate the region of attraction of a nonlinear dynamical system describing the interaction between a tumor, the immune system and combined therapies of cancer. Then, we use the characterization of the RoA for a given model parameters vector, in order to derive an estimation of an outer approximation of the robust RoA under parametric uncertainties.

Probabilistically certified region of attraction of a tumor growth model with combined chemo‐ and immunotherapy

“…We can see that, as the number of cycles increases, the certified RoA gets closer to the robust controlled one denoted $$$ {\Omega}_R $$$. Even if the certified RoA remains smaller, it is important to recall that both the nominal and the robust RoAs in Reference 28 have been estimated using continuous infusions of drugs, and do not consider the PK/PD effects. Note also that changing the type of the sets can change the size of the certified RoA.…”

“…In Reference 28, we proposed a methodology to characterize the controlled region of attraction of model (1) with bang‐bang controls (without pharmacokinetics). Then, we used this approach to derive and estimate the robust region of attraction.…”

“…We assume that the set $$$ {\Omega}_0 $$$ to be certified has the same geometry as the estimated nominal uncontrolled region of attraction $$$ {\hat{\Omega}}_0^{p_{nom}} $$$ (derived in Reference 28) that we shrink until (22) is not satisfied given the confidence probability $$$ 1-\delta $$$. There is clearly no guarantee that the set $$$ {\Omega}_0 $$$ that we obtain is the biggest possible certified set, however, in this case, proving the existence of a set $$$ {\Omega}_0 $$$ satisfying (22) is enough, since $$$ {\Omega}_0 $$$ is only used as a target set for the Algorithm 1 allowing therefore to compute the sequence of certified sets.…”

“…Figure 5 shows the 3 certified cycles for $$$ T=5 $$$ days obtained using Algorithm 1, nominal and robust RoAs that have been estimated using the method presented in Reference 28, where bang‐bang control strategies were considered. We can see that, as the number of cycles increases, the certified RoA gets closer to the robust controlled one denoted $$$ {\Omega}_R $$$.…”

Control of tumor growth modeled by system of PDE, numerical analysis of optimality conditions

“…Ordinary differential equation (ODE) models are studied in previous papers. [9][10][11][12][13] The mathematical model in reference 12 uses 12 biological variables (and thus system of 12 ODEs): three cytokines together with nine cell types without spatial component. In reference, 13 a system with delay of three nonlinear ODEs is studied, which include as state variables the density of tumor populations, the density of cytotoxic T-lymphocytes, and the density of T-helper cells in a single tumor-site compartment.…”