Simulating tumor growth using mathematical and agent-based modeling

In this paper, two different approaches for tumor growth modeling are presented and implemented. In the first part of the paper, a macroscopic approach using a PDE model, where the tumor is viewed as a cell mass, is implemented using the level-set method to track the tumor moving boundary in one hand by using Darcy’s law to compute the normal velocity of the free boundary and on the other hand using the shape optimization to draw the normal velocity. In the second part of the paper, a microscopic approach, which focuses on the cellular scale, is presented. A hybrid model using agent-based modeling for the cell behavior and a PDE for the description of the tumor environment is presented. A sensitivity analysis is performed on the hybrid model for a better understanding of its impact on the tumor growth. Numerical experiments are provided for the proposed approaches.

State and Parameter Estimation of a Mathematical Carcinoma Model under Chemotherapeutic Treatment

One challenging aspect of therapy optimization and application of control algorithms in the field of tumor growth modeling is the limited number of measurable physiological signals—state variables—and the knowledge of model parameters. A possible solution to provide such information is the application of observer or state estimator. One of the most widely applied estimators for nonlinear problems is the extended Kalman filter (EKF). In this study, a moving horizon estimation (MHE)-based observer is developed and compared to an optimized EKF. The observers utilize a third-order tumor growth model. The performance of the observers is tested on measurements gathered from a laboratory mice trial using chemotherapeutic drug. The proposed MHE is designed to be suitable for closed-loop applications and yields simultaneous state and parameter estimation.