In this paper we introduce a new mixed-integer linear programming (MIP) model that explicitly integrates the spread of cancer cells into a spatio-temporal reaction-diffusion (RD) model of cancer growth, while taking into account treatment effects. This linear but non-convex model appears to be the first of its kind by determining the optimal sequence of the typically prescribed cancer treatment methods—surgery (S), chemotherapy (C), and radiotherapy (R)—while minimizing the newly generated tumor cells for early-stage breast cancer in a unique three-dimensional (3D) spatio-temporal system. The quadratically-constrained cancer growth dynamics and treatment impact formulations are linearized by using linearization as well as approximation techniques. Under the supervision of medical oncologists and utilizing several literature resources for the parameter values, the effectiveness of treatment combinations for breast cancer specified with different sequences (i.e., SRC, SCR, CR, RC) are compared by tracking the number of cancer cells at the end of each treatment modality. Our results provide the optimal dosages for chemotherapy and radiation treatments, while minimizing the growth of new cancer cells.
- Cancer spread
- Gompertz growth
- Linear mathematical model
- Mixed-integer linear programming (MIP)
- Spatio-temporal treatment optimization
- Three-dimensional (3D) cancer growth