The malignant glioblastoma is a devastating disease of central nerve system of the brain often lethal in few months time. Combining chemotherapy, radiation therapy, genetic therapy may extend significantly life expectation of patients suffering from this disease, provided that the cancer can be destroyed without harming excessively the patient's brain. Therefore, the challenge for designing effective chemotherapy lies in distributing the chemotherapeutic toxin so that it kills cancer cell while at the same time not causing excessive damage to healthy brain tissue. This problem requires the optimal choice of catheter injection position and design parameters like number of outlet ports and port configuration to best discharge the toxins as well as and design parameters like insertion pressure, flow and dilution rate. It is still the state-of-art treatment of this process to rely on surgeon institution in term of catheter's locations for design parameters and flow rates.

In this presentation, we propose a rigorous mathematical optimization problem on a distributed two-dimensional or three dimensional brain to determine the optimal catheter location, injection pressure and other parameters such as number of hole or outlets. We will use as the objective function the maximal coverage cancerous tissue as well as adjacent white matter tracts where reoccurrence is likely, while at the same time minimizing the concentration of toxins to cortical areas responsible for critical brain function such as smell and sight. We formulate a rigorous distributed optimization problem with partial differential equations constraints representing momentum, specie transport phenomena. The finite volume discretization method using unstructured grid is used to solve the resulting non-linear mathematical program with stochastic mathematical programming techniques.

The methodology of the design of catheter locations integrates interdisciplinary expertise from systems biology and engineering optimization. The approach will allow physicians and scientists to design and optimize chemotherapy in a systematic fashion, thus reducing the need for trial-and-error animal experiments.