21. A Mathematical Model for Growing Metastases on Oncologists’s Service

The dual classification of cancer as localized or metastatic disease is one of the key points in the elaboration of the best therapy for each patient. Nevertheless, many studies reveal that part of these localized diseases is already metastatic. The presence of undetectable or micro-metastases explains the necessity of adjuvant chemotherapies after resection of the primary tumor even for some T1N0M0 cancer. There is probably a continuum between these two stages.We expose here how a mathematical model of growing metastases could reflect this continuum of the disease and how such a model could help the oncologists in the choice of the treatment. This phenomenological model is based on a structured transport equations with nonlocal boundary condition describing the evolution of the density of metastasis. Thanks to this model, we forge a new numerical index, that we call Metastatic Index, able to reveal either the micro-metastatic state of the patient, or the visible metastatic one. Numerical illustrations show how this new index can be used.