Can theorems help treat cancer?

Can mathematicians who prove theorems be useful to physicians who treat patients with cancer? Can mathematics have an impact on cancer biology? Is mathematics more than just a tool in the hands of [a few open-minded] physicians willing to understand and treat cancer from an innovative point of view? ...and is mathematical biology more than just a toy in the hands of mathematicians? I imagine that most readers of this journal would join me in answering “yes” to these questions. But let me be more explicit in my questioning: Is there a place for a very abstract theoretical activity, as is proving theorems, in a team of physicians daily committed to treating human diseases (in particular cancer)? Can these physicians see the contribution of mathematicians to their difficult task as more than helping them to “think differently” (as some of them state it), but further as actually taking part in a therapeutic decision? As regards mathematical biology, when it is applied to medical questions, do we mathematicians feel enough committed to go beyond a few publications in the numerous journals dedicated to applications of maths to biology, and to collaborate with medical teams in a long term effort to improve in common medical treatments? Building on recent articles [3, 4] in which I have outlined technical ways by which one can achieve a working collaboration between mathematicians and physicians toward such mutual usefulness, I will explain briefly below why the answers to these questions should be “yes”. At present, biologists and physicians do not look to mathematicians for theorems. An obvious reason for this is the completely different status of knowledge in mathematics and biology and medicine. Unlike mathematics, biology is a field where little is known for certain, for the very nature of the phenomena under study is often inaccessible, and deductive reasoning about uncertain facts of questionable value. Moreover, theorems as such do not exist in biology, although it is not impossible that theorems proven in a mathematical framework were inspired by problems posed initially as biological questions. The clinical relevance of such theoretical results is, however, seldom clear. More precisely, if our aim is to identify optimal schedules for the treatment of cancers by combining different drugs and taking account of cell cycle phase times, it is possible theoretically to determine optimal schedules for the delivery of drugs (with at least a theorem of existence for such an optimal schedule, and numerically determined schedules, as proposed for instance in [1]). In practice it is not easy to establish whether the assumptions that must hold for the theorems and optimal solutions to be valid are fulfilled for they rely on the measurement of