Mathematical Discovery of Natural Laws in Biomedical Sciences: A New Methodology

Abstract

As biomedical sciences discover new layers of complexity in the mechanisms of life and disease, mathematical models trying to catch up with these developments become mathematically intractable. As a result, in the grand scheme of things, mathematical models have so far played an auxiliary role in biomedical sciences. We propose a new methodology allowing mathematical modeling to give, in certain cases, definitive answers to systemic biomedical questions that elude empirical resolution. Our methodology is based on two ideas: (1) employing mathematical models that are firmly rooted in established biomedical knowledge yet so general that they can account for any, or at least many, biological mechanisms, both known and unknown; (2) finding model parameters whose likelihood-maximizing values are independent of observations (existence of such parameters implies that the model must not meet regularity conditions required for the consistency of maximum likelihood estimator). These universal parameter values may reveal general patterns (that we call natural laws) in biomedical processes. We illustrate this approach with the discovery of a clinically important natural law governing cancer metastasis. Specifically, we found that under minimal, and fairly realistic, mathematical and biomedical assumptions the likelihood-maximizing scenario of metastatic cancer progression in an individual patient is invariably the same: Complete suppression of metastatic growth before primary tumor resection followed by an abrupt growth acceleration after surgery. This scenario is widely observed in clinical practice and supported by a wealth of experimental studies on animals and clinical case reports published over the last 110 years. The above most likely scenario does not preclude other possibilities e.g. metastases may start aggressive growth before primary tumor resection or remain dormant after surgery.