Three facets of mathematical cancer biology research

Cancer, as the uncontrollable cell growth, is related to many branches of biology. In this review, we will discuss three mathematical approaches for studying cancer biology: population dynamics, gene regulation, and developmental biology. If we understand all biochemical mechanisms of cancer cells, we can directly calculate how the cancer cell population behaves. Inversely, just from the cell count data, we can use population dynamics to infer the mechanisms. Cancer cells emerge from certain genetic mutations, which affect the expression of other genes through gene regulation. Therefore, knowledge of gene regulation can help with cancer prevention and treatment. Developmental biology studies acquisition and maintenance of normal cellular function, which is inspiring to cancer biology in the opposite direction. Besides, cancer cells implanted into an embryo can differentiate into normal tissues, which provides a possible approach of curing cancer. This review illustrates the role of mathematics in these three fields: what mathematical models are used, what data analysis tools are applied, and what mathematical theorems need to be proved. We hope that applied mathematicians and even pure mathematicians can find meaningful mathematical problems related to cancer biology.