  Despite the remarkable advances in cancer diagnosis, treatment, and
management that have occurred over the past decade, malignant tumors remain a
major public health problem. Further progress in combating cancer may be
enabled by personalizing the delivery of therapies according to the predicted
response for each individual patient. The design of personalized therapies
requires patient-specific information integrated into an appropriate
mathematical model of tumor response. A fundamental barrier to realizing this
paradigm is the current lack of a rigorous, yet practical, mathematical theory
of tumor initiation, development, invasion, and response to therapy. In this
review, we begin by providing an overview of different approaches to modeling
tumor growth and treatment, including mechanistic as well as data-driven models
based on ``big data" and artificial intelligence. Next, we present illustrative
examples of mathematical models manifesting their utility and discussing the
limitations of stand-alone mechanistic and data-driven models. We further
discuss the potential of mechanistic models for not only predicting, but also
optimizing response to therapy on a patient-specific basis. We then discuss
current efforts and future possibilities to integrate mechanistic and
data-driven models. We conclude by proposing five fundamental challenges that
must be addressed to fully realize personalized care for cancer patients driven
by computational models.
