Mean-field dynamics of tumor growth and control using low-impact   chemoprevention
  Cancer poses danger because of its unregulated growth, development of resistant subclones, and metastatic spread to vital organs. Although the major transitions in cancer development are increasingly well understood, we lack quantitative theory for how chemoprevention is predicted to affect survival. We employ master equations and probability generating functions, the latter well known in statistical physics, to derive the dynamics of tumor growth as a mean-field approximation. We also study numerically the associated stochastic birth-death process. Our findings predict exponential tumor growth when a cancer is in its early stages of development and hyper-exponential growth thereafter. Numerical simulations are in general agreement with our analytical approach. We evaluate how constant, low impact treatments affect both neoplastic growth and the frequency of chemoresistant clones. We show that therapeutic outcomes are highly predictable for treatments starting either sufficiently early or late in terms of initial tumor size and the initial number of chemoresistant cells, whereas stochastic dynamics dominate therapies starting at intermediate neoplasm sizes, with high outcome sensitivity both in terms of tumor control and the emergence of resistant subclones. The outcome of chemoprevention can be understood in terms of both minimal physiological impacts resulting in long-term control and either preventing or slowing the emergence of resistant subclones. We argue that our model and results can also be applied to the management of early, clinically detected cancers after tumor excision. 
Andrei R. Akhmetzhanov, Michael E. Hochberg
q-bio.PE