Predicting success or failure of immunotherapy for cancer: insights from a clinically applicable mathematical model

The objective of this study was to create a clinically applicable mathematical model of immunotherapy for cancer and use it to explore differences between successful and unsuccessful treatment scenarios. The simplified predator-prey model includes four lumped parameters: tumor growth rate, g; immune cell killing efficiency, k; immune cell signaling factor, λ; and immune cell half-life decay, μ. The predator-prey equations as functions of time, t, for normalized tumor cell numbers, y, (the prey) and immunocyte numbers, ×, (the predators) are: dy/dt = gy - kx and dx/dt = λxy - μx. A parameter estimation procedure that capitalizes on available clinical data and the timing of clinically observable phenomena gives mid-range benchmarks for parameters representing the unstable equilibrium case in which the tumor neither grows nor shrinks. Departure from this equilibrium results in oscillations in tumor cell numbers and in many cases complete elimination of the tumor. Several paradoxical phenomena are predicted, including increasing tumor cell numbers prior to a population crash, apparent cure with late recurrence, one or more cycles of tumor growth prior to eventual tumor elimination, and improved tumor killing with initially weaker immune parameters or smaller initial populations of immune cells. The model and the parameter estimation techniques are easily adapted to various human cancers that evoke an immune response. They may help clinicians understand and predict certain strange and unexpected effects in the world of tumor immunity and lead to the design of clinical trials to test improved treatment protocols for patients.