An Alternative Exact Solution of the Two-Stage Clonal Growth Model of Cancer

The approximate solution of the two-stage clonal expansion model of cancer may substantially deviate from the exact solution, and may therefore lead to erroneous conclusions in particular applications. However, for time-varying parameters the exact solution (method of characteristics) is not easy to implement, hampering the accessibility of the model to nonmathematicians. Based on intuitive reasoning, Clewell et al. (1995) proposed an improved approximate solution that is easy to implement whatever time-varying behavior the parameters may have. Here we provide the mathematical foundation for the approximation suggested by Clewell et al. (1995) and show that, after a slight modification, it is in fact an exact solution for the case of time-constant parameters. We were not able to prove that it is an exact solution for time-varying parameters as well. However, several computer simulations showed that the numerical results do not differ from the exact solution as proposed by Moolgavkar and Luebeck (1990). The advantage of this alternative solution is that the hazard rate of the first malignant cell can be evaluated by numerically integrating a single differential equation.