A lumped ODE model for metastatic cancer treatment

In [J Theor Biol 203(2) (2000):177-186], a size-structured PDE population model has been proposed for characterizing the growth (in number of cells) of metastatic tumors.Recently, such simple transport PDE models were carefully validated through laboratory experiments with tumor-bearing mice.Many efforts have been devoted to developing more efficient numerical algorithms for solving such interesting PDE models, but its computational cost remains high in the framework of optimal control for seeking optimized treatment strategy due to a huge spatial domain.In particular, the computed cell-level metastatic density from PDE model is not of direct biological interest, instead, its weighted integration (e.g., the total number of tumor cells) is of more clinical importance in practice.In this work, we reformulate such a transport PDE model into a lumped ODE model that involves a Volterra integral equation of convolution type which is independent of the control variable.Such a reformulation can significantly reduce the computational cost by only computing the lumped (or aggregated) quantity without spatial dependence.Moreover, for better practicality, we incorporate the nonlinear Pharmacokineticv and Pharmacodynamic effects of treatment into our lumped ODE model.Based on the open-source nonlinear optimal control software ICLOCS2, numerical examples are presented to illustrate some interesting findings on optimal treatment that may inspire clinical practice.