Abstract
We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.
Highlights
Tumor anti-angiogenesis is an indirect cancer treatment approach with the aim to limit the tumor’s growth, possibly even shrink the tumor, by depriving it of the vasculature it needs for a steady supply with nutrients
In [19] we have shown that the monotherapy solution becomes a basis upon which the optimal solution for a combination with chemotherapy can be built
For the monotherapy problem [M] this solution is based on a complete analytical solution given earlier [20], for the other versions we presented numerically optimal solutions whose structures are in agreement with existing partial theoretical results
Summary
Tumor anti-angiogenesis is an indirect cancer treatment approach with the aim to limit the tumor’s growth, possibly even shrink the tumor, by depriving it of the vasculature it needs for a steady supply with nutrients. The same structure of optimal combination therapy solutions is valid as it was verified numerically for a wide range of initial conditions in [8] for the model by Hahnfeldt et al [12]: optimal controls for the anti-angiogenic agent follow the optimal angio-monotherapy and at a specific time, chemotherapy becomes active and is given in one full dose session.
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