Abstract
We describe optimal protocols for a class of mathematical models for tumor anti-angiogenesis for the problem of minimizing the tumor volume with an a priori given amount of vessel disruptive agents. The family of models is based on a biologically validated model by Hahnfeldt et al. and includes a modification by Ergun et al, but also provides two new variations that interpolate the dynamics for the vascular support between these existing models. The biological reasoning for the modifications of the models will be presented and we will show that despite quite different modeling assumptions, the qualitative structure of optimal controls is robust. For all the systems in the class of models considered here, an optimal singular arc is the defining element and all the syntheses of optimal controlled trajectories are qualitatively equivalent with quantitative differences easily computed.
Highlights
A growing tumor needs a steady supply of oxygen and nutrients for cell duplication
Assuming a Gompertzian tumor growth model, for a wide class of mathematical models for tumor anti-angiogenesis, including models that make significant differences in their modeling assumptions, there exists a unique path in (p, q)-space along which optimal tumor reductions are achieved. This path is given by a so-called singular arc whose precise analytical structure can be determined explicitly, but depends on the functions entering the dynamics of the models
These approximations are highly insensitive towards the initial value q0 of the carrying capacity and only show strong dependence on the initial tumor volume p0
Summary
A growing tumor needs a steady supply of oxygen and nutrients for cell duplication. Initially, during avascular growth, it is provided through the surrounding environment. Vascular endothelial growth factors (VEGF) are released that stimulate the formulation of new blood vessels and capillaries in order to supply the tumor with needed nutrients The models we consider are minimally parameterised and population based They incorporate the spatial aspects of the underlying consumption-diffusion processes that stimulate and inhibit angiogenesis into a nonspatial 2-compartment model with the primary tumor volume, p, and its carrying capacity, q, as variables. The latter defines the ideal tumor volume sustainable by the vascular network. Where ξ denotes a tumor growth parameter and F is a function of the scalar variable x
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