Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 322

Yet, these optimal solutions have bi-fold relevance for designing practical protocols
for these therapies: (1) Clearly, the optimal solutions define benchmark values
to which other protocols—simple, heuristically chosen and implementable—can
be compared and thus they determine a measure for how close to optimal a
given general protocol is. (2) Equally important, the structure of optimal controls
directly indicates simple, piecewise constant, and thus easily realizable protocols
that approximate the optimal solutions well, so-called suboptimal protocols. In the
papers [ 27 , 31 ], an extensive analysis of suboptimal protocols for the anti-angiogenic
monotherapy problem was undertaken, and it was shown that piecewise constant
suboptimal protocols with a very small number of switchings exist that are able to
replicate the optimal values within 1 %. Similar results are valid for the modification
of the underlying model by Ergun et al. given in [ 8 ]. Since optimal controls for the
chemotherapeutic agent in combination with anti-angiogenic treatments are bang-
bang, there is no need to approximate these and thus these results directly carry over
to these problems.
These excellent approximation properties remain valid if the model is made
more realistic by including pharmacokinetic equations for the anti-angiogenic
and chemotherapeutic agents [ 32 ]. The models considered so far identify their
dosages with their concentrations in the plasma. The controls u and v ,asthey
were used, actually represent the concentrations of these agents and linear terms
of the type qu model the pharmacodynamics of the drugs. If a standard linear

exponential growth/decay model is added for the concentrations of the agents,
e.g.,

− γ

u , then indeed there exist qualitative changes in the optimal
solutions that are due to the fact that the so-called intrinsic order of the singular
control changes from 1 to 2 [ 28 ]. These make the solutions even more complicated
from a mathematical point of view. However, the added complexity disappears
if only suboptimal protocols are considered. Then, as in the simplified models,
the structure of the theoretically optimal controls immediately suggests how to
choose excellent simple approximating protocols [ 32 ]. On the level of suboptimal
realizable protocols, the simplified modeling that ignores a linear pharmacokinetic
model for the therapeutic agents is fully justified.
Similar investigations are ongoing for the model including radiotherapy.
Radiation doses are commonly administered in daily fractionated doses (of short
durations) and this does not agree with the model considered above. Mathematically,
this leads to a more complex, hybrid optimization problem that generally is solved
with numerical methods. These methods and solutions, however, do not provide
much insight into the underlying principles. A continuous-time formulation,
as it was presented here, gives this information about the structure of optimal
controls (and why they look the way they do) and this is what makes it rather
straightforward (e.g., by averaging) to compute approximating fractionated doses.
But our investigations on this topic are still in their preliminary stages.

c = − α c +