Biomedical Engineering Reference
In-Depth Information
11
Conclusion and Discussion
For various treatment strategies, we have considered optimal control problems to
minimize the tumor volume when the overall amounts of agents are limited a priori.
This may be simply because only a limited amount of the agents is available, like
in the case of anti-angiogenic inhibitors which still are very expensive and thus
are only used in limited quantities, or it may be because these treatments have severe
side effects that need to be limited and thus a priori decisions are made to limit
the total amount of drugs or radiation to be given, a standard medical approach.
Then the question how to schedule this agents in time arises naturally. Here we
have considered treatment strategies that combine anti-angiogenic therapies with
the classical approaches of chemo- and radiotherapy. Our main conclusions are
that singular controls (which can be computed analytically using Lie-derivative
based calculations) are at the center of optimal solutions for both the mono- and
combination therapy treatments. Although these controls are feedback functions,
and thus cannot be directly applied, they point the way to simple realizable
approximations that are excellent.
From a general biomedical point of view, our results suggest that in order to
optimally treat a highly dynamical disease such as a cancer, a highly dynamical
schedule of drug delivery may be needed. This remark, which could seem trivial
from a mathematical point of view, has some deep implications in medical oncology.
Indeed, the paradigm of dynamical drug scheduling requires a substantial rethinking
of the concept of clinical trial, which in its current form is largely linked to a static
vision of tumors.
Acknowledgements We would like to thank an anonymous referee for his careful reading of
our chapter and several suggestions that we incorporated into the final version. The research
of A. d'Onofrio has been done in the framework of the Integrated Project “p-medicine—
from data sharing and integration via VPH models to personalized medicine,” project identifier:
270089, which is partially funded by the European Commission under the 7th framework program.
The research of U. Ledzewicz and H. Schattler has been partially supported by the National Science
Foundation under collaborative research grant DMS 1008209/1008221.
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