Geometric methods in optimal control theory applied to problems in biomedicine


Joint work with; Heinz Schaettler, Department of Electrical and Systems Engineering, Washington University, St. Louis, USA.

Optimal control theory initiated as a discipline in the fifties with space exploration and military applications. Later it was introduced into many other fields including various branches of engineering as well as economics and, more recently, biomedicine. Geometric methods of optimal control have proven to be especially effective in the analysis of models for the treatment of diseases also including cancer.

In these lectures, we shall present some examples where a nontrivial mathematical analysis based on the tools of geometric optimal control has led to interesting and relevant biomedical conclusions concerning cancer treatments. The mathematical models for cancer under investigations range from models taking into account cell cycle specificity and tumor heterogeneity to models for tumor-immune interactions and tumor angiogenesis. These models will be formulated as optimal control problems with the control variables representing the dosage of various treatments ranging from classical methods like chemotherapy and radiotherapy to more novel approaches like antiangiogenic treatment and immunotherapy. The objective in these problems is to minimize the tumor volume or, more generally, the cancer cell count while at the same time minimizing the side effects of the drugs. All these treatments, modeled as monotherapy, gives rise to interesting medical questions about timing and dosage of the drug, whereas, when applied in combination to obtain the so desired synergy, generate additional challenges concerning the sequencing.

In these lectures, the tools of geometric optimal control including Lie bracket computations and high-order conditions for optimality the Legendre-Clebsch condition will be introduced and shown “in action” in the analysis of optimal controls for these biomedical problems. Deep mathematical results like the construction of a full synthesis of optimal solutions (the ultimate, but highly nontrivial goal in solving optimal control problems) will be presented for some of these models, which, in addition to their intrinsic theoretical value provide insights and give benchmarks for determining realizable drug dosage protocols. On the other hand, for some of the models, when more than one drug is involved, only partial results are available and some challenges concerning the analysis of these multi-input optimal control problems will be outlined. Throughout the lectures, the relations between the obtained mathematical results and medical data concerning the results of clinical trials will be made. In particular, a connection between theoretically obtained singular controls and the clinical benefits of metronomic chemotherapy will be outlined. Both mathematical results and medical evidence indicate that for some anticancer drugs “more is not necessarily better,” but instead a properly calibrated lower dosage can provide a more beneficial overall effect. Along these lines the classical term of MTD (maximum tolerable dose) is replaced with the OBD (optimal biological dose). The lectures will be concluded with outlining some open problems and challenges, like finding the proper form of the OBD and how optimal control and, more generally, mathematics can aid with the solutions of these important biomedical problems.


  1. H. Schaettler, U. Ledzewicz Geometric Optimal Control, Theory Methods and Examples, Springer-Verlag, 2012.
  2. U. Ledzewicz, H. Schaettler Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (3), (2007), pp. 1052-1079.
  3. A, d’Onofrio, U. Ledzewicz H. Schaettler, and H. Maurer, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222, (2009), pp. 13-26, doi:10.1016/j.mbs.2009.08.004
  4. U. Ledzewicz, H.Schaettler, Multi-Input optimal control problems for combined tumor antiangiogenic and radiotherapy treatments, Journal of Optimization Theory and Applications, (2012), Vol.153, pp.195-224, 10.1007/s10957-011-9954.
  5. U. Ledzewicz, M. Naghneian, H. Schaettler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, Journal of Mathematical Biology, (2011), pp. 557- 577, 10.1007/s00285-011-0424-6



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