Optimizing Chemotherapeutic Anti-cancer Treatment and the Tumor Microenvironment: An Analysis of Mathematical Models

  • Urszula LedzewiczEmail author
  • Heinz Schaettler
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 936)


We review results about the structure of administration of chemotherapeutic anti-cancer treatment that we have obtained from an analysis of minimally parameterized mathematical models using methods of optimal control. This is a branch of continuous-time optimization that studies the minimization of a performance criterion imposed on an underlying dynamical system subject to constraints. The scheduling of anti-cancer treatments has all the features of such a problem: treatments are administered in time and the interactions of the drugs with the tumor and its microenvironment determine the efficacy of therapy. At the same time, constraints on the toxicity of the treatments need to be taken into account. The models we consider are low-dimensional and do not include more refined details, but they capture the essence of the underlying biology and our results give robust and rather conclusive qualitative information about the administration of optimal treatment protocols that strongly correlate with approaches taken in medical practice. We describe the changes that arise in optimal administration schedules as the mathematical models are increasingly refined to progress from models that only consider the cancerous cells to models that include the major components of the tumor microenvironment, namely the tumor vasculature and tumor-immune system interactions.


Mathematical modeling Optimal control Optimization of cancer treatments Tumor microenvironment Anti-angiogenic therapy Tumor immune system interactions Metronomic chemotherapy 



This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA
  2. 2.Institute of MathematicsLodz University of TechnologyLodzPoland
  3. 3.Department of Electrical and Systems EngineeringWashington UniversitySt. LouisUSA

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