On the Dynamics of Tumor-Immune System Interactions and Combined Chemo- and Immunotherapy

  • Alberto d’Onofrio
  • Urszula Ledzewicz
  • Heinz Schättler
Part of the SIMAI Springer Series book series (SEMA SIMAI)


Tumor-immune system interplay is extremely complex, and, as such, it represents a big challenge for mathematical oncology. Here we investigate a simple general family of models for this important interplay by considering both the delivery of a cytotoxic chemotherapy and of immunotherapy. Then methods of geometrical optimal control are applied to a special case (the Stepanova model) in order to infer (under suitable constraints) the best combination of drugs scheduling to transfer — through therapy —the system from an initial condition in the malignant region of the state space into a benign region. Our findings suggest that chemotherapy is always needed first to reduce a large tumor volume before the immune system can become effective.


Optimal Control Problem Stable Manifold Singular Control Control Trajectory Immune Boost 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This material is based upon research supported by the National Science Foundation under collaborative research grants DMS 1008209 and 1008221 (U.L. and H.S.), and by the EU project “p-Medicine: Personalized Medicine” (FP7-ICT-2009.5.3-270089) (A. d’O.). We also would like to thank our students Mohamad Naghnaeian and Mozhdeh Faraji for carrying out the numerical computations and making the figures used in the paper.


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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Alberto d’Onofrio
    • 1
  • Urszula Ledzewicz
    • 2
  • Heinz Schättler
    • 3
  1. 1.Department of Experimental OncologyEuropean Institute of OncologyMilanItaly
  2. 2.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA
  3. 3.Department of Electrical and Systems EngineeringWashington UniversitySt. LouisUSA

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