The role of the vasculature and the immune system in optimal protocols for cancer therapies

1. Heinz Schättler Dept. of Electr. and Systems Engr. Washington University St. Louis, USA The role of the vasculature and the immune system in optimal protocols for cancer therapies UT Austin – Portugal Workshop on Modeling and Simulation of Physiological Systems December 6-8, 2012 Lisbon, Portugal Urszula Ledzewicz Dept. of Mathematics and Statistics Southern Illinois University Edwardsville Edwardsville, USA

3. Heinz Schättler and Urszula Ledzewicz, Geometric Optimal Control – Theory, Methods, Examples Springer Verlag, July 2012 Urszula Ledzewicz and Heinz Schättler, Geometric Optimal Control Applied to Biomedical Models Springer Verlag, 2013 Mathematical Methods and Models in Biomedicine Urszula Ledzewicz, Heinz Schättler, Avner Friedman and Eugene Kashdan, Eds. Springer Verlag, November 2012 Forthcoming Books

5. Main Collaborators and Contacts Alberto d’Onofrio European Institute for Oncology, Milano, Italy Helmut Maurer Rheinisch Westfälische Wilhelms-Universität Münster, Münster, Germany Andrzej Swierniak Silesian University of Technology, Gliwice, Poland Avner Friedman MBI, The Ohio State University, Columbus, Oh

6. External Grant Support Research supported by collaborative research NSF grants DMS 0405827/0405848 DMS 0707404/0707410 DMS 1008209/1008221

7. Components of Optimal Control Problems min or max objective dynamics (model) response disturbance (unmodelled dynamics) control

8. Outline – An Optimal Control Approach to … • model for drug resistance under chemotherapy • a model for antiangiogenic treatment • a model for combination of antiangiogenic treatment with chemotherapy • a model for tumor-immune interactions under • chemotherapy and immune boost • conclusion and future work: model for tumor microenvironment and • metronomic chemotherapy

9. Optimal Drug Treatment Protocols Main Questions QUESTION 1:HOW MUCH? (dosage) QUESTION 2:HOW OFTEN? (timing) QUESTION 3:IN WHAT ORDER? (sequencing)

10. Heterogeneity and Tumor Microenvironment

11. Tumors are same size but contain different composition of chemo-resistant and –sensitive cells. Chemo-sensitive tumor cell Tumor stimulating myeloid cell Fibroblast Chemo-resistant tumor cell Endothelia Surveillance T-cell

12. aS(t),cR(t) outflow of sensitive/resistance cells u – cytotoxic drug dose rate, 0≤u≤1 Model for Drug Resistance under Chemotherapy Mutates SR (1-u)aS p(1-u)aS aS division uaS - killed (2-p)(1-u)aS remains sensitive one-gene forward gene amplification hypothesis, Harnevo and Agur

13. cR(t) – outflow of resistant cells dynamics Mutates back RS division rcR cR remains resistant (2-r)cR

14. Mathematical Model: Objective minimize the number of cancer cells left without causing too much harm to the healthy cells: let N=(S,R)T Weighted average of number of cancer cells at end of therapy Toxicity of the drug (side effects on healthy cells) Weighted average of cancer cells during therapy

15. From Maximum Principle: Candidates for Optimal Protocols bang-bang controls singularcontrols umax T T treatment protocols of maximum dose therapy periods with rest periods in between continuous infusions of varying lower doses MTD BOD

16. If pS>(2-p)R, then bang-bang controls (MTD) are optimal If pS<(2-p)R, then singular controls (lower doses) become optimal Passing a certain threshold, time varying lower doses are recommended Results [LSch, DCDS, 2006] From the Legendre-Clebsch condition

17. “Markov Chain” Models

18. Tumor Anti-angiogenesis

19. Tumor Anti-Angiogenesis avasculargrowth angiogenesis metastasis http://www.gene.com/gene/research/focusareas/oncology/angiogenesis.html

20. Tumor Anti-angiogenesis suppress tumor growth by preventing the recruitment of new blood vessels that supply the tumor with nutrients (indirect approach) done by inhibiting the growth of the endothelial cells that form the lining of the new blood vessels therapy “resistant to resistance” Judah Folkman, 1972 • anti-angiogenic agents are biological drugs (enzyme inhibitors like endostatin) – very expensive and with side effects

21. - tumor growth parameter - endogenous stimulation (birth) - endogenous inhibition (death) - anti-angiogenic inhibition parameter - natural death Model [Hahnfeldt,Panigrahy,Folkman,Hlatky],Cancer Research, 1999 p – tumor volume q – carrying capacity u – anti-angiogenic dose rate p,q – volumes in mm3 Lewis lung carcinoma implanted in mice

22. Optimal Control Problem For a free terminal time minimize over all functions that satisfy subject to the dynamics

23. begin of therapy an optimal trajectory end of “therapy” final point – minimum of p Synthesis of Optimal Controls [LSch, SICON, 2007] u=a u=0 p q typical synthesis: umax→s→0

24. An Optimal Controlled Trajectory for [Hahnfeldt et al.] u maximum dose rate q0 lower dose rate - singular averaged optimal dose no dose robust with respect to q0 Initial condition: p0 = 12,000 q0 = 15,000, umax=75

25. Anti-Angiogenic Treatment with Chemotherapy

26. A Model for a Combination Therapy [d’OLMSch, Mathematical Biosciences, 2009] with d’Onofrio and H. Maurer Minimize subject to angiogenic inhibitors cytotoxic agent or other killing term

27. Questions: Dosage and Sequencing Chemotherapy needs the vasculature to deliver the drugs Anti-angiogenic therapy destroys this vasculature In what dosages? Which should come first ?

28. Optimal Protocols optimal angiogenic monotherapy

29. Controls and Trajectory [for dynamics from Hahnfeldt et al.]

30. Medical Connection Rakesh Jain, Steele Lab, Harvard Medical School, “there exists a therapeutic window when changes in the tumor in response to anti-angiogenic treatment may allow chemotherapy to be particularly effective”

31. Tumor Immune Interactions

32. Mathematical Model for Tumor-Immune Dynamics STATE: - primary tumor volume - immunocompetentcell-density (related to various types of T-cells) Stepanova,Biophysics,1980 Kuznetsov, Makalkin, Taylor and Perelson, Bull. Math. Biology, 1994 de Vladar and Gonzalez, J. Theo. Biology,2004, d’Onofrio,Physica D, 2005

33. Dynamical Model - tumor growth parameter - rate at which cancer cells are eliminated through the activity of T-cells - constant rate of influx of T-cells generated by primary organs - natural death of T-cells -calibrate the interactions between immune system and tumor - threshold beyond which immune reaction becomes suppressed by the tumor

34. Phaseportrait for Gompertz Growth • we want to move the state of the system into the region of attraction of the benign equilibrium minimize

35. Formulation of the Objective • controls • u(t) – dosage of a cytotoxic agent, chemotherapy • v(t)–dosage of an interleukin type drug, immune boost • side effects of the treatment need to be taken into account • the therapy horizon T needs to be limited minimize ( (b,a)T is the stable eigenvector of the saddle and c, d and s are positive constants)

36. For a free terminal time Tminimize over all functionsand subject to the dynamics Optimal Control Problem [LNSch,J Math Biol, 2011] Chemotherapy – log-kill hypothesis Immune boost

37. Chemotherapy with Immune Boost • “cost” of immune boost is high and effects are low compared to chemo • trajectory follows the optimal chemo monotherapy and provides final boosts to the immune system and chemo at the end 1s01 010 * * “free pass” - chemo - immune boost * * *

38. Summary and Future Direction: Combining Models Which parts of the tumor microenvironment need to be taken into account? • cancer cells ( heterogeneous, varying sensitivities, …) • vasculature (angiogenic signaling) • tumor immune interactions • healthy cells Wholistic Approach ? • Minimally parameterized metamodel • Multi-input multi-target approaches • Single-input metronomic dosing of chemotherapy

40. Future Direction: Metronomic Chemotherapy How is it administered? • treatment at much lower doses ( between 10% and 80% of the MTD) • over prolonged periods Advantages • lower, but continuous cytotoxic effects on tumor cells • lower toxicity (in many cases, none) • lower drug resistance and even resensitization effect • antiangiogenic effects • boost to the immune system Metronomics Global Health Initiative (MGHI) http://metronomics.newethicalbusiness.org/