1. 3 Lecture 3 - Part 1: Realizable Suboptimal Protocols for Tumor Anti-Angiogenesis Urszula Ledzewicz Department of Mathematics and Statistics Southern Illinois University, Edwardsville, USA May 11-15, 2009 Department of Automatic Control Silesian University of Technology, Gliwice

2. Heinz Schättler Dept. of Electrical and Systems Engineering Washington University, St. Louis, Missouri, USA Collaborators Helmut Maurer Rheinisch Westfälische Wilhelms-Universität Münster, Münster, Germany John Marriott Dept. of Mathematics and Statistics, Southern Illinois University, Edwardsville, USA

3. Research supported by NSF grants DMS 0205093, DMS 0305965 DMS 0205093, DMS 0305965 and collaborative research grants DMS 0405827/0405848 DMS 0707404/0707410 Research Support

4. References U. Ledzewicz and H. Schättler, Optimal and Suboptimal Protocols for Tumor Anti-Angiogenesis, J. of Theoretical Biology, 252, (2008), pp. 295-312, U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, The scheduling of angiogenic inhibitors minimizing tumor volume, J. of Medical Informatics and Technologies, 12, (2008), pp. 23-28 U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Med. And Biology, (2009), to appear

5. Synthesis of Optimal Controls for [Hahnfeldt et al.] 0asa0 as0 Full synthesis 0asa0 typical synthesis - as0 an optimal trajectory begin of therapy final point – minimum of p end of “therapy” p q u=a u=0

6. An Optimal Controlled Trajectory for [Hahnfeldt et al.] Initial condition: p 0 = 12,000 q 0 = 15,000 Optimal terminal value: 8533.4 time: 6.7221 Terminal value for a0-trajectory: 8707.4 time: 5.1934 full dose no dose partial dose - singular singular arc

7. full dose protocolfull dose protocol: give over time Suboptimal Protocols for [Hahnfeldt et al.] half dose protocol half dose protocol: give over time averaged optimal dose protocol averaged optimal dose protocol: give over time where is the time when inhibitors are exhausted along the optimal solution and e.g., for p 0 =12,000 and q 0 =15,000

8. Minimum tumor volumes Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support full dose averaged optimal dose optimal control p min q0q0 averaged optimal dose u q0q0

9. Minimum tumor volumes Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support full dose averaged optimal dose u optimal control averaged optimal dose half dose p min q0q0 q0q0

10. Minimum tumor volumes Values of the minimum tumor volume for a fixed initial tumor volume as functions of the initial endothelial support full dose averaged optimal dose u optimal control averaged optimal dose half dose p min q0q0 q0q0

11. Comparison of Trajectories averaged optimal dose full dose optimal control 0 0 singular arc half dose

12. Optimal Constant Dose Protocols

13. Minimal Tumor Size dosages from u=10 to u=100 blow-up of the value for dosages from u=46 to u=47

14. Optimal 2-Stage Protocols

15. Cross-section of the Value

17. Optimal 1- and 2-Stage Controls

18. Optimal Daily Dosages

19. An Optimal Controlled Trajectory Initial condition: p 0 = 12,000 q 0 = 15,000 Optimal terminal value: 8533.4 time: 6.7221 Terminal value for a0-trajectory: 8707.4 time: 5.1934 full dose no dose partial dose - singular singular arc

20. minimize For a free terminal time minimize over all measurable functions that satisfy subject to the dynamics [Ergun, Camphausen and Wein], Bull. Math. Biol., 2003

21. Synthesis for Model by [Ergun et al.] Full synthesis 0asa0, typical synthesis - as0

22. Example of optimal control and corresponding trajectory for Model by Ergun et al. Initial condition: p 0 = 8,000 q 0 = 10,000

23. Value of tumor for one dose protocols dosages from u=0 to u=15 blow-up of the value for dosages from u=8 to u=12 minimum at u=10.37, p(T)=2328.1

24. Cross-section of the Value

25. Optimal trajectory corresponding to 2-Stage Protocol

26. Optimal Daily Dosages

27. Conclusions The optimal control which has a singular piece is not medically realizable (feedback), but it provides benchmark values and can become the basis for the design of suboptimal, but realistic protocols. The averaged optimal dose protocol gives an excellent sub-optimal protocol, generally within 1% of the optimal value. The averaged optimal dose decreases with increasing initial tumor volume and is very robust with respect to the endothelial support for fixed initial tumor volume Optimal piecewise constant protocols can be constructed that essentially reproduce the performance of the optimal controls