Challenges in Synthesis of Optimal Controls For Problems Arising in Biomedicine

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1 Challenges in Synthesis of Optimal Controls For Problems Arising in Biomedicine ITN SADCO meeting January 21-25, 2013, Funchal, Portugal Urszula Ledzewicz Dept. of Mathematics and Statistics Southern Illinois University Edwardsville Edwardsville, USA Heinz Schättler Dept. of Electr. and Systems Engr. Washington University St. Louis, USA

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3 Forthcoming Books 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

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5 Alberto d Onofrio Main Collaborators and Contacts 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 Collaborators cont.: Eddy Pasquier Children Cancer Resarch Center of Australia

7 External Grant Support Research supported by collaborative research NSF grants DMS / DMS / DMS /

8 Outline Synthesis for cell-cycle specific models for multi-drug chemotherapy - multi-input optimal control problem bang-bang solutions - local synthesis model for antiangiogenic treatment as monotherapy - bang-singular-bang full global synthesis - incorporating pharmacokinetics chattering controls (Fuller phenomenon) Open problems and challenges in synthesis for models for combinations of antiangiogenic treatment - with chemotherapy - with radiotherapy tumor-immune interactions Conclusion and future work model for 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 Model for Multi-Drug Chemotherapy

11 A 3-Compartment Model with Cytotoxic and Recruiting Agents (Leukemia) COMPARTMENTS : clusters of phases of cell-cycle A. Swierniak, 1993, Cell Proliferation STATES : N = ( N 0, N 1, N 2 ) represent the numbers of cancer cells in the corresponding compartments CONTROLS: u, w represent the drug dosage of: p 1-p G 0 G 1 S+G 2 /M u w - killing agent - recruiting agent N 0 N 1 N 2 u = 0 w = 0 u = 1 w = w max no dose full dose w u

12 Mathematical Model: Dynamics Dynamics: describes the rate of change in the numbers N of cancer cells at various stages of the cell cycle under the therapy mathematical models based on underlying biology

13 Mathematical Model: Objective minimize the number of cancer cells left without causing too much harm to the healthy cells Weighted average of number of cancer cells at end of therapy Weighted average of cancer cells during therapy Toxicity of the drug (side effects on healthy cells)

14 Candidates for Optimal Protocols bang-bang controls singular controls u max T T treatment protocols of maximum dose therapy periods with rest periods in between MTD continuous infusions of varying lower doses OBD

15 Singular Controls is singular on an open interval on all time derivatives must vanish as well allows to compute the singular control order : the control appears for the first time in the derivative Legendre-Clebsch condition (minimize)

16 Singular Controls are NOT Optimal Locally maximizing NOT optimal

17 Example of an Optimal Control [ LSch, CDC, 2012] 1 6 control u u control w 5 w Φ 1 0 Φ time time 1 cancer cells N(t) N N N time Optimal protocols for killing and recruiting agents and their effect on the evolution of cancer cells

18 N transversal fold S 2 p p transversal crossing t S 1 T

19 Bang-Bang Flows N t N t

20 N T t

21 Tumor Antiangiogenesis avascular growth angiogenesis metastasis

22 Tumor Anti-angiogenesis Judah Folkman, 1972 suppress tumor growth by preventing the recruitment of new blood vessels that supply the tumor with nutrients (indirect approach) resistant to resistance anti-angiogenic agents are biological drugs (enzyme inhibitors like endostatin) very expensive and with side effects

23 Model [Hahnfeldt,Panigrahy,Folkman, Hlatky],Cancer Research, 1999 p tumor volume q carrying capacity u anti-angiogenic dose rate p,q volumes in mm 3 - tumor growth parameter - endogenous stimulation (birth) - endogenous inhibition (death) Lewis lung carcinoma implanted in mice - anti-angiogenic inhibition parameter - natural death

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

25 Dynamics revisited Write the system as where Switching function

26 Singular control order 1 singular control strengthened Legendre-Clebsch condition is satisfied

27 Singular Control feedback control psi x

28 Admissible Singular Arc p tumor volume,p carrying capacity of the vasculature, q q x 10 4

29 Synthesis of Optimal Controls [LSch, SICON, 2007] p u=0 u=u max tumor cells end of therapy 6000 an optimal trajectory begin of therapy final point minimum of p endothelial cells q typical synthesis: u max s 0

30 Optimal Synthesis near a Saturation Point u=u max singular arc s u=0

31 [Ergun, Camphausen and Wein] Bull. Math. Biol., tumor growth parameter - endogenous stimulation (birth) - endogenous inhibition (death) - anti-angiogenic inhibition parameter p tumor volume q carrying capacity u anti-angiogenic dose rate - natural death

32 Synthesis for the Anti-Angiogenic Monotherapy Model [LS, Discr.&Cont. Dyn. Syst, 2009] tumor volume, p full dose beginning of therapy 8000 partial dose, singular control 6000 no dose 4000 (q(t),p(t)), point where minimum is realized carrying capacity of the vasculature, q x 10 4 typical synthesis - u max -s-0

33 Extension of the model with with PK (pharmacokinetics) u PK c dynamics for p dosage concentration p and q To what extent is previous analysis preserved? old system with the control replaced by the output of a first order linear system with control as input

34 The Ergun et al. Model with PK [ Springer 2010] For a free terminal time minimize with H. Maurer over all measurable functions subject to the dynamics and terminal condition

35 Singular Control

36 Optimal and Suboptimal Controls (H.Maurer) optimal chattering control and corresponding concentration sub-optimal control and corresponding concentration

37 Multi-input Control: Antiangiogenic Treatment with Chemotherapy

38 A Model for a Combination Therapy [CDC 2009, Math Biosciences 2009, MBE 2011] Minimize subject to with A. d Onofrio and H.Maurer angiogenic inhibitors cytotoxic agent or other killing term

39 No simultaneously singular controls these 4 vector fields are linearly independent, so both switching functions cannot vanish simultaneously on an interval

40 u - singular control order 1 singular control strengthened Legendre-Clebsch condition is satisfied

41 Optimal Protocols (H.Maurer) tumor volume, p optimal angiogenic monotherapy carrying capacity of the vasculature, q

42 Some Examples of Optimal Controls

43 Medical Connection Rakesh Jain, Steele Lab, Harvard Medical School, there exists a therapeutic window when changes in the tumor in response to antiangiogenic treatment may allow chemotherapy to be particularly effective

44 Tumor Anti-Angiogenesis And Radiotherapy

45 A Model for Anti-Angiogenic Treatment with Radiotherapy [LSch, JOTA, 2012] Model based on Ergun et al., Bull. Math. Biology, 2003 For a free terminal time T, minimize the tumor volume p(t) over all measurable functions and subject to the dynamics and terminal constraints and

46 Totally Singular Controls and Surface a system of 2 linear equations: totally singular controls optimal term for angiogenic monotherapy the totally singular vector field is only optimal on the hyper-surface does not depend on the variables y and z

47 Singular Flow on Surface S 30 5 anti-angiogenic dose rate u anti-angiogenic dose rate radiation dose w radiation schedule time time

48 Tumor Immune Interactions

49 Mathematical Model for Tumor-Immune Dynamics 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 STATE: - primary tumor volume - immunocompetent cell-density (related to various types of T-cells)

50 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

51 Phaseportrait of uncontrolled dynamics we want to move the state of the system into the region of attraction of the benign equilibrium minimize

52 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)

53 Optimal Control Problem [LNSch,J Math Biol, 2011] For a free terminal time T minimize over all functions and subject to the dynamics Chemotherapy log-kill hypothesis Immune boost

54 Legendre-Clebsch Condition for Control u (Chemotherapy) The Legendre-Clebsch condition is satisfied if and only if

55 Legendre-Clebsch Condition for Control v (Immunotherapy) Suppose the control v is singular on an interval I. For optimality we need that But in this case singular controls are maximizing, so not optimal the control v, i.e., immune boost, should be bang-bang

56 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 s chemo - immune boost * * * * * free pass

57 Future Direction: Metronomics

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59 Future Direction: Metronomic Chemotherapy How is it administered? treatment at much lower doses ( between 10% and 80% of the MTD) over prolonged periods Advantages 1. lower, but continuous cytotoxic effects on tumor cells lower toxicity (in many cases, none) lower drug resistance and even resensitization effect 2. antiangiogenic effects 3. boost to the immune system Metronomics Global Health Initiative (MGHI)

60 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 Wholistic Approach? Minimally parameterized metamodel Single input multi target metronomic dosing of chemotherapy

61 Summary and Future Direction: Combining Models (?) x(t) primary tumor volume x s (t) sensitive, x r (t) resistant y(t) carrying capacity of the tumor vasculature z(t) immunocompetent cell density u(t) dose rate of chemotherapy c(t) - concentration

62 OBRIGADO!