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

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 Mathematical Methods and Models in Biomedicine Urszula Ledzewicz, Heinz Schättler, Avner Friedman and Eugene Kashdan, Eds. Springer Verlag, November 2012 Forthcoming Books

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

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

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

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 metronomic chemotherapy Outline – An Optimal Control Approach to …

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

Heterogeneity and Tumor Microenvironment Tumor Microenvironment

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

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

cR(t)cR(t) – outflow of resistant cells dynamics cR division cR rcR Mutates back RS R S cR (2-r)cR remainsresistant

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 Weighted average of cancer cells during therapy Toxicity of the drug (side effects on healthy cells)

From Maximum Principle: Candidates for Optimal Protocols bang-bangbang-bang controls singularsingular controls treatment protocols of maximum dose therapy periods with rest periods in between continuous infusions of varying lower doses u max TT MTDBOD

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

“Markov Chain” Models

Tumor Anti-angiogenesis

Tumor Anti-Angiogenesis avascular growth angiogenesis metastasis

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 endothelial cells 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

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

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

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

An Optimal Controlled Trajectory for [Hahnfeldt et al.] Initial condition: p 0 = 12,000 q 0 = 15,000, u max =75 maximum dose rate no dose lower dose rate - singular averaged optimal dose u q0q0 robust robust with respect to q 0

Anti-Angiogenic Treatment with Chemotherapy

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

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 ?Which should come first ?

Optimal Protocols optimal angiogenic monotherapy

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

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

Tumor Immune Interactions

Mathematical Model for Tumor-Immune DynamicsSTATE: - primary tumor volume - immunocompetent cell-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

- 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 Dynamical Model

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

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

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

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

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

Future Direction: Metronomic Chemotherapy treatment at much lower doses ( between 10% and 80% of the MTD) over prolonged periods How is it administered? 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)