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Heinz Sch ättler 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

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

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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

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

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

External Grant Support

Research supported by collaborative research NSF grants

DMS 0405827/0405848

DMS 0707404/0707410

DMS 1008209/1008221

Components of Optimal Control Problems

min or max






(unmodelled dynamics)


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

Optimal Drug Treatment Protocols

Main Questions



QUESTION 3:IN WHAT ORDER? (sequencing)

Heterogeneity and

Tumor Microenvironment

Tumors are same size but contain different composition of chemo-resistant and –sensitive cells.


tumor cell

Tumor stimulating

myeloid cell



tumor cell


Surveillance T-cell

aS(t),cR(t) outflow of sensitive/resistance cells

u – cytotoxic drug dose rate, 0≤u≤1

Model for Drug Resistance under Chemotherapy







uaS - killed




one-gene forward gene amplification hypothesis,

Harnevo and Agur

cR(t) – outflow of resistant cells


Mutates back








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

From Maximum Principle: Candidates for Optimal Protocols

bang-bang controls





treatment protocols of maximum dose therapy periods with rest periods in between

continuous infusions of varying lower doses



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

“Markov Chain” Models

Tumor Anti-angiogenesis

Tumor Anti-Angiogenesis





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

- 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

Optimal Control Problem

For a free terminal time minimize

over all functions that satisfy

subject to the dynamics

begin of therapy

an optimal trajectory

end of “therapy”

final point – minimum of p

Synthesis of Optimal Controls [LSch, SICON, 2007]





typical synthesis: umax→s→0

An Optimal Controlled Trajectory for [Hahnfeldt et al.]


maximum dose rate


lower dose rate - singular

averaged optimal dose

no dose

robust with respect to q0

Initial condition: p0 = 12,000 q0 = 15,000, umax=75


Treatment with


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

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 ?

Optimal Protocols

optimal angiogenic monotherapy

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

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”

Tumor Immune Interactions

Mathematical Model for Tumor-Immune Dynamics


- primary tumor volume

- immunocompetentcell-density

(related to various types of T-cells)


Kuznetsov, Makalkin, Taylor and Perelson,

Bull. Math. Biology, 1994

de Vladar and Gonzalez, J. Theo. Biology,2004,

d’Onofrio,Physica D, 2005

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

Phaseportrait for Gompertz Growth

  • we want to move the state of the system into the region of attraction of the benign equilibrium


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


( (b,a)T is the stable eigenvector of the saddle and c, d and s are positive constants)

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

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





“free pass”

- chemo

- immune boost




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

Future Direction: Metronomic Chemotherapy

How is it administered?

  • treatment at much lower doses ( between 10% and 80% of the MTD)

  • over prolonged periods


  • 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)


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