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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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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

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


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

Forthcoming Books


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

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


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

External Grant Support

Research supported by collaborative research NSF grants

DMS 0405827/0405848

DMS 0707404/0707410

DMS 1008209/1008221


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

Components of Optimal Control Problems

min or max

objective

dynamics

(model)

response

disturbance

(unmodelled dynamics)

control


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

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


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

Optimal Drug Treatment Protocols

Main Questions

QUESTION 1:HOW MUCH? (dosage)

QUESTION 2:HOW OFTEN? (timing)

QUESTION 3:IN WHAT ORDER? (sequencing)


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

Heterogeneity and

Tumor Microenvironment


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

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


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

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


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

cR(t) – outflow of resistant cells

dynamics

Mutates back

RS

division

rcR

cR

remains

resistant

(2-r)cR


Mathematical model objective

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

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


Results lsch dcds 2006

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

“Markov Chain” Models


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

Tumor Anti-angiogenesis


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

Tumor Anti-Angiogenesis

avasculargrowth

angiogenesis

metastasis

http://www.gene.com/gene/research/focusareas/oncology/angiogenesis.html


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


Model hahnfeldt panigrahy folkman hlatky cancer research 1999

- 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


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

Optimal Control Problem

For a free terminal time minimize

over all functions that satisfy

subject to the dynamics


Synthesis of optimal controls lsch sicon 2007

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


An optimal controlled trajectory for hahnfeldt et al

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


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

Anti-Angiogenic

Treatment with

Chemotherapy


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

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

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 Protocols

optimal angiogenic monotherapy


Controls and trajectory for dynamics from hahnfeldt et al

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


Medical connection

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”


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

Tumor Immune Interactions


Mathematical model for tumor immune dynamics

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


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

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

Phaseportrait for Gompertz Growth

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

minimize


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

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)


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

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


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

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

*

*

*


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

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


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

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/


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