The role of the vasculature and the immune system in optimal

protocols for cancer therapies

Heinz SchHeinz SchättlerättlerDept. of Electr. and Systems Engr.Dept. of Electr. and Systems Engr.

Washington University Washington University St. Louis, USASt. Louis, USA

Urszula LedzewiczUrszula LedzewiczDept. of Mathematics and StatisticsDept. of Mathematics and StatisticsSouthern Illinois University Edwardsville Southern Illinois University Edwardsville Edwardsville, USAEdwardsville, USA

UT Austin – Portugal Workshop onModeling and Simulation of Physiological Systems

December 6-8, 2012 Lisbon, Portugal

• Heinz Schättler and Urszula Ledzewicz,

Geometric Optimal Control – Theory, Methods, ExamplesGeometric Optimal Control – Theory, Methods, Examples

Springer Verlag, July 2012Springer Verlag, July 2012

• Urszula Ledzewicz and Heinz Schättler,

Geometric Optimal Control Applied to Biomedical ModelsGeometric Optimal Control Applied to Biomedical Models

Springer Verlag, 2013Springer Verlag, 2013

• Mathematical Methods and Models in BiomedicineMathematical Methods and Models in Biomedicine

Urszula Ledzewicz, Heinz Schättler, Avner Friedman and Eugene Kashdan, Eds.

Springer Verlag, November 2012Springer Verlag, November 2012

Forthcoming Books

Main Collaborators and ContactsAlberto d’OnofrioEuropean Institute for Oncology, Milano, Italy

Helmut MaurerRheinisch Westfälische Wilhelms-Universität Münster, Münster, Germany

Andrzej SwierniakSilesian University of Technology, Gliwice, Poland

Avner FriedmanMBI, The Ohio State University, Columbus, Oh

Research supported by collaborative research Research supported by collaborative research NSF grantsNSF grants

DMS 0405827/0405848DMS 0405827/0405848

DMS 0707404/0707410DMS 0707404/0707410

DMS 1008209/1008221DMS 1008209/1008221

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 chemotherapymetronomic chemotherapy

Outline – An Optimal Control Approach to …

Optimal Drug Treatment Protocols

Main QuestionsMain QuestionsQUESTION 1:QUESTION 1: HOW MUCH? (dosage)HOW MUCH? (dosage)

QUESTION 2:QUESTION 2: HOW OFTEN? (timing)HOW OFTEN? (timing)

QUESTION 3:QUESTION 3: IN WHAT ORDER? IN WHAT ORDER? (sequencing) (sequencing)

Heterogeneity andHeterogeneity and

Tumor MicroenvironmentTumor Microenvironment

Tumor stimulating myeloid cell

Surveillance T-cell

Fibroblast

EndotheliaChemo-resistant tumor cell

Chemo-sensitivetumor cell

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

aS(t),aS(t), cR(t) cR(t) outflow of sensitive/resistance cellsu – cytotoxic drug dose rate, 0≤u≤1

aSaS(1-u)aSaS

uaSaS - killed

divisionp(1-u)aSaS

Mutates

SS RR

(2-p)(1-u)aSaS

remains

sensitivesensitiveone-gene forward gene amplification hypothesis,

Harnevo and AgurHarnevo and Agur

Model for Drug Resistance under Chemotherapy

• cR(t)cR(t) – outflow of resistant cells

• dynamics

cRcRdivision rcRcR

Mutates back

RR SS

(2-r)cRcRremains

resistantresistant

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

umax

T T

MTDMTD BODBOD

• 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 recommendedlower doses are recommended

Results [LSch, DCDS, 2006]

From the Legendre-Clebsch condition

“Markov Chain” Models

Tumor Anti-angiogenesisTumor Anti-angiogenesis

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

Tumor Anti-Angiogenesis

avasculargrowth angiogenesis

metastasis

Tumor Anti-angiogenesis• suppress tumor growth bypreventing the recruitment of newblood vessels that supply the tumor with nutrients

(indirect approach)

• done by inhibiting the growth of the endothelial cellsendothelial cells that form the lining of the new blood vesselstherapy “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 mm3

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

For a free terminal time minimizeminimize

over all functions that satisfy

subject to the dynamics

Optimal Control Problem

Synthesis of Optimal Controls [LSch, SICON, 2007]

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

6000

8000

10000

12000

14000

16000

18000

endothelial cells

tum

or c

ells

an optimal trajectorybegin of therapy

final point – minimum of p

end of “therapy”

p

q

u=au=0

typical synthesis: umax→s→0

An Optimal Controlled Trajectory for [Hahnfeldt et al.]

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

0 1 2 3 4 5 6 7

0

10

20

30

40

50

60

70

time

optim

al c

ontr

ol u

maximum dose rate

no dose

lower dose rate - singular

averaged optimal dose

u

q0

robustrobust with respect to q0

Anti-Angiogenic Anti-Angiogenic

Treatment with Treatment with

ChemotherapyChemotherapy

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

4000 6000 8000 10000 12000 14000 16000

7000

8000

9000

10000

11000

12000

13000

carrying capacity of the vasculature, q

tum

or v

olum

e, p

optimal angiogenic monotherapy

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

0 1 2 3 4 5 6 7

0

10

20

30

40

50

60

70

time (in days)

dosa

ge a

ngio

4000 6000 8000 10000 12000 14000 16000

7000

8000

9000

10000

11000

12000

13000

carrying capacity of the vasculature, q

tum

or v

olum

e, p

0 1 2 3 4 5 6 7-0.2

0

0.2

0.4

0.6

0.8

1

dosa

ge c

hem

o

time (in days)

Medical Connection

Rakesh Jain, Steele Lab, Harvard Medical School,

“there exists a therapeutic windowtherapeutic window when changes in the tumor in response to anti-angiogenic treatment may allow chemotherapy to be particularly effective”

Tumor Immune InteractionsTumor Immune Interactions

Mathematical Model for Tumor-Immune Dynamics

STATE:STATE: - 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

For a free terminal time T minimizeminimize

over all functions and

subject to the dynamics

Chemotherapy – log-kill hypothesis

Immune boost

Optimal Control Problem [LNSch,J Math Biol, 2011]

0 200 400 600 8000

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

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”

1s01 010

- 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?How is it administered?

AdvantagesAdvantages1. 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)

http://metronomics.newethicalbusiness.org/