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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
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• 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
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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
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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
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Components of Optimal Control Problems
dynamics
(model)
min or max
objective
control
response
disturbance
(unmodelled dynamics)
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• 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 …
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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)
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Heterogeneity andHeterogeneity and
Tumor MicroenvironmentTumor Microenvironment
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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.
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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
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• cR(t)cR(t) – outflow of resistant cells
• dynamics
cRcRdivision rcRcR
Mutates back
RR SS
(2-r)cRcRremains
resistantresistant
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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)
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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
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• 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
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“Markov Chain” Models
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Tumor Anti-angiogenesisTumor Anti-angiogenesis
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http://www.gene.com/gene/research/focusareas/oncology/angiogenesis.html
Tumor Anti-Angiogenesis
avasculargrowth angiogenesis
metastasis
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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
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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
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For a free terminal time minimizeminimize
over all functions that satisfy
subject to the dynamics
Optimal Control Problem
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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
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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
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Anti-Angiogenic Anti-Angiogenic
Treatment with Treatment with
ChemotherapyChemotherapy
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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
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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 ?
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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
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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)
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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”
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Tumor Immune InteractionsTumor Immune Interactions
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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
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- 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
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Phaseportrait for Gompertz Growth
• we want to move the state of the system into the region of attraction of the benign equilibrium
minimize
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• 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
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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]
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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
**
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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
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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/