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
Dept. of Electr. and Systems Engr.
Washington University
St. Louis, 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
Main Collaborators and Contacts
Alberto d’Onofrio
European Institute for Oncology, Milano, Italy
Helmut Maurer
Rheinisch Westfälische WilhelmsUniversität 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
objective
dynamics
response
(model)
disturbance
control
(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 1: HOW MUCH?
QUESTION 2: HOW OFTEN?
(dosage)
(timing)
QUESTION 3: IN WHAT ORDER?
(sequencing)
Heterogeneity and
Tumor Microenvironment
Tumors are same size but contain different
composition of chemo-resistant and –sensitive cells.
Chemo-sensitive
tumor cell
Fibroblast
Tumor stimulating
myeloid cell
Chemo-resistant
tumor cell
Endothelia
Surveillance T-cell
Model for Drug Resistance under Chemotherapy
aS(t), cR(t) outflow of sensitive/resistance cells
u – cytotoxic drug dose rate, 0≤u≤1
Mutates
division
p(1-u)aS
S
R
(1-u)aS
aS
(2-p)(1-u)aS
uaS - killed
remains
sensitive
one-gene forward gene amplification hypothesis,
Harnevo and Agur
• cR(t) – outflow of resistant cells
division
rcR
Mutates back
R
cR
(2-r)cR
S
remains
resistant
• dynamics
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-bang controls
• singular controls
umax
T
treatment protocols of
maximum dose therapy
periods with rest periods
in between
MTD
T
continuous infusions of
varying lower doses
BOD
Results [LSch, DCDS, 2006]
From the Legendre-Clebsch condition
• 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
“Markov Chain” Models
Tumor Anti-angiogenesis
Tumor Anti-Angiogenesis
avascular
growth
angiogenesis
metastasis
http://www.gene.com/gene/research/focusareas/oncology/angiogenesis.html
Tumor Anti-angiogenesis
Judah Folkman, 1972
• 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”
• 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 – tumor volume
q – carrying capacity
u – anti-angiogenic
dose rate
p,q – volumes in mm3
- tumor growth parameter
- endogenous stimulation (birth)
- endogenous inhibition (death)
- anti-angiogenic inhibition parameter
- natural death
Lewis lung
carcinoma
implanted
in mice
Optimal Control Problem
For a free terminal time
over all functions
subject to the dynamics
minimize
that satisfy
Synthesis of Optimal Controls
[LSch, SICON, 2007]
18000
p
u=a
16000
u=0
14000
tumor cells
12000
10000
8000
an optimal trajectory
end of “therapy”
begin of
therapy
6000
4000
final point – minimum of p
2000
0
0
2000
4000
6000
8000
10000
endothelial cells
12000
typical synthesis:
14000
16000
q 18000
umax→s→0
An Optimal Controlled Trajectory
for [Hahnfeldt et al.]
Initial condition: p0 = 12,000 q0 = 15,000, umax=75
u
70
maximum dose rate
60
optimal control u
50
40
q0
30
lower dose rate - singular
20
averaged optimal dose
no dose
10
0
0
1
2
3
4
5
6
7
time
robust with respect to q0
Anti-Angiogenic
Treatment with
Chemotherapy
A Model for a Combination Therapy
[d’OLMSch, Mathematical Biosciences, 2009]
Minimize
subject to
angiogenic inhibitors
cytotoxic agent or other killing term
with d’Onofrio and H. Maurer
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
tumor volume, p
13000
12000
optimal
angiogenic
monotherapy
11000
10000
9000
8000
7000
4000
6000
8000
10000
12000
14000
carrying capacity of the vasculature, q
16000
Controls and Trajectory [for dynamics from Hahnfeldt et al.]
dosage angio
70
60
50
13000
40
tumor volume, p
30
20
10
0
0
1
2
3
4
5
6
7
time (in days)
12000
11000
10000
dosage chemo
9000
8000
1
0.8
7000
0.6
4000
6000
8000
10000
12000
14000
carrying capacity of the vasculature, q
0.4
0.2
0
-0.2
0
1
2
3
4
5
6
time (in days)
7
16000
Medical Connection
Rakesh Jain, Steele Lab, Harvard Medical
School,
“there exists a therapeutic window when
changes in the tumor in response to antiangiogenic treatment may allow
chemotherapy to be particularly effective”
Tumor Immune Interactions
Mathematical Model for
Tumor-Immune Dynamics
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
STATE:
-
primary tumor volume
-
immunocompetent cell-density
(related to various types of T-cells)
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
minimize
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)
Optimal Control Problem [LNSch,J Math Biol, 2011]
For a free terminal time T minimize
over all functions
and
subject to the dynamics
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
3
1s01
1
010
2.5
0.8
- chemo
0.6
*
*
“free pass”
2
- immune boost
1.5
0.4
*
1
0.2
0.5
*
*
0
0
2
4
6
8
10
12
0
0
200
400
600
800
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
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)
http://metronomics.newethicalbusiness.org/