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
Dept. of Electr. and Systems Engr.
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
Dept. of Mathematics and Statistics
Southern Illinois University Edwardsville
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 2012Forthcoming Books
European Institute for Oncology, Milano, Italy
Rheinisch Westfälische Wilhelms-Universität Münster, Münster, Germany
Silesian University of Technology, Gliwice, Poland
MBI, The Ohio State University, Columbus, Oh
Research supported by collaborative research NSF grants
min or max
QUESTION 1:HOW MUCH? (dosage)
QUESTION 2:HOW OFTEN? (timing)
QUESTION 3:IN WHAT ORDER? (sequencing)
Tumors are same size but contain different composition of chemo-resistant and –sensitive cells.
aS chemo-resistant and –sensitive cells.(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 chemo-resistant and –sensitive cells.(t) – outflow of resistant cells
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
treatment protocols of maximum dose therapy periods with rest periods in between
continuous infusions of varying lower doses
If chemo-resistant and –sensitive cells.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 recommendedResults [LSch, DCDS, 2006]
From the Legendre-Clebsch condition
Tumor Anti-angiogenesis chemo-resistant and –sensitive cells.
Tumor Anti-Angiogenesis chemo-resistant and –sensitive cells.
suppress tumor growth by
preventing the recruitment of new
blood vessels that supply the
tumor with nutrients
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
- tumor growth parameter chemo-resistant and –sensitive cells.
- endogenous stimulation (birth)
- endogenous inhibition (death)
- anti-angiogenic inhibition parameter
- natural deathModel [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 chemo-resistant and –sensitive cells.
For a free terminal time minimize
over all functions that satisfy
subject to the dynamics
begin of therapy chemo-resistant and –sensitive cells.
an optimal trajectory
end of “therapy”
final point – minimum of pSynthesis of Optimal Controls [LSch, SICON, 2007]
typical synthesis: umax→s→0
maximum dose rate
lower dose rate - singular
averaged optimal dose
robust with respect to q0
Initial condition: p0 = 12,000 q0 = 15,000, umax=75
Anti-Angiogenic chemo-resistant and –sensitive cells.
A Model for a Combination Therapy chemo-resistant and –sensitive cells. [d’OLMSch, Mathematical Biosciences, 2009]
with d’Onofrio and H. Maurer
Minimize subject to
cytotoxic agent or other killing term
Chemotherapy needs the vasculature to deliver the drugs
Anti-angiogenic therapy destroys this vasculature
In what dosages?
Which should come first ?
optimal angiogenic monotherapy
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 chemo-resistant and –sensitive cells.
- primary tumor volume
(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 chemo-resistant and –sensitive cells.
- 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
Formulation of the Objective chemo-resistant and –sensitive cells.
( (b,a)T is the stable eigenvector of the saddle and c, d and s are positive constants)
For a free terminal time chemo-resistant and –sensitive cells.Tminimize
over all functionsand
subject to the dynamics
Optimal Control Problem [LNSch,J Math Biol, 2011]
Chemotherapy – log-kill hypothesis
Chemotherapy with Immune Boost chemo-resistant and –sensitive cells.
- immune boost
Summary and Future Direction: Combining Models chemo-resistant and –sensitive cells.
Which parts of the tumor microenvironment need to be taken into account?
Wholistic Approach ?
Future Direction: Metronomic Chemotherapy chemo-resistant and –sensitive cells.
How is it administered?
Metronomics Global Health Initiative (MGHI)