Optimal Control for Combination Therapy in Cancer

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Optimal Control for Combination Therapy in
Cancer
47th IEEE Conference on Decision and Control
Cancun, Mexico December 9-11, 2008
Heinz Schättler Dept. of Electrical and Systems
Engineering Washington University St. Louis,
Missouri, USA
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Collaborators and Support
Urszula Ledzewicz Dept. of Mathematics and
Statistics Southern Illinois University
Edwardsville, Edwardsville, Illinois, USA
Research supported by Collaborative NSF grant DMS
0707404/0707410
Alberto dOnofrio European Institute for
Oncology, Milano, Italy Helmut Maurer
Rheinisch Westfälische Wilhelms Universität
Münster, Münster, Germany
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Cancer Timeline I. Avascular Growth

• initially, a growing tumor gets sufficient supply
  of oxygen and nutrients from the surrounding host
  blood vessels to allow for cell duplication and
  tumor growth.
• at the size of about 1-3 mm in diameter, this no
  longer is true and tumor cells enter the dormant
  stage G0 in the cell cycle.
• these cells then produce vascular endothelial
  growth factor (VEGF) to start the process of
• angiogenesis

Scientific American, 2003
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Cancer Timeline II. Angiogenesis
The tumor develops its own network of capillaries
which provide nutrients and oxygen to the tumor
and connect it with the blood vessels of the host
http//www.gene.com/gene/research/focusareas/oncol
ogy/angiogenesis.html
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Cancer Timeline III. Metastasis

• (gr. change of the state)
• the spread of cancer from its primary site to
  other places in the body
• cancer cells break away from a primary tumor,
  penetrate into lymphatic and blood vessels,
  circulate through the bloodstream, and grow in
  normal tissues elsewhere in the body
  (metastasize)

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Tumor Anti-Angiogenesis

• the treatment of tumors by
• preventing the recruitment of
• new blood vessels
• this is done by inhibiting the
• Growth/recruitment of
• endothelial cells that form the
• lining of the new blood vessels
• that supply the tumor with
• nutrients
• therapy resistant to resistance

J. Folkman, 1971/72
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Models for Tumor Anti-Angiogenesis
Hahnfeldt, Panigrahy, Folkman and Hlatky,
Cancer Research, 1999 Ergun, Camphausen
and Wein, Bull. Math. Biology, 2003 dOnofrio
and Gandolfi, Math. Biosciences, 2004 Anderson
and Chaplain, 1998 Arakelyan, Vainstain, and
Agur, 2003
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Mathematical Model for Tumor Growth under
Angiogenic Inhibitors

• STATE
• - primary tumor volume
• - carrying capacity of the vasculature
• endothelial or vascular
  support
• CONTROL
• - anti-angiogenic drug dose rate

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Hahnfeldt,Panigrahy,Folkman,Hlatky,Cancer
Research, 1999
Gompertzian with variable carrying capacity q
?
p,q volumes in mm3
where the parameters represent - tumor
growth parameter - angiogenic stimulation
(birth) - inhibition (death) parameters
- anti-angiogenic inhibition parameter -
natural death
Lewis lung carcinoma implanted in mice
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Optimal Control Problem

• For a free terminal time minimize
• over all measurable functions that satisfy
• subject to the dynamics

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Synthesis of Optimal Controls Led Sch, SICON,
2007
ua
u0
p
q
Full synthesis 0asa0 typical synthesis - as0
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Combination Therapies

• tumor anti-angiogenesis is an indirect approach
  that does not kill cancer cells tumor grows
  back once treatment is halted
• need to combine anti-angiogenesis with
  traditional treatments like radiotherapy or
  chemotherapy
• "simultaneously targets two compartments, the
  cancer cells and the vascular cells that support
  the tumor" ( Dr. Qian, John Hopkins Kimmel Cancer
  Center, 2004).

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A Model for a Combination Therapy
with dOnofrio and H. Maurer
Minimize subject to
angiogenic inhibitors
cytotoxic agent or other killing term
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Reformulation

• For a free terminal time minimize
  subject to
• over all measurable functions
• and that satisfy
• and

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4-Dimensional Dynamics
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Two Switching Functions
for
for
expect optimal controls consist of bang-bang and
singular pieces
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u - singular control
strengthened Legendre-Clebsch condition is
satisfied
order 1 singular control
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Formula for u - singular controls
we have
write
optimal singular control for angiogenic
monotherapy
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u - singular arc
The multiplier
vanishes against the vector fields
and
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• for v0 the vector fields , , and
  must be
• linearly dependent singular arc

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• for these conditions only
  determine the multiplier up to a positive multiple

u-singular feedback flow in
(p,q)-space
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Theoretical Tidbits
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Numerical Calculations (H. Maurer)

• based on a discretization scheme the numerically
  optimal solution for a modified objective with
  extra quadratic penalty term on the objective was
  computed numerically

• this numerical solution indicates that the
  optimal drug dosage is bang-bang with a
  switch from
• to

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Proposed Protocol
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Minimum value as function of
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Controls and Trajectory
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Structure of Optimal Protocols (?)
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Other Biological Aspects - Pruning

• Rakesh Jain, Steele Lab, Harvard Medical School,
• some anti-angiogenesis drugs are able to prune
  immature, inefficient blood vessels created
  during angiogenesis and fortify the remaining
  ones
• when judiciously used, anti-angiogenic drugs,
  can make the tumors blood vessels more efficient
  for delivering chemotherapy
• there exists a therapeutic window when
  changes in the tumor in response to
  anti-angiogenic treatment may allow chemotherapy
  to be particularly effective