Urszula Ledzewicz

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Lecture 2 Optimal Protocols for Models of
Tumor Anti-Angiogenesis, Part 1 Analysis of
Models
May 11-15, 2009 Department of Automatic
Control Silesian University of Technology, Gliwice

• Urszula Ledzewicz
• Department of Mathematics and Statistics
• Southern Illinois University, Edwardsville, USA

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Main Collaborator - Heinz Schaettler Dept. of
Electrical and Systems Engineering Washington
University in St.Louis, USA
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Research Support
Research supported by NSF grants DMS 0205093,
DMS 0305965 and collaborative research grants
DMS 0405827/0405848 DMS 0707404/0707410
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Mathematical Models for Cancer Treatments

• Models for traditional treatments (chemotherapy,
  radiotherapy)
• Models for novel treatments
• Immunotherapy
• e.g., Kirschner and Panetta, 1995,
• de Pillis and Radunskaya, 2001,
• Castiglione and Piccoli, 2006, ...
• Tumor anti-angiogenesis new hope for
  cancer treatment

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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 2 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

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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 of the endothelial
• cells that form the lining of
• the new blood vessels that
• supply the tumor with
• nutrients
• therapy resistant to resistance

J. Folkman, 1972
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Models for Tumor Anti-Angiogenesis
Hahnfeldt, Panigrahy, Folkman and Hlatky,
Cancer Research, 1999 Modifications 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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Outline of Lecture 2

• Part 1 Model for anti-angiogenesis by
• Hahnfeldt, Panigrahy, Folkman and Hlatky Cancer
  Research, 1999
• is considered as optimal control problem -
  synthesis of optimal solutions
• Part 2 Modifications of the models
  similarities and differences in solutions

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References for Talk 2, Part 1

• P.Hahnfeldt, D. Panigrahy, J. Folkman and L.
  Hlatky,
  Tumor development under angiogenic
  signaling a dynamical theory of tumor growth,
  treatment response, and postvascular dormancy,
  
  Cancer Research, 59, 1999, pp. 4770-4775

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References for Talk 2, Part 1

• U. Ledzewicz and H. Schättler, Application of
  optimal control to a system describing tumor
  anti-angiogenesis, Proc. of the 17th Int. Symp.
  on Mathematical Theory of Networks and Systems
  (MTNS) , Kyoto, Japan, 2006, pp. 478-484
• U. Ledzewicz and H. Schättler, Anti-Angiogenic
  Therapy in Cancer treatment as an Optimal
  Control Problem, SIAM J. on Control and
  Optimization, 46 (3), pp. 1052-1079, 2007

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Mathematical Model for Tumor Growth under
Angiogenic Inhibitors

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

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Hahnfeldt,Panigrahy,Folkman,Hlatky,Cancer
Research, 1999
Gompertzian with variable carrying capacity
?
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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Growth Models on the Tumor Volume p
Some growth function on the cancer cells p
dependent on the variable carrying capacity q
Gompertzian
logistic growth
F is twice continuously differentiable, strictly
decreasing, and F(1)0

• - Hahnfeldt et al., 1999, - dOnofrio and
  Gandolfi, 2003,
• Agur et al., 2003, - Forys et al., 2004,
  - Swierniak et al., 2006

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Models for the Dynamics of the Carrying Capacity q
inhibition due to administered inhibitors
endogenous inhibition
stimulatory capacity of tumor
spontaneous loss
Hahnfeldt et al., Cancer Research, 1999

• inhibitor from tumor will impact endothelial
  cells in a way that grows like

• the stimulator term will tend to grow at a rate
  of slower than the inhibitor term with

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Specifications of the q-Dynamics
Hahnfeldt et al., Cancer Research, 1999
dOnofrio and Gandolfi, Math. Biosciences, 2004
Ergun,Camphausen and Wein, Bull. Of Math.
Biology, 2003
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Hahnfeldt,Panigrahy,Folkman,Hlatky,Cancer
Research, 1999
Gompertzian with variable carrying capacity
?
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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Phaseportrait of the uncontrolled dynamics, u0
For the system has a globally
asymptotically stable node at
dOnofrio and Gandolfi, Math. Biosciences, 2004
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Phaseportrait of the controlled dynamics, ua
p

• there is no equilibrium for the controlled
  system with ua
• all trajectories converge to (0,0) for

dOnofrio and Gandolfi, Math. Biosciences, 2004
q
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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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Biologically Relevant Region

• the region

is positively invariant for any admissible
control, i.e. if ,
then the solution to the system exists for all
times and remains in .
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Reformulation

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

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Maximum Principle
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Adjoint equations
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Basic Properties (i)
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Degenerate Cases
p
smallest possible value
also matters
q
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Basic Properties (ii)
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Switching function
With
optimal controls satisfy
we expect that optimal controls consist of
bang-bang and singular pieces
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Maximum Principle Candidates for Optimal
Protocols

• bang-bang controls

• singular controls

a
T
T
treatment protocols of full dose therapy periods
with rest periods in between
continuous infusions of varying partial doses
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Singular Controls

• is singular on an open interval
• on
• all time derivatives must vanish as well
• allows to compute the singular control
• order the control appears for the first time
  in the derivative
• Legendre-Clebsch condition (minimize)

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Dynamics revisited
Write the system as
where
Switching function
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A fundamental lemma
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Proof
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Proof
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Proof
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Proof
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Derivative of the switching function
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Second derivative of the switching function
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Singular control
strengthened Legendre-Clebsch condition is
satisfied
order 1 singular control
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Basis
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Singular control (2)
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Singular Trajectory (Arc)
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Geometry of Singular Curve

• make a blow-up in the variables as

• in the singular curve can be expressed
  as
• where

• strictly convex minimum value at
  

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x
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Singular curve
p
q
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Singular control
Hence
Along the singular arc, the singular control only
depends on !
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Admissible Singular Control

• the singular control
• needs to take values in

• there exists a unique
• interval with
• where the control is
• admissible

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Admissible Singular Arc
p
q
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Properties of Optimal Controls

• optimal trajectories do not cross over from
  into
• segments corresponding to the control
  can only be
• at the beginning or at the end

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Synthesis of Optimal Controls LSch, SICON,
2007
ua
u0
p
q
Full synthesis 0asa0 typical synthesis - as0
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Explanation of Synthesis

• most typical structure of solutions
• as0 (full dose, partial dose, no dose)
• realistic initial conditions lie in a region
  where full dose is given immediately
• typically along singular arc available inhibitors
  are exhausted
• if saturation occurs before the amount of drug is
  exhausted another arc with ua is inserted
• due to after effects in the dynamics the tumor
  volume still decays for some time after all
  inhibitors have been given

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An Optimal Controlled Trajectory for Hahnfeldt
et al.
Initial condition p0 12,000 q0 15,000
Optimal terminal value 8533.4 time 6.7221
Terminal value for a0-trajectory 8707.4 time
5.1934
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Comments

• solutions show the maximum tumor reduction
  possible with the fixed amount of inhibitors
• the optimal control is not medically realizable
  (feedback) realistic suboptimal
  protocols
• a measure for the actual treatment schedules to
  be judged against
• in the absence of treatment the tumor grows to
  equilibrium for u0 repeat treatment is
  necessary to maintain tumor volumes low
• no developing drug resistance allows repeated
  treatment