Urszula Ledzewicz

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Lecture 1 Optimal Control of Compartmental
Models in Cancer Chemotherapy, 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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Andrzej Swierniak Department of Automatic
Control Silesian University of Technology, Gliwice
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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
• e.g., Eisen, 1979, Swierniak and Kimmel, 1982,
  1984
• Swan, 1988, Murray, 1990, Martin, 1992,
• Swierniak, 1995, Ledzewicz and Schättler, 2002,
  2003,
• Swierniak, Ledzewicz and Schättler, 2003
• Radiotherapy
• Models for novel treatments
• Tumor anti-angiogenesis, Immunotherapy

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Outline of Lecture 1, Part 1

• Compartmental Models for Cancer Chemotherapy
• General n-compartmental model as optimal control
  problem
• Analysis of 2- and 3-compartment models
• bang-bang and singular controls
• results, simulations

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References

• A. Swierniak and M. Kimmel, O pewnym zadaniu
  sterowania optymalnego zwiazanym z optymalna
  chemioterapia bialaczek, Zeszyty Naukowe
  Politechniki Slaskiej, Z. 65, s. Automatyka,
  1982.
• A. Swierniak and M. Kimmel, Zastosowanie teorii
  i metod sterowania optymalnego do wyznaczania
  protokolow chemioterapii bialaczki, Zeszyty
  Naukowe Politechniki Slaskiej, Z. 73, s.
  Automatyka, 1984.
• A. Swierniak, Optimal treatment protocols in
  leucemia-modeling the proliferationg cycle,
  Proceedings 12th IMACS World Congress, Paris,
  v.4., 1988, 170-172.
• A. Swierniak, Cell cycle as an object of
  control, Journal of Biological Systems, V.3, n.1,
  1995, 41-54.

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References

• U. Ledzewicz and H. Schättler, Optimal
  bang-bang controls for a 2-compartment model in
  cancer chemotherapy, J. of Optimization Theory
  and Applications (JOTA), 114 (3), 2002, pp.
  609-637
• A. Swierniak, U. Ledzewicz and H. Schättler,
  Optimal control for a class of
  compartmental models in cancer chemotherapy,
  Int. J. of Applied Math. and Comp. Sci., 13 (3),
  2003, pp. 357-368
• U. Ledzewicz and H. Schättler, Optimal control
  for a bilinear model with recruiting agent in
  cancer chemotherapy, Proc. of the 42nd IEEE
  Conference on Decision and Control (CDC), Maui,
  Hawaii, 2003, pp. 2762-2767

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Cell-cycle specific models for cancer
chemotherapy

• CELL CYCLE SPECIFICITY
• drugs act at various stages
• of the cell cycle
• Killing agents in G2/M
• (Taxol, Spindle poisons,)
• Blocking agent in S
• (hydroxyurea,)
• (gastro-intestine cancers)
• Recruiting agent in G0
• (interleukin-3, granulate
• colony stimulation factors, ..)
• (luekemia)

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General mathematical structure of compartmental
models Dynamics

• DYNAMICS describes the
• changes in the average number
• of cancer cells in the
• compartments

• COMPARTMENTS
• clusters of phases of cell-cycle
• STATES N ( N1 , , Nn )
• represent the numbers of (cancer) cells in the
  corresponding compartments
• CONTROLS u ( u1 , , ur )
• represent the drug dosages/ effects of various
  drugs
• values in compact intervals
• (Swierniak, Ledzewicz, Schaettler 2003 )

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Two-compartment model with a killing agent

• STATES N (N1 , N2 )
• represent the numbers of cancer cells in G1/S
  and G2/M
• CONTROL u
• drug dosage of the killing agent
• u 0 no dose
• u 1 full dose
• DYNAMICS

Swierniak and Kimmel, 1984
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Dynamics for 2-compartment in Matrix Form
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Bone-marrow model

• STATES N (P , Q )
• numbers of bone marrow cells in proliferating
  and quiescent stage
• CONTROL u
• drug dosage of the killing agent
• u 0 no dose
• u 1 full dose
• DYNAMICS

R. Fister J. Panetta, SIAM J. of Appl. Math.,
2000
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Three-compartment model with recruiting agent
(leukemia)

• STATES N ( N0 , N1 , N2 )
• represent the numbers of cancer cells in the
  corresponding compartments
• CONTROLS u , w
• represent the drug dosage of
• u - killing agent
• w - recruiting agent
• u 0 no dose / u 1 full dose
• w 0 no dose / w wmax lt1 full
  dose
• b0 and b1 - probabilities that a daughter
  cell enters G0 and G1

Swierniak, Kimmel 1984
SG2/M
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Dynamics for 3-compartment Model with Recruiting

• mathematical models based on underlying biology

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General mathematical structure of compartmental
models Objective

• minimize the number of cancer cells left without
  causing too much harm to the healthy cells

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
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Positive Invariance

• In the model formulation, it is implicitly
  assumed that the region

is positively invariant for any admissible
control, the solution to the system exists for
all times and remains in
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Positive Invariance
This property depends on the matrices and
. One easily verifiable sufficient condition
is the following property
References on topic e.g., Kaczorek (1995),
Swierniak (2005)
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Maximum Principle
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Switching functions

• m separate minimization problems
• define the switching functions as

then
bang-bang control
singular control
bang-bang control
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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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2-compartment model

• Minimize
• over all Lebesgue measurable functions ,
• , subject to

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Maximum Principle for 2-compartment Model
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Switching function

• the switching function is
• optimal controls satisfy
• Singular controls - on an
  interval

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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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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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Proof
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Singular control
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Singular Controls are NOT Optimal
Singular controls exist, are of order 1, but are
maximizing
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Dynamics for the 3-compartment Model with
Recruiting
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Singular Controls are NOT Optimal
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Locally maximizing
NOT optimal
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u is locally maximizing
Singular controls are NOT optimal in all cases
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2-compartment modelBang-Bang Controls
need to analyze bang-bang switchings backward in
time
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Optimal and Non-Optimal Switchings
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Parameterization of Controls and Trajectories
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Flow of Parameterized Trajectories
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Bang-Bang Flows
N
t
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Algorithmic Determination
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N
t
T
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Bang-Bang Solutions for 2-comp
transversality condition satisfied
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Bang-Bang Extremal for 2-comp
transversality condition violated
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Simulations for 3-compartment Model
Control u
Control w

• Optimal protocols for killing and recruiting
  agents and their effect
• Transversality conditions satisfied locally
  optimal

States N0, N1, N2
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Conclusions

• Methods used High-order conditions of
  optimality (Legendre-Clebsch condition)
• Results singular controls eliminated as
  candidates for optimality in all models (2-comp.
  3-comp. for cancer)
• Method used Method of characteristics
  (construction of the field of extremals)
• Results easily verifiable transversality
  conditions determining optimality of bang-bang
  controls at switchings