Urszula Ledzewicz

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Lecture 1 Optimal Control of Compartmental
Models in Cancer Chemotherapy, Part 2 PK/PD
and Drug Resistance
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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Outline Lecture 1, Part 2

• Pharmacokinetics PK
• linear, bilinear
• Pharmacodynamics PD
• linear, Emax model, sigmoidal models
• Analysis of a 2-dimensional model with PK/PD
• Drug Resistance
• Simple 2-dimensional model for drug resistance
• analysis and results

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References

• U. Ledzewicz and H. Schättler, The influence of
  PK/PD on the structure of optimal control in
  cancer chemotherapy models, Mathematical
  Biosciences and Engineering (MBE), 2, (3),
  (2005), pp. 561-578
• U. Ledzewicz and H. Schättler, Optimal controls
  for a model with pharmacokinetics maximizing
  bone marrow in cancer chemotherapy, Mathematical
  Biosciences, 206, (2007), pp. 320--342
• U. Ledzewicz, H. Schättler and A. Swierniak,
  Finite dimensional models of drug resistant and
  phase specific cancer chemotherapy, J. of
  Medical Informatics and Technologies, 8, (2004),
  pp. 5-13

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Scope of the talk

• pharmacokinetics (PK)
• pharmacodynamics (PD)
• drug resistance

Does incorporating these aspects into the
models change the qualitative structure of
solutions?
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Pharmacokinetics and Pharmacodynamics (PK/PD)

• in previous models dosage concentration
  effect

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PK/PD
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Linear Model for PK

• Often unknown specifics
• Common approach
• linear model (exponential growth/ decay)
• clearance rate
• first order linear controller
• continuous infusion

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Bilinear Model for PK

• linear model
• different rates
• fg concentration builds up with maximum dose
• f concentration clears with no dose
• maximum concentration

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Bone Marrow Model Fister Panetta

• dynamics
• parameter values dynamics
• transition rate from proliferating to
  quiescent
• transition rate from quiescent to
  proliferating
• growth rate of proliferating cells
• death rate of proliferating cells
• rate at which bone marrow enters blood stream
• objective

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Model with PK

• Maximize
• (1)
• or
• (2)
• over all Lebesgue measurable functions ,
• , subject to

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Maximum Principle (with PK)

• Suppose is an optimal control with
  corresponding trajectory . Then
  there exist absolutely continuous functions
  and ,
• satisfying the adjoint equation
• such that the control maximizes the
  Hamiltonian over 0,1 along

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Switching function

• define the switching function as
• optimal controls satisfy
• if
• hence
  near
• Bang-bang controls
• Singular controls - on an open interval

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Maximizing 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

(maximize)
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Analysis of singular controls
Legendre-Clebsch condition

• if then singular controls are of
  order 1
• - if g gt 0 the LC-condition is violated,
  singular controls are NOT optimal
• - if g lt 0 the LC-condition is satisfied, but
  this case is not relevant

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Analysis of singular controls ctd.

• In the linear case, g 0
• the order of the singular arc is 2
• L - C condition is violated,
• singular arcs are NOT optimal.

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Bone Marrow Model Steady-State

• let
• then
• the Riccati-equation has a unique locally
  asymptotically stable equilibrium in the open
  interval which has the closed interval
  in its region of attraction
• For the numerical values

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Simulations Controls and States
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Pharmacodynamics
saturation models
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Models for PD Linear Model

• effectiveness
• simple, but only valid over small range of
  concentration
• saturation makes model non-smooth undesired
  effect

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Models for PD Michaelis-Menten

• smooth saturation at maximum effect
• immediate effects

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Models for PD Sigmoidal Model

• smooth upper saturation
• delay effect

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2-compartment model with PK/PD

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

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Maximum Principle (with PK/PD)

• Suppose is an optimal control with
  corresponding trajectory . Then there
  exist absolutely continuous functions and ,
  
• satisfying the adjoint equation
• such that the control minimizes the
  Hamiltonian over 0,1 along

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Switching function

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

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Minimizing Singular Controls

• Legendre-Clebsch condition

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Analysis of singular controls

• Switching function

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Linear PD

• If (bilinear PK, linear PD)
  singular of order 1
• For L-C condition violated -
  
• singular controls not
  optimal
• For L-C condition is satisfied
  -
• singular controls tend
  to be optimal

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Linear PK, linear PD

• If , then
• singular controls are of order 2,

L-C condition is violated singular controls are
not optimal
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Linear PK, nonlinear PD

• Linear PK g0
• If s is strictly convex, the L-C condition is
  violated, i.e. singular controls are not optimal
• If s is strictly concave, the L-C condition is
  satisfied, i.e. singular controls tend to be
  optimal

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Example

• sigmoidal model for PD with linear PK
• convex for low concentrations
• singular not optimal
• concave at high concentrations
• singular tend to be optimal
• suggested therapy initially full dose to get the
  concentration high, then partial doses to
  maintain an effective level

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Summarizing

• Parameters of PK and geometric properties
  (convexity/concavity) of the function
  in PD determine the local optimality of
  singular controls
• Linear models for PK and PD do not change the
  optimality properties of singular controls
• Nonlinear models for PK and PD may change the
  qualitative structure of optimal solutions
  (singular controls which are non-optimal without
  PK/PD become optimal with PK/PD).

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Drug Resistance

• one of many possible mechanisms for drug
  resistance is Gene Amplification
• extra copies of genes are acquired which aid
  metabolization or removal of the drug
• Gene Deamplification loss of genes
• one copy forward gene amplification model
  (Agur and Harnevo)
• At least one of the two daughter cells in cell
  division will be an exact copy of the mother cell
  while there is a positive probability that the
  second cell undergoes gene amplification/deamplifi
  cation

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The model

• S average number of cells in sensitive
  compartment
• R average number of cells in resistant
  compartment
• - probability of a daughter cell
  of a sensitive cell to become resistant
• - probability of a daughter cell
  of a sensitive cell to become resistant
• r0 stable gene amplification
• rgt0 unstable gene amplification

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Dynamics

• Drug dosage
• no drug used / full
  dose
• aS(t) outflow of sensitive cells

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• cR(t) outflow of resistant cells
• dynamics

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Results

• Maximum Principle bang-bang and singular
  controls
• Further analysis of singular controls
• L-C condition

positive
determines optimality
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• If then the
  Legendre-Clebsch condition is violated and
  singular controls are not optimal
• If the
  Legendre-Clebsch condition is satisfied and one
  expects that singular controls will at least be
  locally optimal
• Interpretation as resistance builds up,
  singular controls become candidates for
  optimality (partial doses recommended)

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Conclusion

• Both PK/PD and developing drug resistance may
  change the qualitative structure of optimal
  solutions, hence they should be taken into
  account
• medically, partial doses, which are not optimal
  in simplified models may become optimal once this
  factors are taken into account