Models of cancer population evolution combining multidrug chemotherapy and drug resistance'

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Models of cancer population evolution combining
multidrug chemotherapy and drug resistance.

• J. SMIEJA, A. SWIERNIAK
• Dept. of Automatic Control, Silesian University
  of Technology
• Akademicka 16, 44-101 Gliwice
• Poland
• Fax [phone] e-mail jsmieja_at_ia.polsl.gli
  wice.pl

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Presentation outline

• A little bit of history how it started

• The original mathematical model

• Generalized model and its applications

• partial sensitivity of the resistant subpopulation

• multidrug protocols

• phase-specific chemotherapy and drug resistance

• Model decomposition

• Dynamical properties and optimization of
  chemotherapy protocols

• Final remarks

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Some history
Main phenomena Increase of drug resistance by
gene amplification
Harnevo L.E., Agur Z., Drug resistance as a
dynamic process in a model for multistep gene
amplification under various levels of selection
stringency, Cancer Chemother. Pharmacol., 1992,
Vol. 30, ss. 469476.
Mathematical model - branching random walk
Kimmel M., Stivers D.N., Time-continuous
branching walk models of unstable gene
amplification, Bull. Math. Biol., 1994, Vol. 56,
ss. 337357.
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Problems addressed using presented model

• Dynamical behaviour analysis transient states
  corresponding to given initial conditions

• Stabilization treatment using constant dosage
  of a drug

• Optimization finding optimal treatment
  protocols minimizing given performance index

• Optimization finding suboptimal periodical
  treatment protocols

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Drug resistance model (1)
Cells of type 0 sensitive subpopulation
a - probability ratio of the primary single
mutational event
d1
bi, di amplification and deamplification ratios
Cells of type i ? 1 drug resistant subpopulation
...
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Drug resistance model (2)
Ni(t) number of cells with i additional gene
copies responsible for drug removement and
metabolisation,li cell lifespans
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Drug resistance model (3)
ui(t) drug effect on the i-th subpopulation
(fraction of ineffective cell divisions)
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Drug resistance model (4)
Model of cancer cells evolution, taking into
account increasing drug resistance
Ni(t) number of cells with i additional
oncogene copies (increasing i means increasing
resistance to drug) ui(t) drug effect on the
sensitive subpopulation (fraction of ineffective
cell divisions), 0 ? ui(t) ? umax ? 1 li cell
lifespans bi, di amplification and
deamplification probabilities a - probability
ratio of the primary single mutational event
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Drug resistance model (5)
Model of cancer cells evolution, taking into
account increasing drug resistance
Simplifying assumptions
ui(t) 0, i gt 0

• the resistant cells are insensitive to drug's
  action

• there are no differences between parameters of
  cells of different type

li l, bi b, di d
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The simplified mathematical model
Model of cancer cells evolution, taking into
account increasing drug resistance
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A model taking into account partial sensitivity
of the resistant subpopulation
where 0 ? mi ? 1 mi are efficiency factors,
0 ? mi ? mi-1 ? 1, i  1,2,..., l-1.
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Multidrug protocols (1)
d20
a01
d32
a13
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Multidrug protocols (2)
0 ? ui(t) ? umax ? 1
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Multidrug protocols (3)
b0, b1 are efficiency factors, b0, b1 ? 1 0 ?
ui(t) ? umax ? 1
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Phase-specific control of cancer population (1)
A. Swierniak, A. Polanski, Z. Duda, M. Kimmel,
Phase-Specific Chemotherapy of Cancer
Optimisation of Scheduling and Rationale for
Periodic Protocols, Biocybernetics and
Biomedical Engineering, 16, 1997, 13-43
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Phase-specific control of the drug-sensitive
cancer population (2)
Cells of type i  0 are in the phase G1 Cells
of type i  1, are in the phase SG2M
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Phase-specific control of cancer population (3)
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1 g
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Phase-specific control of the drug-sensitive
cancer population (4)
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Problems addressed using presented model

• Dynamical behaviour analysis transient states
  corresponding to given initial conditions

• Stabilization treatment using constant dosage
  of a drug

• Optimization finding optimal treatment
  protocols minimizing given performance index

• Optimization finding suboptimal periodical
  treatment protocols

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General mathematical model (1)
a3 gt a1 gt 0
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General mathematical model (2)
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Model decomposition
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Analysis of the infinite dimensional subsystem
(1)
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Analysis of the infinite dimensional subsystem (2)
Hence, the stability conditions for the infinite
dimensional (positive) subsystem are given by
( b lt d )
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Analysis of the infinite dimensional subsystem (3)
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Analysis of the full model (1)
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Analysis of the full model (2)
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Optimal chemotherapy scheduling
0 ? ui(t) ? umax
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Transformation of the system description
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Transformation of the system description
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Optimal solution
pi(T )  1, i  0,1,...,l-1
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Numerical example
Without gene amplification
With gene amplification
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Conclusions (1)

• The model is general enough to accommodate
  different interpretations

• The decomposition of the infinite dimensional
  system enables addressing the stability problems

• Trasformation of the system description into
  integro-differantial model makes it possible to
  effectively address optimization problem

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Conclusions (2)

• For proper modelling, gene amplification must be
  taken into account

• It adds only one compartment to the basic model

• The method does not allow to design chemotherapy
  protocols however, it makes it possible to
  analyze qualitatively different approaches to
  this problem

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Future work

• Connecting current control u to drug
  concentration

• Different mechanisms of drug resistance