modeling imatinib-treated chronic myeloid leukemiarvbalan/teaching/amsc663fall2015/โฆย ยท bcr-abl1...
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Modeling Imatinib-Treated Chronic Myeloid Leukemia
Mid Year Presentation
Cara [email protected]
Advisor: Dr. Doron [email protected]
Department of MathematicsCenter for Scientific Computing and Mathematical Modeling
IntroductionCML โ cancer of the bloodโฆ Genetic mutation in hematopoietic
stem cells โ Philadelphia Chromosome (Ph)
โฆ Increase tyrosine kinase activity allows for
uncontrolled stem cell growth
Treatment โโฆ Imatinib: tyrosine kinase inhibitor
โฆ Controls population of mutated cells in two ways
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Figure: Chronic Myelogenous Leukemia Treatment. National Cancer Institute. 21 Sept. 2015. Web.
Cell State Diagram (Roeder et al., 2006)
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Stem cellsโฆ Non-proliferating (A)
โฆ Proliferating (ฮฉ)
Precursor cells
Mature cells
Circulation between A and ฮฉ based on cellular affinityโฆ High affinity: likely to stay in/switch to A
โฆ Low affinity: likely to stay in/switch to ฮฉ
Assume fixed and known lifespans for Precursor and Mature cells
Figures: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
Project GoalsMathematically model clinically observed phenomena of three non-interacting cell populationsโฆ Nonleukemia cells (Ph-)
โฆ Leukemia cells (Ph+)
โฆ Imatinib-affected leukemia cells
Three model types based on cell state diagramโฆ Model 1: Agent Based Model (Roeder et al., 2006)
โฆ Model 2: System of Difference Equations (Kim et al., 2008)
โฆ Model 3: PDE (Kim et al., 2008)
Parameter values based on clinical data
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Model 2: Kim et al., 2008System of Deterministic Difference Equationsโฆ Time, affinity and cell cycle discretized
โฆ Transitions between A and ฮฉ given by binomial distributions
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Figures: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
Modeling CML Genesis and TreatmentHealthy cells (Ph-)โฆ Equations same as original
โฆ Transition probabilities governed by sigmoidal functions ๐๐ผ/๐ with corresponding Ph- parameters
Leukemic cells (Ph+)โฆ Equations for A, P and M compartments remain the same as the original
โฆ During treatment, ฮฉ cells may become Imatinib affected or die at each time step
โฆ ฮฉ๐,๐+/๐
๐ก = ฮฉ๐,๐+ ๐ก โ ฮฉ๐,๐
+/๐ผ๐ก
โฆ Transition probabilities governed by sigmoidal functions ๐๐ผ/๐ with corresponding Ph+ parameters
Affected cells (Ph+/A)โฆ Equations for A, P and M compartments remain the same as the original
โฆ ฮฉ cells may undergo apoptosis at each time step
โฆ Transition probabilities governed by sigmoidal functions ๐๐ผ/๐ with corresponding Ph+/A parameters
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Implementation and SimulationImplemented in Matlab. Vectorized difference equations for efficiency.
Initialize Ph- population: ฮฉ0,32(0) = 1For t = 1: Steady Stateโฆ Update Ph- population
Initialize Ph+ population: ฮฉ0,32(0) = 1For t = 1: Genesisโฆ Update Ph- population
โฆ Update Ph+ population
For t = 1: Treatmentโฆ Update Ph- population
โฆ Update Ph+ population
โฆ Update Ph+/A population
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Results: Steady State Profile
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Simulation of healthy cell population for 1 year
Number of cells that transfer between stem cell compartments at time t given by:
ฮ๐ ๐ก ~ ๐ต๐๐ ฮ๐ ๐ก , ๐ ฮฉ ๐ก , ๐โ๐๐
ฮจ๐,๐ ๐ก ~ ๐ต๐๐ ฮฉ๐,๐ ๐ก , ๐ผ ฮ ๐ก , ๐โ๐๐ ๐ = 0, . . 31
Figure: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
0 20 40 60 80 100 120 1400
100
200
300
400Steady State Profiles for Ph- stem cells
k (affinity a=e-k)
Num
ber
of
Cells
A
Results: Steady State Profile
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Simulation of healthy cell population for 1 year
Mean of binomial random variable used to smooth curves
๐ต๐ ๐ก = ฮ๐ ๐ก โ ๐ ฮฉ ๐ก , ๐โ๐๐
ฮจ๐,๐ ๐ก = ฮฉ๐,๐ ๐ก โ ๐ผ ฮ ๐ก , ๐โ๐๐
Figure: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
0 20 40 60 80 100 1200
100
200
300
400
k (affinity a=e-k)
Num
ber
of
cells
Steady State Profile for Ph- Stem Cells
A
Results: CML GenesisMature cell progression for Ph- and Ph+ populations over 15 year span.
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Figure: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
0 5 10 150
2
4
6
8
10
12
14
16x 10
10
Time (years)
Num
ber
of
cells
Healthy
Leukemic
Results: Treatment
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BCR-ABL1 Ratio for duration of treatment (400 days)
๐ต๐ถ๐ โ ๐ด๐ต๐ฟ ๐ ๐๐ก๐๐ =๐๐๐ก๐ข๐๐ ๐โ+๐๐๐๐๐
๐๐๐ก๐ข๐๐ ๐โ+๐๐๐๐๐ +2โ๐๐๐ก๐ข๐๐ ๐โโ๐๐๐๐๐
Figure: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
0 50 100 150 200 250 300 350 400-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Time (days)
BC
R-A
BL1 r
atio (
%)
Model 1: Roeder et al., 2006Single cell-based stochastic model
Complexity based on number of agentsโฆ ~105 cells
โฆ Down-scaled to 1/10 of realistic values
Validate using figures from Kim et al., 2008
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Model 3: Kim et al., 2008Transform model into a system of first order hyperbolic PDEsโฆ Consider the cell state system as a function of three
internal clocks
โฆ Real time (t)
โฆ Affinity (a)
โฆ Cell cycle (c)
โฆ Each cell state can be represented as a function of 1-3 of these variables
Numerical Simulation โฆ Explicit solvers
โฆ Upwinding
โฆ Composite trapezoidal rule
โฆ First order time discretization
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Figures: Kim et al. in Bull. Math. Biol. 70(3), 1994-2016 2008
Project Schedule Phase 1โฆ Implement difference equation model
โฆ Improve efficiency and validate
Phase 2: End of Decemberโฆ Implement ABM
โฆ Improve efficiency and validate
Phase 3: January โ mid Februaryโฆ Implement basic PDE method and validate on simple test problem
Phase 4: mid February โ Aprilโฆ Apply basic method to CML - Imatinib biology and validate
โฆ Test models with clinical data
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ReferencesRoeder, I., Horn, M., Glauche, I., Hochhaus, A., Mueller, M.C., Loeffler, M., 2006. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nature Medicine. 12(10): pp. 1181-1184
Kim, P.S., Lee P.P., and Levy, D., 2008. Modeling imatinib-treated chronic myelogenous leukemia: reducing the complexity of agent-based models. Bulletin of Mathematical Biology. 70(3): pp. 728-744.
Kim, P.S., Lee P.P., and Levy, D., 2008. A PDE model for imatinib-treated chronic myelogenous leukemia. Bulletin of Mathematical Biology. 70: pp. 1994-2016.
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Thank you
Questions?
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