Modeling Imatinib-Treated Chronic Myeloid Leukemiarvbalan/TEACHING/AMSC663Fall2015/β¦Β Β· BCR-ABL1...
Transcript of Modeling Imatinib-Treated Chronic Myeloid Leukemiarvbalan/TEACHING/AMSC663Fall2015/β¦Β Β· 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
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IntroductionCML β cancer of the bloodβ¦ Genetic mutation in hematopoietic
stem cells β Philadelphia Chromosome (Ph)
β¦ Increase tyrosine kinase activity allows for
uncontrolled stem cell growth
Treatment ββ¦ Imatinib: tyrosine kinase inhibitor
β¦ Controls population of mutated cells in two ways
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Figure: Chronic Myelogenous Leukemia Treatment. National Cancer Institute. 21 Sept. 2015. Web.
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Cell State Diagram (Roeder et al., 2006)
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Stem cells⦠Non-proliferating (A)
β¦ Proliferating (Ξ©)
Precursor cells
Mature cells
Circulation between A and Ω based on cellular affinity⦠High affinity: likely to stay in/switch to A
β¦ Low affinity: likely to stay in/switch to Ξ©
Assume fixed and known lifespans for Precursor and Mature cells
Figures: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
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Project GoalsMathematically model clinically observed phenomena of three non-interacting cell populations⦠Nonleukemia cells (Ph-)
β¦ Leukemia cells (Ph+)
β¦ Imatinib-affected leukemia cells
Three model types based on cell state diagram⦠Model 1: Agent Based Model (Roeder et al., 2006)
β¦ Model 2: System of Difference Equations (Kim et al., 2008)
β¦ 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 Equations⦠Time, affinity and cell cycle discretized
β¦ Transitions between A and Ξ© given by binomial distributions
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Figures: Kim et al. in Bull. Math. Biol. 70(3), 728-744 2008
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Modeling CML Genesis and TreatmentHealthy cells (Ph-)β¦ Equations same as original
β¦ Transition probabilities governed by sigmoidal functions ππΌ/π with corresponding Ph- parameters
Leukemic cells (Ph+)β¦ Equations for A, P and M compartments remain the same as the original
β¦ During treatment, Ξ© cells may become Imatinib affected or die at each time step
β¦ Ξ©π,π+/π
π‘ = Ξ©π,π+ π‘ β Ξ©π,π
+/πΌπ‘
β¦ Transition probabilities governed by sigmoidal functions ππΌ/π with corresponding Ph+ parameters
Affected cells (Ph+/A)β¦ Equations for A, P and M compartments remain the same as the original
β¦ Ξ© cells may undergo apoptosis at each time step
β¦ Transition probabilities governed by sigmoidal functions ππΌ/π with corresponding Ph+/A parameters
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β πΏπ
ππ‘π = ππ](https://reader035.fdocuments.us/reader035/viewer/2022071106/5fe064ce4522dc28f27bbad2/html5/thumbnails/7.jpg)
Implementation and SimulationImplemented in Matlab. Vectorized difference equations for efficiency.
Initialize Ph- population: Ω0,32(0) = 1For t = 1: Steady State⦠Update Ph- population
Initialize Ph+ population: Ω0,32(0) = 1For t = 1: Genesis⦠Update Ph- population
β¦ Update Ph+ population
For t = 1: Treatment⦠Update Ph- population
β¦ Update Ph+ population
β¦ Update Ph+/A population
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β πΏπ
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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:
Ξπ π‘ ~ π΅ππ Ξπ π‘ , π Ξ© π‘ , πβππ
Ξ¨π,π π‘ ~ π΅ππ Ξ©π,π π‘ , πΌ Ξ π‘ , πβππ π = 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
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β πΏπ
ππ‘π = ππ](https://reader035.fdocuments.us/reader035/viewer/2022071106/5fe064ce4522dc28f27bbad2/html5/thumbnails/9.jpg)
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
π΅π π‘ = Ξπ π‘ β π Ξ© π‘ , πβππ
Ξ¨π,π π‘ = Ξ©π,π π‘ β πΌ Ξ π‘ , πβππ
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
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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
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Results: Treatment
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BCR-ABL1 Ratio for duration of treatment (400 days)
π΅πΆπ β π΄π΅πΏ π ππ‘ππ =πππ‘π’ππ πβ+πππππ
πππ‘π’ππ πβ+πππππ +2βπππ‘π’ππ πββπππππ
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 (
%)
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Model 1: Roeder et al., 2006Single cell-based stochastic model
Complexity based on number of agents⦠~105 cells
β¦ 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 PDEs⦠Consider the cell state system as a function of three
internal clocks
β¦ Real time (t)
β¦ Affinity (a)
β¦ Cell cycle (c)
β¦ Each cell state can be represented as a function of 1-3 of these variables
Numerical Simulation β¦ Explicit solvers
β¦ Upwinding
β¦ Composite trapezoidal rule
β¦ First order time discretization
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Figures: Kim et al. in Bull. Math. Biol. 70(3), 1994-2016 2008
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β πΏπ
ππ‘π = ππ](https://reader035.fdocuments.us/reader035/viewer/2022071106/5fe064ce4522dc28f27bbad2/html5/thumbnails/14.jpg)
Project Schedule Phase 1β¦ Implement difference equation model
β¦ Improve efficiency and validate
Phase 2: End of December⦠Implement ABM
β¦ Improve efficiency and validate
Phase 3: January β mid Februaryβ¦ Implement basic PDE method and validate on simple test problem
Phase 4: mid February β Aprilβ¦ Apply basic method to CML - Imatinib biology and validate
β¦ 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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