Modeling Imatinib-Treated Chronic Myeloid Leukemiarvbalan/TEACHING/AMSC663Fall2015/… · BCR-ABL1...

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Modeling Imatinib-Treated Chronic Myeloid Leukemia Mid Year Presentation Cara Peters [email protected] Advisor: Dr. Doron Levy [email protected] Department of Mathematics Center for Scientific Computing and Mathematical Modeling

Transcript of Modeling Imatinib-Treated Chronic Myeloid Leukemiarvbalan/TEACHING/AMSC663Fall2015/… · 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 – cancer of the bloodβ—¦ Genetic mutation in hematopoietic

stem cells – Philadelphia Chromosome (Ph)

β—¦ Increase tyrosine kinase activity allows for

uncontrolled stem cell growth

Treatment –◦ 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 cellsβ—¦ Non-proliferating (A)

β—¦ Proliferating (Ξ©)

Precursor cells

Mature cells

Circulation between A and Ξ© based on cellular affinityβ—¦ 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 populationsβ—¦ Nonleukemia cells (Ph-)

β—¦ Leukemia cells (Ph+)

β—¦ Imatinib-affected leukemia cells

Three model types based on cell state diagramβ—¦ 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 Equationsβ—¦ 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 𝑓𝛼/πœ” 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‘˜,𝑐+/𝑅

𝑑 = Ξ©π‘˜,𝑐+ 𝑑 βˆ’ Ξ©π‘˜,𝑐

+/𝐼𝑑

β—¦ Transition probabilities governed by sigmoidal functions 𝑓𝛼/πœ” 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 𝑓𝛼/πœ” with corresponding Ph+/A parameters

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Implementation and SimulationImplemented in Matlab. Vectorized difference equations for efficiency.

Initialize Ph- population: Ξ©0,32(0) = 1For t = 1: Steady Stateβ—¦ Update Ph- population

Initialize Ph+ population: Ξ©0,32(0) = 1For t = 1: Genesisβ—¦ Update Ph- population

β—¦ Update Ph+ population

For t = 1: Treatmentβ—¦ Update Ph- population

β—¦ Update Ph+ population

β—¦ 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:

Ξ’π‘˜ 𝑑 ~ 𝐡𝑖𝑛 Ξ‘π‘˜ 𝑑 , πœ” Ξ© 𝑑 , π‘’βˆ’π‘˜πœŒ

Ξ¨π‘˜,𝑐 𝑑 ~ 𝐡𝑖𝑛 Ξ©π‘˜,𝑐 𝑑 , 𝛼 Ξ‘ 𝑑 , π‘’βˆ’π‘˜πœŒ 𝑐 = 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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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‘˜ 𝑑 = Ξ‘π‘˜ 𝑑 βˆ— πœ” Ξ© 𝑑 , π‘’βˆ’π‘˜πœŒ

Ξ¨π‘˜,𝑐 𝑑 = Ξ©π‘˜,𝑐 𝑑 βˆ— 𝛼 Ξ‘ 𝑑 , π‘’βˆ’π‘˜πœŒ

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)

𝐡𝐢𝑅 βˆ’ 𝐴𝐡𝐿 π‘…π‘Žπ‘‘π‘–π‘œ =π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘’ π‘ƒβ„Ž+𝑐𝑒𝑙𝑙𝑠

π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘’ π‘ƒβ„Ž+𝑐𝑒𝑙𝑙𝑠+2βˆ—π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘’ π‘ƒβ„Žβˆ’π‘π‘’π‘™π‘™π‘ 

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 agentsβ—¦ ~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 PDEsβ—¦ 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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Project Schedule Phase 1β—¦ Implement difference equation model

β—¦ Improve efficiency and validate

Phase 2: End of Decemberβ—¦ Implement ABM

β—¦ Improve efficiency and validate

Phase 3: January – mid Februaryβ—¦ Implement basic PDE method and validate on simple test problem

Phase 4: mid February – 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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