Transcript Presentation

Modeling Tumor Growth
Mathematics Clinic
Prof. Lisette de Pillis
Dr. Yi Jiang
Cris Cecka, Alan Davidson, Tiffany Head,
Dana Mohamed, and Liam Robinson
Los Alamos National Lab
• Operated by:
University of
California
• For: Department of
Energy
• Location: Northern
New Mexico
• Missions:
– National Security
– Scientific Research
Social Implications
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Cancer - the 2nd leading cause of
death in the U.S.
Chemotherapy harmful to patient
Better tumor models can help to
develop more effective treatments
The Main Goal
Scanning Electron Micrograph
• Given:
– Model of a tumor spheroid
• No blood vessels
• Very small
• Goals:
– To extend model to include blood vessels
• Different vasculature structures
– To study chemotherapy treatments
*http://www.vet.purdue.edu/cristal/sem-spheroid1-black.gif
Model Description
• The three cell types within the model are:
– proliferating cells: alive, can divide and grow
– quiescent cells: alive, but dormant
– necrotic cells: dead
Model Description
• Tumor described on 3 biological levels:
– Cellular:
• 3D grid of ‘sites’ created
• Cells can grow and occupy multiple sites
– Extracellular:
• Nutrients, waste, chemicals diffuse through
tumor cells
– Subcellular:
• Chemical concentrations cause the cells to
respond
Simulated Model Cross-Section
• Grid site
• Tumor cell
Initialize
*Adapted from a flow chart in: Yi Jiang
et. al. “A Multiscale Model for
Avascular Tumor Growth”
Monte Carlo Movement
Solve Diffusion Equation
Determine Protein Expression
Chemicals/Volume Favorable?
Quiescent/Necrotic
Time to Divide?
Divide into 2 Cells
Possible Cell Shedding
if on Surface
Monte Carlo
• A stochastic algorithm
• Strategy
– Make a random
change
• Find a border
• Change cell
ownership
– Calculate the
difference in energy
– Accept/Reject
change
• Boltzmann factor
Chemical Diffusion
• Chemicals the cells use in this model:
– O2, Glucose, Waste, Growth Factors, and Inhibitory Factors
• Modeling the time-dependent chemical diffusion
equation:
– Finite Difference Approximations
• yields a linear system of equations
Cell Cycle
Proliferating Cells
Quiescent Cells
GSK3b
TGFb
SCF
SMAD
P15
P27
CyCD, CDK4
P21
CyCE, CDK2
Rb
E2F
S phase
*www.bmb.psu.edu/courses/biotc/489/biointeract.htm
*Adapted from a flow chart in Yi Jiang et. al.
“A Multiscale Model for Avascular Tumor Growth”
Addition of Vasculature
• New blood vessel ‘cell’ type added:
– can occupy sites
– constant chemical concentrations
• Reasonable as the speed of the relevant chemical
diffusion is slow compared to the rate of blood
flow through the vessel.
Vasculature Structure
• Can select one of three different vasculature
structures
– Single Vein
– Grid Lattice Structure
– Hexagonal Lattice Structure
• Have been observed in biological tumors
• Adds a greater degree of flexibility to the model
• Allows for more structural options to be added
later
Extending the Monte Carlo
• Extend the J-matrix to include vasculature
• Vasculature should be static
• Other cells should not encroach upon
vasculature
• The vasculature should not grow
PDE Solver
•
Recall the time-dependent diffusion equation:
•
To solve with arbitrary boundary conditions
– We use a Backwards Euler approximation
– Stable Linear System
•
Solve linear system with Gauss-Seidel method
– Iterative method
– Stable, guaranteed convergence for our system
• Strictly (but weakly) Diagonally Dominant
Vasculature and BCs
• Treat vasculature as boundary conditions
– Can be in an arbitrary geometry
• PDE solver supports this
applying the identity iteration.
Grid Sites
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Constant Line
Vasculature
Grid Sites
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No
Vasculature
Grid Sites
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Avascular vs Vascularized Tumor
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Grid Sites
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Grid Sites
Delayed
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Grid Sites
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Constant
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Line
Grid Sites
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Delayed vs Constant Vasculature
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Grid Sites
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Grid Sites
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Delayed
Grid Sites
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Constant
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Square Grid
Grid Sites
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Delayed vs Constant Vasculature
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Grid Sites
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Grid Sites
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Delayed
Grid Sites
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Constant
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Hexagonal Grid
Grid Sites
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Delayed vs Constant Vasculature
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Grid Sites
200
Types of Chemotherapeutic Agents
• Cell Cycle Specific vs. Non Cell Cycle Specific
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Alkylating Agents
Nitrosoureas
Antimetabolites
Anthracyclines
Topoisomerase II Inhibitors
Mitotic Inhibitors
Corticosteroid Hormones
Apoptosis vs. Necrosis
• Two types of cell death:
– Apoptosis
– Necrosis
While necrosis leaves debris after cell death
occurs, apoptosis does not.
This has implications for the diffusion of
chemicals.
Drug Pharmacokinetics
Cancerboard.ab.ca, www.Canceractive.com
• Route of administration
• Dose administered
• Dosing interval
• Plasma drug concentrations
Modeling Chemotherapy
• Added cyclophosphamide as a new chemical
• Regularly scheduled doses once the vasculature is
created
• Blood plasma concentration
– Constant boundary condition during each step
– Changes from step to step to simulate AUC profile
• Stochastic model determines if cells become
apoptotic based on drug concentration
• Apoptotic cells replaced by medium
Limitations of the Model for Chemo
• Hardware constraints
• Patient toxicity
• Chemotherapy drug cocktails
Chemotherapy Treatments
37 MCS
Low Dose
Chemotherapy
Grid Sites
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0
200
Grid Sites
Grid Sites
Grid Sites
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High Dose
Chemotherapy
200
200
No Treatment
Grid Sites
200
0
Grid Sites
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Chemotherapy Treatments
40 MCS
Low Dose
Chemotherapy
Grid Sites
200
0
200
Grid Sites
Grid Sites
200
Grid Sites
0
High Dose
Chemotherapy
200
No Treatment
Grid Sites
200
0
Grid Sites
200
Chemotherapy Treatments
50 MCS
Low Dose
Chemotherapy
Grid Sites
200
0
200
Grid Sites
Grid Sites
200
Grid Sites
0
High Dose
Chemotherapy
200
No Treatment
Grid Sites
200
0
Grid Sites
200
Chemotherapy Treatments
60 MCS
Low Dose
Chemotherapy
Grid Sites
200
0
200
Grid Sites
Grid Sites
200
Grid Sites
0
High Dose
Chemotherapy
200
No Treatment
Grid Sites
200
0
Grid Sites
200
Chemotherapy Treatments
64 MCS
Low Dose
Chemotherapy
Grid Sites
200
0
200
Grid Sites
Grid Sites
200
Grid Sites
0
High Dose
Chemotherapy
200
No Treatment
Grid Sites
200
0
Grid Sites
200
Chemotherapy Treatments
No Treatment
Low Dose
Chemotherapy
High Dose
Chemotherapy
Future Work
• Chemotherapy experiments that allow the tumor to
reach a detectable size
• Inclusion of multiple chemotherapy drugs, including
cell cycle specific varieties
• Patient toxicity simulation
• Optimal control
– Treatment schedule
– Dose level
Acknowledgments
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Prof. Lisette DePillis, Advisor
Dr. Yi Jiang, Liason
Prof. Michael Raugh, Clinic Director
Los Alamos National Lab, Sponsor
Barbara Schade, Administrative Assistant
Claire Connelly, System Administrator
Questions