1. Modeling Tumor Growth
Mathematics Clinic
Prof. Lisette de Pillis
Dana
Dr. Yi Jiang
Cris Cecka, Alan Davidson, Tiffany Head, Dana Mohamed, and Liam Robinson

2. Los Alamos National Lab
Operated by: University of California
For: Department of Energy
Location: Northern New Mexico
Missions:
National Security
Scientific Research
Dana

3. Social Implications
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
Dana

4. The Main Goal Given: Goals: Model of a tumor spheroid
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
Dana *

5. 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
Dana

6. 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
Dana

7. Simulated Model Cross-Section
Grid site
Tumor cell
Dana

8. Solve Diffusion Equation
*Adapted from a flow chart in: Yi Jiang et. al. “A Multiscale Model for Avascular Tumor Growth”
Initialize
Monte Carlo Movement
Solve Diffusion Equation
Determine Protein Expression
Chemicals/Volume Favorable?
Quiescent/Necrotic
Dana
Time to Divide?
Possible Cell Shedding
if on Surface
Divide into 2 Cells

9. 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
Alan

10. 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
We write “Glucose” and “Waste” cause we don’t care what they actually are, we just diffuse them.
Alan

11. Cell Cycle Proliferating Cells Quiescent Cells GSK3b TGFb SCF SMAD P15
CyCD, CDK4
CyCE, CDK2
Alan Rb
E2F
S phase
*Adapted from a flow chart in Yi Jiang et. al.
“A Multiscale Model for Avascular Tumor Growth” *

12. 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.
Alan

13. 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
Alan

14. 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
Cris

15. PDE Solver We use a Backwards Euler approximation Stable Linear System
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
Cris

16. Vasculature and BCs Treat vasculature as boundary conditions
Can be in an arbitrary geometry
PDE solver supports this
applying the identity iteration.
Cris

17. Avascular vs Vascularized Tumor
Grid Sites
Grid Sites
Grid Sites
Grid Sites
No Vasculature
Constant Line Vasculature
Tiff

18. Delayed vs Constant Vasculature
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Line
Delayed
Constant
Tiff

19. Delayed vs Constant Vasculature
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Square Grid
Delayed
Constant
Tiff

20. Delayed vs Constant Vasculature
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Hexagonal Grid
Delayed
Constant
Tiff

21. Types of Chemotherapeutic Agents
Cell Cycle Specific vs. Non Cell Cycle Specific
Alkylating Agents
Nitrosoureas
Antimetabolites
Anthracyclines
Topoisomerase II Inhibitors
Mitotic Inhibitors
Corticosteroid Hormones
Liam

22. 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.
Liam

23. Drug Pharmacokinetics
Cancerboard.ab.ca,
Route of administration
Dose administered
Dosing interval
Plasma drug concentrations
Liam

24. 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
Liam

25. Limitations of the Model for Chemo
Hardware constraints
Patient toxicity
Chemotherapy drug cocktails
Liam

26. Chemotherapy Treatments
37 MCS
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Tiff

27. Chemotherapy Treatments
40 MCS
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Tiff

28. Chemotherapy Treatments
50 MCS
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Tiff

29. Chemotherapy Treatments
60 MCS
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Tiff

30. Chemotherapy Treatments
64 MCS
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Grid Sites
Tiff

31. Chemotherapy Treatments
Low Dose Chemotherapy
High Dose Chemotherapy
No Treatment
Tiff

32. 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
Tiff

33. Acknowledgments 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
Tiff

34. Questions
Tiff