Modeling Tumor Growth Mathematics Clinic Prof. Lisette de Pillis Dana 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 Dana

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

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 *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 Dana

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

Simulated Model Cross-Section Grid site Tumor cell Dana

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

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

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

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” *www.bmb.psu.edu/courses/biotc/489/biointeract.htm

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

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

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

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

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

Avascular vs Vascularized Tumor 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 No Vasculature Constant Line Vasculature Tiff

Delayed vs Constant Vasculature 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Line Delayed Constant Tiff

Delayed vs Constant Vasculature 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Square Grid Delayed Constant Tiff

Delayed vs Constant Vasculature 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Hexagonal Grid Delayed Constant Tiff

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

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

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

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

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

Chemotherapy Treatments 37 MCS Low Dose Chemotherapy High Dose Chemotherapy No Treatment 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Tiff

Chemotherapy Treatments 40 MCS Low Dose Chemotherapy High Dose Chemotherapy No Treatment 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Tiff

Chemotherapy Treatments 50 MCS Low Dose Chemotherapy High Dose Chemotherapy No Treatment 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Tiff

Chemotherapy Treatments 60 MCS Low Dose Chemotherapy High Dose Chemotherapy No Treatment 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Tiff

Chemotherapy Treatments 64 MCS Low Dose Chemotherapy High Dose Chemotherapy No Treatment 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 0 Grid Sites 200 Grid Sites 200 Tiff

Chemotherapy Treatments Low Dose Chemotherapy High Dose Chemotherapy No Treatment Tiff

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

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

Questions Tiff