The Initial ModelLos Alamos provided our team with a working 3-dimensional model for simulating tumor spheroids, which are small lab-grown tumors. The model consists of a three-dimensional grid of small regions called grid sites. Tumor cells within the model grow and occupy multiple sites on the grid, with a typical tumor cell occupying between 40 and 60 grid sites. The model describes the tumor spheroids on three biological levels: the extracellular level, at which chemicals diffuse through the grid; the cellular level, at which cells grow and change volume; and the subcellular level, at which tumor cells use a protein expression model to simulate the cell cycle, divide, and change between active (proliferating), dormant (quiescent), or dead (necrotic) states.

Code Addition and OptimizationAddition of Vasculature

In order to include vasculature in the model, a new blood vessel ‘cell’ type was added that can occupy grid sites. The blood vessels are modeled with constant chemical concentrations because the speed of the relevant chemical diffusion is slow compared to the rate of blood flow through the vessel. Several vasculature structures have been modeled in order to demonstrate the flexibility of the vasculature system, which is important in order to incorporate it into the second, angiogenic, stage of the multi-stage model. Also, growth interactions between the vasculature and the cells were incorporated into the model.

Code Optimization

In order to run the most biologically significant trials, we wished to run larger simulations. However, the code with which Los Alamos provided us used large amounts of memory. We attempted to optimize the memory usage of the code. To accomplish this, we changed several data structures, and began indexing cells with pointer arithmetic instead of ID numbers. Further changes were made by refactoring large functions into smaller functions.

Diffusion With VasculatureThe addition of vasculature changes the chemical diffusion equation of the model. We rewrote the PDE solver to use the Jacobi iterative method with a backward Euler finite difference approximation. This solver accepts Dirichlet boundary conditions in any geometry.

Simulation of Chemotherapy

To simulate chemotherapy, we added a model of cyclophosphamide, a common chemotherapy drug. In this model, cyclophosphamide diffuses into the tumor from the vasculature and the boundary surrounding the tumor, and is slowly metabolized by the cells. We use a probabilistic model to determine if a cell is killed by chemotherapy, based on the concentration of cyclophosphamide within the cell and a reference concentration. If a cell is killed by chemotherapy, it becomes apoptotic (undergoes biologically programmed cell death) and after a prespecified amount of time will revert back to being a part of the medium surrounding the tumor.

2005-2006 Mathematics Department Clinic

Vascularized Tumor Growth and TreatmentVascularized Tumor Growth and Treatment

Tumor Modeling OverviewDeveloping more accurate and efficient tumor models improves our understanding of cancer and allows more effective and efficient treatments to be developed.

Tumor Growth

Tumor growth occurs in three stages: diffusion-limited growth, angiogenesis, and vascularized growth. In the first stage, no blood vessel structure exists within the tumor, and the size that the tumor can attain is limited by chemical diffusion. When the second stage, angiogenesis, is entered, blood vessels grow within the tumor, allowing for new methods of nutrient delivery and waste removal. Finally, in the vascularized growth stage, the tumor can grow larger due to the added influx of nutrients from the newly developed blood vessel system.

Three-Stage Model

Los Alamos is developing a three-stage model of tumor growth. Although many researchers have developed tumor models that address aspects of tumor growth related to particular stages of growth, little work has been done to develop a multi-stage model that encompasses all three stages. Los Alamos is developing such a model, and the purpose of this Clinic project is to develop the third, vascularized stage. Our contribution will be incorporated into the larger model developed by Los Alamos.

Chemotherapy

After including a blood vessel system in the tumor model, a study of various chemotherapy treatments was conducted. Since tumor lethality greatly increases once angiogenesis has occurred, modeling a tumor that has passed this stage and conducting treatment research has the potential to help many cancer patients.

Picture of a lab-grown tumor spheroid. The model that our team was provided with was developed to describe such spheroids, which are avascular, i.e., they lack blood vessels.

Vascularized Tumor Growth & Effects of Chemotherapy

We ran two experiments using different levels of chemotherapy. Each experiment was conducted on a linear vasculature. The high dose is 2.0 grams of the chemotherapeutic agent and the low dose is 1.2 grams of the chemotherapeutic agent.

The chemotherapeutic agent is added to the vasculature at the same time the vasculature is added. Immediately, a majority of the proliferating cells become apoptotic (they begin biologically programmed cell death) and soon dissolve into the medium. After the chemotherapy dose stops, tumor growth immediately resumes as we would expect.

Results

The number of proliferating cells after simulating 48 hours with no chemotherapy ranges from 6196 to 6652 cells and is 3121 with a low dose of chemotherapy and 1204 with a high dose of chemotherapy. The variation between the chemotherapy and non-chemotherapy runs is significant. The quiescent cell counts have an even larger disparity: the non-chemotherapy runs ranging from 6331 cells to 6557 cells, while a low dose chemotherapy count is 1533 and a high dose chemotherapy count is 256.

DeliverablesWe will deliver the following to Los Alamos:

Optimized model code including the addition of a vasculature and a model of chemotherapy

A final report detailing code development and analysis

AcknowledgmentsLos Alamos Liaison:Yi Jiang

Team Members:Cris Cecka, Alan Davidson, Tiffany Head (Team Leader Fall 2005), Dana Mohamed (Team Leader Spring 2006), Liam Robinson

Faculty Advisor:Lisette de Pillis

A two-dimensional cross-section of waste concentration in a vascularized tumor. The contours of the vasculature, proliferating cells, and quiescent cells can also be seen.

Two-dimensional cross-sections of tumors undergoing different chemotherapy treatments showing cell types. Dark blue is medium, bright blue is proliferating cells, light blue is quiescent cells, yellow is apoptotic cells and red is vasculature.

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A two-dimensional cross-section of a modeled tumor in the diffusion-limited growth stage