Step by Step, Math Models Unlock Secrets of Cancer Biology

October 21, 2006

Michelle Sipics

Cancer is one of the most intensely studied areas of biomedical research. In the U.S., the National Cancer Institute---the segment of the National Institutes of Health responsible for cancer research---has requested $6 billion for 2007, 87% of which is to be allocated directly to some form of research. (For comparison, the 2007 budget for all 27 components of NIH would be $28 billion.) One of the most important goals of cancer research, at NCI and elsewhere, is to understand the nature and development of tumors. An improved understanding of the complexity and growth stages of tumors could allow doctors to make better treatment decisions as well as provide patients with more accurate prognoses.

Yi Jiang of the Los Alamos National Laboratory is using mathematics in an attempt to achieve just such an understanding. Along with colleagues from Los Alamos, MIT, and the University of Tennessee, Jiang is modeling the microenvironment of tumors at three scales. She described the group's model in an invited talk at the SIAM Annual Meeting in Boston.

In the earliest---avascular---stage of tumor growth, a tumor does not have its own network of blood vessels, and must obtain nutrients from the environment's supply via diffusion. Previous avascular tumor models investigated the role of external nutrients in a tumor's growth rate, as well as the role of growth factors and other aspects of tumor development. Building on earlier efforts, Jiang's group also addresses the issue of tissue mechanics and integrates multiple processes at different scales.

Starting with a single tumor cell, the group's model produces an avascular tumor that mimics results of mouse mammary tumor experiments. The researchers model growth at the cellular, subcellular, and extracellular levels using multiple mathematical concepts and techniques. With parameters derived from experimental data, they base their efforts on tumor spheroids: aggregates of tumor cells that receive nutrients through diffusion from the surface.

Previous and still prevalent approaches for modeling spheroid growth frequently rely on one or more ordinary differential equations; in such models, the ODEs themselves model tumor growth. The multiscale model developed by Jiang and her group, by contrast, has three integrated parts, with reaction–diffusion partial differential equations used as an element of the overall attempt to model avascular tumor growth.

Running the model begins at the cellular level, where the researchers use a lattice Monte Carlo model to describe cellular dynamics, including proliferation, adhesion, and viability.

"The Monte Carlo procedure is one way to ‘integrate the equation,' so to speak," says Jiang, "or to evolve the cell configuration on the lattice in our cellular model."

Based on the cellular Q Potts model---which was developed by Francis Graner and Jiang's PhD adviser, James Glazier---the cellular-level model partitions a three-dimensional space into domains of cells and cell mediums, and then assigns each cell a unique identification number, which occupies the lattice sites within the cell domain.

"The cellular Potts model has proved to be a good model for cell-level dynamics," says Jiang. "[It] can retain the individuality of each cell and keep track of their fates, and at the same time each cell is simple enough that it's possible to model the tissue-level dynamics."

From that point, a standard Monte Carlo technique selects a random lattice site and changes its identification number to the value of one of its neighbors---similar to a more complex and backward version of the strategy board game Reversi, or Othello. The probability that the change will be accepted is dependent on the energy difference resulting from the change and the effective cell temperature.

For continuation to the extracellular level, Jiang explains, the model re-quires some simplifying assumptions, such as constant diffusion coefficients and chemically homogeneous cells. These and other assumptions allow the equations for the chemicals to be solved on a coarser lattice than the lattice for cells, the group explains in their paper ("A Multiscale Model for Avascular Tumor Growth," Biophysical Journal, December 2005).

The diffusion–reaction equation for the model, based on the chemical diffusion and metabolism, is generic for all of the chemicals in the model. The extracellular chemical equations output the local concentrations of growth and inhibitory factors, which in turn influence protein expression and, by doing so, affect the state of cell proliferation.

On the subcellular level, the researchers use a Boolean network divided into stages to simulate the regulation of proteins that control the cell cycle. "At a given time corresponding to a given stage of the cell cycle, the corresponding proteins are turned on or off with a probability," Jiang explains. That probability is dependent on the local growth and inhibitory factors.

During what Jiang refers to as the "G1 phase," the results of the Boolean network determine whether the cell will become quiescent or die, or whether it will move to the next phase of the cell cycle, the "S phase." From that point on, proliferating cells either fulfill the division requirements and divide into two cells, or simply progress through the cell cycle. The model then returns to its starting point and repeats the iterations.

Simulating about a month of tumor growth, from a single cell to millions of tumor cells, initially took about one day, Jiang says. But she points out that the time is shrinking: "It's now taking only a few hours to grow from a single cell to a tumor reaching its saturation size." The research team uses a package called BoxMG, developed by Los Alamos scientists in the same group as Jiang. "The package is a parallelized multigrid solver for PDEs, effective for Laplacian-type equations, which is exactly what we have for the tumor model," she explains. Further advances may lie ahead once the group succeeds in integrating the Potts model (which, with only local interactions, parallelizes easily) with the BoxMG package.

The group studied tumor cells from mice, and not humans, but Jiang believes that once their results are validated by experimental data, the potential applications and implications of the research are vast.

"The model can be used to predict tumor growth in different chemical environments that haven't been tried in experiments, including adding drugs to see how well the anti-tumor chemotherapy affects the tumor growth dynamics," she says. That extends to the potential for modeling the effects of varying dosages, of varying schedules for drug administration---and the list can go on. "But more important," she says, "is that during the validation process of the model, we would learn something about the cancer biology, and improve our understanding of the mechanisms of cancer progression."

Jiang's interests are not limited to avascular tumor growth, the initial stage of cancer. She is also studying the subsequent stages of cancer development: angiogenesis, vascular tumor growth, and tumor invation.

"The future plan would be, once these components are developed, to combine all three into a coherent model of cancer development," she says. The resulting model could cover development from initiation to metastasis---and further expand the potential applications of the research.

Michelle Sipics is a contributing editor at SIAM News.

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