Because cancer is complicated system with many nonlinear agents, its emergence and proliferation continues to plague the public health community. “Cancer turns out to be a really complex systems problem,” Paul Macklin of Indiana University said. “You have processes going on at all different scales, and at the molecular, single-cell, and systems level.” The spread of cancerous cells throughout the body involves single-cell behaviors, cell-to-cell communication, physics-imposed constraints, and systems of systems; many of these systems become dysregulated in cancer patients. Mathematical modeling can help researchers better understand a system’s overall function and develop more targeted treatment plans. “The goal of a clinician is to tinker with the system and get a better outcome,” Macklin said. “Our goal as scientists, engineers, and mathematicians is to determine how we can engineer this at the systems level.”
Hypoxia—during which the cells comprising the body’s tissues fail to receive an adequate oxygen supply—contributes to invasion and metastasis in breast cancer patients. Most breast cancers are hypoxic, as tumor tissues have much lower oxygen levels than healthy tissues. For normal breast tissue, pO2 is about 65 mmHg; cancerous tissue has a pO2 level of 10 mmHg or less. Hypoxia also drives a plethora of phenotype changes. “Lots of things happen when you get to low oxygen environments,” Macklin said. For example, hypoxic cells experience increased motility, glycolysis, and acidosis. They are also less proliferative (divisive) and adhesive.
During a minisymposium presentation at the 2018 SIAM Conference on the Life Sciences, currently taking place in Minneapolis, Minn., Macklin used an agent-based modeling framework called PhysiCell to explore the role of hypoxia in breast cancer propagation. Daniele Gilkes’ lab, which is part of the Breast and Ovarian Cancer Program at the Johns Hopkins School of Medicine, employs a novel marker to track hypoxia that also benefits Macklin’s research. In Gilkes’ lab, all healthy cells emit a red fluorescent protein (RFP). But after their pO2 levels drop below 10 mmHg—a sure sign of hypoxia—the RFP becomes a green fluorescent protein (GFP). This color change is permanent. “When you look at cancer cells, there’s a lot of green,” Macklin said.
PhysiCell simulations reveal the growth of a hypoxic plume.
He presented sample observational data containing both red and green lung metastases. While Macklin and his team expected to observe green cells at the perinecrotic boundary, they were surprised to detect them around the edges of the tumors as well — an indication that hypoxic cells were somehow escaping. When an invasive cell successfully metastasizes, it must leave the primary tumor, arrive at a new location, and colonize that location by reverting to a non-motile, proliferative phenotype. This observation led Macklin to the following research questions:
- Are red cells ever motile?
- Are all green cells motile, or just some of them?
- How long do cells keep their motile phenotype?
- Do cells act independently, or do they coordinate?
Addressing these questions requires a diverse computational toolbox. To account for diffusion, Macklin and his team designed BioFVM, a finite volume method for biological problems that simulates three-dimensional biotransport. “We use this for the chemical part of our environment,” Macklin said. “It’s all open-source.”
In addition to BioFVM, Macklin also spearheaded the development of PhysiCell, an open-source multicellular simulator that uses C++ code to simulate 106 or more cells in three dimensions. Significant features include off-lattice cell positions, mechanics-based cell movement, cell processes like cycling and motility, signal-dependent phenotypes, and the ability to attach custom data and functions on a cell-by-cell basis. “If you put these things together,” said Macklin of BioFVM and PhysiCell, “you can simulate these cells in a chemical environment.”
He and his colleagues use PhysiCell to model the shift from red to green fluorescence—non-hypoxic to hypoxic—in genes and proteins. “We’re bootstrapping to explore lots of ideas in two-dimensional simulations,” he said. “If something is interesting we’ll go back and do it in three dimensions. We can see the same behaviors in three dimensions, but it takes different parameter values to get there. That’s worth keeping in mind.”
While a gene switch from healthy to hypoxic takes only a few hours, several days may pass before the fluorescence reflects the change with a corresponding color shift. This time delay means that Macklin must account for lag time when studying green hypoxic cells. “There’s a bit of a disconnect between what you observe and the underlying reality,” he said.
Macklin employed various models to investigate the persistence of phenotypic switching, with each model producing a more desirable result. For example, when modeling permanent phenotype change, green hypoxic cells managed to tumble through the red cells and escape the designated region. “This accomplishes one observation,” Macklin said. “Green cells reach the outside. They’re proliferating, but they’re not forming colonies.” Although simulations of phenotypic persistence did yield the anticipated microcolonies, the desired necrotic core was still absent. Another model depicts what appears to be collective motion as green cells exit the tumor, but instead is purely mechanics; because cells are motile, if one cell discovers a gap the others can more easily find that same gap. The growth of a hypoxic plume—visible both in vivo and clinically—is present in an additional simulation. “Novel imaging reveals a hidden structure, and computational models can explain that structure,” Macklin said. “Because we have a combination of novel imaging techniques, we can see things that we didn’t see before.”
As a final step, Macklin modeled phenotypic persistence in so-called “leader” cells. In this simulation, 10 percent of cells are designated leaders — and only the leaders migrate. “This is getting us closer to the real story,” he said. Macklin continued to conduct preliminary modeling of the extracellular matrix, which directly impacts cells, and observed that leader cells are anisotropic (direction-dependent). He also formulated hypothesis testing as an optimization problem to simultaneously simulate hundreds or thousands of hypothesis sets and evaluate them against an error metric. This will permit future high-throughput investigation, which may reveal further details about coordinated phenotypic switching and lend insight into cancer’s ultimate eradication.
|| Lina Sorg is the associate editor of SIAM News.