Equations Against Cancer: Using Math to Predict a Tumor's Path

Newswise — Hassan Fathallah-Shaykh, M.D., Ph.D., believes that math can transform medicine, and he has the numbers to prove it. In the clinic, this UAB neurologist specializes in treating brain tumors. In his lab at the Comprehensive Cancer Center, Fathallah-Shaykh, who is also a professor of mathematics at UAB, wields equations as well as petri dishes. His mathematical models of cancer behavior are offering new insights on tumor growth. Eventually, they could be used to personalize treatment based on the unique characteristics of each patient’s cancer cells and anatomy.

Fathallah-Shaykh is one of a growing number of researchers worldwide exploring the field of mathematical biology, which “uses mathematical tools to generate models of biological problems,” he said. Building mathematical models based on the current understanding of a disease, for example, allows researchers to “test whether the assumptions are accurate,” Fathallah-Shaykh said.

Models can also be used “to test a treatment strategy, understand why it fails or works, and optimize therapy,” he added. The results of these tests can also generate new insights and hypotheses that can be investigated in the laboratory. “None of these goals can be achieved by traditional methods,” Fathallah-Shaykh said.

Model Behavior

Working with colleagues at the University of Bordeaux, and UAB graduate student Elizabeth Scribner, Fathallah-Shaykh has created an elegant model of the aggressive brain cancer glioblastoma multiforme (GBM). It produces simulations on the scale of clinical MRI scans, so that its predictions can be tested directly against patient data. In a paper published on Dec. 15 in PLOS ONE, the researchers demonstrated that their model can reproduce the typical GBM growth patterns seen on patient scans. They also revealed its value as a research tool.

The model predicted a previously unknown pattern of tumor growth in patients with recurrent GBM treated with the anti-angiogenesis drug bevacizumab. This growth, powered by a cycle of proliferation and brain invasion, is characterized by an expanding area of invasive cells and dead cells known as necrosis, the researchers say. A subsequent search of 70 patient MRI scans by the researchers turned up the same pattern in 11 cases.

That pattern explains the disappointing results of recent Phase III clinical trials of anti-angiogenesis therapies against GBM, the researchers say. Anti-angiogenesis drugs attempt to kill tumors by preventing them from growing new blood vessels. But the model demonstrated how GBM cells can flee from the oxygen-depleted treatment area — and quickly begin expanding again as soon as therapy stops or the tumor becomes resistant to the drugs. (For more on the model and these findings, see “SimTumor.”)

“We’ve shown that we can predict new insights on cancer behavior,” Fathallah-Shaykh said. The results have already spurred Fathallah-Shaykh to pursue new therapies in his lab to limit tumor mobility. Ultimately, the researchers hope to use their model to personalize therapy to the unique characteristics of a patient’s tumor. They could do that by analyzing the existing growth pattern of a tumor and building that into the model’s parameters. Then they could simulate its future behavior on a virtual MRI slice that reproduces the unique anatomy of the patient’s brain. “We hope to tailor radiation therapy, chemotherapy and other treatments based on a personalized model of a patient’s tumor,” said Fathallah-Shaykh.

From Flies to Colon Cancer

Since he joined the UAB faculty in 2008, Fathallah-Shaykh has been developing ever more advanced models to predict the behavior of biological networks. He began by building a model of the molecular clock in a fruit fly’s brain. Despite the fly’s tiny size, it’s a challenging puzzle. The clock is a tangled web of positive and negative feedback loops, with five different genes producing proteins that inhibit and activate one another (as well as themselves, in some cases) in a regular cycle.

First, Fathallah-Shaykh and his collaborators “showed we can replicate everything the clock is known to do,” he said. Then they proved it was a useful research tool, answering a perplexing question about the fruit-fly gene Clockwork Orange that had stumped biologists for years.

The researchers next adapted their model to track the developing neural networks in fruit-fly embryos. To do this, they utilized the Kalman filter, a mathematical technique to analyze and predict changes that helps track planes in flight. Now, “we’re using the model to study molecular networks in colon cancer,” Fathallah-Shaykh said.

Coping with an Information Explosion

Fathallah-Shaykh has always been fascinated with math. “It’s like a symphony; it’s beautiful,” he said. “But it’s also very applicable.” He cemented the connection between medicine and math as a neurologist at Rush University Medical Center in Chicago when he enrolled in a doctoral program in mathematics at the nearby University of Illinois–Chicago. “I would go to class in between patients,” he said.

Math is essential to making progress against the toughest questions in medicine, Fathallah-Shaykh contends. To illustrate the problems that researchers face, he points to a chart of all the known molecular pathways involved in Alzheimer’s disease. It’s a mass of interlocking loops and tangles that fills an entire page. Researchers specialize in tiny sections of this wiring diagram, but understanding how it all works together is another problem entirely. Even worse, these networks are intertwined in such a way that multiple paths can lead to the same destination. That may help explain why treatments that work beautifully in isolated cell lines in a lab so often fail when they encounter the complex networks of the body.

There’s another wrinkle. “Cells migrate, they communicate, they interact with one another over time,” said Fathallah-Shaykh. The waves of mutations, which are a hallmark of cancer, make the problem particularly complex, he noted. “Whole pathways are deleted and new connections start turning up.” It’s a perfect example of a nonlinear dynamic system, like the weather or the stock market, in which slight changes in one parameter can lead to wildly diverging outcomes.

The good news, said Fathallah-Shaykh, is that “mathematics has very rich tools” to model just these types of systems, as he has demonstrated with his cancer simulations. But this work has another exciting element for Fathallah-Shaykh as a mathematician: It opens new horizons in math theory. “Equations have already been developed from biological problems,” he said, “and there is very strong evidence that they will produce spectacular advances in mathematics.”

SimTumor

At the heart of Hassan Fathallah-Shaykh’s new mathematical model of glioblastoma multiforme (GBM) are 10 partial differential equations. Here’s how it works — and what it has revealed about GBM behavior.

Formula 10

Equations track each of four different cell types, with unique rules of behavior.

Proliferative GBM cells (P), which make up the bulk of the tumor, divide but don’t move.

Invasive GBM cells (I), found on the fringes of the tumor, move but don’t divide.

Healthy brain cells (B) neither divide nor move, although they are displaced by the growing tumor.

Cells in the center of the tumor, cut off from nourishing blood vessels, are starved of oxygen (hypoxia) and die, becoming necrotic cells (N).

The remaining six equations track angiogenesis (new blood vessel formation), oxygen levels, and rates of necrosis and cell division.

Built for Speed

Fathallah-Shaykh’s first GBM model, published in August 2014 in the Bulletin of Mathematical Biology, consisted of many more equations. It required a supercomputer, and several days, to run. The model published in PLOS One can run in 50 seconds on a typical desktop computer.

And They’re Off!

The simulation begins with a tiny clump of tumor cells surrounded by healthy brain. As the program continues over several virtual weeks, this mass expands in the characteristic manner seen on patient MRIs, with a dark region of necrotic cells in the center, surrounded by a large group of proliferative cells and an outer rim of invasive cells.

Grow or Go

The model’s main assumption is that proliferative cells can turn into invasive cells in hypoxic conditions. This is in keeping with the “grow or go” hypothesis of GBM behavior, which says that low oxygen levels spur GBM cells to flee the dying core of the tumor. When these new invasive cells reach healthy, oxygenated areas of brain, they switch back into proliferative mode and start growing again.

How GBM Escapes Anti-Angiogenesis Therapy

As tumors grow, cells at the core lose contact with nourishing blood vessels and die.

To get around this problem, tumors release VEGF (vascular endothelial growth factor), which induces the body to create new blood vessels (a process known as angiogenesis). In fact, the well-known Folkman Hypothesis states that tumors must be able to induce blood vessel growth in order to keep growing.

Clinicians had high hopes that anti-angiogenesis medications such as bevacizumab (Avastin), could keep tumor growth in check. But two high-profile Phase III clinical trials, which released results in early 2014, found that bevacizumab therapy did not prolong overall survival in patients with recurrent GBM, although it did extend progression-free survival and patient quality of life.

Fathallah-Shaykh’s model, programmed to simulate the effects of anti-angiogenesis therapy, reveals an explanation for this “unusual clinical finding.” When bevacizumab therapy causes oxygen levels to drop, proliferative cells turn into invasive cells and flee the scene. When they reach an area with sufficient oxygen, they convert back into proliferative cells and begin a new cycle of growth. This sets up the tumor for rapid “rebound” growth as soon as it becomes resistant to bevacizumab or therapy is discontinued. That explains why patients treated with bevacizumab in the recent trials didn’t experience any increase in overall survival rates over those who were not treated.

Toward New Treatment Approaches

The model underlines the importance of better understanding the molecular mechanisms of brain cell invasion, particularly the active transport of invasive cells toward healthy brain regions, says Fathallah-Shaykh.

There are currently no available biomarkers to identify the quantity of invasive cells in a patient’s tumor. But finding such a biomarker, and drugs that can target these cells to prevent tumor migration, is a current research focus in Fathallah-Shaykh’s lab. “If we’re going to kill these tumors,” he said, “we have to target the cells that are invading.”