Mathematical Modeling Gains Days for Brain Cancer Patients

By Matthew R. Francis

Editor's Note: This article originally appeared in SIAM News on July 1, 2019 (

Glioblastoma, or glioblastoma multiforme, is a particularly aggressive and almost invariably fatal type of brain cancer. It is infamous for causing the deaths of U.S. Senators John McCain and Ted Kennedy, as well as former U.S. Vice President Joe Biden’s son Beau. Though glioblastoma is the second-most common type of brain tumor—affecting roughly three out of every 100,000 people—medicine has struggled to find effective remedies; the U.S. Food and Drug Administration has approved only four drugs and one device to counter the condition in 30 years of research. The median survival rate is less than two years, and only about five percent of all patients survive five years beyond the initial diagnosis.

Given these terrible odds, medical researchers strive for anything that can extend the effectiveness of treatment. The nature of glioblastoma itself is responsible for many obstacles; brain tumors are difficult to monitor noninvasively, making it challenging for physicians to determine the adequacy of a particular course of therapy.

Figure 1. Magnetic resonance imaging scan of the brain. Public domain image.
Kristin Rae Swanson and her colleagues at the Mayo Clinic believe that mathematical models can help improve patient outcomes. Using magnetic resonance imaging (MRI) data for calibration (see Figure 1), they constructed the proliferation-invasion (PI) model — a simple deterministic equation to estimate how cancer cells divide and spread throughout the brain. Rather than pinpoint every cell’s location, the model aims to categorize the general behavior of each patient’s cancer to guide individualized treatment.

During her presentation at the American Association for the Advancement of Science 2019 Annual Meeting, which took place in Washington, D.C., earlier this year, Swanson noted that every glioblastoma patient reacts differently to the same treatment. She hopes that use of the PI model might help predict patient response to a given regimen. “The model is able to provide a sort of virtual control,” Andrea Hawkins-Daarud, Swanson’s collaborator at the Mayo Clinic, said. “With a virtual control, you can consider how the size of the tumor changes over time. Then you can begin thinking through a lot of different possible response metrics.”

The team discovered that absolute tumor size was a less important metric than tumor position on the growth curve. Swanson and her colleagues use the term “days gained” to describe the result: does the treatment turn back the clock on cancer proliferation and buy the patient more time? Estimating days gained requires an understanding of the time-dependent growth kinetics pertaining to the individual’s cancer, which is precisely what the PI model attempts to do.

A Model for Tumor Growth

As for many other tumors, neurosurgeons commonly begin glioblastoma treatment by surgically removing as much of the cancer as possible before following up with chemotherapy and radiation. However, glioblastoma is more diffuse than most cancers; because the tumor extends into healthy tissue, it is nearly impossible for surgeons to remove all cancer cells without damaging the brain.

To make matters worse, the degree of diffusivity varies widely among patients, and MRI scans alone are not particularly good at distinguishing the nuances of these cases. “Doctors don’t really have a clean way of knowing the difference between one patient’s tumor being really diffuse and another patient’s tumor being really nodular, or which tumor is growing faster than another,” Hawkins-Daarud said. “MRI detects what the cancer cells have done to the environment, but it can’t specifically say ‘this is a tumor.’ It can’t identify the boundary [of the glioblastoma].”

The uncertainty in measuring that boundary means that clinicians struggle to determine which treatments are working and which require adjustment. However, glioblastoma’s diffusivity also makes it amenable to a reaction-diffusion model — a common type of equation in mathematical biology. The PI model approximates the tumor’s growth in space and time by treating it as a continuous fluid [3]:

\[\frac{\partial c}{\partial t}=\triangledown \cdot (D\triangledown c)+\rho c(1-c),\]

where \(c\) is the tumor cell density. The free parameters \(D\) and \(\rho\) respectively quantify the cancer cells’ diffusion and rate of proliferation. Assuming a spherical tumor, the solution to the PI equation far from the tumor center takes the form of a traveling wave with velocity \(2 \sqrt{D\rho}\) and steepness \(D/ \rho\). 

Figure 2. Estimated radial size of a tumor before and after treatment, where the treated tumor size corresponds to an earlier stage of growth according to the model. This allows researchers to estimate the days gained with a particular treatment. Figure courtesy of [2].
These parameters are not directly measurable. To infer their values, Swanson’s team used MRI measurements for 160 glioblastoma patients [1]. They obtained an estimate of tumor growth and proliferation by comparing two MRI scans for each patient, then applied a Bayesian framework [2] to quantify uncertainties in both the data and model. These efforts yielded a means of classifying patient responses to treatment in terms of days gained (see Figure 2).

“I don’t think [the PI model] is good at giving precise boundaries of tumor cell density throughout the brain,” Hawkins-Daarud said. “However, it is good with helping us conceptually ‘bin’ patients into categories.” The days-gained metric identified via the PI model proved to be a much better predictor than tumor size alone, thanks to incorporation of cancer kinetics.

“The difference in overall survival for patients with a larger days-gained value was statistically significant over those who had the smaller days-gained value,” Hawkins-Daarud continued. “Our hope is that [the model] will be able to identify when a therapy is truly failing and you should change it, or when a therapy is being useful and you should stay on it — even though it looks like it may not be as good as you might expect.”

Hope is a Thing with Equations

The PI model is deterministic and treats tumors as continuous fluids, whereas real glioblastoma consists of discrete cells that spread more haphazardly. For this reason, Swanson, Hawkins-Daarud, and many other researchers are combining forces to create better models that incorporate cancer kinetics, machine learning, and cellular automata, along with a wider range of medical data. The preliminary results of these efforts are not yet published, but Hawkins-Daarud believes that they hold a great deal of promise.

Even so, the problem can still seem insurmountable. Cancer is not a single disease, but rather a large set of conditions with many causes and a number of common features. The PI model enables better understanding of glioblastoma’s specific traits; however, this does not work for most cancers, which metastasize and are non-diffuse. Yet hope is a relative thing in cancer research — for mathematical oncologists as much as for doctors and patients.

“The math isn’t going to cure the cancer,” Hawkins-Daarud said. “But I think that math can certainly help optimize the process of finding a cure. We are actually in the midst of talking to various drug companies to try and incorporate our response metrics into the clinical trials to see if we can speed up the proceedings.”

Even a few months of extra time acquired from improved treatments is significant to glioblastoma patients and their loved ones. While math alone will not provide this time, the PI model shows that it can help gain some valuable days.


[1] Baldock, A.L., Ahn, S., Rockne, R., Johnston, S., Neal, M., Corwin, D., …Swanson, K.R. (2014). Patient-specific Metrics of Invasiveness Reveal Significant Prognostic Benefit of Resection in a Predictable Subset of Gliomas. PLoS One, 9(10), e99057.
[2] Hawkins-Daarud, A., Johnston, S.K., & Swanson, K.R. (2019). Quantifying Uncertainty and Robustness in a Biomathematical Model-Based Patient-Specific Response Metric for Glioblastoma. JCO Clin. Cancer Inform., 3, 1-8. 
[3] Rockne, R., Alvord Jr., E.C., Rockhill, J.K., & Swanson, K.R. (2009). A mathematical model for brain tumor response to radiation therapy. J. Math. Biol., 58(4-5), 561-578.

About the Author

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is

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