Core Concept Mathematics and Economics Published: July 17, 2026

Using Math to Help Treat Brain Cancer

Abstract

Glioblastoma is the most common form of brain cancer in adults and remains one of the most lethal forms of cancer. Treating glioblastoma is uniquely challenging due to its aggressive nature and its critical location, the brain, which must remain largely undamaged. Due to these difficulties, there is a lack of effective treatment options. Computer models simulating tumors and their response to therapy, known as digital twins, provide a promising new strategy. These model systems are mathematical equations that capture the biological knowledge of how cells interact during tumor growth and in response to treatment. By solving these equations, we can create virtual patients that provide accurate predictions of real patients’ tumor progression. This approach has the potential to improve outcomes by designing therapies for the specific needs of each patient, moving beyond the limitations of the “one-size-fits-all” approach that is currently used.

What Is A Digital Twin?

Before building a new aircraft, engineers use computer simulations to test how it will perform. Scientists use similar approaches to study everything from climate change to space travel. Today, computer simulations are also being used in medicine through the development of “digital twins”—virtual versions of patients that can help medical teams design more personalized treatments.

A digital twin is a mathematical model of a physical system. This could be a structural object like a car or bridge, a biological structure like an organ, or even a complete human. Think of a digital twin as an avatar in a video game. When you create your avatar, some common features are “standard” across all avatars. This provides a helpful starting point for the character design that can be built upon and personalized. These common features represent the underlying physical or biological rules that a system follows. These rules can be turned into mathematical equations that make up the digital twin.

When designing your avatar, you can make several individual choices about the character’s appearance and abilities. These choices, or parameters, allow you to make your avatar individual to you. For an avatar, the parameters may be health or stamina, while for a digital twin of cancer, parameters may be growth rate or sensitivity to treatment. Parameters differ between individuals, allowing us to calibrate the mathematical model without changing the underlying rules, so that every patient has their own distinct digital twin.

A Digital Twin For Glioblastoma

Cancer occurs when genetic mutations damage cells’ control mechanisms, causing uncontrolled cell division that leads to the development of a tumor. The most common type of brain cancer in adults is called glioblastoma. Unfortunately, this cancer is almost always fatal, and the average survival time remains low, at around 15 months.

While cancer is a hugely complex disease, a simplified definition of a brain tumor is often given as: “uncontrolled proliferation (division) of cells with the capacity to invade” [1]. This simplified definition represents the fundamental biological rules that our model must capture: proliferation (how fast the tumor cells grow) and invasion (how fast they move). Thus, our digital twin for glioblastoma is a mathematical model called the proliferation–invasion (PI) model [2]. Writing down an equation in words is a useful first step when developing any mathematical model. We can write down a word equation for this model as follows:

rate of change of tumor cell density =rate of cell proliferation                                                                     + rate of cell movement

Now that we have a digital twin model, we must work out the values of the model parameters for each individual patient. To do this, we can use a special type of medical imaging called magnetic resonance imaging (MRI), which uses magnetic fields to see inside a person’s skull to observe the location and size of the tumor. Using MRIs taken at different times, we can calculate the increase in tumor size and use that to estimate the value of the proliferation and invasion parameters. This allows us to calibrate our digital twin for each individual patient. Once we have calibrated our model, we can use it to predict the likely future growth of the tumor for specific patients (Figure 1).

Four-panel figure for Patient X showing brain tumor progression over time. First two panels display MRI scans at diagnosis and pre-operation, both with a highlighted tumor. Next two panels depict digital twin brain simulations at day thirty and day sixty, illustrating the tumor in color with a visible increase in size. A horizontal purple timeline bar with an arrow points from left to right, representing the progression across days.
  • Figure 1 - Example digital twin simulation of a brain tumor.
  • The digital twin model is calibrated using two pre-treatment MRIs: an initial MRI taken when the patient was diagnosed (day 0) and a follow-up scan taken 2 weeks later (day 14). The model can then simulate forward in time to predict the future growth of the tumor. In the virtual digital twin brain, red represents areas with lots of tumor cells per unit volume, while blue represents areas with fewer tumor cells.

When A Patient Is Not Average

One of the biggest challenges in treating glioblastoma is that every patient is unique. Even when patients that appear similar (same age, sex, and ethnicity) are given the same treatment, they can respond very differently. This variability is known as inter-patient heterogeneity and it can make assessing tumor progression and response to treatment surprisingly difficult.

For example, imagine a patient with a particularly aggressive tumor: even if treatment is working very effectively, follow-up MRIs may show a slight increase in tumor size. On the other hand, a patient who has a much less aggressive tumor but is not responding to treatment may show a similar small increase in tumor size. With only this limited data, it may appear that both tumors responded similarly to treatment, when the reality is quite different (Figure 2). Understanding this difference is vital to improving treatment, as we want patients that respond to treatment to remain on it, while those who do not respond should be switched to alternative treatments as soon as possible.

Dual-panel data visualization comparing tumor volume over time in two patients, X and Y, with graphs depicting true tumor size (solid line), virtual control (dashed line), treatment periods marked by green shading, brain scan insets indicating tumor location, and annotations showing days gained from treatment: sixty-seven days for Patient X and two hundred twenty-two days for Patient Y.
  • Figure 2 - Our digital twin helps us determine which patients are responding to treatment.
  • Patients X and Y have similar-sized tumors at day 100 and both undergo the same treatment, starting at day 200. At day 500, both patients still have similar-sized tumors as shown by MRI. However, by considering the virtual controls from our digital twin (dashed lines), we see that treatment was much more effective for Patient Y. This is represented in the days gained scores for both patients, where treatment delayed tumor growth by over 200 days for patient Y, while for patient X it only delayed tumor growth by 67 days.

We can use our digital twin to create a new personalized measure of tumor progression, called days gained, that can help distinguish the overlapping cases. Days gained is calculated as follows:

1) Calibrate the patient-specific digital twin using a patient’s MRIs.

2) Virtually simulate the untreated tumor growth.

3) Find the time when the untreated tumor is the same size as the real tumor after therapy.

4) Subtract this calculated time from the actual time of the post-treatment observation.

The final difference is the days gained score (Figure 2) [3], which is a personalized score that tells us how effectively therapy reduced the potential untreated tumor growth. Our digital twin acts as a personalized virtual control, allowing us to observe how much the tumor would have grown if we did not intervene with treatment. We then compare the predicted size to the actual size after treatment, to get an idea of how much treatment altered the tumor’s growth.

Simulating Treatments, Enter The Multiverse!

Currently, most patients with glioblastoma receive what is known as the standard of care. This is treatment that is accepted by medical experts as the proper treatment (and schedule) for any specific disease. In glioblastoma, standard of care consists of an initial surgery followed by simultaneous radiotherapy and chemotherapy for 6 weeks. This standard of care has been developed over many years and many clinical trials, which have shown on average it leads to better survival. However, this does not account for the large inter-patient heterogeneity that we know exists in glioblastoma and therefore does not guarantee the best outcome for each individual patient. Using digital twins, we aim to personalize treatment and improve outcomes. We can extend our mathematical model to include the effects of standard of care treatments like radiotherapy and chemotherapy by adding additional terms that account for cell death due to therapy [4].

This allows us to enter the multiverse (virtually)! Not only can we simulate the predicted effect of standard of care, but we can also virtually test many different scenarios with alternative treatment schedules or new therapies (Figure 3). This can only be achieved with a digital twin, as it allows us to simulate the same (virtual) patient simultaneously undergoing multiple different alternative treatments, which would be impossible in reality.

Side-by-side data visualizations compare tumor volume changes over time in two patients, X and Y, using line graphs, MRI images, and 3D tumor models. Plots show true tumor size versus virtual control, effects of multiple treatments, and a green shaded region for simulated treatment periods. MRI insets highlight tumor locations, and 3D models depict tumor growth at different time points.
  • Figure 3 - Patients X and Y have similar-sized tumors at 100 days.
  • On day 150, we virtually simulate three potential treatments (see key) as well as a virtual control with no treatment (dashed lines). Patient X is predicted to respond best to treatment C, while Patient Y is predicted to respond best to treatment A. This information can then be taken to the medical team to help them make the best treatment decisions for each patient.

Once we have simulated a variety of possible treatments, we can select the option that is best suited to the individual patient, which may be different from the standard of care due to unique characteristics of each patient’s tumor. For example, patients with a high proliferation rate parameter will likely benefit from radiotherapy that targets dividing cells, while patients with low proliferation and high invasion parameters may benefit more from treatments that reduce cell movement. Think of this as simulating a virtual clinical trial for each individual, to identify the best treatment for every patient uniquely rather than basing treatment decisions on what is best for the “average” patient [5].

Conclusion

Digital twins offer a revolutionary new approach for treating glioblastoma by enabling personalized medicine. By creating a virtual model of a patient’s tumor, we can simulate its growth and response to various treatments, allowing us to tailor therapies to the individual. This approach is not limited to glioblastoma; it has also been used in many other types of cancer, such as prostate and breast cancer, as well as other areas of medicine, including cardiology. In all these uses, digital twins may enhance patient outcomes by moving away from the traditional one-size-fits-all treatment currently used, toward more personalized treatment options. As we continue to refine and expand these models, digital twins could become a crucial tool in the fight against glioblastoma, offering hope for better, more effective, and more personalized treatments in the future.

Glossary

Parameter: A number in an equation that varies (between patients) in equations of the same general form.

Calibrate: To choose the parameters that best fit the data from an individual patient.

Inter Patient Heterogeneity: Differences in tumor characteristics (i.e., size, growth rate, tumor location, sensitivity to treatment) between patients.

Virtual Control: A model that predicts what would happen if no action was taken.

Radiotherapy: A type of cancer treatment that uses high-energy rays (radiation) to destroy cancer cells by damaging their DNA, causing them to die.

Chemotherapy: A type of cancer treatment that uses powerful, cell-killing drugs to destroy rapidly dividing cancer cells. By targeting dividing cells, these drugs stop cancer from growing and spreading.

Clinical Trials: Carefully planned research studies involving human volunteer patients to test if a new medical treatment is safe and effective.

Multiverse: The hypothetical set of all alternative universes.

Acknowledgments

The authors thank Kayla Swanson (Kristin’s daughter) for her valuable input in reviewing the manuscript. The authors would also like to thank the National Cancer Institute (U01CA250481, U54CA274504, U01CA220378, U54CA14397, and U54CA193489), the James S. McDonnell Foundation, the Ben & Catherine Ivy Foundation and Brain Tumour Research (BTR-A-0145). The authors gratefully acknowledge financial support from a PhD studentship provided by the University of Nottingham.

AI Tool Statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Conflict of Interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


References

[1] Harpold, H. L. P., Alvord, E. C., and Swanson, K. R. 2007. The evolution of mathematical modeling of glioma proliferation and invasion. J. Neuropathol. Exp. Neurol. 66:1–9. doi: 10.1097/nen.0b013e31802d9000

[2] Swanson, K. R., Bridge, C., Murray, J. D., and Alvord, E. C. 2003. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurol. Sci. 216:1–10. doi: 10.1016/j.jns.2003.06.001

[3] Hawkins-Daarud, A., Johnston, S. K., and 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. doi: 10.1200/CCI.18.00066

[4] Rockne, R., Alvord, E. C., Rockhill, J. K., and Swanson, K. R. 2009. A mathematical model for brain tumor response to radiation therapy. J. Math. Biol. 58:561. doi: 10.1007/s00285-008-0219-6

[5] Swanson, K. R., Rostomily, R. C., and Alvord, E. C. 2008. A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle. Br. J. Cancer. 98:113–9. doi: 10.1038/sj.bjc.6604125