Mathematical oncology—the application of mathematical models and computational simulations to understand cancer—is fundamentally changing how we comprehend this complex disease.
For decades, cancer research has been dominated by a molecular, reductionist approach—intensively studying individual genes, proteins, and cells. While invaluable, this method often misses the forest for the trees, failing to capture the complex interactions and system-level behaviors that drive cancer progression 1 .
Mathematical oncology addresses this gap. It is founded on two powerful ideas: first, that mathematics can significantly improve our biomedical understanding of cancer; and second, that the fascinating complexities of biology themselves generate new mathematical challenges, pushing the development of sophisticated analytical tools 1 .
The ultimate goal is to create a "digital twin" of a patient's cancer. By calibrating a mathematical model with a specific patient's data, clinicians could, in theory, test countless treatment regimens in silico to identify the most effective, personalized strategy before a single drug is ever administered 2 .
This field uses a diverse arsenal of mathematical frameworks—from differential equations and game theory to stochastic models and agent-based simulations—to create virtual tumors that can be studied, manipulated, and understood in ways that are often impossible with physical experiments 3 4 .
Mathematics provides a unique lens to examine cancer, with different tools suited to unravel specific aspects of the disease.
Scientists typically characterize mathematical models along four key dimensions 3 :
Simple models focus on one aspect of cancer (e.g., tumor growth), while hybrid models combine multiple factors (e.g., growth, immune response, and drug delivery) for a more comprehensive picture.
Homogeneous models treat cancer cells as uniform, whereas heterogeneous models account for the vast genetic and behavioral diversity among cells within a single tumor—a major driver of treatment resistance.
Spatially averaged models provide an overview of the whole tumor, while spatially resolved models capture how cell behavior and interactions vary across different locations within the tumor microenvironment.
Deterministic models always produce the same output from a given set of inputs, while stochastic models incorporate randomness, crucial for simulating unpredictable events like genetic mutations.
One of the most intriguing applications is game theory, which analyzes strategic decision-making among competing players. When applied to oncology, cancer cells become the "players" 5 .
Researchers use games like the "hawk-dove" game to model how cells decide to act aggressively (hawk) or passively (dove) to acquire resources like nutrients and space 5 .
For example, in clear cell renal cell carcinoma, game theory has helped explain the fundamentals of intratumour heterogeneity—the coexistence of different cell subpopulations within a single tumor 5 . This understanding is vital because a highly heterogeneous tumor is often more adaptable and treatment-resistant.
Mathematics thus supports a revolutionary therapeutic idea: instead of always using the maximum tolerable dose (which eliminates sensitive cells and leaves only resistant ones), a therapy aimed at preserving some heterogeneity could slow overall cancer growth and lead to longer patient survival 5 .
A cutting-edge frontier is mechanistic learning, the powerful combination of knowledge-driven mathematical modeling and data-driven machine learning 2 .
| Feature | Knowledge-Driven (Mechanistic Modeling) | Data-Driven (Machine Learning) |
|---|---|---|
| Basis | Approximation of biomedical understanding 2 | Extraction of information from data patterns 2 |
| Data Needs | Can work with limited data for validation 2 | Requires large amounts of high-quality training data 2 |
| Strengths | High interpretability; can simulate unobserved scenarios 2 | Powerful for finding complex, unknown patterns in big data 2 |
| Weaknesses | Relies on simplifying assumptions that may miss complexity 2 | "Black box" nature can make decisions hard to interpret 2 |
| Example | Modeling radiotherapy fractionation schemes 2 | Predicting drug response from genomic data 2 |
Mechanistic learning merges these two worlds. It uses mathematical models based on biological principles to guide and constrain machine learning algorithms, making them more transparent and reliable. Conversely, machine learning can help calibrate complex mathematical models from multi-faceted patient data 2 . This synergy is essential for creating clinically trustworthy tools that can tackle issues like data sparsity and model interpretability.
A compelling example of how advanced mathematics can refine our models of reality is the application of fractional calculus to tumor growth prediction. Classical models like Exponential, Logistic, and Gompertz growth have been used for decades but often lack the flexibility to perfectly capture the observed behavior of real tumors 6 .
Fractional calculus generalizes the standard concept of a derivative from integer orders (first derivative, second derivative) to non-integer orders. This endows the models with a unique "memory" property, meaning the current state of the system depends on its entire history—a characteristic that is particularly fitting for complex biological processes like cancer evolution 6 .
A landmark 2023 study set out to prove the superiority of fractional-order models 6 . Researchers generalized four classic tumor growth models (Exponential, Logistic, Gompertz, and Bertalanffy) into their fractional equivalents.
They then tested these new models against the classical versions using real tumor volume measurements from patients, both untreated and undergoing therapy. The critical metric for comparison was the Mean Squared Error (MSE)—the lower the MSE, the better the model fits the real data 6 .
The results were striking. As shown in the visualization below, the fractional-order models reduced the MSE by at least half compared to their integer-order counterparts across the board 6 .
The profound implication is that tumor growth is indeed a process with memory, where past states significantly influence future progression.
Average improvement in model accuracy with fractional calculus
By providing a more accurate and predictive model, fractional calculus offers a better starting point for designing personalized treatment schedules and forecasting how a specific patient's tumor might respond to therapy 6 .
The modern mathematical oncologist relies on a suite of specialized tools and concepts to build and test their virtual cancer models.
Describe the rates of change in system variables over time (e.g., population growth of cancer cells) 4 .
Provides models with "memory," leading to a more accurate description of complex biological processes like tumor growth 6 .
Models strategic interactions between different cancer cell subpopulations to understand competition and cooperation 5 .
Simulates individual cells (the "agents") and their rules of interaction to observe emergent tumor-level behavior 4 .
A mechanistic learning technique that constrains a neural network with physical laws (equations), improving its predictive power with less data 2 .
A patient-specific, virtual replica of a tumor that is continuously updated with patient data to simulate and forecast treatment outcomes 2 .
The journey of mathematical oncology from a niche theoretical field to a clinically relevant discipline is well underway. The 2024 Fields Thematic Program in Mathematical Oncology highlighted this growth, showcasing how mathematical models are improving the precision of treatments ranging from chemotherapy and radiation to cutting-edge CAR T-cell therapies and oncolytic virotherapies 7 .
Furthermore, the field is actively fostering an inclusive and collaborative environment, recognizing that the toughest problems require diverse minds 7 .
Predicted improvement in treatment personalization
The future of this field is not about replacing oncologists with computers, but about empowering them with predictive, personalized insights.
As these models become more refined and integrated into clinical workflows, they hold the promise of a future where each patient's treatment is not just based on population averages, but on the predicted behavior of their unique disease.