Study of Mathematical Models for Tumor Growth – Exploring Glioma Extension Using Longitudinal Magnetic Resonance Imaging

A recent article published in the IEEE Transactions on Biomedical Engineering presents a systematic study on the mathematical modeling and growth patterns of gliomas (glioma). This research was conducted by Birkan Tunç, David A. Hormuth II, George Biros, and Thomas E. Yankeelov, primarily evaluating three different mathematical models’ performance in simulating tumor growth and mass effect using longitudinal magnetic resonance imaging (MRI) data.

Research Background

Glioblastoma multiforme (GBM) is the most common primary brain tumor with poor patient prognosis. A notable feature of GBM is the severe deformation effect on surrounding brain tissues, also known as the “mass effect.” Although many mathematical models have been used to simulate tumor growth and predict clinical progression and treatment outcomes, many existing models do not explicitly consider the mass effect, limiting their ability to describe the important phenomenon of surrounding tissue deformation during tumor expansion.

Paper Source and Author Background

This paper was authored by Birkan Tunç (Department of Mechanical Engineering, Istanbul Yeditepe University), David A. Hormuth II (Oden Institute for Computational Engineering and Sciences and Livestrong Cancer Institutes, University of Texas at Austin), George Biros (Oden Institute for Computational Engineering and Sciences, University of Texas at Austin), and Thomas E. Yankeelov (Imaging Physics Department, University of Texas at Austin and MD Anderson Cancer Center). It was published in the IEEE Transactions on Biomedical Engineering in December 2021.

Research Process

This study used a mouse model, longitudinally collected MRI data, and individually calibrated and compared the performances of three mathematical models in predicting tumor growth and mass effect. These three models include:

1. Reaction-Diffusion-Advection Model (RDAM): A reaction-diffusion-advection model considering both coupled interaction and mass effect.
2. Reaction-Diffusion Model (RDM): A reaction-diffusion model coupled with linear elasticity but considering the mass effect only in the case of small deformations.
3. Reaction-Diffusion Model (RD): A baseline model that does not include the mass effect.

The study annotated the tumor cell volume fraction and mass effect extracted from MRI data and used calibration data from different time points to forecast and validate each model’s accuracy.

Experimental Methods and Data

Data Collection:

• Nine Wistar rats were selected, with tumors induced in their neocortex by injecting C6 glioma cells.
• T2-weighted and contrast-enhanced T1-weighted MRI data were scanned at six different time points within ten days.

Image Segmentation and Processing:

• Images were manually segmented to distinguish the tumor region and the mass effect caused by the tumor (mainly by observing the displacement of the corpus callosum).
• The level set method was used to track deformation of tissue boundaries.

Model Calibration:

• Three calibration scenarios were used: data from the first four time points, only the first two, and only the last two time points.
• Using finite element computation software (FEniCS), parameters were estimated to optimize the models and perform numerical solutions.

Research Results

Parameter Estimation and Model Prediction Accuracy

Data results from the nine experimental mice indicated that the diffusion coefficient of the Reaction-Diffusion-Advection Model (RDAM) was significantly lower than the other two models, implying that not considering advection, a critical factor, would lead to overestimated diffusion coefficients. Specific data analyses are as follows:

• The diffusion coefficient median value for the RD model was 17.46×10^(-3) mm²·d^(-1), with a range from 6.33×10^(-3) to 78.41×10^(-3) mm²·d^(-1).
• The diffusion coefficient median value for the RDM model was 19.38×10^(-3) mm²·d^(-1), with a range from 4.97×10^(-3) to 53.80×10^(-3) mm²·d^(-1).
• The diffusion coefficient median value for the RDAM model was 10.65×10^(-3) mm²·d^(-1), with a range from 0.92×10^(-3) to 26.17×10^(-3) mm²·d^(-1).

In terms of predictive accuracy, the RDM model performed best in tumor volume fraction error and tumor volume error, while the RDAM model had the smallest diffusion coefficient and proliferation rate estimation errors. However, the increased coupling in the RDAM model made its parameter space more limited, increasing the prediction error.

Error and Statistical Analysis

Statistical analysis of the calibration results for the three models revealed that the RD and RDM models had significantly lower errors and Dice coefficients in tumor volume prediction compared to the RDAM model. In mass effect prediction, the RDAM model’s error was significantly higher than the RDM model.

Conclusion

This study demonstrates that, although the Reaction-Diffusion-Advection Model (RDAM) overall performed worse in predictive accuracy compared to the Reaction-Diffusion Model (RDM), its mass conservation characteristic and smaller parameter variation range suggest it is worth further optimization and improvement in future studies. Using data including multiple time points (calibration #1) for model calibration can effectively reduce prediction errors and increase robustness in practical applications.

Research Significance

This study not only provides a novel mathematical model form but also demonstrates, through detailed experimental and numerical analyses, that the mass effect and advection should be considered in computations of tumor growth and mass effect processes. This offers new approaches and methods for clinical tumor growth prediction.

The results of this study can more accurately describe the growth process of gliomas and provide a basis for personalized medical treatments based on imaging data, having significant implications for clinical decision-making. Future studies can further optimize this model by incorporating more complex nonlinear elasticity models to enhance its accuracy and reliability in practical clinical applications.