The field of medical imaging has seen tremendous advances in recent years, from higher resolution scans to sophisticated AI-powered analysis. But one of the most exciting frontiers in medical imaging today comes from an unexpected source - the mathematical field of topology. By focusing on the fundamental shapes and structures in medical images, rather than just pixel-level details, topology is enabling more accurate and insightful analysis of complex biological structures.

At its core, topology is concerned with properties of shapes that remain unchanged under continuous deformations. In other words, topology cares about the essential structure of an object, rather than its exact geometry. This makes it particularly well-suited for large-scale analysis (e.g., organ systems) as well granular details (e.g., small vessels and bronchi). For example, imagine a diagram of a face drawn with shapes, where the eyes are circles, the nose is a triangle, the mouth is a straight line, and the head is a larger circle encasing the other shapes. Using topology, the “distance” (perceived similarity) between connected points (or shapes in this example) influences how they will appear “connected”. Using a short distance, the two circles (serving as the eyes in this example) would be connected. Increasing the distance, could include either the nose (triangle) or head (larger circle). In this way, topology examines adjacent structures to determine their connectedness, which can be influenced by proximity, attenuation, shape, etc.

One important way topoogy is changing medical imaging is by improving how we separate complex structures in images. Traditional methods often have trouble with thin or branching structures, which can result in broken connections or missing parts. But topology-aware methods can capture the underlying connectedness of structures like blood vessels or neurons, ensuring segmentations preserve crucial topological properties.

For example, researchers have developed topology-preserving neural networks that learn to segment images while maintaining the correct topology of structures. By incorporating topological constraints into the training process, these networks produce more anatomically plausible segmentations, avoiding issues like broken connections in blood vessel networks. This can be critical for tasks like surgical planning or disease progression monitoring.

Beyond just improving segmentation accuracy, topology is also helping us find new ways to analyze medical images. Tools from topological data analysis, like persistent homology, allow researchers to extract meaningful shape descriptors from complex imaging data. Persistent homology is a tool that can help clinicians and researchers understand if a feature identified using topology (e.g., tumor, aneurysm, parenchymal change) is a real, clinically actionable finding. Persistent homology captures topological features across different scales, which could be changes in distance between topology-derived points or across multiple images. These topological signatures can reveal subtle structural differences that may indicate disease or other biological phenomena.

For instance, topological analysis applied to brain computed tomography (CT) and functional magnetic resonance imaging (fMRI) has been used to find structural and functional patterns linked with neurological disorders like Alzheimer's disease, respectively. By quantifying how the topology of brain networks changes over time, researchers can spot early signs of cognitive decline that might not be visible with traditional imaging methods.

Topology is also useful for measuring uncertainty in medical image analysis. Types of uncertainty that can be quantified, and potentially reduced, include measurements, model construction, parameter

optimization, statistical calculations, changes in the environment where the images are obtained, and human factors, such as movement during scanning. Topology can help to quantify uncertainty in a variety of ways, including tasks involving branching and loop structures (such as bronchi and vasculature) where topology excels at connecting these continuous structures and through expanding image interpretation beyond 2D image slices (such as nth dimensional connections across and within image slices). As AI systems play a larger role in medical imaging, it is essential to know how reliable they are in their results. Topology-based uncertainty estimation can highlight areas where a model is less confident, helping humans focus on the most challenging regions.

This is especially useful for segmentation and annotation tasks. By identifying topologically uncertain structures, these methods can guide expert human annotators to efficiently refine AI-generated segmentations. This human-in-the-loop approach combines the speed of AI with the expertise of human annotators.

Looking to the future, topology has the potential to provide new insights in many areas of medical imaging:

• Improved tracking of tumor growth and treatment response by analyzing topological changes over time – detailed morphological changes that go beyond gross changes in size.
• More accurate segmentation and analysis of complex vascular networks for cardiovascular health assessment – finding non-calcified plaques and clots earlier.
• Enhanced detection of subtle structural brain changes for early diagnosis of neurodegenerative diseases – Excelling in both functional and structural characterization.
• Topology-aware image registration to better align scans from different modalities or time points – synchronizing staging scans over multiple years.

As imaging technology continues to advance, producing ever higher resolution and more complex data, topology will likely play an increasingly important role in extracting meaningful information. The ability to focus on essential structural properties, rather than getting lost in pixel-level details, makes topology a powerful tool for medical image analysis.

Of course, realizing the full potential of topology in medical imaging will require close collaboration between mathematicians, computer scientists, and medical experts. Translating topological insights into clinically relevant tools remains an active area of research and development.

The fundamental power of topology to capture the essence of complex shapes and structures makes it a promising approach for solving modern medical imaging challenges. As we continue to push the boundaries of what is possible in medical imaging and analysis, topology could be the key to discovering new frontiers in understanding the shapes of health.

Additional resources on Topology in Medical Imaging and AI:

1. Hu, X., Li, F., Samaras, D., & Chen, C. (2019). Topology-preserving deep image segmentation. Advances in Neural Information Processing Systems, 32.
2. Xu, F. H., Gao, M., Chen, J., Garai, S., Duong-Tran, D. A., Zhao, Y., & Shen, L. (2024). Topology-based Clustering of Functional Brain Networks in an Alzheimer’s Disease Cohort. AMIA Summits on Translational Science Proceedings, 2024, 449.
3. Clough, J. R., Oksuz, I., Byrne, N., Schnabel, J. A., & King, A. P. (2019). Explicit topological priors for deep-learning based image segmentation using persistent homology. In International Conference on Information Processing in Medical Imaging (pp. 16-28). Springer, Cham.
4. Levenson, R. M., Singh, Y., Rieck, B., Hathaway, Q. A., Farrelly, C., Rozenblit, J., ... & Sarkar, D. (2024). Advancing precision medicine: algebraic topology and differential geometry in radiology and computational pathology. Laboratory Investigation, 104(6), 102060.
5. Chung, M. K., Hanson, J. L., Ye, J., Davidson, R. J., & Pollak, S. D. (2015). Persistent homology in sparse regression and its application to brain morphometry. IEEE transactions on medical imaging, 34(9), 1928-1939.
6. Singh, Y., Farrelly, C. M., Hathaway, Q. A., Leiner, T., Jagtap, J., Carlsson, G. E., & Erickson, B. J. (2023). Topological data analysis in medical imaging: current state of the art. Insights into Imaging, 14(1), 58.
7. Hu, X., Samaras, D., & Chen, C. (2023). Learning probabilistic topological representations using discrete Morse theory. In International Conference on Learning Representations.

Yashbir Singh, ME, PhD | Assistant Professor, Mayo Clinic | Rochester, Minnesota