Probabilistic numerics for statistical-shape models

The goal of this project is to develop fully probabilistic shape models, which are essential tools for understanding complex variations and patterns in anatomical structures, such as bones, organs, or soft tissues. These models hold significant potential to improve clinical diagnosis and treatment with enhanced accuracy and efficiency. In the field of computational anatomy, the proposed models will be built using landmarked X-ray data, employing statistical analysis to identify key shape differences associated with disease progression.

As a practical application, the project will focus on creating models to describe the anatomical structures of bones affected by psoriatic arthritis (PsA), an autoimmune inflammatory disease that causes significant changes in bone shape. These models will offer a more comprehensive understanding of how bone shapes evolve in patients with PsA.

The project will explore the use of probabilistic numerics in state-of-the-art flow-based statistical shape models, which are designed to track and predict bone shape evolution over time in a flexible and biologically plausible manner. A key feature of this approach is its ability to quantify the uncertainty arising from data noise, computations, and predictions, resulting in more robust and interpretable models.

The shape deformations will be described using mathematical models, specifically differential equations that govern how shapes change over time and across space. A key focus of this research will be the development of probabilistic solvers for these equations. These solvers approximate solutions as probability distributions, offering greater flexibility and incorporating uncertainty into the model to provide more reliable predictions.

Additionally, the project will reinterpret shape discrepancies, represented as integrals, within a probabilistic framework. Using a technique called Bayesian quadrature, the shape representations and their associated measurements will be rigorously approximated. This will enable the creation of a nonlinear probabilistic shape model that maintains the topological integrity of the bones, ensuring that the model remains anatomically realistic and produces accurate predictions with quantifiable confidence.

To handle large-scale data, modern computational techniques such as JAX and KeOps will be employed to improve computational efficiency and memory usage.

In summary, this project aims to develop reliable, flexible, and interpretable models that can better capture complex shape variations in bones caused by PsA. The resulting generative models and algorithms will be valuable for clinical applications, such as the detection, segmentation, and classification of disease status, particularly when working with noisy or incomplete medical imaging data. Ultimately, this research has the potential to lead to improved diagnostic tools and more personalised treatments for patients, enhancing clinical outcomes and advancing our understanding of PsA.