Diffeomorphic Statistical Shape Models and Damage Quantification in Psoriatic Arthritis

The study of shape and variations in shape, is an exceedingly important part of computational anatomy and is useful in study of disease progression in various conditions; e.g. Psoriatic Arthritis. Standard Statistical Shape models in the computer vision literature do not preserve topology or regularity of the underlying family of contours, especially for large variations in shape. This is a drawback in medical applications where accurate models of shape and variations in shape, are an important tool in disease stratification.

The proposed research project will develop Diffeomorphic Statistical Shape models. This uses ideas from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) theory. This theory is typically used in nonrigid registration, computing minimal energy diffeomorphisms aligning two geometric structures of interest.

We will use LDDMM to model variations in shape via action of diffeomorphism on an initial template. This guarantees topological consistency, as well as regularity of the deformed template. Shapes can be represented as point clouds, contours or measures in the more general case. This representation has links to the optimal transport literature.

We combine this with Gaussian processes, and Gaussian Process Latent Variable Models to construct a generative model for shape. We place Gaussian process priors, on time dependent velocity fields that generate diffeomorphic flows. This prior will be conditioned on an observed shape family, to fit variations in shape to variations in fields.

To regularise the shape generation process, we will extend this initial model using conservation properties and PDE arising from the Hamiltonian formulation of the LDDMM theory. In particular, momentum and energy conservation energy properties can be used to place GP priors over time dependent momentum and initial momentum.

Our primary application of the proposed shape models will be to medical imaging and diagnosis. We will work with a dataset of shapes of hand bones with Psoriatic Arthritis or PsA (collected by the NHS). The shapes are in the form of landmarked X-Rays, annotated and scored by an expert clinician.

Our generative shape model, will be used to model shape and shape variations in bones with PsA in a topologically consistent manner. In the long term, we aim to build a probablistic scoring tool for damage in PsA using the labelled dataset as training data.

Upon success, such a tool will improve the diagnostic process in PsA and result in better understanding of the relationships between shape variation and disease trajectories in PsA.

Healthcare technologies that 'optimise patient-specific illness prediction, accurate diagnosis and effective intervention' is one of the key themes in recent research supported by EPSRC. Therefore, this research is highly relevant to the research council.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.