Diffusion Tensor Imaging

DTI has been applied to reveal subtle abnormalities in a large number of brain diseases and disorders including multiple sclerosis, stroke, schizophrenia, dyslexia, etc. Another promising application of DTI is fibre tracking. The ability of fibre tracking to visualise anatomical connections between different parts of the brain, in vivo, non-invasively and on an individual basis, has emerged as a major breakthrough for neurosciences.

Although DTI has been widely used to study the white matter of the brain, Musculoskeletal (MSK) DTI is still in its infancy. There is an urgent need for statistical methods and computing tools for DTI image analysis. The work of Loughborough University's Dr Diwei Zhou on Statistics for DTI has substantially improved diffusion tensor data analysis from MRI experiments.

The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the nonEuclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.

• Dryden IL, Koloydenko AA, Zhou D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. The Annals of Applied Statistics 3(3): 1102-1123. DOI: 10.1214/09-AOAS249

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First we describe a family of anisotropy measures based on a scale invariant power Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples.

Secondly we discuss weighted Procrustes methods for DTI interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal square root Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation, and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable.

Thirdly we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a dataset of human brain diffusion weighted MRI, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.

• Zhou D, Dryden IL, Koloydenko AA, Audenaert KMR, Bai L. (2015). Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics. Journal of Applied Statistics 43(5). DOI: 10.1080/02664763.2015.1080671

• Ma, Y, Zhou, D, Ye, L, Housden, RJ, Fazili, A, Rhode, K. (2021). A tensor-based catheter and wire detection and tracking framework and its clinical applications. IEEE Transactions on Biomedical Engineering, ISSN: 0018-9294. DOI: 10.1109/tbme.2021.3102670

Dr Diwei Zhou, Senior Lecturer in Statistics in The Department of Mathematical Sciences at Loughborough University, specialises in the development of statistical methodology for large and highly-structured data analysis with applications to neuroscience, sport science, chemistry, engineering and computer science.

One of Dr Zhou’s research focuses is the development of novel statistical methods and tools for analysing the structure of brain white matters and skeletal muscle tissues using data from Diffusion Tensor Imaging (DTI).

Dr Zhou would be happy to share future research plans and collaborate with other researchers.