Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI

Area of Science:

• Medical Imaging
• Biophysics
• Computational Neuroscience

Background:

• Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) commonly uses 2nd order tensors to model water diffusion in tissues.
• This 2nd order approximation is insufficient for complex structures like crossing fibers.
• Existing higher-order tensor approximations lack guarantees for the essential positive semi-definite (PSD) property.

Purpose of the Study:

• To develop and present a novel technique for estimating 4th order symmetric positive semi-definite (PSD) tensors from DW-MRI data.
• To address the limitations of 2nd order tensor approximations in capturing intricate local tissue architectures.
• To ensure the physical meaningfulness of diffusion tensor coefficients by enforcing the PSD constraint.

Main Methods:

• Employed a 4th order symmetric PSD tensor approximation for the diffusivity function.
• Utilized the Gram matrix method and Hilbert's theorem on ternary quartics to parameterize 4th order tensors as sums of squares.
• Applied a nonlinear-least squares formulation for estimating PSD tensors of order 4 from DW-MRI data.
• Introduced a metric for higher-order tensors and applied lattice-wide regularization.

Main Results:

• Successfully estimated 4th order PSD tensors from DW-MRI data, guaranteeing the positive semi-definite constraint.
• Demonstrated the model's capability to represent complex local tissue structures more accurately than 2nd order methods.
• Validated the performance on both synthetic datasets and real DW-MRI data from a rat hippocampus.

Conclusions:

• The proposed 4th order tensor approximation with guaranteed PSD property offers a more accurate representation of DW-MRI data, especially for complex microstructural environments.
• This novel technique advances DW-MRI processing by providing a physically constrained and robust method for estimating diffusion properties.
• The findings have significant implications for improving the analysis and interpretation of diffusion MRI data in neuroscience and clinical applications.