Enforcing necessary non-negativity constraints for common diffusion MRI models using sum of squares programming

Tom Dela Haije,Evren Özarslan,Aasa Feragen

doi:10.1016/j.neuroimage.2019.116405

Tom Dela Haije, Evren Özarslan + Show 1 more

Open Access

https://doi.org/10.1016/j.neuroimage.2019.116405

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Abstract

In this work we investigate the use of sum of squares constraints for various diffusion-weighted MRI models, with a goal of enforcing strict, global non-negativity of the diffusion propagator. We formulate such constraints for the mean apparent propagator model and for spherical deconvolution, guaranteeing strict non-negativity of the corresponding diffusion propagators. For the cumulant expansion similar constraints cannot exist, and we instead derive a set of auxiliary constraints that are necessary but not sufficient to guarantee non-negativity. These constraints can all be verified and enforced at reasonable computational costs using semidefinite programming. By verifying our constraints on standard reconstructions of the different models, we show that currently used weak constraints are largely ineffective at ensuring non-negativity. We further show that if strict non-negativity is not enforced then estimated model parameters may suffer from significant errors, leading to serious inaccuracies in important derived quantities such as the main fiber orientations, mean kurtosis, etc. Finally, our experiments confirm that the observed constraint violations are mostly due to measurement noise, which is difficult to mitigate and suggests that properly constrained optimization should currently be considered the norm in many cases.

Highlights

Diffusion-weighted magnetic resonance imaging (MRI) captures local micro-structural information by observing diffusing molecules probing their surroundings at a microscopic scale
These results show quite clearly that violated constraints are commonplace in all models and orders except for the K 1⁄4 2 cumulant expansion (DTI)
In this work we have proposed to use sum of squares optimization to enforce non-negativity of the displacement probability density function in diffusion MRI model reconstruction problems

Summary

Introduction

Diffusion-weighted magnetic resonance imaging (MRI) captures local micro-structural information by observing diffusing (water) molecules probing their surroundings at a microscopic scale. In order to analyze this type of data one can either estimate parameters that describe the diffusion itself, which provides a somewhat abstract but accurate description of the observed stochastic motion, or one can use a model of the ambient structure that re-expresses the observed diffusion in terms of more intuitive structural parameters. Both cases generally rely on optimization to reconstruct the descriptive parameters from diffusion-weighted images, and in this work we introduce a specific set of basic constraints to improve such model reconstructions. In diffusion tensor imaging (DTI) (Basser et al, 1994a) for example, a non-positive-definite diffusion tensor can result in unphysical ‘negative’ apparent diffusion or extreme fractional anisotropy values

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