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

Area of Science:

• Medical Imaging
• Computational Neuroscience
• Applied Mathematics

Background:

• Diffusion-weighted magnetic resonance imaging (dMRI) models estimate water diffusion in biological tissues.
• Accurate estimation of the diffusion propagator is crucial for quantitative analysis.
• Existing methods often struggle to enforce non-negativity of the diffusion propagator, a fundamental physical property.

Purpose of the Study:

• To investigate the use of sum of squares (SOS) constraints for enforcing strict, global non-negativity of the diffusion propagator in various dMRI models.
• To formulate and verify these constraints for the mean apparent propagator (MAP) model and spherical deconvolution (SD).
• To assess the impact of constraint enforcement on model parameter accuracy and derived quantities.

Main Methods:

• Formulation of SOS constraints for MAP and SD models to guarantee diffusion propagator non-negativity.
• Derivation of auxiliary, necessary (but not sufficient) constraints for the cumulant expansion model.
• Verification and enforcement of constraints using semidefinite programming (SDP) at reasonable computational costs.

Main Results:

• SOS constraints successfully guarantee strict non-negativity for MAP and SD diffusion propagators.
• Current weak constraints are largely ineffective in ensuring non-negativity.
• Failure to enforce strict non-negativity leads to significant errors in model parameters and derived quantities (e.g., fiber orientation, mean kurtosis).
• Constraint violations are primarily attributed to measurement noise, suggesting the need for constrained optimization.

Conclusions:

• Sum of squares constraints provide a robust method for enforcing diffusion propagator non-negativity in dMRI.
• Constrained optimization should be considered standard practice for dMRI model reconstruction to ensure accurate results.
• Noise mitigation remains a challenge, highlighting the importance of robust constrained fitting.