Bi-tensor Free Water Model With Positive Definite Diffusion Tensor and Fast Optimization

Diffusion tensor imaging is a widely used imaging methodology to infer the microstructure of brain tissues. When an image voxel contains partial volume of brain tissue with free water, the traditional one tensor model is not appropriate. A bi-tensor free water elimination model has been proposed to correct for the mixing effects. Moreover, recent studies have shown that the free water volume derived from this model could be a biomarker for brain aging and numerous brain disorders such as Parkinson’s and Alzheimer’s disease. However, the problem of fitting this model is ill-posed without additional assumptions. Models by adding spatial constraints or using data from multi-shell acquisition are proposed to stabilize the fitting, but none of them restricts the diffusion tensor D to be positive definite, which is a necessary condition. In this work, we formulate the bi-tensor model fitting as an optimization problem over the space of symmetric positive definite matrices and show that the objective function is a ratio of two geodesically convex functions. We also demonstrate by simulation that the estimation may be highly biased with single-shell data in the presence of noise, so multi-shell data are needed for the fitting of the bi-tensor free water elimination model. Inspired by the Cholesky decomposition, we treat the diffusion tensor D as the product LLT where L is a lower triangular matrix. The optimization is performed on L which guarantees the positive definiteness of D. Our model are evaluated with both simulations and real human brain data. Simulation results show that the model is computationally efficient and the two-shell acquisition gives the best estimation.