Positive definiteness of Diffusion Kurtosis Imaging

Shenglong Hu; Zheng-Hai Huang; Hong-Yan Ni; Liqun  Qi

doi:10.3934/ipi.2012.6.57

Article Contents

2012, Volume 6, Issue 1: 57-75. Doi: 10.3934/ipi.2012.6.57

This issue Previous Article Identification of obstacles using only the scattered P-waves or the scattered S-waves Next Article Inverse obstacle scattering with limited-aperture data

Positive definiteness of Diffusion Kurtosis Imaging

1.
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072
2.
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P.R.
3.
Tianjin First Central Hospital, Tianjin 300192
4.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon

Received: September 2009

Revised: October 2011

Published: February 2012

Abstract / Introduction Related Papers Cited by

Abstract

Diffusion Kurtosis Imaging (DKI) is a new Magnetic Resonance Imaging (MRI) model to characterize the non-Gaussian diffusion behavior in tissues. In reality, the term $bD_{app}-\frac{1}{6}b^2D_{app}^2K_{app}$ in the extended Stejskal and Tanner equation of DKI should be positive for an appropriate range of $b$-values to make sense physically. The positive definiteness of the above term reflects the signal attenuation in tissues during imaging. Hence, it is essential for the validation of DKI.
In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.

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Mathematics Subject Classification: Primary: 90C25, 90C33; Secondary: 35P30.

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