# American Institute of Mathematical Sciences

February  2012, 6(1): 57-75. doi: 10.3934/ipi.2012.6.57

## Positive definiteness of Diffusion Kurtosis Imaging

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

Received  September 2009 Revised  October 2011 Published  February 2012

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.
Citation: Shenglong Hu, Zheng-Hai Huang, Hong-Yan Ni, Liqun Qi. Positive definiteness of Diffusion Kurtosis Imaging. Inverse Problems & Imaging, 2012, 6 (1) : 57-75. doi: 10.3934/ipi.2012.6.57
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