# American Institute of Mathematical Sciences

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
##### References:
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Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy,", Oxford University Press, (1993). [7] G. Chesi, A. Garulli, A. Tesi and A. Vicino, "Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems,", Lecture Notes in Control and Information Sciences, 390 (2009). [8] D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,", Graduate Texts in Mathematics, 185 (1998). [9] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Apparent diffusion coefficients from hogh angular resolution diffusion imaging: Estimation and applications,, Magnetic Resonance in Medicine, 56 (2006), 395. doi: 10.1002/mrm.20948. [10] L. Frank, Characterization of anisotropy in high angular resolution diffusionweighted MRI,, Magnetic Resonance in Medicine, 47 (2002), 1083. doi: 10.1002/mrm.10156. [11] A. Ghosh, M. Descoteaux and R. Deriche, Riemannian framework for estimating symmetric positive definite 4th order diffusion tensors,, in, (2008), 858. [12] H. Jensen, J. Helpern, A. Ramani, H. Lu and K. Kaczynski, Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging,, Magnetic Resonance in Medicine, 53 (2005), 1432. doi: 10.1002/mrm.20508. [13] M. Lazar, J. Jensen, L. Xuan and J. Helpern, Estimation of the orientation distribution function from diffusional kurtosis imaging,, Magnetic Resonance in Medicine, 60 (2008), 774. doi: 10.1002/mrm.21725. [14] C. Liu, R. Bammer, B. Acar and M. Moseley, Characterizing non-gaussian diffusion by using generalized diffusion tensors,, Magnetic Resonance in Medicine, 51 (2004), 924. doi: 10.1002/mrm.20071. [15] C. Liu, R. Bammer and M. Moseley, Generalized diffusion tensor imaging (gdti): A method for characterizing and imaging diffusion anisotropy caused by non-gaussian diffusion,, Israel Journal of Chemistry, 43 (2003), 145. doi: 10.1560/HB5H-6XBR-1AW1-LNX9. [16] C. Liu, S. Mang and M. Moseley, In vivo generalized diffusion tensor imaging (GDTI) using higher-order tensors (HOT),, Magnetic Resonance in Medicine, 63 (2010), 243. [17] H. Lu, H. Jensen, A. Ramani and J. Helpern, Three-dimensional characterization of non-Gaussian water diffusion in humans using diffusion kurtosis imaging,, NMR in Biomedicine, 19 (2006), 236. [18] E. Ozarslan and T. Mareci, Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution imaging,, Magnetic Resonance in Medicine, 50 (2003), 955. [19] L. Qi, Eigenvalues of a real supersymmetric tensor,, Journal of Symbolic Computation, 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007. [20] L. Qi, D. Han and E. Wu, Principal invariants and inherent parameters of diffusion kurtosis tensors,, Journal of Mathematical Analysis and Applications, 349 (2009), 165. doi: 10.1016/j.jmaa.2008.08.049. [21] L. Qi, Y. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors,, Journal of Computational and Applied Mathematics, 221 (2008), 150. doi: 10.1016/j.cam.2007.10.012. [22] L. Qi and Y. Ye, Space tensor conic programming,, Technical Report, (2009). [23] L. Qi, G. Yu and E. Wu, Higher order positive semidefinite diffusion tensor imaging,, SIAM Journal on Imaging Sciences, 3 (2010), 416. doi: 10.1137/090755138. [24] E. Sigmund, M. Lazar, J. Jensen and J. Helpern, In vivo Imaging of Kurtosis Tensor Eigenvalues in the Brain at 3 T,, in, (2009). [25] E. Stejskal and J. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient,, Journal of Chemical Physics, 42 (1965), 288. doi: 10.1063/1.1695690. [26] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 625. doi: 10.1080/10556789908805766. [27] K. Toh, M. Todd and R. Tütüncü, SDPT3-a Matlab software package for semidefinite programming, version 1.3. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 545. doi: 10.1080/10556789908805762. [28] D. Tuch, Q-ball imaging,, Magnetic Resonance in Medicine, 52 (2004), 1358. doi: 10.1002/mrm.20279.

show all references

##### References:
 [1] E. Ahrens, D. Laidlaw, C. Readhead, C. Brosnan, S. Fraser and R. Jacobs, MR microscopy of transgenic mice that spontaneously acquire experimental allergic encephalomyelitis,, Magnetic Resonance in Medicine, 40 (1998), 119. doi: 10.1002/mrm.1910400117. [2] Y. Assaf and O. Pasternak, Diffusion tensor imaging (DTI)-based white matter mapping in brain research: A review,, Journal of Molecular Neuroscience, 34 (2008), 51. doi: 10.1007/s12031-007-0029-0. [3] A. Barmpoutis, B. Jian, B. Vemuri and T. Shepherd, Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI,, in, (2007), 308. [4] P. Basser and D. Jones, Diffusion-tensor MRI: Theory,experimental design and data analysis: A technical review,, NMR in Biomedicine, 15 (2003), 456. [5] P. Basser, J. Mattiello and D. LeBihan, Estimation of the effective self-diffusion tensor from the NMR spin echo,, Journal of Magnetic Resonance Series B, 103 (1994), 247. doi: 10.1006/jmrb.1994.1037. [6] P. Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy,", Oxford University Press, (1993). [7] G. Chesi, A. Garulli, A. Tesi and A. Vicino, "Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems,", Lecture Notes in Control and Information Sciences, 390 (2009). [8] D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,", Graduate Texts in Mathematics, 185 (1998). [9] M. Descoteaux, E. Angelino, S. Fitzgibbons and R. Deriche, Apparent diffusion coefficients from hogh angular resolution diffusion imaging: Estimation and applications,, Magnetic Resonance in Medicine, 56 (2006), 395. doi: 10.1002/mrm.20948. [10] L. Frank, Characterization of anisotropy in high angular resolution diffusionweighted MRI,, Magnetic Resonance in Medicine, 47 (2002), 1083. doi: 10.1002/mrm.10156. [11] A. Ghosh, M. Descoteaux and R. Deriche, Riemannian framework for estimating symmetric positive definite 4th order diffusion tensors,, in, (2008), 858. [12] H. Jensen, J. Helpern, A. Ramani, H. Lu and K. Kaczynski, Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging,, Magnetic Resonance in Medicine, 53 (2005), 1432. doi: 10.1002/mrm.20508. [13] M. Lazar, J. Jensen, L. Xuan and J. Helpern, Estimation of the orientation distribution function from diffusional kurtosis imaging,, Magnetic Resonance in Medicine, 60 (2008), 774. doi: 10.1002/mrm.21725. [14] C. Liu, R. Bammer, B. Acar and M. Moseley, Characterizing non-gaussian diffusion by using generalized diffusion tensors,, Magnetic Resonance in Medicine, 51 (2004), 924. doi: 10.1002/mrm.20071. [15] C. Liu, R. Bammer and M. Moseley, Generalized diffusion tensor imaging (gdti): A method for characterizing and imaging diffusion anisotropy caused by non-gaussian diffusion,, Israel Journal of Chemistry, 43 (2003), 145. doi: 10.1560/HB5H-6XBR-1AW1-LNX9. [16] C. Liu, S. Mang and M. Moseley, In vivo generalized diffusion tensor imaging (GDTI) using higher-order tensors (HOT),, Magnetic Resonance in Medicine, 63 (2010), 243. [17] H. Lu, H. Jensen, A. Ramani and J. Helpern, Three-dimensional characterization of non-Gaussian water diffusion in humans using diffusion kurtosis imaging,, NMR in Biomedicine, 19 (2006), 236. [18] E. Ozarslan and T. Mareci, Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution imaging,, Magnetic Resonance in Medicine, 50 (2003), 955. [19] L. Qi, Eigenvalues of a real supersymmetric tensor,, Journal of Symbolic Computation, 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007. [20] L. Qi, D. Han and E. Wu, Principal invariants and inherent parameters of diffusion kurtosis tensors,, Journal of Mathematical Analysis and Applications, 349 (2009), 165. doi: 10.1016/j.jmaa.2008.08.049. [21] L. Qi, Y. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors,, Journal of Computational and Applied Mathematics, 221 (2008), 150. doi: 10.1016/j.cam.2007.10.012. [22] L. Qi and Y. Ye, Space tensor conic programming,, Technical Report, (2009). [23] L. Qi, G. Yu and E. Wu, Higher order positive semidefinite diffusion tensor imaging,, SIAM Journal on Imaging Sciences, 3 (2010), 416. doi: 10.1137/090755138. [24] E. Sigmund, M. Lazar, J. Jensen and J. Helpern, In vivo Imaging of Kurtosis Tensor Eigenvalues in the Brain at 3 T,, in, (2009). [25] E. Stejskal and J. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient,, Journal of Chemical Physics, 42 (1965), 288. doi: 10.1063/1.1695690. [26] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 625. doi: 10.1080/10556789908805766. [27] K. Toh, M. Todd and R. Tütüncü, SDPT3-a Matlab software package for semidefinite programming, version 1.3. Interior point methods,, Optimization Methods and Software, 11/12 (1999), 545. doi: 10.1080/10556789908805762. [28] D. Tuch, Q-ball imaging,, Magnetic Resonance in Medicine, 52 (2004), 1358. doi: 10.1002/mrm.20279.
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