Synopsis

To enhance the robustness of the fiber-radial diffusional kurtosis computation, the signal decay in the Q-space can be averaged among the fiber-radial orientations. By using the radial basis functions with Gaussian basis, it is shown that the fiber-radial signal decay can be represented in a pseudo-analytic form with the Bessel function. By using in-vivo diffusion MRI data, the computation results were presented in comparison with those by diffusion kurtosis tensor of 4th order to prove the effectiveness of the proposed method.

INTRODUCTION

METHODS

(1) Fiber-Radial Signal Decay Averaging

To obtain the fiber-radial kurtosis value, we first reconstruct a smooth series of signal decays by line integral in Q-space (Fig.1). That is, we average the signal decay values on the circle with radius $$$q$$$ perpendicular to fiber direction in Q-space, similarly to the Funk-Radon transform of QBI [3]. If the fiber orientation is given as $$${{\bf{v}}_f}$$$, we can define a unique and smoothed signal decay $$${E_{\bot {\bf{v}}_f}}$$$ for a q-value $$${q}$$$ as follows.

$$E_{\bot {\bf{v}}_f}(q) =\frac{1}{{2\pi q}} {\oint_{\bf{q}\bot{{\bf{v}}_\it{f}},\parallel\bf q\parallel=\it {q} } E(\bf q)}$$

$$$E({\bf q})$$$ is the signal decay function in Q-space, which is interpolated with measured signal decays by using radial basis functions in our study.

(2) RBF Representation of Signal Decay

When a Q-space data set is given as a series of pairs of q-vector $$${\bf{q}}_\it{i}$$$ and measured signal decay value $$$E_i$$$ at $$${\bf{q}}_\it{i}$$$ $$$(i=0,…,N)$$$, we can obtain signal decay value at arbitrary location $$$\bf{q}$$$ by using radial basis functions [2] as below.

$$E({\bf q}) =\sum_{i=0}^N w_{i}\cdot \phi(\parallel\bf{q}-\bf{q}_{\it i}\parallel)$$

Note that $$${\bf q}_0={\bf{O}}, E_0=1$$$ and $$$N$$$ is the number of the measured signals duplicated by a point-symmetry assumption $$$E(-{\bf q})=E({\bf q})$$$. The weights $$$w_i$$$ are simply estimated by the matrix methods of linear least squares, and $$$\phi(r)$$$ is the basis function.

(3) Gaussian Basis and Fiber-Radial Signal Decay with the Bessel Functions

When we use the Gaussian basis $$$\phi(r)=e^{-(\epsilon \cdot r)^2}$$$, we get $$$E_{\bot {\bf v}_{\it f} }(q)$$$ in a pseudo-anlaytic form as follows.

$$E_{\bot {\bf{v}_\it{f}}}(q) =\frac{1}{{2\pi q}} {\oint_{\bf{q}\bot {\bf{v}_\it{f}},\parallel\bf{q}\parallel=\it {q} } \sum_{i=0}^N w_{i}\cdot e^{-(\epsilon\cdot \parallel\bf{q}-\bf{q_{\it i}}\parallel)^{2}}}\\=\frac{1}{{2\pi q}} \sum_{i=0}^N w_{i}\cdot e^{-\epsilon^2(q^2+{\parallel\bf{q_{\it i}}\parallel}^2)}\int_{0}^{2\pi} e^{2\epsilon^2\sqrt{\parallel\bf{q_{\it i}}\parallel^2-({\bf{v}_\it{f}}\cdot \bf{q_{\it i}})^2}\cos\theta}d\theta\\=\frac{1}{q} \sum_{i=0}^N w_{i}\cdot e^{-\epsilon^2(q^2+{\parallel\bf{q_{\it i}}\parallel}^2)}\cdot I_0(2\epsilon^2 \small {\sqrt{\parallel\bf{q_{\it i}}\parallel^2-({\bf{v}_\it{f}}\cdot \bf{q_{\it i}})^2}})$$

$$$I_0(\cdot)$$$ is the 0-th order modified Bessel function of the first kind, that is, $$$I_0(z)=\sum_{k=0}^{\infty}\frac{(\frac{1}{4}z^2)^k}{(k!)^2}$$$. Finally, we can obtain the kurtosis value with series of signal decays $$$E_{\bot \bf{v}_\it{f}}(q)$$$ by using the closed-form expression [4] with conversion of q-values to b-values. For fiber orientation estimation, a diffusion tenor model of the 2nd order is employed in this study.

RESULTS AND DISCUSSION

CONCLUSION

Acknowledgements

References

1. Jensen JH, Helpern JA, Ramani A, et al. Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med 2005; 53: 1432–1440.

2. Buhmann MD. Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, UK, 2003.

3. Tuch DS. Q-ball imaging. Magn Reson Med. 2004 Dec;52(6):1358-72.

4. Masutani Y and Aoki S. Fast and robust estimation of diffusional kurtosis imaging (DKI) parameters by general closed-form expressions and their extensions. Magn Reson Med Sci 2014;13:97–115.