Yoshitaka Masutani^{1} and Koh Sasaki^{1,2}

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.

(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.

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.