Diffusion kurtosis imaging (DKI) provides a quantitative measure of the “tailed-ness” of diffusion. It is computed as follows: $$$S(\textbf{b})=S_{0}e^{-\textbf{bD}+\frac{1}{6}\textbf{b}^2\textbf{D}^2\textbf{K}}$$$, and is based on the Taylor series expansion of the diffusion signal under a narrow pulse assumption, truncated at the second term. The quadratic structure of the exponent means that the model has a minimum at, and is applicable only up to, b=3/DK s/mm². In this work, we show that the anisotropic DKI model corresponds to a 6-D Gaussian distribution of diffusivities, and that the undesirable behaviour at high b-values arises from the tails of this distribution with negative diffusivity. We propose instead to model diffusion as a truncated Gaussian. Our model approximates the DKI model at low b-values, but avoids the undesirable quadratic behaviour. This is demonstrated in Figure 1.

The model is an extension of the statistical model of Yablonskiy et al.$$$^2$$$, modified to allow fitting of diffusion MRI data in 3-D and account for anisotropic diffusion. It can be used to estimate diffusion kurtosis without the need for setting a b-value threshold, which may be difficult to estimate, particularly in highly anisotropic tissue. Additionally, fitting models using all acquired data will allow for more noise robustness, compared to fitting only a subset.

Let the diffusivity $$$\textbf{D}=[D_{xx}, D_{xy}, ..., D_{zz}] \rm $$$ be represented by a 6-D PDF $$$P(\textbf{D})$$$. The signal defined as $$$S(\textbf{b})=S_0\int{P(\textbf{D})} \it e^{-\textbf{bD}}d\textbf{D}$$$ where $$$\textbf{b}$$$ is the b-matrix. Assuming that $$$P(\textbf{D})$$$ follows a multivariate normal distribution yields the diffusion kurtosis model:

$$S(\textbf{b})=S_0\int{\frac{1}{(2\pi)^3\mid\bf{\Sigma}\mid^{\frac{1}{2}}}e^{-\frac{1}{2}\bf{(D-\mu)'\Sigma^{-1}}(D-\mu)}e^{-\textbf{bD}}d\textbf{D}}=S_0e^{-\bf{b'\mu}+\rm\frac{1}{2}\bf{b'\Sigma b}}$$

We propose to instead model P(D) as a truncated multivariate normal distribution, and instead fit the following model to the signal:

$$S(\textbf{b})=S_0 \frac{\Phi(\bf\mu-\Sigma b, \Sigma\rm)}{\Phi(\bf\mu,\Sigma\rm)}e^{-\bf{b'\mu}+\rm\frac{1}{2}\bf{b'\Sigma b}}$$

where $$$\Phi(\bf\mu,\Sigma\rm)$$$ is the cumulative distribution function of $$$P(\textbf{D})$$$ over $$$D_{xx}, D_{yy}, D_{zz}\in\mathbb{R}^+$$$, $$$D_{xy}, D_{xz}, D_{yz}\in\mathbb{R}$$$. The kurtosis of the truncated normal model is similar to the 1D case$$$^3$$$, and is given by

$$\bf K=\rm\frac{3(\bf \Sigma \rm - \bf\bar{D}\rm^T \bf \bar{D}\rm + \bf \bar{D}^TD \rm)}{\bf \bar{D}\rm^T\bf\bar{D}}$$

where $$$\bf \bar{D} \rm = \int \bf D \rm . \it P(\bf D) \it d\bf D$$$ is the mean value of $$$\bf D$$$ (analogue of the apparent diffusion coefficient).

First, we demonstrate the stability of the proposed model on simulated data. A bi-exponential curve, $$$S(\bf b \rm)=\it ve^{-\bf b \it D_1}+(1-v)e^{-\bf b \it D_2}$$$ was generated using the following parameters: $$$v=0.7$$$, $$$D_1=1.1*10^{-3}$$$, $$$D_2=0.7*10^{-3}mm^2/s$$$. The b-values were arrayed between 0 and 10 000 $$$s/mm^2$$$ in steps of 100. The diffusion kurtosis and proposed models were fit to the data, truncated to a maximum b-value $$$b_m$$$. Kurtosis was computed using each model.

One heart from a female Sprague-Dawley rat was excised, fixed and embedded in gel for MRI. Non-selective 3-D fast spin echo diffusion spectrum imaging (DSI) data were acquired on a 9.4T preclinical MRI scanner (Agilent, CA, USA). Data were sampled in q-space in a Cartesian grid over a half-sphere ($$$bmax = 10 000 s/mm^2$$$). The diffusion kurtosis model for this data was fit only to b-values up to $$$5 000 s/mm^2$$$. The proposed model was fit to all data.

The stability of the proposed model is shown in Figure 2. For b-values greater than $$$4000s/mm^2$$$, the kurtosis derived from the diffusion kurtosis model decreases with increasing $$$b_m$$$, whereas the kurtosis measurement from the proposed model is less sensitive to $$$b_m$$$.

Kurtosis maps of the cardiac diffusion data in the directions of the principal eigenvectors are shown in Figure 3. In general, the kurtosis values derived from the two models are in good agreement, but the kurtosis of the primary eigenvector is higher for the proposed model.

In this work, we have demonstrated that the diffusion kurtosis model assumes a multivariate normal distribution, and proposed a physically-constrained model that avoids undesirable quadratic behaviour by enforcing positivity on diffusivity values.

Several other distributions have been proposed for modelling diffusivity within a voxel, including the Poisson, Gamma, and log-normal distribution$$$^4$$$. However, these distributions do not consider diffusion anisotropy (i.e. they have been proposed using a univariate distribution). The truncated normal distribution has the key advantages that: (i) it permits certain variables ($$$D_{xy}, D_{xz}, D_{yz}$$$) to have negative values while constraining positivity on $$$D_{xx}, D_{yy}, D_{zz}$$$; (ii) the model approximates the DKI model at low b-values, and (iii) through bootstrap analysis, it has been shown that the majority of tissue fits a Gaussian distribution$$$^5$$$. The model allows for the accurate computation of diffusion kurtosis parameters without the need for truncation of data above a certain b-value threshold.