Diffusion kurtosis imaging (DKI)¹ is an advanced neuroimaging modality that extends the well-known diffusion tensor imaging (DTI) model by incorporating the kurtosis of the water diffusion probability distribution function. DKI parameters and their derived metrics are known to be sensitive to certain brain physiological changes, especially compared to DTI^2-3. However, in diffusion MRI, image voxels are relatively large (2 to 3 mm), making them susceptible to partial volume effects. This is especially true for brain structures close to cerebrospinal fluid (CSF) regions since the diffusivity of water in CSF can easily be up to 3 times larger than the water diffusivity in white matter brain tissue. Ignoring these effects may lead to large biases in the estimation of diffusion properties of brain tissue⁴.

In this work, we extend the DKI model to a bi-exponential model to incorporate a free water signal fraction. Earlier work in which the DTI model is extended to include a free water component indicates promising results^5-7. However, the fitting problem of the bi-tensor models turned out to be ill-conditioned, which calls for parameter estimation methods that include prior information⁸.

We propose a DKI+CSF model that describes the diffusion weighted signal $$$S_i$$$ with a bi-exponential function consisting of a tissue compartment and a CSF compartment linked by a relative volume fraction $$$f$$$:

$$S_i=S_0\left(\left(1-f\right)\exp\left(-b\sum_{i,j=1}^3g_ig_jD_{ij}+\frac{b^2}{6}\left(\sum_{i,j=1}^{3}\frac{D_{ii}}{3}\right)^2\sum_{i,j,k,l=1}^{3}g_ig_jg_kg_lW_{ijkl}\right)+f\exp\left(-bd\right)\right)$$

with $$$S_0$$$ the signal without diffusion weighting, $$$b$$$ and $$$\bf g$$$ the diffusion weighting strength and unit gradient, respectively, $$$\bf D$$$ the diffusion tensor, $$$\bf W$$$ the kurtosis tensor and $$$d=3\hspace{2mm}\mu m^2/ms$$$ the diffusivity of free water at body temperature. We propose to estimate the parameters of this model using a Bayesian approach with a shrinkage prior (BSP)⁹. This prior assumes that the spatial variation of the DKI+CSF model parameters can be described by a Gauss distribution. The prior distribution parameters are determined from the data, omitting the need for any user-defined parameters. Furthermore, a Rician data likelihood function is assumed and a Markov chain Monte Carlo implementation is used to compute the posterior mean estimates.

Our BSP based estimator was compared to a non-linear least squares (NLS) and a maximum likelihood estimator (MLE), both with fixed constraints, in a Monte Carlo simulation and a real data experiment. A weighted linear least squares (WLLS)¹⁰ fit of the standard DKI model is also included. The simulation consisted of 1000 voxels, each with 151 diffusion weighted images disturbed by Rician noise. The underlying true parameters were distributed according to values taken from a WM region in a real data set. For the real human data, a healthy volunteer was scanned with the following number of directions per respective b-value shell: ($$$ms/\mu m^2$$$): $$$6\times b=0;\hspace{2mm}25\times b=0.7;\hspace{2mm}45\times b=1.2;\hspace{2mm}75\times b=2.8$$$.

The results of the simulation experiment suggest that the proposed BSP estimator leads to a more accurate, precise and robust estimation of the DKI+CSF model parameters (see Fig. 1-2). This can be seen in the fractional anisotropy (FA), mean diffusivity (MD) and mean kurtosis (MK) error plots where in general the BSP error distribution is narrower, more centered around 0 and has fewer estimates near the user-defined constraints of NLS and MLE. In the real data FA maps (see Fig. 3), NLS and MLE also produce many FA values that are unrealistically high with respect to what can be reasonably expected in those regions, as opposed to BSP estimates which appear much smoother.

When we compare the DKI parameter estimates from BSP with the DKI+CSF model to a WLLS fit of the standard DKI model, it is clear that by modeling the CSF fraction, the true underlying tissue diffusion parameters can be estimated more accurately and precisely (see Fig. 1-2). Ignoring the CSF fraction leads to large biases towards a higher FA and lower MD and MK.