Physicists, technologists and clinicians, who wish to understand the basics and more advanced concepts and applications of diffusion MRI.

- Explore the non-Gaussian properties of the diffusion signal in biological tissue by both varying the diffusion wave vector q and diffusion time t

- Comprehend the concept of q-space imaging, diffusion tensor imaging (DTI), diffusion kurtosis imaging (DKI), and other high-order methods

- Understand the difference between diffusion signal representations and biophysical models

- Appreciate how the tissue microstructure and corresponding relevant length scales can be probed directly by varying the diffusion time

In contrast to simple liquids, which exhibit Gaussian and isotropic diffusion, the diffusion signal in biological tissue is restricted and anisotropic. In this lecture, we will explore the non-Gaussian diffusion signal by varying both the gradient wave vector q and the diffusion time t, the time over which the molecules diffuse. While both q and t are typically collectively grouped together as the diffusion weighting, b-value in 1D or b-matrix in 3D, we will explore both avenues here separately to illustrate their different specific sensitivities, as illustrated in Figure 1.

q-space imaging (QSI), pioneered by Callaghan in 1988 [1], measures the diffusion signal as function of the wave vector q=δg, where g is the gradient vector corresponding to the direction and magnitude of the diffusion gradient, and δ is the gradient pulse duration. (q is here analogous to the vector k which is at the basis of MRI theory.) if the diffusion gradients are infinitely narrow ($$$\delta \rightarrow 0 $$$), the signal attenuation $$$\it{S}_{\bf{q}}(\it{t})$$$ of the diffusion experiment is given by

$$\frac{S_{\mathbf{q}}(t)}{S_{q=0}(t)} = \int d\boldsymbol{\mathbf{r}} \, e^{-i\mathbf{qr}} \int \frac{d\mathbf{r}_{0}}{V} {\cal G}_{\mathit{t};\mathbf{r}_{0}+\mathbf{r},\mathbf{r}_{0}}\equiv \mathit{G}\mathit{_{\mathit{t},\mathbf{q}}} \qquad (1)$$

where $$${\cal G}_{\mathit{t};\mathbf{r}_{0}+\mathbf{r},\mathbf{r}_{0}}$$$ is the diffusion propagator around $$$\bf{r}_{0}$$$, being the conditional probability that a molecule (initially at $$$\bf{r}_{0}$$$) is displaced over r in time t, a.k.a. the probability density of molecular displacement. In other words, the diffusion-weighted signal is the Fourier transform of the voxel-averaged propagator $$$G_{t,{\bf r}} = \int\! \frac{d{\bf r}_0}{V}\, {\cal G}_{t; {\bf r}+{\bf r}_0, {\bf r}_0}$$$.

Mathematically, the diffusion propagator admits a regular expansion of $$$\ln S_{\bf{q}}(t)$$$ in terms of $$$q^2$$$, a.k.a. the cumulant expansion [2]:

$$\mathit{G}_{\mathit{t},\mathbf{q}}=e^{-\mathit{bD(t)}+\frac{1}{6}\mathit{K}[\mathit{bD(t)}]^{2}+\vartheta (\mathit{b}^{3})}\,, \qquad \it{b}=\it{q}^2\it{t}\,, \qquad (2) $$

whereby the 1^st and 2^nd-order yields the diffusion coefficient and kurtosis, as illustrated in Figure 2, and $$$b$$$ is the b-value (the diffusion weighting) in the narrow pulse limit.

Diffusion tensor imaging (DTI): At low diffusion weighting (bD<< 1/K) the diffusion signal can be well represented by the lowest-order term in Eq (2), $$$\ln S \simeq –bD(t)$$$. The diffusion coefficient $$$D(t) \equiv \langle x^2(t)\rangle/2t$$$ is a measure of how fast mean squared molecular displacement grows with time. Since biological tissues such as white matter are anisotropic, this Gaussian part is, in general, a rank-2 diffusion tensor D, characterized by a $$$3\times3$$$ symmetric matrix with 6 independent parameters. The linear estimation problem of D, referred to as diffusion tensor imaging (DTI), has been solved by Basser et al in 1994 [3]. It requires measurements for at least one b ≠0 value (often set at 1000 s/mm² for brain) along at least six non-collinear directions in addition to the b = 0 (unweighted) image. The diffusion tensor is usually visualized by an ellipsoid, whereby the axes of the ellipsoid coincide with the eigenvectors and the corresponding eigenvalues are taken as radii. A number of scalar indices can be derived from the diffusion tensor, including the widely used mean diffusivity (MD) and fractional anisotropy (FA).

DTI is commonly used to visualize white matter anisotropy and has shown to be very useful in clinical applications [4, 5]. Though, it is important to note that DTI estimates only the lowest-order term of the cumulant series (2), thereby probing the Gaussian properties of the propagator, yielding no information about higher-order terms. Practically, one should always choose the b-range such that the higher-order terms do not bias the estimation of D. Similarly, the diffusion tensor is not able to accurately describe fiber geometries in voxels containing multiple fiber tracts in multiple directions, e.g., crossing fibers.

Diffusion kurtosis imaging (DKI) extends conventional DTI by estimating the kurtosis of the water diffusion probability distribution function based on the diffusion signal acquired at slightly higher diffusion weighting (b-value ≤ 2000 s/mm²). In case of diffusion anisotropy, DKI requires the introduction of a rank-4 diffusional kurtosis tensor in addition to the rank-2 diffusion tensor used in DTI. The linear estimation problem of both the diffusion and kurtosis tensors, via the expansion up to b², referred to as diffusion(al) kurtosis imaging (DKI) , has been solved by Jensen et al in 2005 [6]. The number of parameters is now 6 (diffusion tensor) + 15 (kurtosis tensor) = 21, hence at least 21 q-space points spread over at least two b ≠ 0 values (often set at 1000 and 2000 s/mm² for brain) and at least 15 non-collinear directions in addition to the b = 0 (unweighted) image are required. The weights for unbiased estimation of diffusion and kurtosis tensors for non-Gaussian NMR noise were recently proposed [7].

Qualitatively, a large diffusional kurtosis suggests a high degree of diffusional heterogeneity and/or microstructural complexity [8]. The extra information provided by DKI can also resolve intra-axonal fiber crossings and thus improve fiber tractography of white matter [9].

Diffusion signal representations versus biophysical models: While both DTI and DKI are popular methods that empirically have shown sensitivity to pathological changes in the brain and body, it is important to note that both methods are examples of diffusion signal representations (i.e., convenient mathematical functions which fit the data well). To become specific to microstructure, many biophysical models for specific tissue types have been proposed, as will be discussed in the next lecture. These models have concrete assumptions about tissue parameters and have, per definition, limited range of applicability.

Relation between q-space and the fiber orientation distribution function (ODF): By sampling the signal attenuation $$$\it{S}_{\bf{q}}(\it{t})$$$ for a series of q-space points, the measured profile can be directly related to molecular displacement probability distribution $$$G_{\mathit{t};\mathbf{r}}$$$ by an inverse fourier transform (DSI-method [10]). The anisotropy of the diffusional ODF (dODF), derived from $$$G_{\mathit{t};\mathbf{r}}$$$, and originating from the underlying tissue anisotropy, can be represented, e.g., using spherical harmonics expansion, and is useful for fiber tracking [11].

Ultimately, one would want to obtain the fiber ODF (fODF) directly from the signal $$$\it{S}_{\bf{q}}(\it{t})$$$. Since the signal $$$\it{S}_{\bf{q}}(\it{t})$$$ is a convolution of the fODF with the response of an elementary well-aligned fiber tract, one has to assume (or determine) a specific fiber response, from which the fiber ODF can be obtained using deconvolution methods [12]. The elementary fiber response can be found [13] or postulated empirically, or modeled [14].

Aside from DTI and DKI sampling protocols, there are several comprehensive q-space sampling methods, which involve both incrementing the gradient strength and changing its direction such that several spherical shells are acquired in q-space. This acquisition method is known as the high angular resolution diffusion imaging (HARDI) technique [15]. In the case of Q-ball imaging, only one spherical shell is sampled, which reduces the scan time [16].

The unique advantage of diffusion MRI arises from the sensitivity of water diffusion to the tissue microstructure, and holds the promise of quantifying relevant length scales, such as the cell size and packing correlation length. Rather than by probing q space, where 1 μm diameter axons and dendrites require q values prohibitively large for in vivo human measurements, relevant micrometer-level length scales can be clinically measured at low q by varying t. Since the diffusion coefficient in a given direction is a measure of the mean squared displacement, i.e. , we can vary the length scale probed by the water molecules by varying t. With increasing t, water molecules encounter more hindrances and restrictions to their diffusion paths, such as cellular walls and myelin, and therefore the resultant measured diffusion coefficient will decrease [17-19]. The biophysical origin of observed time-dependence reflects the non-Gaussian nature of diffusion in at least one tissue compartment, and thereby potentially enables novel microstructural contrasts, as recently demonstrated for time dependent diffusion observed in vivo in brain white matter [20].