Jürgen Finsterbusch^{1}

The signal of diffusion-weighted MR reflects the tissue structure on a cellular or microstructure scale. Many different approaches have been proposed that aim to characterize diffusion-weighted data or derive diffusion or microstructural tissue properties from it. They could be divided into model- and data-driven approaches. Data-driven approaches either derive diffusion or structural properties directly from the data or, in a broader sense, approximate the data with equations that are “borrowed” from diffusion in simple physical systems or motivated mathematically. The most important of these approaches will be covered in this presentation.

MR experiments with diffusion weighting are sensitive to the translational mobility of the molecules contributing to the signal [1], usually unbound water in imaging or metabolites in spectroscopy. On the time scales accessible in vivo, cell membranes significantly impact this mobility, and the signal amplitude depends on the tissue structure on a cellular or microscopic scale, i.e. features that are about two orders of magnitude smaller than the spatial resolution of the experiment. This intriguing property has a significant clinical impact, e.g. for the diagnosis of acute stroke [2], but also has stimulated research to characterize microstructural properties in healthy and pathological tissue like the organization of muscle fibers in the myocard (e.g. [3]), cell shapes in cancer tissues [4], and the orientation of nerve fibers in brain white matter [5] enabling fiber tracking [6] and the mapping of the human “connectome” (e.g. [7]).

Due to the complexity of tissue and because the diffusion weighting has several degrees of freedom (gradient pulse amplitude, duration, and direction; diffusion time) of which only a few can be sampled rather sparsely in an in vivo experiment, many different approaches have been proposed that aim to characterize diffusion-weighted data or derive parameters from it that describe diffusion or microstructural properties. The approaches differ regarding the diffusion weighting parameters they are appropriate for and the physical meaning of their parameters and could be divided into model- and data-driven approaches. Model-driven approaches rely on a very simplistic model of the microstructure with only a few free parameters, e.g. cell diameters and volume fractions, for which the diffusion-weighted signal can be calculated (e.g. [8-11]). Data-driven approaches either derive diffusion or structural properties directly from the data or, in a broader sense, approximate the data with equations that are “borrowed” from diffusion in simple physical systems or motivated mathematically (e.g. [12-21]). They will be considered in more detail in the following.

The approaches solely based on the data usually allow a straight physical interpretation of diffusion properties like for q-space [12] and q-ball imaging [13]. Q-space imaging is based on a 3D sampling of the q space defined by the time integral of the diffusion gradient vector. For short gradient pulses, the Fourier transformation of these data is proportional to the diffusion displacement distribution that reflects features of the tissue microstructure. This information can also be derived from experiments sampling only the surface of a sphere (q-ball imaging). These approaches do not make any assumptions about the specific signal behavior but assume pulse durations that may not be realistic for in vivo acquisitions.

Other approaches assume a specific signal behavior and fit the corresponding equations to the data. Important examples are the estimation of diffusion coefficients (e.g. [17]), often for a mixture of two or more compartments (e.g. [18]), the diffusion tensor [5], and the diffusion kurtosis [19,20]. The equations involved usually describe diffusion in simple systems for which their free parameters have a physical meanings, typically diffusion coefficients or volume ratios. However, in a few cases, equations or parameters are more descriptive and lack a clear physical interpretation like the kurtosis [19].

Diffusion coefficients are usually estimated from acquisitions with at least two diffusion weightings under the assumption of free diffusion, i.e. an exponential signal decay (e.g. [17]) and are still the clinically most widely used parameters, e.g. to detect acute stroke [2]. For experiments covering multiple diffusion weightings over a larger range, deviations from the simple exponential decay are often considered by assuming two (or more) compartments with different volume fractions and diffusion coefficients (e.g. [18]). so-called “bi-“ or “multi-exponential” approaches.

The diffusion tensor [5] is the simplest generalization of a diffusion coefficient that considers a direction dependency (anisotropy) as present in tissues with directed structures like muscles or brain white matter. It is a symmetric rank-2 tensor represented by a 3x3 matrix with six independent elements. The average of the diagonal elements, the mean diffusivity, is rotationally invariant and the target in most clinical applications. From all elements, measures of the diffusion anisotropy can be derived, the most prominent being the fractional anisotropy (FA) [22]. It vanishes for isotropic diffusion and has its maximum of 1 for diffusion occurring in only a single direction. It depends on the integrity and the degree of orientational coherence of the directed structures, e.g. of muscle or nerve fibers. Of particular importance in such tissues is the coordinate system for which the diffusion tensor is diagonal: for a voxel with a bundle of coherent fibers, the axis with the largest diagonal value (eigenvalue) corresponds to the dominating fiber orientation which, e.g., can be used to track the course of nerve fibers in the brain (“fiber tracking”) [6].

Diffusion kurtosis is an approach explicitly considering that the diffusion in tissue is not free, i.e. restricted or hindered [19]. It appears in the next relevant, i.e. fourth, order when expanding the signal of diffusion-weighted acquisitions. The (excess) kurtosis parameter describes the degree of deviation of the diffusion displacement distribution from a Gaussian function, i.e. free diffusion. It vanishes for free diffusion and is negative for distributions that have reduced tails and a less sharp maximum. In the presence of anisotropic diffusion, the kurtosis term yields a tensor of rank 4 from which a mean kurtosis can be derived [20].

Almost all data-driven approaches are based on assumptions with respect to the parameters of the diffusion weighting or the expected signal behavior that are more or less significantly violated in vivo. As a consequence, the parameters are not reliable, quantitative measures but usually depend significantly on the parameters chosen for the diffusion weighting making identical acquisition protocols mandatory for a profound comparison of values. Furthermore, with very few exceptions, the parameters do not provide direct insight into properties of the tissue microstructure, and their physical meaning may have been lost or significantly weakened. Nevertheless, they are widely used and have been shown to provide sensitive markers for microstructural changes or differences although their physical meaning should not be over-interpreted.