The phenomenon of diffusion was first described by Robert Brown in the 19^th century, after observing particles of pollen moving in a water solution [1]. This random translational motion arises from the thermal energy carried by particles. As demonstrated by Einstein [2], in the absence of obstacles (free diffusion), the mean square displacement is proportional to the diffusion time τ and the diffusion coefficient D of the medium:

$$\langle r^2 \rangle =6D\tau$$

where r represents the three-dimensional displacement during the period τ. To give an idea, since the diffusion coefficient of water molecules at room temperature is 2.3×10⁻³ mm²/s [3], the root mean square displacement for a diffusion time of 10 millisecond (on the order of that used in MR experiments) is on the order of 10μm. This length is much smaller than the typical size of imaging voxels (about a millimeter), and comparable to cellular dimensions. The reason why diffusion-weighted imaging (DWI) is such a powerful tool is that it allows us to use water molecules to probe tissue structures much smaller than the achievable voxel sizes.

Hahn first noted the effects of diffusion on MR signals in 1950 [4]. To describe them, in 1956 Torrey added an extra term to the Bloch Equations, creating the well-known Bloch-Torrey equation [5]. Stejskal and Tanner were the first to propose the use of a pulsed-gradient spin echo sequence (PGSE) for diffusion measurements in 1965 [6]. The Stejskal-Tanner diffusion module consists of a pair of gradients with the same amplitude G and duration δ with an 180° RF pulse applied in between – Figure 1A. The interval between application of the two diffusion gradients is often labelled as Δ.

By solving the Bloch-Torrey equation it is possible to demonstrate that the MR signal will be multiplied by an additional diffusion term:

$$S(b)=S_0.exp(-bD)$$

where S₀ is the T2-weighted signal measured at the echo time TE in the absence of diffusion gradients, and b describes the sequence's sensitivity to diffusion which can be calculated from:

$$b=\gamma^2\int_{0}^{TE} (\int_{0}^{t'}G(t')dt')^2dt$$

which in the case of the PGSE module results in:

$$b=(\gamma G \delta)^2(\Delta - \frac{\delta}{3})$$

Note that the MR signal is only sensitive to diffusion occurring along the direction of the diffusion gradient as shown in Figure 2.

One of the main challenges of DWI results from the fact that a higher diffusion contrast implies stronger signal attenuation. Additionally, as achieving a higher b-value typically requires longer diffusion gradients, longer echo times are needed when using the PGSE module. DW images thus typically suffer from a low signal-to-noise ratio and this is further aggravated with increasing b-values.

An alternative to PGSE is to use a stimulated-echo diffusion-sensitisation module (Figure 1B). By storing the magnetization in the longitudinal axis after applying the first diffusion gradient, it is possible to increase the b-value by increasing the time interval between the two gradients instead of prolonging them, thus limiting T2-decay. The disadvantage of this approach is the 50% signal penalty inherently associated to stimulated echoes.

Given that the DW signal needs to detect very minute molecular motion, it is extremely sensitive also to any other type of motion. Subject motion during the PGSE module affects the phase of the transverse magnetization, resulting in a spatially varying phase pattern [7].

By sampling k-space following a single excitation, inconsistencies between k-space segments can be avoided. Single-shot Echo-Planar Imaging (EPI) is therefore still the sequence most commonly used to acquire DW images [8]. An additional advantage of single-shot EPI is that it enables the acquisition of several images, with varying b-values and/or diffusion directions, within reasonable scanning times.

On the other hand, the T2* signal decay occurring during the long EPI readout (of the order of 100 ms) limits the achievable image resolution. To increase it further, multi-shot approaches can be used, but require calibration of the motion-induced phase patterns. The use of navigator echoes was initially proposed [9,10] but recently a promising alternative based on parallel imaging has been suggested [11].

Due to its long readout window, single-shot EPI is also extremely sensitive to eddy currents. As diffusion-sensitisation requires high amplitude gradients applied for a long period, eddy current fields are produced which may linger during the readout period and result in geometric image distortions [12]. The doubly-refocused module (Figure 1C) is often used as an alternative to the PGSE module since by adjusting the duration of the diffusion gradients it is possible to null dominant eddy current fields [13].

The first application of DWI, in which it continues to have a significant impact, was in stroke imaging. Moseley et al. demonstrated that diffusivity is significantly decreased minutes after an infarct, with the affected areas appearing brighter in DW images with no visible changes in conventional images [14]. Since it allows probing tissue microstructure, DWI is often used for lesion characterization both in the brain (e.g. in neurodegenerative diseases [15] and tumours [16]) and in other parts of the body. A recent review of DWI body imaging can be found in [17].