Diffusion Weighted Imaging

Although my PhD work involved functional MRI, I have not got a chance to work on diffusion-weighted imaging (DW). But more recently, more and more studies combine both MRI and DW together for more comprehensive measures.

Studies have shown that many developmental, aging and pathological processes influence the microstructural composition and architecture of the brain. As a primary consequence, water diffusion within the tissues will be altered by changes in the tissue microstructure and organization. Diffusion-weighted imaging (DW) becomes a popular tool to be used along with functional MRI. Another term closely associated with DW is diffusion tensor imaging (DTI). Whereas DWI is the raw data, containing diffusion maps in different directions, DTI is the process by which the raw data is transformed to calculate tensors (matrices that summarize the diffusion pattern in each voxel). In practice, DTI can be used for tractography as useful estimates of white matter fibers, albeit it does not capture or represent the actual myelinated fibers of the brain.

Water Diffusion and Tensor
Generally speaking, a diffusion model describes a transport phenomenon of a molecule from one location to another over time. In DW imaging, we are modeling the diffusion of water molecules, which are abundant inside a biological system. Water diffusion is very much influenced by the interactions with extracellular space, cellular membranes, and organelles. Cellular membranes hinder the diffusion of water, thus decreasing the mean squared displacement.

In fibrous tissues including white matter, water diffusion is relatively unrestricted along the fiber orientation. Conversely, it is highly restricted and hindered in the directions perpendicular to the fibers. In other words, the diffusion inside fibrous tissue is said to be directional-dependent or anisotropic. This phenomenon has been modelled using multivariate Gaussian distribution by Basser et al. (1994). In that model, a 3 × 3 variance-covariance matrix is used to capture diffusivity in a 3D space. The matrix is also known as a diffusion tensor D. The diagonal elements denote the diffusion variances along the X, Y and Z axes, and the off-diagonal elements are the covariance terms. For example, variability along X and Y directions are denoted by D_xy. Diagonalization of the diffusion tensor yields two outputs:

1. Eigenvalues λ, which describes the directions of the diffusion. Isotropic diffusion is represented with equal eigenvalues in the three directions. Conversely, the diffusion is anisotropic when the eigenvalues differ significantly.
2. Corresponding eigenvectors (e), apparent diffusivities along the axes of principal diffusion. The diffusion tensor can be visualized in Fig-1, with the eigenvectors defining the directions of the principal axes, with the radius defined by the eigenvalues. 

The solid shape shown in Fig-1 is also known as an ellipsoid, whose principal axes are in the direction of maximum diffusivity. In an isotropic condition (e.g. in CSF and gray matter), the ellipsoid will resemble a ball. However, in the neural tissues (e.g. white matter), the eigenvectors are more or less parallel to the tract orientation. Thus, diffusion tensors have been shown to be a sensitive tool to detect abnormality in the neural tissues.

Fig-1: (Top left) An illustration of fiber tracts with a certain orientation w.r.t scanner coordinate system. The fiber tissues exhibit directional dependence (anisotropy) on water diffusion. (Top right) The 3D diffusivity is modeled as a tensor, and visualized as an ellipsoid whose orientation is characterized by 3 eigenvectors (ϵ) and whose magnitude or length is characterized 3 eigenvalues (λ). The eigenvectors represent the major, medium, and minor axes of the ellipsoid and the eigenvalues represent the diffusivities in these three directions, respectively. (Bottom) This ellipsoid model is shown as a tensor, requiring a procedure known as matrix diagonalization. The major eigenvector (associated with the largest of the three eigenvalues), reflects the direction of maximum diffusivity, which, in turn, reflects the orientation of fiber tracts. Ref: Jellison et al, AJNR (2004).





On Image Acquisition
If we apply a certain MRI sequence that can manipulate this diffusion gradient, then we are able to see how the water diffuse in space. DW image acquisition is based on a specific but modified spin-echo sequence called the pulse gradient spin-echo (PGSE). In a way, it is a kind of a combination of spin-echo and gradient-echo sequence where there is a 90° and a 180° RF pulse, and a pair of diffusion gradient fields (a specific magnetic field with varying strength spatially) delivered on both sides of the 180° pulse. The purpose of this diffusion gradient is to allow the detection of phase differences in molecule spins.

Thus to produce DW imaging along the X-axis direction, for example, we apply strong magnetic field gradients along X. If molecules diffuse along X during the specified time interval, a signal attenuation will be observed compared to the signal without gradient.

For stationary water molecules, there will be no phase difference between before and after the gradient fields and no signal attenuation will occur. However, if there is a coherent flow in the direction of the applied gradient, a net phase difference will be introduced by the different amounts for each gradient field, resulting in signal attenuation. This phase difference is proportional to the gradient strength G, the duration of each gradient pulse 𝛿, and the time difference between the two gradient pulses Δ. It is well-known that PGSE is very sensitive to head motion. Because of this, a single shot echo pulse is used to acquire the signal readout in a very swift and efficient manner. The echo time (TE) is usually 100 msec in the DW sequence; with the repetition time (TR) between successive RF pulses to be rather long (6-7 sec).

Fig-2: Illustration of a pulse-gradient spin-echo MRI sequence used in the DW image acquisition (1 repetition).

The attenuated signal for moving molecules can be modeled as an exponential form containing the diffusivity and a b factor. This b factor depends only on the acquisition or MRI sequence parameters. Here the diffusivity is represented by the apparent diffusion coefficient (ADC) to indicate that the diffusion process is not free in the fibrous tissues, but restricted or modulated by many biological and anatomical factors. In the end, images are "weighted" by the diffusion process. This means that the signal is more attenuated the faster the diffusion and the larger the b factor is.

where S is the DW signal (or image), S₀ is the signal without any DW gradients, ADC is the apparent diffusion coefficient, and b factor is the diffusion-weighting that solely depends on the properties of the MRI sequence (gradient strength, time spacing). The optimum diffusion-weighting (b factor) for the brain is usually ~ 700 and 1300 s/mm² with 1000 s/mm² being the most common in use. From here, the 6 independent elements of the diffusion tensor D, i.e. 3 diagonal and 3 off-diagonal elements, may be estimated from the apparent diffusion coefficients using multiple linear least-squares methods or nonlinear modeling.

Fig-3: Illustration of different natures of anisotropy of the DW image inside the brain (from: FSL training slides).

Measurements tend to be quite sensitive to image noise, which can bias the anisotropy estimates. The accuracy of DTI measures (see below) may be improved by either: increasing the number of encoding directions or increasing the number of averages. Unfortunately, this will also affect the scan time during data collection. The image SNR can also obviously be improved by using larger voxels, although this will increase partial volume averaging of tissues, which can lead to errors in the diffusion tensor model. Moreover, image quality and thus spatial resolution may also depend on the application (clinical vs research). 

What is HARDI?

As stated above, the imaging process is usually obtained using b = 1000 s/mm2, N = 12 - 32 directions, parallel imaging with a very long repetition time (TR > 6 sec). DTI assumes water diffusion to have a Gaussian distribution. The estimation in 3D has six unknown coefficients to reconstruct, and thus a minimum of six diffusion-weighted images is required in addition to a baseline S0 image.

However, one great disadvantage of DTI imaging is dealing with crossing fibers. A technique called High Angular Resolution Diffusion Imaging or HARDI (Wedeen & Tuch, 2000) has the ability to discriminate multiple fiber populations crossing within the same voxel. To achieve that, HARDI requires the acquisition of > 50 gradient directions at a high b-value. In contrast, the traditional DTI only requires 6 directions at lower b-values. The higher angular resolution provides a more accurate representation of the 3D pattern of water diffusion within a voxel. All methods have in common the ability to provide the orientation of multiple white matter tracts within each voxel.

Two Main DTI Measures
In DTI, two most common measures are the mean diffusivity and anisotropy metric of the diffusion tensor. Mean diffusivity (MD, equivalent to ADC) is the average magnitude of diffusion in three main axes and quantified as the trace of the tensor divided by 3, tr(D)/3. Here, the trace is a sum of the diagonal elements of the tensor, which is equivalent to the average of the eigenvalues, λ. Many measures of anisotropy have been described, but the most widely used invariant measure of anisotropy is the fractional anisotropy (FA) (Basser and Pierpaoli, 1996). Both MD and FA are rotationally invariant.

In healthy people, WM regions have a wide range of FA values, ranging between 0.1 - 1.0, peaking at ~0.3, and much of this variation is due to crossing WM fibers (anisotropic!). Unfortunately, using FA should be handled with care. Although FA is a very sensitive and popular measure, it is not specific enough and does not capture the full shape of the diffusion tensor. Using this measure as a biomarker of WM integrity can be tricky. Some recent studies have suggested that the eigenvalues or their combinations demonstrate more specific relationships to white matter pathology. For example:
      (a) Radial diffusivity, D_R = (λ₂ + λ₃)/2, which appears to be modulated by WM myelin.
      (b) Axial diffusivity, D_A = λ₁, is more specific to axonal degeneration.
Some use cases of DTI in neuropathology is as follows. Example: demyelination could cause an increase in the radial diffusivity D_R, with little influence on the axial diffusivity measure. Increased tissue water in edema will increase the MD, whereas abnormal tissue growth may decrease the MD. In the case of ischemic stroke, FA and MD are found to increase and decrease during the acute phase. In the later chronic phase, FA and MD properties will reverse (decrease and increase).

Fiber orientation in the brain can be visualized using DW imaging. The strong assumption here is that the direction of maximum diffusivity in anisotropic voxels is an estimate of the major fiber orientation. Through some calculus operations, we can define the path taken by the white matter fibers. Applications of such tractography will be for future blog posts.

Motor Skill Learning
While there are ample studies examining the effect of motor learning on plasticity in the grey matter areas using functional neuroimaging, lesser studies have discussed changes in the white matter tracts as found using DWI. Results from those studies are not conclusive. Increase in FA values around the intraparietal sulcus has been observed after repeated juggling (Scholz, et al, 2009), and around the primary motor cortex related to motor adaptation (Landi, et al, 2011). Another study, however, reported significantly lower FA in both the left and the right CST in the musician group. Some recent studies have even made a claim to observe structural changes after a few sessions of motor skill learning (e.g. sequence tapping). 

Reference: Alexander AL, et al, Diffusion Tensor Imaging of the Brain (2007).