To overcome the limitations of C-MRI, several advanced MR modalities have been developed such as diffusion, functional MRI (fMRI), and spectroscopy. Diffusion imaging modalities are of the most important advanced MRI techniques in neuroradiology (Huisman, 2010), which will also be investigated in this study alongside with conventional MR techniques. The most important types of diffusion MRI are diffusion weighted (DWI) and diffusion tensor imaging (DTI). The following sections are dedicated to description of the concept of diffusion and how MR images are generated based on the diffusion of water molecules in the brain structures.
2.4.1 Magnetic Resonance Diffusion
Diffusion is defined as the result of microscopic random thermal movements, which are also known as Brownian motion. The principle of diffusion imaging is based on the diffusion of water molecules in the brain which is determined by several factors such as type of tissue, temperature and microenvironmental structure of the surrounding area (Huisman, 2010). The magnitude and direction of diffusion of a water molecule is determined by the geometry of the environment which includes the water mass. In the case of brain microstructure, this is useful since they have very small size boundaries which can be determined by calculating the water tensor. These boundaries cannot be identified by conventional MRI.
Diffusion has two different types: isotropic and anisotropic. The displacement of molecules can be modelled by a sphere when the water molecules move randomly in all directions. The
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diffusion can be described by a single value, D, which is known as the isotropic constant (Figure 2-9 (a)). In the case the molecules are restricted by their surrounding environment, their diffusion occurs along axes (anisotropic). Therefore, the molecular displacements can be model by an ellipsoid (Figure 2-9 (b)).
Diffusion-weighted (DW) sequences are generated by adding two magnetic field gradients while varying the magnetic field linearly across the tissue. The precessional frequency is related to the magnetic field strength, therefore the gradient will impose a precessional frequency which is position dependent. The spins acquire a phase while they are precessing over a duration of time. By applying a gradient with the same size and duration but with 180o (refocusing RF pulse), the phase will be reversed. Figure 2-10 shows a diffusion pulse sequence which is used to detect the diffusion signal.
a
b
Figure 2-9 Schematic illustration for isotropic and anisotropic diffusion of water molecule in: a) the free space (isotropic), and b) restricted to tissue (anisotropic)
Figure 2-10 Spin-echo sequence in diffusion weighted imaging. The shaded rectangles show the gradient pulses which induce (left block) and reverse (right block) the phase shift. The image is recreated from (Winston, 2012).
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Phase depression depends on the strength (G), duration (Ξ΄), and time intervals (Ξ) of the gradient field which are presented in Figure 2-10. The measure of sensation to diffusion, b- value (in s/mm2), is calculated using the diffusion gradient characteristics and the Stejskal- Tanner equation
π = (πΎπΊπΏ)2(β βπΏ
3) (2-1)
where Ξ³ is the gyromagnetic ratio. The diffusion is measured by comparing the voxel signal intensity between low and high DW which corresponds to low and high b-values, respectively. The following section will explain the details of diffusion-weighted imaging.
2.4.2 Diffusion Weighted Imaging
The diffusion images are generated by comparing the images acquired with low and high b- values, which are often 0 and 1000 s/mm2, respectively (Huisman, 2010). The image acquired with b = 0 s/mm2 is not sensitised for diffusion and is termed S
0. This image is equivalent to a
T2-weighted image without diffusion-weighting. The DW image, Sb, is acquired with a known
b-value (e.g. b = 1000 s/mm2) and the same TE. The diffusion constant, D, of a voxel is then calculated using
ππ π0
= πβππ· (2-2)
The calculated value of D in Equation (2-2) corresponds to diffusion in one direction. By measuring the diffusion in several directions, the signal loss versus S0 is measured and reflects
the displacement in the corresponding direction of gradient.
2.4.3 Imaging Using the Diffusion Tensor
A tensor of size 3 Γ 3 is used to characterise the diffusion ellipsoid by modelling the coefficient along different directions and is called the diffusion tensor, D.
π«π,π= [
π·π₯π₯ π·π₯π¦ π·π₯π§
π·π¦π₯ π·π¦π¦ π·π¦π§
π·π§π₯ π·π§π¦ π·π§π§
] . (2-3)
The diffusion tenor is symmetric with diagonal elements representing the mobility rate in each direction and non-diagonal elements representing the correlation between the orthogonal directions.
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When the water molecules are moving freely, their motion will be random in all directions. Therefore, the displacement of the molecules is modelled by a sphere with a diameter which is determined by D. The centre of the sphere remains in the same position without moving. The anisotropic diffusion ellipsoid is described by three principal axes which are perpendicular to each other which are also called eigenvectors. The magnitudes and the directions of these axes are determined by the corresponding eigenvalues and eigenvectors, respectively. Therefore, six measurements are required to describe the anisotropic ellipsoid which are Ξ»i iΟ΅{1,2,3} and vi, iΟ΅{1,2,3}.
a b
Figure 2-11 Isotropic and anisotropic diffusion, their eigenvalues and corresponding directions. a) isotropic diffusion, and b) anisotropic diffusion.
2.4.4 DTI Measures
The diffusion parameters of the voxels are represented by 3D ellipsoids, which are difficult to be visualised for the whole brain. For the purpose of visualisation and to make them comprehendible by human observers, 2D scalar isotropic and noninotropic maps are used. Isotropic Diffusion Maps
The magnitude of diffusion can be measured by the summation of the diagonal elements of D, or trace of the tensor
ππ (π«) = π·π₯π₯+ π·π¦π¦+ π·π§π§ . (2-4)
Tr(D) is orientation invariant which means that it is not sensitive to the orientations of microstructures or cells.
The most basic measure of the diffusion is the mean diffusivity (MD) which is also known as the apparent diffusion coefficient. MD is calculated using
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ππ· =π‘π (π«)
3 , (2-5)
which can be considered equivalent to
ππ· =π1+π2+π3
3 . (2-6)
The map of isotropic diffusion provides a measure for the content of diffusivity without considering the directional properties of the diffusion.
Anisotropic Diffusion Maps
Measuring the diffusion anisotropy provides the directional characteristics of DW images. Several formulations have been proposed for measuring the diffusion anisotropy (Pierpaoli et al., 1996). A simple representation of the anisotropy is the ratio of the longest and shortest axes of the ellipsoid (Ξ»1/Ξ»3) which shows the elongation of the ellipsoid. The most common measure for anisotropy is fractional anisotropy (FA) which is calculated using
πΉπ΄ = β3 2
β(π1βπ2)2+(π2βπ3)2+(π1βπ3)2
βπ12+π22+π32
. (2-7)
which is a scalar value in the range [0, 1]. The zero value represents pure isotropic while 1 is related to pure anisotropic diffusion.
The scalar map of the whole brain can be generated by calculating MD and FA for each voxel. Therefore, the diffusion images are visualised similar to C-MRI with greyscale 2D slices.
2.4.5 Decomposition of Tensor
A tensor has nine elements in 3D space. To reduce the dimensionality of tensor, transformation is used to create a single scalar value. Also, more detailed measure of the tensor can be extracted by decomposing the diffusion tensor into its components, i.e. isotropic and anisotropic. An advanced decomposition technique for visualising more scalar measures for the tensor is proposed by PeΓ±a et al. (PeΓ±a et al., 2006) which will be explained in the following. The tensor in Equation (2-3) is firstly decomposed using
π·ππ = π·πΌππ+ [π·ππβ π·πΌππ] , (2-8)
where Iij is the identity tensor. Regarding to Equation (2-8), the diffusion tensor is decomposed
into two separate tensors. By considering the components as P and Q terms, Equation (2-8) can be rewritten as
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π·ππ= πππ+ πππ , (2-9)
where P and Q represents the isotropic tensor and the deviatoric tensor, respectively. In other word, the decomposed tensors are
π = π·πΌππ , (2-10)
π = π·ππβ π·πΌππ . (2-11)
The magnitude of these tensors represents the isotropic (p) and anisotropic (q) components of the tensor which are calculated using
π = β3ππ· , (2-12)
π = β(π1β ππ·)2+ (π2β ππ·)2+ (π3β ππ·)2 , (2-13)
where MD is calculated from Equation (2-6).
Figure 2-12 shows DTI protocols (i.e. p- and q-map) for the same patient which was shown in Section 2.3.6 (Figure 2-8).
a b
Figure 2-12 MRI different protocols: a) p-map and b) q-map.