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The basic idea of diffusion tensor imaging is the same as for Phase Contrast Angiographic MRI. The following sections are mainly a summary of different articles [22].
(3.1) 
which leads together with the equation of conservation of mass
(3.2) 
to the diffusion equation:
(3.3) 
To be able to determine the diffusivity in vivo the diffusion process itself has to be monitored, i.e., the random motions of an ensemble of particles, rather than solving the equation for some initial and boundary conditions. Einstein [10] showed that the diffusion coefficient measured in the nonequilibrium concentration cell experiments is the same quantity that appears in the variance of the conditional probability distribution , the probability of finding a molecule at a position at a time which was originally at position . For free diffusion this conditional probability distribution obeys the same diffusion equation as the particle concentration given. The expectancy is then
In the case of molecular displacements in tissues, in which diffusion is an
anisotropic process with different molecular mobility in x, y and z
directions, the diffusion constant has to be replaced by a diffusion
tensor. Equation 3.4 shows that the diffusivity can be inferred
directly by measuring the second moment of the conditional probability
distribution of the diffusing species.
The basic principles of diffusion imaging can be understood from a simple bipolar pulsed gradient experiment (see Figure 3.1). The purpose of these gradient pulses is to magnetically label spins carried by molecules. Here denotes the gradient strength, as the gradient duration and as the time interval between the pulses. The first gradient pulse induces a phase shift of the spin transverse magnetization, which depends on the spin position. If the gradient is along , then:
(3.5) 
is the spin position supposed to be constant during the short duration of the gradient pulse. is the gyromagnetic ratio, which is a nuclear specific factor. Its value is 42 for hydrogen protons. After the 180 radio frequency (RF) pulse, is transformed into . Similarly, the second pulse after a delay will produce a phase shift :
(3.6) 
where is the spin position during the second pulse. The resulting net dephasing is:
(3.7) 
It can be seen that for static spins, i.e. not moving molecules , the bipolar gradient pair produces no net dephasing. For moving spins there is a net dephasing that will depend on the spin history during the time interval between the pulses, and which will affect the transverse magnetization. The position of the two gradient pulses in each half of the spinecho sequence does not matter; it is the time elapsed between them that affects the net phase. In Nuclear Magnetic Resonance (NMR) the total magnetization is measured, the vector sum of the magnetic moments of the individual nuclei, which may have different motion histories:
(3.8) 
where is equilibrium magnetization in the direction of the static applied magnetic field . This sum can be evaluated once the net phase distribution is known. Assuming free diffusion in a homogeneous domain, the probability of finding a spin at position is a constant. If is the conditional probability of finding a spin initially at between positions and after a time interval , the amplitude attenuation is:
For free diffusion in one dimension, the conditional probability is given by:
where is the diffusion coefficient. Combining equations 3.9 and 3.10 leads to:
(3.11) 
This last equation relates the measured signal attenuation to the diffusivity, and is the basis for diffusion measurement using NMR.
Taking several pulses and the fact that may not be negligible as compared to , into account, Equation 3.10 has to be solved for a general pulse sequence. This leads to the following relation for an isotropic medium in a spinecho experiment (for a detailed derivation of these equations see [22] p. 8, 9).
(3.12) 
where
(3.13) 
Introducing the gradient factor b
(3.14) 
which characterizes the sensitivity of NMR sequences to diffusion, the signal attenuation can be represented by the simpler expression
Diffusion is a threedimensional process. However molecular mobility may not be the same in all directions. This anisotropy may be due to the physical arrangement of the medium or the presence of obstacles that limit diffusion (restricted diffusion) or both. Moreover structures that exhibit anisotropic diffusion at the molecular level can be isotropic at the microscopic level.
As mentioned earlier, in anisotropic diffusion the effective diffusion coefficient is replaced by an effective diffusion tensor. The echo attenuation then becomes
(3.16) 
where is a bmatrix and is an effective diffusion tensor. Its diagonal terms , and represent correlations between molecular displacements in the same directions, whereas its offdiagonal terms , , reflect correlations between molecular displacements in orthogonal directions.
To obtain the different diffusion coefficients at each voxel position, different echo and gradient sequences have been proposed [23].
As can be seen from Table 3.1, diffusion in white matter is extremely variable. The value of the diffusion coefficient directly depends on the relative orientation of the fibers and the magnetic field gradients, which is known as ''anisotropic diffusion.'' Water diffusion in gray matter does not exhibit anisotropy or restriction by impermeable walls [26], [27]. White matter on the other hand is extremely anisotropic, the results of the measurements depending on the respective orientation of the myelin fiber tracts and the gradient direction at each different image location. It appears that diffusion coefficients are significantly decreased when the myelin fiber tracts are perpendicular to the direction of the magnetic field gradient used to measure molecular displacements.
Figure 3.2 shows adjacent myelinated fibers and the diffusion of water. The diffusion coefficient measured parallel to the myelin fiber direction is about three times larger mms than the diffusion coefficient perpendicular to fibers mms.
Since MRI methods in general always obtain a macroscopic measure of a
microscopic quantity which necessarily entails intravoxel averaging,
the voxel dimensions influence the measured diffusion tensor at any
particular location in the brain.
Factors which would affect the shape of the apparent diffusion tensor (i.e., the shape of the diffusion ellipsoid) in the white matter include the density of fibers, the degree of myelination, the average fiber diameter, and the directional similarity of the fibers in the voxel. The geometric nature of the measured diffusion tensor within a voxel is thus a meaningful measure of fiber tract organization.
Although the individual axons and the surrounding myelin sheaths cannot be revealed with the limited spatial resolution of in vivo imaging, distinct bands of white matter fibers with parallel orientation may be distinguished from others running in different directions. Figure 3.3 shows how two crossing fibertracts would ideally be represented by a diffusion tensor image.
Although there is no doubt that diffusion is anisotropic in white matter, controversies about the origin of this anisotropy remain. The diffusiontime dependence of the measured diffusion coefficient is the crucial experimental test for the presence and dimension of diffusive barriers. If diffusion is restricted by impermeable barriers the diffusion coefficient decreases when the diffusion distance reaches the dimension of the available volume. Water in gray and white matter diffuses without encountering significant barriers  at least on the distance range of 8  10 microns, which exceeds the dimensions of most cellular compartments [26]. Anisotropy also exists in brains of neonates before the histological appearance of myelin [18]. This leads to the conclusion that myelination is not essential for the diffusion anisotropy of nerves. Nevertheless myelin is widely assumed to be the major barrier to diffusion in myelinated fiber tracts. Therefore the demonstration of anisotropic diffusion in the brain by magnetic resonance has opened the way to explore noninvasively the structural anatomy of the white matter in vivo [16].
In summary, diffusion measurements in vivo reflect complicated
pathways of water molecules in the tissue.
(3.17) 
so that the loss of signal intensity is (Stejskal Tanner formula)
The data was acquired at the Brigham and Women's Hospital on a GE Signa 1.5 Tesla Horizon Echospeed 5.6 system with standard 2.2 Gauss/cm field gradients. The time required for acquisition of the diffusion tensor data for one slice was 1 min; no averaging was performed. Imaging parameters were: effective TR=2.4s, TE=65ms, =750 s/mm, =5 s/mm, field of view 18 cm, 6 kHz readout bandwidth, acquisition matrix .
Usually one coronal and one axial slice with effective voxel dimensions mm were acquired. Newer acquisitions (September 1998 to present) with multiple slices (between 9 and 14 slices) have an effective voxel size of mm. Axial and coronal multislice acquisitions locations are chosen to include the majority of the white matter. Figure 3.4 shows roughly in a sagittal view the positions from where axial and coronal, multiple slice diffusion weighted scans were taken. The factor of inplane to interslice resolution is still very large (about 7.1 respectively 6.25) so that most processing is done separately for each slice.
