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Image Acquisition

This chapter assumes basic knowledge of magnetic resonance imaging (MRI), i.e. how nuclear magnetic resonance is used to obtain information from the tissue under inspection. Excellent introductions into the principles of MRI are available [8], [9].

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].

Molecular Diffusion and Nuclear Magnetic Resonance

Diffusive transport is observed in steady-state, non-equilibrium systems, such as in cells. A concentration difference is established between two compartments (cells) and a macroscopic diffusive flux can be observed between them. Fick's law describes how the molecular flux density J depends on the molecular concentration gradient $\nabla C$

$$\mathbf{J} = -D \nabla C$$ (3.1)

which leads together with the equation of conservation of mass

$$\frac{\partial C}{ \partial t} = -\nabla \mathbf{J}$$ (3.2)

to the diffusion equation:

$$\frac{\partial C}{ \partial t} = -\nabla \mathbf{J} = \nabla ( D \nabla C)$$ (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 non-equilibrium concentration cell experiments is the same quantity that appears in the variance of the conditional probability distribution $P(\mathbf{r}\vert \mathbf{r}_0, t)$, the probability of finding a molecule at a position $\mathbf{r}$ at a time $t$ which was originally at position $\mathbf{r}_0$. For free diffusion this conditional probability distribution obeys the same diffusion equation as the particle concentration given. The expectancy is then

$$\langle(\mathbf{r} - \mathbf{r}_0)(\mathbf{r} - \mathbf{r}_0)\rangle = 6Dt$$ (3.4)

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 $D$ 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 $G$ denotes the gradient strength, $\delta$ as the gradient duration and $\Delta$ as the time interval between the pulses. The first gradient pulse induces a phase shift $\phi_1$ of the spin transverse magnetization, which depends on the spin position. If the gradient is along $z$, then:

$$\phi_1 = \gamma \int_0^\delta Gz_1dt = \gamma G \delta z_1$$ (3.5)

Figure 3.1: Bipolar pulsed gradient experiment (adapted from [22] and [9])

$z_1$ is the spin position supposed to be constant during the short duration $\delta$ of the gradient pulse. $\gamma$ is the gyromagnetic ratio, which is a nuclear specific factor. Its value is 42 for hydrogen $\mathrm{^1H}$ protons. After the 180$^\circ$ radio frequency (RF) pulse, $\phi_1$ is transformed into $-\phi_1$. Similarly, the second pulse after a delay $\delta$ will produce a phase shift $\phi_2$:

$$\phi_2 = \gamma \int_\Delta^{\Delta+\delta} Gz_2dt = \gamma G \delta z_2$$ (3.6)

where $z_2$ is the spin position during the second pulse. The resulting net dephasing $\delta(\phi)$ is:

$$\delta(\phi)=\phi_2 - \phi_1 = \gamma G \delta(z_1 - z_2)$$ (3.7)

It can be seen that for static spins, i.e. not moving molecules $z_1 = z_2$, 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 $\Delta$ between the pulses, and which will affect the transverse magnetization. The position of the two gradient pulses in each half of the spin-echo 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 $M$ of the individual nuclei, which may have different motion histories:

$$\frac{M}{M_0} = \sum_{j=1}^N e^{i\delta(\phi_j)}$$ (3.8)

where $M_0$ is equilibrium magnetization in the direction of the static applied magnetic field $\mathbf{B}_0$. 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 $z_1$ is a constant. If $P(z_2, z_1, \Delta)dz_2$ is the conditional probability of finding a spin initially at $z_1$ between positions $z_2$ and $z_2 + dz_2$ after a time interval $\Delta$, the amplitude attenuation is:

$$\frac{M}{M_0} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{i\gamma G \delta(z_1 - z_2)} P(z_2, z_1, \Delta)dz_1 dz_2$$ (3.9)

For free diffusion in one dimension, the conditional probability is given by:

$$P(z_2, z_1, \Delta) = \frac{1}{\sqrt{4 \pi D \Delta}} e^{\frac{-(z_1 - z_2)^2}{4D\Delta}}$$ (3.10)

where $D$ is the diffusion coefficient. Combining equations 3.9 and 3.10 leads to:

$$\frac{M}{M_0} = e^{-(\gamma G \delta)^2 D \Delta}$$ (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 $\delta$ may not be negligible as compared to $\Delta$, 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 spin-echo experiment (for a detailed derivation of these equations see [22] p. 8, 9).

$$\frac{M(TE)}{M_0} = e^{-D \int_0^{TE} \mathbf{k}(t)\mathbf{k}(t)dt}$$ (3.12)

where

$$\mathbf{k}(t) = \gamma \int_0^t G(t')dt'$$ (3.13)

Introducing the gradient factor b

$$b = \int_0^{TE} \mathbf{k}(t)\mathbf{k}(t)dt$$ (3.14)

which characterizes the sensitivity of NMR sequences to diffusion, the signal attenuation can be represented by the simpler expression

$$\frac{M(TE)}{M_0} = e^{-bD}$$ (3.15)

Diffusion is a three-dimensional 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

$$\frac{M(TE)}{M_0} = e^{-\sum_{i=1}^3 \sum_{j=1}^3 b_{ij} D_{ij} }$$ (3.16)

where $b_{ij}$ is a b-matrix and $D_{ij}$ is an effective diffusion tensor. Its diagonal terms $D_{xx}$, $D_{yy}$ and $D_{zz}$ represent correlations between molecular displacements in the same directions, whereas its off-diagonal terms $D_{xy}$, $D_{xz}$, $D_{yz}$ 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].

Diffusion in the Normal Brain - Anisotropic Diffusion in White Matter

The diffusion coefficient of water in tissues was found to be 2-10 times less than that of pure water [24], [25]. This is understandable, given that water molecules are obliged to move tortuously around obstructions presented by fibers, intracellular organelles or macromolecules. In addition, there is a continual exchange between free water molecules and water molecules which spend some of their time associated with the much more slowly moving macromolecules. Diffusion is thus more likely to be ''hindered'' by random obstacles than strictly ''restricted'' in close spaces by walls. Table 3.1 shows some diffusion coefficients of water in the human brain. The diffusion in the cerebrospinal fluid (CSF) is similar to that of pure water at the same temperature $(2.5 * 10 ^ {-3}$ mm$^2/$s @ 37.5$^\circ$ C$)$

Table 3.1: Diffusion coefficients of water in the human brain [22]

	Diffusion coefficient
Tissue	$\cdot10^{-3}$mm$^2/$s
CSF	2.94 $\pm$ 0.05
Gray matter	0.76 $\pm$ 0.03
White matter:
Corpus callosum	0.22 $\pm$ 0.22
Axial fibers	1.07 $\pm$ 0.06
Transverse fibers	0.64 $\pm$ 0.05

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 $D_\Vert$ is about three times larger $(1.2 * 10^{-3}$ mm$^2/$s$)$ than the diffusion coefficient perpendicular to fibers $D_\perp (0.4 * 10 ^{-3}$mm$^2/$s$)$.

Figure 3.2: Diffusion in myelinated fibers (adapted from [22])

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.

Figure 3.3: Diffusion tensor overlying two crossing fibertracts

Although there is no doubt that diffusion is anisotropic in white matter, controversies about the origin of this anisotropy remain. The diffusion-time 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.

Baby Brain Data

The data used in this study was generated using a modified version of the Line Scan Diffusion Imaging (LSDI) technique [28]. In this technique a bipolar gradient pulse echo is used. This sequence has been shown in Figure 3.1. In this case $b$ in Equation 3.15 becomes

$$b = \gamma^2G^2\delta^2(\Delta - \frac{\delta}{3})$$ (3.17)

so that the loss of signal intensity is (Stejskal Tanner formula)

$$\ln(M) = \ln(M_0) - \gamma^2G^2\delta^2(\Delta - \frac{\delta}{3})D$$ (3.18)

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, $b_{high}$=750 s/mm$^2$, $b_{low}$ =5 s/mm$^2$, field of view 18 cm, 6 kHz readout bandwidth, acquisition matrix $128\times128$.

Usually one coronal and one axial slice with effective voxel dimensions $5\times0.703125\times0.703125$ mm$^3$ were acquired. Newer acquisitions (September 1998 to present) with multiple slices (between 9 and 14 slices) have an effective voxel size of $4.4\times0.703125\times0.703125$ mm$^3$. 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 in-plane to inter-slice resolution is still very large (about 7.1 respectively 6.25) so that most processing is done separately for each slice.

Figure 3.4: Sagittal view showing the positions from where axial and coronal, multiple slice diffusion weighted scans were taken.