x3
x2
x1 M
(3)
x3
x2
x1 M
Figure 2.1.: Above: Behaviour of hydrogen nuclei, represented as blue dots, in an MR scanner with static magnetic fieldB0. The arrows pointing up and down indicate the protons’ spin axes at different energy states (here, up: low, down: high), the circular arrows indicate precession around the axes aligned with the direction of B0. (1) The protons in the MR scanner are in equilibrium state. They are aligned with the magnetic field and in precession. (2) An RF-pulse results in the excitation of spins from low to high energy state. (3) Protons move back from high to low energy state while emitting electromagnetic energy. Below: Corresponding net magnetization.
2.1. Basic Principles of MRI
Human tissue consists of about 60% water, and the hydrogen nuclei thereof constitute more than 90% of the overall hydrogen nuclei in the human body.
Hydrogen nuclei consist of single protons, which are spinning around their own axis. Due to its spin, each proton has anangular momentumand amagnetic mo-ment. These two properties lay the foundation for MR measurements. Though not solely inherent in hydrogen but in all isotopes with an odd number of protons and/or neutrons, the behaviour of hydrogen nuclei is mainly what is measured in MRI due to the abundance of water in the body. In the following, we discuss the basic ideas of MRI. For a more detailed description, we refer to the books by Slichter (1990) and Haacke et al. (1999).
6
2.1. Basic Principles of MRI In an MR scanner where a static magnetic field with strength B0 is applied, protons can be viewed as being in precession, which means that they rotate - in addition to their spin - with their rotation axis around the direction of the B-field.
The precession has a certain frequency calledLarmor frequency, ω=γB0, which depends only on the field strength and the gyromagnetic ratioγ. In the presence of a static magnetic field, the spins are aligned with the magnetic field and are in one of two different states, the parallel (or low-energy) or the antiparallel (or high-energy) state. In fact, a little more than half of the protons are in the low-energy state, whereas the remaining protons are in the high-energy state.
This so called initial or equilibrium state is exemplarily illustrated in Figure 2.1 (1). The space and time dependent net magnetization M : R3 × R → R3 is proportional to the difference of the overall amount of parallel and antiparallel spins in the scanner. Without a magnetic field, the spins are oriented randomly resulting inM = 0. In the equilibrium state where the spins are aligned with the magnetic field, spins with opposite alignment cancel out, but |M| > 0 as more spins are in the low-energy state. We define the MR scanner coordinate system such that thex3-axis is aligned with the direction of the static magnetic field. M initially points in x3-direction, also called longitudinal direction. Then, a short radio frequency pulse (RF-pulse) with Larmor frequency is applied to the static magnetic field. As shown in Figure 2.1 (2), the electromagnetic energy emitted from the RF coils results in excitations of the protons from low to high energy states. Correspondingly, the direction of the net magnetization precesses with Larmor frequency around the x3-axis while spiraling out into the x1-x2-plane perpendicular to the direction of the magnetic field, also called the transverse plane.
After the RF-pulse has been switched off again, the protons move back to their initial states while emitting electromagnetic energy, as illustrated in Figure 2.1 (3). This restoration of the equilibrium is calledspin relaxation. A distinction is made between longitudinal and transverse relaxation. Longitudinal relaxation, or T1 recovery, is the restoration process of the net magnetization in direction of the magnetic field due to spins returning to the low-energy state. At the same time, transverse relaxation, or T2 decay proceeds, which can be described as a decrease of transverse magnetization due to phase decoherence of spins. In the following, time constants T1 and T2 indicate the time required for T1 recovery and T2 decay, respectively. Let B : R3×R → R3 denote the sum of the static magnetic field with field strength B0 and a time-dependent, spatially varying gradient field
BG :R3×R→R3, BG(x, t) = (G(t)>x)e3
with G : R → R3 and e3 = (0,0,1)>, that is necessary to encode spatial infor-mation in the MR signal. The change in net magnetization with time explained
2. Diffusion Magnetic Resonance Imaging and Fiber Tractography
above is summarized by the Bloch equation dM(x, t)
that was derived in the article by Bloch (1946). Here, ×denotes the cross prod-uct, M0 denotes the initial magnetization where all spins are at the equilibrium state, and M1, M2, M3 are the three components of M.
Basically, the signal at time t that is measured in MRI is given by the total transverse magnetization,
S(t) = Z
R3
M⊥(x, t)dx, where the transverse magnetization M⊥ is defined by
M⊥(x, t) :=M1(x, t) +iM2(x, t). Solving the partial differential equation (2.1) yields
M⊥(x, t) =M⊥(x,0)e−t/T2e−γB0te−iγR
t
0BG(x,t)dt
for the transverse magnetization. Note, that M⊥(x,0) is proportional to the density of hydrogen atoms. Omitting the scaling parameters e−t/T2 and e−γB0t we obtain for the MR signal the commonly used representation
S(t) = Z
R3
M⊥(x,0)e−iγR
t
0G(t)>x dtdx.
In practice, the MR signal is usually measured as 2D slices, and ink-space(Twieg, 1983). For instance, let z denote the x3−coordinate of the scanned slice. Then, the MR signal of a 2D slice in the transverse plane can be represented as
S(z, t) = Z
where G is chosen such that k is sampled on a uniform grid. A 2D inverse Fourier transform of (2.2) yields the respective MR image slice. Different MR
8