Читать книгу Essential Concepts in MRI - Yang Xia - Страница 21

The usefulness of M is not its equilibrium state but its time evolution after M is tipped away by an external perturbation (another radio-frequency field) from its thermal equilibrium along the z axis. The evolution of M produces the NMR signal, which reveals the environment of the molecules. In a classical description, the time evolution of the macroscopic magnetization in the presence of a magnetic field can follow the same approach that we used before in deriving Eq. (2.4). By equating the torque to the rate of change of the angular momentum of the macroscopic magnetization M, we have

(2.9)

Equation (2.9) is a vector equation, which states that the rate of the change of the magnetization has a direction that is at right angles to both the magnetization vector and the magnetic field vector (by the right-hand rule in the cross product of vector analysis, Appendix A1.1). When B is parallel with the z axis as B₀k, the above equation has the same solution as given before in Eq. (2.5), which corresponds to a precessional motion about k at the rate ω₀.

A time evolution of M can be introduced by the application of a small linearly polarized radio-frequency (rf) field B₁(t) that is oscillating in the transverse plane. This B₁ field is actually a superposition of two counter-rotating fields in the transverse plane (Figure 2.5):

Figure 2.5 Two counter-rotating fields (right) can form a B₁ field precessing in the transverse plane of the laboratory reference frame (left).

(2.10)

One of the two B₁(t) components (which rotates in the same direction as ω₀) will have strong influence on the nuclei despite its small magnitude compared with B₀, while the other component, which rotates in the opposite direction, will have negligible effect on the nuclei (provided |B₁| ≪ B₀).

It will be very useful to introduce a rotating frame of reference, as shown in Figure 2.6, where the xyz frame is the laboratory frame of reference (stationary) and the x′y′z′ frame is the rotating frame of reference at an angular frequency ω′ in the laboratory frame, in which z and z′ are parallel. In addition, one of the B₁(t) components appears stationary in the rotating frame; let’s set that stationary component of B₁(t) along the x′ axis.

Figure 2.6 (a) A magnetic moment µ precessing in the stationary (xyz) and rotating (x’y’z’) frames. (b) A macroscopic magnetization vector M in the two frames.

When a magnetic moment is precessing at a rate of ω₀ in the xyz frame, a stationary observer in the x′y′z′ frame should see the magnetic moment precessing at a reduced rate ω₀ – ω′. Since ω = γB, a reduced precession rate implies that the magnetic moment is experiencing a reduced B₀, as

(2.11)

In the presence of both B₀ and B₁(t) fields, the total vector field B_total is the sum of all fields and the effective field B_eff varies with the frequency of B₁(t), as

(2.12)

and

(2.13)

where Eq. (2.12) is for the stationary xyz coordinates and Eq. (2.13) is for the rotating x′y′z′ coordinates. The directions of these vector fields are shown schematically in Figure 2.7.

Figure 2.7 In the rotating frame that has a frequency ω′, the external magnetic field B₀ appears to be reduced in magnitude, where the reduction is a function of ω′. This reduction results in the tipping of B_eff towards the x’ axis, which is in parallel with B_1.

In a typical NMR system, |B₁| is several orders of magnitude smaller than B₀, which means that B_total is almost along the z axis. However, the magnitude and the direction of B_eff will drastically depend upon the frequency of the rotating frame ω′. When ω′ approaches the value of ω₀, the effective magnetic field B_eff tips more towards the transverse plane. When ω′ = ω₀ (i.e., the frequency of B₁(t), ω′, equals the Larmor frequency ω₀), the second term in Eq. (2.13) becomes zero. Hence B₁(t) becomes the only field in B_eff to interact with the nuclei. M will therefore respond to the effect of B₁(t), which is set along the x′ axis (Figure 2.7). This condition, ω′ = ω0, is termed as the resonance condition.

Under this resonance condition, the magnetization will be tipped away from its equilibrium orientation towards the transverse plane under the influence of B₁(t), as ω₁ = γB₁ (Figure 2.8). Since the effective component of B₁(t) has been set along the +x′ axis, the magnetization will rotate about B₁ in the y′z′ plane of the rotating frame. Following the notation for rotation as shown in Figure 1.3b, the direction of the magnetization towards the transverse plane under the influence of a B₁ field set along the +x′ axis follows the circular trajectory in Figure 2.8b, towards the +y′ axis.

Figure 2.8 Motion of the magnetization in the rotating frame and the laboratory frame, under the influence of a B₁ field set along the x’ axis. (a) M is at the thermal equilibrium, along the z axis. (b) M tips towards the transverse plane when B₁ is turned ON. (c) M reaches the transverse plane (which is to say that M has been rotated by 90˚).

Note that this rotation of the magnetization towards the +y′ axis under a B₁ field set at the +x′ axis is determined by the convention of positive rotation that is set earlier in Figure 1.3. As we mentioned in Chapter 1.2, different textbooks and academic papers contain inconsistencies in the notation for rotation of M, to either the –y′ axis or +y′ axis upon a B₁ field set along the +x′ axis, depending upon which direction is labeled as the positive rotation. Although this discrepancy seems problematic for the graphical illustration of vectoral motion of magnetization at the first appearance, it does not matter as long as one chooses one notation and keeps it for the entire analysis of spin evolution.

Since B₁ is several orders of magnitude smaller than B0, ω₁ is typically tens to hundreds of kilohertz (instead of a typical ω0 at tens to hundreds of megahertz). The time to tip the magnetization from the thermal equilibrium towards the transverse plane is typically several to tens of microseconds (µs), depending upon the magnitude of the B₁ field. A particular B₁ that is able to tip the magnetization from its z axis to the transverse plane (to the +y′ axis, as shown in Figure 2.8c in the rotating frame x′y′z′) is termed in practice as a 90˚ B₁ field (or more commonly a 90˚ B₁ pulse, or a π/2 pulse). The term pulse in the last sentence refers to the short duration of the B₁ field; see Section 2.10 for the definition of rf pulses and Chapter 7 for more description on spin manipulation under various pulse sequences. The motions of the magnetization in the laboratory frame (xyz) are also shown in Figure 2.8, which graphically is a spiraling vector away from its equilibrium position over the envelope of a dome. When the magnetization M reaches the transverse plane, the longitudinal component (Mz) of M becomes zero.

In general, the equation of motion for M in the presence of B₁(t) is given by

(2.14a)

or

(2.14b)