Essential Concepts in MRI. Yang Xia

      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)

      2.6 SPIN RELAXATION PROCESSES

      After M has been tipped to the transverse plane, if we switch off the B₁(t) field and sit there watching, what happens to the spin system? As soon as B₁ is switched off, two processes will happen to M, which would eventually lead to the return of M to thermal equilibrium (i.e., Mz = M₀, and the zero transverse components of M).

      The processes that return the magnetization M to the thermal equilibrium are termed as relaxation, which may be described by two time-constants in the following equations:

(2.15a)

      and

(2.15b)

      where M⊥ refers to the transverse component, defined by Eq. (2.8).

      T₁ is known as the spin-lattice (or longitudinal) relaxation time because the relaxation process involves an energy exchange between the spin system and its surrounding thermal reservoir, known as the “lattice.” The term “longitudinal” comes from the fact that this relaxation process restores the disturbed magnetization to its thermal equilibrium, being along the longitudinal direction k. T₁ in simple liquids is usually in the range of several seconds.

      When the B₁ field rotates the magnetization entirely to the transverse plane (i.e., Mz = 0), the magnetization is said to be rotated by 90˚ (i.e., π/2). The solution to Eq. (2.15a) becomes

       (2.16a)

      If the B₁ field is sufficiently powerful or its duration is sufficiently long, the magnetization can be inverted (i.e., Mz = –M₀). Such a B₁ field is said to rotate the magnetization by 180˚ (a π pulse). Under this condition, the solution to Eq. (2.15a) becomes

       (2.16b)

Figure 2.9 shows schematically the motion of the longitudinal magnetization with these two different initial conditions, where t = 0 marks the moment that the B₁ field is turned off.