Essential Concepts in MRI. Yang Xia

Figure 2.15 (a) A vector M rotates in the xy plane at a constant angular frequency ω and with a constant amplitude M, where the tip of the vector draws a sinusoidal wave in a plot of the amplitude vs. time, as in (b). (c) Four different initial orientations of the rotating vector give rise to four different phases of the wave, as shown in (d). The constant-amplitude oscillation implies no attenuation of the vector length. (e) When the oscillation is attenuated by an exponential decay, the motion of the rotating vector generates a damped oscillatory signal in the transverse plane (the FID). The FID signal will have the same decay envelope, regardless of which of the four initial orientations of the rotating vector shown in (c).

       (2.35a)

       (2.35b)

       (2.35c)

      where m is the amplitude of the motion, M is the maximum amplitude (i.e., the length of the vector), θ is the rotation angle in radian (rad), ω is the angular frequency in rad/s, T is the period of the wave (i.e., the time to complete one cycle) in seconds (s), and f is the frequency of the wave in hertz (Hz).

      If no friction slows down the rotation of the disc and nothing changes the length of the vector, the disc will rotate at a constant frequency indefinitely and the wave will oscillate between +M and –M continuously as the function of time without any attenuation to its amplitude, as shown in Figure 2.15b. If the vector M in Figure 2.15a does not start the motion from being parallel with the x axis but with another axis, the oscillation will have exactly the same frequency and period, only with an extra phase shift (Figure 2.15c and d).

      Similar visualization has in fact been used in the illustration of the motion of the magnetization in the laboratory and rotating frames (cf. Figure 1.3, Figure 2.14), except the FID signal has an attenuation term, which modulates the oscillation amplitude in the time domain. So instead of the constant amplitude sinusoidal waves as in Figure 2.15d, the amplitude of the FID oscillation will decay with time, as in Figure 2.15e, which generates the NMR signal. The only difference among the four NMR signals in Figure 2.15e is the four initial orientations of the magnetization vectors in the transverse plane (Figure 2.15c), that is, the initial phases of the magnetization. Even when M does not start from being parallel with any axis in the transverse plane, the real and imaginary parts of the FID both start with some initial values but still oscillate and decay in the same manner as in Figure 2.15e.

With this in mind, we can make some further comments on the laboratory and rotating frames. In the laboratory (stationary) frame, the z axis is firmly defined by the direction of B₀ (as shown in Figure 1.2), while the x and y axes are commonly defined by the typical 3D Cartesian coordinates in the 2D transverse plane, which is perpendicular to the z axis (Figure 1.3). In the rotating frame, the direction of the x′ axis is quite arbitrary, defined at the instant when the B₁(t) field is switched on. For a single B₁(t) field, whether B₁ is along the x′ or y′ or –x′ or –y′ axis does not matter at all, since the variation of the B₁ direction only results in the exchange of the absorption and dispersion terms in Eq. (2.34). Even when the B₁ direction is at an arbitrary angle between the x′ and y′ axes in the transverse plane, it results in only slightly complicated quadrature signals, where the mixed terms can be and are always phase-adjusted in the phasing process after the signal acquisition [with the use of a simple 2D rotation, as Eq. (A1.23)]. The phase of a B₁ field/pulse becomes critical only when this pulse is among a train of B₁ pulses, that is, the relative phases among the B₁ pulses in a pulse train do matter.

      References

      1 1. Harris RK, Becker ED, Cabral De Menezes SM, Goodfellow R, Granger P. NMR Nomenclature. Nuclear Spin Properties and Conventions for Chemical Shifts (IUPAC Recommendations 2001). Pure Appl Chem. 2001; 73(11):1795–818.

      2 2. Callaghan PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Oxford University Press; 1991.

      3 3. Hennel JW, Klinowski J. Fundamentals of Nuclear Magnetic Resonance. Essex: Longman Scientific & Technical; 1993.

      4 4. Harris RK. Nuclear Magnetic Resonance Spectroscopy – A Physicochemical View. Essex: Longman Scientific & Technical; 1983.

      5 5. Bovey FA. Nuclear Magnetic Resonance Spectroscopy. 2nd ed. San Diego, CA: Academic Press; 1988.

      6 6. Meadows M. Precession and Sir Joseph Larmor. Concepts in Magnetic Resonance. 1999; 11(4):239–41.

      7 7. Bloch F. Nuclear Induction. Phys Rev. 1946;70(7–8):460–74.

      8 8. Bracewell R. The Fourier Transform and Its Applications. New York: McGraw-Hill Book Company; 1965.

      9 9. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes. Cambridge: Cambridge University Press; 1989.

      10 10. Hoult DI, Richards RE. The Signal-to-Noise Ratio of the Nuclear Magnetic Resonance Experiment. J Magn Reson. 1976; 24:71–85.

      Notes

      1 1 Larmor precession is named after Irish physicist Joseph Larmor, who in 1897 first described the circular motion of the magnetic moment of an orbiting charged object about an external magnetic field.

3 Quantum Mechanical Description of Magnetic Resonance

Although the visualization of a vector M moving under the direction of a B₁ pulse is useful for the understanding of the simplest cases in NMR and MRI, a deep understanding of magnetic resonance [1–6] requires the aid of quantum mechanics, where the essential information of the nuclear systems is contained in the complex wave functions (labeled with Greek letters Ψ, Φ, φ, ψ). These wave functions can be described by the use of a vectoral term called a ket and written as |φ>. For each |φ>, one further defines a conjugated vector of a different nature, called a bra and written as <φ|. (The terms bra and ket come from truncations of the word bracket.)

      In modern physics, the energies (E ) and wave functions (ψ) for a molecular or atomic system can be investigated by the use of the Schrödinger equation (ℋ ψ = Eψ), where the operator ℋ is called the Hamiltonian and commonly contains the differential operator ∇2. (A spin term is usually neglected for the computation of atomic and molecular orbitals because its influence, in terms of energy shift, is negligibly small in the absence of a magnetic field.)