Читать книгу Essential Concepts in MRI - Yang Xia - Страница 23

When it can be assumed that the change of the magnetization following excitation is independently caused by external magnetic fields and relaxation processes, the equation of motion of M can be written by combining Eq. (2.14) and Eq. (2.15), in the laboratory frame, as

(2.18)

This is the well-known Bloch equation [7]. The first term is due to precessional motion and the second term is due to relaxation. While a precise evaluation of the spin system requires a quantum mechanical treatment, the Bloch equation provides a classical, phenomenological description for liquids and liquid-like systems where the Hamiltonian is of a simple magnetic (vector) form, for example, protons of water molecules in non-viscous liquids and many biological tissues.

Now we are ready to solve the Bloch equation under various conditions. First rewrite the vector equation into the component form, as

(2.19a)

(2.19b)

(2.19c)

The above equations have the usual setup for the magnetic fields as

(2.20a)

(2.20b)

Thermal equilibrium ensures the initial condition of M in Eq. (2.19) as

(2.21)

Note that as soon as M is tipped away from its thermal equilibrium state, relaxation processes start. In most analyses, however, we consider only one event at a time – that is, when we use the B₁ field to tip the magnetization, we do not consider the relaxation of the magnetization during the tipping process.

In order to better examine the solution of the Bloch equation (more precisely, to examine the spectral shapes of the waveform solutions), we will describe the magnetization in a rotating frame with an angular velocity ω about the z axis. In this x′y′z′ frame, we have the component u in the direction of x′ and v in the direction of y′. We can use the common rotation matrix in linear algebra to rewrite the transform matrix as

(2.22a)

(2.22b)

Note that Eq. (A1.23) is used in this clockwise rotation, which is consistent with the convention specified in Figure 1.3. (For a counterclockwise rotation, keep both terms of v positive and use a minus sign for the second term of u; see Appendix A1.1).

By writing the Bloch equation in this rotating frame and by setting the time derivatives in the equation equal to zero, we can solve the Bloch equation in the rotating frame. The solutions are

(2.23a)

(2.23b)

(2.23c)