This presentation will provide an overview of the typical forms of the Bloch Equations, the physical mechanisms of relaxation phenomena as well as the basis of typical MRI contrasts.
Understanding:
In 1946 Felix Bloch introduced a set of differential equations describing the temporal evolution of the net magnetization of a spin system $$$ \vec{M} = M_x \cdot \hat{x} + M_y \cdot \hat{y} + M_z \cdot \hat{z}$$$ under the influence of external fields and relaxation effects [1]. These macroscopic and entirely phenomenological “equations of motion of the nuclear magnetization” serve as an excellent basis for describing a wide range of effects in magnetic resonance imaging and spectroscopy.
An elegant formulation can be achieved by using a reference system rotating around a static magnetic field $$$B_0$$$ along $$$\hat{z}$$$ (e.g. the main field of an MR-scanner: $$${\vec{B}}_0 = B_0\hat{z}$$$, which is usually switched on for all t, $$$\omega_0 = \gamma B_0$$$) at a frequency $$$\omega$$$ equally to the resonance frequency:
$$ \frac{d}{dt} \begin{bmatrix}M_x \\ M_y \\ M_z \end{bmatrix} = \begin{bmatrix} -1/T_2 & \omega_0 - \omega & -\gamma B_{1,y} \\ \omega - \omega_0 & -1/T_2 & \gamma B_{1,x} \\ \gamma B_{1,y} & -\gamma B_{1,x} & -1/T_1 \end{bmatrix} \begin{bmatrix}M_x \\ M_y \\ M_z \end{bmatrix} + \begin{bmatrix}0 \\ 0 \\ M_0 / T_1 \end{bmatrix} \qquad (1)$$
$$${\vec{B}}_1$$$ is
a magnetic field oscillating in the $$$\hat{x}$$$-$$$\hat{y}$$$-plane with amplitude $$$B_1$$$ and frequency $$$\omega_0$$$ which typically embodies RF-excitation, only
active for short periods of time. $$$T_1$$$ and $$$T_2$$$ function as damping terms caused by relaxation
phenomena driving the spin system back into an equilibrium state. By finding solutions of equation (1), MR-experiments can be simulated and ultimately, MR-signal behavior can be
predicted.
If a spin system in a static magnetic field $$${\vec{B}}_0 = B_0 \hat{z} $$$ has been disturbed from its equilibrium state, e.g. by some RF-excitation, $$$M_z$$$ exhibits an exponential relaxation towards its initial magnitude $$$M_0$$$ aligned with $$${\vec{B}}_0$$$. The time constant $$$T_1$$$ of this process is called longitudinal or spin-lattice relaxation time. The underlying process is an exchange of the energy, which was brought into the system by the perturbation, to other atoms in the molecule or the molecules as a whole [2].
After exciting the spin system, a component of the magnetization can be measured in the $$$\hat{x}$$$-$$$\hat{y}$$$-plane. Spins are initially in phase, however, the magnetic dipoles of the spins interact with each other, which immediately results in a dephasing of the spin system and ultimately in a decay of the signal. This transverse relaxation is also of exponential behavior, now governed by the time constant $$$T_2$$$ (transverse or spin-spin-relaxation time). Usually, $$$T_2$$$ is considerably smaller than $$$T_1$$$. Besides intrinsic inhomogeneities (i.e. interactions internal to the proton system) also external ones like inhomogeneities in $$$B_0$$$ or those caused by susceptibility interfaces drive the transversal relaxation. If the latter effects are incorporated, $$$T_2^*<T_2$$$ commonly represents the corresponding time constant.
For monomolecular (pure) substances, the Bloembergen-Purcell-Pound (BPP, [3]) theory provides a straightforward explanation of the underlying mechanisms of the $$$T_1$$$- and $$$T_2$$$-relaxation phenomena. Local magnetic field fluctuations evoked by molecular motion are modelled for a quantitative description. Refinements to this theory are necessary to reliably predict the relaxation times in more complicated, biological tissues.
[1] Nuclear Induction. Bloch F. Physical Review 70:460-474 (1946)
[2] Magnetic Resonance Imaging. Bushong SC. Mosby (1996)
[3] Relaxation Effects in Nuclear Magnetic Resonance Absorption. Bloembergen N, Purcell EM, Pound RV. Physical Review 73:679-746 (1948)