From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 193

and to return to their low energy configuration or the thermal equilibrium. This is
known as the relaxation phase. The signal is created as the spins precess tilted away
from B 0 , and it decays as the spins relax, dissipating the absorbed energy. The
longitudinal relaxation and the transverse relaxation of M are governed by different
phenomena and are characterized by different time signatures.
The longitudinal relaxation is known as the T1 relaxation since it is described
using a time signature denoted T1. The T1 relaxation occurs as the spin ensemble
radiates the energy it had absorbed from the RF pulse to the surrounding thermal
reservoir or lattice and regains its thermal equilibrium with the lattice. Therefore,
the T1 relaxation is also known as the spin-lattice relaxation. In this process the
spins realign themselves with B 0 . In terms of the net magnetization vector M ,
this implies that the longitudinal component M z progressively regains its initial
magnitude, while the transverse component M xy progressively becomes null again.
The transverse relaxation involves the phenomenon of the spins regaining their
thermal equilibrium amongst themselves, and is characterized by the time signature
T2. Therefore it is also known as the spin-spin relaxation or the T2 relaxation. In the
transverse plane this is interpreted by the spins losing their initial coherence. From
an initial coherent transverse magnetization vector M xy , they progressively dephase
as they radiate the energy they had absorbed to neighbouring spins. Transverse
relaxation is, however, a complex phenomenon. Although theoretically B 0 is sup-
posed homogeneous, in reality minor inhomogeneities exist. These inhomogeneities
are relevant enough to also contribute to spins dephasing in the transverse plane,
though this is not a true relaxation. Transverse relaxation is therefore a combination
of spin-spin relaxation and field inhomogeneity dephasing. The pure spin-spin
relaxation time is known as T2. The combined transverse relaxation time is known
as T2 ∗ .
The Bloch equations are a coupled set of three differential equations that combine
the effects of NMR and describe the evolution of the net magnetization vector M
over time. These are macroscopic and phenomenological equations that include the
effects of Larmor precession and T1 and T2 relaxations. They are written in the
fixed frame of reference in terms of the relaxation time constants as:

⎛ ⎞ ⎛ ⎞

− T 2 00
0 − T 2 0
00 − T 1

0
0 M 0
T 1

d M ( t )

⎝ ⎠ ⎝ ⎠

γ M t

) ×

B t M t ,

= ( ( )+ ( )+ (6.1)

dt

where B ( t ) is the total external magnetic field.

6.3.1 The Hahn Spin Echo Experiment

Erwin L Hahn was the first to notice the effects of diffusion when he conceived the
spin echo experiment to remove the effects of field inhomogeneities or T2 ∗ from the