RELAXATION (NMR)


In Nuclear magnetic resonance (NMR) spectroscopy and Magnetic resonance imaging (MRI) the term 'relaxation' describes several processes by which
nuclear magnetization prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins "forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.

Contents
T1 and T2
T1
T2
T2
★ and magnetic field inhomogeneity
The reason that T1 is slower than T2
Common relaxation time constants in human tissues
Microscopic mechanism
References
See also

T1 and T2


Different physical processes cause different rates of spin relaxation in different directions with respect to the magnetic field, conventionally referred to as the z axis. The dominant two relaxation rates are described by the 'relaxation times' T_1 and T_2:

★ The 'longitudinal relaxation time T1' is the decay constant for the part of the magnetization vector 'M' that is parallel to the main magnetic field 'B'0, designated as Mz. The process proceeds at a rate governed by a time constant T1.
:M_z(t) = M_z(0) cdot left( 1 - e^{-t/T_1}
ight) ,

★ The 'transverse relaxation time T2' is the decay constant for the part of the magnetization vector 'M' that is perpendicular to the main magnetic field 'B'0 designated as 'M'xy, 'M'T, or M_{perp}. The process proceeds at a rate a governed by time constant T2.
:M_{xy}(t) = M_{xy}(0) cdot e^{-t/T_2} ,
T1

In an ideal environment where strict conservation of angular momentum is true for the nuclei being observed, T1 would not exist. When the magnetization of a nucleus in the experimental pulse is altered, it should maintain its precession. So the bulk magnetization which is set into a disequilibrium cannot equilibrate. However, in a real system, there is spin transfer between the observed nuclei and the environment. This allows for "forbidden" transitions to occur, and "relaxation" from "excited" state back to equilibrium.
T1 is by definition, the component of relaxation which occurs in the direction of the ambient magnetic field. This generally comes about by interactions between the nucleus of interest and unexcited nuclei in the environment, as well as electric fields in the environment (collectively known as the 'lattice'). Therefore, T1 is known as "spin-lattice" relaxation.
T1 is measured as the time required for the magnetization vector M to be restored to 63% of its original magnitude. It varies with the magnetic field B.
T2

Main articles: Spin-spin relaxation time

In an idealized system, T2 would also not exist. However, in real systems, there is spin transfer amongst excited nuclei which disperses magnetization that is out of equilibrium.
T2, by definition, is the component of 'true' relaxation (see T2
★ ) to equilibrium that occurs perpendicular to the ambient magnetic field. Because of this, the relaxation is dominated by interactions between spinning nuclei which are already excited. For this reason, T2 relaxation is called "transverse" or "spin-spin" relaxation.
Since T2 processes follow an exponential decay, the quantity T2 is defined as the time required for the transverse Magnetization vector to drop to 37% of its original magnitude after its initial excitation.
Unlike T1, T2 is much less susceptible to variations of field strength B.
T2
★ and magnetic field inhomogeneity

In an idealized system, all nuclei in a given chemical environment in a magnetic field spin with the same frequency. However, in real systems, there are minor differences in chemical environment which can lead to a distribution of resonance frequencies around the ideal. Over time, this distribution can lead to a dispersion of the tight distribution of magnetic spin vectors, and loss of signal (Free Induction Decay). In fact, for most magnetic resonance experiments, this "relaxation" dominates. This results in intra-voxel dephasing.
However, decoherence because of magnetic field inhomogeneity is not a true "relaxation" process; it is not random, but dependent on the location of the molecule in the magnet. For molecules that aren't moving, the deviation from ideal relaxation is consistent over time, and the signal can be recovered by performing a spin echo experiment.
The corresponding transverse relaxation time constant is thus T2
★ , which is usually much smaller than T2. The relation between them is:
: rac{1}{T_2^
★ }= rac{1}{T_2}+ rac{1}{T_{inhom}} = rac{1}{T_2}+gamma Delta B_0
where γ represents gyromagnetic ratio, and ΔB0 the difference in strength of the locally varying field.
Unlike T2, T2
★ is influenced by magnetic field gradient irregularities. The T2
★ relaxation time is always shorter than the T2 relaxation time and is typically milliseconds for water samples in imaging magnets.
The reason that T1 is slower than T2

As a general rule, the following always holds true: T1 > T2 > T2
★ .
In order to get magnetization transfer, the energies and orientations of spins with magnetic entities in the lattice must be matched. In most setups, this is a relatively rare condition, compared to spin-spin interactions, which ''a priori'' are aligned with each other.
More simply, if T2 were to be slower than T1, then the magnetizations perpendicular to the initial direction would have not dephased by the time the sample had returned to equilibrium. This is physically impossible, as once the sample has returned to equilibrium, there is no magnetization perpendicular to the original direction. Hence, T1 must be greater than or equal to T2.

Common relaxation time constants in human tissues


Following is a table of the approximate values of the two relaxation time constants for nonpathological human tissues, just for simple reference.


'At a main field of 1.5 T'
Tissue Type Approximate T1 value in ms Approximate T2 value in ms
Adipose tissues 240-250 60-80
Whole blood (deoxygenated) 1350 50
Whole blood (oxygenated) 1350 200
Cerebrospinal fluid (similar to pure water) 2200-2400 500-1400
Gray matter of cerebrum 920 100
White matter of cerebrum 780 90
Liver 490 40
Kidneys 650 60-75
Muscles 860-900 50

Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human brain magnetic resonance spectroscopy (MRS) studies, physiologically or pathologically.


'At a main field of 1.5 T'
Signals of Chemical Groups Relative resonance frequency Approximate T1 value (ms) Approximate T2 value (ms)
Creatine (Cr) and Phosphocreatine (PCr) 3.0 ppm gray matter: 1150-1340,
white matter: 1050-1360		 gray matter: 198-207,
white matter: 194-218
N-Acetyl group (NA),
mainly from N-Acetylaspartate (NAA)		 2.0 ppm gray matter: 1170-1370,
white matter: 1220-1410		 gray matter: 388-426,
white matter: 436-519
—CH3 group of
Lactate		 1.33 ppm
(doublet: 1.27 & 1.39 ppm)		 (To be listed) 1040

Microscopic mechanism


In 1948, Nicolaas Bloembergen, Edward Mills Purcell, and R.V. Pound proposed the so-called Bloembergen-Purcell-Pound theory (BPP theory) to explain the relaxation constant of a pure substance in correspondence with its state, taking into account the effect of tumbling motion of molecules on the local magnetic field disturbance . The theory was in good agreement with the experiments for pure substance, but not for complicated environment such as human body.
From this theory, one can get T1、T2:
: rac{1}{T_1}=K[ rac{ au_c}{1+omega_0^2 au_c^2}+ rac{4 au_c}{1+4omega_0^2 au_c^2}]
: rac{1}{T_2}= rac{K}{2}[3 au_c+ rac{5 au_c}{1+omega_0^2 au_c^2}+ rac{2 au_c}{1+4omega_0^2 au_c^2}],
where omega_0 is the Larmor frequency in correspondence with the strength of the main magnetic field B_0. au_c is the correlation time of the molecular tumbling motion. K= rac{3mu^2}{160pi^2} rac{hbar^2gamma^4}{r^6} is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, hbar= rac{h}{2pi} the reduced Planck constant, γ the gyromagnetic ratio of such species of nuclei, and r the distance between the two nuclei carrying magnetic dipole moment.
Taking for example the H2O molecules in liquid phase without the contamination of oxygen 17, the value of K is 1.02×1010 sec-2 and the correlation time au_c is on the order of ps = 10^{-12} sec, while hydrogen nuclei 1H (protons) at 1.5 tesla carry an Larmor frequency of approximately 64 MHz. We can then estimate using τc = 5×10-12 sec:
:omega_0 au_c = 3.2 imes 10^{-5} (dimensionless)
:T_1=(1.02 imes 10^{10}[ rac{ 5 imes 10^{-12} }{1 + (3.2 imes 10^{-5} )^2} + rac{ 4cdot 5 imes 10^{-12} }{1 + 4cdot (3.2 imes 10^{-5} )^2}])^{-1} = 3.92 sec
:T_2=( rac{1.02 imes 10^{10}}{2}[3cdot 5 imes 10^{-12} + rac{5cdot 5 imes 10^{-12} }{1 + (3.2 imes 10^{-5} )^2} + rac{ 2cdot 5 imes 10^{-12} }{1 + 4cdot (3.2 imes 10^{-5} )^2}])^{-1} = 3.92 sec,
which is close to the experimental value, 3.6 sec. Meanwhile, we can see that at this extreme case, T1 equals T2.

References


# 'Chemicals of brain relaxation time at 1.5T.' Kreis R, Ernst T, and Ross BD "Absolute Quantification of Water and Metabolites in the Human Brain. II. Metabolite Concentrations" ''Journal of Magnetic Resonance'', Series B 102 (1993): 9-19
# 'Lactate rexalation time at 1.5 T'. Isobe T, Matsumura A, Anno I, Kawamura H, Muraishi H, Umeda T, Nose T. "Effect of J coupling and T2 Relaxation in Assessing of Methyl Lactate Signal using PRESS Sequence MR Spectroscopy." ''Igaku Butsuri'' (2005) v25. 2:68-74.
# 'BPP theory'. Bloembergen, E.M. Purcell, R.V. Pound "Relaxation Effects in Nuclear Magnetic Resonance Absorption" ''Physical Review'' (1948) v73. 7:679-746

See also



★ MRI - Magnetic resonance imaging Animation made by bigs.eu; contents are: spin, spin modification, induction, relaxation and precession, spin echo sequence, gradient echo sequence, inversion recovery sequence

★ [1] Relaxation in high-resolution NMR spectroscopy

★ Relaxometry

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