Relaxation (NMR)
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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
T_{1} and T_{2}
Different physical processes are responsible for the relaxation of the components of the nuclear spin magnetization vector M parallel perpendicular to the external magnetic field, B_{0} (which is conventionally oriented along the z axis). These two principal relaxation processes are termed T_{1} and T_{2} relaxation respectively.
T_{1}
The longitudinal relaxation time T_{1} is the decay constant for the recovery of the z component of the nuclear spin magnetization, M_{z}, towards its thermal equilibrium value, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_{z,\mathrm{eq}} . In general,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_z(t) = M_{z,\mathrm{eq}} - [M_{z,\mathrm{eq}} - M_z(0)]e^{-t/T_1}
In specific cases:
- If M has been tilted into the xy plane, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_z(0)=0 and the recovery is simply
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_z(t) = M_{z,\mathrm{eq}}\left( 1 - e^{-t/T_1} \right)
i.e. the magnetisation recovers to 63% of its equilibrium value after one time constant T_{1}.
- In the inversion recovery experiment, commonly used to measure T_{1} values, the initial magnetisation is inverted, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_z(0)=-M_{z,\mathrm{eq}} , and so the recovery follows
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_z(t) = M_{z,\mathrm{eq}}\left( 1 - 2e^{-t/T_1} \right)
T_{1} relaxation involves redistributing the populations of the nuclear spin states in order to reach the thermal equilibrium distribution. By definition this is not energy conserving. Moreover, spontaneous emission is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of T_{1} relaxation. However, a variety of relaxation mechanisms allow nuclear spins to exchange energy with their surroundings, the lattice, allowing the spin populations to equilibrate. The fact that T_{1} relaxation involves an interaction with the surroundings is the origin of the alternative description, spin-lattice relaxation.
Note that the rates of T_{1} relaxation are generally strongly dependent on the NMR frequency and so may vary considerably with magnetic field strength, B.
T_{2}
The transverse relaxation time T_{2} is the decay constant for the component of M perpendicular to B_{0}, designated M_{xy}, M_{T}, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_{\perp} . For instance, initial xy magnetisation at time zero will decay to zero (i.e. equilibrium) as follows:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): M_{xy}(t) = M_{xy}(0) e^{-t/T_2} \,
i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant T_{2}.
T_{2} relaxation is a complex phenomenon, but at its most fundamental level, it corresponds to a decoherence of the transverse nuclear spin magnetization. Random fluctuations of the local magnetic field lead to random variations in the instantaneous NMR precession frequency of different spins. As a result, the intial phase coherence of the nuclear spins is lost, until eventually the phases are disordered and there is no net xy magnetization. Because T_{2} relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation.
T_{2} values are generally much less dependent on field strength, B, than T_{1} values.
T_{2}* 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 T_{2}^{*}, which is usually much smaller than T_{2}. The relation between them is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{T_2^*}=\frac{1}{T_2}+\frac{1}{T_{inhom}} = \frac{1}{T_2}+\gamma \Delta B_0
where γ represents gyromagnetic ratio, and ΔB_{0} the difference in strength of the locally varying field.
Unlike T_{2}, T_{2}* is influenced by magnetic field gradient irregularities. The T_{2}* relaxation time is always shorter than the T_{2} relaxation time and is typically milliseconds for water samples in imaging magnets.
The reason that T_{1} is slower than T_{2}
As a general rule, the following always holds true: T_{1} > T_{2} > T_{2}*.
If T_{2} were to be slower than T_{1}, 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, T_{1} must be greater than or equal to T_{2}.
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.
Tissue Type | Approximate T_{1} value in ms | Approximate T_{2} 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.
Signals of Chemical Groups | Relative resonance frequency | Approximate T_{1} value (ms) | Approximate T_{2} 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 |
—CH_{3} 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 T_{1}、T_{2}:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{T_1}=K[\frac{\tau_c}{1+\omega_0^2\tau_c^2}+\frac{4\tau_c}{1+4\omega_0^2\tau_c^2}]
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{1}{T_2}=\frac{K}{2}[3\tau_c+\frac{5\tau_c}{1+\omega_0^2\tau_c^2}+\frac{2\tau_c}{1+4\omega_0^2\tau_c^2}] ,
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega_0 is the Larmor frequency in correspondence with the strength of the main magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): B_0 . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tau_c is the correlation time of the molecular tumbling motion. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): K=\frac{3\mu^2}{160\pi^2}\frac{\hbar^2\gamma^4}{r^6} is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \hbar=\frac{h}{2\pi} 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 H_{2}O molecules in liquid phase without the contamination of oxygen 17, the value of K is 1.02×10^{10} s^{-2} and the correlation time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tau_c is on the order of picoseconds = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 10^{-12} s, while hydrogen nuclei ^{1}H (protons) at 1.5 teslas carry an Larmor frequency of approximately 64 MHz. We can then estimate using τ_{c} = 5×10^{-12} s:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega_0\tau_c = 3.2\times 10^{-5} (dimensionless)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T_1=(1.02\times 10^{10}[\frac{ 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 4\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1} = 3.92 s
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T_2=(\frac{1.02\times 10^{10}}{2}[3\cdot 5\times 10^{-12} + \frac{5\cdot 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 2\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1} = 3.92 s,
which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T_{1} equals T_{2}.
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 relaxation 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
- [5] Relaxation in high-resolution NMR spectroscopy
- Relaxometry