Nuclear Magnetic Resonance
By Bodo Kaiser
Introduction
Usage, Examples, Content
Examples
From Siemens Healthcase
Examples
Content
- Introduction
- Overview
- Measurement
- Classic Description
- Quantum Description
- Refinements
- Prospects
- Closing
Overview
Apparatus, Setup
MRI Apparatus
From USRE32619 (1988)
NMR Apparatus
From Agilent
Summary
- What is the setup of a NMR apparatus?
- What is the measurement procedure?
- What magnetisations do we observe?
Measurement
RF Pulse, Free Induction Decay, Spectra, Bloch Equations, Time Constants
RF Pulse
Free Induction Decay
Spectra
Bloch Equations
$$\begin{align} M_L(t)=M_0\left(1-e^{-t/T_1}\right) && M_T(t)=M_0e^{-t/T_2} \end{align}$$
Time Constants
| Tissue Type | $T_1$ in ms | $T_2$ in ms |
|---|---|---|
| Fat | 240-250 | 60-80 |
| Blood (deoxygenated) | 1350 | 50 |
| Blood (oxygenated) | 1350 | 200 |
| Gray Matter | 920 | 100 |
| White Matter | 780 | 90 |
| Liver | 490 | 40 |
| Kidney | 650 | 60-75 |
| Muscles | 860-900 | 50 |
From Wikipedia for magnetic field $B=1.5T$
Image Weighting
From CWR University
Summary
- What is the FID and how is it measured?
- What time constants are there and what do they describe?
Classic Description
Magnetic Moment, Nuclei, Precession, Rotating Frame, Magnetisation
Magnetic Moment
The magnetic moment $\mu$ is defined by $$\boldsymbol{\tau}=\boldsymbol{\mu}\times\boldsymbol{B}.$$
Moving charges possess a magnetic moment $$\boldsymbol{\mu}=\frac{q}{2m}\boldsymbol{l}.$$
Gyromagnetic constant $$\gamma=\frac{q}{2m}.$$
Relevant Nuclei
| Nucleus | $\gamma$ $\left[10^6\frac{rad}{s T}\right]$ | $f$ $[10^6Hz]$ | $m/n$ |
|---|---|---|---|
| $^1H$ | 267.5 | 42.6 | 99.98 |
| $^2H$ | 40.8 | 6.5 | 0.02 |
| $^{10}B$ | 28.3 | 4.6 | 18.8 |
| $^{11}B$ | 86.1 | 13.7 | 81.2 |
| $^{13}C$ | 67.2 | 10.7 | 1.11 |
| $^{19}F$ | 252.0 | 40.1 | 100 |
| $^{23}Na$ | 70.8 | 11.3 | 100 |
| $^{31}P$ | 108.1 | 17.2 | 100 |
From Wikipedia1, Wikipedia2 and Bergmann und Schaefer (1992) for magnetic field $B=1T$
Larmor Precession
Torque changes angular momentum $$\dot{\boldsymbol{l}}=\boldsymbol{\tau}.$$
Magnetic moment precesses in magnetic field $$\dot{\boldsymbol{\mu}}=\gamma\boldsymbol{\mu}\times\boldsymbol{B}_0.$$
Magnetic Pulse
Equation of motion in rotatin frame $$\dot{\boldsymbol{\mu}}_\varphi =\boldsymbol{\mu}_\varphi\times\left(\gamma\boldsymbol{B}_0-\boldsymbol{\omega}_0\right).$$
Magnetic field with pulse in rotating frame $$\boldsymbol{B}_\varphi=B_0\boldsymbol{n}_\parallel +R(-\omega_0t)R(\omega_1t)B_1\boldsymbol{n}_\perp.$$
New equation of motion in rotating frame $$\dot{\boldsymbol{\mu}}_\varphi =\boldsymbol{\mu}_\varphi\times\boldsymbol{B}_1(t).$$
Energy Distribution
Potential energy stored in magnetic field $$V(\theta)=-\mu B_0\cos\theta.$$
Boltzmann distribution of energy in equilibrium $$P(\theta)=\frac{\exp(-V(\theta)/k_B T)} {\int_0^\pi d\theta\sin\theta\exp(-V(\theta)/k_B T)}.$$
Boltzmann factor of antiparallel magnetic state $$P_{\uparrow\downarrow}=\exp\left(-\frac{\Delta V}{k_B T}\right) \approx 1+\frac{2\mu B_0}{k_B T} \overset{\text{H}}{\underset{B_0=3T\\T=38°C}{\approx}} 1+2{\cdot}10^{-5}.$$
Magnetisation I
Net magnetisation for $I=\frac{1}{2}$:
$$\begin{align} \boldsymbol{M}_0 &=\mu(N_\uparrow-N_\downarrow)\boldsymbol{n}\\ &=\mu N\frac{1-\exp(-\mu B_0/kT)}{1+\exp(-\mu B_0/kT)}\boldsymbol{n}\\ &=\mu N\tanh\left(\mu B_0/2kT\right)\boldsymbol{n}\\ &\approx\frac{N\mu^2}{2kT}\boldsymbol{B}_0 =\frac{N\gamma^2l^2}{2kT}\boldsymbol{B}_0 =\chi_0\boldsymbol{B_0} \end{align}$$
Magnetisation II
Summary
- What did we do?
- What assumptions did we make?
- What orders of magnitude do we have?
Quantum Description
Angular Momentum, Magnetic Moment, Hamiltonian, Precession, Magnetisation
Nuclear Angular Momentum
The nucleus possesses total angular momentum $$\hat{\boldsymbol{I}} =\sum_{i=1}(\hat{\boldsymbol{l}}_i+\hat{\boldsymbol{s}}_i)$$
with eigenvalues $$\begin{align} \hat{\boldsymbol{I}}^2\vert I,m\rangle&=I(I+1)\hbar^2\vert I,m\rangle\\ \hat{\boldsymbol{I}}_z\vert I,m\rangle&=m\hbar\vert I,m\rangle \end{align}$$
From Hyperphysics
Nuclear Magnetic Moment
Magnetic moment $$\hat{\boldsymbol{\mu}}= \frac{e_0}{2m_p}\sum_{i=1}(g_{li}\hat{\boldsymbol{l}}_i+g_{si}\hat{\boldsymbol{s}}_i).$$
| Nucleon | $g_l$ | $g_s$ |
|---|---|---|
| Proton | 1 | 5.58 |
| Neutron | 0 | -3.82 |
Expectation value $$\langle\hat{\boldsymbol{\mu}}\rangle= g_I\frac{e_0}{2m_p}\langle\hat{\boldsymbol{I}}\rangle=\gamma\langle\hat{\boldsymbol{I}}\rangle.$$
| Nucleus | $I$ |
|---|---|
| $^1H$ | $1/2$ |
| $^2H$ | $1$ |
| $^{10}B$ | $3$ |
| $^{11}B$ | $3/2$ |
| $^{13}C$ | $1/2$ |
| $^{19}F$ | $1/2$ |
| $^{23}Na$ | $3/2$ |
| $^{31}P$ | $1/2$ |
From Wikipedia
Energy Eigenvalues
Hamiltonian of nucleus $$\hat{H}= -\gamma\hat{\boldsymbol{\mu}}{\cdot}\boldsymbol{B}_0= -\gamma\hat{\boldsymbol{I}}{\cdot}\boldsymbol{B}_0.$$
has energy eigenstates $$\begin{align} \hat{H}\vert I,m\rangle &=-\gamma\hat{\boldsymbol{I}}{\cdot}\boldsymbol{B}_0\vert I,m\rangle\\ &=-\gamma B_0\hat{I}_z\vert I,m\rangle\\ &=-\gamma B_0\hbar m\vert I,m\rangle \end{align}$$
therefore enery levels of $$\Delta E=\hbar\gamma B_0=\hbar\omega_0.$$
From Hyperphysics
Larmor Precession
Total angular momentum changes $$\begin{align} i\hbar\frac{d}{dt}\hat{I}_i &=\left[\hat{I}_i, -\gamma B_j \hat{I}_j\right]\\ &=-\gamma\left[\hat{I}_i, \hat{I}_j\right]B_j\\ &=-i\hbar\gamma\epsilon_{ijk}\hat{I}_kB_j \end{align}$$
equals classical movement $$\frac{d}{dt}\langle\hat{\boldsymbol{\mu}}\rangle =\gamma\langle\hat{\boldsymbol{\mu}}\rangle\times\boldsymbol{B}(t).$$
From Hyperphysics
Magnetisation
Energy stored in magnetic field now discrete $$E_m=-\gamma\hbar m B_0.$$
Boltzmann distribution of energy in equilibrium $$P_m=\frac{\exp(-E_m/k_B T)} {\sum^I_{m=-I} \exp(-E_m/k_B T)}.$$
Net magnetisation ($I=\frac{1}{2}$) $$\boldsymbol{M}_0 =\mu N\tanh\left(\hbar\gamma B_0/2kT\right)\boldsymbol{n} \approx\frac{N\gamma^2\hbar^2}{4kT}\boldsymbol{B}_0.$$
Summary
- How does quantum differ from classical description?
- Where do we need to be careful with the quantum description?
- What effects do we still neglect?
Refinements
Relaxation, Spin Sequences
Revision Relaxation
Spin-Lattice Relaxation
Caused by thermal energy dissipation.
Responsible for intensity loss in FID (determines recycle time).
Strongly depends on magnetic field and frequency.
Spin-Spin Relaxation
$$\frac{1}{T_2^\ast}=\frac{1}{T_2}+\frac{1}{T_2^\prime}.$$
$T_2$ caused by tissue structure.
$T_2^\prime$ caused by local inhomogeneities.
$$2T_1\geq T_2^\ast\geq T_2^\prime\geq T_2$$
Spin-Echo
From O. Dietrich (2001), see Literature
Stimulated Spin-Echo
From O. Dietrich (2001), see Literature
Inversed Spin-Echo
From O. Dietrich (2001), see Literature
Summary
- What are marcoscopic effects behind $T_1,T_2,T_2^\prime$?
- What is the basic idea behind spin echos?
- What pulse sequences can we use for specific $T_1,T_2,T_2^\prime$?
- What tissue type are $T_1,T_2,T_2^\prime$ sensitive to?
- What is the general time inequation that connects $T_1,T_2,T_2^\prime$?
Prospects
In Vivo MR Spectroscopy, Transformations
In Vivo MR Spectroscopy
From M. Kim et al, 2001., see Literature
CT Transformation
From D. Nie et al, 2001., see Literature
US Transformation
Summary
- How does In Vivo NMR differ from MRI?
- What is the best medical imaging technique?