Comprehensive Guide to MRI Physics
Table of Contents
- Quantum Mechanical Foundations
- Magnetization and Magnetic Moments
- Relaxation Processes
- Bloch Equations
- RF Pulses and Excitation
- Spatial Encoding and K-space
- Imaging Sequences and Contrast Mechanisms
- Advanced MRI Techniques
- MRI Hardware Physics
- Safety Considerations in MRI Physics
1. Quantum Mechanical Foundations
At its core, Magnetic Resonance Imaging (MRI) is rooted in the principles of quantum mechanics, specifically the quantum mechanical property of spin. To understand MRI physics, we must start with these fundamental concepts.
1.1 Spin and Magnetic Moment
Atomic nuclei with an odd number of protons or neutrons possess an intrinsic angular momentum, known as spin. The hydrogen nucleus (a single proton) is of particular interest in MRI due to its abundance in biological tissues. The spin of a proton gives rise to a magnetic moment (μ), which can be thought of as a tiny bar magnet.
The magnetic moment is related to the spin angular momentum (S) by the gyromagnetic ratio (γ):
μ = γS
For hydrogen, the gyromagnetic ratio is approximately 42.58 MHz/T.
1.2 Zeeman Effect
When placed in an external magnetic field (B0), the magnetic moments of protons align either parallel (low energy state) or antiparallel (high energy state) to the field. This splitting of energy levels is known as the Zeeman effect. The energy difference between these states is given by:
ΔE = γℏB0
Where ℏ is the reduced Planck constant.
1.3 Larmor Precession
In the presence of B0, the magnetic moments precess around the axis of the external field at a frequency known as the Larmor frequency (ω0):
ω0 = γB0
This relationship is fundamental to MRI, as it allows for the selective excitation of protons using radiofrequency (RF) pulses at the Larmor frequency.
2. Magnetization and Magnetic Moments
While individual protons behave according to quantum mechanics, in MRI we deal with large populations of spins, which can be described using classical electromagnetic theory.
2.1 Net Magnetization
In thermal equilibrium, slightly more spins align parallel to B0 than antiparallel, resulting in a net magnetization (M0) parallel to the external field. The magnitude of M0 is given by:
M0 = ρ(γℏ)2B0 / (4kT)
Where ρ is the proton density, k is Boltzmann's constant, and T is the absolute temperature.
2.2 Boltzmann Distribution
The ratio of spins in the parallel (N↑) and antiparallel (N↓) states follows the Boltzmann distribution:
N↑ / N↓ = eΔE/kT
At body temperature and typical field strengths, this ratio is very close to unity, resulting in a small but detectable net magnetization.
2.3 Classical Description of Precession
The behavior of the net magnetization vector can be described classically using the equation of motion:
dM/dt = γM × B
This equation describes the precession of M around B at the Larmor frequency.
3. Relaxation Processes
After excitation by an RF pulse, the magnetization returns to its equilibrium state through two primary relaxation processes: longitudinal (T1) and transverse (T2) relaxation.
3.1 T1 Relaxation (Spin-Lattice Relaxation)
T1 relaxation describes the recovery of the longitudinal component of magnetization (Mz) to its equilibrium value. It is characterized by the time constant T1:
Mz(t) = M0(1 - e-t/T1)
T1 relaxation involves energy exchange between the spin system and the surrounding lattice. It is influenced by molecular motion and the local magnetic environment.
3.2 T2 Relaxation (Spin-Spin Relaxation)
T2 relaxation describes the decay of the transverse component of magnetization (Mxy). It is characterized by the time constant T2:
Mxy(t) = Mxy(0)e-t/T2
T2 relaxation results from interactions between neighboring spins, causing a loss of phase coherence in the transverse plane.
3.3 T2* Relaxation
In practice, the observed transverse relaxation occurs faster than predicted by T2 alone due to local magnetic field inhomogeneities. This combined effect is described by T2*:
1/T2* = 1/T2 + γΔB0
Where ΔB0 represents local field inhomogeneities.
| Tissue | T1 (ms) at 1.5T | T2 (ms) |
|---|---|---|
| Gray Matter | 950 | 100 |
| White Matter | 600 | 80 |
| Cerebrospinal Fluid | 4500 | 2200 |
| Fat | 250 | 60 |
4. Bloch Equations
The behavior of magnetization in the presence of external magnetic fields and relaxation processes is described comprehensively by the Bloch equations:
dMx/dt = γ(MyBz - MzBy) - Mx/T2
dMy/dt = γ(MzBx - MxBz) - My/T2
dMz/dt = γ(MxBy - MyBx) + (M0 - Mz)/T1
These equations form the foundation for understanding and simulating MRI experiments.
5. RF Pulses and Excitation
Radiofrequency pulses are used to manipulate the net magnetization and generate detectable MR signals.
5.1 Resonance and the Rotating Frame
When an RF pulse is applied at the Larmor frequency, it appears stationary in a reference frame rotating at ω0. This rotating frame simplifies the analysis of RF pulse effects.
5.2 Flip Angle
The flip angle (α) describes the degree of rotation of the magnetization vector due to an RF pulse:
α = γB1τ
Where B1 is the amplitude of the RF field and τ is the pulse duration.
5.3 Types of RF Pulses
- 90° (π/2) Pulse: Rotates the magnetization into the transverse plane.
- 180° (π) Pulse: Inverts the magnetization or refocuses transverse magnetization.
- Small Flip Angle Pulse: Used in gradient echo sequences.
5.4 Shaped Pulses
RF pulses can be shaped in time and frequency to achieve specific excitation profiles. Common shapes include sinc, Gaussian, and adiabatic pulses.
6. Spatial Encoding and K-space
Spatial encoding is crucial for creating images from MR signals. It involves the use of magnetic field gradients to encode spatial information into the frequency and phase of the MR signal.
6.1 Slice Selection
A slice-select gradient is applied during the RF pulse to excite a specific slice. The thickness and position of the slice are determined by the gradient strength and RF bandwidth:
Δz = BW / (γGz)
Where BW is the bandwidth of the RF pulse and Gz is the slice-select gradient strength.
6.2 Frequency Encoding
A frequency-encoding gradient is applied during signal readout, causing protons to precess at different frequencies depending on their position along the gradient axis.
6.3 Phase Encoding
Phase-encoding gradients are applied between excitation and readout to induce position-dependent phase shifts.
6.4 K-space
K-space is a conceptual space in which MRI data is acquired. Each point in k-space corresponds to a specific combination of frequency and phase encoding. The relationship between k-space and image space is described by the Fourier transform:
S(kx, ky) = ∫∫ ρ(x,y) e-i2π(kxx + kyy) dx dy
Where S(kx, ky) is the signal in k-space and ρ(x,y) is the spin density in image space.
7. Imaging Sequences and Contrast Mechanisms
MRI sequences are specific combinations of RF pulses and gradients designed to highlight particular tissue properties or pathologies.
7.1 Spin Echo Sequence
The spin echo sequence uses a 90° excitation pulse followed by a 180° refocusing pulse to generate an echo. It is less sensitive to magnetic field inhomogeneities and provides excellent T2 contrast.
Key parameters:
- TR (Repetition Time): Time between consecutive 90° pulses, affects T1 weighting.
- TE (Echo Time): Time between 90° pulse and echo formation, affects T2 weighting.
7.2 Gradient Echo Sequence
Additional parameters:
- Flip Angle: Affects the balance between T1 and T2* contrast.
- Spoiling: Techniques to eliminate residual transverse magnetization before the next RF pulse.
The signal intensity in a gradient echo sequence is given by:
S ∝ ρ(1 - e-TR/T1)sin(α) / (1 - cos(α)e-TR/T1) * e-TE/T2*
7.3 Inversion Recovery Sequence
Inversion recovery sequences begin with a 180° inversion pulse, followed by a 90° excitation pulse after an inversion time (TI). These sequences are useful for suppressing specific tissue signals (e.g., fat or fluid).
The signal intensity in an inversion recovery sequence is given by:
S ∝ ρ(1 - 2e-TI/T1 + e-TR/T1)
7.4 Echo Planar Imaging (EPI)
EPI is an ultra-fast imaging technique that acquires multiple echoes following a single excitation. It's widely used in functional MRI and diffusion-weighted imaging but is susceptible to artifacts due to magnetic field inhomogeneities.
7.5 Contrast Mechanisms
MRI contrast can be manipulated by adjusting sequence parameters to emphasize different tissue properties:
- T1-weighted: Short TR, short TE (e.g., TR = 500ms, TE = 15ms)
- T2-weighted: Long TR, long TE (e.g., TR = 4000ms, TE = 90ms)
- Proton Density-weighted: Long TR, short TE (e.g., TR = 4000ms, TE = 15ms)
| Tissue | T1-weighted | T2-weighted | PD-weighted |
|---|---|---|---|
| Fat | Bright | Bright | Bright |
| Water | Dark | Bright | Dark |
| White Matter | Bright | Dark | Intermediate |
| Gray Matter | Gray | Light Gray | Bright |
8. Advanced MRI Techniques
8.1 Diffusion-Weighted Imaging (DWI)
DWI measures the random Brownian motion of water molecules in tissue. It's based on the application of strong diffusion-sensitizing gradients that cause signal attenuation proportional to the degree of water diffusion.
The signal attenuation in DWI is given by the Stejskal-Tanner equation:
S = S0 e-bD
Where S0 is the signal without diffusion weighting, b is the b-value (diffusion sensitization factor), and D is the apparent diffusion coefficient.
8.2 Perfusion Imaging
Perfusion imaging assesses tissue blood flow. Two main techniques are used:
- Dynamic Susceptibility Contrast (DSC): Tracks the first pass of a gadolinium-based contrast agent.
- Arterial Spin Labeling (ASL): Uses magnetically labeled arterial blood water as an endogenous tracer.
8.3 Functional MRI (fMRI)
fMRI detects changes in blood oxygenation and flow related to neural activity. It relies on the Blood Oxygenation Level Dependent (BOLD) effect, where deoxyhemoglobin acts as an endogenous contrast agent.
The BOLD signal change is approximated by:
ΔS/S ≈ TE * ΔR2*
Where ΔR2* is the change in the effective transverse relaxation rate.
8.4 Magnetic Resonance Spectroscopy (MRS)
MRS provides information about the chemical composition of tissues. It relies on chemical shift, the slight frequency differences in the Larmor precession of nuclei in different molecular environments.
The chemical shift (δ) is typically expressed in parts per million (ppm):
δ = (ν - νref) / νref * 106
Where ν is the frequency of the metabolite of interest and νref is the reference frequency.
9. MRI Hardware Physics
9.1 Main Magnet
The main magnet produces the strong, static magnetic field B0. Most clinical MRI systems use superconducting magnets, which require liquid helium cooling to maintain superconductivity.
The magnetic field strength is related to the current in the magnet coils by:
B = μ0nI
Where μ0 is the permeability of free space, n is the number of turns per unit length, and I is the current.
9.2 Gradient Coils
Gradient coils produce linear magnetic field gradients for spatial encoding. The slew rate, a measure of how quickly the gradients can be switched, is limited by physiological considerations (peripheral nerve stimulation) and hardware capabilities.
The magnetic field due to a gradient is given by:
Bz(x,y,z) = B0 + Gxx + Gyy + Gzz
9.3 RF System
The RF system consists of transmit and receive coils. Transmit coils generate the B1 field for spin excitation, while receive coils detect the MR signal induced by the precessing magnetization.
The signal induced in a receive coil is given by Faraday's law of induction:
ε = -dΦ/dt
Where Φ is the magnetic flux through the coil.
9.4 Shimming
Shimming improves B0 field homogeneity using additional coils or ferromagnetic materials. Active shimming uses current-carrying shim coils, while passive shimming uses strategically placed pieces of ferromagnetic material.
10. Safety Considerations in MRI Physics
10.1 Static Magnetic Field Effects
The strong static magnetic field can exert forces on ferromagnetic objects, potentially causing them to become projectiles. It can also affect implanted medical devices and cause vertigo in some patients due to magnetohydrodynamic effects on the vestibular system.
10.2 Time-Varying Magnetic Fields
Rapidly switching gradient fields can induce electric fields in the body, potentially causing peripheral nerve stimulation or cardiac stimulation. The rate of change of the magnetic field (dB/dt) is limited to prevent these effects.
10.3 RF Energy Deposition
RF pulses deposit energy in tissue, which can lead to heating. This is quantified by the Specific Absorption Rate (SAR):
SAR = σ|E|2 / (2ρ)
Where σ is the tissue conductivity, |E| is the magnitude of the electric field, and ρ is the tissue density.
SAR limits are set to prevent excessive tissue heating:
- Whole body average: 4 W/kg for 15 minutes
- Head: 3.2 W/kg for 10 minutes
- Local (e.g., extremities): 10 W/kg for 10 minutes
10.4 Acoustic Noise
The rapid switching of gradient coils produces loud acoustic noise. Sound pressure levels can exceed 100 dB, necessitating hearing protection for patients and staff.
10.5 Cryogen Safety
Superconducting magnets use liquid helium as a coolant. In the event of a quench (loss of superconductivity), rapid boil-off of helium can displace oxygen in the scanner room, posing an asphyxiation risk.
Conclusion
MRI physics is a complex and multifaceted field that combines principles from quantum mechanics, classical electromagnetism, and nuclear magnetic resonance. Understanding these fundamental concepts is crucial for optimizing image quality, developing new MRI techniques, and ensuring patient safety. As MRI technology continues to advance, with developments such as ultra-high field systems and hyperpolarized MRI, the importance of a deep understanding of MRI physics will only grow.