Lecture 2: Relaxation Times, Pulse Sequences & MRI Contrast
Learning Objectives
Upon understanding this lecture, you should be able to:
Explain nuclear precession and its consequences for the net magnetisation vector.
Define the longitudinal (Mz) and transverse (Mxy) components of the net magnetisation vector.
Describe what is meant by phase coherence of nuclear spins.
Explain how a Larmor-frequency RF-pulse alters the voxel magnetisation.
Describe the production of flip angles by such a pulse and state the factors which determine the size of the flip angle produced.
Describe the effect of such excitation upon the longitudinal and transverse components of the magnetisation.
Explain what the signal detected in NMR represents.
Relate the idea of relaxation to the behaviour of the net magnetisation following excitation.
Explain the physical processes which give rise to the relaxation mechanisms quantified by T1 and T2.
Describe the microscopic tissue properties which determine T1 and T2.
Understand what is meant by a pulse sequence.
Describe the application and utility of the saturation-recovery pulse sequence.
Understand the spin-echo pulse sequence.
Describe the effect of the spin-echo parameters TR and TE upon the sensitivity weighting of the NMR signal.
3.1 Overview of This Lecture
Recap from Last Week:- Introductory understanding of NMR phenomena.
Modeled a spinning nucleus as a magnetic dipole.
Investigated behavior of a single magnetic dipole in a strong external field.
NMR observed for a collection of nuclei when irradiated with Larmor frequency radiation.
This Lecture's Focus:- Delving deeper into NMR for MRI application, requiring revision of some oversimplified notions from the previous lecture.
Reformulating the description of NMR in more thorough terms.
Considering the form of the detected signal and its interpretation for tissue type and physiological status.
Introducing basic NMR pulse sequences.
3.2 Nuclear Precession
Revision of Two Spin States:- Last week: Spinning nucleus in a strong magnetic field (B0) exists in two states:
- Low-energy state: Magnetic dipole moment parallel to the field.
- High-energy state: Magnetic dipole moment antiparallel to the field.Correction/Revision: While these states exist and their energy levels are correct, the visualization of the nucleus's orientation needs revision based on quantum mechanics.
Quantum Mechanical Analysis:- Describes the behavior of very small physical systems like nuclei.
Supersedes classical mechanics and classical electromagnetism for such systems.
Predicts the existence of the low-energy and high-energy states.
Key Difference: The magnetic dipole moment (µ) of the nucleus does not point exactly along the field direction in either state.
Instead, it lies inclined to the field at an angle of 54.7°. (Illustrated in Figure 3.2a).
Precession:- A dipole inclined to the field in this manner does not remain stationary.
The direction of the magnetic dipole moment vector of the nucleus changes over time.
The tip of the vector traces a circle around the direction of the magnetic field (illustrated in Figure 3.2b).
This form of motion is called precession.
Precessional Frequency:- The rate of this precession is such that the precessional frequency is exactly equal to the Larmor frequency (f0).
Larmor frequency formula: f0 = (gamma / 2pi) * B0
Summary of Revised Model for Single Spinning Nucleus:1. In both low-energy and high-energy states, the µ-vector of the spinning nucleus is inclined to the direction of the field at an angle of 54.7°.
The nucleus undergoes precession, with the tip of its µ-vector tracing circles around the direction of the external field.
The rate of this precession means the precessional frequency is equal to the Larmor frequency.
3.3 Net Magnetisation
Definition: The net magnetisation (M) of a voxel is the vector sum of all individual magnetic moments for the spins within that voxel.
In Absence of External Field (B0 = 0): M = 0 because all spins are randomly oriented.
In Presence of External Field (B0 != 0): M != 0 due to a net alignment of nuclear spins and a slight preference for the low-energy state. M points in the same direction as the external field.
Effect of Updated Ideas on Net Magnetisation:- Even with individual µ-vectors inclined at 54.7°, the net magnetisation vector at thermal equilibrium is still as predicted (aligned with B0).
Why?- z-direction: More spins are in the low-energy (parallel) state, so the sum of their z-components still gives a resultant in the direction of the field.
x- and y-directions: Since the tips of the µ-vectors are randomly distributed in their cycle of precession, their individual x and y components cancel each other out vectorially, yielding no net resultant in the transverse plane.
Therefore, the overall resultant M points straight along the direction of the external field (the +z-axis) at thermal equilibrium (Figure 3.3b).
Thermal Equilibrium Magnetisation: Denoted as M0. This is the magnetisation vector at thermal equilibrium.
Components of the Net Magnetisation Vector: For convenience in understanding NMR as a rotation of M:
Longitudinal Magnetisation (Mz): The component of M along the direction of the field (z-direction).
Transverse Magnetisation (Mxy): The component of M in the xy-plane.
In general, M = Mz + Mxy (vector sum) (Figure 3.3c).
At thermal equilibrium: Mz = M0 and Mxy = 0.
3.4 The Rotating Reference Frame
Laboratory Frame: The initial coordinate system (x, y, z) where the strong external field (B0) is along the z-axis.
Rotating Frame: A second, alternative coordinate system (x', y', z').
Origin is centered on the origin of the laboratory frame.
The z'-axis is identical to the z-axis of the laboratory frame.
The rotating frame itself rotates around the z-axis at the Larmor frequency (illustrated in Figure 3.4a).
Utility: This frame simplifies the analysis of what happens during excitation.
3.5 Excitation
Application: A voxel is irradiated with Larmor frequency RF radiation.
RF Radiation Components: Comprises both an electric-field and a magnetic-field component. The magnetic field component is significant in NMR/MRI.
Magnetic Field Component in NMR/MRI:- Consists of two superimposed magnetic fields.
Both magnetic field vectors are oriented in the xy-plane.
Both rotate at the RF frequency, but in opposite directions.
View from the Rotating Frame (assuming Larmor frequency RF radiation):- One of these magnetic field vectors (the one rotating in the opposite direction to the rotating frame) will appear to rotate at twice the Larmor frequency (2f0).
The other vector (the one rotating in the same direction as the rotating frame) will appear stationary. This stationary field is called the B1 field.
If the RF radiation is at any other frequency, both vectors would appear to rotate in the rotating frame.
Effect of B1 Field on Net Magnetisation (M):- Any component of the RF field that appears to rotate in the rotating frame has no effect on M.
The stationary B1 field (considered oriented along the x'-axis, Figure 3.5a) exerts a force (torque) that changes the orientation of M.
Mechanism: Just as spins precess around B0, the presence of B1 causes them to also attempt to precess around B1 for the duration of the RF radiation.
Since B1 is much weaker than B0, this secondary precession is much slower.
The result is that M is twisted away from the z-axis (Figure 3.5a).
Precessional Frequency around B1 Field:- f1 = (gamma / 2pi) * B1
Flip Angle (theta):- The angle through which M is twisted.
Formula: theta = gamma * B1 * t
Where t is the time the Larmor frequency radiation is applied.
Factors Determining Flip Angle: Proportional to both the strength of the B1 field and the time for which it is applied.
Longer application time = larger flip angle.
For flip angles up to 180°, this radiation is applied for short durations (e.g., 10s to 100s of microseconds in clinical MRI), hence it's called an RF-pulse.
The duration of the RF-pulse is chosen to achieve the desired flip angle.
Excitation: This twisting of M away from equilibrium by an RF pulse is a manifestation of NMR. It occurs because the RF frequency matches the Larmor frequency, allowing spins to absorb energy.
Consequence: Phase Coherence:- Beyond twisting M, the Larmor frequency radiation causes the precessing spins to synchronize.
At thermal equilibrium, spins have randomized phases (at random points in their precession cycle).
Following RF pulse, they obtain phase coherence (precess in sync, at the same point in their cycle).
Effect on Magnetisation Components:- Decrease in longitudinal magnetisation (Mz).
Increase in transverse magnetisation (Mxy).
A 90° flip angle produces the maximum transverse magnetisation (and Mz = 0).
A 180° flip angle produces no transverse magnetisation but reverses the direction of longitudinal magnetisation. It takes twice as long as a 90° flip angle.
3.6 Relaxation and the Detected Signal
Relaxation: Once the RF-pulse ceases, M immediately begins to return to its equilibrium value. This process is called relaxation.
Significance: The precise way the M-vector undergoes relaxation provides information about the nature of the tissues within the voxel.
Detected Signal in NMR/MRI:- The signal is caused only by the transverse magnetisation (Mxy).
When M points along the z-axis (at equilibrium), Mxy = 0 and there is no signal.
When M is rotated away from the z-axis, Mxy != 0 and a signal can be detected.
Coil Arrangement (Figure 3.6a):- Transmission coil: Acts as an antenna to generate Larmor frequency RF-pulses.
Detection coil: Acts as an antenna to detect the NMR signal emitted by the spins during relaxation. A voltage is induced in this coil, proportional to the rate of change of the y-component of Mxy.
Form of the Detected Voltage (FID Signal):- At Thermal Equilibrium: M points along the z-axis, Mxy = 0, so detected voltage is zero.
Following Excitation (e.g., 90°-pulse): M is rotated into the transverse plane, Mxy is at its maximum possible value. Spins precess in phase, so M itself precesses about B0 at the Larmor frequency.
The y-component of the transverse magnetisation (Mxy) varies sinusoidally, causing the detected signal to oscillate at the Larmor frequency.
Hypothetically (no relaxation): The voltage would be a continuous sinusoid (Figure 3.6b(a)).
Realistically (with relaxation): As M relaxes back to its equilibrium value, the signal oscillates at Larmor frequency but its amplitude decreases over time (Figure 3.6b(b)). This decaying oscillatory signal is called the free induction decay (FID) signal.
Information from FID: The shape of the FID signal depends on specific relaxation processes, which are determined by the atomic and molecular content of the voxel. Thus, tissue properties can be extracted.
Relaxation Mechanisms: To describe relaxation, Mz and Mxy are considered separately.
4.1 Longitudinal Relaxation: T1
Focus: The recovery of the longitudinal magnetisation (Mz).
Excitation Effect: Depletes Mz, reducing its value from the equilibrium maximum.
T1 Relaxation (Spin-Lattice Relaxation): The component of the relaxation process that causes Mz to recover to its equilibrium value. It occurs as excess spins, promoted to the high-energy state by the RF-pulse, return to the low-energy state.
Equation for Recovery of Longitudinal Magnetisation:- Mz(t) = M0 * cos(theta) * e^(-t/T1) + M0 * (1 - e^(-t/T1))
theta is the flip angle.
T1 (Longitudinal Relaxation Time / Spin-Lattice Relaxation Time): A parameter describing the rate at which Mz is restored.
After a time t = T1, 63% of the longitudinal magnetisation has been restored following a 90°-pulse (Figure 4.1a).
Microscopic Processes (Spin-Lattice Interactions):- At room temperature, random thermal motions and vibrations of molecules within the voxel generate magnetic fields that oscillate at a range of frequencies.
If some of these molecular oscillations occur at the Larmor frequency, they can induce spins to drop from the high-energy to the low-energy state, releasing energy to the surrounding