3.3 Spatial Localization and Gradients

Spatial Encoding and Gradient Coils

Without a mechanism to localize signals, a receiver coil would detect the sum of all signals emitted from the patient's body, making it impossible to construct an image. Spatial localization in MRI is achieved by using gradient coils, which are electromagnets placed inside the scanner bore. When an electric current is passed through these coils, they superimpose a linear variation onto the main magnetic field (B0).

This linear variation changes the magnetic field strength along a specific axis:

B(x) = B0 + Gx × x

According to the Larmor equation, the precessional frequency of hydrogen protons is directly proportional to the magnetic field strength. Therefore, when a gradient is active, the precessional frequency of the protons becomes a function of their spatial position:

ω(x) = γ × (B0 + Gx × x)

By measuring these frequencies and their phases, the MRI scanner can determine the exact location of the signals in three-dimensional space using three orthogonal gradients:

1. Slice-Select Gradient (G_ss): Identifies the slice position and thickness.
2. Frequency-Encoding Gradient (G_fe): Localizes signals along one axis within the slice.
3. Phase-Encoding Gradient (G_pe): Localizes signals along the perpendicular axis within the slice.

Slice Selection (G_ss)

The slice-select gradient determines which section of the patient's body is excited and imaged. It is turned on concurrently with the radiofrequency (RF) excitation pulse.

Because the gradient is active, the precessional frequency of the protons varies along the slice-select axis. The RF excitation pulse is transmitted with a specific range of frequencies, known as the transmit bandwidth. Only the protons located in the region where their precessional frequency matches the frequency range of the RF pulse will absorb energy and undergo resonance. Protons outside this region are unaffected.

The slice thickness is determined by two factors:

• Gradient Slope (Amplitude): A steeper gradient slope (higher amplitude) creates a larger frequency difference across a short distance, resulting in a thinner slice. A shallower slope creates a smaller frequency difference, resulting in a thicker slice.
• RF Transmit Bandwidth: A narrower RF bandwidth excites a smaller range of precessional frequencies, resulting in a thinner slice. A wider RF bandwidth excites a larger range of frequencies, resulting in a thicker slice.

Slice Thickness = RF Transmit Bandwidth / (γ × Gradient Amplitude)

The slice position (where the slice is located along the axis) is determined by the center frequency of the RF excitation pulse. Shifting this frequency shifts the location of the excited spins.

Frequency Encoding (Readout)

Once a slice is selected, the signals must be localized within the slice. The first step is frequency encoding. The frequency-encoding gradient (also known as the readout gradient) is turned on during the collection of the echo signal (the readout period).

When the frequency gradient is active, it causes the protons to precess at different frequencies along the frequency axis. For example, protons on the left side of the slice will precess slower than protons on the right side. The receiver coil detects a complex waveform containing a mixture of all these different frequencies. The scanner uses a mathematical algorithm called the Fourier Transform to analyze the raw signal, separate the individual frequency components, and map them to their corresponding spatial positions.

The range of frequencies collected during readout is known as the receiver bandwidth (rBW). The rBW is a user-selectable parameter that impacts the signal-to-noise ratio (SNR) and chemical shift artifact.

Phase Encoding

The third dimension of spatial localization is phase encoding, which occurs along the axis perpendicular to both slice selection and frequency encoding. The phase-encoding gradient is turned on briefly after the RF excitation pulse and slice-select gradient, but before the frequency-encoding gradient.

When the phase-encoding gradient is active, it causes the protons to precess at different speeds along the phase axis. Protons in a higher magnetic field precess faster and gain phase, while protons in a lower magnetic field precess slower and lose phase. When the phase-encoding gradient is turned off, all protons return to precessing at the same Larmor frequency. However, they retain a permanent phase shift (or phase angle difference) that corresponds to their position along the phase axis.

Because only one phase-encoding value can be applied per repetition time (TR) interval, the phase-encoding gradient must be stepped through different amplitudes (negative to positive) to fill each line of k-space (the raw data matrix). The number of phase-encoding steps directly determines the number of TR cycles needed, which is a major factor in the total scan time:

Scan Time = TR × Number of Phase Encodings × NEX

Physical Gradient Coils and Scan Planes

The MRI scanner contains three physical gradient coils, named after the physical axes of the scanner:

• Z-Gradient: Produces a gradient along the longitudinal axis of the bore (head-to-toe). It is created using circular Helmholtz/Maxwell coils.
• X-Gradient: Produces a gradient along the horizontal axis (left-to-right). It is created using saddle-shaped coils.
• Y-Gradient: Produces a gradient along the vertical axis (anterior-to-posterior). It is created using saddle-shaped coils.

Depending on the desired imaging plane (axial, sagittal, or coronal), these physical coils are assigned to perform slice selection, frequency encoding, or phase encoding:

(Note: Phase and frequency axes can be swapped by the user to avoid artifacts, but slice-select is fixed by the plane.)

Gradient Performance Parameters

The performance of the gradient system is characterized by four key parameters:

• Gradient Amplitude: The maximum strength of the gradient magnetic field, measured in millitesla per meter (mT/m) or gauss per centimeter (G/cm). Typical clinical amplitudes range from 30 to 45 mT/m.
• Rise Time: The time required for a gradient to ramp up from zero to its maximum amplitude, measured in microseconds (µs). Typical rise times are 100 to 300 µs.
• Slew Rate: The speed and strength of the gradient, representing how fast the gradient can switch on and off. Slew rate is calculated as amplitude divided by rise time:

Slew Rate = Amplitude / Rise Time

Slew rate is measured in Tesla per meter per second (T/m/s) or millitesla per meter per millisecond (mT/m/ms). High slew rates (e.g., 120–200 mT/m/ms) are critical for fast imaging sequences but increase the risk of inducing peripheral nerve stimulation (PNS) in patients.

• Duty Cycle: The percentage of time a gradient can run at maximum amplitude without overheating.