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Imaging

Signal Localization using gradients, initially PR but Fourier better (why?)

Field Gradients

Creating an image of a sample requires a process to localize the NMR signal. This can be done by superimposing a small, spatially varying magnetic field onto the main field, making the resonant frequency a function of position. These applied fields are called field gradients.

Consider a magnetic field $B_{grad}$ aligned to the z-axis, the magnitude of which varies linearly in the x direction

$B_{grad}=x\cdot G_{x}{\hat {z}}\,$

where $G_{x}$ is a constant. Applying the gradient field to the sample causes the Larmor frequency vary linearly as a function of position along the x-axis

$\omega (x)=\gamma (B_{0}+x.G_{x})\,$

An MRI system typically has three independent orthogonal gradient coils, each designed to produce a field gradient $G_{x}$, $G_{y}$ and $G_{z}$ varying linearly along its respective axis, producing a total gradient field

$G=G_{x}{\hat {z}}+G_{y}{\hat {z}}+G_{z}{\hat {z}}\,$

The magnetic field at point r is then

$B(r)=(B_{0}+G\cdot r){\hat {z}}\,$

and the resonant frequency is

$\omega (r)=\omega _{0}+\gamma G\cdot r$

Slice Selection

Slice selection is a technique used to excite a slab of spins in a sample and forms the first stage of most imaging sequences. Applying a linear field gradient across the sample makes the spin resonant frequency a function of distance along the gradient axis. If the slice select gradient is applied along the z-axis,

$\omega (z)=\gamma (B_{0}+z.G_{sl})\,$

If an RF pulse is now applied to the sample, only those spins resonant at or close to the frequency of the RF pulse will be affected (such that $B_{0}<B_{1}$ [16]), so that only a slice of the sample is excited (fig. 2.4). The position of the excited slice along $G_{sl}$ can be changed by changing the frequency of the RF pulse. The thickness of the slice is determined by the gradient amplitude.

Frequency Encoding

Frequency encoding uses the same principle as slice selection, but the field gradient is applied during image readout rather than excitation and so only influences the precession frequency of spins that have already been excited. Applying a linear field gradient $G_{fr}$ across the sample makes the precession frequency a function of position along the gradient axis

$\omega (x)=-\gamma (B_{0}+G_{\textit {fr}}x)=\omega _{0}+\gamma G_{\textit {fr}}x\,$

The observed signal is then

$s(t)=\int _{sample}\rho (x)\exp(-i(\omega _{0}+\gamma G_{\textit {fr}}x)t)dx\,$

where $\rho (x)$ is the projection of the spin density function along the x-axis

$\rho (x)=\int \int \rho (\mathbf {r} )dydz\,$

The signal is then demodulated to remove the RF carrier frequency $\omega _{0}$, leaving the envelope

$s(t)=\int _{sample}\rho (x)\exp(-i(\gamma G_{\textit {fr}}t)x)dx\,$

which is the Fourier transform of $\rho (x)$. Fourier transforming the acquired signal produces a projection of the sample magnetisation along the gradient axis (fig.1d_localisation). If the frequency encoding gradient is orthogonal to the slice select gradient, the two techniques can be combined to image the spin density of a column through the sample (fig.select_pencil).

Phase Encoding

Frequency encoding acquires a series of time points during a single echo (fig.phase_encode_a). The same result may be achieved by exciting the sample a number of times and applying a constant gradient for a different duration $T_{\textit {ph}}$ each time before sampling a single data point (fig.phase_encode_b)~\cite{kumar75}. If the $y$ gradient is used, the acquired signal is

$s(T_{\textit {ph}})=\int _{sample}\rho (y)\exp(-i\gamma G_{\textit {ph}}T_{\textit {ph}}y)dy\,$

Repeating the experiment at many different values of $T_{\textit {ph}}$ samples the complete signal. The Fourier transform of the acquired signal is then the projection of signal intensity along the gradient axis as before.

While requiring more acquisitions than frequency encoding (one per sample point), phase encoding can be separately applied to all three orthogonal axes, localising the signal to a point and making 3D imaging possible. Phase encoding can also be used in combination with slice selection and frequency encoding, forming the basis for most common imaging sequences.

Varying $T_{\textit {ph}}$ allows the complete echo signal to be acquired. However, it also means that the time between the excitation pulse and the sampled point varies. During this time the echo signal undergoes $T_{1}$ and $T_{2}^{*}$ decay.

Returning to eqn.~\ref{eqn:phase_encode} it can be seen that the location of the sampled point is determined by the product $G_\textit{ph} T_\textit{ph}$. However, as the gradient is switched off by the time the signal is sampled, it is the sum effect rather than the exact waveform that is important. Eqn.~\ref{eqn:phase_encode} can be written more generally as

$s(T_{\textit {ph}})=\int _{sample}\rho (y)\exp(-i\phi (T_{\textit {ph}}))dy\,$

where

$\phi (T_{\textit {ph}})=\gamma \int _{0}^{T_{\textit {ph}}}G_{\textit {ph}}\,d\tau$

For a constant gradient amplitude $\phi (T_{\textit {ph}})$ reduces to $G_{\textit {ph}}T_{\textit {ph}}$. By varying the amplitude of the gradient, rather than the duration, exactly the same sampling of the echo is achieved but each point has undergone the same degree of $T_{1}$ and $T_{2}^{*}$ relaxation (fig.phase_encode_c)~\cite{edelstein80}.

Previous Text

A number of schemes have been devised for combining field gradients and radiofrequency excitation to create an image. One involves 2D or 3D reconstruction from projections, much as in Computed Tomography. Others involve building the image point-by-point or line-by-line. One even uses gradients in the rf field rather than the static field. Although each of these schemes is occasionally used in specialist applications, the majority of MR Images today are created either by the Two-Dimensional Fourier Transform (2DFT) technique with slice selection, or by the Three-Dimensional Fourier Transform (3DFT) technique. Another name for 2DFT is spin-warp. What follows here is a description of the 2DFT technique with slice selection.

Slice selection is achieved by applying a magnetic gradient in addition to the external magnetic field during the radio frequency pulse. Only one plane within the object will have protons that are on–resonance and contribute to the signal.

A real image can be considered as being composed of a number of spatial frequencies at different orientations. A two–dimensional Fourier transformation of a real image will express these waves as a matrix of spatial frequencies known as k–space. Low spatial frequencies are represented at the center of k–space and high spatial frequencies at the periphery. Frequency and phase encoding are used to measure the amplitudes of a range of spatial frequencies within the object being imaged.

The frequency encoding gradient is applied during readout of the signal and is orthogonal to the slice selection gradient. During application of the gradient the frequency differences in the readout direction progressively change. At the midpoint of the readout these differences are small and the low spatial frequencies in the image are sampled filling the center of k-space. Higher spatial frequencies will be sampled towards the beginning and end of the readout filling the periphery of k-space.

Phase encoding is applied in the remaining orthogonal plane and uses the same principle of sampling the object for different spatial frequencies. However, it is applied for a brief period before the readout and the strength of the gradient is changed incrementally between each radio frequency pulse. For each phase encoding step a line of k–space is filled.

Either a spin echo or a gradient echo can be used to refocus the magnetisation.

The 3DFT technique is rather similar except that there is no slice selection and phase-encoding is performed two separate directions.

Another scheme which is sometimes used, especially in brain scanning or where images are needed very rapidly, is called echo-planar imaging (EPI): in this case each rf excitation is followed by a whole train of gradient echoes with different spatial encoding.

Example of a Pulse Sequence

Simplified timing diagram for two-dimensional-Fourier-transform (2DFT) pulse sequence

In the timing diagram, the horizontal axis represents time. The vertical axis represents: (top row) amplitude of radiofrequency pulses; (middle rows) amplitudes of the three orthogonal magnetic field gradient pulses; and (bottom row) receiver analog-to-digital converter (ADC). Radiofrequencies are transmitted at the Larmor frequency of the nuclide to be imaged: for example for ¹H in a magnetic field of 1T, a frequency of 42578100 Hz would be employed. The three field gradients are labeled G_X (typically corresponding to a patient's Left-to-Right direction and colored red in diagram), G_Y (typically corresponding to a patient's Front-to-Back direction and colored green in diagram), and G_Z (typically corresponding to a patient's Head-to-Toe direction and colored blue in diagram). Where negative-going gradient pulses are shown, they represent reversal of the gradient direction, i.e. Right-to-Left, Back-to-Front or Toe-to-Head. For human scanning gradient strengths of 0.001-0.1 T.m^-1 are employed: higher gradient strengths permit better resolution and faster imaging. The pulse sequence shown here would produce a transverse (axial) image.

The first part of the pulse sequence, SS, achieves Slice Selection. A shaped pulse (shown here with a sinc modulation) causes a 90° (π/2 radian) nutation of longitudinal nuclear magnetization within a slab, or slice, creating transverse magnetization. The second part of the pulse sequence, PE, imparts a phase shift upon the slice-selected nuclear magnetization, varying with its location in the Y direction. The third part of the pulse sequence, another Slice Selection (of the same slice) uses another shaped pulse to cause a 180° (π radian) rotation of transverse nuclear magnetization within the slice. This transverse magnetisation refocuses to form a spin echo at a time TE. During the spin echo, a frequency-encoding (FE) or readout gradient is applied, making the resonant frequency of the nuclear magnetization vary with its location in the X direction. The signal is sampled n_FE times by the ADC during this period, as represented by the vertical lines. Typically n_FE of between 128 and 512 samples are taken.

The longitudinal relaxation is then allowed to recover somewhat and after a time TR the whole sequence is repeated n_PE times, but with the phase-encoding gradient incremented (indicated by the horizontal hatching in the green gradient block). Typically n_PE of between 128 and 512 repetitions are made.

The negative-going lobes in G_X and G_Z are imposed to ensure that, at time TE (the spin echo maximum), phase only encodes spatial location in the Y direction.

Typically TE is between 5msec and 100msec, while TR is between 100msec and 2000msec.

After the two-dimensional matrix (typical dimension between 128x128 and 512x512) has been acquired, producing the so-called K-space data, a two-dimensional Fourier transform is performed to provide the familiar MR image.