From Measured Electrical Current to Image in MRI

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From Measured Electrical Current to Image in MRI

All methods of forming MR images rely on the application of magnetic field gradients. The appropriate placement of extra coils around an MR image scanner creates magnetic fields that vary linearly in any direction, by definition producing a spatial gradient in the magnetic field as well as the corresponding Larmor frequency of spin precession. The magnitude and direction of these field gradients can be changed in less than a millisecond. These field gradients are used to isolate the signal for image formation in two ways: slice selection and wave formation.

Slice selection is the limitation of spin excitation to a single plane. The excitation RF pulse contains a range of frequencies, approximately (but not limited to) 1000 Hz wide, that is centered around 64 MHz (for B0 = 1.5 Tesla). Because of the requirement for resonance, this pulse only excites spins precessing within that range of frequencies. When a field gradient is applied in a particular direction, the changing magnetic field in that direction creates a corresponding change in the Larmor frequency of precession. Only a limited range of spins in the direction of the applied field gradient will have Larmor frequencies corresponding to the frequency range of the RF pulse. The result is a plane of excitation orthogonal to the direction of the applied field gradient (Fig. 6). The plane can be translated by changing the frequency of the RF pulse and rotated by changing the direction of the field gradient.

Figure 7.  An image (H) can be broken into a series of spatial waves that vary in their amplitude, peak-to-peak spacing, and direction.  Each has an integer number of cycles in the horizontal and vertical directions.  The wave patterns can then be represented by a number pair equal to the number of horizontal and vertical cycles in that pattern; waves corresponding to (A) {-3,2}, (B) {13,-40}, and (C) {5,9}, respectively.  (D) Summation of the 25 (5 x 5) wave patterns centered around {5,9} (i.e., {3 through 7, 7 through 11}); in this image, the waves add together in some areas and cancel in others like ripples in a pond.  (E) The sum of 25 x 25 waves centered at {0, 27}({-12 through 12, 15 through 40}).  The sum of 50 x 50 waves centered at (F) {-37,0} and (G) {0,0}.  (H) The final image is the sum of 256 x 256 waves, each corresponding to a particular measurement during the MR image scan.

Once a plane of spins has been excited, the current induced in the RF coil is proportional to the magnetization from all the excited spins in the slice, which is measured as a single number. In contrast, the separate detectors in an array of detectors in digital radiography each simultaneously measures the signal at a single location in the imaging plane. In MR imaging a single detector (the RF coil) records consecutive measurements, each measurement providing information about the entire imaging plane. MR imaging takes advantage of the ability to separate any image into a series of waves, or spatial frequencies (Fig. 7).

Figure 8. Illustration of excited spins in the plane perpendicular to B0. After excitation, (A) the spins begin parallel to each other. With the application of a magnetic field gradient from left to right, the spins on the left precess more slowly than those on the right. Over time (B and C) phase 'twists' are created in the direction of the applied field gradient. These twists correspond to the waves in Figure 7 A through C and Figure 9.

When a magnetic field gradient is turned on in a particular direction across an excited slice, spins on one side of the slice precess faster than those on the other side. A `twist' of the spin phases is thereby formed across the image (Fig. 8). The longer the gradient field is on, the greater this twist becomes. The amount of twist can be characterized by the number of 360º cycles the phase turns across the prescribed field of view. If an excited plane of spins is defined by two directions x and y, the scanner is programmed to create many twists in the directions of both x and y after the spins are excited. Thus, each MR image signal measured corresponds to the amplitude of a particular (known) wave function across the entire image (Fig. 7). After a sufficient number of these waves have been measured, they are added together—weighted by the corresponding measured amplitude—to form the image. Mathematically, this summation of wave patterns is a Fourier transform .

Figure 9. Example of waves collected during a single data acquisition period during one TR of a conventional MR image.  The phase-encoding direction is vertical and frequency-encoding direction is horizontal.  The number of phase cycles in the vertical direction remains the same (four), and the number of horizontal phase cycles progresses from -128 to +128 twists under the influence of the frequency-encoding gradient.

Typical image collection involves a series of measurements for each TR, in which some number of phase twists are induced in one direction (the phase-encoded direction). Data are measured as a magnetic field gradient is applied in the other in-plane direction, creating an evolving set of phase twists in that direction (the frequency-encoded direction; Fig. 9). For each TR the amount of phase twist in the phase-encoded direction is changed slightly, and the entire set of phase twists in the frequency-encoded direction is measured again. After a sufficient number of TRs, a `complete' set of waves with phase twists in both the x and y directions is acquired. The matrix term associated with an MR image (e.g., 256 x 192) refers to the total number of waves measured in each direction.

Figure 10.  The waves in Figure 7 represented in k-space.  The center of each figure {0,0} corresponds to no phase twists in any direction.  The brightness of the point represents the strength of the corresponding measured MR signal, which in turn is used to weight that wave during reconstruction.  Single waves are seen as dots at (A) {-3,2}, (B) {13,-40}, and (C) {5,9}.  Groups of waves are centered at (D) {5,9}, (E) {0,37}, (F) {-37,0}, and (G) {0,0}.  (H) All 2562 k-space measurements used to form the entire image.

An entity called k-space further describes how these different data, each corresponding to a particular wave, are collected. As measurements are collected, they are stored in a two-dimensional array in which the row and column of each measurement correspond to the amount of horizontal and vertical phase twists in the excited slice, respectively. This array, a k-space map of the collected data, can be considered an `image' of the spatial frequencies contained in that slice and is a bookkeeping device used to relate each measured current value to the particular wave it represents (Fig. 10). The values in one row of k-space correspond to the series of measurements made during a single TR with increasing phase twists in the frequency-encoded direction. Individual rows correspond to the different amount of phase twists achieved in the phase-encoded direction.