07-04 Filling k-Space with Data and Image Reconstruction

In most MR imaging sequences applied in clinical routine today, raw data are placed in a rectangular k-space grid [⇒ Edelstein].

In a standard spin-echo sequence, each 90° pulse creates a new line (Figure 07-08). The length of the line is determined by the strength of the frequency-encoding gradient and the sampling time, its position by the strength of the phase-encoding gradient.

Figure 07-08: Graphic depiction of a spin-echo pulse sequence.

The position of the line is determined as follows. After the initial 90° ex­ci­ta­tion pulse, the spins evolve in the direction given by the phase-encoding gradient G_y and the frequency-encoding gradient G_x (yellow arrow in Figure 07-09a).

They are then turned around by the 180° pulse (red/magenta arrow). Then the frequency encoding gradient is switched on again and sampling starts. This is repeated for different amplitudes of the phase-encoding gradient until k-space is filled (Figure 07-09b).

Figure 07-09: Mapping of k-space in a spin-echo pulse sequence. (a) Positioning of a single line. (b) Filling of the entire k-space. Phase di­rec­tion: blue arrow, frequency direction: red arrow. In conventional pulse sequences, such as the spin-echo sequence, one line of k- space is filled per repetition time (TR) cycle (commonly there are 256 cycles per ima­ging experiment, not only ten as in this fi­gure).

The time needed for such an imaging experiment is the number of phase-encoding steps (NG_y) multiplied by the repetition time (TR) and the number of excitations (NEX):

NGy × TR × NEX

Now we have filled the data matrix with each row containing information from one echo. Each data point is then Fourier-transformed in the x-direction, which leads to a new data matrix where every point in each column contains in­for­ma­tion stemming from a certain frequency; the phase information differs point-by-point per row. The second Fourier transform is performed in the y-direction to extract phase information. This again leads to a new data matrix containing combined phase and frequency information. The output is a matrix showing a modulus or magnitude image which corresponds to the bulk of MR signals from each point. Phase correction might be necessary to correct for phase jumps bet­ween 0° and 360°.

Among the main parameters influenced by k-space are the speed of ac­qui­si­tion, spatial resolution, field-of-view, contrast, and artifacts. Details can be found in some dedicated papers [Review articles:⇒ Hennig; ⇒ Mezrich; ⇒ Pelc; ⇒ Peters].