## Magnetic resonance image formation (encoding)

Two-dimensional Fourier transform technique enabling spatial encoding

A significant issue in MRI is to determine from which point the signal originates. To this purpose, it is necessary to use a technique that encodes different space points by linking them to a coordinate system (x,y axes). This is mediated by the procedure of spatial encoding.

The standard MRI image formation technique used today is the two-dimensional Fourrier transform technique (2D FT) (ref.) which enables spatial encoding. It consists of the sequential application of two magnetic field gradients, that is, the phase encoding gradient, followed by the frequency encoding gradient (Figure 1).

Figure 1: Frequency encoding and phase encoding. Image: Allen D. Elster, MRIquestions.com (ref).

Application of a "phase encoding gradient" on the y axis

Initially, a magnetic field gradient termed "phase encoding gradient" is applied along the y axis. This makes the spins precess at different velocities depending on the magnetic field stength at the specific point of the gradient on the y axis. When the gradient is stopped, the spins will be on different positions on the unit circle (cf. first image from https://en.wikipedia.org/wiki/Unit_circle) and will therefore have acquired a different phase depending on their position on the y axis. In other words, a phase shift will have occured. We will therefore have columns of spins with the same phase. If we calculate the phase, we can determine where the specific point is located in the y axis. In this way, we can differentiate between points in the y axis depending on their phase. This is evaluated with a Fourier transform of the data (1st dimension).

The first video at this page and the animation at https://www.cis.rit.edu/htbooks/mri/chap-7/g2-3.htm illustrate the principle. (By pausing the video at the end, we can appreciate that the spins have acquired different phases.)

The phase shift (Φ) is proportional to the strength of the gradient (G), the time (t) that the gradient is applied, as well as the gyromagnetic ratio (γ). It is calculated from the equation (ref.):
Φ = γ * G * t

Please refer to Fig. 2. (It is noted that the x and y introduced magnetic field gradients are added (accumulated) to the z magnetic field of the scanner.)

Figure 2: Two protons, a reference proton and a proton subjected to a gradient, after ten cycles/peaks have the signal that is represented by the dot. The phase difference Φ is indicated by the horizontal arrow. Image: Allen D. Elster, MRIquestions.com (ref).

Application of a "frequency encoding gradient" on the x axis

Then, a magnetic field gradient termed "frequency encoding gradient" is applied along the x axis. This makes the spins precess at different velocities depending on the magnetic field stength at the specific point of the gradient on the x axis. We will therefore have rows of spins with the same frequency. We then excite the spins and during relaxation we acquire the signal. We will obtain a signal that will be the sum of all the different amplitudes corresponding to the different frequencies. This is provided by a Fourier trasform of the data in this dimension (second dimension), hence the name two-dimensional Fourier transform. The second video at this page and the animation at https://www.cis.rit.edu/htbooks/mri/chap-7/g2-2.htm illustrate the principle.

From the NMR signal to image formation

Frequency encoding requires only a few milliseconds of signal reading, while phase encoding necessitates repetition of the imaging sequence. For a classic spin echo sequence, one phase encoding step is performed during each repetition time (TR). Since repetition times can be up to 3 seconds, phase encoding is much longer.

The 2D Fourier procedure will provide a matrix of magnetic resonance signal intensities. These will be associated with x and y locations. Where the signal is highest, the pixels will be displayed as white, while where the signal is lowest, the pixels will be displayed as black. For intermediate signal values, pixels will have a certain greyscale value.

The most important signal intensity parameter is the proton number (proton density). A large number of protons, as in the case of soft tissues for instance, will provide a strong net magnetization M (Mz rotated to Mxy) and therefore an intense signal. Another important parameter is the tissue relaxation time. Fat and muscle have similar proton density but fat has a more intense signal. This is due to the fact that fat has a shorter T1 and a longer T2. Longer T2 means more persistent magnetization in the xy plane and therefore increased signal due to the Mxy magnetization and shorter T1 means quick recovery of magnetization to the z axis and therefore increased signal due to Mz.

Storing MRI data in the k-space, a Fourier transform space

The data aquired from an MRI experiment are stored in a special data matrix termed k-space (ref. https://magnetic-resonance.org/ch/07-02.html) which has two coordinate axes: kx which refers to the gradient strength of the frequency-encoding gradient and ky which refers to the gradient strength of the phase-encoding gradient (Fig. 2). Their values range from -1 (lowest strength) to +1 (highest strength) and they cross at the center at 0 which represents zero strength.

The kx and ky correspond to k(FE) and k(PE) respectively which are provided by the equations:

k(FE) = γ * G(FE) * m * Δt

k(PE) = γ * n * G(PE) * τ

where G refers to magnetic field gradient, FE to frequency encoding, PE to phase encoding, Δt is the sampling time (the reciprocal of sampling frequency), termed Dwell time (typically of 10 µs), τ is the duration of G(PE), γ (gamma bar) is the gyromagnetic ratio, m is the sample number in the FE direction and n is the sample number in the PE direction (Wikipedia).

Figure 3: k-space. Image from

There are different ways by which data are acquired i.e. sampling is conducted and the k space is filled. Traditionally, one (horizontal) line is filled for every repetition time (TR)  and one line is filled after the other. This consitutes a linear trajectory. This reference (https://magnetic-resonance.org/ch/07-03.html) and the second video at https://www.imaios.com/en/e-Courses/e-MRI/The-Physics-behind-it-all/K-space describe a typical experiment.

New sampling schemes for high-field MRI: the SPARKLING technique

Researchers at NeuroSpin (France) have developed an algorithm for compressed sampling to be used with high-field MRI. It is termed "SPARKLING" (Spreading Projection Algorithm for Rapid K-space sampLING).

Details:

http://joliot.cea.fr/drf/joliot/en/Pages/news/Science/2019/High-resolution-MRI-Sparkling.aspx

Publication: Lazarus C, Weiss P, Chauffert N, Mauconduit F,  El Gueddari L, Destrieux C, Zemmoura I, Vignaud A, Ciuciu P. SPARKLING: variable-density k-space filling curves for accelerated T2* -weighted MRI. Magn Reson Med. 2019;81(6):3643-3661.

Video slides (narrated presentation): https://youtu.be/itT3qRQRXCc

Related Neurospin post:

Algorithmes d'acquisition IRM permettant de gagner un facteur 20 en imagerie 2D et 70 en imagerie 3D

Dr. Philippe Ciuciu (CEA) https://youtu.be/ms7k6IWnUzs?t=116: "En IRM on n’acquiert pas les données point par point, on les acquiert le long de trajectoires." "Cet algorithme a consisté a concevoir de nouvelles formes de trajectoire pour accélérer le processus d'acquisition"

Dr. Philippe Ciuciu (CEA) https://youtu.be/gQh6D_vpkSo?t=88: "Les images IRM ne sont pas collectés pixel à pixel comme l'image d'un appareil photo, mais fréquence par fréquence. Et par souci d'efficacité, ces informations sont acquises en fait, le long de trajectoire, par exemple des segments rectilignes.

Les informations acquises dans les basses fréquences rendent compte du fond de l'objet, de l'organe que l'on cherche à imager. Tandis que les informations collectées dans les hautes fréquences vont nous informer sur les détails, les contours de l'organe à imager.

Les approches classiques pour réduire le temps d'examen IRM consistent à substituer à ce trajectoire rectiligne des formes géométriques plus complexes tels que des spirales. Tout notre travail a consisté à proposer de nouveaux schémas d'échantillonnage qui sont polyvalents, qui permettent d'optimiser la construction de ce trajectoire, tout en répondant aux contraintes matérielles de l'imageur, en repoussant les limites de cet imageur et en s'adaptant à n'importe quel type d'examen IRM qu'on veut réaliser.

Les trajectoires que nous avons développés permettent de réduire le temps d'acquisition d'un facteur 20 sans perte de qualité image dans les examens IRM à l'instar des résultats que nous avons obtenus chez l'homme sur un imageur 7 Tesla de Neurospin.

La puissance de cette nouvelle approche réside dans le fait qu'elle est purement logicielle, elle ne nécessite pas l'installation d'un nouvel équipement sur l'imageur. En ce sens, son coût est faible et son impact potentiellement majeur puisqu'elle peut être déployée dans n'importe quel environnement IRM comme à l' hôpital.

Donc tout notre travail, tous nos efforts ont consisté à améliorer les stratégies d'acquisition.

Pour aller plus loin, nous avons lancé un projet "DRF Impulsions" du CEA avec l'équipe CosmoStat dirigé par Jean Louis Starck à l'IRFU dont l' expertise est justement dans la reconstruction des images astrophysiques."