MRI from k-Space to Parallel Imaging 1

O. Dietrich, PhD

Department of Clinical Radiology, University Hospitals – Gross hadern, Ludwig Maximilian University of Munich, Marchioni nistr. 15, 81377 Munich, Germany

C O N T E N T S

1.1 Introduction 3

1.2 Fourier Analysis and Fourier Transform 4 1.3 Spatial Encoding, Gradients, and Data Acquisition 5 1.4 k-Space and Spatial Frequencies 8

1.5 Parallel Imaging 13 1.6 Signal and Noise 15 1.7 Conclusion 16 References

MRI from k-Space to Parallel Imaging 1

Olaf Dietrich

1.1

Introduction

Magnetic resonance imaging (MRI) is based on the prin- ciples of nuclear magnetic resonance (NMR), i.e., the excitation of the spin of the atomic nucleus by resonant electromagnetic radio frequency (RF) fi elds (Wehrli 1992; Gadian 1995). This physical phenomenon was discovered in the 1940s (Bloch et al. 1946; Bloch 1946;

Purcell et al. 1946) and has been applied since then in a multitude of experiments and measurements, e.g., in physics, inorganic chemistry, biochemistry, biology, and medical research. A simple NMR experiment requires a strong, static magnetic fi eld, B

, a short RF pulse (also referred to as the B

fi eld), and a sample inside the static fi eld whose spins are excited by the RF pulse. After the excitation, the sample emits a quickly decaying RF signal, which can be received by an antenna or RF coil (Fig. 1.1); this signal can be used to analyze the chemical or physical properties of the sample.

Both RF signals, the one used to excite the spins and the one emitted by the sample afterwards, are charac-

terized by a certain frequency, f, which typically is in the MHz range. This precession frequency, or Larmor frequency, f, is proportional to the fi eld strength, B

; thus, e.g., by doubling the static fi eld strength, the Larmor frequency is also doubled. This relation can be written f = γ × B

with the gyromagnetic ratio, γ, as a constant of proportionality. The gyromagnetic ratio, γ, depends on the kind of nucleus and on the chemical environment of the nucleus such that nuclei in differ- ent chemical compounds can be differentiated by ana- lyzing the frequencies, f, they are emitting. This is the basic principle of MR spectroscopy.

B

RF trans- mitter

RF receiver

B

RF trans- mitter

t

RF receiver

B

RF trans- mitter

t RF

receiver

a

b

c

Fig. 1.1a–c. Illustration of a simple NMR experiment. a The sample (three red spheres) is placed in the static magnetic fi eld,

. b The RF transmitter excites the spins of the sample by sending a short resonant RF pulse with the Larmor frequency.

c The exponentially decaying RF signal emitted by the precess- ing spins of the sample is received.

a

b

c

An important limitation of the NMR experiment described above is that the received RF signal cannot be assigned to a spatial location within the sample.

Only in the 1970s were the methods developed for localized NMR, i.e., for magnetic resonance imaging (Wehrli 1992; Lauterbur 1973; Mansfi eld and Granell 1973). The crucial extension to the simple NMR experiment described above is the introduc- tion of additional magnetic fi elds whose fi eld strength varies linearly with the spatial position. These fi elds are called magnetic fi eld gradients, or simply gradi- ents, G, and they are used to distinguish the RF signals emitted at different locations. In a very simple MRI experiment, we may excite a sample as before, but then switch on a gradient during the data acquisition or readout (Fig. 1.2). Because of this gradient, the Larmor frequencies of the RF-emitting spins now depend on their location and, thus, their location can be deter- mined by analyzing the frequencies received by our antenna. The mathematical procedure to calculate the different frequencies mixed into the received signal of

all spins is called the Fourier transform and is the very basis of almost all image reconstruction in MRI.

The relation of received RF signals on the one hand and the reconstructed image data on the other hand is essential in order to understand the techniques of parallel imaging presented in this book. Especially important is the concept of signal data in the frequency domain or k-space (Paschal and Morris 2004) and the image reconstruction by Fourier transformation that are introduced in the following sections.

1.2

Fourier Analysis and Fourier Transform Fourier analysis and the Fourier transform can be illustrated using oscillating processes such as sound waves in acoustics or RF signals in MRI. The most basic oscillation (at least from a mathematical point of view) is a sine (or cosine) wave, which is also called a harmonic oscillation. Harmonic oscillations are characterized by their frequency, f; all other periodic oscillations can be described as a mixture of several harmonic oscillations with different frequencies and different amplitudes. The frequencies and amplitudes that are required to describe an arbitrary oscillation are known as the frequency spectrum of the oscil- lation; this relationship is illustrated in Fig. 1.3. The oscillating signal on the left-hand side is described by one or several harmonic oscillations with different amplitudes. The mathematical tool to calculate the frequency spectrum of a signal is the (discrete) Fou- rier transform: the signal is transformed into a series of Fourier coeffi cients describing the amplitude of the oscillation for each frequency, e.g., the rectan- gular wave with frequency f shown at the bottom of Fig. 1.3 can be composed from sine waves with the frequencies f, 3 f, 5 f, 7 f… and with the amplitudes 1, 1/3, 1/5, 1/7…, respectively.

An important (but certainly not obvious) property of the discrete Fourier transform is that it can be cal- culated itself as a superposition of harmonic oscil- lations. This means, in order to determine the har- monic components of a given signal, the time course of this signal is interpreted as a series of coeffi cients to calculate a new mixture of oscillations. This idea is illustrated for a rectangular wave in Fig. 1.4. The right-hand side of this fi gure is simply a visualization of the concept of the frequency spectrum: each point in the frequency domain (shown in red) refers to an

Fig. 1.2a–c. Illustration of a simple MRI experiment. a The spins of the sample (three red spheres) placed in the static magnetic fi eld, B

, are excited by an RF pulse. b A gradient is switched on, causing a linear magnetic-fi eld dependence and, thus, linearly varying precession frequencies in the sample.

The superposition of the of the RF signals is received. c A one- dimensional “image” of the sample is reconstructed by Fourier transforming the received signal.

B

RF trans- mitter

b

B

RF trans- mitter

RF receiver

a

t RF

receiver

B

z

t

‘Image’ reconstruction (Fourier transform)

f

c

t

c

b

a

oscillation in the time domain (small blue graphs), and the frequency spectrum defi nes how to super- pose these temporal oscillations to yield the original signal. The (perhaps) surprising part of this fi gure is shown on the left-hand side and illustrates that the original signal in the time domain can also be inter- preted as a spectrum of waves in the frequency space.

The mathematical background of this property is that the Fourier transform is its own inverse, i.e., it can be inverted by a second Fourier transform (neglecting normalization factors).

Finally, it should be noted that every signal that is restricted to a certain time interval, i.e., every signal of fi nite duration, can be regarded as a periodic signal simply by repeating it after its duration, as shown in grey in Fig. 1.4. In particular, signals in MRI are restricted either to the duration of the readout inter- val or to the reconstructed fi eld of view and, thus, the properties of the discrete Fourier transform of peri- odic data described above are valid for MRI data.

1.3

Spatial Encoding, Gradients, and Data Acquisition

As mentioned above, a single receiving RF coil cannot distinguish where a received MR signal is emitted, but

“sees” only the superposition of all radiation emitted from a sample. MRI uses magnetic fi eld gradients to overcome this limitation. By adding a gradient fi eld to the static B

fi eld, the resulting magnetic fi eld varies linearly in space. Hence, the Larmor frequencies, being proportional to the fi eld, vary as well; e.g., in the one-dimensional example of Fig. 1.2, spins at the left- hand side have a lower Larmor frequency than spins at the right-hand side. All spins now emit electro- magnetic radiation with a frequency corresponding to their spatial position and intensity proportional to

Fourier transform

t f

Periodic oscillation Periodic frequency spectrum

f f

f f

f

Harmonic oscillations in frequency domain

t t

t t

t

Harmonic oscillations in time domain (RF) weighted

sum

weighted sum

Fig. 1.4. Calculation of the Fourier transform. The discrete Fourier transform of a periodic oscillation (blue oscillation on the

the (inverse) Fourier transformation of the frequency spectrum (red signal on the right-hand side) can also be calculated as weighted sum of oscillations, i.e., the Fourier transform of a Fourier transform yields the original signal (neglecting normali- zation factors).

Fig. 1.3. Periodic oscillations can be described by their fre- quency spectrum. The amplitudes of the frequencies are the coeffi cients of the Fourier series of a periodic oscillation.

t sin(2S t/f

)

t sin(2S t/f

) + sin(2S t/f

)

t triangular wave

t rectangular wave

Periodic oscillation

f f

f f

f

f

f

Frequency spectrum

the density of spins at the different locations. Thus, spatial positions can be distinguished by means of different Larmor frequencies that contribute to the resulting RF signal.

Hence, the “analysis of the received frequency spectrum,” i.e., the determination of all frequencies that are superposed within the resulting RF signal, is the fundamental problem of MR image recon- struction as illustrated in Fig. 1.5. Fortunately, this is exactly what the Fourier transform of a signal does:

A mixture or superposition of harmonic oscillations with different frequencies and different amplitudes is transformed into a series of Fourier coeffi cients that describe the amplitude of the oscillation for each fre- quency. Thus, by Fourier transforming the received RF signal, the spin density is determined for each received frequency, and a one-dimensional “image”

can be calculated by assigning the spatial position to each frequency. This is illustrated in Fig. 1.5a for a

very simple “image” with spins at only two positions in space. In this case, only two corresponding fre- quencies are superposed, and the Fourier transform of the mixture of these two frequencies consists of only two non-zero coeffi cients. A more complicated image with a continuous spectrum of frequencies and more non-zero Fourier coeffi cients is shown in Fig. 1.5b. Since spatial encoding is achieved by a gradient during readout that assigns different fre- quencies to different spatial locations, this technique is called frequency encoding, and the gradient is referred to as the frequency-encoding gradient or readout gradient.

While the concept of one-dimensional spatial encoding by a frequency-encoding gradient during the data acquisition is relatively uncomplicated, the two- or three-dimensional spatial encoding of image data requires more explanation. First, it is important to note that all data acquisition in MRI is always dis-

t t

f = J ×B

z

z U

t Receiver +

Analysis of frequency spectrum

„Fourier transform“

f S

f = J ×B

z

z U

t Receiver

…

Analysis of frequency spectrum

„Fourier transform“

f S

Fourier coefficients (spectrum) Fourier coefficients (spectrum)

a b

t t

RF

Fig. 1.5a,b. The Fourier transform of the RF signal describes the frequency spectrum of the signal and thus the spin distribution in space. a Simple object (red) with spins only at two positions in space. The Larmor frequencies of these two positions vary due to an applied gradient. The superposition of two oscillations with different frequencies is received. The calculated frequency spectrum (Fourier coeffi cients) corresponds to the original object. b The object (red) emits an RF signal with a continuous distribution of Larmor frequencies.

As before, the calculated frequency spectrum corresponds to the original object.

a b

crete, i.e., data consist of a series of separate values and not of a smooth, continuous curve (we will neglect for now that data in MRI are almost always complex data with a real and an imaginary part and show only the real part). For one-dimensional imag- ing, these discrete data points can be acquired during the application of the readout gradient. Thus, for each data point the readout gradient has been switched on for a certain time before its value is actually meas- ured.

As a consequence, instead of acquiring all data points while the readout gradient is applied, it would also be possible to acquire only a single value at a time after the application of correspondingly timed gradients (see Fig. 1.6). To acquire the same signal curve as before, this procedure must be repeated after the sequence repetition time, TR, for every data point of the readout. Although it may be less obvi- ous, the separately acquired data points describe the same superposition of different Larmor frequencies corresponding to different spatial locations as before.

Thus, the spatial distribution of spins can again be calculated by Fourier transforming the signal assem- bled from several separate measurements. With the described technique, spins at different locations are not distinguished by their current frequency during the (now very short) readout, but by the accumu- lated phase acquired in the gradient interval before the readout. This technique is therefore called phase encoding and the gradient is referred to as the phase- encoding gradient.

Obviously, this technique is dramatically slower than the acquisition during a single gradient, since instead of a single readout interval of, e.g., 8 ms for all 256 data points, now 256 readouts separated by a TR of, e.g., 600 ms are required, resulting in a scan time increased from 8 ms to 153 600 ms. However, the described technique is exactly what we need in two- or three-dimensional imaging to encode the second and third spatial direction, since the fast data acquisition during the readout gradient can only be applied for a single spatial direction. The only differ- ence in real life is that usually the amplitude of the phase-encoding gradient is modifi ed for each phase- encoding step instead of the duration of the gradi- ent (Fig. 1.7). The resulting data points are the same (neglecting relaxation effects), since the “area under the gradient,” i.e., the product of gradient amplitude and duration is the same in both cases.

Thus, using a combination of frequency encoding for one spatial direction and phase encoding for the others, the acquired raw data have identical proper- ties for all directions and can be reconstructed by applying Fourier transforms for all three axes. A sim- plifi ed version of a three-dimensional non-selective gradient-echo pulse sequence scheme is displayed in Fig. 1.8. The displayed pulse sequence scheme acquires a single frequency-encoded line of raw data and must be repeated for each required line with dif- ferent combinations of the two phase-encoding gra- dients. For a three-dimensional data set with a matrix size of 256×256×64, this results in 256×64=16,384 rep-

Fig. 1.6a,b. a Frequency encoding: The signal is acquired while the readout gradient is applied. The smooth signal is discretized during acquisition. b Phase encoding: Instead of acquisition of the complete signal during a single readout, the discrete data points can be acquired separately after applying gradients of increasing duration. The resulting series of signal intensities is the same as before.

Readout t gradient Acquired (discrete) data Smooth signal

t

t t t etc.

Data point #1

Data point #2 Data point #3 Data point #4 All data points (full readout)

a _a b _b

etitions. In the case of a fast gradient-echo sequence with a short repetition time of TR=10 ms, the total acquisition time for the three-dimensional data set is thus 163.84 s.

1.4

k-Space and Spatial Frequencies

The complete process of one-dimensional data acqui- sition and image reconstruction is summarized in Fig. 1.9a. The object to be imaged is shown in red in the image space as a function ρ(x) of the spatial coordinate, x. The frequency-encoding gradient is applied such that the Larmor frequency, f, of the spins becomes proportional to the spatial position, f ~ x;

the different frequencies are illustrated in blue above the object. The received RF signal (shown in blue on the right hand side) is the sum of all Larmor frequen- cies weighted by the image density, ρ(x). Mathemati- cally, the received signal is the Fourier transform of the image function ρ(x). At this point, the process of data acquisition is completed.

Image reconstruction is performed as the inverse process of data acquisition. The RF signal is to be

Fig. 1.8. Pulse diagram of a simplifi ed three-dimensional gra- dient-echo sequence. Frequency and phase encoding are com- bined to acquire three-dimensional image data. The shown di- agram must be repeated for all combinations of amplitudes of the two phase-encoding gradients. Not shown in this diagram is an additional “dephasing” gradient in frequency-encoding direction, which is usually inserted before the data acquisi- tion to shift the maximum of the RF signal to the center of the readout.

repeat n

× n

n

steps

n

steps RF events

Excitation pulse RF signal

Freq.-enc.

gradient

Phase-enc.

gradient 1

Phase-enc.

gradient 2

Fig. 1.7a,b. Phase-encoding gradient. a The signal is sampled by applying phase-encoding gradients of increasing duration. b The same signal can be sampled by applying phase-encoding gradients of increasing amplitude.

t etc.

Data point #2

t Data point #3

t Data point #4 t

Data point #1

t etc.

Data point #2

t Data point #3

t Data point #4 t

Data point #1

t All data points

t All data points

a

b

a

b

decomposed into harmonic oscillations, which again can be achieved with a Fourier transform as described above in Sect. 1.2. In summary, MR data acquisition corresponds to a Fourier transform, and MR image reconstruction corresponds to a second Fourier transform, which is the inverse transforma- tion of the fi rst one.

In the explanations above, one-dimensional raw data have been presented as an RF signal time course or, equivalently, as a series of intensity values as shown in Fig. 1.6. However, as demonstrated in Fig. 1.7, the signal is in general not fully characterized by a gra- dient duration alone, but rather by the product of duration, t, and amplitude, G, of the encoding gradi- ent. Therefore, it has proven convenient to use the spatial frequency, k =γ×G×t, which is proportional to the product G×t, instead of t as a coordinate for the acquired signal. According to this defi nition, k is

proportional to the area under the gradient of ampli- tude G and duration t. The received RF signal is now no longer described as a signal time course, but as a function of the coordinate k. Obviously, this defi ni- tion provides coordinates not only for the frequency- encoding direction, but also for the phase-encoding directions where the amplitude (and not the dura- tion) of the gradient is varied. Thus, each data point of a one-dimensional readout is assigned to a coordi- nate k between –k

and +k

in k-space, and each data point of a two- or three-dimensional acquisition is described by a pair (k

, k

) or a triple (k

, k

k

) of coordinates in k-space, respectively (Pascal and Morris 2004).

The name spatial frequency has been chosen because each coordinate, k, i.e., each point in k- space, corresponds to a certain spatial oscillation in the image space as demonstrated in Fig. 1.9b. This

Fig. 1.9a,b. Complete illustration of data acquisition and image reconstruction in one-dimensional MRI. a Frequency encoding: Different spatial positions of the imaged object (shown in red on the left-hand side) correspond to differ- ent RF frequencies (blue); data acquisition is the process of acquiring the superposition of all these RF oscillations.

Conversely, different points in the time domain can be identifi ed with oscillations in the “frequency domain,” and a superposition of these oscillations results in the reconstructed image in frequency (or image) space. Frequency domain and time domain are transformed into each other using the Fourier transform. b Generalized k-space de- scription: Points in image space are identifi ed with oscillations in k-space (instead of the time domain) and points in k-space with oscillations in image space (instead of the frequency domain). The latter explains why k is referred to as spatial frequency.

x ~ f Image space with

frequency-encoding gradient

0 Time domain t

with RF signal 0 weighted

sum

weighted sum Frequency encoding

General k-space description

t t

t t

t

f f

f f

f

k_x k_x

k_x k_x

k_x

x x

x x

x

x

Image space 0 ^k-space 0 k_x

Fourier transform

Fourier transform

Harmonic oscillations in time domain (RF) Harmonic oscillations in frequency domain

Harmonic oscillations in k-space Harmonic oscillations in image space Acquisition

process Reconstruction

process

a

b

U

a

b

fi gure is a generalization of Fig. 1.9a with RF signals as functions of k

instead of t. As mentioned above, each coordinate of the RF signal corresponds to har- monic oscillation, and in the generalized description, these are oscillations in image space (in contrast to oscillations in the frequency domain for frequency encoding or oscillations in an “inverse gradient”

domain for phase encoding). Thus, the description of MR raw data and image reconstruction can be uni- fi ed for all spatial directions and all kinds of spatial encodings by introducing k-space. It should fi nally be noted that the spatial frequency is given in units of an inverse length, e.g., in 1/cm, in analogy to the more common temporal frequencies with units of inverse time: 1 Hz=1/s.

In MRI we usually deal with two- or three-dimen- sional data sets and their representation in k-space.

Again, each point in two-dimensional k-space cor- responds to a harmonic oscillation as shown in Fig. 1.10, and image reconstruction aims at adding all these oscillations weighted with the amplitude of k- space data at this point. Mathematically, this is done by calculating the two- or three-dimensional Fourier transform of the k-space data.

Raw data in k-space have some essential proper- ties that can be derived from Fig. 1.10 and are dis- cussed using the MR image shown in Fig. 1.11. A fi rst important observation is that most of the intensity of k-space data is usually found in the center of k- space. The center corresponds to low spatial fre- quencies, i.e., to slowly varying image intensity in

the image space. This slowly varying portion usu- ally determines the overall impression of an image, as illustrated in Fig. 1.12. Although only a very small fraction of less than 1% of all k-space data were used for reconstruction, the overall shape and contrast of

Fig. 1.10. Illustration of spatial frequencies: Each square il- lustrates a single point in (two-dimensional) k-space. These points correspond to those spatial oscillations in image space that are shown in the squares. Low spatial frequencies are found in the center of k-space; image intensity varies slowly in image space. High spatial frequencies correspond to periph- eral points in k-space.

kx= 0 1 2 3 5

-1 -2 -3

-5 -4 4

k_y= 0 1 2 3 5

-1 -2 -3

-5 -4 4

Fig. 1.11a,b. a Raw data in k-space (magnitude presentation of complex data) and b corresponding reconstructed MR image.

Highest intensities in k-space are found in the center, corresponding to slowly varying image intensities in b.

b

a

the image is still recognizable. The remaining 99%

of k-space contains information about the image details, in particular about the shape of sharp edges as illustrated in Fig. 1.13. High spatial frequencies are required to describe rapid changes of signal intensity in very small areas, and the maximum spatial reso- lution of an image, 'x, is determined by the highest spatial frequencies acquired, i.e., by the largest coor- dinates, k

, in k-space.

Another essential property of the Fourier trans- form can now be deduced from the symmetry of the Fourier transform. Since we have just seen that the image resolution, 'x, depends on the maximum value of the spatial frequency, k

, in k-space, it must be also be true that the resolution in k-space, 'k, is con- nected with the maximum image coordinate, x

. The meaning of this is illustrated in Fig. 1.14. If the resolution in k-space is reduced by doubling 'k,

Fig. 1.12a,b. a Raw data in the center of k-space and b corresponding reconstructed MR image. The image appears fi ltered; only large-scale changes of intensity corresponding to low spatial frequencies remain.

a b

Fig. 1.13a,b. a Raw data in the periphery of k-space and b corresponding reconstructed MR image. The image con- tains information complementary to Fig. 1.12a. Only changes of intensity in small spatial scales corresponding to high spatial frequencies remain.

a b

then the maximum fi eld of view of the reconstructed image is reduced to 50%. In the shown example, the fi eld of view is now smaller than the head in the ante- rior-posterior direction, resulting in typical aliasing or fold-over artefacts. Although the shown image still has the same nominal size as before, it contains information in only 50% of all data rows, i.e., the right half of the image is completely identical to the left

half. Further reduction of k-space sampling density is shown in Fig. 1.15, where only every fourth k-space line is acquired. Aliasing is much more severe in the reconstructed image with an effective fi eld of view of only a fourth of the original one.

A further property of raw data is its (Hermitian) point symmetry that is employed for partial Fourier imaging as described in Chapter 7. Finally, it should

Fig. 1.15a,b. a Raw data after removing three of four neighboring lines and b corresponding reconstructed MR image. The effective fi eld of view of the image is reduced to 25%, resulting in severe aliasing artefacts in anterior- posterior direction.

b a

Fig. 1.14a,b. a Raw data after removing every other line in k-space and b corresponding reconstructed MR image. The effective fi eld of view of the image is reduced to 50%, resulting in aliasing artefacts in anterior-posterior direction.

b

a

be noted that each point in k-space corresponds to an oscillation over the complete image space as dem- onstrated in Fig. 1.10. Thus, an erroneous data value at a single point in k-space will infl uence all pixels in image space, as illustrated in Fig. 1.16. The artifi cially added point artefact in the raw data results in a global artefact in image space.

The k-space is important to understand image reconstruction, but also to distinguish different kinds of data acquisition schemes. The simplifi ed three- dimensional gradient-echo sequence shown in Fig. 1.8 acquires k-space data line-by-line; all lines are paral- lel, and their orientation is defi ned by the frequency- encoding gradient. However, other k-space sampling schemes can also be employed, although they are less frequently used than conventional Cartesian acquisi- tions. The most important alternatives are radial and spiral k-space sampling (Lauzon and Rutt 1996;

Block and Frahm 2005). Both techniques typically start data acquisition in the center of k-space and cover k-space either with straight radial lines (like the spokes of a wheel) or with a single or with sev- eral spiral trajectories. To avoid image artefacts, the maximum distance, 'k, between neighboring sample points in k-space is required to be suffi ciently small, similarly as in Cartesian imaging. Further details about these non-Cartesian techniques can be found in Chapter 6.

1.5

Parallel Imaging

As mentioned above, MRI acquisitions can be very time consuming since raw data are typically acquired line by line, and the pulse sequence must be repeated for each of these lines in order to build up a full data set in k-space. Even with the minimum possible echo times and repetition times, the total acquisition time of a data set may be unacceptably long for many state-of-the-art MRI applications such as dynamic MR angiographies, perfusion MRI, or MR imaging of the cardiac function. Therefore, reducing the number of k-space lines required for an acquisition is often desirable. However, it is usually not acceptable simply to acquire fewer lines in the k-space periphery since this decreases the spatial resolution of the image data, although a reduction of lines of up to almost 50% can be achieved with partial-Fourier techniques where the k-space periphery is sampled asymmetrically (King 2004). Partial-Fourier imaging and several other specifi c techniques to increase imaging speed are discussed in detail in Chapter 7.

Most of these techniques (with the exception of partial-Fourier imaging) are limited to a few spe- cifi c MRI applications. In contrast to these specifi c approaches, an idea for accelerated acquisitions pro-

Fig. 1.16a,b. a Raw data after adding a single point with artifi cially increased intensity (arrow) and b corresponding reconstructed MR image. The image is affected from artefacts extending over the complete fi eld of view.

a d

posed in the second half of the 1990s has found wide acceptance in virtually all areas of MRI (Dietrich et al. 2002; Heidemann et al. 2003; Bammer and Schoenberg 2004): this approach is now known as parallel imaging, parallel MRI, (partially) parallel acquisition (PPA), or parallel acquisition techniques (PAT).

The basic idea of parallel imaging is to employ several independent receiver coil elements in parallel to reduce the number of phase-encoding steps. Thus, a certain amount of the spatial encoding originally achieved by the phase-encoding gradients is now substituted by evaluating data from several coil ele- ments with spatially different coil sensitivity profi les.

The reduction of phase-encoding steps is achieved by reducing the sampling density, 'k, in k-space as illus-

Fig. 1.17a–c. Data acquisition and image reconstruction in parallel MRI. a Data is acquired by four coil elements with differ- ent spatial sensitivity profi les in parallel. b Four sets of reduced raw data are available for image reconstruction. Each of these data sets corresponds to an aliased image from one of the coil elements. c Specifi c reconstruction algorithms are required to reconstruct the image in parallel MRI.

a Acquisition with four coil elements (coil sensitivity profi les shown)

b Raw data and temporary images

c Reconstructed image

trated in Fig. 1.17. Raw data reduction factors between 2 and 6 can typically be achieved with parallel imag- ing in a single direction; a combination of reduced sampling densities in two phase-encoding direc- tions is possible in 3D MRI and results in higher total reduction factors. The maximum reduction factor is limited by the number of independent coil elements or separate receiver channels of an MRI system. Thus, the most important precondition for the applicabil- ity of parallel imaging is a multi-channel MRI system with several parallel receiver channels as well as appropriate multi-channel coil systems.

Data acquisition for parallel MRI is very similar to acquisition schemes of conventional pulse sequences.

Most relevant techniques of MRI can be adapted rela-

tively easily to parallel imaging, since the data acqui-

sition is essentially equivalent to an acquisition of a reduced (rectangular) fi eld of view in the phase-encod- ing direction. Image reconstruction of parallel imaging raw data on the other hand is much more demanding than conventional Fourier-transform reconstruction, and considerable efforts have been made to develop optimized reconstruction algorithms. Today, several reconstruction techniques are used for parallel imaging (Blaimer et al. 2004); the most common approaches are known by the acronyms SENSE (Pruessmann et al.

1999), SMASH (Sodickson and Manning 1997), and GRAPPA (Griswold et al. 2002). All these approaches aim for the best compromise between minimizing localized image artefacts on the one hand and image noise on the other hand. Details about the different techniques used for parallel imaging as well as about hardware and software requirements are presented in Chapters 2 and 3.

The main motivation for the development of parallel imaging and its application in the clinical routine is the acceleration of the image acquisi- tion. Acceleration factors of two, three, or four are achieved by sampling only every second, third, or fourth k-space line, respectively. However, several

other areas of MRI also benefi t from parallel imag- ing; in particular, parallel imaging can be effi cient to reduce several kinds of image artefacts as discussed in Chapter 5.

1.6

Signal and Noise

Image noise is present in practically all MR images and is one of the major limiting factors for MRI pro- tocols. The effect of image noise is demonstrated in Fig. 1.18: small low-contrast structures, such as, e.g., in the cerebellum, become more diffi cult to differ- entiate with increasing noise levels. The noise level is described by the noise standard deviation, σ, and its statistical distribution. The visual impression of an MR image depends on the signal-to-noise ratio (SNR), i.e., the ratio of the signal intensity and the noise standard deviation:

SNR=(signal intensity)/(noise standard deviation).

Fig. 1.18a–d. Illustration of image noise. The same MR image with increasing noise levels (from a to d) is shown. High noise levels reduce the visibility of small details and of low-contrast changes.

c

b a

d

Typically, noise becomes a limitation in fast imag- ing techniques, since the noise level of an MR image is connected with its acquisition time. The dependence of the signal-to-noise ratio on the acquisition time and several other MR imaging parameters can be summarized in the formula (Edelstein et al. 1986;

Haake et al. 1999):

SNR C ≈ × ∆ x × ∆ y × ∆ z × T

× B

where ' x, 'y and 'z describe the dimensions size, T

is the total time of receiving RF data, B

is the static fi eld strength, and C is a factor describing all other infl uences such as image contrast, relaxation effects, coil geometry and effi cacy, coil fi ll factor, tem- perature, etc. Most important for fast imaging is the dependence on the square root of T

. The time T

is calculated as the product of the duration of a single readout, T

, and the number of all rea- douts of the complete raw data set, i.e., the number of phase-encoding steps, N

, times the number of averages, N

:

T

= N

× N

× T

The duration of the single readout, T

, depends on the receiver bandwidth, BW, of the pulse sequence, T

~ 1/BW. Thus, by doubling the receiver band- width, the SNR is reduced to 1 2 | 71% of the original value. On the other hand, by increasing the number of averages from one to four, the SNR can be doubled. Similarly, increasing the fi eld of view in a phase-encoding direction while leaving the voxel size constant, i.e., increasing the number of phase- encoding steps, N

, results in a higher SNR as well.

It should be noted that the total receiving time, T

, is not the total acquisition time, T

, i.e., the duration of the pulse sequence. The latter depends on the rep- etition time, TR, and the pulse sequence design (e.g., single-echo vs. multi-echo sequence, etc.).

Parallel imaging reduces the number of phase- encoding steps without changing the geometry (fi eld of view) or the voxel size of the reconstructed image.

Thus, the total acquisition time, T

, and simultane- ously the total receiving time, T

, can be reduced to, e.g., 50 or 25% of its original value. Consequently, the SNR is also reduced to at least 71 or 50%, respec- tively. This is an unavoidable side effect of parallel imaging that is discussed in detail in the following chapters. Consequently, quantifi cation of SNR is an important issue in parallel imaging studies; however, due to the specifi c properties of the reconstruction

algorithms used for parallel imaging, this quantifi ca- tion can usually not be performed in a straightfor- ward manner (Dietrich et al. 2005). The potential diffi culties and several appropriate techniques to determine the SNR in parallel imaging are presented in Chapter 4.

1.7

Conclusion

Parallel imaging is a relatively new MRI technique that employs parallel data acquisition from several receiver coil elements in order to reduce the number of phase-encoding steps and, thus, to increase imaging speed. To discuss acquisition techniques and image reconstruction algorithms used in parallel MRI, the concepts of k-space and of the Fourier transform introduced in this chapter are indispensable. Based on these concepts, parallel imaging from its historic origins to current state-of-the-art techniques is pre- sented in the following chapters of the fi rst part of this book.

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