Radio frequency (RF) pulses are used to bring the macroscopic magnetization ! out of thermal equilibrium and turn it into the transversal plane (excitation pulse, flip angle = 90°) or to turn it with a flip angle of 180° (refocusing and inversion pulse). The simplest RF pulse is the block pulse. The flip angle ! of ! depends on the amplitude !! and duration ! of the RF pulse:
! = !!!!
The bandwidth of an RF pulse is inversely related to the pulse length !. The shorter the pulse, the wider is its bandwidth. To still obtain the desired flip angle, high bandwidth RF pulses require a large !! amplitude, leading to a
high RF power absorption. This can result in long acquisition times due to SAR restrictions.
The frequency of an RF pulse has to be equal to the Larmor frequency of spins to achieve the required flip angle. If spins resonate at a different frequency, due to e.g. !! inhomogeneities, they will experience a different rotation angle.
Also inhomogeneous !! fields caused by inhomogeneous profiles of transmit
coils can cause undesired flip angles. Adiabatic RF pulses are designed to be less sensitive to !! inhomogeneities.
Specific absorption rate (SAR)
Next to the energy absorbed by nuclear spins, the electromagnetic waves of the RF pulses induce electric currents in the tissue of which the major part is converted to heat, increasing the tissue temperature. The amount of power absorbed per unit of mass is called specific absorption rate (SAR), measured in W/kg. In case of field inhomogeneities and coil couplings the deposited power can lead to local burnings at worst. To prevent heating of the body, the SAR is limited to 2 W/kg for the whole body, 3.2 W/kg for the head, 10 W/kg for head
35
and trunk (local) and 20 W/kg for extremities (local), all values averaged over 6
minutes (IEC 60601-2-33 (2002)). Since the Larmor frequency increases with the external magnetic field !!, also
the RF frequency has to be increased to excite the spins. The deposited power is proportional to the square of the electric field of the RF pulse (12), making the SAR increase quadratically with the !! field. Therefore the allowed SAR is,
especially at high field strengths, a limiting factor with regard to the minimal allowed TR, the RF pulse amplitude and the number of RF pulses. This makes pulses with reduced RF amplitude advantageous at high !! fields, like e.g.
gradient modulated RF pulses. Adiabatic pulses
Conventional RF pulses have a modulated !! amplitude and a constant carrier
frequency, mostly applied at the center of the spectral bandwidth of interest. Adiabatic pulses, in contrast, have a time varying carrier frequency !!" ! .
Across the spectral bandwidth of interest !" spins with different resonance frequencies !!, called isochromats, are rotated at different points in time, as
the frequency sweep !!" ! approaches the resonance frequency !! of each
isochromat. Therefore the range of this frequency sweep usually determines the spectral bandwidth of an adiabatic pulse. For spins within the bandwidth !" the flip angle will be uniform. Hence, the bandwidth of adiabatic pulses is no longer inversely dependent on the duration of the pulse, allowing for higher bandwidths.
If !! is not exactly applied at the Larmor frequency but off-resonance, the
magnetization ! does not precess around !!, but around the effective
magnetic field !!"". !!"" is the vector sum of !! and ∆!/!, with ∆! = !!−
!!" being the difference between the Larmor frequency !! and the applied carrier frequency:
!!""= !!+
!" !
In the adiabatic passage the carrier frequency !!" ! is swept, making the
effective magnetic field !!"" ! change in time as well, with an angular velocity
!"/!", where
Chapter 2
36
! ! = !"#$!% !!! ! ∆! !As long as the orientation of the effective magnetic field !!"" ! changes more
slowly than the rotation of ! around !!"" ! , ! will follow !!"" ! to its final
position and therefore all spins with !! are rotated by a constant flip angle.
This requirement is called the adiabatic condition: !!!"" ! ≫ !"/!"
The adiabatic condition is fulfilled if !! is high enough (above a certain
threshold) or the frequency sweep is slow. Therefore the classical adiabatic rotation is independent of small !! amplitude changes, making adiabatic
pulses advantageous in combination with surface coils. (13–16). Offset-independent adiabaticity (OIA)
The offset-independent adiabaticity (OIA) approach is an optimization procedure to fulfill the adiabatic condition. In the OIA approach the RF power is distributed uniformly over the requested bandwidth, but sequentially in time. Hence, a uniform rotation over the bandwidth ∆! is obtained.
Many adiabatic modulation functions solely consider the center frequency of isochromats !!= !! (! = 0) , but lose efficiency for isochromats with an
additional frequency offset ! = !!− !!. Here we include a constant frequency
offset !, e.g. caused by a chemical shift or a constant gradient. This frequency offset has to be within the bandwidth ∆! of the frequency sweep:
! ≤!" 2
The adiabatic condition can be given as a ratio !, which is a function of ! and !:
! !, ! = !!!""
! (!)
!" !"
For the most critical moment ! = !!, when the isochromat at ! is on resonance
and the effective field !!"" ! crosses the transverse plane, the adiabatic
condition is given by:
! !! !"
2 !! !! = (!!!!!! !! )!
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Every combination of amplitude modulation (AM) function !! ! and the
frequency modulation (FM) function !! ! fulfilling this condition will lead to
uniform adiabaticity for all isochromats within the bandwidth ∆!. (13,15) Gradient-modulated offset-independent adiabaticity (GOIA)
Gradient-modulated offset-independent adiabaticity (GOIA) pulses have a time-dependent resonance offset ! ! , as consequence of a modulated magnetic field produced by a changing gradient field. Next to the AM function !! ! and the FM function !! ! , GOIA pulses also have a gradient modulation function !! ! (Figure 13). Because of the additional gradient modulation GOIA
pulses require less !!amplitude to fulfill the adiabatic condition, which is given
by: ! !! !" 2 !! !! − !! !! !! !! !! !! = (!!! !! ! !! )!
Every combination of !! ! , !! ! and !! ! fulfilling this condition will lead to
uniform adiabaticity for all isochromats within the bandwidth ∆! (13,15). The constant adiabatic factor over the entire spectral bandwidth and the low demand for RF power make GOIA pulses optimal for MRS, especially when large bandwidth are needed and SAR limits are low.
Frequency offset corrected inversion (FOCI)
FOCI pulses are more regularly used in clinical MRS studies than GOIA pulses. The slice profile they provide is very flat with sharp edges, a little better than that of GOIA pulses (Figure 14). However, FOCI pulses need higher RF power to realize the adiabatic condition (Figure 13) and therefore their use is cumbersome in combination with an ERC or at high field strength. Furthermore it has been found that FOCI pulses are more prone to eddy current artifacts as a consequence of their large gradient modulation slew rate (17).
Chapter 2
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Figure 13 Adiabatic pulses. (A) Modulations of RF amplitude (B1(t)), gradient (G(t)), carrier frequency (ω(t)) and phase (φ(t)) for GOIA-W(16,4) (solid line), GOIA-HS(8,4) (dotted line) and FOCI (dashed line). Source: (17).
Figure 14 Measured slice profile of GOIA-W(16,4) (solid line), GOIA-HS(8,4) (dotted line) and FOCI pulse (dashed line). The duration of all pulses is 3.5 ms and they have a bandwidth of 20 kHz. Their RF amplitudes are: 0.817 kHz for GOIA-W(16,4), 0.711 kHz for GOIA-HS(8,4), and 1.463 kHz for FOCI. Source: (17).
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k-space
To understand MR imaging and spectroscopy acquisition techniques, it is important to understand the concept of k-space. In Fourier imaging the raw MR data is collected in k-space [1/m]. This k-space data then is converted to an image by Fourier transformation. Therefore k-space (!!, !!, !!) is the Fourier
conjugate of the spatial domain (!, !, !) . Time dependent gradient fields (!!, !!, !!) within a pulse sequence are used to describe a path in k-space:
! ! = !
2! ! !! !"′
! !
The most basic method to fill k-space is Cartesian sampling, which fills a Cartesian grid in k-space (Figure 15). Points in k-space (!!, !!, !!) do not
correspond one-by-one to the individual voxels (!, !, !) in the final image, but each point in k-space contains information about every pixel in the corresponding image. In the center of k-space (low frequencies) mainly signal contrast is stored, whereas in the outer parts of k-space (high frequencies) detailed image information is deposited. An image reconstructed from only the central k-space points has a high SNR, but it appears blurry and lacks detail. Only sampling the outer part of k-space would result in an image with sharp details, but very low signal. Hence, the way of sampling k-space influences the quality of the resulting image.
Figure 15 k-space and the corresponding image space, transformed by Fourier transformation.
Chapter 2
40
In classical MRSI, k-space has four dimensions (!!, !!, !!, !) in which in the time
! the spectral frequencies are encoded. Fourier transformation gives signal intensity in the spatial and frequency domains. For MRSI Cartesian sampling is very time consuming, because spectral information has to be acquired. Therefore the time signal of each k-space point has to be sampled separately, resulting in a measurement time TA of:
!" = !!∙ !!∙ !!∙ !"
The distance between points sampled in k-space defines the FOV of the image:
∆! = 1
!"# ⇔ !"# = 1 !"
The pixel size of the image is given by the size of sampled k-space !!"#:
∆! = 1
!!"# ⇔ !!"!=
1 ∆!
Spiral k-space sampling
There are several techniques to fill k-space, like filling it in a different order, with a different trajectory or just parts of it and calculating the missing parts. Applying advanced k-space sampling techniques can save measurement time. One technique is spiral k-space acquisition (10). Instead of sampling line by line or point by point of k-space, the x- and y-dimensions of k-space can be sampled simultaneously with a spiral trajectory within one acquisition. Therefore the x- and y-gradients traverse (!!, !!)-space at the same time. The
spiral trajectories can be interleaved in (!!, !!)-space to achieve a larger FOV
(Figure 16a).
For MRSI also spectral information has to be acquired. To do so several spiral gradients are repeated to acquire the time component, which is necessary for the spectral information. For sufficient spectral bandwidth the basic spiral trajectories can be interleaved in time, by repeating the spirals !! times in
subsequent repetitions with different delays (Figure 16b). The measurement time for a 3D MRSI with spatial and temporal interleaves is:
41
!" = !!∙ !!∙ !!∙ !"
with !! the number of slices in z-direction, !! the number of spatial interleaves
and !" the repetition time.
Figure 16 Spiral k-space sampling. (a) Two spatial interleaved spiral trajectories. (b) To get spectral information in MRSI, several temporal interleaves have to be executed. Source (10).