Three-Dimensional Tailored RF Pulses for the Reduction of Susceptibility Artifacts in T2* Weighted Functional MRI

Abstract

A three-dimensional tailored RF pulse method for reducing intravoxel dephasing artifacts in T2* weighted functional MRI is presented. A stack of spirals k-space trajectory is employed to excite a disk of magnetization for small tip angles. Smaller disks with a linear through-plane phase are inserted into the disk to locally refocus regions which are normally dephased due to susceptibility variations. Numerical simulations and imaging experiments which use the tailored RF pulses are presented. Limitations of the method and improvements are also discussed.

Rapid imaging methods, such as spiral imaging [1–3] or echo planar imaging [4, 5], are of current interest in MRI. In particular, these methods are widely implemented in brain functional MRI (fMRI) due to their high sensitivity to blood oxygen level dependent (BOLD) T2* contrast [6, 7] and due to their high temporal and spatial resolution of brain activity. However, the long echo times required for BOLD T2* contrast make the images acquired with these modalities more prone to intravoxel dephasing from magnetic susceptibility variations, particularly near air/tissue boundaries. Many brain regions which are significant for psychiatric and basic nueroscientific research, including the orbitalfrontal and the inferior and medial temporal cortices, are plagued by susceptibility artifacts in BOLD fMRI. Therefore, the development of methods which can reduce susceptibility artifacts and still preserve sensitivity to BOLD T2* contrast with high temporal and spatial resolution is of importance.

Numerous methods have been proposed to mitigate susceptibility artifacts in T2* weighted imaging, including automatic shimming [8], data post processing [9–12], gradient compensation [13–19], and tailored RF (TRF) pulses [20–22]. These methods, however, either provide limited correction or demand scan times that are unreasonably long for fMRI. Gradient compensation techniques, for example, work through the implementation of an extra phase encoding gradient to refocus spins that are out of phase due to susceptibility variations. However, the spins normally in-phase are now incoherent as a result of the extra gradient lobe and as many as eight [17] or sixteen scans are required for the acquisition of a complete artifact-free image at 1.5 T and 3 T, respectively. An even larger number would be required for a true 3D correction. The inclusion of extra scans is impractical for fMRI due to reduced temporal resolution and, furthermore, these techniques only correct for susceptibility induced inhomogeneities that are linear. Tailored RF pulses which produce a quadratic phase across the slice [20] have also been suggested as a means to mitigate the influence of field inhomogeneities in T2* weighted imaging. Although this method does recover lost signal, there is a global reduction in SNR.

Recently, it has been suggested [21, 22] that fMRI in regions of high field non-uniformity can be accomplished through the implementation of TRF pulses that are constructed within the small tip angle approximation [23]. Although the small tip angle approximation is technically only valid for flip angle of 30° or less, it is found to hold well for angles up to 90°. Tailored RF methods offer the potential for a phase correction with a spatial dependence designed specifically for the region of interest. A major limitation of these techniques, however, is that they are one-dimensional, providing phase correction to the artifact affected regions while introducing dephasing to normal regions. This communication extends these ideas by presenting preliminary results on a method which uses three-dimensional (3D) TRF pulses [24] to provide phase correction only to the regions which are corrupted by susceptibility induced intravoxel dephasing. This method allows for the possibility for an anatomically accurate 3D correction with one or a few number of excitations.

THEORY

In the small tip angle approximation [23] there is a direct correspondence between the Fourier transform of the RF pulse and the desired magnetization profile excited by the RF pulse. If the susceptibility induced phase distribution throughout the slice volume at TE is known, one can build a 3D map of the negative of this phase with a corresponding uniform magnitude, Fourier transform, and build a 3D TRF pulse which traverses an appropriate k-space trajectory. In this manner it is possible to design a 3D TRF pulse which excites a slice volume with a phase distribution through the slice that cancels the phase distribution created at TE by susceptibility variations.

The 3D TRF pulses implemented in this work were based on the spin-echo inversion pulse design of Pauly et al. [24] which uses a “stack of spirals” k-space trajectory. The stack of spirals trajectory was created from 2D spiral trajectories in k_x-k_y and gradient “blips” along k_z such that a cylindrical k-space volume was covered. Appropriate RF waveforms were obtained by representing the desired magnetization profile by mathematical functions with analytical Fourier transformations. Exploiting the cylindrical symmetry of the stack of spirals, the k-space functions for a disk of magnetization were created from a cylindrical Bessel function for the radial k_r direction and a Gaussian function for the k_z direction:

$$M\left(k\right)=\text{exp}(-\pi k_z^2)J_1\left(\pi k_r\right)∕2k_r.$$ [1]

The Fourier transform of the above equation produces a cylinder or disk. To approximate the phase distribution created by susceptibility variations above the sagital sinus, for example, a smaller disk of adjustable size was subtracted from the main disk and an identical small disk with a linear phase along z was added in its place. The spatial location of the small disk and the magnitude and sign of the through-plane phase of the small disk could be tuned by adjusting a k_r phase factor and a k_z shift, respectively. Although the utilization of a measured anatomical map of the desired magnetization profile, including susceptibility induced phase variations at TE, is more desirable than this technique, the use of this approximation reduces complexity and is adequate for a demonstration of the 3D TRF method.

Figure 1 shows an “axial” pulse 3D TRF pulse designed for the excitation of a slice above the sagital sinus. In this pulse, a 10 cm diameter small disk was located 10 cm off-center with approximately π radians of phase across the z half width. The pulse was constructed by implementing a stack of 48 spirals such that a 22 cm diameter disk with an in-plane resolution of 2.1 cm would be excited. The Gaussian profile along the z direction had a 2 cm thick half width. These parameters were based on the constraint that the maximum length for a symmetric pulse was 60 ms such that TE would be approximately 30 ms, a reasonable TE for an fMRI acquisition at 3 T. The pulse was designed to be intentionally longer than necessary because the introduction of linear phase variations along z for artifact correction will require shifting of the RF in time. The spirals were generated with the analytical spiral algorithm of Glover [25], designed for use on a system which has a maximum gradient slew rate of 200 T/m/sec and a peak gradient strength of 35 mT/m. The flip angle was defined as being proportional to the integral of the magnitude of the RF with respect to time.

Figure 1.

Three-dimensional tailored RF pulse for the excitation of a 2 cm thick, 22 cm diameter disk with a 10 cm diameter linear through-plane phase located 10 cm off-center. Rows from top to bottom are the real and imaginary part of the magnetization, and the x, y, and z gradients, respectively.

Sampling limitations will produce sidelobes of the excited magnetization [24]. For an axial slice, the in-plane sidelobes can be easily placed outside of the head. However, replicas of the disk will be excited within the head along z due to the limited sampling of the k_z blips and the corresponding small FOV provided by the pulse. For the pulse shown in Fig. 1, the FOV along z was such that sidelobes were created in multiples of 10 cm above and below the desired slice location. These sidelobe structures produced by the pulse shown in Fig. 1 can be simulated by numerical integration of the Bloch equations. Although the effects of T2* relaxation will certainly affect the pulse profiles due to the long pulse lengths, these effects were excluded for simplicity. Figure 2 (a) shows a mesh diagram representing the magnetization profile for |M_xy| in the x–z plane from the Bloch equation simulation. Figure 2 (b) displays the profile of the central lobe of |M_xy| along the z or “slice-select” direction. Figure 3 (a) and (b) show |M_xy| in the x–y plane and as a profile along the x-direction, respectively.

Figure 2.

(a) Simulated profile of |M_xy| in the x–z plane produced by the 3D tailored RF pulse shown in Figure 1. The profile is Gaussian along the z-direction and a first-order Bessel function along the x-direction. (b) Profile of the central lobe of |M_xy| along z.

Figure 3.

(a) Simulated profile of |M_xy| in the x–y plane produced by the 3D tailored RF pulse shown in Figure 1. The profile is a first-order Bessel function along the radial direction. (b) Profile of the central lobe of |M_xy| along x.

The effects of susceptibility variations and the appropriate amount of through-plane phase necessary to cancel the destructive interference effects can be estimated with a simulation model [26]. This method uses two whole brain 192×256×128 (1 mm³ isotropic voxel resolution) 3D gradient echo volumes acquired at 3 T with TE's of 9 ms and 10 ms, respectively. An estimate of the rate of phase accrual, dψdt, can be determined from the difference between the image phases at the two echo times. Images of thicker slices at arbitrary echo times can be simulated from each of the thinner 3D slices M_z collected at TE = 9 ms through the relationship,

$$I={\Sigma }_z{W}_z{M}_z\text{exp}\left[i{\varphi }_z\left(\text{TE}\right)\right].$$ [2]

Here the summation is along the slice select direction, ψ_z(TE) = (dψ / dt)TE, W_z is a weighting factor representative of the RF excitation profile, and I is the final thicker slice. Susceptibility artifacts are created in the final slices by the phase incoherence during the summation at TE. To simulate how effective a TRF pulse will be at mitigating the artifacts produced by susceptibility induced dephasing, Eq. [2] can be modified to include the phase ${\varphi }_z^{RF}$ and magnitude $W_z^{RF}$ of the transverse magnetization produced from numerical integration of the Bloch equations using the TRF pulse as input. Eq. [2] can then be written,

$$I={\Sigma }_z{W}_z{M}_z^{RF}{M}_z\text{exp}\left[i({\phi }_z\left(TE\right)+{\phi }_z^{RF})\right].$$ [3]

METHODS

The 3D TRF pulses were constructed using the Matlab (The MathWorks Inc., Natick, MA) software package. The parameters for the pulses were identical to those used to create the pulse shown in Fig. 1. A 3D TRF pulse with identical parameters, except with a 1.05 cm in-plane resolution and a corresponding 175 ms pulse length, was also constructed for comparison purposes (not shown). Simulations were performed using these pulses and the model described above with Eqs. [2] and [3]. During scanner operation, the TRF pulse generation programs could be run simultaneously on the same SGI workstation (Silicon Graphics Inc. Mountain View, CA) that the scanner used such that the parameters for the 3D TRF pulses could be adjusted in real-time with an approximate delay of 30–60 seconds for pulse generation, file transfer, and pulse sequence download. The pulse sequence implemented for image acquisition was a spiral sequence [2, 3, 25] optimized for fMRI on the General Electric (Milwaukee, WI) LX 3 T scanner. Each image was generated from a single-shot spiral acquisition with a 24 cm FOV and a 64×64 matrix size. The maximum gradient amplitude and slew rate for the scanner were 35 mT/m and 200 T/m/sec, respectively.

As a proof-of-concept experiment, an axial slice location was chosen directly above the sagital sinus in three normal human volunteers such that a large signal void was seen in a 2 cm thick spiral image acquired with a TE of 32 ms. Implementing 3D TRF pulses with parameters identical to the one displayed in Fig. 1 in the same spiral sequence, images were acquired at the same location. Up to four spatial saturation pulses could be applied prior to the 3D tailored RF pulse to crush the pairs of first and second order sidelobes along the z-direction. However, in practice only two spatial saturation pulses needed to be applied below the central lobe for elimination of sidelobes. Saturation pulses for the sidelobes above the slice were not necessary because these lobes were located above the subject's head. The time required for one pair of saturation pulses, including the crusher gradient, was approximately 10 ms. This produced a minimum time of 108 ms for the acquisition of one image utilizing the 3D TRF pulse and a pair of spatial saturation pulses. The time required for the acquisition of one standard spiral image without TRF was 55 ms. During image acquisition, the first step was to view the slice profile and sidelobes along the slice-select direction by switching the y-gradient for the spiral readout to the z-direction. Then the location, spatial widths, and frequency bandwidths of the spatial saturation pulses were adjusted until all the signal from the sidelobes was reduced below approximately 5% of the central lobe signal magnitude. After switching the spiral gradient back to the y-direction, the center of the small phase correction disk was positioned in the middle of the image and both the through-plane phase and the size of the disk were adjusted until a 3–5 cm hole was introduced in the middle of the slice. The small disk was subsequently moved towards the center of the susceptibility induced signal void above the sinus. The magnitude and sign of the phase of the small disk were then modified until signal was recovered in the void above the sinus.

RESULTS

Figure 4 shows images produced from the simulation model described above in the Methods section. A 2 cm thick slice was constructed using either Eqs. [2] or [3] from a position in the high-resolution 3D volumes which was located above the sagital sinus. Figure 4 (a) and (b) shows the simulated slice at echo times of 9 ms and 32 ms, respectively, produced from Eq. [2]. The slice profile was a Gaussian function with a 2 cm half-width. The image in (b) shows a large signal void above the sinus created by the summation of the 1 mm thick slices and the incoherence of the accrued phase of each 1 mm slice at TE. This result is in good qualitative agreement with what is observed in actual T2* weighted images acquired with a 32 ms TE at 3 T (see Fig. 6 (b) below, for example). Figure 4 (c) shows the simulated image at a 32 ms TE created with Eq. [3] where the RF weighting and phase were generated by numerical integration of the Bloch equations using a TRF pulse with a 1.05 cm in-plane resolution as input. Figure 4 (c) shows recovered signal above the sinus region. Figure 4 (d) shows a simulated image at a 32 ms TE created with Eq. [3] and an RF weighting and phase generated using a 2.1 cm in-plane resolution TRF pulse. Although Fig. 4 (d) also shows recovered signal above the sinus, there is a small reduction in signal around the corrected region compared to (c) due to the imperfect resolution of the TRF pulse.

Figure 4.

Simulated axial images constructed with a 2 cm half-width Gaussian slice profile at (a) 9 ms and (b) 32 ms echo times. Simulated 2 cm thick images constructed at a 32 ms echo time using RF weightings and phases produced by numerical integration of the Bloch equations with a (c) 1.05 cm and (d) 2.1 cm in-plane resolution tailored RF pulse as input.

Figure 6.

Human brain images acquired using a sinc RF pulse at 3 T with a (a) 5 ms TE and a (b) 32 ms TE and a 2 cm slice thickness. The arrow in (b) shows the region of signal loss due to the susceptibility variation between brain tissue and air in the sinus cavity. The images in (c)–(f) were acquired at a 32 ms TE using a 2.1 cm in-plane resolution 3D tailored RF pulse with through plane phase corrections of approximately π, π, 2π, and 3π radians, respectively. Image (c) shows signal loss (white arrow) in a slice where the phase correction was located in the center of the x–y plane. Images (d)–(f) show signal recovery (white arrow) in slices where the phase correction was located above the sinus cavity.

Figure 5 (a) and (b) shows the through-plane phase difference (32 ms TE) and magnitude (9 ms TE) of the simulated slice shown in Fig. 4. The windowing of the phase difference image was set between 0 and π such that black represents no through-plane phase and white represents π radians of phase or greater. There is a direct correspondence between the location of large through plane phase difference an the artifact seen in Fig. 4 (b). Figure 5 (c) and (d) shows the through-plane phase difference and magnitude of the transverse component of the magnetization generated from numerical integration of the Bloch equations with a 1.05 cm in-plane resolution TRF pulse used as input. It can be seen that the 3D TRF pulse method excites a slice with a localized, linear through-plane phase as predicted. Figure 5 (e) and (f) shows the through-plane phase difference and magnitude of the magnetization generated by a 2.1 cm in-plane resolution TRF pulse. It is evident from comparing (c) and (d) with (e) and (f) that the higher in-plane resolution produces a sharper magnetization profile, more accurately modeling the anatomical phase distribution shown in (a) and (b).

Figure 5.

(a) Simulated phase difference at 32 ms TE through a 2 cm thick slice at 3 T. (b) Corresponding 2 cm thick simulated magnitude image at 9 ms TE. (c) Through plane phase difference of the transverse component of the magnetization obtained from numerical integration of the Bloch equations using a 1.05 cm in-plane resolution 3D tailored RF pulse as input. (d) Corresponding magnitude at the center of the slice. (e) Through plane phase difference and (f) magnitude of transverse magnetization using 2.1 cm in-plane resolution 3D tailored RF pulse. The phase difference images are windowed between 0 and π.

Figure 6 shows 2 cm thick axial brain images acquired above the sagital sinus in one of the human volunteers at 3 T. Figure 6 (a) shows an axial slice above the sagital sinus acquired with a standard 2 cm thick “sinc” slice-select RF pulse at a TE of 5 ms and (b) shows the same slice acquired at a TE of 32 ms. In Fig. 6 (b) it is clear that there is a large signal void compared to (a) in the region above the sinus cavity. This signal void is indicated by the white arrow in (b). Figure 6 (c)–(f) shows the same slice at TE of 32 ms acquired with a 2.1 cm in-plane resolution 3D TRF pulse similar to that shown in Fig. 1 with through plane phase corrections of π, π, 2π, and 3π, respectively. The location of the phase correction for the image in (c) was positioned such that intravoxel dephasing was produced in the center of the slice (as indicated by white arrow). In (d)–(f) the phase correction was moved anterior such that it was above the sinus artifact. Pulse parameters including the location, size, and magnitude of phase correction, were modified in-vivo by visual inspection until a slice with most uniform signal recovery was obtained. Much of the signal lost above the sagital sinus in (b) is recovered with the 3D TRF pulse, as indicated by the white arrows in (d)–(f). Decreasing or increasing the magnitude of the through-plane phase produced images with less or more signal magnitude in the sinus void than the surrounding tissue, respectively. Images were also acquired with the sign of the through-plane phase correction reversed (not shown). These images showed no signal recovery above the sinus.

DISCUSSION

The above results show that it is possible to implement 3D TRF pulses to reduce susceptibility induced intravoxel dephasing in T2* weighted imaging modalities. The advantage these methods have over 1D methods, such as gradient compensation, is that the correction is applied only to the affected regions. However, numerous challenges need to be overcome before their full potential is met. A major limitation of the method, as it is presented here, is a lack of adequate sampling resolution. Problems from inadequate resolution along the z-direction include the presence of sidelobes, unreasonably large slice widths, and long pulse lengths. Although long pulse lengths are manageable due to the long TE's required for fMRI, they could lead to increased scan times and increased sensitivity to off-resonance. The pulse shown in Fig. 1 was left intentionally longer than necessary, as seen by regions where the RF is zero, to leave room for shifting the RF in either direction such that enough through-plane phase correction was provided. Truncation of the ends of these pulses where the RF is zero will reduce their length by approximately a third. The effects of inadequate resolution in the in-plane direction are seen near the edges of the images due to the slow fall-off of the cylindrical Bessel functions implemented in the pulse design. In particular, the low resolution of the small disk used for the phase correction produces artifacts in both the simulation shown in Fig. 4 (d) and in the actual images shown in Fig.6 (d)–(f). The artifacts in these images result both from undesired phase being introduced to spins outside of the sinus void region and from the uneven subtraction in the magnitudes of the slice profiles. The simulation using the high resolution pulse, displayed in Fig. 4 (c), however, shows a more localized correction and a more uniform signal intensity.

Pauly et al. [24] raise the issue of off-resonance effects which are a concern due to both in-plane blurring [27] of the excited magnetization and to shifting of the slice location. Due to the short duration of the spirals, in-plane blurring is less of a concern than the shifting of the slice location. If a spatially uniform phase accumulation through the slice is assumed, then n cycles of phase accrued during the pulse will shift the slice location by an amount equal to, where N is the number of z blips and FOV is the field of view along z (reciprocal of the blip area). For the pulses presented above, the FOV along the slice-select direction was 10 cm and n was 48, giving a slice shift of approximately 2 mm for every cycle of phase accumulated. The simulation model which used the high-resolution images estimated rates of phase accrual between zero to approximately ±.3 radians per millisecond, depending on location. This suggests shifts over 60 ms on the order zero to ±6 mm, well within the 2 cm slice thickness. Shorter pulse lengths with the same number of z blips will decrease sensitivity to off-resonance, however, larger through-plane FOV's (desirable for greater sidelobe displacement) will produce an increased sensitivity.

There are numerous avenues that can be pursued to decrease pulse lengths and increase sampling efficiency. One such direction is to implement multi-shot TRF pulses. This could provide thin slices at high resolution and have a low sensitivity to off-resonance due to shorter pulse lengths. Decreased pulse lengths will also produce a more spatially uniform T2* weighting, an important issue for an accurate determination of functional activation. A 1 cm thick could have been obtained with the pulse used in this work, therefore, a 5 mm thick slice can be obtained with two shots. This is effectively a factor of four if the increase due to TRF pulse length and the spatial saturation train are included, giving sixteen 5 mm thick slices every 3.5 seconds. This is an adequate temporal resolution and coverage for a large number of fMRI studies. Even a factor of four is still advantageous compared to gradient compensation techniques where we find (not shown) approximately sixteen steps are needed at 3 T. Although longer TR's will require flip angles on the order of 90°, numerical simulations show no noticeable difference between the magnetization profiles at a 90° flip angle and those shown in Figs. 2 and 3. Motion, however, will also be more of a concern with multi-shot methods. Another means to increase sampling efficiency could involve variable sample density methods [28]. By increasing the density of the low spatial frequency points it may be possible to reduce the intensity of the sidelobe structures. Finally, actual phase maps of susceptibility variations measured from the subject's brain in-vivo will improve this method. Phase maps would truly set the sampling requirements for the TRF pulses, including the necessary resolution in all dimensions, the pulse lengths, and the number of shots. The choice of mathematical functions with sharp boundaries may have actually been too restrictive compared to the true anatomically required phases, which have smoother and more amorphous transitions. Furthermore, measured phase maps will also naturally provide quadratic and higher order corrections along each of the x, y, and z axes. A true 3D correction of phase variations with arbitrary functionality should prove to be more effective than only a linear correction along z.

CONCLUSIONS

This work shows that it is possible to design 3D TRF pulses with the low tip angle approximation which can be used to eliminate intravoxel dephasing from susceptibility variations. The use of the low tip angle approximation is practical for fMRI due to its reasonable behavior for flip angles up to 90°. Current susceptibility artifact mitigation methods, which use either gradients or RF phases to refocus spins, apply only a 1D correction which introduces a global phase to the images. This global phase refocuses part of the artifact affected regions but defocuses other regions. The 3D nature of this TRF method, however, locally refocuses the spins in the artifact affected regions and leaves the phase of the rest of the image unaltered. This is advantageous because it allows for the possibilities of both single-shot correction and the utilization of anatomically accurate maps of the susceptibility related phase variations. Tailored RF methods which implement anatomical maps measured in-vivo will have the potential for the correction of both linear and higher order susceptibility induced phase variations, along all three spatial dimensions. Although the current 3D TRF technique presented here is limited by sampling requirements and hardware restrictions, with more efficient sampling schemes and multi-shot approaches 3D TRF may prove to be useful for reducing susceptibility artifacts in T2* weighted fMRI.