Provided that the apodization function (Apo(t)) is described explicitly, time-dependent expression for a tailored sinc pulse is given by where t₀ is the time of the first zero-crossing and is one-half the width of the central lobe in the sinc function. The apodization function typically tapers the sinc pulse at its edges in order to minimize the truncation artifacts. Common apodization functions including the Hanning and Hamming windows are based on the cosine function, which is (3), where N is the total number of zero-crossings in the sinc pulse and α is the parameter used to determine the type of apodization. The Hanning and Hamming windows are generated by setting α = 0.5 and 0.46, respectively. When α = 0, Apo(t) = 1 and Eq. [1] then reduces to the original sinc function with no apodization.

While common tailored sinc pulses employ a cosine function as a basic constituent (Eq. [2]), the proposed new apodization function is given by combining two cosine-damped functions, which are represented inside the parentheses in the following:

Here η and β were empirically determined to be 0.2 and 0.3 while optimizing the apodizing performance of Eq. [3] in terms of transition sharpness, side lobes, and bandwidth. It is interesting to note that common apodization functions in Eq. [2] are basically bell shaped (Fig. 1a), whereas the new apodization function has quite a different shape from them as shown in Figure 1b, having two sharp bipolar peaks the positions of which correspond to the two first zero-crossings of the sinc pulse to be apodized (Fig. 2a). This new type of weighting form is mainly attributed to the first cosine-damped function inside the brackets in Eq. [3]. Since β does not exist in the denominator, the first cosine-damped function behaves differently with respect to t from the second cosine-damped function outside the brackets, which is bell-shaped like the conventional apodization functions, and makes the peculiar shape shown in Figure 1b especially having two sharp bipolar peaks at the positions of the two first zero-crossings of the sinc pulse to be apodized (Fig. 2a). As a result of this special weighting, the first zero-crossings of the newly apodized sinc pulse are slightly shifted outward from the central peak of the sinc pulse (Fig. 2b), thereby producing a narrower bandwidth and a sharper transition of the excited-band profile than the most common Hanning-windowed sinc pulse (Fig. 3a, b). These appealing features of the proposed sinc pulse might be expected from the fact that the bandwidth of a sinc pulse is inversely proportional to the width of the first zero-crossing through Fourier transform to a good approximation in the low-flip angle regimes.

Recently, as one of the attractive fast imaging methods, simultaneous multi-slice imaging techniques have been proposed using a multiband pulse for simultaneous excitation of multiple slices (456). There are several ways of designing a multiband pulse including a Sinnar-Le-Roux (SLR) algorithm (7). Among them, the approaches that provide an analytical expression for pulse generation are easy to implement and flexible in changing the pulse parameters such as phase, bandwidth, and the number of bands (8910). They simply write a multiband pulse as a sum of single-band pulses with the phase of each pulse shifted in a scheduled way. For example, when based on a sinc pulse, a multiband pulse can be expressed as where the MB in the subscript was used as an abbreviation of multiband and φ_n is the phase of the n^th sinc pulse. According to Eq. [4], φ_n determines the location of the n^th excitation band in the frequency domain due to the linear and frequency-shift properties of Fourier transform, whereas the sinc function determines the shape of the slice (or slab) profile. To minimize the truncation effects on multi-slice profiles, therefore, the sinc function in Eq. [4] needs to be tailored with proper apodization. In this case, the newly apodized sinc pulse proposed in Eqs. [1] and [3] can be utilized and Eq. [4] is then rewritten for the newly tailored multiband pulse as

Simulation was carried out for testing the performance of the proposed sinc pulse with new apodization and the four-band multiband pulse designed with it. Excitation profiles were obtained by Bloch simulation in MATLAB (Mathworks, R2011a, Natick, MA, USA) using three sinc pulses with no apodization, Hanning apodization, and new apodization, respectively, and corresponding three four-band multiband pulses. A three-lobe sinc pulse with four zero-crossings was used with 4-ms pulse length and thus 1-kHz pulse bandwidth.

For the simulation using the single-band tailored sinc pulses for spin excitation, the entire frequency range of all the assumed spin isochromats was set to 3 kHz so that it can fully cover the excitation bandwidth which approximately corresponds to the pulse bandwidth. The flip angle for spin excitation was assumed to be 90°. The full-width-half-maximum (FWHM) of the excitation profiles was measured to see whether the apodization-induced broadening of the excitation profile affects the excitation bandwidth. The area of the excitation profile outside the FWHM was also calculated by numerical integration to evaluate the effects of side lobes and broadening of the excitation profile simultaneously, which can be a good indicator of how much unwanted signal existing outside the excitation bandwidth smears into a slice through slice selection because the FWHM is used to determine the magnitude of the slice-selective gradient in practice. Lastly, the transition sharpness of the excitation profile was measured by calculating the width between the frequencies at 5% and 95% of the maximum amplitude of the excitation profile.

In the case of using the multiband pulses for spin excitation, each excitation bandwidth and the separation width between two consecutive excitation bands were set to be same as 1 kHz. The whole frequency range across all the assumed spin isochromats was assumed to be 10 kHz. The pulse length was set to 4 ms as was in the simulation using the single-band sinc pulse.

To confirm the Bloch simulation results of the proposed sinc pulse and the multiband pulse designed with it, experiments were performed using a spherical water phantom on a 3T whole-body scanner manufactured by ISOL technology of Korea.

First, the newly apodized sinc pulse was used for slab selection in conventional 3D gradient-echo imaging. For comparison, the Hanning-windowed and unapodized sinc pulses were also tested with the same experimental setup. Scan parameters were: TE/TR = 4.8/9.1 ms, FOV = 250 × 250 × 120 mm³, flip angle = 8°, matrix size = 512 × 256 × 120, and slab thickness = 50 mm.

Next, a four-band pulse designed with the newly apodized sinc pulse was applied for slab selection in the same 3D gradient-echo imaging sequence. Four-band pulses based on the Hanning-windowed and unapodized sinc pulses were also tested for comparison purposes. Scan parameters were: TE/TR = 20/30 ms, FOV = 256 × 256 × 32 mm³, flip angle = 14°, matrix size = 64 × 64 × 104, and slab thickness = 26 mm. While the image was highly defined along the slab-selective direction with a spatial resolution of 0.31 mm in order to clearly demonstrate the apodization effects on the excitation profile, especially on the region between two consecutive bands as well as the transition region of each band, the data in other two directions were obtained with a relatively low spatial resolution of 4 mm.

Figure 3 shows the excitation profiles produced by the single-band tailored sinc pulses (Fig. 3a, b) and the four-band multiband pulses designed with them (Fig. 3c, d) with no apodization (black), Hanning apodization (green), and the new apodization (red). As shown in Figure 3a and 3b, while the newly apodized sinc pulse delivered the excitation profile with side lobes of much less amplitude than the unapodized sinc pulse but a slightly larger amplitude than the Hanning-windowed sinc pulse, it produced sharper transition (0.36 kHz) and narrower FWHM (1.02 Hz) in the excited-band profile than the Hanning-windowed sinc pulse which yielded 0.5 kHz and 1.10 kHz for the transition width and FWHM, respectively (Table 1). The area of the excitation profiles outside the FWHM was 131.6, 127.9, and 106.3 [a.u.] for the cases with no apodization, Hanning apodization, and new apodization, respectively, implying that unwanted signals existing outside the excitation bandwidth least contributes to the inside of a slice in the case of the new tailored sinc pulse.

On the other hand, as shown in Figure 3c and 3d, the performance of a single-band pulse was well inherited to a multiband pulse designed with it. In other words, each excitation-band profile of the multiband pulse is almost as same as that of the single-band pulse used for designing the multiband pulse. Therefore, as was the case in the single-band sinc pulses, the multiband pulse designed with the new tailored sinc pulse demonstrated more desirable performance in simultaneous four-slice selection than the one generated by the Hanning-windowed sinc pulse.

The phantom experiments well confirmed the results of Bloch simulation. In Figures 4 and 5, phantom images were demonstrated for the three cases with Hanning apodization, new apodization, and no apodization, having the slab-selective orientation as a horizontal direction. To better illustrate the difference between the three cases, 1D magnitude profiles were obtained along the horizontal lines drawn with yellow color and plotted as black, green, and red lines for the three cases of no apodization, Hanning apodization, and new apodization, respectively.

As shown from the images as well as the 1D profiles in Figure 4, the new tailored sinc pulse yielded a sharper excitation profile than the Hanning-windowed one, well suppressing the side-lobe artifacts in comparison to the unapodized one. These experimental results of the single-band sinc pulses are in exact agreement with the simulation results shown in Figure 3a and 3b.

In the case of the multiband pulses, the phantom imaging results were also consistent with the simulation results. As expected from the number of bands of the multiband pulse used, four regions were excited and visualized corresponding to the four pass-bands of the multiband pulse (Fig. 5a). With of the multiband pulse made of unapodized sinc pulses, side-lobe effects as banding artifacts are clearly seen between the excited regions, whereas the side-lobe artifacts are barely recognized but at the expense of broadening of the excited regions when using the multiband pulse designed with the Hanning-windowed sinc pulse. The most desirable situation occurred in the use of the multiband pulse based on the new tailored sinc pulse, showing that the side-lobe artifacts were well suppressed with a sharper transition of each excitation band than the Hanning-window-based multiband pulse.