Velocity-selective excitation: principles and applications

Abstract

Velocity-selective (VS) excitation is a relatively new type of excitation that can be useful for generating image contrast based on spin’s motion. This review aims to explain the principles of VS excitation and their utilization for clinical applications. We first review the generalized excitation k-space formalism, which reveals a Fourier relationship between sequence parameters and excitation profiles for spins with arbitrary spatial location, off-resonance and velocity. Based on the k-space framework, we analyzed practical velocity-selective excitation pulse sequences, which yield sinusoidal or sinc-shaped velocity profiles. Then we demonstrated how these two types of velocity-selective excitation can be used as magnetization preparation for clinical applications, including saturation-based or inversion-based arterial spin labeling and black or bright blood angiography. We also discussed practical considerations and issues for each application, including the determination of design parameters and the effects of MR system errors, such as magnetic field offsets and eddy currents.

Graphical Abstract

We review the generalized excitation k-space formalism which reveals Fourier relationship between pulse sequence parameters and excitation profiles. Based on the k-space framework, we analyze practical velocity-selective excitation pulses and show how they can be used as magnetization preparation for arterial spin labeling and angiography applications. We also discuss practical considerations and issues including the determination of design parameters and the effects of MR system errors.

1. Introduction

Excitation of spin magnetization is an important step in MRI for creating an electromagnetic signal during data acquisition (readout excitation) or for preparatory manipulation of magnetization before the acquisition (magnetization preparation). While spatially selective excitation is in dominant use for signal readout, more diverse excitation techniques are used for magnetization preparation, including spatially selective, non-selective, and frequency-selective excitation.

Velocity-selective (VS) excitation is another type of excitation useful for generating image contrast based on spin’s motion. About two decades ago, several VS excitation pulse sequences were developed to distinguish between flowing blood and stationary tissues for angiographic applications.^1–5 While these methods successfully generated sinusoidal velocity profiles using a sequence of 90° tip-down, a bipolar gradient, and a 90° tip-up (denoted by 90°_x−bipolar−90°_−x) or its variants, only smooth transition between velocity pass- and stopbands handicapped a clear separation between arterial blood and background tissues. There were attempts to improve VS excitation toward an ideal excitation profile with a sharp transition band through a binomial or numerically determined expansion of the 90°_x−bipolar−90°_−x sequence.^6,7 However, these approaches lacked an explicit relationship between the desired excitation profile and pulse sequence parameters, which hinders the intuitive design of VS excitation pulse sequences.

The design of VS excitation pulse sequences can be systematized through a generalized excitation k-space formalism which reveals the Fourier transform relationship between pulse sequence parameters and excitation profiles for spins with arbitrary location, off-resonant frequency and velocity.^8,9 This explicit Fourier relationship has since enabled the analytic design of new VS pulse sequences for specific target VS excitation profiles and intuitive interpretation of earlier versions of VS pulse sequences. In this article, we first review the generalized excitation k-space formalism including the effect of refocusing pulses that are often used to cancel off-resonance effect. We utilize the k-space formalism to analyze VS pulse sequences that generate sinusoidal or sinc-shaped velocity profiles, which have been successfully used for magnetization preparation for distinguishing flowing blood from stationary tissues. Then we describe how these two types of VS magnetization preparations can be used for arterial spin labeling (ASL) and angiography, which appear to be promising clinical applications. We also discuss practical considerations and issues for each application including the determination of design parameters and the effects of MR system errors, such as magnetic field offsets and eddy currents.

2. Generalized excitation k-space formalism

The transverse component of spin magnetization resulting from a set of B₁ field and gradient field waveforms can be written as a function of spin’s location, velocity and off-resonant frequency as follows (see the Appendix for the derivation):

$$\begin{array}{l}{M}_{xy}\left(x,v,f\right)=j\gamma M_0\underset{K}{\int }p(k)e^{-j2\pi k\cdot x}dk\\ \text{where }x={\left[xyz{v}_x{v}_y{v}_z\text{f}\right]}^T,\\ k(t)={\left[\begin{array}{ccc}{k}_x(t)& k_y(t)& k_y(t)\end{array}{k}_{v_x}(t)k_{v_y}(t)k_{v_z}(t)k_f(t)\right]}^T,\\ k_{x,y,z}(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_{x,y,z}(s)ds,k_{v_{x,y,z}}(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_{x,y,z}(s)(s-T)ds\text{, }{k}_f(t)=T-t,\\ p(k)={\int }_0^T{B}_1(t)\delta (k(t)-k)dt\end{array}$$ (1)

That is, the transverse magnetization resulting from an excitation pulse sequence is the Fourier transform of the B₁ field deposited along a trajectory in multi-dimensional k-space where the trajectory is determined by the applied gradient waveforms as functions of time. It should be noted that k_x,y,z is proportional to the area of the gradient waveform; $k_{v_{x,y,z}}$ is proportional to the first-moment of the gradient waveform; and k_f is proportional to time.

2.1. Extension of k-space analysis with refocusing RF pulses

Excitation pulse sequences often include 180° refocusing pulses to reduce off-resonance effects especially when used for magnetization preparation. Here we present a new approach to interpreting the effects of refocusing RF pulses within the excitation k-space framework which originally allows for excitation RF pulses only.¹⁰ We will use the term RF subpulses to refer to the non-refocusing RF pulses (i.e., for excitation) such that they can be distinguished from 180° refocusing pulses. Suppose that an RF waveform B₁(t) with a total duration of T contains one refocusing RF pulse at time T₀ and multiple RF subpulses before and after the refocusing pulse. Immediately before the refocusing pulse (at time T₀⁻), M_xy can be written using Eq. 1:

$$M_{xy}\left(T_0^{-}\right)=j\gamma M_0{\int }_K^{}{p}_0\left(k\right)e^{-j2\pi k\cdot x}dk,$$

where

$$p_0\left(k\right)={\int }_0^{T_0^{-}}{B}_1\left(t\right)\delta \left(k\left(t\right)-k\right)dt.$$ (2)

For refocusing rotation along the x-axis, the phase of M_xy, ϕ at time T₀⁻ is changed to −ϕ at time T₀⁺ (time immediately after the refocusing). Therefore, M_xy(T₀⁺) equals the complex conjugate of M_xy(T₀⁻).

$$\begin{array}{l}{M}_{xy}\left(T_0^{+}\right)=M_{xy}^{*}\left(T_0^{-}\right)\\ =\left(-j\right)\gamma M_0{\int }_{\text{K}}^{}{p}_0^{*}\left(k\right)e^{j2\pi k\cdot x}dk.\end{array}$$ (3)

Assuming that the RF subpulses at 0 < t < T₀ are real, p₀*(k) = p₀(k). Also by substituting k′=−k, Eq (3) can be rewritten as

$$\begin{gathered} M_{xy}\left(T_0^{+}\right)=j\gamma M_0{\int }_{-K}^{}-p_0\left(-k^{\prime }\right)e^{-j2\pi k^{\prime }\cdot x}\left(-d{k}^{\prime }\right) \\ =j\gamma M_0{\int }_K^{}-p_0\left(-k\right)e^{-j2\pi k\cdot x}dk \end{gathered}$$ (4)

This is still a form of the Fourier transform of weighted k-space where the k-space weight is now $p_0\left(-k\right)={\int }_0^{T_0^{-}}{B}_1\left(t\right)\delta \left(k\left(t\right)+k\right)dt$, which corresponds to the origin symmetry of p₀(k) in k-space. Note that −1 is multiplied to p₀(−k) in Eq. 4, which indicates the sign change of the B₁(t) deposited for 0 < t < T₀. The rest of B₁(t) (T₀ < t < T) can continue to be deposited in k-space after the origin symmetric conversion of the B₁(t) deposited for 0 < t < T₀. However, after the refocusing pulse, magnetization is near the south pole (M_z < 0), and the effect of the excitation flip angle of α on M_xy is now -sin(α) instead of sin(α). Therefore, the k-space weight for T₀ < t < T should be considered to be -B₁(t). Together with −1 multiplied to B₁(t) for 0 < t < T₀, this leads to multiplication of −1 to the entire B₁(t) (0 < t < T) in the k-space analysis. Eventually, M_xy at the end of excitation pulse sequence can be rewritten as:

$$M_{xy}\left(T\right)=-j\gamma M_0{\int }_K^{}p\left(k\right)e^{-j2\pi k\cdot x}dk,$$

where

$$p\left(k\right)={\int }_0^T{\widehat{B}}_1\left(t\right)\delta \left(k\left(t\right)-k\right)dt,$$

${\widehat{B}}_1\left(t\right)=B_1\left(t\right)$ without refocusing pulse,

$$k\left(t\right)={\left[k_x\left(t\right)\hspace{1em}{k}_y\left(t\right)\hspace{1em}{k}_z\left(t\right)\hspace{1em}{k}_{v_x}\left(t\right)\hspace{1em}{k}_{v_x}\left(t\right)\hspace{1em}{k}_{v_x}\left(t\right)\hspace{1em}{k}_f\left(t\right)\right]}^T,$$

$$k_{x,y,z}\left(t\right)=\frac{\gamma }{2\pi }{\int }_t^T{\widehat{G}}_{x,y,z}\left(s\right)ds,$$

$$k_{v_{x,y,z}}\left(t\right)=\frac{\gamma }{2\pi }{\int }_t^T{\widehat{G}}_{x,y,z}\left(s\right)\left(s-T\right)ds,$$

$${\widehat{G}}_{x,y,z}\left(t\right)=\{\begin{array}{c}-G_{x,y,z}\left(t\right) t\le T_0\\ G_{x,y,z}\left(t\right) t>T_0\end{array},$$

$$k_f\left(t\right)=-\left|t-T_0\right|+T-T_0.$$ (5)

For excitation pulse sequences with multiple refocusing pulses, the origin symmetric conversion should be performed whenever refocusing occurs. M_xy of RF pulse with n refocusing pulses will include a multiplier of (−1)ⁿ at the end of excitation. This sign change, however, may be ignored because the sign of M_xy is usually not included in design parameters. We find that this term is also multiplied to M_z, which is described in the next section.

When a refocusing pulse is applied along the y-axis, the phase of M_xy, ϕ at time T₀⁻ is changed to π-ϕ at time T₀⁺. Therefore,

$$\begin{array}{l}{M}_{xy}\left(T_0^{+}\right)=e^{j\pi }{M}_{xy}^{*}\left(T_0^{-}\right)\\ =-M_{xy}^{*}\left(T_0^{-}\right).\end{array}$$ (6)

Note that there is an additional term of −1 in Eq. 6 compared to Eq. 3. This is different from the sign change described above. The resultant M_xy profile will still include the aforementioned multiplier of (−1)ⁿ, which again can be ignored, but the additional term of −1 in Eq. 6 imposes another sign change only on the pre-deposited B₁(t) at the time of refocusing pulse. This sign change cannot be neglected because it reverses the sign of only parts of the RF subpulses (not the entire RF subpulses) and substantially affects the shape of M_xy profile. In general, if n refocusing pulses along the y-axis occur at time t₁, t₂, … t_n, the sign of B₁(t) deposited in k-space will be reversed for t_n > t ≥ t_n-1, t_n-2 > t ≥ t_n-3, t_n-4 > t ≥ t_n-5, and so on.

2.2. Practical analysis method for VS pulses with refocusing

The extended k-space analysis described in Section 2.1 can be used for a more convenient analysis or design of VS excitation pulse sequences with refocusing pulses. The idea is to find the equivalent pulse sequence that yields the same excitation profile without refocusing pulses for on-resonance spins. Assuming that refocusing pulses are correctly timed to achieve spin-echo (e.g., at t = ½TE for one refocusing pulse, or at t = ¼TE and ¾TE for two refocusing pulses), k_f can be ignored in k-space analysis because RF subpulses will be deposited along the k_f-axis anyway. This approach includes two parts, each of which is applied to gradient pulses and RF subpulses, respectively. First, in order to compensate for the origin symmetric conversion, the parts of gradient pulses that precede an odd number of refocusing pulses until the end of the excitation pulse sequence, should have their signs inverted. Note that the parts of gradient pulses that precede an even number of refocusing pulses will experience the same even number of the origin symmetric conversion and should remain unchanged. This gradient modification applies to refocusing pulses along both x- and y-axes. Second, the parts of RF pulses that precede an odd number of y-axis-refocusing pulses should have their signs inverted to reflect the multiplication by −1 demonstrated in Eq. 6. This approach may also be used to design new VS excitation pulse sequences because sequence design is relatively straightforward for on-resonance spins. More examples of k-space analysis with refocusing pulses follow in the next section.

3. K-space analysis of VS excitation

A baseline design of a VS excitation pulse sequence is in a form of multiple RF subpulses separated by bipolar gradients which vary k_v the most time efficiently⁹ Due to relatively long pulse duration resulting from large bipolar gradients, VS excitation has been used for magnetization preparation rather than readout excitation. Although the Fourier transform relationship directly applies to the transverse magnetization M_xy, the longitudinal magnetization M_z can be easily deduced from the shape of M_xy profile using the relationship $M_z=\sqrt{M_0^2-M_{xy}^2}$ under the assumption of no T₁ and T₂ relaxations. Mangetization preparation typically involves large flip angles which violate the small-tip assumption in k-space formalism and leads to inaccurate estimation of M_xy profiles. Nevertheless, it turns out that M_z profiles are largely maintained regardless of flip angle and can still be estimated using the k-space formalism. This will be verified by two types of practical VS excitation pulses, described in the following sub-sections.

3.1. Sinusoidal excitation profile

3.1.1. Baseline design

The simplest case of VS excitation is two RF subpulses separated by a bipolar gradient as shown in Fig. 1. When the total flip angle (i.e. 2α in Fig. 1) is small, a sinusoidal velocity profile is obtained based on the Fourier transform of two delta functions deposited in k-space (illustrated in Fig. 1B). Since the net change of k_z made by the bipolar gradient is zero, the two RF pulses are deposited only when k_z = 0, which ensures no spatial selectivity. When viewed from the perspective of $k_{v_z}$ without considering k_f encoding, the Fourier transform pair determining the excitation profile is (ignoring a scaling factor):

$$-\delta \left(k_{v_z}\right)+\delta \left(k_{v_z}-\Delta k_{v_z}\right)\iff \text{sin}\left(\pi \Delta k_{v_z}{v}_z\right)e^{-i\pi \Delta k_{v_z}{v}_z}$$ (7)

where $\Delta k_{v_z}$ is the $k_{v_z}$ change generated by the bipolar gradient and can be calculated using Eq. 1. However, as α increases, the small-tip assumption is violated and the M_xy profile deviates from the profile derived based on the k-space formalism (Fig. 1C). When α reaches 90° (i.e., the total flip angle = 180°), the period of |M_xy| profile becomes halved compared to the |M_xy| profiles with small flip angles. This is evidence of a breakdown of the Fourier relationship between the RF deposit in k-space and M_xy for a larege flip angle. In contrast, the overall shape of M_z is maintained regardless of the flip angle, with changes in the amplitude of M_z. When VS excitation is used for magnetization preparation where only M_z is the focus of pulse design, the k-space formalism based on Fourier transform can still be used to design pulse sequences for large flip-angle excitation such as inversion.

Figure 1.

VS excitation using two hard RF subpulses separated by a bipolar gradient for sinusoidal velocity profile. A: Pulse sequence diagram of the RF and G_z gradient. Spacing between the gradient and RF for reducing eddy current artifacts is not included for simplicity. B: K-space trajectory and RF deposit. The blue line shows the overall trajectory and red dots represent the RF deposits. C: Velocity profiles acquired using Bloch simulation with α = 20°, 45°, and 90°. D: Simulated M_z profile in the v_z-f plane with α = 90°.

The use of a bipolar gradient also creates off-resonance encoding in addition to velocity encoding due to the change of k_f = T-t during the bipolar gradient. Let T_bp denote the time duration of the bipolar gradient, which corresponds to the k_f increment. Then the two impulses in Eq. 7 will be tilted by the angle $\theta \triangleq ta{n}^{-1}\left(\frac{T_{bp}}{\Delta k_{v_z}}\right)$ with respect to the $k_{v_z}$ -axis (Fig. 1B). Accordingly, the M_z profile will also be tilted by the same angle in the v_z-f space; that is, the M_z profile is shifted in proportion to off-resonance (Fig. 1D).

3.1.2. Refocused designs

To overcome the off-resonance sensitivity, one can use two unipolar gradients separated by a refocusing RF pulse instead of a bipolar gradient (Fig. 2A). Based on the description in Section 2.2, its equivalent refocusing-free pulse sequence for on-resonance spins can be found in Fig. 2B. Figure 2C illustrates the evolution of the k-space trajectory of the original pulse sequence at intermediate times simulated based on the extended k-space analysis. As described in Section 2.1, origin symmetric conversion occurs when refocusing pulse is applied and the sign of the first RF subpulse of α is inverted with the conversion because the refocusing pulse is along the y-axis. Note that the final configuration of the RF deposit (rightmost in Fig. 2C) matches the deposit that would be obtained from Fig. 2B ignoring k_f encoding. M_xy profile can be estimated from the following Fourier transform pair (ignoring a scaling factor):

$$-\delta \left(k_{v_z}\right)-\delta \left(k_{v_z}-\Delta k_{v_z}\right)\iff \text{cos}\left(\pi \Delta k_{v_z}{v}_z\right)e^{-i\pi \Delta k_{v_z}{v}_z}$$ (8)

where $\Delta k_{v_z}$ can be calculated using Eq. 5. M_xy deviates from this relationship with large flip angles, but the overall shape of M_z is maintained (Fig. 2D), which is similar to the baseline design of Fig. 1. One important difference from the baseline design is that the sign of M_z is now inverted, which is apparent in the case of α = 20° in Fig. 2D. In general, the additional sign change for M_z is determined by (−1)ⁿ (where n is the number of total refocusing pulses), which is the same factor multiplied to the M_xy profile as described in Section 2.1. Unlike M_xy, this sign change is important for M_z. Note that both pulse sequences of Figs. 1 and 2 yield the same M_z profile for on-resonance spins when α is 90°. Figure 2E demonstrates that the M_z profile is insensitive to off-resonance.

Figure 2.

VS excitation for sinusoidal profile using a refocusing pulse for reduced sensitivity to off-resonance. A: Pulse sequence diagram of the RF and G_z gradient. B: Equivalent refocusing-free pulse sequence for on-resonance spins. C: K-space trajectory and RF deposit at intermediate times. The blue line shows the overall trajectory and red dots represent the RF deposits. D: Velocity profiles from Bloch simulation using α = 20°, 45°, and 90°. E: Simulated M_z profile in the v_z-f plane with α = 90°.

The single-refocused design described above is still susceptible to other MR hardware imperfections such as B₁ inhomogeneity and eddy current. The double-refocused design¹¹ shown in Fig. 3A is very effective in reducing artifacts induced by these errors, and is currently used in many applications of diffusion-weighted imaging and ASL. Figures 3B and 3C show its equivalent refocusing-free pulse sequence and the resultant k-space deposit at the end of the pulse sequence, respectively. Unlike the single-refocused design, two refocusing pulses, in this case, do not lead to flipping of the M_z profile (n = 2). Although the simulated M_z profile in the v_z-f plane with α = 90° will be the same as the single-refocused case, the benefit of the double-refocused design includes a reduced sensitivity to B₁ variation due to an even number of refocusing pulses and reduced eddy current artifacts due to alternating gradient polarity. Of note, sequences with α = 20° and 45° are illustrated in Figs. 1C, 2D, and 3D to demonstrate the gradual deviation from the Fourier transform relationship in M_xy as the flip angle increases; however, α = 90° is used for most of sinusoidal VS excitation in practice.

Figure 3.

VS excitation that achieves a sinusoidal velocity profile similar to Fig. 3 but uses double-refocused design for improved immunity to B₁ inhomogeneity and eddy current. A: Pulse sequence diagram of RF and G_z gradient. B: Equivalent refocusing-free pulse sequence for on-resonance spins. C: K-space trajectory and RF deposit at the end of the RF pulse. The blue line shows the overall trajectory and red dots represent the RF deposits. D: Velocity profiles from Bloch simulation using α = 20°, 45°, and 90°. Note that the rotation axis of refocusing pulses can be in any direction to achieve the same excitation profile as long as it is consistent for the two refocusing pulses.

3.2. Arbitrary excitation profile

The shape of VS excitation profile can be further generalized beyond sinusoids by repeating the basis velocity encoding step in a form of α_i−bipolar−α_i+1. In this case, the RF envelope function of α_i should be designed to be the inverse Fourier transform of a target excitation profile over velocity. In this section, we demonstrate a sinc-shaped excitation profile over velocity using a rectangular function for the RF envelope along the $k_{v_z}$ -axis. This pulse sequence excites stationary tissues (around zero velocity) and suppress their M_z while conserving the M_z of the moving spins.

3.2.1. Baseline design

The example pulse sequence consists of five hard RF subpulses interleaved with four identical bipolar gradients on the z-axis (Fig. 4A). With each RF subpulse contributing to 18° excitation, the entire sequence has a flip angle of 90°. With $\Delta k_{v_z}$ and T_bp denoting the $k_{v_z}$ increment and the time duration of each bipolar gradient respectively, the five RF subpulses are deposited in a discrete fashion along a line tilted by tan^{−1 $\left(T_{bp}/\Delta k_{v_z}\right)$} with respect to the k_vz-axis in the k_vz-k_f plane (Fig. 4B). That is, the resultant k-space deposit is a tilted and sampled rectangular function. The Fourier transform of this function is a titled (by the same angle) and replicated sinc function and matches the simulated M_xy in the v_z-f plane (Fig. 4C top). The sinc-shaped M_xy profile is linearly dephased along both v_z- and f-axes since the center of the rectangular RF deposit is offset from the k-space origin; however, this dephasing does not affect the M_z profile that is important in magnetization preparation. The corresponding M_z profile should have passband and stopband swapped compared to the M_xy profile and should therefore be notch-shaped, which matches the simulated M_z in the v_z-f plane (Fig. 4C bottom). Fig. 4D shows the 1D profiles of M_xy and M_z over velocity for on-resonance spins.

Figure 4.

VS excitation using five RF subpulses to obtain a sinc-shaped M_xy and notch-shaped M_z profiles. A: Diagram of RF and gradient waveforms. B: The five hard RF subpulses are deposited in a discrete fashion along a line in the k_vz-k_f plane. C: Simulated M_xy profile in the v_z-f plane which is equivalent to the 2D Fourier transform of the RF deposit in the k_vz-k_f space shown in B, and corresponding M_z profile in the v_z-f plane. D: On-resonance velocity profiles of M_xy and M_z.

3.2.2. Refocused designs

Similar to sinusoidal VS excitation, refocusing pulses can be inserted to mitigate the off-resonance effects in arbitrary VS excitation.¹² Figure 5A shows a single-refocused version of the excitation pulse sequence that achieves the same velocity selectivity as the one shown in Fig. 4 for on-resonance spins. According to the practical rules described in Section 2.2, the equivalent refocusing-free pulse sequence can be obtained by inverting the phase of RF subpulses denoted by yellow circles and the unipolar gradient pulses denoted by green circles. This equivalent pulse sequence will be the same as Fig. 4A. Figure 5B shows the resultant M_z profile in the v_z-f plane, which verifies the off-resonance insensitivity of the refocused design. Figure 5C shows a double-refocused design (using two refocusing pulses between consecutive RF subpulses) which achieves the same velocity selectivity as in Fig. 4 with less sensitivity to B₁ inhomogeneity than the single-refocused design. Figure 5D shows the equivalent refocusing-free sequence for on-resonance spins, obtained by applying the practical rule explained in Section 2.2. This sequence appears to be virtually the same as the baseline design shown in Fig. 4A once the two consecutive (positive or negative) unipolar gradient pulses are combined into one.

Figure 5.

VS preparation with the same on-resonance velocity profile as Fig. 4 but with off-resonance sensitivity removed using refocusing pulses. A: Pulse sequence diagram of single-refocused VS preparation. B: Simulated M_z response in the v-f plane showing a replicated notch-shaped profile insensitive to off-resonance. C: Pulse sequence diagram of double-refocused VS preparation. D: Equivalent refocusing-free pulse sequence on resonance appears similar to Fig. 4A once the two consecutive unipolar gradient pulses are combined into a single one.

Recall that an additional sign change of M_z profile is determined by the factor (−1)ⁿ (n is the number of total refocusing pulses). To prevent undesired flipping of target M_z profiles, the total number of refocusing pulses should be even, which is satisfied by both single- and double-refocused designs in Fig. 5 using four and eight refocusing pulses, respectively. The double-refocusing strategy ensures an even number of total refocusing pulses regardless of the number of RF subpulses.

4. Applications

VS excitation has only relatively recently been applied in practical applications compared to spatially selective or spectrally selective excitation. Because of the long pulse duration, VS excitation has proven useful largely as magnetization preparation for distinguishing moving blood from stationary tissues. Here we present two such applications of VS magnetization preparation, ASL and angiography. Both sinusoidal and sinc-shaped velocity profile excitations have been utilized in each of the two categories.

4.1. Arterial spin labeling

Velocity-selective ASL (VSASL) was introduced to address the errors of conventional brain ASL in the presence of slow or delayed blood flow, which leads to long arterial transit delay, hence either underestimation of cerebral blood flow (CBF) or reduced signal-to-noise ratio (SNR).¹³ In this technique, blood inflow is labeled based on its velocity regardless of spatial location, eliminating arterial transit delay in principle. VSASL has the potential for more accurate measurements of CBF than conventional ASL methods in patient cohorts with vascular disease such as Moyamoya (Fig. 6).^14,15 Recently, VSASL has also been used in body applications such as the heart^16,17, kidney^18,19, placenta^20–23, breast²⁴, and lung²⁵ for more efficient arterial labeling for these organs with complex vessel geometry or limited water exchange (Fig. 7A–D). Besides perfusion measurements, VSASL has been utilized to assess oxygen extraction fraction^26,27, blood volume²⁸, and functional activation^29,30 in the brain (Fig. 7E–F). There are two different approaches to VS labeling: saturation- and inversion-based labeling.

Figure 6.

Digital subtraction angiogram (DSA) (A) and perfusion images of conventional pulsed ASL (B) and VSASL (C) acquired in a patient with moyamoya disease. DSA indicates delayed anterograde filling in the left hemisphere through a proximal M1 middle cerebral artery (MCA) stenosis (red arrowhead) at a post-injection time of 2 s. While conventional ASL imaging with an inflow time of 2 s shows underestimated perfusion in the left MCA territory, VSASL imaging demonstrates intact perfusion in the same region. Reprinted from Ref.¹⁵.

Figure 7.

VSASL images of the heart (A), kidney (B), placenta (C), lung (D), and breast (E). Mapping of oxygen extraction fraction (F) and cerebral blood volume (G) based on VSASL. A: Left ventricular myocardium was divided into six segments per the American Heart Association’s 17-segment model.⁸⁸ Myocardial perfusion measured in this healthy volunteer was comparable to that measured using pulsed ASL. B: Renal VSASL demonstrated clear contrast between the highly perfused cortex and low perfused medulla in a healthy volunteer. C: Placental VSASL was performed in a pregnant woman complicated by fetal heart disease at gestational age of 29 weeks, showing higher placental perfusion than in healthy pregnancies. The placenta is delineated by the red dotted line. D: The lung has very limited water exchange between the intra- and extravascular space, which makes ASL quantification inaccurate using conventional methods. VSASL demonstrated more homogeneous and reasonable perfusion signals afforded by spatially uniform arterial transit delay. E: VSASL demonstrated an increased signal in an invasive carcinoma lesion in the right breast. F: The oxygen extraction fraction was relatively uniform across the entire brain in a healthy subject. G: Measured cerebral blood volume values were comparable to those reported in the literature using other imaging modalities. Reprinted from Refs.^{16,18,20,24–26,28}.

4.1.1. Saturation labeling

VS magnetization preparation in most of VSASL studies is based on sinusoidal excitation pulses, similar to the pulse sequence of Fig. 3A with α = 90°.^{13–16,18,20–27,30–36} The resultant M_z profile follows a cosine function of velocity (see Fig. 3D, rightmost column). With the assumption of laminar flow distributions within a voxel coupled with a long delay time until data acquisition, this profile becomes a sinc function, effectively serving as a low-pass filter of velocity (Fig. 8A).¹³ In practice, to reduce sensitivity to B₀ and B₁ inhomogeneities and eddy current, either all or part of RF pulses can be implemented using adiabatic pulses and the number of refocusing pulses can be further increased.^13,31,33 Ultimately, these variations share a similar baseline design because they all deposit two ±90° RFs along the k_v-axis in the k_f-k_v plane. For example, in VSASL with double-refocused hyperbolic secant (DRHS) design¹³, RF weights of {90°, −90°} are deposited in k-space, whereas in VSASL with B₁-independent rotation (BIR)-8^31,33, RF weights of {90°, 90°} are deposited (Fig. 9). M_z profiles of these methods can be inferred from their M_xy profiles, which are the inverse Fourier transform of RF deposit in k-space. After M_z profile flipping in BIR-8-based VSASL due to the odd number (three) of refocusing pulses, it can be seen that both methods have the same M_z velocity profile.

Figure 8.

Spin-wise (black) versus voxel-wise (red) M_z response over velocity generated by the two types of VS preparation in Section 3. With larminar flow distributions within a voxel as typically assumed in ASL applications, the cosinusoidal spin-wise profile (black in A) changes to a sinc-shaped voxel-wise function (red in A) and the notch-shaped spin-wise profile (black in B) has a reduced amplitude for moving spins in the voxel-wise response (red in B). When convolved with a kernel 5 cm/s wide, which would be more appropriate for angiography applications, the two spin-wise profiles changes to a sinc-like profile (red in C) and blurred notch profile (red in D), respectively.

Figure 9.

Labeling pulse sequences of VSASL based on DRHS (top) and BIR-8 (bottom) designs. Actual pulses (left) can be shown as a simplified diagram (middle) and further converted to the equivalent refocusing-free pulse for on-resonance spins (right) to provide intuition about the velocity profiles. In DRHS, RF weights of {90°, −90°} are deposited in k-space whereas in BIR-8, RF weights of {90°, 90°} are deposited. After M_z profile flipping in BIR-8 design due to the odd number (three) of refocusing pulses, both designs have M_z profiles of the same shape.

4.1.2. Inversion labeling

Recently, VS magnetization preparation with more than two RF subpulses has been demonstrated in VSASL to directly achieve notch-shaped M_z velocity profiles, as shown in Section 3.2 (Fig. 10).^{17,28,29,37–41} Compared to previous VSASL studies that achieve saturation-based labeling, this approach can attain inversion labeling by increasing total flip angle to 180° in VS excitation, which can improve the SNR of ASL scans. With laminar flow modeling, however, the notch-shaped M_z velocity profile of this method shows somewhat reduced M_z amplitude for moving blood flow, preventing a doubled SNR compared to saturation-based labeling (Fig. 8B). Also, it would be difficult to deploy adiabatic RF pulses for this design, as they would incur prohibitively long VS pulse duration with the prolonged number of RF subpulses. Instead, an increased number of RF refocusing pulses allows for the use of Malcolm-Levitt (MLEV) phase cycling⁴² to increase B₀ and B₁ immunity and reduce eddy current artifacts.^28,29,37,38 Therefore, in addition to SNR differences, saturation- and inversion-based schemes of VSASL may show different sensitivities to magnet imperfections. These aspects should be evaluated carefully before choosing the VS labeling method for each application.^19,43

Figure 10.

CBF measured in a healthy volunteer using PCASL (pseudocontinuous ASL; current standard ASL method for CBF measurement) (top), saturation-based VSASL (DRHS) (middle), and inversion-based VSASL (bottom). All three methods demonstrated comparable perfusion contrast and quantified CBF levels. Reprinted from Ref.³⁷.

4.1.3. Further application of k-space formalism

Excitation k-space formalism can be further exploited in RF pulse design or interpretation beyond basic profile designs. For example, a recent study demonstrated velocity selectivity combined with spatial selectivity in ASL by changing nonselective hard RF subpulses of 90° and −90° in DRHS sequences to spatially-selective RF pulses.⁴⁴ This may be desirable for ASL applications where VS labeling must be limited to particular regions. The excitation profile of this RF pulse sequence can be viewed using multidimensional k-space analysis (k_v, k_f, and k_z), which provides intuition for a more comprehensive velocity profile along the axes of velocity, off-resonance, and spatial location. Another example is a modification to invert the velocity profile. It has been shown that additional phase of π can be included in the second 90° RF subpulse of sinusoidal excitation to tip down rather than tip up magnetization of static tissues.³² This serves as VS labeling combined with an inversion pulse, leading to a reduced number of inversion pulses in ASL background suppression and therefore a reduced specific absorption rate. This inverted velocity profile can be easily found using k-space analysis as RF deposit changes from {90°, 90°} to {90° −90°}. In inversion labeling, there are many more ways to utilize the k-space formalism to accomplish specific design parameters. In the case of a rectangular velocity profile, the envelope of RF subpulses can be modulated to apply windowing or achieve a specific time-bandwidth product of the velocity profile.¹⁷ RF design methods can even be extended to Shinnar-Le Roux algorithm with various filter designs, such as minimum-phase RF.¹⁷ Also, based on the Fourier transform shifting property, the stopband of the velocity profile can be moved to a particular target velocity range, which may vary depending on applications. For example, in a myocardial VSASL study¹⁷, shifted inversion excitation was used to invert magnetization of blood in the coronary arteries, rather than static myocardial tissues. The RF phase change for desired velocity shifting can be calculated based on the following equation:

$$\begin{gathered} \text{For }{M}_{xy}\left(r,v-v_0\right)=j\gamma M_0{\int }_t^T{B}_1(t)e^{j2\pi k_v(t)v_0}{e}^{-j2\pi \left[k_r(t)r+k_v(t)v\right]}dt \\ {\widehat{B}}_1(t)=B_1(t)e^{j\theta (t)},\theta (t)=\gamma v_0{\int }_t^T\left(s-T\right)G(s)ds \end{gathered}$$ (9)

Lastly, excitation k-space provides information about the cutoff velocity for both saturation- and inversion-based VSASL. The cutoff velocity is the velocity above which arterial blood is labeled. It is defined in saturation-based VSASL as the first zero crossing of M_z profile,¹³ which can be explicitly determined by:

$$v_c=\frac{1}{4\Delta k_v}$$ (10)

where Δk_v is the distance between two RF deposits along the k_v-axis. Despite a lack of consensus on its definition that is valid for both saturation- and inversion-based approaches, cutoff velocity is a critical parameter in VSASL and must be determined considering blood velocity and motion in the organ of interest. Larger VS gradient pulses increase the gaps between RFs deposited on the k_vz-axis, as shown in Figs. 1 – 4 (and vice versa). Based on the Fourier transform scaling property, increasing the RF gaps on the k_vz-axis shrinks the overall profile along the v_z-axis. This lowers the cutoff velocity and increases sensitivity to blood flow at the cost of an increased diffusion effect and T₂ decay during VS labeling.

4.1.4. Practical considerations

While VS excitation pulses can be designed or interpreted using excitation k-space, there are several practical issues surrounding VSASL that require optimization. First, eddy current error has been identified as a primary issue of VSASL.^{13,31,33,38,39,43} Eddy currents are generated by the large gradients required to obtain high sensitivity to flow, deforming gradient pulse shapes slightly. This subtle change in gradients creates a nonzero spatial encoding between RF subpulses and results in a mismatch in the signal between the control and labeling imaging for the static tissues that are distant from the isocenter. A simple solution to lessen this effect is to insert a delay after each gradient segment and/or reduce the maximum gradient strength. In saturation-based VSASL, DRHS labeling pulse with alternating gradient polarity afforded by double-refocusing was shown to greatly reduce the eddy current artifacts compared to the single-refocused design with the same polarity of the two gradient segments.¹³ BIR-8-based labeling pulse with triple-refocusing and with a gradient scheme modified accordingly was found to further reduce the artifacts without prolonging the pulse duration as shown in the simplified pulse sequence diagram in Fig. 9.³¹ In inversion-based VSASL with more than two RF subpulses, additional method to reduce eddy current artifacts is MLEV phase cycling.^28,29,37,38 The benefits increase with a larger number of RF subpulses, which indicates that this scheme is well suited for inversion-based VSASL. Second, B₁/B₀ inhomogeneity is another concern for VSASL, which diminishes labeling efficiency and leads to SNR reduction.^{13,31,33,38,39,43,45} Similar to the eddy current, B₁/B₀ sensitivity can be substantially reduced by using DRHS or BIR-8 labeling methods in saturation-based VSASL because B₁ and B₀ errors are decreased by the adiabatic and refocusing pulses in these methods, respectively. As described above, adiabatic pulses are not realistic options for inversion-based VSASL because of an increased number of RF subpulses. Instead, MLEV phase-cycled composite RF for refocusing pulses can be used to improve the immunity to B₀/B₁ variations.^37–39 Figure 11 compares errors caused by the eddy current and B₁/B₀ variations for different VSASL methods estimated in Ref.⁴³. In this study, BIR-8-based saturation labeling and inversion labeling with a sinc RF envelope showed lower sensitivities to these systemic errors compared to other schemes. Lastly, the time duration of VS labeling pulses has been prolonged in the course of technical developments to reduce artifacts of the eddy current and B₁/B₀ inhomogeneity or to enhance SNR of ASL, leading to increased T₂ decay of blood during VS labeling. Unlike eddy current and B₁/B₀ inhomogeneity, relaxation-related issues may not create spatially varying errors but may yield a global SNR reduction, which is a much less severe problem.

Figure 11.

Comparison of simulated sensitivities to eddy current and B₁/B₀ inhomogeneity between different VSASL approaches. Top: Magnetization of static tissues in labeling imaging for different eddy current time constants and different distances to the isocenter. Bottom: SNR efficiency of VSASL for different B₁ and B₀ variations. BIR-8-based saturation labeling and both inversion labeling methods demonstrated substantially reduced static tissue errors from eddy current compared to the DRHS approach. BIR-8-based labeling showed the most reduced sensitivity to B₁/B₀ variation while inversion-based labeling with a sinc RF showed the highest SNR with somewhat increased sensitivity to B₁/B₀ compared to the BIR-8 method. Reprinted from Ref.⁴³.

4.2. Angiography

VS magnetization preparation has also been used to generate contrast in angiography, including both black blood vessel wall imaging and bright blood arteriography without the use of contrast agents which involve the risk of nephrogenic systemic fibrosis.^46–48 This approach separates blood and stationary tissues based on velocity rather than the spatial location of spins and therefore can mitigate the issues of poor suppression (vessel wall imaging) or depiction (arteriography) of blood flow that is slow and/or in-plane. This has been an inherent problem of conventional non-contrast-enhanced techniques, such as double-inversion-prepared imaging (vessel wall imaging), time-of-flight (carotid and cerebral angiography), and slab-selective inversion-prepared imaging (renal angiography).^49–55

4.2.1. Black blood preparation

VS magnetization preparations with cosinusoidal velocity profiles of M_z (Figs. 2 and 3 with α = 90°) have been used for black blood vessel wall imaging.^56–58 With large VS gradient pulses that ensure sufficiently high frequency of the cosinusoidal response, arterial blood with a range of velocity distribution experiences intravoxel magnetization cancellation and results in reduced signal whereas stationary tissues (e.g. vessel wall) with zero velocity yield high signal (Fig. 8C). Single- or double-refocused designs with 90°−180°−90° composite pulses have been most often used for this approach. In combination with black-blood VS preparation, fast spin-echo sequences are typically used as readouts, which further suppress the blood signal due to inherent flow-spoiling properties.^59,60 While carotid vessel wall imaging has been the main application (Fig. 12), this technique has been successfully used for other vasculatures, including intracranial arteries, thoracic aorta, heart chamber, coronary arteries, and peripheral arteries.^56,61–65

Figure 12.

Vessel wall images obtained using single-refocused VS black blood preparations with small (A; gradient first moment = 950 mTms²/m) and large (B; gradient first moment = 1545 mTms²/m) VS gradients, and using double-refocused preparations with small (C) and large (D) VS gradients. While both single- and double-refocused designs better suppress slow flow using larger VS gradients that allow higher flow sensitivity, the double-refocused design yields higher wall signals due to lower sensitivity to B₁ offset and eddy current. Reprinted from Ref.⁵⁸.

The same black blood VS preparation can also be used for bright lumen imaging (i.e. with image contrast of standard angiography).^66,67 In addition to the black blood image, one can acquire another image (often called reference image) with the same imaging parameters but with the gradient pulses in VS preparation turned off. The image obtained from this gradient-free preparation would have the same image contrast, except moving blood that appears brighter than in the black blood image. When the black blood image is subtracted from the reference image, a background-suppressed, artery-only image is obtained. Due to the high sensitivity to slow flow, this technique is appealing particularly for distal arterial territories such as the lower extremities, feet and hands (Fig. 13).^68–72 The subtractive nature offers excellent background suppression and superior artery-to-background contrast, but requires a doubled scan time and increases motion sensitivity.

Figure 13.

Comparison of VS black blood-prepared subtractive angiography (A, C) and contrast-enhanced angiography (B, D) in hands (A, B) and feet (C, D). Compared to contrast-enhanced angiography, VS subtractive angiography offers a comparable depiction of relatively large vessels and a superior depiction of distal arteries, which are challenging to visualize in conrast-enhanced angiography due to difficulty determining optimal imaging time post injection. Reprinted from Refs.^69,72.

4.2.2. Bright blood preparation

VS magnetization preparation used in inversion-based VSASL was also deployed to generate bright artery angiography.^12,73,74 Notch-shaped velocity profiles of M_z using a rectangular RF envelope were particularly effective for achieving a sharp transition between stopband and passband (shown in Figs. 4–5).⁷³ In this magnetization preparation, fast-moving blood spins are preserved while stationary tissue spins are either saturated or inverted using a flip angle of 90° or 180°, respectively. Single, double, or quadruple refocused VS preparation pulses have been investigated with 90°−180°−90° composite pulses for refocusing. In this type of VS angiography,⁷⁵ 3D encoding can be used to achieve high spatial resolution in all three dimensions as opposed to inflow-based 2D approaches. Also, positive angiographic contrast can be generated with a single acquisition, which is advantageous over subtractive approaches based on black-blood preparation^66,67 in terms of scan time and potential subtraction errors. VS-MR angiography has been shown to be feasible for various vascular beds, including the peripheral, renal, abdominal, pedal, and cerebral arteries (Fig. 14).^{12,73,76–78}

Figure 14.

Representative VS-angiography images of peripheral arteries in a 56-year-old male patient with severe bilateral claudication (A), pedal arteries in a 71-year-old female patient with dorsalis pedic aneurysm (B) and abdominal arteries in a 68-year-old male patient with stenosis on the left renal artery and occlusion on the right (unpublished data) (C). Reprinted from Refs.^76,77.

The primary design parameters of this notch-shaped magnetization preparation using a rectangular RF envelope include the extent of RF deposit and RF spacing on the k_v-axis. A larger k-space extent narrows the excitation bandwidth (i.e., the width of velocity stopband) and therefore widens the velocity passband for flowing spins. RF spacing along the k_v-axis determines the distance between the replicas of excitation on the v-axis or “velocity FOV” (e.g., 80 cm/s in Figs. 4 and 5). Smaller RF spacing increases the upper bound of the velocity passband as velocity FOV increases. For a fixed number of RF subpulses, the k-space extent is inversely proportional to RF spacing, leading to a trade-off between reducing the stopband bandwidth and increasing the velocity FOV. Another design parameter to consider is shifting the excitation profile along the velocity axis such that stopband includes slightly negative velocities as well as zero velocity. This is useful for suppressing venous blood which is typically moving slowly in the opposite direction of the arterial blood. These design parameters should be empirically determined depending on arterial territories which involve varying ranges of arterial flow velocity. For peripheral angiography, for instance, nine RF subpulses, velocity FOV ranging from 50 – 140 cm/s (from the ankle through the pelvis) and velocity shifting of ~ 1cm/s (in the superior direction) are reasonable. For more distal regions, such as pedal or intracranial arteries, 9–11 RF subpulses and velocity FOV of 30~40 cm/s would be desirable.⁷⁶

In vascular regions inferior to the heart, such as the abdominal, peripheral, and pedal arteries, appropriate combinations of velocity FOV and velocity shifting could suppress venous blood as well as stationary tissues. However, in cerebral and carotid regions, venous blood moves faster (in the negative velocity direction) than the lower bound of the velocity stopband and therefore difficult to suppress. A recent study on cerebral angiography proposed to apply slab selective inversion in the superior part of imaging FOV followed by a delay period and VS saturation preparation without velocity shifting. During the delay period, the inverted venous blood recovers only partially while fresh arterial blood unaffected by the inversion preparation flows into the imaging volume, resulting in artery-only images.⁷⁹ The combination of slab selective inversion and VS saturation was also applied to carotid angiography with improved visualization of small vessels compared to time-of-flight.⁸⁰

4.2.3. Practical considerations

The primary issues surrounding VS black-blood preparation include insufficient suppression of slow arterial flow and undesired signal loss in stationary tissues (vessel wall), which may be related. One needs to use larger VS gradient pulses to increase sensitivity to slow blood flow (similar to decreasing the cutoff velocity in saturation-based VSASL). However, larger gradient pulses increase the distortion of gradient waveforms due to eddy currents and may cause signal reduction in stationary tissues far from the isocenter. Experimental studies on this trade-off found reasonable first-moment values of VS gradients ranging from 500 – 1,500 mT·ms²/m^58,81 while the optimal value may vary depending on the range of arterial flow velocity and the degree of eddy currents. B₁ inhomogeneity is another source of signal drop in stationary tissues, as it degrades the accuracy of 180° rotation of a composite pulse. Double refocusing pulses with a 90°-shifted phase with respect to RF subpulses are significantly more robust to B₁ offset due to the complementary effects of two refocusing pulses and yield higher vessel-wall SNR than single-refocused designs.⁵⁷ The severity of the T₂ decay effect will depend on the total pulse duration, which is largely proportional to the size of the VS gradient pulse. Our numerical simulations of black blood preparation with a gradient first moment of ~1,000 mT·ms²/m and a pulse duration of 17 ms show that the M_z value of the vessel wall (assumed to have T₁ = 700 ms and T₂ = 70 ms) decreases from the ideal value of M₀ to 0.81M₀.

Bright blood VS preparation pulses also have potentially high sensitivity to MR systems errors due to the use of multiple RF pulses interleaved with multiple gradient pulses. Initial technical development studies focused on improving the immunity to B₀ and B₁ field errors. Single-refocused designs are robust to B₀ offset compared to baseline designs but suffer from the arterial signal loss in the presence of a large B₁ offset (Fig. 15).⁸² Increasing the number of refocusing pulses to two or four in each velocity encoding step significantly reduces the B₁ sensitivity, especially when combined with MLEV phase cycling (Fig. 15).^73,83 Though the velocity passband for highlighting arterial blood can be made robust through the multi-refocusing schemes, the velocity stopband for suppressing stationary tissues is still vulnerable to B₁ inhomogeneity due to the use of hard RF subpulses for excitation. A recent study resolved this issue by replacing the hard RF subpulses with numerically optimized RF pulses using the optimal control theory.^79,84 A 1.9-ms-long RF pulse generated based on target ranges of the immunity to B₀ offset of ±250 Hz and B₁ scale of ±0.3 achieved more uniform suppression of stationary tissues, improving the artery-to-background contrast. Another issue in bright blood VS preparation is that artifactual stripes may occur in the direction of the VS gradient pulses when the 180° rotation of a refocusing pulse is not perfect (Fig. 16A).⁸² Using the extended phase graph analysis with an approximation of the imperfect refocusing as a small erroneous excitation,^85,86 the stripes were shown to contain up to 2^nd order harmonics of the fundamental frequency determined by each unipolar gradient (Fig. 16B). Based on this characterization, a recent study proposed a method of alternately applying four VS preparations whose excitation profiles are spatially shifted by a quarter of the fundamental period of the stripes (Figs. 16C and 16D).⁸³

Figure 15.

At 1.5T, baseline VS preparation yields a high background signal (A) in the regions of large off-resonance (C), whereas single-refocused preparation achieves uniform background suppression (B). At 3T, single-refocused VS preparation suffers from arterial signal loss (D) in the regions of large B₁ inhomogeneity (F), which is reduced using a double-refocused VS preparation (E). Reprinted from Ref.^12,82.

Figure 16.

Maximum-intensity-projection images of VS angiography (top row) and k-space data (bottom row) obtained using double-refocused VS preparations in the thigh. As the unipolar gradient designed for a velocity FOV of 80 cm/s yielded the fundamentatl frequency k₁ of 1.9 cm⁻¹, the 2k₁ harmonics (3.8 cm⁻¹) were also included which appear as erroneous impulses in the k-space data (A and B). By alternately applying four VS preparations whose excitation profiles are shifted by a quater of the fundamental period (0.25k₁⁻¹), the stripes are significantly suppressed (C). This is demonstrated by the corresponding k-space, which barely shows the impulses (D). Reprinted from Ref.⁸³.

Refocused VS preparation with multiple RF subpulses become lengthy and prone to the effects of spin relaxation. In Ref⁸², Bloch simulations considering tissue relaxations for a double-refocused design with five RF subpulses and a total duration of ~19 ms showed that artery-to-muscle contrast is reduced by 6% compared to the ideal situation (arterial M_z = M₀ and muscle M_z = 0). When the number of RF subpulses increases to nine and the pulse duration becomes ~38 ms, our simulations show that the contrast reduction rate increases to 11% which is marginal for angiography applications. Eddy current effect is another potential concern, but turns out to be marginal, due to the use of small gradient pulses. Our simulations similar to those in Refs.^33,87 show that the maximum possible reduction of artery-to-muscle contrast by eddy currents is less than 5% within the practical ranges of eddy-current time constants (10⁻⁴-10⁰ sec) and spatial offsets from the isocenter (−30 – 30 cm).

5. Conclusion

Design and interpretation of VS excitation pulses can be facilitated by the generalized excitation k-space formalism, which shows the explicit Fourier relationship between pulse sequence parameters and target excitation functions of velocity. Sinusoidal and sinc-shaped VS excitation profiles translate to cosinusoidal and notch-shaped M_z profiles, respectively, when used for magnetization preparation, and are useful for separating blood flow from stationary tissues. The two types of VS magnetization preparation have shown great promise for saturation- or inversion-based ASL and black-blood or bright-blood angiography in various vascular territories. While significant technological advances have been made to make VS excitation robust to magnetic field offsets and eddy currents, further technical optimization and clinical validation are warranted for this relatively new technique.

Abbreviations:

ASL

arterial spin labeling

BIR

B₁-independent rotation

CBF

cerebral blood flow

DRHS

double-refocused hyperbolic secant

MLEV

Malcolm-Levitt

SNR

signal-to-noise ratio

VS

velocity-selective

Appendix

A.1. Derivation of generalized excitation k-space formalism

We derive the expression for the transverse magnetization resulting from a set of B₁ field and gradient field waveforms, as a function of spin’s location, velocity and off-resonant frequency. The key strategy is to split the B₁ waveform into incremental segments (thin gray bar in Fig. A1A) and sum the resultant M_xy created by all B₁ segments in the relative frame rotating at Larmor frequency as follows:

1. Tip-down by the B₁ segment during the infinitesimal time of dt (Fig. A1B): Using the general relationship ω = γ B (B = magnetic field, γ = the gyromagnetic ratio, and ω = resultant angular frequency), the rotation generated by the B₁ segment along the x-axis will be α = γ B₁(t) dt and result in transverse magnetization written as a complex number dM_xy (t) = jM₀ sin(α), where j indicates the direction of M_xy aligned with the y-axis and M₀ denotes the initial longitudinal magnetization at equilibrium. Since sin(α) ≈ α with small-tip approximation, dM_xy(t) becomes

   $$d{M}_{xy}(t)=j\gamma M_0{B}_1(t)dt.$$ (A1)
2. In-plane phase accrual during the time from t to T (Fig. A1C): Assuming a spin of interest is moving at a velocity v_z in the z-direction and reaches position z at the end of the excitation pulse (i.e., at time T) with a gradient field played on the z-axis only (G_z), the spin’s location at time s (t < s < T) will be written as z + v_z(s - T). Since the magnetic field strength induced by the gradient field G_z at time s will be B(s) = G_Z(s)(Z + v_z(s − T)), the in-plane phase accrued by G_z will be ${\theta }_g=\gamma {\int }_t^T{G}_z\left(s\right)\left(z+v_z\left(s-T\right)\right)ds$. The effect of off-resonant frequency f is simply additional phase determined as ${\theta }_f=2\pi {\int }_t^{T}fds=2\pi f\left(T-t\right)$. Combining the effects of the gradient field and off-resonance on phase accrual, the transverse magnetization induced by the B₁ increment at the end of the pulse will be:

   $$d{M}_{xy}(t)=j\gamma M_0{B}_1(t)e^{-j\left(\gamma {\int }_t^T{G}_z(s)\left(z+v_z(s-T)\right)ds+2\pi f(T-t)\right)}dt$$ (A2)

Taking into account the contributions from all B₁ segments,

$$\begin{gathered} M_{xy}=j\gamma M_0{\int }_0^T{B}_1(t)e^{-j\left(\gamma {\int }_t^T{G}_z(s)\left(z+v_z(s-T)\right)ds+2\pi f(T-t)\right)}dt \\ =j\gamma M_0{\int }_0^T{B}_1(t)e^{-j2\pi \left(k_z(t)z+k_{v_z}(t)v_z+k_f(t)f\right)}dt \\ \text{where }{k}_z(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_z(s)ds\text{, }{k}_{v_z}(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_z(s)(s-T)ds\text{, }{k}_f(t)=T-t \end{gathered}$$ (A3)

If the gradient field is played in all three dimensions and three-dimensional position and velocity are considered, the result can be generalized to:

$$\begin{array}{l}{M}_{xy}=j\gamma M_0{\int }_0^T{B}_1(t)e^{-j2\pi k(t)\cdot x}dt\\ \text{where }x={\left[\begin{array}{ccc}x& y& z\end{array}{v}_x{v}_y{v}_z\text{ f}\right]}^T,\\ k(t)={\left[\begin{array}{ccc}{k}_x(t)& k_y(t)& k_y(t)\end{array}\begin{array}{ccc}{k}_{v_x}(t)& k_{v_y}(t)& k_{v_z}(t)\end{array}{k}_f(t)\right]}^T,\\ k_{x,y,z}(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_{x,y,z}(s)ds\text{, }{k}_{v_{x,y,z}}(t)=\frac{\gamma }{2\pi }{\int }_t^T{G}_{x,y,z}(s)(s-T)ds\text{, }{k}_f(t)=T-t,\end{array}$$ (A4)

This can be rewritten as:

$$\begin{gathered} M_{xy}\left(x,v,f\right)=j\gamma M_0{\int }_0^T{B}_1(t)\left(\underset{K}{\int }\delta (k(t)-k)e^{-j2\pi k\cdot x}dk\right)dt \\ =j\gamma M_0\underset{K}{\int }p(k)e^{-j2\pi k\cdot x}dk \\ \text{where }p(k)={\int }_0^T{B}_1(t)\delta (k(t)-k)dt \end{gathered}$$ (A5)

Here δ(k) denotes a seven-dimensional delta function, but the dimension can be reduced depending on how many variables are included in k.

Figure A1.

A: Timing diagram of B₁ and G_z field waveforms for selective excitation. B: B₁ field during infinitesimal time dt yields a rotation (around the x-axis) of α = γ B₁(t) dt, leading to a transverse component of dM_xy (t) = jM₀ sinα≈ jM₀ α = jγ M₀ B₁(t) dt. C: Assuming a spatial position of z, velocity of v_z, and off-resonance f for a spin of interest, dM_xy (t) will experience a precession by an amount of $\theta =\gamma {\int }_t^T{G}_z\left(s\right)\left(z+v_z\left(s-T\right)\right)ds+2\pi \left(T-f\right)$ during the time interval from t to T.