Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2764012

Formats

Article sections

Authors

Related links

Magn Reson Med. Author manuscript; available in PMC 2010 June 1.

Published in final edited form as:

PMCID: PMC2764012

NIHMSID: NIHMS135971

Magnetic Resonance Systems Research Laboratory, Department of Electrical Engineering, Stanford University, Stanford, California, USA

The publisher's final edited version of this article is available free at Magn Reson Med

See other articles in PMC that cite the published article.

Variable-rate selective excitation (VERSE) is a radio frequency (RF) pulse reshaping technique. It is most commonly used to reduce the peak magnitude and specific absorption rate (SAR) of RF pulses by reshaping pulses and gradient waveforms to reduce RF magnitude while preserving excitation profiles. In this work, a general time-optimal VERSE algorithm for multidimensional and parallel transmit pulses is presented. Time optimality is achieved by translating peak RF limits to gradient upper bounds in excitation *k*-space. The limits are fed into a time-optimal gradient waveform design technique. Effective SAR reduction is achieved by reducing peak RF subject to a fixed pulse length. The presented method is different from other VERSE techniques in that it provides a noniterative time-optimal multidimensional solution, which drastically simplifies VERSE designs. Examples are given for 1D and 2D single channel and 2D parallel transmit pulses.

In magnetic resonance imaging (MRI), spatially-selective excitation radio frequency (RF) pulses are often limited by hardware and pulse sequence constraints, such as RF power, peak RF magnitude, and SAR. In most imaging scenarios it is desirable to excite sharp slice profiles. However, such target profiles often result in oscillatory RF waveforms with high peak RF and unacceptable levels of SAR. High peak RF magnitude and SAR effectively set lower limits on the pulse length. In some fast imaging applications, the time required to perform excitation can reduce the available data acquisition time, resulting in SNR degradation (1). In other fast imaging applications, shorter TRs can be achieved by reducing SAR (2). Parallel transmission has recently gained the interest of researchers (3–9) due to the additional degrees of freedom afforded by excitation with multiple independent transmit coils. This can be used to shorten pulse duration, increase spatial resolution, and control RF power deposition (4). Numerical optimization-based pulse design methods are widely used to design parallel transmit pulses (3–9). Numerical optimization is necessary to meet the unique requirements of parallel RF pulse design such as the incorporation of coil sensitivities, regions of interest (ROIs), and other constraints such as low integrated RF power. In practice, numerical approaches often yield pulses with relatively high RF magnitude. The reasons for it are (i) the peak RF magnitude constraints are usually ignored due to the difficulty of solving constrained optimization problems and (ii) target excitation profiles are usually specified prior to pulse design, without consideration of their influence on RF magnitude and (iii) accelerated parallel transmission pulses lead to higher peak RF magnitude (6), which can diminish the benefit of parallel transmission.

The simplest way to deal with RF magnitude violations is to uniformly stretch out the original RF and gradient waveforms in time. But, this is often undesirable if not unfeasible due to the associated time cost. In particular, the excitation profiles of long RF pulses are more prone to distortions caused by relaxation and off-resonance. An alternative to this is variable-rate selective excitation (VERSE) (1,2,10–17), which is a general RF and gradient reshaping principle. VERSE jointly reshapes RF and gradient pulses to meet temporal or peak RF constraints, without changing the pulses’ on-resonance excitation profiles. It has been used to (i) reduce RF Power (2,10–13), (ii) overcome peak RF power limitations (1, 10, 14), and (iii) shorten pulse duration (1, 10, 15), though these three categories are not exclusive.

Three algorithms were introduced in the original VERSE paper (10). Among these, the minimum-time formulation is especially important because it gives the shortest RF pulse that does not violate peak RF limits. This can be used to reduce SAR by temporally dilating high peak RF regions of a pulse (1).

Various approaches to VERSE can be found in the literature. In particular, near-optimal solutions exist for one-dimensional (1D) VERSE (1, 10, 16, 17). However, these do not guarantee time-optimality. In the case of multidimensional excitation, Hardy and Cline (15) used VERSE principle to shorten 2D selective pulses using maximum amplitude and slew-rate spiral gradient waveforms. Recently, Xu et al. reported a reduction of peak RF amplitude using a properly chosen variable slew-rate spiral trajectory while avoiding uniform dilation of the RF waveform when the RF limit was reached (14). Wu et al. adopted the VERSE principle to reduce SAR in parallel transmission (12). These methods are all specific to spiral excitation and their target is trajectory design. To our knowledge, a generalized time-optimal design method for multidimensional and parallel transmit VERSE has not been presented.

In this paper, we propose a general, noniterative, and time-optimal VERSE RF pulse design method. We first show that re-casting VERSE to the excitation *k*-space domain guarantees identical spin rotation in both multidimensional and parallel excitation schemes. Therefore, applying VERSE does not change the pulse’s on-resonance excitation profile. We then develop our method based on the following key concepts: (i) peak RF constraints can be equivalently translated to maximum allowable gradient amplitudes in excitation *k*-space; (ii) these gradient limits can be used as an input to a time-optimal gradient waveforms design algorithm (18); (iii) identical spin rotation is achieved by adjusting the RF pulse according to the time optimal gradient amplitude. In addition, a strategy for SAR reduction is presented. Finally, the design methods are validated with examples.

The goal of RF pulse design is to produce a time-varying complex-valued RF pulse *B*_{1} (*t*) = *B*_{1,x}(*t*) + *iB*_{1,y}(*t*) and gradient field G(*t*) = [*G _{x}*(

$$\mathbf{k}(t)\equiv -\gamma {\displaystyle {\int}_{t}^{T}\mathbf{G}(\tau )d\tau ,}$$

[1]

where *T* is the RF pulse duration, and γ is the gyromagnetic ratio. Even beyond the STA regime, this formalism is useful in representing cumulative gradient area as a path in excitation *k*-space. Most RF pulse design techniques design *B*_{1} along a pre-defined gradient pattern that is determined by a desired path through *k*-space. The RF excitation pulse sequence is then defined by the waveform pair {*B*_{1}(*k*), G(k)}. *k*-Space trajectories are 1D curves in a multidimensional space and can be described as a function of a parameter such as time, k(*t*) = [*k _{x}*(

$$\begin{array}{c}s(t)\equiv \gamma {\displaystyle {\int}_{0}^{t}|\mathbf{G}(\tau )|d\tau}\hfill \\ {B}_{1}(s(t))={B}_{1}(t)\hfill \\ \mathbf{G}(s(t))=\mathbf{G}(t).\hfill \end{array}$$

[2]

The arc-length *s* describes points along RF and gradient waveforms as a function of total distance traversed in excitation *k*-space, or k(s) = [*k _{x}*(

VERSE dynamically dilates a pulse and its gradient waveform in a manner that preserves the original rotation. As a result, the effect of the reshaped pulse is identical to that of the input pulse. To extend the 1D version of VERSE to multidimensional and parallel excitation pulses, we need to show that it preserves rotations under these more complicated conditions.

We start with the rotation of spins at a spatial coordinate r under piece-wise constant *B*_{1} and G. For convenience, we define the following relationships at the *j ^{th}* sample:

$$\begin{array}{c}\mathrm{\Delta}{s}_{j}\equiv \gamma \mathrm{\Delta}t\left|{\mathbf{G}}_{j}\right|:\phantom{\rule{thinmathspace}{0ex}}\text{incremental arc-length at time point}j\hfill \\ {W}_{j}\equiv {B}_{1,j}/\left|{\mathbf{G}}_{j}\right|:\phantom{\rule{thinmathspace}{0ex}}\text{ratio of RF and gradient amplitude}\hfill \\ {\text{\hspace{1em}}\mathbf{g}}_{j}\equiv {\mathbf{G}}_{j}/\left|{\mathbf{G}}_{j}\right|:\phantom{\rule{thinmathspace}{0ex}}\text{unit gradient field vector}.\hfill \end{array}$$

[3]

Neglecting relaxation, the spin rotation is obtained by solving the Bloch equation, yielding:

$$\begin{array}{cc}\mathrm{\Delta}{\varphi}_{j}\hfill & =-\gamma \mathrm{\Delta}t\sqrt{|{B}_{1,j}{|}^{2}+{({\mathbf{G}}_{j}\cdot \mathbf{r})}^{2}}\hfill \\ \hfill & =-\mathrm{\Delta}{s}_{j}\sqrt{|{W}_{j}{|}^{2}+{({\mathbf{g}}_{j}\cdot \mathbf{r})}^{2}}\hfill \\ {\mathbf{n}}_{j}\hfill & =\frac{\gamma \mathrm{\Delta}t}{\left|\mathrm{\Delta}{\varphi}_{j}\right|}({B}_{1,x,j},{B}_{1,y,j},{\mathbf{G}}_{j}\cdot \mathbf{r})\hfill \\ \hfill & \propto (\frac{{B}_{1,x,j}}{\left|{\mathbf{G}}_{j}\right|},\frac{{B}_{1,y,j}}{\left|{\mathbf{G}}_{j}\right|},{\mathbf{g}}_{j}\cdot \mathbf{r}),\hfill \end{array}$$

[4]

where Δϕ* _{j}* is the incremental rotation angle about the axis of rotation

We can obtain expressions for the incremental rotation in the continuous *s*-domain by taking the limiting value of the ratio of Δϕ/Δ*s* as Δ*s* becomes infinitely small in Eq. [4]:

$$\begin{array}{cc}\varphi \prime (s)\hfill & =-\sqrt{|W(s){|}^{2}+{(\mathbf{g}(s)\cdot \mathbf{r})}^{2}}\hfill \\ \mathbf{n}(s)\hfill & \propto (\frac{{B}_{1,x}(s)}{|\mathbf{G}(s)|},\frac{{B}_{1,y}(s)}{|\mathbf{G}(s)|},\mathbf{g}(s)\cdot \mathbf{r}).\hfill \end{array}$$

[5]

To distinguish between pre- and post-VERSE waveform pairs, we henceforth denote {*B*_{1}(*s*), G(*s*)} as the pre-VERSE pair and $\{{B}_{1}^{\upsilon}(s),{\mathbf{G}}^{\upsilon}(s)\}$ as the post-VERSE pair in the *s*-domain.

It is important to note that the control variables in multidimensional VERSE are $|{B}_{1}^{\upsilon}(s)|$ and |G^{υ}(*s*)|, the RF and gradient *magnitudes*, respectively. Because VERSE only adjusts the net gradient magnitude and not the relative sizes of *G _{x}, G_{y}* and

The spin rotation is preserved for parallel excitation if the magnitude of all transmit channels is scaled by the same value. In that case, we have:

$$\left|{\displaystyle \sum _{n=1}^{N}C(s){d}_{n}(\mathbf{r}){B}_{1}^{n}(s)}\right|=C(s)\left|{\displaystyle \sum _{n=1}^{N}{d}_{n}(\mathbf{r}){B}_{1}^{n}(s)}\right|=C(s)|{B}_{1}(\mathbf{r},s)|,$$

[6]

where *N* is the number of transmit channels, *C*(*s*) is the common scaling factor, and *d _{n}*(

In summary, an RF pulse can be completely described in the *s*-domain. The spin rotation in this domain is preserved as long as the ratio *B*_{1}(*s*)*/|G*(*s*)| is unchanged when VERSE is applied. Therefore, time-optimal VERSE can be divided into three separate problems: (i) translation of the RF magnitude constraints into gradient magnitude constraints in the *s*-domain, (ii) designing a time-optimal gradient amplitude that satisfies the constraints, (iii) modification of the RF magnitude according to the new gradient magnitude.

In previous works, (1,11,15) time-dilation functions τ (*t*) have been used to scale the {*B*_{1}(t), *G*(*t*)} pair. Time dilation functions are difficult to optimize since in general there is no known closed-form solution (1). Moreover, for multidimensional pulses there is an additional slew-rate dependency (14, 15, 23, 24). Therefore, we take a different approach. Rather than optimizing the function τ (*t*) directly, we transform the problem into the *s*-domain, where the solution is greatly simplified.

Given the original RF pulse described by the pair {*B*_{1}(*t*), *G*(*t*)} we transform it to generate the pair {*B*_{1}(*s*)*, G*(*s*)} in the *s*-domain. Our aim is to design a new pair $\{{B}_{1}^{\upsilon}(s),{\mathrm{G}}^{\upsilon}(s)\}$ that is time-optimal and satisfies the following constraints,

$$\begin{array}{c}|{B}_{1}^{\upsilon}(s)|\le {B}_{1,\text{max}}\hfill \\ {|\mathbf{G}}^{\upsilon}(s)|\le {G}_{\text{max}}\hfill \\ {|\dot{\mathbf{G}}}^{\upsilon}(s)|\le {S}_{\text{max}}\hfill \\ \frac{{B}_{1}^{\upsilon}(s)}{|{\mathbf{G}}^{\upsilon}(s)|}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{{B}_{1}(s)}{|\mathbf{G}(s)|}=W(s).\hfill \end{array}$$

[7]

Here *G*_{max}, S_{max}, and B_{1,max} denote the maximum gradient, slew-rate, and RF magnitudes, respectively, and are set according to the hardware limits of the specific system; the dot denotes a time derivative. The physical consequence of these constraints is that time-reductions can not result in violations of hardware limitations.

Combining the first two constraints and rearranging terms we get,

$$\begin{array}{cc}{|\mathbf{G}}^{\upsilon}(s)|\hfill & \le \text{min}\{\frac{{B}_{1,\text{max}}}{|W(s)|},{G}_{\text{max}}\}\equiv {G}_{u}(s)\hfill \\ |{\dot{\mathbf{G}}}^{\upsilon}(s)|\hfill & \le {S}_{\text{max}}\hfill \\ \hfill {B}_{1}^{\upsilon}(s)& =W(s)|{\mathbf{G}}^{\upsilon}(s)|.\hfill \end{array}$$

[8]

Now, the peak RF constraint has been transformed and merged into a gradient constraint. The result is a pure time-optimal gradient design problem with a maximum gradient constraint set by *G _{u}*(

In summary, the process of reshaping {*B*_{1}(*t*), *G*(*t*)} into $\{{B}_{1}^{\upsilon}(t),{\mathbf{G}}^{\upsilon}(t)\}$ is as follows.

- Find the time-optimal
*s*^{υ}(*t*) and gradient waveforms G^{υ}(*t*). - Recalculate the RF waveform according to Eq. [8].

The gradient design method introduced by Lustig et al. (18) is a convenient way to efficiently design time-optimal gradient waveforms for arbitrary *k*-space trajectories. This method solves the problem by expressing the gradient and slew-rate constraints in the *s*-domain, then solving for the time optimal gradient magnitude in the *s*-domain, and finally transforming the result to the time domain and deriving the optimal gradient waveforms.

In Eq. [8], we expressed the RF constraints as gradient limits in the *s*-domain. We modify the time-optimal gradient design algorithm to include this additional gradient limit in the design. The modified algorithm is described as follows:

$$\begin{array}{c}\text{Constant}:\hfill \\ \text{\hspace{1em}\hspace{1em}}L\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{total arc-length}\hfill \\ \text{Variables}:\hfill \\ \text{\hspace{1em}\hspace{1em}}s\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{Euclidian arc-length}\hfill \\ \text{\hspace{1em}\hspace{1em}}T\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{RF pulse duration}\hfill \\ \text{\hspace{1em}\hspace{1em}}\kappa (s)\equiv \left|\frac{{d}^{2}\mathbf{k}(s)}{d{s}^{2}}\right|\hfill \\ \text{Constraint}:\hfill \\ \text{\hspace{1em}\hspace{1em}}|\mathbf{G}(s)|\le {G}_{u}(s)\le {G}_{\text{max}}\hfill \end{array}$$

[9]

$$\begin{array}{cc}\text{minimize}\hfill & T\hfill \\ \text{subject to}\hfill & \dot{s}\le \text{min}\phantom{\rule{thinmathspace}{0ex}}\{\gamma {G}_{u}(s),\sqrt{\gamma {S}_{\text{max}}/\kappa (s)}\}\hfill \\ \hfill & \left|\ddot{s}\right|\le \sqrt{{\gamma}^{2}{S}_{\text{max}}^{2}-{\kappa}^{2}(s){\dot{s}}^{4}}\hfill \\ \hfill & s(0)=0,\phantom{\rule{thinmathspace}{0ex}}\dot{s}(0)=0,\phantom{\rule{thinmathspace}{0ex}}s(T)=L\hfill \end{array}$$

[10]

This is effectively solved by Eq. [18–Eq. 21] of Ref. (18) in the phase-plane ( vs. *s*) using optimal control theory. The resultant solution denoted by s*(*t*) leads to ${\mathbf{G}}^{*}(t)={\gamma}^{-1}\frac{d\mathbf{k}({s}^{*}(t))}{dt}$.

There is a major difference between time- and *s*-domain approaches regarding the switching between acceleration/deceleration/gradient-limited/RF-limited regions. In the time-based differential equation, one must explicitly find the switching times. In the *s*-domain, the switching locations are found implicitly by solving the differential equations forward and backward and taking the minimum solution. This approach will not work for a time-based differential equation; a solution in one direction can lead to a different total time than a solution in the other. Therefore, these solutions cannot be compared. In the *s*-domain, the two solutions are aligned in space, and can be compared thereby avoiding the need to iterate (18).

SAR is proportional to the power of the RF waveform. The minimum-SAR RF waveform is the solution to the problem:

$$\begin{array}{cc}\text{minimize}\hfill & {\displaystyle {\int}_{0}^{T}|{B}_{1}^{\upsilon}(t){|}^{2}dt}\hfill \\ \text{subject to}\hfill & {|\mathbf{G}}^{\upsilon}(t)|\le {G}_{\text{max}}\hfill \\ \hfill & {|\dot{\mathbf{G}}}^{\upsilon}(t)|\le {S}_{\text{max}}\hfill \\ \hfill & {s}^{\upsilon}(t)=\gamma {\displaystyle {\int}_{0}^{t}|{\mathbf{G}}^{\upsilon}(\tau )|}d\tau \hfill \\ \hfill & \gamma {\displaystyle {\int}_{0}^{t}{\mathbf{G}}^{\upsilon}}(\tau )d\tau =\mathbf{k}({s}^{\upsilon}(t))\hfill \\ \hfill & {B}_{1}^{\upsilon}(t)=W({s}^{\upsilon}(t))|{\mathbf{G}}^{\upsilon}(t)|.\hfill \end{array}$$

[11]

Conolly et, al. showed that when the slew-rate constraint is neglected, the minimum-SAR solution is the minimum-peak RF solution (10). When the slew-rate constraint is included, the minimum-peak RF is not necessarily the optimal minimizer of Eq. [11], but is near optimal and significantly reduces the SAR objective. Therefore, to reduce SAR we solve the problem:

$$\begin{array}{cc}\text{minimax}\hfill & |{B}_{1}^{\upsilon}(t)|,\phantom{\rule{thinmathspace}{0ex}}t\in [0,T]\hfill \\ \text{subject to}\hfill & {|\mathbf{G}}^{\upsilon}(t)|\le {G}_{\text{max}}\hfill \\ \hfill & {|\dot{\mathbf{G}}}^{\upsilon}(t)|\le {S}_{\text{max}}\hfill \\ \hfill & {s}^{\upsilon}(t)=\gamma {\displaystyle {\int}_{0}^{t}|{\mathbf{G}}^{\upsilon}(\tau )|}d\tau \hfill \\ \hfill & \gamma {\displaystyle {\int}_{0}^{t}{\mathbf{G}}^{\upsilon}}(\tau )d\tau =\mathbf{k}({s}^{\upsilon}(t))\hfill \\ \hfill & {B}_{1}^{\upsilon}(t)=W({s}^{\upsilon}(t))|{\mathbf{G}}^{\upsilon}(t)|.\hfill \end{array}$$

[12]

Conveniently, we can use the time-optimal VERSE method to solve this problem. It is possible to find a minimum peak RF bound for which the time-optimal solution gives the desired pulse length since the time-optimal pulse length is a non-increasing function of the peak RF, i.e., higher peak RF results in shorter or equal RF pulse duration. This will result in a pulse that is bounded and tightly compressed in time for a fixed pulse duration. Given a waveform pair {*B*_{1}(s), G(s)} and a target pulse length, *T*, the minimum-peak RF solution is found using bisection in the following manner:

- ${B}_{1,\text{SAR}}^{-}=0,\phantom{\rule{thinmathspace}{0ex}}{B}_{1,\text{SAR}}^{+}=\phantom{\rule{thinmathspace}{0ex}}{B}_{1,\text{max}}$
- ${B}_{1,\text{SAR}}=({B}_{1,\text{SAR}}^{-}+{B}_{1,\text{SAR}}^{+})/2$
- Solve for the time-optimal VERSE with
*B*_{1,max}=*B*_{1,SAR}to get {*B*(^{v}*t*),*G*(^{v}*t*)} and*T*^{v} - if
*T*≤ : return {^{υ}–T*B*(^{υ}*t*), G^{υ}(*t*)}else:- if (
*T*<^{υ}*T*− ε): ${B}_{1,\text{SAR}}^{+}={B}_{1,\text{SAR}}$ - else: ${B}_{1,\text{SAR}}^{-}={B}_{1,\text{SAR}}$
- goto 2

The parameter should be set according to the discretizing time of the RF sequencer. In step 4, we tighten the search region according to the equality test by exploiting the monotonic relationship between peak RF and RF pulse duration.

We showed in the previous section that spin rotations are preserved in VERSE for parallel excitation. Since RF limits must be observed in all *N* channels, RF and gradient limits can be forced similarly by

$$\begin{array}{c}{|\mathbf{G}}^{\upsilon}(s)|\le {G}_{u}^{n}(s)\text{\hspace{1em}for\hspace{1em}}n=1,\dots ,N\hfill \\ \iff |{\mathbf{G}}^{\upsilon}(s)|\le \text{min}\{{G}_{u}^{1}(s),\dots ,{G}_{u}^{N}(s)\}\equiv {G}_{u}(s).\hfill \end{array}$$

[13]

Thus, peak RF constraints in parallel transmission can be incorporated via a global gradient upper bound *G _{u}*(s). Redesigning each ${B}_{1}^{n,\upsilon}(t)\phantom{\rule{thinmathspace}{0ex}}\text{from}\phantom{\rule{thinmathspace}{0ex}}{B}_{1}^{n}(s),{W}^{n}(s)$, and

In this section, we present several examples to validate the proposed time-optimal VERSE method. A time-optimal gradient design tool (http://www-mrsrl.stanford.edu/~mlustig/ software) (18) was used in the gradient design step. The tool’s MATLAB (The MathWorks, Inc., Natick, MA) source code was modified to accommodate the additional peak RF constraints as described in the Theory section. We used spline interpolation to calculate *W*(s) for RF pulses designed with numerical optimization methods. Hardware constraints of *G*_{max} = 40 mT/m, *S _{max}* = 150 T/m/s, and

Slab-selective excitation with a sharp transition band requires high peak *B*_{1}. We used the same pulse specification as in Ref. (1) for comparison. A pre-VERSE slab-selective RF (slab thickness = 40 mm, time-bandwidth product = 10, and flip-angle = 60°) was designed using the SLR pulse design technique (21). *W*(s) was first computed (Fig. 1c) and G^{υ}(s) was designed (Fig. 1d). Note that the resulting gradient waveforms are always either peak RF-, maximum gradient-, or slew rate-limited, which is a necessary condition for time-optimality. VERSE decreased the pulse duration by 72.2% compared to the original pulse with the same peak RF (Fig. 1a,b). The result is in good agreement with previous results (1).

For comparison, we used the same excitation profile specification as in the first example. The minimum-peak RF solution was found with the bisection search on RF upper bound within the range of hardware limit (0 < *B*_{1,u} ≤ 15 µT) to match the duration of 2.63 msec. As shown in Fig. 2, the minimum-peak RF solution had *B*_{1,u} = 2.6 µT and 71% lower SAR than the original. This was made possible by the optimal gradient control demonstrated in Fig. 2b.

We designed a 2D spatially-selective pulse with a spiral-in excitation *k*-space trajectory (diameter of circular excitation = 9 cm, resolution = 0.5 cm, FOV = 12 cm, and flip-angle = 60°) using the STA approximation method (19). The original and reshaped pulses are shown in Fig. 3. RF amplitude violations (grey region in Fig. 3d) were efficiently corrected (grey region Fig. 3g). Note the tight optimal gradient slew-rate constraint in Fig. 3i compared to the variable slew-rate spiral design presented in Ref. (14). The peak RF magnitude of the VERSE pulse was decreased from 35 µT to 15 µT (the hardware limit), whereas the pulse duration was increased by only 3% from 6.39 msec to 6.58 msec.

To validate parallel excitation VERSE, we applied our algorithm to an inversion pulse with a rectangular excitation profile designed by the additive angle method (6). This method is a numerical optimization-based approach to designing large flip-angle parallel RF pulses. A 2D RF pulse with a single-shot spiral-out excitation *k*-space trajectory (resolution = 0.75 cm, FOV = 24 cm, excitation FOV = 5 cm; speedup factor = 4.8) was designed assuming a 4 µsec discretizing time for the discrete form of STA approximation adopted in the additive angle method. Figure 4a shows the simulated ${B}_{1}^{+}$ magnitude maps of an eight element active rung transmit array (25, 26), that were used to design π-pulses with the method of Ref. (6). Without VERSE, the large flip-angle of this pulse combined with a high acceleration factor results in very high peak RF. We set *B*_{1,max} = 15 (in arbitrary unit) for our VERSE constraint which is about 30% of the original peak value. The global gradient upper bound *G _{u}*(s) was computed by taking the minimum among upper bounds of all 8 channels (black solid line in Fig. 4c). The post-VERSE gradient waveforms were found (Fig. 4i) using the new limit. The RF pulses were scaled globally as described previously. The resulting RF pulses were effectively limited by the predefined upper bound (Fig. 4h). Fig. 4g,k show that the Bloch simulated profile of the post-VERSE pulse is in good agreement with the pre-VERSE pulse. The resulting time increase was 17%, which is small compared to the 230% time increase required by uniform time-dilation to achieve the same peak magnitude.

As opposed to other near optimal solutions for 1-D VERSE problem, our approach is time-optimal and, most importantly, is generalizable to multidimensional excitation and parallel transmit; this drastically simplifies multidimensional and parallel transmit VERSE pulse design while achieving time-optimality and SAR reduction. It can be applied to any type of *k*-space trajectory. SAR can be controlled via simple control of the RF upper bound.

The most time-consuming step in our method is finding time-optimal gradient waveforms. The computational complexity is linear with respect to the total arc-length of trajectory (18). In the case of parallel transmission, the time cost does not scale with the number of channels unlike the original parallel excitation design problem. This is because we only need to design a single gradient waveform set. For example, it only takes a couple of seconds to solve our parallel transmission example with *L* = 5.76 cm^{−1}, when all routines are written in MATLAB and run on 2.13 GHz processor with 2 GB memory. Thus, our method can be used as an on-the-fly reshaping tool.

When RF and gradient waveforms are defined in continuous time, the post-VERSE pulse will achieve the exact same excitation profile as the pre-VERSE counterpart. This assumption does not hold for real MRI systems; discrete update times as well as nonlinear behavior of RF and gradient systems will introduce mismatch in RF and gradient waveforms, thereby distorting the spin rotation depending on the integral effect, which is governed by Eq. [5]. This deviation, though very small, will accrue during pulse playout and may distort the excitation profile. One manifestation would be erroneous excitation due to harmonic distortions introduced by VERSE, whose frequency (and therefore location) are determined by the RF update time; this may exist inside an object within the sensitivity of receiver coils if no limit is set on the temporal compression of the waveforms. Thus, it additionally limits the gradient strength since the residual excitation can be pushed out in space by lowering gradient amplitude. In short, the compressibility of VERSE is limited by the RF update time as well as *G*_{max} and *B*_{1,max}. This problem will not affect non-compressed regions.

The RF-to-gradient amplitude ratio *W*(s) is preserved by VERSE and can be exactly calculated when the original pulse is analytically designed. For most numerically-designed pulses, *W*(s) can be calculated using spline interpolation. For very high bandwidth pulses such as those common to parallel transmission, it may be necessary to use a better interpolation scheme since interpolation errors can have an adverse effect on pulse performance. In STA excitation, *W*(s) has a special interpretation. In that regime, it is the same as the spatial weighting function *W*(k(*t*)) = *B*_{1}(*t*)/γG(*t*) defined in Ref. (19). When this time-independent function *W*(k) is multiplied by a spatial frequency sampling function *S*(k), it becomes the Fourier transform pair of the excitation profile *M _{xy}*(r) (19). Thus,

As an alternative method for SAR reduction, variable-density *k*-space trajectories have been proposed (13). These trajectories have the additional benefit of reduced coherent aliased energy. However, it is difficult to optimize variable density sampling schemes to guarantee a desired level of SAR and aliased energy for a given excitation profile, since the waveform will depend on the flip angle, shape, and transition width of the profile. VERSE can be used in conjunction with variable-density trajectories to overcome this difficulty, making variable density trajectories more robust in practice. When VERSE is applied after pulse design, a reasonably good trajectory will suffice for a wide range of design scenarios. The same trajectory can be reused and RF reshaping will be minimal.

In numerical optimization-based pulse design, peak RF can be limited via local regularization (27), however this constraint may compromise the fidelity of the target excitation profile. In addition, it may require increased complexity since non-convex optimization techniques are required to accommodate RF magnitude constraints. In this context, the VERSE principle can be used to efficiently redesign gradient waveforms for which the pulse design algorithm produces accurate pulses with lower peak RF. The gradient waveform found by VERSE improves pulse realizability, so we can redesign RF waveforms with or without peak RF constraints, e.g., hardware limits, and are more likely to achieve realizable pulses. This alternating joint design of gradient waveforms and RF pulses can be performed iteratively. Local regularization can then be excluded to achieve faster optimization. In particular, this approach should be used when main field inhomogeneities are incorporated in the original pulse design, due to the time-dependent effect of field inhomogeneities.

We have introduced a noniterative time-optimal design method for VERSE, and validated it with representative examples. The VERSE interpretation in excitation *k*-space was studied to find conditions of identical spin rotation in multidimensional and parallel excitation. Time-optimality was achieved by translating peak RF limits into gradient upper bounds in excitation *k*-space that are fed into a time-optimal gradient waveform design. We also provided an algorithm for reducing the SAR of fixed duration pulses that yields a minimum-peak RF solution. The method can be applied to parallel transmission via a global gradient upper bound, without increasing algorithm complexity.

This work was supported by NIH-41RR09784, NIH-R21EB007715, NIH-R01EB005307, and GE Healthcare.

1. Hargreaves BA, Cunningham CH, Nishimura DG, Conolly SM. Variable-rate selective excitation for rapid MRI sequences. Magn Reson Med. 2004;52:590–597. [PubMed]

2. van den Bos IC, Hussain SM, Krestin GP, Wielopolski PA. Extending slice coverage for breathhold fat-suppressed T2-weighted fast spin-echo of the liver at 3.0T: application of variable-rate selective-excitation (VERSE) RF pulses. J Magn Reson Imaging. 2008;27:110–116. [PubMed]

3. Katscher U, Börnert P, Leussler C, van den Brink JS. Transmit SENSE. Magn Reson Med. 2003;49:144–150. [PubMed]

4. Zhu Y. Parallel excitation with an array of transmit coils. Magn Reson Med. 2004;51:775–784. [PubMed]

5. Grissom W, Yip CY, Zhang Z, Stenger VA, Fessler JA, Noll DC. Spatial domain method for the design of RF pulses in multicoil parallel excitation. Magn Reson Med. 2006;56:620–629. [PubMed]

6. Grissom WA, Yip CY, Wright SM, Fessler JA, Noll DC. Additive angle method for fast large-tip-angle RF pulse design in parallel excitation. Magn Reson Med. 2008;59:779–787. [PubMed]

7. Xu D, King KF, Zhu Y, McKinnon GC, Liang ZP. A noniterative method to design large-tip-angle multidimensional spatially-selective radio frequency pulses for parallel transmission. Magn Reson Med. 2007;58:326–334. [PubMed]

8. Xu D, King KF, Zhu Y, McKinnon GC, Liang ZP. Designing multichannel, multidimensional, arbitrary flip angle RF pulses using an optimal control approach. Magn Reson Med. 2008;59:547–560. [PMC free article] [PubMed]

9. Setsompop K, Wald LL, Alagappan V, Gagoski BA, Adalsteinsson E. Magnitude least squares optimization for parallel radio frequency excitation design demonstrated at 7 Tesla with eight channels. Magn Reson Med. 2008;59:908–915. [PMC free article] [PubMed]

10. Conolly S, Nishimura D, Macovski A, Glover G. Variable-rate selective excitation. J Magn Reson. 1988;78:440–458.

11. Conolly S, Glover G, Nishimura D, Macovski A. A reduced power selective adiabatic spin-echo pulse sequence. Magn Reson Med. 1991;18:28–38. [PubMed]

12. Wu X, Akgun C, Vaughan JT, Ugurbil K, van de Moortele PF. SAR reduction in transmit SENSE using adapted excitation k-space trajectories. Proceedings of the 15th Annual Meeting of ISMRM; Berlin, Germany. 2007. p. 673.

13. Liu Y, Feng K, McDougall MP, Wright SM, Ji J. Reducing SAR in parallel excitation using variable-density spirals: a simulation-based study. Magn Reson Imaging. 2008;26:1122–1132. [PubMed]

14. Xu D, King KF, Liang ZP. Variable slew-rate spiral design: theory and application to peak B1 amplitude reduction in 2D RF pulse design. Magn Reson Med. 2007;58:835–842. [PubMed]

15. Hardy CJ, Cline HE. Broadband nuclear magnetic resonance pulses with two-dimensional spatial selectivity. J Appl Phys. 1989;66:1513–1516.

16. Gai ND, Zur Y. Design and optimization for variable rate selective excitation using an analytic RF scaling function. J Magn Reson. 2007;189:78–89. [PubMed]

17. Cunningham CH, Wright GA, Wood ML. High-order multiband encoding in the heart. Magn Reson Med. 2002;48:689–698. [PubMed]

18. Lustig M, Kim SJ, Pauly JM. A fast method for designing time-optimal gradient waveforms for arbitrary *k*-space trajectories. IEEE Trans Med Imaging. 2008;27:866–873. [PMC free article] [PubMed]

19. Pauly J, Nishimura D, Macovski A. A *k*-space analysis of small-tip-angle excitation. J Magn Reson Imaging. 1989;81:43–56.

20. Pauly J, Nishimura D, Macovski A. A linear class of large-tip-angle selective excitation pulses. J Magn Reson Imaging. 1989;82:571–587.

21. Pauly J, Le Roux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Trans Med Imaging. 1991;10:53–65. [PubMed]

22. Pauly J, Spielman D, Macovski A. Echo-planar spin-echo and inversion pulses. Magn Reson Med. 1993;29:776–782. [PubMed]

23. King KF, Foo TKF, Crawford CR. Optimized gradient waveforms for spiral scanning. Magn Reson Med. 1995;34:156–160. [PubMed]

24. Glover GH. Simple analytic spiral *k*-space algorithm. Magn Reson Med. 1999;42:412–415. [PubMed]

25. Kurpad KN, Boskamp EB, Wright SM. A parallel transmit volume coil with independent control of currents on the array elements. Proceedings of the 13th Annual Meeting of ISMRM; Miami Beach. 2005. p. 16.

26. Wright SM. 2D full-wave modeling of SENSE coil geometry factors at high-fields. Proceedings of the 10th Annual Meeting of ISMRM; Honolulu. 2002. p. 854.

27. Yip CY, Fessler JA, Noll DC. Iterative RF pulse design for multidimensional, small-tip-angle selective excitation. Magn Reson Med. 2005;54:908–917. [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |