NOTE Magnetic Resonance in Medicine 00:00–00 (2014)
Peripheral Nerve Stimulation-Optimal Gradient Waveform Design Rolf F. Schulte1* and Ralph Noeske2 ern whole-body MRI scanners with high-performance gradient systems are often more limited by human physiology in the form of PNS than the actual gradient hardware limitations, which are mainly the maximum gradient strength (Gmax) and slew rate (Smax). In other words, PNS often becomes a more dominating factor during gradient waveform design than Gmax and Smax. A typical gradient trajectory design starts by setting up the sequence with Gmax and Smax adhering to hardware limitations, and then iteratively derating Smax until the calculated maximum PNS is within permissible limits. The most accurate and least restrictive mathematical model to calculate PNS is the so-called convolution model, wherein a decaying nerve response function is convolved with the slew rate of the gradient trajectory (2). The trajectories most restricted by PNS are those with long durations, such as Cartesian echo-planar imaging (EPI) (3) or spiral trajectories (4). Derating Smax for the whole trajectory leads to considerably longer readout durations. The goal of the current work was to minimize this stretching of trajectories by including the PNS convolution model in the design of the gradient waveform, hence derating Smax only where necessary. This leads to a time-varying Smax instead of a constant global one. The approach was demonstrated for both spiral and EPI trajectories.
Purpose: Modern magnetic resonance imaging scanners with high-performance gradient systems have high maximum gradient strength (Gmax) and slew rate (Smax). Peripheral nerve stimulation (PNS) is often a more limiting factor for gradient waveform design than Gmax and Smax. Traditionally, the slew rate is derated globally to adhere to PNS limitations. Methods: In this work, the PNS limitation is already included in the gradient waveform design in the form of a time-varying slew rate, hence shortening the overall gradient duration. Results: Spiral and echo-planar imaging trajectories were designed with a multitude of parameters, and it was demonstrated that trajectory durations from conventional to PNSoptimal design can be shortened by 8 and 3%, respectively. Conclusion: Including PNS-limits in the gradient waveform design can shorten the duration of gradient trajectories, thereby reducing associated artifacts. Magn Reson Med C 2014 Wiley Periodicals, Inc. 000:000–000, 2014. V Key words: peripheral-nerve stimulation; peripheral nerve stimulation convolution model; echo-planar imaging; spiral imaging; time-optimal gradient-trajectory design
INTRODUCTION Rapidly switching magnetic field gradients can lead to an electric depolarization and hence stimulation of nerve cells (1). The effect is noticeable predominantly in peripheral nerves, hence it is commonly called peripheral-nerve stimulation (PNS). PNS ranges from small stimulations at the sensation threshold up to intolerable pain. Potentially much more harmful than PNS is a stimulation of heart cells, which could lead to arrhythmia. The threshold for cardiac stimulation is about an order of magnitude higher as compared to PNS, and thus adhering to the PNS limit also prevents cardiac stimulation with a large safety margin (1). The PNS limits are governed by an international engineering standard (IEC 60601-2-33) (2), and every pulse sequence on all magnetic resonance imaging (MRI) scanners has to adhere to this standard for human scans. Mod-
THEORY Peripheral-Nerve Stimulation The most accurate and least restrictive PNS model is given by the convolution of the slew rate Si ðtÞ ¼ dGdti ðtÞ of the gradient trajectory Gi ðtÞ with the nerve response (2) Ri ðtÞ ¼
Smin
Z
t 0
Si ðuÞc du; ðc þ t uÞ2
[1]
where i ¼ ðx; y; zÞ denotes the respective gradient axis. The stimulation slew rate Smin ¼ ar is the threshold at which an infinitely long ramp with this slew rate leads to a detectable stimulation for every second person on average. The effective coil length a, rheobase r, and chronaxie time c are gradient-coil specific constants, which are determined experimentally for each gradient coil model by the respective manufacturers. The overall PNS threshold in percentage for all the gradient axes combined is given by
1
GE Global Research, Munich, Germany. GE Healthcare, Berlin, Germany. €r Bildung und Forschung (BMBF); Grant sponsor: Bundesministerium fu Grant number: 13EZ1114. *Correspondence to: Rolf F. Schulte; GE Global Research, Freisinger Land€nchen, Germany. strasse 50, 85748 Garching bei Mu 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pthresh ¼ 100 R2x þ R2y þ R2z :
Additional Supporting Information may be found in the online version of this article. Received 9 April 2014; revised 10 August 2014; accepted 12 August 2014 DOI 10.1002/mrm.25440 Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V
1
[2]
The permissible maximum is Pthresh ¼ 80% for the normal and 100% for the first and second controlled modes, respectively (2). 1
2
Schulte and Noeske
Spirals Spiral trajectories are a highly efficient approach to sampling k-space. Many different algorithms to design the actual trajectories were proposed over the years, and a recent overview of these algorithms can be found in the introduction of (5). Most common are Archimedian spiral trajectories, which exhibit equidistant revolutions, yielding an approximately uniform sampling density. In the present work, we use a design algorithm developed by Meyer et al. and Hargreaves (6,7), which is available for download from the internet (http://www-mrsrl.stanford.edu/ brian/vdspiral/). The spiral trajectory can be parameterized by kðtÞ ¼ kx þ iky ¼ auðtÞeiuðtÞ :
[3]
The rate of increase a is given for a uniform sampling density by
trajectory. Therefore, we introduce a third, PNS-limited regime directly in the design of the function h(t). For this, the global, constant Smax limit, as used for Eq. [6] is modified to a time-varying function Smax ðtÞ. During each time step in the design of h(t), PNS is also calculated according to Eqs. [1] and [2]. If PNS reaches its threshold (normally 80 or 100%), the maximum slew rate used for Eq. [6] is derated for the corresponding and following time steps in such a manner that PNS remains within limits. As shown in Figure 1 (left column), a typical spiral trajectory is composed of an Smax-limited regime, possibly followed by a PNS and a Gmax-limited regime. In general, these PNSoptimal spiral trajectories are shorter in duration as compared to trajectories in which Smax is derated globally to adhere to PNS limits, but longer as compared to trajectories for which PNS is ignored. Echo-Planar Imaging
N a¼ ; 2pFOV
[4]
where N denotes the number of spiral interleaves and FOV the spatial field of view. The design problem is essentially how to choose h(t) as a function of time while sampling k-space as rapidly as possible under given constraints. Differentiating Eq. [3] and scaling it to the gyromagnetic ratio c yields the following gradient trajectory (7) GðtÞ ¼
2pa iu _ e u þ iuu_ g
[5]
A second differentiation yields the expression for the slew rate SðtÞ ¼
2 i 2pa iu h€ 2 e u uu_ þ i 2u_ þ u€u : g
[6]
The physical limits of the gradient hardware constrain both the gradient strength and slew rate to Gmax and Smax, respectively. Furthermore, the sampling rates 1=t of receiver systems are limited, possibly constraining gra2p . Also, Smax is dients further to the condition jGj gtFOV reduced in the conventional design if the PNS limits are exceeded. The parameter h can be designed as a function of time iteratively, starting with h ¼ 0 and u_ ¼ 0. The spiral design algorithm (6,7) involves checking for each new time time step whether Smax or Gmax is limiting, and determines h(t) according to Eq. [6] or [5], respectively. In general, the first part of the spiral lies in an Smax-limited regime. The gradient strength continually increases until it possibly reaches Gmax, as shown in Figure 1. From there on, the trajectory is in the Gmax-limited regime, while the slew rate continually decreases. PNS, as determined through Eq. [1], is continually increasing in the Smax-limited regime, and it only decays when the slew rate is derated, as is the case in the Gmaxlimited regime or at the end of the trajectory (Fig. 1). Derating Smax over the whole trajectory duration is suboptimal, because the PNS curve starts at zero and the PNS threshold is typically exceeded only later during the
The typical design of a bidirectional Cartesian EPI trajectory involves going to the desired k-space position with a triangular or trapezoidal prewinder gradient. The main part of EPI encoding is composed of a train of bidirectional trapezoids with phase blips along the second dimension to sample 2D k-space line by line in one readout, as shown in Figure 2. The second dimension can be split up into separate excitations (interleaves) to shorten the readout duration accordingly. The trapezoidal gradients are constrained by system limitation for Gmax and Smax, in combination with possibly sampling and PNS limitations on Gmax and Smax, respectively. For the EPI trajectory, only ramps and blips introduce PNS, while it recovers during the plateaus of the trapezoids with its constant gradient amplitude. To reduce PNS in a similar fashion as shown above for the spiral trajectory design, we can again introduce a time-varying Smax ðtÞ function. However, there is an additional constraint that the first moment of the ramps must be nulled (i.e., ascending and descending ramps must have the same absolute area) in order to start always from the same k-space position along the first dimension. That implies that the ramps have to be symmetric functions. Different functions were investigated empirically and a suitable function proved to be tanh, for which the width was optimized individually for each trajectory. The phase-encoding blips along the other dimension are relatively small and their contribution to PNS is negligible. Hence, the tanh function was only applied to the ramps of the readout trajectory. The ramp function RðtÞ is given by RðtÞ ¼ Gmax tanh ðtÞ=tanh ðt0 Þ
[7]
with the discretized time s running from t0 to s0. The parameter s0 was optimized by an exhaustive search for each trajectory, yielding values of t0 0:5. The maximum slew rate was optimized by an exhaustive search as well, decreasing its value by 1 T/m/s if the respective PNS limit was exceeded. METHODS Many different spiral and EPI trajectories were designed to investigate the possible reduction in the readout
PNS-Optimal Gradient Waveform Design
3
FIG. 1. Typical spiral trajectories designed without adhering to PNS limits (left column), with the conventional global slew-rate derated design (middle column) and with the PNS-optimal design (right column). In the conventional design, PNS generally increases in the Smax-limited regime and subsequently decreases in the Gmax-limited regime. The PNS-optimal design introduces a third, PNS-limited regime. One can observe, that PNS levels exhibit a maximal peak; and this peak can be spread out into wider area, thereby optimally utilizing PNS limits. (Gx is indicated in blue, Gy in green, and jGj in red. FOV ¼ 100 100 mm2, mtx ¼ 128 128, interleaves ¼ 4, Gmax ¼ 50 mT/m, Smax ¼ 200 T/m/s, Smin ¼ 70.27 T/m/s, c ¼ 334 ms.)
trajectory duration. All trajectories were designed for a nominal pixel size of 1 1 mm2, and FOVs of 100, 150, 200, and 250 mm along both directions. The trajectories encoded k-space in 1, 2, 4, 8, and 16 number of excitations. Note that the term number of excitations is identical to spiral arms or EPI interleaves. All trajectories were designed for the parameters of the MR750 3T whole-body scanner (GE Healthcare, MilwauT kee, WI) with Gmax ¼ 50 mT m and Smax ¼ 200 ms (physical limitation of each gradient axis), an effective coil length of a ¼ 0:333m, a rheobase of r ¼ 23:4 Ts, a stimulation T , and a chronaxie time conslew rate of Smin ¼ 70:27 ms stant of c ¼ 334ms. To further investigate the PNS limitations on gradient sets in general, another set of spiral and EPI trajectories was designed with Gmax varying from 10 to 100 mT/m and Smax from 10 to 1000 T/m/s (FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16). One set of exemplary trajectories to experimentally compare conventional and PNS-optimal design with PNS thresholds of 100 and 80% was designed with FOV ¼ 200 mm, 1-mm nominal pixel size and eight arms. The resulting trajectory durations are: PNS ¼ 100%: 14.8 ms (conventional), 13.6 ms (PNS-optimal); PNS ¼ 80%: 17.1 ms (conventional), 15.4 ms (PNS-optimal). The trajectories were implemented on a MR750 scanner to measure images in the human brain of a healthy volunteer
(written informed consent was obtained prior to scanning). All design routines were implemented in MATLAB (The MathWorks, Natick, MA). RESULTS It is possible to shorten the spiral arm duration from the conventional, global-slew rate derated to the PNSoptimal design by 8.04 6 0.45% and 10.20 6 0.70% for PNS thresholds of 100 and 80%, respectively. This relative amount of shortening is very similar, despite the fact that many different parameters (FOVs ¼ 100, 150, 200, 250 mm; arms¼ 1, 2, 4, 8, 16) were used, leading to widely varying spiral arm durations from 3 up to 170 ms (Supporting Information Table S1). The unconstrained design is 13.30 6 0.43% and 25.19 6 0.52% shorter as compared to the PNS-optimal design with 100 and 80% thresholds, respectively. However, the PNS levels are usually exceeded with Pthresh ¼ 111:8611:1%, and hence these trajectories are forbidden for human scans. Typical spiral gradients are shown in Figure 1. Images of the human brain acquired with exemplary spiral trajectories are shown in Figure 5. A small improvement in susceptibility artifacts in areas near the sinoids is visible for trajectories with shorter durations. The EPI trajectory durations can be shortened from the conventional to the PNS-optimal design by 2.70 6 0.01%
4
Schulte and Noeske
FIG. 2. Typical EPI trajectories with the conventional designs (left and middle columns) and the PNS-optimal design (right column). One can observe that by rounding off the slew rate in the PNS-optimal design, PNS is also smoothed; hence, higher Smax values can be used in the design, which consequently shorten the trajectory duration. (Gx is indicated in blue, Gy in green, and jGj in red; the ramp function is magnified in the insets in the gradient row; FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16, Gmax ¼ 50 mT/m, Smax ¼ 200 T/m/s, Smin ¼ 70.27 T/m/s, c ¼ 334 ms.)
and 3.03 6 0.003% for PNS thresholds of 100 and 80%, respectively. This relatively small reduction is probably due to the fact that the ramps are symmetric; the effective time is increased because time is lost during the first half of the ramp. Trajectory durations ranged from 7 to 463 ms, and the PNS levels were exceeded severely for the unconstrained design with Pthresh ¼ 159:669:0%. The unconstrained design is 34.51 6 0.09% and
47.29 6 0.10% shorter as compared to the PNS-optimal design with 100 and 80% thresholds, respectively. Typical EPI trajectories are shown in Figure 3. The trajectory durations and possible time savings for a varying range of Gmax and Smax are shown in Figures 3 and 4. For larger values of Gmax and Smax, the possible savings in spiral arm durations converge to 13%. The PNS limitations for the EPI trajectories are even more
FIG. 3. Spiral arm duration of the PNS-optimal trajectory design and its possible time savings plotted over varying Gmax and Smax. (Pthresh ¼ 100%, Smin ¼ 70.27 T/m/s, c ¼ 334 ms, FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16.). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]
FIG. 4. EPI trajectory duration of the PNS-optimal design and its possible time savings plotted over varying Gmax and Smax. (Pthresh ¼ 100%, Smin ¼ 70.27 T/m/s, c ¼ 334 ms, FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16.). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]
PNS-Optimal Gradient Waveform Design
5
trajectory durations, it would be necessary to improve PNS properties by, for instance, increasing Smin (which could be achieved with a head-only gradient set). DISCUSSION AND CONCLUSIONS In summary, it is possible to shorten the duration of trajectories with a PNS-optimal design as opposed to derating Smax globally. PNS is often a neglected aspect in research sequences. However, it is necessary to consider PNS and prevent its occurrence in human scans particularly when using modern MRI scanners with high-performance gradient systems. We demonstrated in this work that inclusion of the mathematical PNS model directly in the design can not only prevent stimulation, but can also lessen the negative effects in the form of longer durations. Furthermore, it is important to educate researchers and raise awareness in the MRI community with regards to PNS. ACKNOWLEDGMENT The authors are responsible for the contents of this publication. FIG. 5. Human head images measured with exemplary spiral trajectories for the purpose of comparing conventional and PNSoptimal designs for PNS thresholds of 100 and 80%. Spiral arm durations are (from left to right) 14.8, 13.6, 17.1, and 15.4 ms, respectively. A small reduction of susceptibility artifacts is visible in the area of the sinoids when shorter trajectory durations are used (bottom row; distortions in between the eyes).
severe. Interestingly, reducing Gmax to 20 mT/m in order to stay within PNS limits leads to shorter trajectories than reducing Smax (see Figure 4). This effect is qualitatively similar for the traditional design and for Pthresh ¼ 80%, hence suggesting that simply derating Smax is not optimal. Possible savings for the new design are with 13% largest for Gmax 35 mT/m. The used gradient set is with Gmax ¼ 50 mT/ms and Smax ¼ 200 T/m/s already stronger than necessary for EPI and spirals, and further increases in Gmax and Smax will not lead to a significant reduction of trajectory duration due to PNS limitations. To utilize stronger gradients for shortening the
REFERENCES 1. Schaefer DJ, Bourland JD, Nyenhuis JA. Review of patient safety in time-varying gradient fields. J Magn Reson Imaging 2000;12:20–29. 2. International Electrotechnical Commission. IEC 60601-2-33. Medical electrical equipment - part 2–33: particular requirements for the basic safety and essential performance of magnetic resonance equipment for medical diagnosis. http://en.wikipedia.org/wiki/IEC_60601. 3. Harvey PR, Mansfield P. Avoiding peripheral nerve stimulation: gradient waveform criteria for optimum resolution in echo-planar imaging. Magn Reson Med 1994;32:236–241. 4. King KF, Schaefer DJ. Spiral scan peripheral nerve stimulation. J Magn Reson Imaging 2000;12:164–170. 5. Pipe JG, Zwart NR. Spiral trajectory design: a flexible numerical algorithm and base analytical equations. Magn Reson Med 2014;71:278– 285. 6. Meyer CH, Pauly JM, Macovski A. A rapid, graphical method for optimal spiral gradient design. In Proceedings of the 4th Annual Meeting of ISMRM, New York, New York, USA, 1996. p. 392. 7. Hargreaves BA. Spin-manipulation methods for efficient magnetic resonance imaging. Ph.D. thesis. Stanford University; 2001. Available at http://www-mrsrl.stanford.edu/brian/thesis/thesis.html. Accessed August 22, 2014.
Peripheral Nerve Stimulation-Optimal Gradient Waveform Design Rolf F. Schulte1* and Ralph Noeske2 ern whole-body MRI scanners with high-performance gradient systems are often more limited by human physiology in the form of PNS than the actual gradient hardware limitations, which are mainly the maximum gradient strength (Gmax) and slew rate (Smax). In other words, PNS often becomes a more dominating factor during gradient waveform design than Gmax and Smax. A typical gradient trajectory design starts by setting up the sequence with Gmax and Smax adhering to hardware limitations, and then iteratively derating Smax until the calculated maximum PNS is within permissible limits. The most accurate and least restrictive mathematical model to calculate PNS is the so-called convolution model, wherein a decaying nerve response function is convolved with the slew rate of the gradient trajectory (2). The trajectories most restricted by PNS are those with long durations, such as Cartesian echo-planar imaging (EPI) (3) or spiral trajectories (4). Derating Smax for the whole trajectory leads to considerably longer readout durations. The goal of the current work was to minimize this stretching of trajectories by including the PNS convolution model in the design of the gradient waveform, hence derating Smax only where necessary. This leads to a time-varying Smax instead of a constant global one. The approach was demonstrated for both spiral and EPI trajectories.
Purpose: Modern magnetic resonance imaging scanners with high-performance gradient systems have high maximum gradient strength (Gmax) and slew rate (Smax). Peripheral nerve stimulation (PNS) is often a more limiting factor for gradient waveform design than Gmax and Smax. Traditionally, the slew rate is derated globally to adhere to PNS limitations. Methods: In this work, the PNS limitation is already included in the gradient waveform design in the form of a time-varying slew rate, hence shortening the overall gradient duration. Results: Spiral and echo-planar imaging trajectories were designed with a multitude of parameters, and it was demonstrated that trajectory durations from conventional to PNSoptimal design can be shortened by 8 and 3%, respectively. Conclusion: Including PNS-limits in the gradient waveform design can shorten the duration of gradient trajectories, thereby reducing associated artifacts. Magn Reson Med C 2014 Wiley Periodicals, Inc. 000:000–000, 2014. V Key words: peripheral-nerve stimulation; peripheral nerve stimulation convolution model; echo-planar imaging; spiral imaging; time-optimal gradient-trajectory design
INTRODUCTION Rapidly switching magnetic field gradients can lead to an electric depolarization and hence stimulation of nerve cells (1). The effect is noticeable predominantly in peripheral nerves, hence it is commonly called peripheral-nerve stimulation (PNS). PNS ranges from small stimulations at the sensation threshold up to intolerable pain. Potentially much more harmful than PNS is a stimulation of heart cells, which could lead to arrhythmia. The threshold for cardiac stimulation is about an order of magnitude higher as compared to PNS, and thus adhering to the PNS limit also prevents cardiac stimulation with a large safety margin (1). The PNS limits are governed by an international engineering standard (IEC 60601-2-33) (2), and every pulse sequence on all magnetic resonance imaging (MRI) scanners has to adhere to this standard for human scans. Mod-
THEORY Peripheral-Nerve Stimulation The most accurate and least restrictive PNS model is given by the convolution of the slew rate Si ðtÞ ¼ dGdti ðtÞ of the gradient trajectory Gi ðtÞ with the nerve response (2) Ri ðtÞ ¼
Smin
Z
t 0
Si ðuÞc du; ðc þ t uÞ2
[1]
where i ¼ ðx; y; zÞ denotes the respective gradient axis. The stimulation slew rate Smin ¼ ar is the threshold at which an infinitely long ramp with this slew rate leads to a detectable stimulation for every second person on average. The effective coil length a, rheobase r, and chronaxie time c are gradient-coil specific constants, which are determined experimentally for each gradient coil model by the respective manufacturers. The overall PNS threshold in percentage for all the gradient axes combined is given by
1
GE Global Research, Munich, Germany. GE Healthcare, Berlin, Germany. €r Bildung und Forschung (BMBF); Grant sponsor: Bundesministerium fu Grant number: 13EZ1114. *Correspondence to: Rolf F. Schulte; GE Global Research, Freisinger Land€nchen, Germany. strasse 50, 85748 Garching bei Mu 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pthresh ¼ 100 R2x þ R2y þ R2z :
Additional Supporting Information may be found in the online version of this article. Received 9 April 2014; revised 10 August 2014; accepted 12 August 2014 DOI 10.1002/mrm.25440 Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V
1
[2]
The permissible maximum is Pthresh ¼ 80% for the normal and 100% for the first and second controlled modes, respectively (2). 1
2
Schulte and Noeske
Spirals Spiral trajectories are a highly efficient approach to sampling k-space. Many different algorithms to design the actual trajectories were proposed over the years, and a recent overview of these algorithms can be found in the introduction of (5). Most common are Archimedian spiral trajectories, which exhibit equidistant revolutions, yielding an approximately uniform sampling density. In the present work, we use a design algorithm developed by Meyer et al. and Hargreaves (6,7), which is available for download from the internet (http://www-mrsrl.stanford.edu/ brian/vdspiral/). The spiral trajectory can be parameterized by kðtÞ ¼ kx þ iky ¼ auðtÞeiuðtÞ :
[3]
The rate of increase a is given for a uniform sampling density by
trajectory. Therefore, we introduce a third, PNS-limited regime directly in the design of the function h(t). For this, the global, constant Smax limit, as used for Eq. [6] is modified to a time-varying function Smax ðtÞ. During each time step in the design of h(t), PNS is also calculated according to Eqs. [1] and [2]. If PNS reaches its threshold (normally 80 or 100%), the maximum slew rate used for Eq. [6] is derated for the corresponding and following time steps in such a manner that PNS remains within limits. As shown in Figure 1 (left column), a typical spiral trajectory is composed of an Smax-limited regime, possibly followed by a PNS and a Gmax-limited regime. In general, these PNSoptimal spiral trajectories are shorter in duration as compared to trajectories in which Smax is derated globally to adhere to PNS limits, but longer as compared to trajectories for which PNS is ignored. Echo-Planar Imaging
N a¼ ; 2pFOV
[4]
where N denotes the number of spiral interleaves and FOV the spatial field of view. The design problem is essentially how to choose h(t) as a function of time while sampling k-space as rapidly as possible under given constraints. Differentiating Eq. [3] and scaling it to the gyromagnetic ratio c yields the following gradient trajectory (7) GðtÞ ¼
2pa iu _ e u þ iuu_ g
[5]
A second differentiation yields the expression for the slew rate SðtÞ ¼
2 i 2pa iu h€ 2 e u uu_ þ i 2u_ þ u€u : g
[6]
The physical limits of the gradient hardware constrain both the gradient strength and slew rate to Gmax and Smax, respectively. Furthermore, the sampling rates 1=t of receiver systems are limited, possibly constraining gra2p . Also, Smax is dients further to the condition jGj gtFOV reduced in the conventional design if the PNS limits are exceeded. The parameter h can be designed as a function of time iteratively, starting with h ¼ 0 and u_ ¼ 0. The spiral design algorithm (6,7) involves checking for each new time time step whether Smax or Gmax is limiting, and determines h(t) according to Eq. [6] or [5], respectively. In general, the first part of the spiral lies in an Smax-limited regime. The gradient strength continually increases until it possibly reaches Gmax, as shown in Figure 1. From there on, the trajectory is in the Gmax-limited regime, while the slew rate continually decreases. PNS, as determined through Eq. [1], is continually increasing in the Smax-limited regime, and it only decays when the slew rate is derated, as is the case in the Gmaxlimited regime or at the end of the trajectory (Fig. 1). Derating Smax over the whole trajectory duration is suboptimal, because the PNS curve starts at zero and the PNS threshold is typically exceeded only later during the
The typical design of a bidirectional Cartesian EPI trajectory involves going to the desired k-space position with a triangular or trapezoidal prewinder gradient. The main part of EPI encoding is composed of a train of bidirectional trapezoids with phase blips along the second dimension to sample 2D k-space line by line in one readout, as shown in Figure 2. The second dimension can be split up into separate excitations (interleaves) to shorten the readout duration accordingly. The trapezoidal gradients are constrained by system limitation for Gmax and Smax, in combination with possibly sampling and PNS limitations on Gmax and Smax, respectively. For the EPI trajectory, only ramps and blips introduce PNS, while it recovers during the plateaus of the trapezoids with its constant gradient amplitude. To reduce PNS in a similar fashion as shown above for the spiral trajectory design, we can again introduce a time-varying Smax ðtÞ function. However, there is an additional constraint that the first moment of the ramps must be nulled (i.e., ascending and descending ramps must have the same absolute area) in order to start always from the same k-space position along the first dimension. That implies that the ramps have to be symmetric functions. Different functions were investigated empirically and a suitable function proved to be tanh, for which the width was optimized individually for each trajectory. The phase-encoding blips along the other dimension are relatively small and their contribution to PNS is negligible. Hence, the tanh function was only applied to the ramps of the readout trajectory. The ramp function RðtÞ is given by RðtÞ ¼ Gmax tanh ðtÞ=tanh ðt0 Þ
[7]
with the discretized time s running from t0 to s0. The parameter s0 was optimized by an exhaustive search for each trajectory, yielding values of t0 0:5. The maximum slew rate was optimized by an exhaustive search as well, decreasing its value by 1 T/m/s if the respective PNS limit was exceeded. METHODS Many different spiral and EPI trajectories were designed to investigate the possible reduction in the readout
PNS-Optimal Gradient Waveform Design
3
FIG. 1. Typical spiral trajectories designed without adhering to PNS limits (left column), with the conventional global slew-rate derated design (middle column) and with the PNS-optimal design (right column). In the conventional design, PNS generally increases in the Smax-limited regime and subsequently decreases in the Gmax-limited regime. The PNS-optimal design introduces a third, PNS-limited regime. One can observe, that PNS levels exhibit a maximal peak; and this peak can be spread out into wider area, thereby optimally utilizing PNS limits. (Gx is indicated in blue, Gy in green, and jGj in red. FOV ¼ 100 100 mm2, mtx ¼ 128 128, interleaves ¼ 4, Gmax ¼ 50 mT/m, Smax ¼ 200 T/m/s, Smin ¼ 70.27 T/m/s, c ¼ 334 ms.)
trajectory duration. All trajectories were designed for a nominal pixel size of 1 1 mm2, and FOVs of 100, 150, 200, and 250 mm along both directions. The trajectories encoded k-space in 1, 2, 4, 8, and 16 number of excitations. Note that the term number of excitations is identical to spiral arms or EPI interleaves. All trajectories were designed for the parameters of the MR750 3T whole-body scanner (GE Healthcare, MilwauT kee, WI) with Gmax ¼ 50 mT m and Smax ¼ 200 ms (physical limitation of each gradient axis), an effective coil length of a ¼ 0:333m, a rheobase of r ¼ 23:4 Ts, a stimulation T , and a chronaxie time conslew rate of Smin ¼ 70:27 ms stant of c ¼ 334ms. To further investigate the PNS limitations on gradient sets in general, another set of spiral and EPI trajectories was designed with Gmax varying from 10 to 100 mT/m and Smax from 10 to 1000 T/m/s (FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16). One set of exemplary trajectories to experimentally compare conventional and PNS-optimal design with PNS thresholds of 100 and 80% was designed with FOV ¼ 200 mm, 1-mm nominal pixel size and eight arms. The resulting trajectory durations are: PNS ¼ 100%: 14.8 ms (conventional), 13.6 ms (PNS-optimal); PNS ¼ 80%: 17.1 ms (conventional), 15.4 ms (PNS-optimal). The trajectories were implemented on a MR750 scanner to measure images in the human brain of a healthy volunteer
(written informed consent was obtained prior to scanning). All design routines were implemented in MATLAB (The MathWorks, Natick, MA). RESULTS It is possible to shorten the spiral arm duration from the conventional, global-slew rate derated to the PNSoptimal design by 8.04 6 0.45% and 10.20 6 0.70% for PNS thresholds of 100 and 80%, respectively. This relative amount of shortening is very similar, despite the fact that many different parameters (FOVs ¼ 100, 150, 200, 250 mm; arms¼ 1, 2, 4, 8, 16) were used, leading to widely varying spiral arm durations from 3 up to 170 ms (Supporting Information Table S1). The unconstrained design is 13.30 6 0.43% and 25.19 6 0.52% shorter as compared to the PNS-optimal design with 100 and 80% thresholds, respectively. However, the PNS levels are usually exceeded with Pthresh ¼ 111:8611:1%, and hence these trajectories are forbidden for human scans. Typical spiral gradients are shown in Figure 1. Images of the human brain acquired with exemplary spiral trajectories are shown in Figure 5. A small improvement in susceptibility artifacts in areas near the sinoids is visible for trajectories with shorter durations. The EPI trajectory durations can be shortened from the conventional to the PNS-optimal design by 2.70 6 0.01%
4
Schulte and Noeske
FIG. 2. Typical EPI trajectories with the conventional designs (left and middle columns) and the PNS-optimal design (right column). One can observe that by rounding off the slew rate in the PNS-optimal design, PNS is also smoothed; hence, higher Smax values can be used in the design, which consequently shorten the trajectory duration. (Gx is indicated in blue, Gy in green, and jGj in red; the ramp function is magnified in the insets in the gradient row; FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16, Gmax ¼ 50 mT/m, Smax ¼ 200 T/m/s, Smin ¼ 70.27 T/m/s, c ¼ 334 ms.)
and 3.03 6 0.003% for PNS thresholds of 100 and 80%, respectively. This relatively small reduction is probably due to the fact that the ramps are symmetric; the effective time is increased because time is lost during the first half of the ramp. Trajectory durations ranged from 7 to 463 ms, and the PNS levels were exceeded severely for the unconstrained design with Pthresh ¼ 159:669:0%. The unconstrained design is 34.51 6 0.09% and
47.29 6 0.10% shorter as compared to the PNS-optimal design with 100 and 80% thresholds, respectively. Typical EPI trajectories are shown in Figure 3. The trajectory durations and possible time savings for a varying range of Gmax and Smax are shown in Figures 3 and 4. For larger values of Gmax and Smax, the possible savings in spiral arm durations converge to 13%. The PNS limitations for the EPI trajectories are even more
FIG. 3. Spiral arm duration of the PNS-optimal trajectory design and its possible time savings plotted over varying Gmax and Smax. (Pthresh ¼ 100%, Smin ¼ 70.27 T/m/s, c ¼ 334 ms, FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16.). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]
FIG. 4. EPI trajectory duration of the PNS-optimal design and its possible time savings plotted over varying Gmax and Smax. (Pthresh ¼ 100%, Smin ¼ 70.27 T/m/s, c ¼ 334 ms, FOV ¼ 200 200 mm2, mtx ¼ 200 200, interleaves ¼ 16.). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary. com.]
PNS-Optimal Gradient Waveform Design
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trajectory durations, it would be necessary to improve PNS properties by, for instance, increasing Smin (which could be achieved with a head-only gradient set). DISCUSSION AND CONCLUSIONS In summary, it is possible to shorten the duration of trajectories with a PNS-optimal design as opposed to derating Smax globally. PNS is often a neglected aspect in research sequences. However, it is necessary to consider PNS and prevent its occurrence in human scans particularly when using modern MRI scanners with high-performance gradient systems. We demonstrated in this work that inclusion of the mathematical PNS model directly in the design can not only prevent stimulation, but can also lessen the negative effects in the form of longer durations. Furthermore, it is important to educate researchers and raise awareness in the MRI community with regards to PNS. ACKNOWLEDGMENT The authors are responsible for the contents of this publication. FIG. 5. Human head images measured with exemplary spiral trajectories for the purpose of comparing conventional and PNSoptimal designs for PNS thresholds of 100 and 80%. Spiral arm durations are (from left to right) 14.8, 13.6, 17.1, and 15.4 ms, respectively. A small reduction of susceptibility artifacts is visible in the area of the sinoids when shorter trajectory durations are used (bottom row; distortions in between the eyes).
severe. Interestingly, reducing Gmax to 20 mT/m in order to stay within PNS limits leads to shorter trajectories than reducing Smax (see Figure 4). This effect is qualitatively similar for the traditional design and for Pthresh ¼ 80%, hence suggesting that simply derating Smax is not optimal. Possible savings for the new design are with 13% largest for Gmax 35 mT/m. The used gradient set is with Gmax ¼ 50 mT/ms and Smax ¼ 200 T/m/s already stronger than necessary for EPI and spirals, and further increases in Gmax and Smax will not lead to a significant reduction of trajectory duration due to PNS limitations. To utilize stronger gradients for shortening the
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