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J Magn Reson Imaging. Author manuscript; available in PMC 2009 October 1.

Published in final edited form as:

PMCID: PMC2590645

NIHMSID: NIHMS76758

Adam C. Zelinski, M.S.,^{1} Leonardo M. Angelone, Ph.D.,^{2} Vivek K Goyal, Ph.D.,^{1} Giorgio Bonmassar, Ph.D.,^{2} Elfar Adalsteinsson, Ph.D.,^{1,}^{3} and Lawrence L. Wald, Ph.D.^{2,}^{3}

Corresponding Author Adam C. Zelinski 77 Massachusetts Avenue Room 36-680 Cambridge MA 02139 617-324-5647 (desk), 617-324-4290 (fax), Email: ude.TIM@iksnilez

The publisher's final edited version of this article is available at J Magn Reson Imaging

See other articles in PMC that cite the published article.

To investigate the behavior of whole-head and local SAR as a function of trajectory acceleration factor and target excitation pattern due to the parallel transmission (pTX) of spatially-tailored excitations at 7T.

FDTD simulations in a multi-tissue head model are used to obtain *B*^{+}_{1} and electric field maps of an eight-channel transmit head array. Local and average SAR produced by 2D-spiral-trajectory excitations are examined as a function of trajectory acceleration factor, *R*, and a variety of target excitation parameters when pTX pulses are designed for constant root-mean-square excitation pattern error.

Mean and local SAR grow quadratically with flip angle and more than quadratically with *R*, but the ratio of local to mean SAR is not monotonic with *R*. SAR varies greatly with target position, exhibiting different behaviors as a function of target shape and size for small and large *R*. For example, exciting large regions produces less SAR than exciting small ones for *R* ≥ 4, but the opposite trend occurs when *R* < 4. Furthermore, smoother and symmetric patterns produce lower SAR.

Mean and local SAR vary by orders of magnitude depending on acceleration factor and excitation pattern, often exhibiting complex, non-intuitive behavior. To ensure safety compliance, it seems that model-based validation of individual target patterns and corresponding pTX pulses is necessary.

Parallel transmission (pTX) of radio-frequency (RF) pulses in the presence of two-dimensional (2D) and three-dimensional (3D) gradient trajectories (1-5) offers a flexible means for volume excitation (6) and the mitigation of main field (*B*_{0}) and *B*^{+}_{1} inhomogeneity (7), the latter of which is particularly prevalent at high field in the body (8) and brain (9,10) due to wavelength interference (9,10) and tissue-conductive RF-amplitude attenuation (11). Parallel transmission systems are able to accomplish these tasks because unlike standard single-channel systems, their RF excitation coils consist of multiple independent transmission elements with unique spatial profiles, all of which may be modulated and superimposed to nearly-arbitrarily tailor the magnitude and phase of the transverse magnetization (12) across a chosen field of excitation (FOX) (subject to gradient and RF hardware constraints).

Parallel transmission systems are promising because they enable one to reduce the duration of an RF pulse even after one has exhausted the ability to do so by increasing the amplitude and slew rates of the system’s gradient coils. Namely, when using a parallel excitation system rather than a conventional single-channel transmit array, one may significantly undersample the excitation *k*-space trajectory, reducing the distance traveled in *k*-space, in turn reducing the duration of the corresponding RF pulse (1-5). The reduction of pulse length is important given the lengthy durations of typical 2D and 3D spatially-tailored excitation pulses and gradient hardware limitations. The ability to “accelerate” in the *k*-space domain arises due to the extra degrees of freedom provided by the system’s multiple transmit elements, analogous to readout-side acceleration where multiple reception channels allow one to undersample readout *k-*space and reduce readout time (13,14).

Unfortunately, high specific absorption rate (SAR)—defined as the average energy deposition in a region of a certain mass over an extended period of time due to the application of an excitation pulse—is a major concern in the parallel transmission of spatially-tailored multidimensional excitation pulses, especially the potential for a relatively high ratio of local SAR to average SAR. When multiple transmit channels are simultaneously employed, the local electric fields generated by each channel undergo local superposition and thus local extremes in electric field magnitude may arise (5), causing spikes in local SAR (“hot spots”). Additionally, using pTX to accelerate a given *k*-space trajectory and reduce pulse duration has the detrimental effect of increasing peak pulse power and thus SAR (15-17). For example, a conventional hard excitation pulse requires a peak power increase by a factor of *C* as its duration is shortened by the same factor, causing global SAR to increase by a factor of *C*^{2}. Furthermore, even when the repetition time (TR) of the pulse is kept constant such that total RF duty cycle decreases by a factor of *C*, SAR still increases linearly with *C* (18).

In this paper we seek to investigate the above concerns and test if global and local SAR obey intuitive scaling rules as a function of excitation *k*-space trajectory acceleration factor and the parameters of an inner-volume excitation pattern excited by 2D spatially-tailored RF excitation pulses. Our goal is to give pulse designers, coil array engineers, and RF safety researchers insight into the SAR characteristics of a high-field pTX system. Specifically, we study whole-head and maximum local 1 gram (1g) and 10 gram (10g) SAR in a multi-tissue head model during constant-fidelity excitation of 2D boxes. Box-shaped excitations are useful because they allow one to exclude moving tissues from a volume undergoing imaging (6) and also permit reduced field-of-view (FOV) imaging, which improves temporal resolution without compromising spatial resolution (19). Such excitations have applications to echo-volumnar imaging (20,21) as well as perfusion territory mapping via arterial spin labeling (22).

The fidelity of each box-shaped excitation is kept constant as measured by normalized root-mean-square error (NRMSE) with respect to a target pattern. The simulations are based on an eight-channel parallel transmission system at 7T; this field strength is chosen because many researchers are focusing on using pTX to mitigate *B*_{1}^{+} inhomogeneity occurring at 7T. Local 1g and 10g SAR are studied in addition to whole-head mean SAR because these correspond to limits specified by the Food and Drug Administration (FDA) (23) and the International Electrotechnical Commission (IEC) (24,25), and will permit observation of local SAR hot spots.

The study uses finite-difference time domain (FDTD) simulations (26) in a multi-tissue high-resolution head model as described by Angelone et al. (27), obtaining electric and *B*_{1}^{+} fields generated by each array element when its corresponding channel is driven by a unit current, as described by Angelone et al. (28). The *B*_{1}^{+} maps are then used to design a variety of pTX box-shaped excitations, the latter of which are validated with Bloch-equation simulations. Finally, the electric field maps are used to calculate whole-head and maximum local 1g and 10g SAR due to each pTX pulse as a function of target flip angle, position, size, smoothness, and orientation, as well as spiral trajectory undersampling (acceleration) factor, *R*. The results demonstrate a wide range of SAR values that arise due to tailored excitations that produce similar spatial patterns. This data, together with the non-intuitive behavior SAR exhibits as a function of the excitation parameters, suggests that explicit SAR calculations will likely be needed on a per-pulse basis, even for relatively minor variations in RF pulse properties, in order to ensure that local SAR values meet regulatory criteria.

SAR characteristics of excitation pulses transmitted through an eight-channel head array are calculated in a high-resolution (1×1×2 mm^{3}) 29-tissue human head model, the latter of which is obtained via segmentation of anatomical MRI data. Each of the tissues in the model is assigned a density, ρ (kg/m^{3}), and electrical conductivity, σ (S/m), using Federal Communications Commission data available at www.fcc.gov/fcc-bin/dielec.sh. Overall, the model consists of ~2.3 million Yee cells (29).

The pTX system is modeled by placing eight copper circular loop elements at 45° increments along a 25-cm-diameter cylindrical surface, the latter of which is centered on the head, as described by Angelone et al. (28). Each loop element is overlapped to null mutual inductance with its neighbors and has a diameter of 15 cm, a trace width of 1 cm, and an input resistance of 50 Ω.

Electric and *B*_{1}^{+} fields produced in the head by each individual transmit channel are needed in order to design and calculate the SAR of pTX excitation pulses. We obtain these via 300-MHz FDTD simulations (XFDTD software, REMCOM Inc., State College, PA, USA), and then evaluate the SAR of any given excitation by superimposing modulated versions of these electric fields (5,18,30,31), avoiding the computationally-intractable scenario of FDTD-simulating each of the 200+ pTX excitations evaluated in this paper.

For *p* = 1, …, 8, we drive the *p*th transmit channel of the eight-channel array with a 1-ampere peak-to-peak 300-MHz sinusoid, leave all other channels dormant, and use the FDTD method to obtain steady-state electric fields *per ampere* of input to the *p*th channel, **E*** _{p}*(

The *B*_{1}^{+} field that arises when channel *p* is driven by the unit ampere input, denoted *S _{p}*(

$${S}_{p}\left(\mathbf{r}\right)=\frac{1}{2}\{{B}_{p,x}\left(\mathbf{r}\right)+{\mathit{jB}}_{p,y}\left(\mathbf{r}\right)\}(\mathrm{T}\u2215\text{ampere}).$$

[1]

Note: “*S _{p}*” is used (rather than “

We now describe one way to design a set of *P* RF pulse shapes to concurrently play through the *P* elements of a *P*-channel parallel transmission array in order to generate a user-defined target excitation, *d*(**r**). Note that “pTX pulse” will be used as shorthand for “a set of *P* concurrently-transmitted RF pulse shapes that yields a box-shaped excitation”. To begin, we will apply the linear formalism of (33) to reduce the design problem to that of solving a linear system. Assume for now that the gradient waveforms, **G**(*t*) = [*G _{x}*(

For all upcoming simulations, we design pTX pulses that form approximations of a 2D box-shaped inner-volume target in the center transverse slice of the head; the desired excitation has zero-degree flip angle at spatial locations outside of the box, a positive flip angle inside the box, and zero phase everywhere. The FOX is the center transverse slice of the head. The *k*-space trajectories are 2D spirals that are radially undersampled (accelerated) by a factor of *R* relative to a 25.6 cm FOV, where *R* = 1, 2, …, 8. Based on this definition, the *R* = 1 spiral is simply a conventional Nyquist-sampled spiral. The gradients are always constrained to amplitude and slew rates of 30 mT/m and 300 T/m/s, such that the *R* = 1, …, 8 spiral trajectories have durations of 6.8 ms, 3.47 ms, 2.36 ms, 1.81 ms, 1.48 ms, 1.26 ms, 1.11 ms, and 0.99 ms.

We first apply the small-tip-angle approximation (12) to the Bloch equation (34), yielding

$$m\left(\mathbf{r}\right)=j\gamma {M}_{0}\sum _{p=1}^{P}{S}_{p}\left(\mathbf{r}\right){\int}_{0}^{L}{a}_{p}\left(t\right){e}^{j\mathbf{r}\cdot \mathbf{k}\left(t\right)}\mathit{dt},$$

[2]

where *m* is the (approximate) complex-valued transverse magnetization (rad) arising due to the transmission of the *P* RF pulse shapes (i.e., one pTX pulse) in the presence of the gradients, γ the gyromagnetic ratio (rad/T/sec), *M _{0}* the steady-state magnetization,

We now discretize Eq. [2] by sampling space at locations **r**_{1}, …, **r*** _{Ns}* within the user-defined FOX and sampling time at

$$\mathbf{m}={\mathbf{S}}_{1}\mathbf{F}{\mathbf{a}}_{1}+\cdots +{\mathbf{S}}_{P}\mathbf{F}{\mathbf{a}}_{P}=[{\mathbf{S}}_{1}\mathbf{F}\cdots {\mathbf{S}}_{P}\mathbf{F}]\left[\begin{array}{c}\hfill {\mathbf{a}}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill {\mathbf{a}}_{P}\hfill \end{array}\right]={\mathbf{A}}_{\mathrm{tot}}{\mathbf{a}}_{\mathrm{tot}},$$

[3]

where **m** is an *N _{s}*-element vector of samples of

To excite a desired pattern, *d*(**r**), *P* pulse shapes are needed. To generate these pulse shapes we first determine *B*_{1}^{+} maps for each of the *P* channels and then decide on a *k*-space trajectory, which lets us generate **S*** _{p}*,

$$\underset{{\mathbf{a}}_{\mathrm{tot}}}{\mathrm{min}}\{\Vert \mathbf{d}-{\mathbf{A}}_{\mathrm{tot}}{\mathbf{a}}_{\mathrm{tot}}{\Vert}_{2}^{2}+\lambda \Vert {\mathbf{a}}_{\mathrm{tot}}{\Vert}_{2}^{2}\}$$

[4]

for **a**_{tot}. In practice, setting λ to a small nonnegative value and solving Eq. [4] results in a reasonably-conditioned solution that produces an excitation close (in the *L _{2}* sense) to the one desired (33). After solving Eq. [4], we extract samples of each

Unfortunately, Eq. [4] does not let us allocate and fix the residual error between the resulting and desired excitation across different spatial regions. For example, for box-shaped excitations, designers are willing to tolerate larger errors outside of the box but require *within-box* error to be small. Equation [4] gives us only one variable, λ, with which to influence overall error, and we are thus unable to design pTX pulses that achieve both a chosen in-box error and a chosen overall (or out-of-box) error.

To circumvent this problem, we pose a novel algorithm that designs a pTX pulse that yields an excitation with both a desired in-box error and a desired overall error. First, we arrange *N*_{in} within-box samples of *d*(**r**) into **d**_{in}, and *N*_{out} out-of-box samples into **d**_{out} (where *N*_{in} + *N*_{out} = *N _{s}*), and then set

$$\underset{{\mathbf{a}}_{\mathrm{tot}}}{\mathrm{min}}\{\Vert \mathbf{W}(\mathbf{d}-{\mathbf{A}}_{\mathrm{tot}}{\mathbf{a}}_{\mathrm{tot}}){\Vert}_{2}^{2}+\lambda \Vert {\mathbf{a}}_{\mathrm{tot}}{\Vert}_{2}^{2}\}$$

[5]

where **W** is an *N _{s}* ×

In this paper, within-box and overall normalized root-mean-square error (NRMSE) are used as fidelity metrics and expressed as percentages. In-box NRMSE, ε_{1}, is defined as $100.(\Vert {\mathbf{d}}_{\text{in}}-{\mathbf{A}}_{\text{in}}{\mathbf{a}}_{\mathrm{tot}}{\Vert}_{2}\phantom{\rule{thickmathspace}{0ex}}\u2215\phantom{\rule{thickmathspace}{0ex}}\Vert {\mathbf{d}}_{\text{in}}{\Vert}_{2})$, whereas overall NRMSE, ε_{tot}, is defined as $100.(\Vert \mathbf{d}-{\mathbf{A}}_{\mathrm{tot}}{\mathbf{a}}_{\mathrm{tot}}{\Vert}_{2}\phantom{\rule{thickmathspace}{0ex}}\u2215\phantom{\rule{thickmathspace}{0ex}}\Vert \mathbf{d}{\Vert}_{2})$. Whenever we design a pTX pulse, we first decide on a desired in-box error and overall error, denoted ε_{1,des} and ε_{tot,des}. We then iteratively search over (α, λ), repeatedly solving Eq. [5] until a solution **a**_{tot} is found such that the resulting ε_{1} and ε_{tot} are close to ε_{1,des} and ε_{tot,des}. Finally, we simulate the waveform samples in **a**_{tot} and compare the simulated excitation to the desired one, ensuring that ε_{1} and ε_{tot} remain close to ε_{1,des} and ε_{tot,des}. Overall, this approach lets us design a variety of pTX pulses across different scenarios while guaranteeing that every excitation has essentially identical in-box and overall error.

After designing *P* pulse shapes to produce a desired excitation, we must determine the global and local SAR they produce in the head model. At this point, we know *N _{t}* time samples of each pulse shape spaced uniformly in time by Δ

We first calculate SAR (W/kg) at each location **r** by superimposing the electric field produced by each transmit channel due to each time sample in the RF pulse shape and then time averaging the net field’s squared magnitude over the pTX pulse duration and weighting by the conductivity and density of that location. Formally,

$$\mathrm{SAR}\left(\mathbf{r}\right)=\frac{\sigma \left(\mathbf{r}\right)}{2\rho \left(\mathbf{r}\right)}\frac{1}{L}{\int}_{0}^{L}\Vert \mathbf{E}(\mathbf{r},t){\Vert}_{2}^{2}\mathit{dt}\cong \frac{\sigma \left(\mathbf{r}\right)}{2L\rho \left(\mathbf{r}\right)}{\Delta}_{t}\sum _{n=0}^{{N}_{t}-1}\Vert \mathbf{E}(\mathbf{r},n{\Delta}_{t}){\Vert}_{2}^{2},$$

[6]

where **E**(**r**, *t*) is the superposition of the electric fields generated by each of the channels scaled by the waveform samples transmitted at each time instant (5), i.e.,

$$\mathbf{E}(\mathbf{r},t)=\sum _{p=1}^{P}{a}_{p}\left(t\right){\mathbf{E}}_{p}\left(\mathbf{r}\right).$$

[7]

Recall that each **E*** _{p}*(

Having obtained SAR(**r**) for all **r** in the head, whole-head global SAR is obtained by averaging the SAR(**r**) values. Likewise, *N*-gram SAR at each **r** is obtained by finding an *N*-gram cube around each **r** and then averaging SAR(**r**) over all **r** within the cube, in line with (35). [The FDA (23) and IEC (24,25) mandate averaging over cubes rather than spheres.]

To find an *N*-gram cube around each **r**, we use a fast algorithm rather than brute-force region growth because the latter is computationally infeasible. We first form a list of spatial positions from a simple cubic lattice in order of distance from the origin and interpret this as a “universal” list of offsets from any given position **r**. We then find the set of voxels that comprise *N* grams of tissue around **r** by choosing the shortest prefix of this list that yields sufficient total mass. Formally, given the mass per voxel around **r** and this universal list, we form a voxel mass vector and search the cumulative sum of this voxel mass vector for the number of voxels needed to form the set; this rapidly determines the cluster of points that comprise an *N*-gram cube around **r**. Note that for a spatial location on the edge of the head bordering air (where σ_{air} = 0), the associated **r** may not be at the center of the corresponding *N-*gram averaging cube, but this is mitigated by the fact that this **r** is often part of the averaging cubes of many adjacent spatial locations, as noted in (36).

When computing SAR, the effect of the trajectory acceleration factor, *R*, is always accounted for to ensure that any SAR differences across *R* reflect only the extra power needed to maintain target fidelity. For example, *R* = 1 pTX pulses (6.8 ms long) have a 100% duty cycle, whereas *R* = 4 pTX pulses (1.81 ms long) have a 26% duty cycle. This is accomplished by fixing the repetition time (TR) to 6.8 ms for each effective sequence in which the pTX pulses are used, regardless of *R*.

We begin by driving the array in a birdcage configuration, transmitting a 2.6-ampere, 3-ms, 100%-duty-cycle rectangular pulse shape through each channel in the absence of gradients and setting the phase of channel *p*’s pulse shape to 45(*p* — 1) degrees. This produces a 90° flip angle in the center transverse slice of the head, analogously to the “90°/3-ms” hard pulse of (37). Figure 2 depicts the inhomogeneous flip angle map that arises in the head when the set of pulse shapes that comprise one pTX pulse undergoes Bloch-equation simulation, along with the resulting SAR values. The image exhibits strong center brightening, resembling brain images collected on actual 7T systems equipped with homogeneous RF excitation volume birdcage coils (9,10,27). In this case, there is approximately a 3-to-1 variation in peak-to-trough flip angle. The qualitative similarity of the simulated birdcage mode shown in Fig. 2 to the simulated and *in vivo* images presented in the lower half of the fifth figure in (10) suggests, to some extent, that the simulation methodology accurately captures the behavior of an eight-channel array at 7T.

The top row of Fig. 3 shows the simulated square inner-volume excitations for *R* = 4, along with ε_{1} and ε_{tot} error with respect to the 28-mm × 28-mm target box pattern; for all cases, ε_{1} = 15 ± 2% and ε_{tot} = 40 ± 1%. Figure 3 also graphs global and maximum local 1g SAR for each acceleration factor (*R* = 1, …, 8) as a function of flip angle (θ = 5, 15, …, 45 degrees). For fixed *R*, global and local SAR scale quadratically with flip angle, whereas for fixed θ, mean and local SAR increase more than quadratically with *R*. Finally, global and local SAR vary strikingly by over five orders of magnitude across the various excitations.

SAR as a function of acceleration factor, *R*, and flip angle, *θ* (fixed excitation quality). Target: 28 mm × 28 mm centered square with in-box flip angle *θ*, *ε*_{1} = 15 ± 2%, *ε*_{tot} = 40 ± 1%. Top row: **...**

In general, the eight RF pulse shapes used to produce each of the excitations in Fig. 3 consistently exhibit low amperages during the time interval when the trajectory proceeds through high-frequency *k*-space regions, and progressively larger amperages as the trajectory spirals inward toward the origin of *k*-space. In other words, all RF pulse shapes deposit most of their energy at low spatial frequencies. For the 15° excitation, the peak RF magnitude across all eight pulse shapes increases from 1.5 amperes for *R* = 1, to 8 amperes for *R* = 4, to 150 amperes for *R* = 7. The *R* = 7 pulse shape has a much higher peak than the *R* = 1 pulse shape, a natural consequence of the fact that with only an *R* = 7 spiral, there are very few degrees of freedom in *k*-space with which to form the excitation (relative to the Nyquist-sampled *R* = 1 spiral), forcing the algorithm to drive the channel profiles intensely with high-amplitude RF pulse shapes in order to form the desired pattern. This lack of *k*-space freedom forces the system to rely heavily upon its degrees of freedom in the spatial domain and partly explains the high SAR values of the highly-accelerated pTX pulses.

Since MR scanners typically monitor mean SAR (in the form of average forward power), it is informative to assess local SAR under the condition where the operator adjusts the sequence to achieve a fixed global average SAR. Figure 4 depicts such a scenario, showing iso-SAR operating contours in the (*R*, θ) parameter space that characterize how maximum local 1g and 10g SAR vary when mean SAR is held constant at 0.15 W/kg. Excitation patterns and residual errors here are the same as in Fig. 3. Figure 4 shows that maximum local 1g SAR is always higher than maximum local 10g SAR, which in turn is always higher than whole-head mean SAR. Across *R*, local 1g and 10g SAR vary from peak-to-trough by factors of 1.64 and 1.34, respectively. Furthermore, Fig. 4’s left panel shows that in order to produce fixed-fidelity excitations with 0.15 W/kg mean SAR, in-box flip angle must decrease rapidly with *R*, e.g., only a 2° flip is achievable for *R* = 6, whereas a 36° flip is possible using an unaccelerated trajectory. Counter-intuitively, Fig. 4’s right panel reveals that maximum local 1g and 10g SAR are not monotonic with *R*, e.g., the *R* = 4 excitation produces 1.3 times less maximum local SAR than the *R* = 1 excitation. Finally, the ratios of local 1g and 10g SAR to mean SAR are erratic: they are roughly constant for *R* = 1 to 3, decrease for *R* = 4, rise for *R* = 5, 6, and then decrease again for *R* = 7, 8.

To assess the effect of excitation symmetry on SAR we design pTX pulses that excite 28 mm × 28 mm, 15° flip angle squares at different locations along the right-left (RL, or “*x*”) axis and the anterior-posterior (AP, or “*y*”) axis. Excitation quality is fixed across all designs. Figure 5 illustrates how global and maximum local 1g SAR behave when exciting boxes centered at different *x* locations. The upper row depicts *R* = 4 excitations, while the next two rows show local 1g SAR maps arising due to *R* = 1 and *R* = 5 excitations. For a given row of local SAR maps, all are displayed using the same dynamic range, permitting local SAR comparisons among the maps of each row. Furthermore, because a 1g local SAR data point exists for each of the voxels in the 3D head model, the 2D local SAR maps have been “collapsed” along the *z*-axis. Namely, for each location (*x*, *y*) in each given map, we have displayed max* _{z}* SAR

SAR as a function of *R* and shift along *x* (fixed excitation quality). Target: 15°, 28 mm × 28 mm square whose center *x*_{0} varies along *x* with *ε*_{1} = 15 ± 1% and *ε*_{tot} = 40 ± 1%. Top row: *R* = 4 excitations. Second **...**

SAR as a function of *R* and shift along *y* (fixed excitation quality). Target: 15°, 28 mm × 28 mm square whose center *y*_{0} varies along *y* with *ε*_{1} = 15 ± 1% and *ε*_{tot} = 40 ± 1%. Top row: *R* = 4 excitations. Bottom **...**

In both Fig. 5 and Fig. 6, SAR as a function of position is roughly convex for low trajectory acceleration factors (*R* = 1 to 4), i.e., centered excitation boxes yield the lowest SAR. At higher acceleration factors, however, centered excitations generally produce the highest SAR. Figure 5 also shows that mean and local SAR do not always behave similarly, e.g., for *R* ≤ 4, the mean SAR vs. *R* curves are generally symmetric about *x*_{0} = 0, whereas the local SAR curves are asymmetric. Figure 6 shows that qualitative SAR differences also arise when shifting the box along *y*. Furthermore, for fixed *R*, all of Fig. 6’s local and mean SAR curves seem to exhibit the same shape, in contrast with the *R* ≤ 4 curves of Fig. 5.

Regarding the local SAR maps of Fig. 5, the local 1g SAR patterns for *R* = 1 change across space in a way that is correlated with the position of the excited box, while the *R* = 5 local SAR maps, to the first order, seem to only scale by a multiplicative constant with box position. Finally, the peaks in both the *R* = 1 and *R* = 5 local SAR maps are not strongly correlated with the box position.

Figure 7 shows the effect of excitation quality on SAR, illustrating global and maximum local 1g SAR for *R* = 1, …, 8 due to exciting 28 mm × 28 mm centered boxes with different in-box fidelities; ε_{1} varies from 10%, 15%, …, 30%, while ε_{tot} is fixed at 40 ± 2%. Figure 7’s top row shows excitation patterns associated with *R* = 4, while the bottom row shows global and maximum local 1g SAR as a function of ε_{1}. We see from the top row of *R* = 4 excitations that the in-box flip angle decreases (the boxes grow darker) as more in-box error is permitted. Further, global SAR and maximum local 1g SAR both increase rapidly with *R*, ranging over three orders of magnitude for each fixed ε_{1}. Finally, SAR decreases fairly regularly with ε_{1} for most accelerations, but this does not hold for *R* = 5 and *R* = 6, e.g., when *R* = 6, the ε_{1} = 15% excitation produces 1.3 and 1.5 times *higher* mean and local SAR than the ε_{1} = 10% excitation.

Figure 8 illustrates the dual of Fig. 7’s experiment, showing how global and maximum local 1g SAR behave across *R* when producing excitations with different *overall* fidelities (ε_{tot} = 20%, 25%, …, 40%) while keeping ε_{1} = 15 ± 1%. Here, global SAR and local SAR vary by over four orders of magnitude across *R* and ε_{tot}. For *R* ≠ 5, global and local SAR decrease monotonically with ε_{tot}, and even for *R* = 5, SAR is nearly monotonic: the lower-fidelity ε_{tot} = 35% excitation produces only 1.007 times higher SAR than the ε_{tot} = 30% case. (Note: there are no data points for ε_{tot} = 20% when *R* = 7 and *R* = 8 because for these cases the pTX pulse design algorithm does not produce well-conditioned solutions.)

Figure 9 illustrates the affect of excitation orientation on global and maximum local 1g SAR across the eight acceleration factors. Here we excite 44-mm × 28-mm rectangles in the center of the head that are rotated by degrees, characterizing SAR as a function of *R* and . Excitation fidelity is fixed as in prior experiments. For *R* ≤ 4, SAR is relatively constant, whereas for *R* > 4, SAR exhibits spikes at particular rotations. For example, when *R* = 5, exciting a 90°-rotated box produces 3.3 W/kg of maximum local 1g SAR, whereas exciting a 135°-rotated box produces 9.54 W/kg, i.e., local SAR varies by a factor of 2.9 when simply rotating the excitation by 45 degrees, even with acceleration factor and excitation quality held constant. Finally, SAR is generally higher when boxes are highly asymmetric with respect to the AP-direction (*y*-axis) of the head (consider the *R* = 6, = 45° excitation).

Figure 10 illustrates how the size of an excitation impacts global and maximum local 1g SAR. Here a series of increasingly-larger boxes of length and width *N* (mm) are excited across various acceleration factors. Excitation fidelity is fixed as in prior experiments, and Fig. 10 is formatted analogously to Figs. Figs.66 to to9.9. Both the log-scaled and normalized data along the middle and bottom rows of Fig. 10 show that mean and local SAR behave quite differently depending on *R*: for *R* ≤ 4, SAR grows rapidly with *N*, whereas for *R* > 4, SAR decreases with *N*. This means that for highly accelerated trajectories, exciting larger regions actually reduces energy deposition. For example, for *R* = 5, exciting a 52 mm box produces 3.2 times less mean SAR than exciting a 12 mm box, yet for *R* = 1, the opposite behavior occurs: exciting a 52 mm rather than 12 mm box leads to ten times higher mean SAR.

Figure 11 shows how sharp excitation pattern edges affect SAR. Here, increasingly-smooth 44 mm × 44 mm centered boxes are excited and the SAR of each is analyzed. The series of desired excitations is generated by applying successively larger *M* mm × *M* mm Gaussian smoothing kernels to the original sharp-edged target pattern and running the design algorithm each time to produce a pTX pulse. The top row of Fig. 11 shows that for *R* = 4, the excitations increase in smoothness with *M* as intended. The bottom row shows that mean and local SAR decrease significantly with target smoothness for all *R*. For example, when *R* = 6, exciting the smoothest box requires 2.8 times less global SAR and 2.7 times less maximum local 1g SAR than does exciting the sharpest-edged box.

Across all experiments, maximum 1g SAR ranges from 3.8 to 13.8 times larger than corresponding mean SAR and on average is 5.6 times larger. Likewise, maximum 10g SAR is always 2.3 to 7.7 times larger than mean SAR and on average is 3.4 times larger. Finally, maximum 1g SAR ranges from 1.1 to 2.1 times larger than maximum 10g SAR and on average is 1.7 times larger.

The high-resolution head model in this study meets or exceeds the requirements for accurate SAR calculation in volumes as small as 1g because each of its voxels contains no more than 0.0037 grams of tissue (every possible 1g region consists of at least 270 cells). Additionally, the model’s high resolution mitigates staircasing artifacts (38) that adversely impact local field and local SAR calculations because the model’s ~1000 cells per 300-MHz wavelength should reduce staircase error to less than one decibel (39).

Considerable effort has been made to keep excitation fidelity constant across all experiments and thus permit SAR comparisons across experiments. One noticeable trend is that a consistent “jump” in mean and local SAR occurs as *R* transitions from 4 to 5, regardless of excitation shape, size, asymmetry, etc., which suggests the array is better conditioned in the *R* ≤ 4 operating region.

Across all experiments, global and local SAR always increase dramatically with trajectory acceleration factor. Highly-accelerated pulses yield extremely high SAR values and may not be practical for *in vivo* imaging. Furthermore, because even moderate trajectory accelerations may lead to order-of-magnitude SAR increases, it seems likely that further innovation beyond or in conjunction with conventional pTX pulse design is necessary in order to enable the use of highly accelerated trajectories. Two recent examples of design innovation are the emerging body of algorithms that explicitly consider SAR during the design of parallel transmission pulses and multi-channel arrays (40-46) and the recent proposal of several methods that monitor and track SAR arising during a scan in real time (47,48). One limitation that seems unavoidable from a design standpoint, however, is that in order to minimize or guarantee local SAR at a particular spatial location during the design stage, one must place a quadratic constraint on the system of equations being solved (41,46). For example, the algorithm of (41) explicitly accounts for global SAR as well as local SAR at a few other locations by incorporating several quadratic constraints into the design. But in order to design truly SAR-optimal pulses—ones where local SAR is guaranteed at *all* spatial locations—one is faced with the computationally-intractable problem of solving a system of equations with millions of quadratic constraints. Finally, one may be able to mitigate SAR by taking a hardware approach, improving the efficiency and spatial encoding capabilities of pTX arrays (49,50).

In Fig. 3, mean and local SAR scale quadratically with flip angle. Fig. 4 shows that the ratio of local to mean SAR does not increase monotonically with acceleration factor. Figures Figures55 and and66 show that mean and local SAR behave non-intuitively when excitation spatial position is varied. For *R* = 1, local SAR varies significantly across space with excitation position, whereas for *R* = 5, to the first order, the local SAR maps simply undergo scalings by a multiplicative constant. Figure 7 shows that it is not always possible to reduce SAR by simply permitting more excitation error within a specific spatial region (e.g., within the box) and that one may in fact significantly increase SAR (e.g., by a factor of 1.5) by generating lower-quality excitations. Figure 8, on the other hand, shows that permitting more *overall* excitation pattern error generally decreases SAR. Figure 9 shows that for large acceleration factors, excitation asymmetry detrimentally impacts SAR. Figure 10 shows that SAR is sensitive to excitation size, suggesting that parallel transmission may complicate the ability to perform and flexibly scale reduced-FOV imaging (19). Finally, Fig. 11 suggests that sharp edges are costly in terms of SAR, revealing that one reliable way to reduce SAR is to excite a smoother version of the desired pattern.

In conclusion, with the exception of dramatic SAR increases with trajectory acceleration factor, global and local SAR do not always exhibit intuitive trends. Nonetheless, it is clear from the experiments that maximum local 1g and 10g SAR are always significantly higher than global SAR. Because both the United States and European Union safety standards impose limits on maximum local SAR (23,24,25), and because the ratio of local to global SAR is often considerably greater than the regulatory ratio required to maintain safety compliance for the human head by monitoring average power alone, it is evident that local SAR, rather than global SAR, is the limiting factor of eight-channel parallel transmission at 7T. Namely, it is likely that the safety limit imposed upon local SAR will preclude the user from utilizing the full limit of mean SAR. Seifert et al. have arrived at this identical conclusion after studying four-channel parallel transmission at 3T (31).

Although the range of excitation patterns studied here is not exhaustive, a sufficient number of variations of the size, shape, position, rotation and smoothness of the excitation pattern have been considered to suggest that mean and local SAR exhibit complex and at times non-intuitive behavior as a function of target excitation pattern and trajectory acceleration factor. In order to ensure patient safety, it seems that model-based validation of individual target patterns and corresponding sets of parallel transmission pulses will be required.

We thank our anonymous reviewers for their useful suggestions.

Grant Support NIH NCRR P41RR14075, NIBIB 1R01EB006847, 1R01EB007942, 1R01EB000790; MIND Institute; Siemens Medical Solutions; US Department of Defense NDSEG Fellowship F49620-02-C-0041; R. J. Shillman Career Development Award

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