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Magn Reson Med. Author manuscript; available in PMC 2013 June 1.

Published in final edited form as:

Published online 2011 November 14. doi: 10.1002/mrm.23140

PMCID: PMC3291736

NIHMSID: NIHMS312707

Joonsung Lee,^{1} Matthias Gebhardt,^{2} Lawrence L. Wald,^{3,}^{4} and Elfar Adalsteinsson^{1,}^{4}

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

See other articles in PMC that cite the published article.

The management of local and global power deposition in human subjects (Specific Absorption Rate, SAR) is a fundamental constraint to the application of parallel transmission (pTx) systems. Even though the pTx and single channel have to meet the same SAR requirements, the complex behavior of the spatial distribution of local SAR for transmission arrays poses problems that are not encountered in conventional single-channel systems and places additional requirements on pTx RF pulse design. We propose a pTx pulse design method which builds on recent work to capture the spatial distribution of local SAR in numerical tissue models in a compressed parameterization in order to incorporate local SAR constraints within computation times that accommodate pTx pulse design during an in vivo MRI scan. Additionally, the algorithm yields a Protocol-specific Ultimate Peak in Local SAR (PUPiL SAR), which is shown to bound the achievable peak local SAR for a given excitation profile fidelity. The performance of the approach was demonstrated using a numerical human head model and a 7T eight-channel transmit array. The method reduced peak local 10g SAR by 14–66% for slice-selective pTx excitations and 2D selective pTx excitations compared to a pTx pulse design constrained only by global SAR. The primary tradeoff incurred for reducing peak local SAR was an increase in global SAR, up to 34% for the evaluated examples, which is favorable in cases where local SAR constraints dominate the pulse applications.

Recent work has demonstrated great utility of parallel transmission (pTx) systems for radio frequency (RF) pulses in magnetic resonance imaging (MRI) for the purpose of generating more flexible magnetization profiles than is feasible with conventional single-channel RF systems (1–9). A strong motivation for the deployment of pTx RF systems is at ultra high field, such as 7 T, where pTx RF excitation benefits have been demonstrated by very effective mitigation of B1+ field inhomogeneities to achieve uniform slice selective excitations and spatially tailored excitations. Although pTx is an excellent tool to achieve demanding flip-angle targets, a critical challenge to routine in vivo application of this technology is the management of the specific absorption rate (SAR), which is a measure of the deposited power per unit mass of tissue due to the RF pulses. Maximum allowable values for SAR are specified by FDA and IEC guidelines (10–12) and must be met both globally, such as for power absorbed by the whole head or whole body, and locally, e.g. power absorbed per 10g of tissue. Recently, several approaches were proposed to estimate the distribution of local SAR in living subjects via B1 mapping under several assumptions that the longitudinal component of magnetic field is negligible and the variations of tissue conductivity is smaller than the variations of electric fields along integration path (13), or the axial-component of the electric field is dominant (14). Conventional approaches rely on numerical models of human anatomy and RF coils to estimate the electric field distribution due to a given pTx RF pulse, which, when combined with tissue conductivity and density can be used to generate maps of local SAR (15–17). Given such a local SAR distribution, it is straightforward to extend many of the proposed parallel RF transmission designs to incorporate penalties for local SAR in iterative optimization schemes, but the resulting computational demands are prohibitive for body models with ~mm resolution and three-dimensional field-of-view (FOV), which can yield voxel counts on the order of ~10^{5}–10^{6}.

Several pTx RF pulse design approaches have been proposed to incorporate SAR penalty in addition to the constraints on the pulse excitation performance, including the RF pulse power (4–6,18–23), which carries minimal computation penalty but naturally imposes no specific minimization of local SAR; the inclusion of global SAR penalty term (9), which incorporates measurements of forward and reflected power (24) but still does not directly penalize local SAR; multiple constraints on peak RF power, average RF power, and average SAR over multiple compartments (25) where peak local SAR over 10g tissue is not directly penalized; a so-called K maximum SAR constraints (26,27) where in an iterative scheme the design penalizes K current local SAR hotspots; and channel-dependent iterative power regularization (28), which is an iterative method to update weighting factors for excitation channels near a current hotspot. These methods can be useful to reduce both the global SAR and peak local SAR; however, they do not reveal any knowledge on the minimum peak local SAR subject to the given constraints on the pulse excitation performance and how close their peak local SAR of the pTx RF pulse to the minimum.

In this work, we describe and demonstrate a pTx RF design that incorporates electric fields from numerical simulations on human body models to jointly arrive at both a local-SAR penalized pTx pulse for a specified excitation performance fidelity, as well as a well-defined lower-bound on peak local SAR for all pTx pulses subject to the same excitation performance constraints when played on the same array for the same subject, which we refer to as the Protocol-specific Ultimate Peak in Local SAR, or the PUPiL SAR. Comparison of the local SAR achieved by an actual pulse design algorithm to the PUPiL SAR calculated for the spatial fidelity achieved provides a useful metric of the pulse design algorithm. While perhaps not needed for robust convex optimization schemes which can be counted on to find the global minimum, we propose that the PUPiL SAR value may be useful for monitoring non-convex optimization algorithms.

The local SAR minimization is general in the sense that many different and previously proposed pTx design methods are amenable to the inclusion of this power penalty and estimation of peak local SAR bounds. We demonstrate our method for three particular pTx design algorithms, RF shimming (20,29,30), spoke design (6,31,32), and arbitrary excitation for spiral trajectory (2,5,7,23,33), but the same peak local SAR constraints can be applied to e.g. spatially selective excitation (2,24), composite pTx pulses for uniform volume excitation (21), spatial-domain design for small-flip-angle approximation (23), linear class of large-tip-angle pulses (19), and methods based on optimal control (18,22).

Our work builds on a critical development by Gebhardt et al. (34) and Eichfelder et al. (35) that decreases the complexity of the prediction of the peak local SAR in a numerical human model through the use of so-called virtual observation points, or VOPs. The VOPs effectively enable a model compression, where a model-wide peak local SAR value can be estimated to within a specified tolerance, by monitoring the virtual observation points instead of searching exhaustively over all voxels in the 3D model. We generalize the VOP model compression method for local SAR estimation to capture more dependencies among the voxels in the model and decrease the number of VOPs required to estimate local SAR. We propose a heuristic, greedy algorithm that generates VOPs to determine both lower and upper bounds on the peak local SAR for all voxels in the numerical model. The VOPs can be pre-computed once for a given model and array configuration and applied in subsequent computations to efficiently estimate peak local SAR due to a given pTx RF pulse. By capturing peaks in the local SAR distribution with VOPs, it becomes feasible to incorporate peak local SAR constraints in pTx RF designs.

Our proposed algorithm for local-SAR penalized pTx design is an iterative method that solves jointly for both a bound on peak local SAR and a pTx pulse that approaches this local SAR bound subject to the given constraints on the pulse excitation performance.

When compared to pTx pulses that only incorporate a global SAR penalty, we demonstrate 14–66% reduction in the peak local SAR (10g) for slice-selective pTx excitation and 2D selective pTx excitation while maintaining a fixed excitation performance. The peak local SAR of the demonstrated pTx designs approaches the estimated lower bound on achievable peak local SAR to within 2–36%. In return for improved peak local SAR performance, the principal trade-offs incurred by the proposed method are increased global SAR, which was less than 34% in our demonstrations, and an increased design computation time, but which is still feasible for in vivo applications (sub minute) for spoke-based pTx designs for B1+ inhomogeneity mitigation.

FDTD numerical simulations using REMCOM (REMCOM Corp., State College, PA) (36) were used to simulate electrical fields and magnetic fields in the Virtual Family model, Ella, (37) for an eight-channel excitation array of loops for brain imaging at 7T (ω_{0}=297.2 MHz). The model includes both head and shoulder of the numerical model as shown in Figure 1, with voxel size of 3×3×3 mm^{3}, field of view in Right-Left (x), Anterior-Posterior (y), and Superior-Inferior (z) directions of 62×44×47 cm^{3}. The total number of voxels in this model was 269,024. The transmit array was modelled as copper loops with a voltage source and an associated pair of tuning and matching capacitors.

Shaded-surface rendition of the 3D numerical head and shoulders model of Ella, a member of the virtual family, placed in an eight-channel excitation loop coil array.

Using the FDTD simulations, each coil was tuned, matched, and decoupled by iteratively updating the values of tuning and matching capacitors. In each iteration, as an input, a broadband RF pulse, centered at 300MHz and with the bandwidth of 100MHz, was played on each coil, one coil at a time having all the coils present. The scattering matrix (S-matrix) was acquired and used to update the values of tuning and matching capacitors until the reflection coefficients, the diagonal elements of S-matrix, are at most −20dB and the voltage transfer ratios, the off-diagonal elements of S-matrix, are at most −40dB at the resonance frequency, 297.2MHz. Starting with the initial guess of the capacitor values, in each broadband simulation, we updated the values of the tuning and matching capacitors at the coil element where the RF pulse was played to tune the element at 297.2MHz and match the coil element with 50Ω. Since the change of capacitor values at any other coil element can modify the reflection coefficient of the current coil element, we simulated several rounds of the eight simulations, where the RF pulse was played on each coil element one at a time.

Then we simulated electric and magnetic fields resulted from an RF pulse at the operating frequency for 7T. An RF pulse, sinusoid at frequency ω_{0}, with peak amplitude of 1V, was played eight times on each individual coil, k, in the array with 8 coils. For all the voxels inside the model, the resulting electric fields and magnetic fields are computed and stored as matrices, *Q _{v}* and

With the rho mass density, ρ (Kg/m^{3}), and the electric conductivity, σ (S/m), of the head model, the sampling period of the RF, Δ*t*, the duration of the RF, *T _{RF}*, the repetition time,

$${\mathit{\text{SAR}}}_{v}=\frac{{\displaystyle \sum _{t}}{\mathit{\text{SAR}}}_{v}(t)\mathrm{\Delta}t}{\mathit{\text{TR}}}=\frac{D{\displaystyle \sum _{t}}\frac{{\sigma}_{v}}{2{\rho}_{v}}{\Vert {E}_{v}(t)\Vert}^{2}\mathrm{\Delta}t}{{T}_{\mathit{\text{RF}}}}=D{\displaystyle \sum _{t}}b(t)\text{'}{S}_{v}b(t),\text{where}{S}_{v}=\frac{{\sigma}_{v}\mathrm{\Delta}t}{2{\rho}_{v}{T}_{\mathit{\text{RF}}}}{Q}_{v}\text{'}{Q}_{v}.$$

The regulatory limits on local SAR are specified for 10-gram tissue volumes in (10–12). For a 10-gram SAR calculation, we pre-calculate the set of voxels, indexed by w, that best approximates a 10-gram region around the voxel *v*, *V _{10g}*, and average the SAR in each 10-gram region,

$${\mathit{\text{SAR}}}_{v,10g}=\frac{{\displaystyle \sum _{v{V}_{10g}}{\mathit{\text{SAR}}}_{v}}{N}_{v}}{=}$$

[1]

where *N _{v}* is the number of voxels in the set,

Global SAR can be obtained by averaging the SAR in the human model as

$${\mathit{\text{SAR}}}_{\mathit{\text{global}}}=\frac{{\displaystyle \sum _{v{V}_{\mathit{\text{all}}}}{\mathit{\text{SAR}}}_{v}}N}{=}$$

[2]

where *V _{all}* is the set of the voxels in the model within the head and shoulder, and

The proposed generalized model compression method reduces the complexity of the prediction of the peak local SAR given a human model, transmit array configuration and a fixed relative geometry between the model and the array. This method captures the peak local SAR of all the voxels, *V _{all}*, in a 3D numerical model by the peak local SAR of a subset of the voxels,

$$\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}}$$

[3]

We note that in the original presentation of VOPs (34,35), the overestimation factor (ε_{G}) was expressed relative to RF power, whereas in the current formulation ε_{G} is a multiplier against global SAR. We chose the latter representation since the overestimating factor, ε_{G}, determines the accuracy conveniently in units of global SAR.

Since *V _{sub}* is a subset of

$$\sum _{t}}b(t)\text{'}{S}_{v,10g}b(t)\le {\displaystyle \sum _{t}}b(t)\text{'}{S}_{w,10g}b(t)+{\epsilon}_{G}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \sum _{t}}b(t)\text{'}{S}_{\mathit{\text{Global}}}b(t)$$

[4]

In the generalized model compression method, not upper-bounded by only one VOP, each voxel in the model is jointly upper-bounded by all the VOPs within the overestimating term: For any voxel *v* in *V _{all}*, there exists a set of voxel-dependent nonnegative coefficients

$$\sum _{t}}b(t)\text{'}{S}_{v,10g}b(t)\le {\displaystyle \sum _{w{V}_{\mathit{\text{sub}}}}{c}_{w,v}\phantom{\rule{thinmathspace}{0ex}}\left({\displaystyle \sum _{t}}b(t)\text{'}{S}_{w,10g}b(t)\right)+{\epsilon}_{G}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \sum _{t}}b(t)\text{'}{S}_{\mathit{\text{Global}}}b(t).$$

[5]

Since the weighted average of local SARs in *V _{sub}* is less than or equal to the maximum of local SARs in

The generalized formulation extends the original VOP method. In the original method, for any voxel v, there exists one VOP which always bounds this voxel for any RF pulse. Whereas, in the generalized method, several VOPs may contribute to bound a voxel v, and which VOP will finally bound voxel *v* in Eq. [3] is allowed to depend on the waveform *b(t)*. The condition used in the original method (Eq. [4]) is a sufficient version of the condition in the generalized method (Eq. [5]), where, for a voxel v, the one of the coefficients *c _{w,v}* is one while all others are zero. Thus, the generalized method captures more dependencies among the voxels in the model than in the original method of generating the subset

To validate the condition in Eq. [5] naively by calculating the both sides for infinite number of RF pulses is infeasible. Instead, the condition can be replaced by its identical condition in terms of matrix-inequalities (39,40):

$${S}_{v,10g}\le {\displaystyle \sum _{w{V}_{\mathit{\text{sub}}}}{c}_{w,v}{S}_{w,10g}+{\epsilon}_{G}{S}_{\mathit{\text{Global}}},}$$

[6]

which is true if all the eigenvalues of the matrix, $\sum _{w{V}_{\mathit{\text{sub}}}}{c}_{w,v}{S}_{w,10g}+{\epsilon}_{G}{S}_{\mathit{\text{Global}}}-{S}_{v,10g}$, are nonnegative. Eigenvalues of matrix differences have been similarly used in (34,35).

To generate the *V _{sub}*, we have developed a heuristic, greedy algorithm as follows:

- As an initialization step, we calculate the maximum eigenvalues of the SAR matrices,
*S*, of all the voxels in the 3D model and reorder the voxels based on the maximum eigenvalues in a descending order. The greedy search process will be done in this order. Then, to trade off the number of voxels in_{v,10g}*V*and the tightness of the bound in Eq. [3], choose an overestimation factor ε_{sub}_{G}> 0. - In step 1, add the 1
^{st}voxel in*V*._{sub} - In step k, consider k
^{th}voxel and check whether the voxel can be upper-bounded (Eq. [6]), by previously determined VOPs, voxels in*V*. If not upper-bounded, then add the k_{sub}^{th}voxel in*V*._{sub} - Finish when all the voxels have been considered.

In each greedy step (step k), to determine whether the k^{th} voxel, *v*, can be upper-bounded or not by previously determined the VOPs, ideally, we need to evaluate all possible choices of the coefficient, *c _{w,v}*. To reduce the computation time, we developed an iterative method searching over the coefficients,

- Initialize the coefficients.
- If all the eigenvalues of the matrix, $P={\displaystyle \sum _{w{V}_{\mathit{\text{sub}}}}{c}_{w,v}{S}_{w,10g}+{\epsilon}_{G}{S}_{\mathit{\text{Global}}}-{S}_{v,10g}}$, are nonnegative, the voxel can be upper-bounded by previously determined VOPs.
- If not, calculate the eigenvector,
*b*, of*P*corresponding to the minimum eigenvalue. If, by vector*b*, the local SAR at the voxel*v*is greater than the maximum local SAR over the VOPs plus the overestimating term, i.e. $b\text{'}{S}_{v,10g}b>\underset{w{V}_{\mathit{\text{sub}}}}{\text{max}}$, the voxel*v*cannot be upper-bounded. - If any of the two conditions in step 2, 3 is not satisfied, update the coefficients,
*c*, to make_{w,v}*b*'*Pb*nonnegative. - Iterate step 2–4. If any of the two conditions in step 2, 3 is not satisfied until the number of iterations exceeds pre-selected the maximum number of iterations, we add the voxel
*v*in*V*._{sub}

After performing this algorithm, we have a subset, *V _{sub}*, for which Eq. [3] holds. This representation will be used to control peak local SAR in our pulse design. The overestimating factor, ε

In the proposed method for pTx pulse design with efficient incorporation of local SAR constraints using the generalized model compression method, instead of minimizing the maximum local SAR of the entire model, we minimize its lower bound in the VOP condition, which is a tight lower bound for small overestimating factor, ε_{G}, and also can be equal to the maximum local SAR of the entire model if the voxel of the maximum local SAR belongs to the subset, *V _{sub}*, i.e. the maximum local SAR in the subset

$$\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}}$$

[7]

The performance of this pTx pulse design is evaluated for pulses with a fixed excitation performance target, which we take to be the root mean square error (RMSE) relative to the desired transverse magnetization profile, *m _{d}*, inside the region of interest (ROI). The transverse magnetization,

$$1.\phantom{\rule{thinmathspace}{0ex}}\text{Least Squares}\phantom{\rule{thinmathspace}{0ex}}(\text{LS}):\mathit{\text{RMSE}}={\left({\displaystyle \sum}{\Vert {m}_{d}-m(b)\Vert}^{2}/{N}_{\mathit{\text{ROI}}}\right)}^{1/2},$$

[8]

$$2.\phantom{\rule{thinmathspace}{0ex}}\text{Magnitude least squares}\phantom{\rule{thinmathspace}{0ex}}(\text{MLS}):\mathit{\text{RMSE}}={\left({\displaystyle \sum}{\Vert {m}_{d}-|m(b)|\Vert}^{2}/{N}_{\mathit{\text{ROI}}}\right)}^{1/2},$$

[9]

where *N _{ROI}* is the number of voxels in the region of interest.

Our iterative pTx pulse design determines the lower bound, *SAR _{low}*, of the peak local SAR in the subset

$${\mathit{\text{SAR}}}_{\mathit{\text{low}}}\le \underset{\mathit{\text{RMSE}}=c}{\text{min}}\phantom{\rule{thinmathspace}{0ex}}(\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}}).$$

[10]

In our design, we only consider pTx RF pulses that minimize the weighted average of the local SAR. Among the pTx RF pulses designed, we determine the lower bound of the peak local SAR over VOPs and a pTx RF pulse that minimizes the peak local SAR over VOPs. Given nonnegative weighting factors, *w _{v}*, whose sum is equal to one, let

$${b}_{w}(t)\underset{b(t)}{\text{arg min}}\phantom{\rule{thinmathspace}{0ex}}\{{\displaystyle \sum _{t}}b(t)\text{'}({\displaystyle \sum _{v{V}_{\mathit{\text{sub}}}}{w}_{v}{S}_{v,10g}})b(t)\}\phantom{\rule{thinmathspace}{0ex}}\text{such that RMSE}=c.$$

[11]

The minimum weighted average of the local SAR, *SAR _{w}*, achieved by

$${\mathit{\text{SAR}}}_{w}\underset{\underset{\mathit{\text{RMSE}}=c}{\text{min}}}{{\displaystyle \sum _{t}}{b}_{w}(t)\text{'}({\displaystyle \sum _{{}^{vV}\mathit{\text{sub}}}{w}_{v}{S}_{v,10g})\phantom{\rule{thinmathspace}{0ex}}{b}_{w}(t)\le}{w}_{v}}$$

[12]

The proof is as follows: Let the *b _{p}(t)* be the RF pulse that minimizes peak local SAR in the subset

$$\underset{\mathit{\text{RMSE}}=c}{\text{min}}(\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}}=\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}}$$

[13]

We define the Protocol-specific Ultimate Peak in Local SAR, or the PUPiL SAR, as the maximum, among all the weighting factors *w _{v}*, of the minimum weighted average of the local SAR,

$$\mathit{\text{PUPiL SAR}}\underset{{w}_{v}}{\text{max}}\phantom{\rule{thinmathspace}{0ex}}\{{\mathit{\text{SAR}}}_{w}\}=\underset{{w}_{v}}{\text{max}}\phantom{\rule{thinmathspace}{0ex}}\{{\displaystyle \sum _{t}}{b}_{w}(t)\text{'}({\displaystyle \sum _{v{V}_{\mathit{\text{sub}}}}{w}_{v}{S}_{v,10g}})\phantom{\rule{thinmathspace}{0ex}}{b}_{w}(t)\}\phantom{\rule{thinmathspace}{0ex}}\text{such that}{\displaystyle \sum _{v{V}_{\mathit{\text{sub}}}}{w}_{v}=1\text{and}{w}_{v}\ge 0.}$$

[14]

We developed an iterative method to estimate the global maxima, PUPiL SAR, by searching over many weighting factors. In our initial implementation, we update weighting factors through a gradient descent method. In each iterative step, we calculate the local SAR in *V _{sub}* deposited by

After determining a lower bound, we then design pTx RF pulses whose peak local SAR in *V _{sub}* approaches the lower bound. In the previous step of determining PUPiL SAR, the goal is to find the maximum lower bound. However, the objective of our local SAR constrained pulse design is minimizing the peak local SAR in

$$b(t)=\underset{{b}_{w}(t)}{\text{arg min}}\phantom{\rule{thinmathspace}{0ex}}(\underset{v{V}_{\mathit{\text{sub}}}}{\text{max}},$$

[15]

where *b _{w}(t)* satisfies Eq. [11]

We have developed an iterative method to determine the pTx RF pulse. As an initial choice to reduce the computation time, we choose as the best pTx RF pulse that has the minimum peak local SAR in *V _{sub}* among the pulses designed in the process of determining a lower bound. In each iterative step, we update the weighting factors, through a gradient descent method, in the direction to reduce the peak local SAR in

The pTx pulse design that minimizes the weighted average of the local SAR, Eq. [11], can be implemented with a minor modification to the procedure described above for many conventional pulse design methods (5,6,18–23).

In summary, Figure 2 shows the flow diagrams that describe our proposed local SAR constrained design. To reduce the number of constraints, we perform our generalized model compression method. Then, we design pTx RF pulses to minimize the peak local SAR over the VOPs, the voxels in the subset, *V _{sub}*. In our method, we only design pTx RF pulses that minimize weighted averages of the local SAR and among the pTx RF pulses designed, we find the maximum weighted average of the local SAR called PUPiL SAR, which is in fact a lower bound of the peak local SAR released by the pTx RF pulse that minimizes the peak local SAR over all VOPs. Ideally, we need to consider all possible weighting factors but in our implementation we only consider several weighting factor that could represent all possible cases. The weighting factors,

We performed the original and the proposed generalized model compression methods for local SAR estimation for several different overestimating factors, ε_{G}, from 0.7 to 10. Figure 3a shows the number of voxels in *V _{sub}*,

(a) The number of voxels in the subset, *V*_{sub}, determined by model compression methods as a function of the overestimating factor, ε_{G}, the proposed generalized method for VOP classification is in green, whereas the original method is shown in blue. **...**

We demonstrate the application of the local SAR reduction in pTx design by designing pTx pulses for two different low-flip-angle applications. The first is to mitigate B1+ magnitude inhomogeneity in the Ella head model at 7T for an axial, 1-cm thick, slice-selective excitation at the S-I center of the RF coil array, while the second design uses a spiral-based excitation k-space trajectory with a square-box target pattern of uniform intensity in dimensions of x×y = 63×63 mm^{2}, with zero excitation outside the box. The desired profiles (Figure 5) are chosen to have the maximum normalized transverse magnetization of 0.5 (equivalent to a 30° flip angle) assuming that the fully relaxed longitudinal magnetization is unity. For the slice-selective application, we considered three different design categories for pTx pulses, namely RF shimming with an MLS criterion, a two-spoke pulse with the MLS criterion (6), and a four-spoke pulse with the LS criterion. For the box excitation, we used a 2D spiral pTx pulse (5) with the MLS criterion with an acceleration factor of four and FOV of 27cm in x and y, and resolution of 3mm. For this demonstration, the simulated B1+ fields were used and B0 was assumed to be homogenous. For these simulations, the duty cycle was assumed to be 100%, i.e. TR equals the duration of the RF pulse. The durations of RF pulses are 1.6ms for MLS RF Shimming, 2.51ms for MLS two spokes, 4.16ms for LS four spokes, and 5.95ms for 2D spiral pTx pulse.

(a) Axial section of conductivity map at the S-I center of the RF coil array in S/m with voxel size of 3×3×3 mm^{3}, Target profiles for transverse magnetization normalized to a maximum of unity within a FOV of 24cm for (b) a slice-selective, **...**

While we constrained the peak local SAR of the voxels in the subset *V _{sub}* generated by the generalized model compression method at an overestimation level of ε

- Power constrained: $b(t)=\underset{b(t)}{\text{arg min}}\left({\displaystyle \sum _{t}}b(t)\text{'}b(t)\right)$ such that RMSE = c,
- Global SAR constrained: $b(t)=\underset{b(t)}{\text{arg min}}\left({\displaystyle \sum _{t}}b(t)\text{'}{S}_{\mathit{\text{Global}}}b(t)\right)$ such that RMSE = c.

For all our calculation of the pTx RF pulses to satisfy the fixed RMSE, we regard the SAR constraints, weighted average of the local SAR in the proposed method, RF power in the power constrained method, and global SAR in the global SAR constrained methods a regularization term, i.e. $b(t)=\underset{b(t)}{\text{arg min}}(\mathit{\text{RMSE}}+\lambda \mathit{\text{SAR}})$. We then design with several values of regularization parameter, λ, and find the pTx RF pulse that satisfy the desired excitation performance, a fixed RMSE.

In Figure 6, the peak local SAR over the entire model is shown: a lower bound (the estimate of the PUPiL SAR, blue), proposed local SAR constrained design (green), an upper bound (the overestimation of the peak local SAR determined by VOPs in Eq. [3], cyan), global SAR constrained design (black), and RF power constrained design (red). For all the cases, the peak local SAR of the proposed method (green) is larger than the lower bound (blue) and is less than the upper bound (cyan). The reduction of the peak local SAR varies with the pulse design method. The ratio of the peak local SAR of proposed local SAR constrained design to the peak local SAR of global SAR constrained design (green/black) was 64–86% for MLS RF Shimming, 53–72% for MLS two spokes, 60–85% for LS four spokes, and 34–48% for MLS 2D spiral. The ratio of the peak local SAR of proposed local SAR constrained design to the lower bound was 102–109% for MLS RF Shimming, 106–135% for MLS two spokes, 104–119% for LS four spokes, and 112–136% for MLS 2D spiral. The ratio of the peak local SAR of proposed local SAR constrained design to the upper bound was 87–90% for MLS RF Shimming, 83–86% for MLS two spokes, 83–89% for LS four spokes, and 81–85% for MLS 2D spiral.

Peak 10g local SAR over the entire model for pulses played with a 100% duty-cycle as a function of Root-mean-square error (RMSE) of pTx excitation performance, showing the lower bound for peak local SAR as estimated by the proposed design algorithm, the **...**

For MLS two spokes, LS four spokes, and MLS 2D spiral designs, the peak local SAR of the RF power constrained design was larger than the peak local SAR of the global SAR constrained. However, for MLS RF Shimming, the peak local SAR of the RF power constrained design was smaller than the peak local SAR of the global SAR constrained. Global SAR constraint design only guarantees that the global SAR is minimized, but it does not guarantee that peak local SAR is minimized. Infinite number of RF pulses with the same excitation fidelity may exist that have a lower peak local SAR. One example is local SAR constrained design. For our case of MLS RF Shimming, RF power constrained design happened to have the lower peak local SAR. Also, we found that even with the large mitigation error, it is possible to have larger peak local SAR for global SAR constrained design and RF power constrained design.

For B1+ mitigation of slice-selective excitation with 30 flip angle, at RMSE = 0.08, the minimum TR required not to violate the FDA limit on the peak local SAR, 10 W/Kg over any 10g tissue, is 6.3 ms (1.6 ms) for MLS RF Shimming, 1.73 ms (2.51 ms) for MLS two spokes, and 1.74 ms (4.16 ms) for LS four spokes.

One of the trade-offs in the proposed method is an increased global SAR. As it is shown in Figure 7, compared to the global SAR constrained design, the global SAR was 110–117% for MLS RF Shimming, 109–119% for MLS two spokes, 106–132% for LS four spokes, and 119–134% for MLS 2D spiral for box excitation. The other trade-off is the increased computation time due to the iterative procedure. The number of iterations in the proposed method was 14–31 for MLS RF Shimming, 23–45 for MLS two spokes, 24–32 for LS four spokes, and 50 for MLS 2D spiral.

Global SAR for pulses played with a 100% duty-cycle as a function of Root-mean-square error (RMSE) of pTx excitation performance. The graphs show the global SAR for the proposed pTx design method (green), a global-SAR constrained design (black), and RF-power-constrained **...**

Figure 8 shows the distribution of 10g local SAR corresponding to the 2D spiral design corresponding with the mitigation error of RMSE = 0.025. In Figure 8a, the 10g local SAR maps, the sagittal, coronal, and axial slices containing the voxel with the peak 10g local SAR in the entire model, as identified by an exhaustive search, are shown. The peak 10g local SAR is decreased by 54% compared to the global SAR constrained design and 63% compared to the power constrained design. The hottest voxel is in a different location in the local SAR constrained pTx RF design method than in the global SAR constrained design and RF power constrained design. In Figure 8b, the 10g local SAR of the entire 3D voxels is shown in a descending order for the proposed method (green) and the global SAR constrained method (black). Compared to the global SAR constrained method, the peak local SAR is reduced by trading off the local SAR of other voxels and the global SAR.

We proposed an iterative pTx pulse design method which incorporates constraints on peak local SAR for a given human model and transmit array configuration. A weighted average of local SAR, which is a lower bound of the peak local SAR, is minimized in each iterative step of the pTx design. The proposed method also determines a lower bound on the peak local SAR, PUPiL SAR, and the pTx RF pulse closest to the lower bound while still satisfying the given excitation performance criterion.

While we demonstrated the utility of our approach with pTx pulses designed by RF shimming (20,29,30), magnitude least square spoke design (6), least-squares spoke design, and arbitrary excitation for spiral trajectory (2,5,7,23,33), the method is general in that it extends to and is compatible with other pTx pulse design methods such as spatially selective excitation (2,24), composite pTx pulse for uniform volume excitation (21), spatial domain design for small flip angle approximation (23), LCLTA (19), and optimal control approaches (18,22).

In a numerical head model, we demonstrated 14–66% of reduction in the peak local 10g SAR for slice-selective pTx excitation and 2D selective pTx excitation compared to the global SAR constrained design. The trade-offs of the proposed method are increased global SAR, within 34% in our demonstrations, and increased computation time by a factor of 14–50.

In Figure 9, the peak local SAR over the entire model (green) and its upper bound (Eq. [3]), which is the peak local SAR of the voxels in the subset, *V _{sub}*, determined by the model compression method plus the overestimation term (cyan), are shown for ε

The peak local SAR of the pTx pulses exceeds the PUPiL SAR by 2–36%. In Figure 10, the peak local SAR over the entire model (green), the peak local SAR of the voxels in the subset *V _{sub}* determined by the model compression method (black), and a lower bound (the estimate of the PUPiL SAR, blue), are shown for ε

Peak local SAR and its lower bounds as a function of overestimation factor, ε_{G} for four different pTx pulse designs. The graphs show the peak local SAR over the entire mode (green), the peak local SAR of the voxels in the subset, *V*_{sub}, determined **...**

Alternatively, by improving the method for updating the weighting factors or considering more combinations of weighting factors, *w _{v}*, the global maximum of weighted average of the local SAR could be determined and the gap between the peak local SAR at the voxels in the subset

Our design method to minimize the peak local SAR over VOPs, voxels in the subset, *V _{sub}*, considers only the pTx RF pulses that minimize the weighted average of the local SAR in Eq. [11]. By doing this, we can address minimization over hundreds of peak local SAR constraints. This approach also can be applied to many conventional pulse design methods (5,6,18–23) and can be implemented with a minor modification in their implementations. The trade-off is the increased computation time, by a factor of the number of pTx RF pulses,

The generalized VOP method and proposed pulse design method were demonstrated for a given a human model, transmit array configuration, and a fixed relative geometry between the model and the array. As a future work, we will generalize the proposed method to impose peak local SAR constraints on several configurations of human model and transmit arrays, which can differ by types of the human model or the relative geometry between the model and the array. This could jointly minimize the peak local SAR by trading off peak local SAR for one configuration compared to the optimized method for one configuration. To proceed, we need to have simulated electric fields on the several configurations.

While the proposed pulse design method is demonstrated for the design of a single pulse, it can be generalized to joint pulse design over a set of pTx pulses to further reduce the time-average peak local SAR (26,41). For instance, in pulse sequences that apply saturation pulses prior to excitation, a joint design of the saturation and excitation pulses with local-SAR penalty could yield better local-SAR performance than if each pTx pulse is designed in isolation, and further, a joint design of a family of slice-selective excitation pulses in multi-slice acquisitions could be performed to minimize peak local SAR over the duration of the multi-slice scan.

The incorporation of peak local SAR constraints in pTx RF design was made feasible by the model compression method for estimation of peak local SAR, which dramatically reduces the number of voxels that need to be considered. The original model compression method (34,35) was generalized to capture additional local-SAR dependencies among voxels. The proposed pTx pulse design minimizes peak local SAR while satisfying the given excitation performance target, and yields a bound on the achievable SAR performance, the PUPiL SAR.

This work is supported by Siemens Healthcare, The Siemens-MIT Alliance, National Institute of Health NIH R01EB006847, NIH R01EB007942, and NCRR P41RR14075. The authors acknowledge Lohith Kini, MEng, Berkin Bilgic, SM, Borjan Gagoski, Ph.D, Kawin Setsompop, Ph.D., and Khaldoun Makhoul, Ph.D., for help with FDTD simulations and beneficial discussions.

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