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
), magnitude least square spoke design (6
), least-squares spoke design, and arbitrary excitation for spiral trajectory (2
), the method is general in that it extends to and is compatible with other pTx pulse design methods such as spatially selective excitation (2
), composite pTx pulse for uniform volume excitation (21
), spatial domain design for small flip angle approximation (23
), LCLTA (19
), and optimal control approaches (18
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 , the peak local SAR over the entire model (green) and its upper bound (Eq. 
), which is the peak local SAR of the voxels in the subset, Vsub
, determined by the model compression method plus the overestimation term (cyan), are shown for εG
= 1–9 and four pTx RF designs. For small overestimating factor, εG
= 1, the peak local SAR is 82~89% of its upper bound in Eq. 
. However, for large overestimating factor, εG
= 9, the peak local SAR is 42~48% of its upper bound.
Figure 9 Peak local SAR and its upper bound (Eq. ) as a function of overestimation factor, εG for four different pTx pulse designs. The graphs show the peak local SAR over the entire mode (green), its upper bound, which is the peak local SAR of the (more ...)
The peak local SAR of the pTx pulses exceeds the PUPiL SAR by 2–36%. In , the peak local SAR over the entire model (green), the peak local SAR of the voxels in the subset Vsub
determined by the model compression method (black), and a lower bound (the estimate of the PUPiL SAR, blue), are shown for εG
= 1–9 and four pTx RF designs. As it is shown in and , for our local SAR constrained pulse design, the peak local SAR over the entire model is much closer to the lower bound than its upper bound in Eq. 
. The gap between the peak local SAR over the entire model and the lower bound can be reduced in two ways. First, by using the smaller overestimating factors, which takes longer to compute VOPs and increases the number of VOPs considered in the computation of the lower bound, the difference between the peak local SAR over the entire model and the peak local SAR at the voxels in the subset Vsub
is reduced. As a future work, we will investigate the limitation of our local SAR constrained design; how many VOPs can our method deal with and how much the number of iteration to converge increases if we use more VOPs.
Figure 10 Peak local SAR and its lower bounds as a function of overestimation factor, εG 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, Vsub, determined (more ...)
Alternatively, by improving the method for updating the weighting factors or considering more combinations of weighting factors, wv, 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 Vsub and the lower bound can be reduced. In our current implementation, which is a gradient descent method, our estimate of the PUPiL SAR may converge to a local maximum.
Our design method to minimize the peak local SAR over VOPs, voxels in the subset, Vsub
, considers only the pTx RF pulses that minimize the weighted average of the local SAR in Eq. 
. 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
) 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, bw(t)
, designed. The computation time can be reduced by parallel computing, designing several pTx RF pulses simultaneously. However, to minimize the peak local SAR further, we might need to consider all possible pTx RF pulses given excitation performance criterion. The methods that minimize the peak local SAR using second order cone programming (25
) may be useful. Before its use, the applicability of the methods, in terms of the number of constraints that the methods can solve and their computation time, should be investigated
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
). 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.