Arian Beqiri^{1}, Anthony N Price^{1,2}, Joseph V Hajnal^{1,2}, and Shaihan J Malik^{1}

**
Balanced
steady-state free precession (bSSFP) cardiac MRI benefits greatly
from reduced repetition time (TR). Minimum TR is often limited by
specific absorption rate (SAR) and hardware constraints. RF shimming
can be used with parallel transmission (PTx) to work within such
constraints, but direct minimization of TR is not straightforward
since the constraints themselves vary as TR is reduced.**

**We
present an extended RF shimming framework in which PTx degrees of
freedom are simultaneously optimised with pulse sequence properties.
The result is minimum TR bSSFP sequences that operate at the SAR
limits and within hardware constraints for 3T cardiac MRI.**

For a PTx system with $$$N_c$$$ channels, RF shimming is the process of determining an optimal set of complex dimensionless weighting $$$\mathbf{w}_j (j=1:N_c)$$$ to apply to each channel.

For a sequence with a given TR and RF pulse shape $$$p(t)$$$ we may write the constraints on $$$\mathbf{w}$$$ as a function of pulse duration ($$$\tau$$$):

\[constraints_{\mathbf{w}}(\tau):\left\{\begin{array}{ll}\max\limits_{i}\left\lbrace\mathbf{w^*Q_iw}\right\rbrace\times LF^2\times\dfrac{\theta_0^2}{\gamma^2}\times\dfrac{\delta_2}{\delta_1^2} \times\dfrac{1}{\tau TR}\leq lSAR_{max}\\\left\lbrace\mathbf{w^*Q_{wb}w}\right\rbrace\times LF^2\times\dfrac{\theta_0^2}{\gamma^2}\times\dfrac{\delta_2}{\delta_1^2} \times\dfrac{1}{\tau TR}\leq wbSAR_{max}\\|\mathbf{w}_j|\leq\tau\dfrac{P_{peak}}{\sqrt{A}} \dfrac{\delta_1\gamma}{\theta_0}\quad\forall j\\|\mathbf{w}_j|\leq\sqrt{\tau TR}\dfrac{\delta_1\gamma}{\theta_0\sqrt{\delta_2}}\sqrt{\dfrac{P_{avg}}{A}}\quad\forall j\end{array}\right.\]

Top-to-bottom these expressions represent local SAR (10g averaged), whole-body SAR, peak power and average power respectively.

$$$\mathbf{Q_{wb}}$$$ and $$$\mathbf{Q_{i}}$$$ are the whole body and local SAR matrices^{4,5} and $$$\theta_0$$$ the target flip angle. $$$\delta_1$$$ and $$$\delta_2$$$ are intrinsic properties of $$$p(t)$$$ relative to a block pulse with the same flip angle and amplitude ($$$\delta_1$$$ is the relative duration, $$$\delta_2$$$ is the relative energy). $$$lSARmax$$$ and $$$wbSARmax$$$ are the local and
whole-body SAR limits; $$$P_{peak}$$$ and $$$P_{avg}$$$ are peak and average RF
power limits. $$$LF$$$ is a scalar ‘loading factor’ to match the SAR model and $$$A$$$ is a scalar that quantifies RF chain efficiency in W/$$$\mu$$$T^{2}.

The minimum TR is set by:

\[TR_{min} = max\left\{\begin{array}{ll}\tau+t_{enc}\\\tau/\delta_0\end{array}\right\}\]

$$$t_{enc}$$$ is time required for image encoding and $$$\delta_0$$$ is the RF amplifier gating duty-cycle limit (50%). Figure 1 illustrates the intersection of these constraints for a single-channel system – the green circle indicates the TR that would be selected for scanning.

The optimization is performed using a nested approach with an outer step that seeks to find the lowest acceptable $$$\tau$$$ to minimize sequence TR:

$$\mathrm{arg\,min}_{\tau}\left\lbrace TR+f(\hat{\theta})\right\rbrace$$

where $$$f(\hat{\theta})$$$ is a function that penalises solutions with high bias $$$(\hat{\theta})$$$ in achieved flip angle within a region of interest (ROI). For each $$$\tau$$$, an inner optimization seeks optimal $$$\mathbf{w}$$$ given the constraints. This was done in two ways – one minimises mean squared error (MSE) in $$$\theta$$$:

$$\begin{align} \mathrm{arg\,min}&_{\mathbf{w}}\||\mathbf{S_{\theta}w}|-\mathbf{\theta_0}\|^2_{ROI}\\ \mathrm{s.t.}&\quad constraints_{\mathbf{w}}(\tau) \end{align}$$where $$$\mathbf{S_{\theta}}$$$ represents spatially dependent RF sensitivities of the transmit channels in units of achieved flip angle. Although MSE improves flip angle homogeneity in the target region, it cannot achieve a zero bias solution (5% bias was allowed). The second optimisation minimizes bias $$$\hat{\theta}$$$ alone:

$$\begin{align}\mathrm{arg\,min}&_{\mathbf{w}}\|\overline{|\mathbf{S_{\theta}w}|}-\theta_0\|^2_{ROI}\\ \mathrm{s.t.}&\quad constraints_{\mathbf{w}}(\tau)\end{align}$$

Experiments
performed on a Philips 3T MRI system with an 8-channel transmit body array^{6}
and 6-channel receiver array. Seven male subjects were scanned; each matched
to one of two pre-calculated SAR models based on their body mass index (a smaller
model (BMI=23.5) for subjects 1-5 and a larger model (BMI=31) for subjects 6&7).

B$$$_1^+$$$ mapping used DREAM^{7} and bSSFP imaging with $$$\theta_0$$$=45° was performed for a 4-chamber cardiac view. The ROI was
manually drawn to only include myocardium. Optimal sequence
parameters for the transmit coil quadrature mode (similar to a whole-body
birdcage) were also determined using the outlined approach (Figure 1).

Figure
2a shows minimum TRs achieved in the three scenarios. Both optimisations substantially reduce TR compared with quadrature mode (TR reduced by
1.35±0.31ms) with the minimum bias optimisation performing better. However, minimum bias optimization leads to unpredictable flip-angle
uniformity (Figure 2b). Figure 3 shows B$$$_1^+$$$ maps from one subject,
illustrating this effect, along with predicted $$$lSARmax$$$ distributions – all limited by 10Wkg^{-1}.
Figure 4 shows bSSFP images
for all subjects.

The
method can be used with different SAR limits – repeating the optimization for $$$\theta_0$$$=60°
with $$$lSARmax$$$=20Wkg^{-1} as in Reference 8 leads to the
results in Figure 5. In this case minimum TRs of 2.7ms are possible, comparing
favourably with the published 3.8ms for the same constraints.

Discussion and Conclusions

An extended RF shimming framework was proposed that reduces TR in bSSFP scans by finding optimal pulse sequence properties in conjunction with RF shimming. Figure 3 shows that while remaining within the maximum local SAR limits, the optimized sequences operate closer to the whole-body SAR limit, effectively making the sequence more efficient. The method is applicable in principle to any gradient-echo sequence, and for a range of different hardware and safety limits.The research was funded by the following grant:

EPSRC (UK research councils) grant number EP/L00531X/1

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Figure
1 – Timing diagram of bSSFP for lSARmax=10Wkg$$$^{-1}$$$, $$$P_{peak}$$$=1kW,
$$$P_{avg}$$$=100W and 50% gating duty cycle limit with $$$t_{enc}$$$=1.7ms. The whole-body SAR limit is not drawn because
it is never the limiting factor.

Figure
2 – **(a) **TR and **(b)** coefficient of variation in B$$$_1^+$$$ shown for
quadrature, minimum bias and mean squared error shimmed solutions. All
sequences had $$$lSAR_{max}$$$ = 10W kg$$$^{-1}$$$ and $$$\theta_0$$$=45°.

Figure 3 – Top row: B$$$_1^+$$$ maps in the 4-chamber
view shown for one subject for quadrature, minimum bias, and MSE shimming. Each
map is normalized to desired B$$$_1^+$$$, so a value of 1.0
is ideal. The region of interest used for optimization is highlighted. Bottom
row: SAR maximum intensity projections for the subject in all scenarios. The maximum local SAR is 10 Wkg$$$^{-1}$$$ in all cases. Note the quadrature solution has an asymmetric SAR distribution
which is more uniform for the optimized solutions. Whole body SAR is on average
28% higher for the RF shimmed solutions compared with quadrature.

Figure
4 – bSSFP imaging data shown for all subjects for quadrature, minimum bias, and
mean squared error shimming. The same
window/level settings were used for all images from each subject.

Figure
5 – Predicted results for lSAR$$$_{max}$$$=20Wkg$$$^{-1}$$$ and $$$\theta_0$$$=60° (these solutions were not used for
imaging). **(a)** TR and **(b)**
coefficient of variation in B$$$_1^+$$$. The achieved minimum TRs
are 3.96±0.55ms
for quadrature mode, 2.81±0.16ms
for minimum bias and 2.92±0.33ms
for MSE shimming.