Introduction

A common method for producing Magnetization Transfer (MT) contrast in tissues with a semisolid proton fraction is to apply off-resonant RF irradiation to saturate their magnetization. The amount of saturation depends on the RF power applied off-resonance. At 7T the high degree of RF magnetic field ($$$B_1^+$$$) inhomogeneity causes saturation to be highly non-uniform spatially because RF power depends on the square of $$$B_1^+$$$. Parallel transmit (pTx) offers additional control of $$$B_1^+$$$ by creating a field that is the linear superposition of different transmit channels. Standard pTx RF pulse design methods¹ are not suitable for designing pulses with uniform saturation properties because they optimize for rotation, not saturation. In this work we propose a pulse design framework for directly homogenizing RF saturation power using pTx, allowing Direct Saturation Control (DSatC) to obtain a more uniform MT contrast.

Theory

Under the small tip angle approximation, the magnetization achieved by a saturation pulse in a parallel transmit system (pTx) is given by¹:
$$m(\mathbf{x})=\imath\gamma{}m_0\sum_{j=1}^{N_{coils}}s_j(\mathbf{x})\int_{0}^{\tau_{sat}}b(t)e^{\imath\mathbf{x}\cdot{}\mathbf{k}(t)}dt\qquad[1]$$
In a two-pool system, the magnetization vector from free water ("f") is extended to include a semisolid compartment ("s") such that $$$\vec{M}=[M_x^f M_y^f M_z^f M_z^s]^\mathsf{T}$$$. Evolution is described by the Bloch-McConnell (BM) equations:$$\frac{d\vec{M}}{dt}=\begin{bmatrix}-R_2^f&&\gamma\Delta{}B_0&&-\gamma{}B_{1,y}&&0\\-\gamma\Delta{}B_0&&-R_2^f&&\gamma{}B_{1,x}&&0\\\gamma{}B_{1,y}&&-\gamma{}B_{1,x}&&-R_1^f-k_{fs}&&k_{sf}\\0&&0&&k_{fs}&&-R_1^s-k_{sf}-W\end{bmatrix}\vec{M}+\begin{bmatrix}0\\0\\R_1^f\\R_1^s\end{bmatrix}M_0\qquad[2]$$$$W=\pi\gamma^2{B_1^{+2}g(\Delta})\qquad[3]$$
where the pulse is applied at off-resonance frequency $$$\Delta$$$ and $$$g(\Delta)$$$ is the absorption lineshape. No transverse magnetization is considered for the semisolid pool due to its very short $$$T_2$$$ ($$$\approx{}10\,\mathrm{\mu{}s}$$$). The key consideration for pulse design is that the lack of transverse components means that the semisolid magnetization does not see any applied gradients.
From [2] and [3], it is the $$$B_1^{+2}$$$ of the saturation pulse that governs the longitudinal magnetisation of the semisolid:$$B_1^{+2}(\mathbf{x},t)={\left\lvert{}\sum_{j=1}^{N_{coils}}s_j(\mathbf{x})b_j(t)\right\rvert{}}^2\qquad[4]$$
where $$$b_j$$$ are different RF pulse waveforms to play on each channel. The effect of the RF pulse can be found by integrating [2] over time; the method proposed for "pulsed" sequences (i.e. short RF pulses)² is to consider the average saturation:$$\langle{}B_1^{+2}\rangle(\mathbf{x})=\frac{1}{\tau_{sat}}\int_{0}^{\tau_{sat}}B_1^{+2}(\mathbf{x},t)dt\qquad[5]$$such that an average saturation rate $$$\langle{}W\rangle$$$ can be applied over the duration of a whole pulse.
The method can further be simplified in the case that the overall RF pulse is made up of a train of $$$N_{sp}$$$ identical sub-pulses $$$b$$$ that are individually weighted (Figure 1). In this case the overall $$$\langle{}B_1^{+2}\rangle$$$ is determined by:$$\langle{}B_1^{+2}\rangle(\mathbf{x})=\frac{1}{N_{sp}}\sum_{i=1}^{N_{sp}}{\left\lvert\sum_{j=1}^{N_{coils}}s_j(\mathbf{x})w_{ij}\right\rvert}^2\langle{}b^2\rangle\qquad[6]$$where $$$w_{ij}$$$ is the complex weight applied to the $$$i^{th}$$$ sub-pulse and the $$$j^{th}$$$ transmit channel.
Uniform saturation power can be obtained by solving the RF optimization problem:$$w_{ij}^{opt}:=\underset{w_{ij}}{\arg\min}\,{\left\lVert\langle{}B_1^{+2}\rangle(\mathbf{x})-\mathrm{Target}\right\rVert}^2\qquad[7]$$subject to peak power, SAR, and average power constraints, where the target defines the desired $$$\langle{}B_1^{+2}\rangle$$$.

Methods

Experiments were performed on a 7T scanner (MAGNETOM Terra, Siemens Healthcare, Erlangen, Germany) in prototype research configuration, with an 8-Tx-channel head coil (Nova Medical, Wilmington MA, USA) using a prototype MT-weighted spoiled gradient-recalled-echo sequence (SPGR). Two phantoms (cross-linked egg white albumin, and water) were scanned using different number of sub-pulses and both circular-polarized (CP) and optimized shimming settings for the saturation pulse. In all cases, readout pulse was performed in CP mode. For all experiments $$$b$$$ was a Gaussian waveform applied at $$$|mathrm{1.2\,kHz}$$$ off-resonance (time-bandwidth-product of 1.92). For implementation reasons the total duration of all sub-pulses was fixed at $$$\tau_{sat}=\mathrm{10\,ms}$$$; as $$$N_{sp}$$$ was increased, the duration of each individual sub-pulse was then $$$\tau_{sat}/N_{sp}$$$. $$$B_1^+$$$ mapping was done using a pre-saturation turbo-FLASH sequence³, prior to shim optimization.
The complex weights were optimized for composite pulse, varying the number of sub-pulses from one to three for a homogeneous $$$\mathrm{Target}=16\,\mathrm{\mu{}T}^2$$$.
To visualize the MT effects, a reference SPGR image with no MT prep-pulse ($$$M_{ref}$$$) was acquired in order to calculate the Magnetization Transfer Ratio (MTR):$$\mathrm{MTR}(\%)=100\times\frac{M_{ref}-M_{sat}}{M_{ref}}\qquad[8]$$where $$$M_{sat}$$$ is the image acquired with the saturation pulse.
Bloch-McConnell simulations were performed using measured $$$B_1^+$$$ maps and optimized complex weights from the phantom scanning to reproduce the experimental findings.

Results

Figure 2 shows the $$$\langle{}B_1^{+2}\rangle$$$ of each sub-pulse for all optimized composite pulses, and the final $$$\langle{}B_1^{+2}\rangle$$$ achieved by each composite pulse, showing a decrease in the coefficient of variation (CoV) with increasing number of sub-pulses. Phantom scanning results (Figure 4) show that virtually no MT effects nor direct effect are observed in a water phantom, while in the MT phantom the MTR maps become more homogeneous for composite pulses with increasing number of sub-pulses. The Bloch-McConnell simulations (Figure 3) show a good agreement with the phantom results, especially for the CP mode and with 1 sub-pulse.

Discussion and Conclusion

The optimization results showed that the $$$\langle{}B_1^{+2}\rangle$$$ produced by a composite pulse with more than one sub-pulse can deliver a much more homogeneous power distribution. The resulting improvement in uniformity of saturation was confirmed with phantom MTR measurements.Although other pTx based saturation pulse designs have been proposed in the past4,5 these still applied to systems with transverse magnetization, so gradients encoding could be employed. Our work aims to provide a framework for the design of uniform saturation of semisolids, which can only be influenced by spatial-temporal modulation of the $$$B_1^+$$$ field. Though this work demonstrates a "saturation only" design, composite pulse designs considering the effect on free water and semisolids together, effectively by jointly performing a classic RF shimming/pulse design with the $$$\langle{}B_1^{+2}\rangle$$$ optimization would be a generalized extension of this.

Acknowledgements

This work was funded by the King’s College London & Imperial College London EPSRC Centre for Doctoral Training in Medical Imaging [EP/L015226/1] and supported by a Wellcome Trust Collaboration in Science Award [WT 201526/Z/16/Z] and the Wellcome/EPSRC Centre for Medical Engineering [WT 203148/Z/16/Z].