Ultra-high field MRI: parallel-transmit arrays and RF pulse design

This paper reviews the field of multiple or parallel radiofrequency (RF) transmission for magnetic resonance imaging (MRI). Currently the use of ultra-high field (UHF) MRI at 7 tesla and above is gaining popularity, yet faces challenges with non-uniformity of the RF field and higher RF power deposition. Since its introduction in the early 2000s, parallel transmission (pTx) has been recognized as a powerful tool for accelerating spatially selective RF pulses and combating the challenges associated with RF inhomogeneity at UHF. We provide a survey of the types of dedicated RF coils used commonly for pTx and the important modeling of the coil behavior by electromagnetic (EM) field simulations. We also discuss the additional safety considerations involved with pTx such as the specific absorption rate (SAR) and how to manage them. We then describe the application of pTx with RF pulse design, including a practical guide to popular methods. Finally, we conclude with a description of the current and future prospects for pTx, particularly its potential for routine clinical use.


Introduction
Magnetic resonance imaging (MRI) plays a powerful role in modern healthcare as a non-invasive diagnostic tool. Unlike other imaging modalities such as nuclear medicine and x-ray computed tomography, MRI does not use ionizing radiation and offers excellent soft tissue contrast. In recent years, member countries of the Organisation for Economic Co-operation and Development have increased MRI scanner prevalence to over 16 per 100 000 people (OECD 2022) with the highest application of clinical exams taking place in the spine, brain, upper extremites, and lower extremities with usage of 26%, 25%, 11%, and 9%, respectively (Rinck 2018).
Until recently, only MRI scanners with a static magnetic field strength of up to 3T had the regulatory approval for clinical use. Since the late 1990s, however, ultra-high field (UHF, 7T) scanners with 7T and even higher static magnetic field strengths such as 8T, 9.4T and 10.5T have been used in neuroscience and clinical research (Robitaille et al 1998, Vaughan et al 2001, Sadeghi-Tarakameh et al 2020. Continuing with this trend, a whole body 11.7T MRI scanner has been commissioned at CEA Neurospin, Paris and even 14.1T magnets for human imaging are currently under consideration (Nowogrodzki 2018). The push towards increasing the B 0 field strength is mainly driven by the inherent supra-linear increase in signal-to-noise ratio (SNR) with respect to the strength of the static magnetic field (Hoult andRichards 1976, Pohmann et al 2016). The SNR increase at UHF enables higher imaging resolution as well as unique imaging contrasts and enhanced opportunities for applications such as functional MRI (fMRI), spectroscopy, and susceptibility-weighted imaging (SWI).
At the same time, there are challenges associated with UHF MRI: the increased radiofrequency (RF) power deposition in tissue, the worsening of susceptibility and other imaging artifacts, and the greater non-uniformity of the RF field (Kraff and Quick 2017). By far the most popular approach to improve RF field homogeneity at UHF is a form of MRI called parallel transmission or 'pTx' (Hoult 2000, Ibrahim et al 2000. Since it's introduction, the technical development of dedicated pTx RF coils, pTx safety studies, and pTx pulse design have been a major theme of UHF MRI research (Padormo et al 2016).
Following the tremendous progress that has been achieved in understanding and mitigating the challenges of UHF, 7T MRI scanners from two vendors have been granted FDA ('FDA Clears GE 7T MRI,' 2020) and FDA and CE approval ('FDA Clears First 7T MRI, ' 2017). Currently, the approval for clinical use concerns single-transmit (sTx) mode for imaging of the head or the knee. Already, close to one hundred 7T or higher field scanners are operational, and the uptake of 7T scanners for clinical use has been increasing since the regulatory approval for 7T MRI (Barisano et al 2019).
The next logical step in this progression is the approval for clinical use of pTx technology because this is essential to overcome RF field non-uniformity and fully exploit the capability of ultra-high field MRI. In addition, a wide range of high-performance RF coil arrays are necessary to develop clinical applications and expand 7T MRI to body parts other than the brain.
The aim of this review paper is to introduce the various components involved with pTx at UHF. First, we will describe the challenges motivating the development of pTx at UHF and provide an overview of the design considerations of dedicated transmit array coils. We then discuss briefly the role of electromagnetic (EM) field simulations in the design of transmit arrays and the management of their safety. Next, we will review applications of pTx and provide a generalized framework for pTx pulse design. Finally, we end with a look towards the future of pTx MRI. Selected references on each topic have been provided for further reading.

RF Challenges at UHF
MRI is based on the physical principles of nuclear magnetic resonance (NMR), and requires dedicated hardware to generate EM fields within the bore of the MRI scanner and to receive RF signals from atomic nuclei within the human body. An RF coil is an essential component responsible for two crucial functions during an MRI examination. Firstly, a transmit coil generates a magnetic field (B 1 + ) perpendicular to the static magnetic field (B 0 ). The pulsed RF energy applied through the transmit coil flips the net magnetization of the atomic nuclei perpendicular to its equilibrium along the direction of B 0 in a circular motion called precession. The precessing magnetization induces a signal on a receive coil, which is the MR signal processed by the scanner to generate the MR image. The abovementioned transmit and receive functions are either performed by the same coil or by two separate coils. These RF coils are known as transceiver (TxRx) coils and transmit-only receive-only (ToRo) coils, respectively. The EM field produced by the transmit coil is expected to be homogeneous to achieve constant excitation flip angle across the imaging volume. This is essential to produce a homogeneous image with uniform tissue contrast.
The RF coils are made of resonant circuits tuned to the Larmor frequency ω 0 , which depends on the nucleus and the strength of the main magnetic field B 0 , where γ is the gyromagnetic ratio. The most widely used atomic nucleus used in MRI is 1 H because it has the highest inherent NMR sensitivity and is abundant in the body in the form of H 2 O. For 1 H, the resonance frequency is 42.58 MHz at 1 tesla. The engineering difficulties as well as the fundamental physics challenges increase with field strength because of the higher Larmor frequency and the decrease in consequent wavelength which becomes comparable to body dimensions. For example, 1 H 7T RF coils are tuned to about 298 MHz and the corresponding RF wavelength in biological tissue is 12 cm, which is about half the largest dimension of the head. This results in subject-dependent spatial variations in the B 1 + field produced by the RF coil and an inhomogeneous image. In fact, the associated fundamental physical problems of RF magnetic field penetration, RF amplitude and phase variations in biological tissue, increase in RF power deposition, and the power required to achieve a 90°pulse were recognized very early in the development of MRI (Bottomley and Andrew 1978), although this study limited the feasibility of MRI to 10 MHz. However, imaging studies performed at a range of field strengths from 4T to 9.4T demonstrated that imaging could be performed at these field strengths and that the fundamental issues such as RF penetration (Robitaille et al 1998), and RF power requirements (Vaughan et al 2001) would not be staggering at UHF. Although measurements at frequencies up to 220 MHz demonstrate that the power absorbed in conductive tissue increases with the square of the operating frequency (Roschmann 1987), numerical calculations of absorbed power and specific absorption rate (SAR) between 200 and 400 MHz ( andSmith 2001a, Ibrahim 2004) suggest that this increase is not as fast as predicted with low frequency approximations. Simulations and experiments provide evidence that dielectric effects in the head overwhelm eddy current shielding and RF penetration issues , Vaughan et al 2001.
While image non-uniformities due to central brightening seen with volume coils at 4T was initially attributed to the presence of dielectric resonances (Bomsdorf et al 1988, Barfuss et al 1990), analytical calculations confirmed that the resonances are dampened at conductivity levels of human tissue, and the central brightening is a field focusing effect (Hoult 2000). Subsequent theoretical and experimental studies at 8T demonstrated that dielectric resonance effects are not sustained at the permittivity and conductivity levels of the biological tissue (Ibrahim et al 2001a). The RF wave is attenuated at higher conductivities as it travels away from the coil and reflected from boundaries leading to a complex constructive and destructive interference pattern (Yang et al 2002. Figure 1 shows the progressively decreasing image homogeneity in simulation, which is caused by these constructive and destructive interference patterns due to RF wavelength effects in the human head for MRI field strengths ranging from to 3T to 12T (Webb and Collins 2010a). Due to the complex interactions between the RF coil and the sample, the spatial distribution of the B 1 + field is determined by the geometry and electrical properties of the sample, the RF coil design itself as well as the amplitude and phase relationships between the input power to the coil elements .
In addition to the B 1 + field inhomogeneity, the electric field produced by the RF coil, which generates electric currents in conductive tissue and causes tissue heating, plays an important role in UHF MRI. The RF power deposition in tissue is regulated by SAR, which is the absorbed RF power per tissue mass. SAR is calculated from the local electric field ( ) E , local tissue conductivity ( ) s , and mass density ( ) r of tissue SAR is quantified by the global and local SAR in Watts per kilogram of tissue, and there are regulatory limits in terms of 10 s and time averaged over 6 min values ('IEC 60601-2-33,' 2010). Global SAR is the SAR averaged across entire body parts (whole body, partial-body, head) and local SAR is averaged over a volume of 10 g of tissue.
At static magnetic field strengths of up to 3T, the relationship between the applied RF power and energy deposition can be calculated by assuming a homogeneous RF field distribution. However, UHF MRI operates in the intermediate region between the near and far-field where the RF field distribution in biological tissue is a function of the subject size and position, and becomes proggessively more inhomogeneous (figure 1). This increases the risk of localized 'RF hot spots' due to elevated power deposition. Hence the concern is the increase in RF power deposition in the exposed anatomy yielding a rise in local temperature.
While SAR is used as a surrogate metric, temperature is the primary safety criterion. This is modelled by the Pennes bioheat equation (Pennes 1948, Le Ster 2021 Here, c is the tissue heat capacity, k is the thermal conductivity, T, is the absolute temperature, T b , is the arterial blood temperature, B is the blood perfusion, and Q m is the heat generated by metabolism. SAR and temperature distribution on similar types of coils at 3T and 7T exhibit very different spatial distribution (Webb and Collins 2010a). Therefore, full-wave EM field simulations which include heterogenous body models, create accurate numerical models of the RF coil, and capture the MRI scanner environment have played an improtant role in safety assessment and verification at UHF.
Thus, the major engineering challenge in the design of an RF coil for ultra-high field MRI is the ability to produce a strong and homogeneous B 1 + field, while managing the RF power deposition and staying within the FDA or IEC SAR guidelines (Delfino 2014, 'IEC 60601-2-33,' 2010.

Parallel RF transmission
In clinical MRI scanners with static magnetic field strengths up to 3T, a large volume coil is installed in the scanner bore. A volume coil encompasses the anatomy of interest, and the most commonly used volume coil is the birdcage resonator . The birdcage resonators installed in the MRI scanner bore are bodysized resonators capable of producing a homogeneous B 1 + field over a large field of view (FOV) and are typically referred to as body coils. The body coil is primarily operated as a transmit coil but can also be operated as a transmit-receive coil. The two orthogonal modes of the birdcage resonator are excited with an RF input of equal amplitude but with a 90°p hase offset (Glover et al 1985) to set up a circularly polarized (CP) RF excitation. This is typically achieved by feeding the output of the RF power amplifier (RFPA) through a high-power quadrature hybrid to generate two outputs. Figure 2(A) shows a functional block diagram of this configuration.
The introduction of phased array technology by Roemer et al revolutionized MR signal detection and has resulted in tremendous advances towards improved sensitivity (Roemer et al 1990) and reduced scan time (Sodickson and Manning 1997, Pruessmann et al 1999, Griswold et al 2002 in MRI examinations. The phased array technology enabled simultaneous detection of MR signal with an array of surface coils, and outperforms a single large coil covering the same area. Shortly after, the phased array concept was extended to volume imaging (Hayes et al 1991). This work further demonstrated that the SNR of a receive array is significantly better than the large body coil. Dedicated receive array coils for the anatomy of interest thus became the most favored method of signal detection, with the built-in body coil being used as a transmit coil for spin excitation.
Body sized volume transmit coils analogous to those at field strengths of 1.5T and 3T are not feasible at UHF due to the B 1 + field inhomogeneity already introduced in the previous section. The spatial distribution of the B 1 + field produced by the coil, which is homogeneous in the absence of a load, becomes highly non-uniform when the coil is loaded with a human body (Vaughan et al 2001). The RF field interferences can cause total cancellation of the B 1 + field as demonstrated in figure 1, when the transmit coil is excited in the conventional CP mode.
pTx was originally introduced in 2000 as a means of mitigating transmit RF field inhomogeneity (Hoult 2000, Ibrahim et al 2000. In pTx, a dedicated RF coil with multiple, independent channels generates individual B 1 + fields. These single-channel fields enable a spatially varying combined B 1 + field that can be optimized for homogeneous imaging even with the reduced RF wavelengths at UHF. Shortly afer, the concept of pTx RF pulses known as Transmit SENSE (Katscher et al 2003, Zhu 2004 was presented. This is analogous to the well known parallel imaging concept on the receive side, in which the undersampled data is reconstructed by leveraging the sensitivies of multiple receive channels (Sodickson and Manning 1997, Pruessmann et al 1999, Griswold et al 2002. Because pTx involves superposition of electromagnetic fields, there are important implications for SAR (Zhu 2004). Following the introduction of a mathematical formulation for pTx RF pulse optimization in the spatial domain (Grissom et al 2006), pTx has taken hold as an important tool in UHF MRI. A parallel-transmit array coil is an essential tool in the mitigation of the B 1 + inhomogeneity because it offers the flexibility to influence the amplitude and phase of the currents to each of the individual array elements. The experimental setup in the early work by Glover et al already had the capability to vary the magnitude and phase of the RF power applied to the two ports of the birdcage (Glover et al 1985). Ibrahim et al demonstrated in a numerical study the potential to achieve homogeneous B 1 + distribution at 8T by driving the multiple ports of a 24-strut TEM volume resonator with variable phase and variable magnitude (Ibrahim et al 2001b). In the first multi-channel transmit array implementation by Adriany et al the volume resonator was split in to decoupled independent coil elements . Figure 2(B) shows the functional block diagram of the setup used by Adriany et al referred to as RF phase shimming. Here, the amplitude of the RF pulse from each channel is equal and the phase of each channel is optimized for B 1 + homogeneity and/or efficiency. In the figure 2(B) setup, the output power from the RFPA is split into 8 outputs of equal amplitude and phase using an 1 × 8 power splitter and the electrical length of the coaxial cable between the power splitter and coil input is varied to modulate the B 1 + distribution. Eight or 16channel versions of this setup were used in most early transmit array implementations.

Definition of static pTx
Modern UHF scanners are equipped with eight or 16 independent pTx channels, offering seamless integration of transmit arrays and providing the flexibility to vary the amplitude and phase of the currents to each of the coil elements (figure 2(C)). This version of pTx is known as static pTx or alternatively, RF/B 1 + shimming. In static pTx the relative weights or shims are optimized between individual transmit channels. RF shimming can be performed on any RF pulse type, for example slice-selective sinc or non-selective adiabatic; the RF envelope for each channel is the same and only the RF weights vary (Padormo et al 2016). Although RF shimming can be broadly used in applications due to its simplicity, there are limits to the capability of static pTx to correct a heterogeous B 1 + field, particularly as the field strength increases (Mao et al 2006).

Definition of dynamic pTx
A further evolution of pTx is dynamic or full-waveform pTx, where B 1 + varies with both space and time. Here, the dynamic pTx pulse waveform is unique on each transmit channel and is typically accompanied by a set of specified excitation gradient fields. For slice-selective pTx, the most popular trajectory is the spokes trajectory (Saekho et al 2005, Setsompop et al 2006 and for non-selective pTx, it is the kT-point trajectory . Dynamic pTx leverages the full benefits of pTx by utilizing all degrees of freedom in RF pulse performance.
4. Coil configurations and transmit arrays for pTx 4.1. UHF coil configurations Transmit arrays used in UHF imaging can be subdivided into two main categories namely TxRx arrays and ToRo arrays. In TxRx arrays, the same coil elements are used to transmit and receive, and each array element is interfaced to the scanner through a TR switch with an integrated preamplifier.
Miniaturized TR switches, assembled inside the coil housing and close to the coil elements, prevents loss in SNR due to cable losses before the preamplification stage. An 8-channel segmented loop 7T TxRx array with integrated TR switches is shown in figure 2(D) (Avdievich et al 2017, Paterson et al 2020. TxRx arrays can be closely confined to the contours of the anatomy, which provides higher transmit efficiency. However, the number of receive channels in a TxRx array is limited by the number of transmit channels, which is up to a maximum of 16 channels in state-of-the-art UHF scanners. In a recent approach aimed at increasing the number of receive channels while maintaining a tight fit, 16 additional receive-only vertical loops were combined with 16 TxRx elements. While higher SNR was measured at centre of the brain, the peripheral SNR and parallel imaging performance was lower compared to conventional receive arrays (Avdievich et al 2019).
Experimental multi-channel transmit system with up to 64-transmit channels have been presented (Feng et al 2012). Massively parallel TxRx coil arrays are feasible with such systems. However, tight-fitting TxRx arrays that are shaped analogous to tight-fitting receive arrays could lead to increased SAR (Avdievich et al 2017). Furthermore, decoupling of non-adjacent coil elements when the coil is in transmit mode could become challenging in this setup. While low impedance preamplfiers can be used to implement preamplifier decoupling (Roemer et al 1990) to decouple non-adjacent elements during receive mode, 50 Ω (ohm) impedance is seen by the coil elements during transmit. This requires strategies to minimize coupling between non-neighboring elements for efficient transfer of power from the RFPA to the coil A ToRo array consists of a separate receive array in combination with a transmit array (Barberi et al 2000. Scanners with 64-receive channels are the current industry standard, and dedicated research scanners with up to 128 receive channels are already operational . The ToRo configuration offers the flexibility to maximize the number of receive channels in the coil setup (Gruber et al 2021, Shajan et al 2021. Increasing the number of receive elements is essential to approach the ultimate intrinsic SNR (Ohliger et al 2003, Wiesinger et al 2004, and also take advantage of the improved parallel imaging performance offered by high density recieve arrays at UHF (Vaidya et al 2016).
The engineering and integration of transmit arrays with high-density receive arrays is complex and challenging. Firstly, the transmit arrays must be large enough to accommodate the receive array, which reduces transmit efficiency in addition to the attenuation of the transmit field due to the RF shielding effect caused by the receive coil elements and its electronics. This further depends on the density of receive elements in the array and its implementation. Transmit efficiency loss of 7% and 10% has been reported in 32  and 64channel implementations (Uğurbil et al 2019), respectively.
In addition, the interaction between the transmit array must be carefully controlled so that the spatial distribution of the B 1 + field is not altered and the transmit coil performance is preserved . This can be verified by acquiring B 1 + maps with and without the actively detuned receive array physically present within the FOV of the transmit array.

Transmit array element types
The fundamental transmit array element in most designs are based on conventional segmented loops, microstrip transmission line elements, different versions of dipole antennas or a combination of loops and dipoles. These building blocks of transmit arrays and methods to decouple them are discussed in the following sections.

Microstrip transmission line
The first experimental transmit array implementation was using microstrip transmission line (MTL) as array elements . Coil arrays using MTL elements were originally introduced as planar strip arrays (PSA) (Lee et al 2001), in which the array elements are either quarter wavelength or half wavelength long and are inherently decoupled. The individual array element consists of a narrow strip of width (w) and a ground plane separated by a low-loss dielectric substrate with thickness (h). In the original 1.5T work, w, h and the spacing between the array elements were chosen to achieve a characteristic impedance of 50 Ω, and a high dielectric substrate (e r = 6.4) was chosen to shorten the physical length because the guide wavelength (l g ) scales by e  .
r In parallel, single and two turn MTL-based coil elements were proposed as alternatives to loop coils for MRI at extremely high field (Zhang et al 2001). The distributed nature of the transmission line allows these l 4 structures to be suitable for high frequency applications. Furthermore, the RF shield which is part of the microstrip element reduces radiation loss, which is a prominent loss factor as the frequency of operation increases. This would result in a compact and easily implementable coils as compared to conventional loop coils which requires an additional layer for RF shielding to minimize the radiation loss at high Larmor frequencies (Ong et al 1995). The work by (Zhang et al 2001) further demonstrated that thinner substrates negatively influences the B 1 + penetration, and substrate thickness of 7 mm was required to achieve good B 1 + penetration.
In lumped-element planar strip array (LPSA) (Lee et al 2004) implementation, the physical length of the MTL elements are shortened by terminating the ends of the microstrip with capacitors . This method allowed the physical length of the MTL elements to be dictated by the anatomy of interest and not by the wavelength as in PSA. Instead, the MTL elements are tuned to the desired frequency by adjusting the capacitors. The strip width and substrate thickness are chosen to optimize the penetration depth and the MTL elements are matched to 50 Ω through an impedance matching network. Several versions of MTL based arrays have been developed for 7T and 9.4T applications (Adriany et al 2010, p. 32, Vaughan et al 2006, Shajan et al 2011, Snyder et al 2012. The MTL elements in these early setups were tuned and matched to individual subjects before each scan session. To reduce the sensitivity of the array tuning to cable routing, the grounds of all the feed cables were shorted together at about 90 degrees from the coil input.
Center-fed MTL array (Brunner et al 2007) was proposed as an improved version of the l 4 MTL array. Its symmetric feed improved the stability of the array tuning and matching, minimized the sensitivity of the array to cable routing and substantially improved the longitudinal coverage because the center-fed MTL element is essentially two l 4 sections. A variant of the center-fed MTL element is the introduction of meanders to enhance decoupling (Orzada et al 2008). The size of the meanders can be further optimized to influence the decoupling and transmit performance of the array (Rietsch et al 2015). MTL elements with meanders is being used as the building block on the feasibility studies for scanner bore integrated massively parallel transmit array systems for body imaging at 7T (

Segmented loop arrays
Loop coils have been the most widely used in MRI signal detection for more than three decades since the introduction of the NMR phased array (Ackerman et al 1980, Roemer et al 1990, but have also been used in parallel MR excitation (Setsompop et al 2006). The size of the individual coil element, and hence its inductance, is determined by the number of available channels, size of the overall coil, and the anatomy of interest. This is irrespective of the field strength of interest. The capacitance value needed to resonate the coil element at high Larmor frequencies can become very small and comparable to the stray capacitance between the coil and the sample. The body becomes part of the coil resonance and as a result, the coil tuning depends on the subject size and position. This capacitive coupling results in dielectric losses and exhibits as a downward shift in coil resonance, whereas inductive coupling to the sample results in an upward shift to the resonant frequency Robinson 1979, Ong et al 1995).
Multiple capacitors are evenly distributed in series, which increases the value of the individual capacitors, causing the voltage to be distributed evenly along the loop (Decorps et al 1985). This approach reduces the capacitive coupling, which in turn improves the robustness of the coil tuning under different loading conditions.
Optimally distributing the capacitors along the loop is a critical consideration in the design of loop coils for UHF. The number of capacitors is empirically determined by the coil designer as the trade off to consider is the increase in equivalent series resistance (ESR) due to multiple capacitors in the loop. The ratio of the unloaded coil Q-factor to the loaded coil Q-factor is a figure of merit that determines the performance of the loop (Gruber et al 2018). The coil input is matched to 50 Ω through a balanced matching network (Murphy-Boesch and Koretsky 1983) and a cable trap is installed close to the coil input to minimize the currents induced in the shields of the feeding cable (Peterson et al 2003, Seeber et al 2004. Cable traps also reduce parasitic coupling between coil elements and reduce sensitivity of the coil parameters to cable routing and adds stability to the S-parameter measurements. There are different types of cable traps such as the solenoid trap, the lattice balun, the sleeve or bazooka balun and the floating trap.
RF shielding is another important consideration in the design of loop coils at UHF because the radiation loss increases with frequency. A distance of 4 cm between the coil and the RF shield was found to be optimal in the study at 4T by (Ong et al 1995). However, UHF coils with varying distances to shields can be found in literature (Avdievich 2011, Sengupta et al 2016, Williams et al 2021a, Chu et al 2022. While placing the RF shield too close to the coil will negatively influence coil performance, a larger distance to the shield is sometimes not feasible especially on coils meant for scanners with head gradient inserts. Numerical simulations of UHF coils offer insights into the influence of RF shielding on coil performance. This further depends on several factors such as the operating frequency, size of the coil, anatomy of interest, sample loading as well as the scanner environment which is different for whole body and head-only gradient systems.
Neuroimaging had been the primary focus for developing UHF technology, motivated by the push to achieve high resolution functional images of the human brain. Most early loop based UHF transmit array implementations has been for imaging the human brain in 7T and 9.4T scanners (Avdievich 2011, Gilbert et al 2012, 2011

Dipole arrays
Conventional loops and MTL elements were used as fundamental building blocks in UHF transmit arrays until dipole antenna elements were introduced (Raaijmakers et al 2011). The main advantage are their increased penetration depth to excite deeply located region such as the prostate which could be outside the near-field region to effectively image with surface coils. Dipole elements exhibits a nearly symmetric B 1 + profile, whereas the conventional loop elements at UHF have a charecteristic double lobe with a strong null in between (Vaidya et al 2016). Radiative antenna, as it was originally named, is made of a l 2 dipole antenna on a thick block of dielectric substrate. The high dielctric substrate was chosen to increase the directivity towards the imaging subject and its permittivity is chosen to match the average permittivity of body tissue to minimize reflections at the substrate-tissue interface.
In a concurrent work on approaching the ultimate intrinsic SNR (UISNR), the electric dipole antenna array  was developed to mimic the ideal current patterns (Lattanzi et al 2010). The dipole elements were built without the thick block of dielectric subtrate. The physical length of the dipole elements was shortened by a folded end and with discrete inductors. Fractionated dipoles, introduced later (Raaijmakers et al 2016), are similar to the radiative antenna but without the dielectric substrate block. The optimal element length for body imaging was found to be about 30 cm, and meanders were introduced to shorten the physical length of the individual array element. The SAR efficiency of the dipole array is shown to be better than loop arrays because of the elevated SAR observed under the overlap region on loop arrays ( , and many more. At high frequencies for large objects, the curl-free current modes corresponding to electric dipole makes a significant contribution to UISNR at the center of the object (Lattanzi and Sodickson 2012). This was demonstrated by mixing loops and dipoles, which achieved significant improvement in central SNR at 7T . Numerical evaluation of combined loop and dipole arrays for brain imaging at 7T also provided lower SAR compared to loop-only arrays . Because of the distinct current distribution pattern of the loop and dipole elements, decoupling can be easily achieved by carefully aligning the two elements along their central longitudinal axis. The promise of combined use of loops and dipoles as transceivers for body imaging was demonstrated (Ertürk et al 2017a) as this combines the better transmit and receive performance of the loop elements at shallow depths with the superior performance of the dipoles at deeper regions.

Decoupling of transmit arrays
Decoupling of array elements is a critical consideration in the design of transmit arrays. The transmit array elements must be well-decoupled from each other to enable them to be operated independently and to be tuned and matched to 50 Ω for maximum power transfer. RF shimming to reduce B 1 + inhomogeneity  and dynamic parallel transmission to accelerate multi-dimensional excitation (Katscher et al 2003, Zhu 2004 or provide efficient SAR management (Eryaman et al 2015 can significantly benefit from the low mutual coupling between the array elements as this enables independent control of the waveforms applied to individual transmit channels. While increasing the number of transmit elements and their arrangements provides additional degrees of flexibility to modulate the B 1 + field, low mutual coupling between the array elements is essential to realize these benefits . The performance and characteristics of coupled transmit array elements will depend on subject size and position, and results in power loss due to impedance mismatch and coupling. Furthermore, the reflected power will also depend on the excitation signal, and it becomes challenging to accurately predict coil behavior of a coupled transmit array (Kazemivalipour et al 2021). Several techniques to decouple transmit array elements have been developed depending on the coil type and geometric arrangement. A rule-of-thumb in terms of S-parameters is to achieve a clear reflection response without any peak split to be able to match the individual coil elements independently to 50 Ω and to minimize power loss due to impedance mismatch and coupling The transmit architecture in the commercial systems requires each coil element to be tuned and impedance matched to transfer maximum power to a nominal load. Hence, the classic preamplifier decoupling technique (Roemer et al 1990) employed in receive arrays, which transforms the low input impedance of the preamplifiers to a high impedance across the coil input and creates sufficiently high isolation between the coil elements, is not feasible in transmit arrays. The transmit elements look into the 50 Ω impedance of the RF amplifier, and an analogous implementation on the transmit side requires low output impedance RF amplifiers which are not commonly available (Chu et al 2009).
Different experimental RF transmit amplifier architectures have been proposed recently to drive multichannel transmit arrays (Hoult et al 2004, Kurpad et al 2006, Heilman et al 2007, Scott et al 2008, Gudino and Griswold 2013. In the on-coil current mode RF amplifier system (Gudino et al 2020), transmit element decoupling of less than −15 dB and high power efficiency was achieved through the high output impedance of the RF power amplifier, without any matching and decoupling circuitry.

Loop arrays
Geometric overlap of adjacent elements is a commonly used approach in loop-based transmit arrays (Roemer et al 1990). In arrays with large sized loops, the coupling between the second-neighboring elements is significant. In a recent implementation, the array loops were nested, and a pair of counter-wound inductors was used to minimize the coupling between the next-neighboring elements (Williams et al 2021a). In another category of loop array designs with gaps between the adjacent elements, capacitive network (Lee et al 2002), counter wound inductors (Kokubunji et al 1994), resonant inductive decoupling (RID) (Avdievich et al 2013), and RF shielding of the individual array elements (Gilbert et al 2010) have been implemented. In shared conductor design, the capacitor distribution in the common conductor between the adjacent loops is balanced to cancel the mutual coupling (Lanz et al 2010, Chen et al 2018. Self-decoupled coils, which utilize an asymmetric distribution of coil capacitance to balance and cancel the magnetic and electric coupling, has been proposed as an efficient and flexible alternative that do not impose geometric constraints (Yan et al 2018).

MTL arrays
The geometric length of PSAs (Lee et al 2001) is quarter or half of the resonant wavelength at the Larmor frequency. The PSA elements are inherently decoupled from each other if the element length is an integer multiple of l 4. The LPSAs (Lee et al 2004) or the capacitively terminated MTLs are significantly shorter than the PSAs, and the condition for intrinsic decoupling is not achieved. The adjacent elements in an MTL array are typically achieved by interconnecting capacitors at one or both ends of the strip (Lee et al 2004. This is analogous to the capacitive decoupling in conventional loop coils (Lian and Roemer 1997). Moderate improvement in decoupling can be achieved by wrapping the ground plane of the microstrip element along the sidewalls of the substrate.

Dipole arrays
Dipole arrays when used for body imaging are tightly wrapped around the anatomy of interest and therefore the array elements are very close to the tissue. Because the antenna is heavily loaded by the tissue, the array elements are well decoupled from each other. Hence, decoupling circuits analogous to the ones used in loop array coils are not required and the array implementation becomes a straightforward arrangement of the fundamental coil elements surrounding the anatomy of interest. In situations when the dipole elements are not heavily loaded by the tissue, for example in brain imaging applications, there is strong coupling between the dipole elements which is detrimental to the transmit efficiency of the array. Passive decoupling dipoles and the RF shield have been used to improve the decoupling between the adjacent elements of the dipole array (Yan et al 2015, Avdievich et al 2020.

Coil losses
The performance of a transmit array is evaluated by the B 1 + measurement in tesla in terms of transmit efficiency for input power P, and SAR efficiency or safety excitation efficiency (SEE) for simulated SAR given by Firstly, the drop in transmit efficiency at higher frequency increases the demand for high power amplifiers (Vaughan et al 2001). However, the output power available from the RFPA is limited. Furthermore, the cable losses between the coil input and the RFPA can be up to 3 dB and nearly half the power is wasted. Hence, engineering of transmit arrays should involve identifying and minimizing the different loss mechanisms within the coil to extract the best performance out of the transmit array. Full-wave EM simulation tools are particularly beneficial in analyzing the efficiency metrics of the transmit array and guide the coil designer in choosing the most suitable coil configuration for the intended application. The flowchart in figure 3 captures the various loss mechanisms within the coil setup.
As shown in the flow chart, power is lost by way of reflected power due to impedance mismatch. In addition, power is lost through neighboring channels due to coupling. These losses can be minimized by achieving robust S-parameters performance in terms of excellent impedance matching and decoupling.
The cut-off frequency of the scanner bore is close to the Larmor frequency at 7T (Brunner et al 2009), and a significant amount of RF power applied to the transmit coil can be lost due to radiation and wave propagation (Kozlov et al 2018). This will result in decreased RF transmit efficiency and increased RF power absorption in the human tissue, which can significantly influence the performance of UHF transmit coils. Incorporating an RF shield at an appropriate distance (Avdievich 2011) from the coil elements reduces the problematic RF radiation. Furthermore, transverse electromagnetic (TEM) coil designs are inherently shielded and are widely used at UHF (Roschmann 1986, Vaughan et al 1994. The RF coil losses consists of resistive losses of the coil conductors, series resistance of the components such as capacitors, inductors, diodes and radiation loss. The coil conductor losses depends on the conductivity of the material and the surface area because the RF currents flows near the surface of the conductor (Kumar et al 2009). The requirement for increased number of series capacitors will increase the RF circuit loss due to the ESR of the individual capacitors. It is important to select components with low ESR to minimize the series resistance as well as the potential risk of component heating due to power dissipation.

Design choice
Transmit array design for UHF MRI applications is an active field of research and it is a challenging task to provide a comprehensive comparison of different designs. A survey of UHF transmit arrays from the literature are provided in table 1. The main performance metrics of transmit array are its SAR efficiency, adequate coverage, robustness in coil performance under different loading conditions, and effective pTx performance. UHF transmit array design choice is mainly driven by the size of the anatomy. Hence, it can be broadly classified into coils for head imaging and coils for body imaging. Furthermore, head coil design principles can be easily extended to coils for extremities like knee and wrist imaging.
Conventional loop based transmit arrays have been the workhorse coils for brain imaging in 7T and 9.4T scanners. This includes both single and dual row designs. Single row designs have been shown to provide adequate whole brain coverage at 7T (e.g., 8Tx32Rx by Nova Medical Inc., MA, United States), but dual-row designs are essential to achieve whole brain excitation in 9.4T applications. 7T and 9.4T dipole based transmit arrays achieve whole brain coverage even at 9.4T with an 8-element single row design , Avdievich et al 2021, 2022, However, it is important to note that the peak SAR 10g values reported at 7T are significantly higher than corresponding values reported for loop arrays (Williams et al 2021a), yet a direct comparison of the SAR efficiency is not currently available. Another factor to consider is the frequency shift in the tuning of dipole elements under different loading conditions (Lakshmanan et al 2020).
In case of body imaging, although loop-based body arrays have been developed earlier (Graessl et al 2013), dipole-based array elements and combinations of loops and dipoles has been the most popular choice recently due to the extended longitudinal coverage, greater penetration depth, and their ease in implementation.
Transmit array designs for extreme high field scanners such as 10.5T and 11.7T is even more challenging because of the increase in wave propagation. It has been shown recently that the coil design must be accompanied by optimization of RF shield design to maximize coil performance (Zhang et al 2021, Chu et al 2022.

Modelling the electromagnetic field of pTx coils
Parallel-transmit arrays in combination with pTx techniques can produce time dependent spatially varying energy deposition in biological tissue. MRI at UHF operates in a regime where quasi-static approximations are invalid and knowledge of the spatial distribution of the magnetic and electric field is essential for reliable safety assessment as well as in the management of energy deposition during an MRI examination Smith 2001, Ibrahim andTang 2007). Numerical simulations using heterogeneous body models are the current approach to obtain the locally varying electric and magnetic field information as well as in the estimation of tissue temperature. EM simulations have thus become an indispensable tool in demonstrating safety compliance while using pTx coils. In addition, the role of EM simulation tools is vital in the optimization of transmit arrays because it provides the necessary information to calculate the SEE (section 4.4), which is the most important performance metric of the transmit array (Avdievich et al 2020).

Numerical modelling of the transmit array
Electromagnetic field solvers depend upon numerical methods to solve Maxwell's equations. Early work was carried out using self-developed tools and methodologies, suited to the application and coil (Ibrahim et al 2000, Collins andSmith 2001).
A thorough review of the commonly available software tools used for EM simulation in MRI is beyond the scope of this paper. However, a detailed discussion can be found in (Fiedler et al 2018). Here we present an example using CST Studio Suite (Dassault Systèmes SE, Vélizy-Villacoublay, France) and its transient finite integration solver in conjuction with a circuit simulation co-simulaion to model the system.
The numerical model of the coil should realistically replicate the constructed coil, and the scanner environment is incorporated into the model to account for the RF wave propagation (Brunner et al 2009). The coil model shall consist of the mechanical model and material properties of the coil housing, the local RF shield if the transmit array is locally shielded, the RF shield in the scanner, and the coil circuits. The coil circuits consist of the input impedance matching circuit, tuning circuits, coil conductors, and decoupling circuits. To enable faster simulations by utilizing the co-simulation feature (Kozlov and Turner 2009), the tuning, matching and decoupling circuit components are defined as ports. The fixed circuit components are defined as lumped elements.
Coil component losses such as the series resistance of the capacitors, inductors, and diodes must be incorporated into the simulation environment to accurately capture the coil losses . Other sources of loss that could be included in the numerical model are the cable losses between the coil feedpoint and coil plug, and the insertion loss of the TR switch in the case of TxRx arrays. Further considerations include local mesh refinements to ensure all ports and lumped elements are properly connected to the coil conductors, and coarser mesh in the periphery of the simulation domain.

Example numerical model of an 8-channel transmit array
In this section we provide a 'how-to' guide to the numerical modelling using an example 8-channel TxRx array. The transmit array is a tight-fit 8-channel segmented loop TxRx array constructed on a 3 mm thick fiberglass tube with an inner cross-section of 21 cm × 24 cm (Avdievich 2011, Paterson et al 2020. A local RF shield is placed concentrically on an outer tube at 30 mm distance from the array elements. Adjacent array elements are decoupled by geometrical overlap, and the coil elements measure 12 cm along the z-direction. Custom-built TR switches with preamps are part of the coil housing. A photograph of the constructed coil array is shown in figure 2(D).
The equivalent circuit of a single array element is shown in figure 4(A). Each coil element consisted of seven fixed capacitors (C 1 to C 7 ; 6.8pF; C series, AT Ceramics, Huntington Station, NY, USA) and one tuning capacitor (C 8 ; 5610; Johanson, Camarillo, CA, USA). The input circuit consists of a balanced matching circuit with two fixed capacitors of equal value (C 9 and C 10 ) and a variable capacitor C 11 . The loop capacitors are connected by 2 mm diameter silver plated copper wire. The coil tuning, matching, and decoupling was adjusted while loading the coil with a head and shoulder phantom filled with tissue equivalent solution (Beck et al 2004. The numerical model of the 8-channel array was created in CST Studio Suite (Dassault Systèmes, Vélizy-Villacoublay, France). A screenshot of the model of a single channel is shown in figure 4(B). The matching and tuning circuits shown in schematic 4A are modelled as ports (red) and the fixed capacitors are modelled as lumped elements (blue). This enables the coil parameters to be adjusted in circuit co-simulation without the need to re-simulate in the 3D domain-effectively saving considerable simulation time (Kozlov and Turner 2009). The schematic of the input circuit included in circuit co-simulation is shown in figure 4(C). The lumped element model included ESR and equivalent series inductance (ESL) values from the component datasheet and the connecting wires are modelled as perfect electrical conductors. The coaxial cables and the cable traps are not included to minimize computational complexity. However, all losses up to the scanner coil plug consisting of the coaxial cable loss and TR switch loss are included in the model as an attenuator (figure 4(C)). This will also shift the reference power from coil input to scanner coil plug.
The numerical model also included the local shield as in the constructed coil, the two concentric fiberglass tubes (ε = 4.3 F m −1 , loss tangent = 0.025), and the scanner bore modelled as a large RF shield (Wolf et al 2013. The coil was loaded with a head-and-shoulder phantom with known electrical properties (ε = 51.1 F m −1 , σ = 0.4 S m −1 ). A screenshot of the numerical model of the coil is shown in figure 4(D). The RF shield and fiberglass tubes are hidden for visualization. Once a numerical model is created, the first step is to ensure that the model is robust and reliable. In this example a hexahedral mesh is used along with afinite integration solver. Although the simulation tools automatically create the mesh, it is critical to ensure that mesh has sufficient resolution for the model to avoid a staircase error. In this example the overlapping elements are sufficiently close enough that the automatic mesh may place a voxel spanning two separate coils and short them. The number of mesh cell across the lumped elements and ports may also need to be carefully considered. This is true for most of the solvers and some care and time should be assigned to improving the mesh. As such, there will be a trade-off between the ideal mesh and a practical value that is adequate, without being computationally prohibitive (Kozlov and Turner 2009). The simulation should also take care to enforce sensible boundary conditions (Fiedler et al 2018). Tangential electric fields should vanish at a perfectly conducting boundary and free space should be modelled using a perfectly absorbing boundary that ensures there is enough space, so that this is located outside the near field region. It is also important that the correct convergence criteria is set for an accurate solution. For example, in the transient solver the simulation will run until the energy in the system falls below a set value. In this example the convergence for the time domain solver was set to −40 dB. This criterion was set in line with the literature (Wolf The next step after creating the numerical model of the transmit array is the optimization of the array S-parameters. This includes selecting suitable circuit component values to achieve S11 values of less than −30dB on each of the array elements and optimizing the overlap distance to minimize the coupling between adjacent elements. The coil tuning and matching are adjusted within a few seconds in co-simulation domain because these circuit components were modelled as ports. Optimization of the overlap distance however is adjusted iteratively and requires a new 3D simulation each time the geometry is changed until the minimum value of S21 is achieved. Figure 5(A) shows the final simulated S11 plots and adjacent element coupling (S21) for the 8-channel TxRx array, while figure 5(C) shows the same measured parameters. The highest S11 was −47 dB in simulation and −34 dB in measurement, while the worst case S21 was −15.7 dB and −18.2 dB for simulation and measurement, respectively.
Once the S-parameters are fully optimized, field maps can be generated from the fully tuned and matched system and the B 1 + and B 1 field maps may be derived from the simulated H-field using the following equations H, and μ is the magnetic permeability. The simulated B 1 + maps in CP excitation mode are shown in figure 5(B) and the measured maps are shown in figure 5(D). The peak B 1 + in simulation was 125.0 nT/V and in measurement was 108.6 nT/V. To achieve CP excitation, each coil element is driven with equal magnitude and the incremental phase offset between the adjacent channels is 45°. These values can be set as a separate excitation for the corresponding simulation ports inside the 3D solver, or if the co-simulation approach is used, they will be set here, once again providing a rapid solution for deriving the fields for an arbitrary signal.
The power budget in figure 5(E) Turner 2010, Kuehne et al 2015) shows the distribution of the power applied to the RF coil and it captures the various losses due to the loss mechanisms outlined in section 4.4. It reveals the amount of outgoing power due to impedance mismatch and coupling (3.2%), amount of power dissipated in the coil components (1.6%), power lost due to radiation (6.6%), and the amount of power absorbed by the dielectric sample (58.1%). In addition to insights into realistic estimation of SAR per input power, the power budget provides the coil engineer with the information necessary to apply design improvements to improve the transmit efficiency of the array by minimizing the losses. This includes measures such as improving the decoupling to minimize losses due to outgoing power, RF shielding to reduce radiation loss, and selection of low loss components to minimize power dissipated in the coil components.

Validation of the numerical model
Establishing confidence in the numerical model of the coil is essential prior to SAR simulations with the coil model . This can be accomplished by: (1) Comparing the values of the circuit components in the EM model with the corresponding component values in the actual coil; (2) comparing the S-parameter matrix of the simulated and constructed transmit array; (3) comparing the simulated and measured B 1 + maps in a phantom in more than one excitation modes (for example, CP and CP 2+ with 90°phase offsets); and (4) comparing the EM simulation-derived temperature with MR thermometry (explained in greater detail below). These simulations and measurements are performed using phantoms that load the coil comparable to the human body. The phantoms are filled with tissue equivalent solution with electrical properties of the average tissue of the anatomy of interest (Beck et al 2004). In the numerical domain, voxel models of the test phantoms are created and the measured electrical properties of the solution (permittivity and conductivity) are assigned for the phantom solution. For head coils, a head-andshoulder phantom is necessary to account for loading effects of the shoulder. The entire EM simulation and validation workflow in digital phantoms up to the point of SAR simulation is shown in the first half of figure 6.

MR thermometry for model validation
Temperature can be both simulated and measured in specific sequences. This process, known as MR thermometry, is a critical step to validating RF coils because it corroborates the accuracy of the EM model with measured experimental data (Hoffmann et al 2016). The temperature rise ΔT within the model can be calculated from the EM simulated SAR (equation (2)), which is calculated from the electric fields. Given the specific heat capacity C p of the phantom material and the pulse time Δt, we have  (3)), this simplified version does not take diffusion and radiation into account. Nevertheless, it is a useful metric for safety evaluation when comparing the temperature rise within the scanner with the predictions on the model. Experimentally, fluoroptic probes provide a means of measuring temperature in an MRI scanner with high precision and are a popular approach for measuring heating of implants (Mattei et al ). However, thermal probes are only capable of measuring temperature change at point source and are furthermore invasive (Shrivastava et al 2011).
Instead it is desirable to map the spatially varying temperature for corroborating the EM simulation. This can be done with MRI using the proton resonance frequencey shift (PRFS) (Ishihara et al 1995). The PRFS method typically is performed by acquiring a series of GRE images with an additional heating pulse applied offresonance (to avoid contaminating the imaging signal). Given a series of phase images Δj(r), the temperature ΔT(r) at location r is given by where B 0 is the main magnetic field strength, TE is the sequence echo time, γ is the gyromagnetic ratio, and α is the PRFS coefficient known to be −0.01 ppm C −1 in water (Ehses et al 2008). The PRFS method is the recommended approach for validating the EM simulation of a particular RF coil, because a matched phantom experiment can be performed to verify the accuracy of the simulation to the physical coil (Hoffmann et al 2016).
One disadvantage of the PRFS method is that it typically lacks high levels of precision. Given an α of −0.01 ppm C −1 , the sensitivity of PRFS at 7 tesla is −3Hz for 1°C (Le Ster et al 2021), but magnetic field fluctuations not attributed to temperature rise are of similar orders of magnitude. This makes it challenging to reliably measure temperature changes below 1°C even at ultra-high field strengths. Performing temperature mapping with fluoroptic probe measurements can help provide additional thermal resolution, yet remains unviable for human studies. Recently, an adapted MRI themometry approach was proposed that used the PRFS technique with improved precision by using field probes to correct for magnetic field perturbations. In conjunction with motion correction, this resulted in successful MRI thermomety performed in a set of healthy volunteers in the brain at 7T (Le Ster et al 2021).

Digital human body models
After the coil model is validated it can be used for SAR simulation, safety validation, and SAR management. At this point the phantom is replaced by heterogeneous human body models, which are vitally important to derive local SAR and tissue temperature distributions. Many of the simulation tools used in MRI come with a diverse range of human models from infants to geriatric, in various body types and conditions. A set of standard models from the IT'IS foundation is widely used in the community, and serve for comparisons between different coil designs (Christ et al 2010, Guérin et al 2019, Carluccio et al 2021, Noetscher et al 2021. These models are of a voxel type, which consist of a regular cuboid grid lattice with each element of the grid constituting a tissue type with corresponding material properties. It is also possible to use more complex polygonal models when using the frequency domain solvers, however, these can be quite complex to construct and may require large computational resources. Voxel models are relatively small in memory since they are represented by a simple matrix. For reliable SAR management, a range of body models in different positions and rotations within the RF coil needs to be considered. The final SAR model can be concatenated into a single dataset (Williams et al 2021a). To minimize the computational burden, the body models can be truncated depending on the application. While simulating a head coil, for example, it is not necessary to include the entire anatomical body model in the simulation (Wolf et al 2013). This study further showed that body models that include only the head and the shoulders provides suffciently accurate results. Furthermore, the resolution in the shoulders can be modelled to be homogeneous without affecting the SAR distribution in the head, reducing the computation demands of the simulation.

SAR simulation and VOP generation
The spatially varying SAR in the case of pTx can be derived from equation (2) for time point t i using the simulated electric field E c for each N c transmit element at all r voxel locations in the simulated object with known electrical conductivity and mass density (Graesslin et al 2012), Note here that SAR involves summing electric fields, which can lead to non-intuitve local SAR patterns. Given a normalized set of individualized N c × 3 field matricesẼ, and a particular complex transmit vector configuration w(t i ), can also be expressed quadractically by reformulating equation (8) in terms of so-called Q-matrices (Graesslin et al 2012) The IEC safety standard for SAR is measured on averaged 10 gram volumes ('IEC 60601-2-33,' 2010). Figure 7 shows the results of SAR simulations in the example 8-channel coil model described previously. SAR still remains the key metric to manage power deposition in MRI. For conventional sTx, the B 1 + configuration is constant (often CP), and is conventionally monitored with a constant safety factor of SAR overestimation (so-called 'k-factor' supervision) (De Zanche et al 2022).
With the spatially varying SAR deposition, pTx necessitates local SAR supervision based on the simulated electromagnetic fields from a validated EM model. For any coil model simulation (or concatenated series thereof), Q-matrices (Graesslin et al 2012) can be formed to calculate SAR quadratically for any B 1 + shim configuration as shown in equation (10).
In practice, the EM simulations can result in hundreds of thousands or millions of voxels in the digital model, making online local SAR calculation computationally challenging. Virtual observation points (VOPs) offer a tractable way of estimating local SAR by exploiting the positive semidefinite properites of Q-matrices and clustering them based on their maximum eigenvalues (Eichfelder and Gebhardt 2011). The number of clusters is governed by the percent overestimation tolerated for the B 1 + shim configuration contributing the 'worst case' or peak local SAR. A secondary work then generalizes this VOP approach to consider both upper and lower bounds of peak SAR and provides even further compression of the EM simulation (Lee et al 2012). Thereby, VOPs provide flexibility whereby local SAR supervision is made more or less conservative with fewer or greater numbers of VOPs, respectively. This process of SAR simulation and VOP generation in heterogeneous body models after EM model validation is shown in the latter half of figure 6. In addition to the percent overestimation with respect to 'worst case' configuration, other safety factors can be considered in generation of VOPs for local SAR mitigation. These safety factors are outlined and described in table 2.
While conventional VOP compression outlined by Eichfelder andGebhardt 2011 andLee et al. 2012 remain the most popular methods for local SAR estimation in pTx, recent work has sought to improve performance. In one method using VOPs, overestimation of SAR is kept constant while the number of VOPs clusters is reduced, enabling faster pTx local SAR estimation . Another method compares overly conservative linear SAR safety factors by deriving a conditional safety factor based on probability theory (Meliadò et al 2020). Finally, a third method proposes the use of temperature VOPs in lieu of local SAR supervision to mitigate the true, relevant safety metric for pTx (Boulant et al 2016). In general, as pTx gains popularity and widespread use, the safety considerations are of continued importance and under continuous research for improvement.

pTx applications and RF pulse design
To complete the discussion of pTx RF coils, it is also necessary to discuss their use. As mentioned, pTx offers additional degrees of freedom to control the transmit RF field, yet careful RF pulse design is often needed to do so. While a larger, more expansive review of pTx pulse design specifically can be found in a separate review paper (Padormo et al 2016), in this section we discuss pTx applications and walk through a generalized pulse design framework. The capacity for B 1 + correction with pTx relies on several factors including MRI field strength, the body region being scanned, SAR, the pTx RF coil design and number of transmit elements, and the pTx RF pulse used. Given these factors, there is a theoertical upper limit to the homogenization potential of pTx (Katscher et al 2004).

Pulse acceleration
Although pTx is most commonly associated with mitigating flip angle inhomogeneity, pulse acceleration using the Transmit SENSE concept is another important application of pTx (Katscher et al 2003). Here, a shorter RF pulse length is achieved by leveraging the redundancy of pTx compared to sTx for when using RF and gradient pulses that provide spatial selection in multiple dimensions (Zhu 2004). This idea is demonstrated in figure 8, where a 2D excitation produced in single-transmit is accelerated using pTx with over a factor three improvement in excitation performance. The use of pTx for pulse acceleration is most notable for reduced field of view (rFOV) or inner volume imaging (Feinberg et al 1985). For rFOV imaging, a 2D or 3D excitation pattern is generated by choosing an appropriately matched excitation gradient trajectory (e.g., spiral and EPI for 2D, stack of spirals and concentric shells for 3D) (Grissom et al 2006, Schneider et al 2013. With rFOV, the best excitation performance can be achieved when the excitation trajectory is optimized jointly with the RF waveforms (Davids et al 2016, Luo et al 2020, Majewski 2021. For the conventional sTx case, these excitation trajectories can be prohibitively long for some applications. This may lead to long RF pulse durations and thereby considerable T2 and T2 * relaxation effects. For 2D rFOV imaging, pTx acceleration factors R > 4 have been explored while considering SAR and performance (Zelinski et al 2008). 2D excitation pulses have also been used in vivo at very high field strengths such as 9.4T (X Wu et al 2010). Meanwhile, the pTx tradeoff benefits of full FOV with local excitation, rFOV with R > 6 reduced imaging time, and rFOV with higher resolution were explored for 3D excitation have been shown (Schneider et al 2013).
Reduced FOV pTx pulses have since been incoporated in RARE (Hennig et al 1986) or turbo spin echo (TSE) sequences at 7T (Mooiweer et al 2018). Furthermore, the recent concept of 'Universal Pulses' (UP) derived from optimization over a database of preacquired B 1 + and B 0 maps (Gras et al 2017b) has been extended to rFOV excitation, and at the UHF strength of 9.4T (Geldschläger et al 2021). Finally, recent work has incorporated deep learning methods into 2D rFOV excitation for single-transmit pulses (Vinding et al 2020) which creates the potential for such powerful artificial intelligence methods in the pTx case.

Low SAR and implant-friendly RF pulses
An additional pTx application that is often less-appreciated is to alleviate SAR conditions in UHF imaging as well as in the scanning of medical device implants. pTx involves the summation of superimposed RF fields, and the local SAR distribution varies with pTx modes (Lee et al 2012). However, another perspective is that the additional degrees of freedom introduced with pTx offer an opportunity for better control of local SAR and associated tissue heating. In a common pTx pulse design scenario, a pulse is optimized with specified local and global SAR constraints (Hoyos-Idrobo et al 2013, Guérin et al 2014). These constraints are especially important at higher field strengths with higher RF power deposition. Alternatively, a pTx pulse optimization can seek to minimize SAR while maintaining a standard flip angle homogeneity enforced as a constraint (Pendse et al 2019). More details of these pulse design formulations are discussed below in section 6.5.
Even at conventional field strengths such as 1.5 and 3 tesla, metallic implants and interventional devices pose signficiant heating risks and may be a contraindication for MRI. This can restrict a patient's healthcare, as is commonly seen in the example of patients with deep brain stimulators who could benefit greatly from MRI neuroimaging (Cabot et al 2013). Implant and device heating is related to the electric field and with a multitransmit coil it is possible to generate a null mode with near-zero electric field while maintaining a desirably homogeneous B 1 + field for imaging (Eryaman et al 2011, Etezadi-Amoli et al 2015. These implant-friendly null modes have even been incorporated into pTx pulse design whereby electric field reduction and B 1 + field homogenization are optimized simultaneously (Eryaman et al 2015). Recent work has sought to rigorously safety check the use of pTx for mitigating implant and device heating before use in patients and has included realistic phantom experiments (Godinez et al 2019, McElcheran et al 2019, in vivo scans in sheep (Godinez et al 2021), and more accurate electromagnetic field device models (Guérin et al 2019). This active work suggests that this strategy of pTx in patients with implants could be realized in the near future.
6.4. Tailored versus universal pulse designs pTx RF pulses require measurement of the B 1 + field from each individual transmit channel, which vary based on the object loaded in the coil and for each new scan occurrence. Therefore, conventional pTx RF pulse design has necessitated the measurement of subject-specific B 1 + maps, and often static field B 0 maps as well. This appraoch to pTx pulse design is referred to as tailored pulse design and is performed online for each new subject scan.
Recently, Universal Pulses have sought to alleviate the requirement of subject-specific field map measurements (Gras et al 2017b). Here, a generalized pTx pulse is designed using a set of pre-acquired B 1 + and B 0 maps from a representative population of subjects. The pulse optimization achieves robust pTx performance if performed globally over the entire test field map dataset. So far, UPs have been applied to the human head and heart (Aigner et al 2022). While UPs do not perform as well to their tailored counterparts, their benefit comes from eliminating scan time with additional measurements and pulse design, their ease of use (aka 'push-andplay' pTx), and their robustness to motion across the total scan session (Gras et al 2019). To-date, UPs have been deployed for a variety of non-selective pulses in 3D sequences (e.g. gradient echo, MPRAGE (Mugler and Brookeman 1990), MP2RAGE (Marques et al 2010), FLAWS (Beaumont et al 2020), SPACE (Mugler 2014)) in the PASTeUR package  and have been refined to overcome high B 0 field off-resonance (Van Damme et al 2020) using the GRAPE algorithm (Khaneja et al 2005). They have been extended to 2D and 3D rFOV excitations (Geldschläger et al 2021) and the concept has been also been used to generate calibrationaless small region shims for single-voxel spectroscopy (Berrington et al 2021). The recent 'Standardized Universal Pulses' have shown to achieve further robustness by employing a quick calibration scan of~10 s ( Another further advancement are Fast-Online Customized or FOCUS pulses , which leverage both universal and tailored pulse designs. In the FOCUS method, a tailored pTx pulse is designed rapidly, initializing the pulse with a UP. While the FOCUS approach still requires the initial B 1 + and B 0 map measurements for each subject, the pulse design time rapidly decreased to previous tailored approaches and also occurs fully online. Furthermore, FOCUS adapts to individual field maps that are unseen in the UP design database, which is particularly important in patient populations with irregular anatomy. FOCUS pulses were originally proposed for low flip angle 3D excitation pulses, but have since been extended to inversion pulses  for MPRAGE. It is anticipated that FOCUS approaches can be extended to a larger class of 3D pulse sequences, similar to UPs. 6.5. How-to guide for pTx RF pulse design Here we will describe a generalized framework for all pTx pulse design which is adaptable-from static RF shimming, to dynamic UPs; from slice-selective spokes to transmit SENSE 3D inner volume excitations. We begin with a simple formulation of an RF pulse design problem and add layers of complexity geared towards particular applications. One of the first introduced methods of pTx RF pulse design was the spatial domain method (Grissom et al 2006). This approach leverages the small-tip angle (STA) approximation, which relates an RF pulse b and the enacted transverse magnetization m xy through a STA system matrix, A, that can be seen as Fourier operator incorporating B 0 -induced off-resonances and a defined excitation k-space trajectory (Pauly et al 1989). In Following this, an efficient way to design a pTx RF pulse is by solving a weighted, regularized least-squares problem Here, m T is the target transverse magnetization, W defines the spatial weights of the design, and λ is the Tikhonov regularization parameter that penalizes the integrated RF power of the pulse, which also reduces global SAR. When the spatial domain is sufficiently small such that A pTx is easily invertible, a closed form exists to solve for b pTx , offering a simple approach for pulse design. For larger problems, iterative methods like the conjugate gradient algorithm can be used (Sutton et al 2003).
In the case of pTx, more advanced cost functions and optimization algorithms are often necessary. For example, consideration of local SAR and per-channel voltage limits can also be included in the cost function in the form of constraints (Hoyos-Idrobo et al 2013, Guérin et al 2014) The first two quadratic constraints enforce limits on local and global SAR and the second two restrict hardware voltage and average power limits, respectively. Matrices Q v are a set of N v positive semidefinite matrices representing the spatial local SAR derived from electromagnetic field simulations of the pTx coil (Graesslin et al 2012) while matrix Q G is a single matrix to compute the global SAR. The infinity-norm (third constraint in equation (14)) limits the per-channel voltage of a pTx RF pulse, while the final 2-norm constraint limits the average per-channel power. Despite being a convex function with convex constraints, solving the cost function in equation (14) requires a solver that can handle a potentially large number of inequality constraints, but fortunately a variety of programs exist in commonly used scientific programming languages (Boyd and Vandenberghe 2004).
Despite the ease of a linear STA pulse designs, this approximation only remains valid in the STA regime, at most up to flip angles of 90° (Pauly et al 1989). Further works have introduced techniques to re-cast the STA approximation in terms of target flip angle (Boulant and Hoult 2012) or even by summing multiple STA designs (Grissom et al 2008) to acheive larger flip angles. However, to more accurately design large flip angle pTx pulses such as inversion and refocusing pulses, the nonlinear Bloch equation has been iteratively compensated for with the introduction of optimal control (Xu et al 2008). Optimal control methods often increase computational burden of large tip-angle (LTA) pulse design, so a popular approach is to neglect T1 and T2 relaxation and reduce the 3 × 3 Bloch equation rotations into the Bloch Spinor domain using Cayley-Klein parameters (Pauly et al 1991) or quaternions (Majewski and Ritter 2015). This generalized LTA design cost function is therefore where the STA approximation in equation (13) has been replaced by a Bloch evaluation (likely via optimal control) as a function of RF and excitation gradients. For the sake of brevity, the explicit SAR and hardware constraints in equation (14) have been summarized generically in subsequent expressions. Excitation gradients play an important role in dynamic pTx, complementing the spatially varying RF field. Most commonly, the gradient waveforms are fixed for the pTx pulse design target. Due to design complexity, full-waveform pTx pulses don't commonly optimize every time point in a particular RF pulse, but rather at particular k-space locations. As mentioned in section 3.3, the most common trajectories for slice-selective pulses are spokes (Saekho et al 2005, Setsompop et al 2006, and for non-selective pulses are kT-points . Both spokes and kT-points are designed by optimizing the per-channel weights of RF subpulses (typically sinc for spokes, rectangular hard pulse for kT-points) and small gradient blips between subpulses. For nonselective pulses, the SPINS trajectory (Malik et al 2012) is also used and offers additional degrees of freedom with respect to the RF waveform optimization compared to kT-points at the expense of increased optimization complexity. Nevertheless, it is also possible to consider the joint design of RF and gradients, where the cost function in equation (15) is alternated between solving for the pTx pulse b pTx and gradients g x,y,z to minimize the difference between Bloch-evaluated magnetization and the target. This nonlinear problem is challenging to solve, although recent approaches with auto-differntiation (Luo et al 2020) or explicit derivatives (Majewski 2021) offer computationally tractable solutions.
The final variation to the pTx pulse design optimization framework considered in this section is the cost function itself. Previously, equations (13)-(15) considered ordinary least-squares minimizations where a particular complex magnetization pattern is targeted. However, for many RF pulses only the magnetization magnitude or in fact the flip angle is important, so the phase can be disregarded. This leads to the formulation of a the non-convex magnitude least-squares (MLS) approach (Setsompop et al 2008), 6.6. Troubleshooting challenges with pTx pulse design One important aspect of pTx pulse design is the challenges the pulse designer might face throughout the implementation process. In this section we mention a few common problems and offer tips on adressing them. As discussed previously, both B 1 + and B 0 mapping is essential for each pTx pulse design in the case of tailored RF pulses, requiring critical scan time. This compounds with the subsequenct design time associated with pulse optimization. For the case of 3D sequences, the Universal Pulse concept can help tackle the need for field mapping for an individual scan, as long as a representative population of maps exist for the initial UP creation, and also eliminates any online pulse design. Secondly, patient motion is of concern with pTx. If using subject-specific B 1 + and B 0 maps, these are acquired at the beginning of a scan session, and the subject can move afterwards which will reduce pulse design accuracy. This is an additional benefit of UPs, which are generally more immune to motion. Additionally, rFOV excitations are inherently less motion-sensitive. Recently, UPs have been applied to rFOV excitations (Geldschläger et al 2021), offering the advantages of both methods. Importantly, motion is not only detrimental to tailored pTx pulse designs, it also has potential safety implications around SAR. In the head, simulation studies of motion have found dramatic increase in SAR estimates for both RF shimming and particular dynamic pTx pulses (Ajanovic et al 2020, Kopanoglu et al 2020. Again, this is due to employing a pTx pulse designed for a set of field maps that have since changed after motion. Nevertheless, an additional solution to motion is to adapatively predict the effects of motion on B 1 + maps using deep learning which could be used to re-design pTx pulses after motion is detected (Plumley et al 2021).
Alongside increasing optimization times, as pTx pulse designs become more complex with nonlinear cost functions and multiple constraints (e.g. equations (14)-(16)) they can also converge to suboptimal solutions in the form of local minima. This makes the initialization of pulse optimizations critical (Sun et al 2016), but recent work has rigorously studied how to enable improved solutions for non-convex pTx pulse optimizations (Eberhardt et al 2020). For the specific case of slice-selective spokes pulses, another recent solution using finitedifference regularization to avoid unwanted flip angle nulling has also been proposed (Paez et al 2021).
Finally, because pTx pulses are designed to pair with complementary excitation gradients, the fidelity of those gradients to their prescribed waveforms is crucial for pulse accuracy. This becomes increasingly more relevant with more dynamic waveforms. For fixed excitation gradients, the gradients can be measured a priori with calibration measurements (Duyn et al 1998) Figure 9 provides a workflow summarizing the general pTx pulse design framework, including a few considerations to make when troubleshooting.

Future direction for pTx
pTx RF coils and their use has been under continuous development for the past two decades and have led to significant advancements in hardware, simulation, and pulse design technology. In this final section we will discuss what lies ahead in the future for pTx including the use of artificial intelligence (AI) and the potential for routine clinical use.

Role of artificial intelligence in pTx
Machine learning and AI have recently proven to be powerful tools for various aspects of pTx. For example, AI methods have helped alleviate the computational burden of electromagnetic field simulations. In one recent work, a convolutional neural network was used to predict subject-specific local SAR given their B 1 + maps using an 8Tx array for prostate imaging (Meliadò et al 2020). Another model generated B 1 + maps given subject motion measurements from an initial B 1 + maps without motion (Plumley et al 2021). A final recent work used generative adversarial networks to synthesize missing B 1 + maps from a reduced subset of maps (Eberhardt et al 2022).
Artificial intelligence has also been introduced to accelerate RF pulse optimization. The 'DeepRF' deep reinforcement learning approach has been proposed to generate slice-selective excitation, slice-selective inversion, and B1-insensitive inversion pulses for single-transmit pulses (Shin et al 2021). Meanwhile the 'DeepControl' method uses convolutional neural networks to design 2D sTx spatial excitations for UHF (Vinding et al 2020). These deep learning approaches for single transmission could be extended in the future for pTx pulse design as more training data (typically B 1 + and B 0 maps) becomes available.
Furthermore, some pTx pulse design using machine learning has already been proposed. For example in the case of static pTx, a projected ridge regression method was created to generate slice-specific shims rapidly with a subset B 1 + mapping data (Ianni et al 2018). Meanwhile, the 'SmartPulse' approach offered a pTx machine learning method where a classifier learned features from the initial localizer scan to select an appropriate class of pre-designed Universal Pulses for 3T abdominal imaging (Tomi-Tricot et al 2019). Recently, B 1 + map data from a 9.4T 16Tx coil was used in a regression model for spokes design in simulation (Eberhardt et al 2022).
To date, there is still some additional validation required for the role of AI in pTx in terms of accuracy and safety before these methods can be fully deployed. Nevertheless, it is clear that the computational benefits machine learning models can provide will complement the progress and development of pTx use.

Towards regulatory approval of UHF pTx and clinical use
At the time of writing, pTx is not yet approved for regulatory use at UHF by major medical regulatory agencies such as the US Food and Drug Administration or the European Medicines Agency. This means that pTx cannot be used for diagnostic purposes, despite the advantages it offers for B 1 + homogenization, control over SAR, and pulse acceleration. However, modern 3 tesla scanners have two-port body coils that enable pTx using two separate transmitters. These systems are most commonly used for RF shimming for imaging in the body (Brink et al 2015) and also pTx applications such as rFOV imaging (Siemens' ZOOMit) (Boada et al 2013). These systems have improved B 1 + homogeneity in clinical 3T imaging of large regions in the body for a variety of applications (Willinek et al 2010). It is expected that regulatory approval of pTx for 7T MRI could have similar or even further clinical benefits. . Workflow summarizing a general framework for pTx pulse design. The flow walks a pulse designer through a series of questions they should ask themselves and points them to relevant approaches to meet their design requirements. This outline can be extended for further pTx design complexities not considered here.
Currently, there are very few research publications exploring pTx in clinical populations for UHF. In the paper introducing FOCUS pulses , an online customized pTx pulse was shown in a brain surgery patient for MPRAGE (Mugler and Brookeman 1990). Recently several works-in-progress pTx sequences have been made available on some vendor systems to support UPs through the PASTeUR package  and also fully-integrated direct signal control (Malik et al 2015) for T2w RARE/TSE/FSE (Tomi-Tricot et al Thalhammer et al 2012. The ease of use of these research sequences have also permitted initial clinical investigations with pTx at 7T. A further example of FOCUS pTx MPRAGE in a patient with multiple sclerosis is compared to sTx MPRAGE in figure 10.

Conclusion
Radiofrequency coils play a key role in magnetic resonance imaging. At ultra-high field strengths, the reduced RF wavelength and increased power requirements make generating uniform excitations and thereby image homogeneity challenging. Fortunately, technological and engineering advancements in pTx RF coils can mitigate these challenges. Here we have reviewed the various transmit arrays used currently for UHF imaging and their design considerations. We have explored how to optimize coil design and to ensure safety with their generated electromagnetic fields through simulations, and have also mentioned how thermometry and local SAR monitoring contribute to the coil validation. We then explain how pTx coils are used with advanced RF pulse design and offer description of push-and-play methods such as UPs. Finally, we have provided a snapshot glimpse of the future of pTx, which is growing in use as UHF MRI becomes increasingly important for clinical imaging.