# Optimal excitation of single mode resonators: demonstration with a 3 T MRI metasolenoid

Dmitrii Tikhonenko, Kaizad Rustomji, Christophe Vilmen, Arnaud Durand, Georges Nouari, Stefan Enoch, David Bendahan, Redha Abdeddaim, Marc Dubois

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Dmitrii Tikhonenko, Kaizad Rustomji, Christophe Vilmen, Arnaud Durand, Georges Nouari, et al. Optimal excitation of single mode resonators: demonstration with a 3 T MRI metasolenoid. Journal of Magnetic Resonance, 2026, 382, pp.107992. 10.1016/j.jmr.2025.107992. hal-05401008

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# Optimal excitation of single mode resonators: demonstration with a 3 T MRI metasolenoid

Dmitrii Tikhonenko<sup>a</sup>, Kaizad Rustomji<sup>b</sup>, Christophe Vilmen<sup>a</sup>, Arnaud Durand<sup>c</sup>, Georges Nouari<sup>c</sup>, Stefan Enoch<sup>b</sup>, David Bendahan<sup>a</sup>, Redha Abdeddaim<sup>b</sup>, Marc Dubois<sup>b,*</sup>

## ARTICLE INFO

### Keywords:

MRI

Transmit efficiency

Passive resonator

Metasolenoid

Metamaterials

## ABSTRACT

Wireless passive resonators have been developed to inductively couple to the birdcage body coil. Such systems have been explored in the form of ceramic resonators with high permittivity but also with metamaterial or metasurface devices that can exhibit resonant behaviour at a given Larmor frequency. The resonant focusing of the radiofrequency field is used to lower the input power during transmission and improve the sensitivity of the body coil during reception. The gain is only obtained in a limited volume located within or close to the resonant structure. Typically, such passive devices do not support parallel imaging and demonstrated limited SNR enhancement compared to dense multichannel receive arrays. Nonetheless, these resonators have seen recent development with applications to wrist or breast MRI mostly in 1.5 T MRI scanners. Here we propose to design, build, and study a metasolenoid resonator operating at 3 T. The metasolenoid was characterized on phantom to validate the B₁ efficiency increase with respect to the birdcage polarization excitation. We reported a high B₁ efficiency gain for circularly (3.2-fold) and linearly (5.8-fold) polarized excitation. Consequently, and according to analytical calculations, we demonstrated that when excited with linearly polarized excitation, the metasolenoid had a B₁ efficiency 26 % higher when excited by the default circularly polarized excitation. Numerical simulations on voxel model showed that in presence of the resonator the B₁ efficiency gain normalized by the maximum local SAR was significantly improved when introducing the metasolenoid but the influence of the excitation polarization was reduced to a few percent.

## 1. Introduction

Over the past decades, magnetic resonance imaging (MRI) has been established as a safe and widely used non-invasive diagnostic method, offering exceptional soft tissue contrast without exposing patients to ionising radiation. MR scanners use a strong static magnetic field to polarize proton magnetic spins in the tissues and excites them with radiofrequency (RF) pulses. The resulting relaxation signals are detected and reconstructed into high-resolution images, enabling detailed visualization of structures such as the brain [1], spine [2], extremities [3], and internal organs [4].

The most used MR scanners operate at field strengths of 1.5 and 3 Tesla, providing a balance between signal-to-noise ratio (SNR), resolution, and patient safety. In these MR scanners, large birdcage coils are typically used for RF signal transmission (Tx coils) with high signal

homogeneity across the body. However, these large birdcage coils are not ideal receiving coils (Rx coils) given their large volume leading to low sensitivity and low SNR. Dedicated Rx coils based on phased array technology [5] have been developed to enhance detection sensitivity and SNR. These Rx coils are usually positioned near the specific region of interest (ROI), such as extremities, breast, or brain [6]. However, standard Rx coils have several drawbacks such as a lack of flexibility, a compatibility limited to specific MRI machines, the presence of active electronic components and bulk wired connections, which complicates usability and maintenance of MR systems [7].

With the aim of overcoming these limitations, studies have been dedicated to the potential advantages of metamaterials and metasurfaces for MRI applications. Metamaterials and metasurfaces are artificial structures with unique electric and/or magnetic responses not found in natural materials [8]. Metamaterial inspired coils offer

<footnote><sup>a</sup> Aix Marseille Univ, CNRS, Centre de Résonance Magnétique Biologique et Médicale, Marseille, France</footnote>

<footnote><sup>b</sup> Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Institut Marseille Imaging, Marseille, France</footnote>

<footnote><sup>c</sup> Aix Marseille Univ, Institut Universitaire de Technologie, Génie Electrique et Informatique Industrielle, Marseille, France</footnote>

<footnote>* Corresponding author at: Institut Fresnel, Faculté des Sciences de Saint Jérôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France. E-mail address: marc.dubois@univ-amu.fr (M. Dubois).</footnote>
significant advantages over conventional RF coils, including wireless usage, the absence of active electronic components, and flexibility in customisation and tuning [9–11]. Wireless coils can be driven as passive devices through inductive coupling with the main birdcage coil [12,13]. In this configuration, resonant passive devices can be used to focus the RF field emitted by the birdcage within a smaller ROI leading to a significant sensitivity increase of the volume coil. Recent studies conducted at 1.5 T [14,15] have shown that a structure consisting in array of split-loop resonators (SLRs), also known as “metasolenoid” [16] could improve image quality for wrist imaging. Similar results have been reported for head imaging using a metamaterial inspired structure [17]. For 3 T MR scanners, metamaterials inspired solutions have been scarcely described. An array of high-permittivity ceramic rings has been used in breast imaging improving birdcage coil efficiency up to 7 times in vivo [18]. However, due to their fragility and their high sensitivity to temperature, ceramic coils cannot be easily used in vivo. Metasolenoid and other types of passive resonators have been numerically assessed for 3 T MRI of breast [19]. To the best of our knowledge, experimental validations have not been reported yet.

In this work, we reported experimental demonstration of an improved transmit efficiency at 3 T using a passive metamaterial inspired device. The design of the structure was based on a metasolenoid previously numerically estimated and used in vivo at 1.5 T [20–22] and numerically tested at 3 T [19]. Our structure was inductively coupled to the birdcage coil and significantly increased the RF field amplitude in a specific ROI for a given input power. We also reported that the performance of the metasolenoid was related to the polarization used to drive the birdcage coil during transmission. Finally, we compared the SNR obtained in reception to the performance of a conventional Rx array. We also conducted numerical simulation of the specific absorption rate in human models as a perspective for future in vivo measurements.

## 2. Theory

The expression of the transverse RF magnetic flux density in the form of two counter rotating components ($B_1^+$ and $B_1^-$) in function of the two cartesian components ($B_x$ and $B_y$) is shown in Eq1. Derivation and discussion can be found in Vaidya et al. [23]. Under the assumption that the $B_1^+$ component rotates in the same direction as the nuclear precession, it will be responsible for the nuclear excitation leading to the tilting of the magnetization vector or flip angle:

$$B_1^+ = \frac{B_x + iB_y}{2}; \quad B_1^- = \frac{(B_x - iB_y)^*}{2} \qquad (1)$$

Based on the expression above, three different type of RF magnetic field polarization can be described. The linear polarization corresponds to any amplitude value of $B_x$ and $B_y$ with either 0° or 180° phase offset. From Eq1, we observe that such polarization will split equally in $B_1^+$ and $B_1^-$ components. Surface loop coil or single mode volume coil will typically produce such polarization. The circular polarization corresponds to equal amplitude and exact phase quadrature between $B_1^+$ and $B_1^-$ component. The birdcage coil is one of the most used RF coils that can be excited with two ports in quadrature to provide a single rotating component of the RF magnetic field allowing significant improvement in transmission efficiency [24,25]. Any other combination of $B_x$ and $B_y$ field amplitude and/or phase offset corresponds to an elliptical polarization. In such elliptical polarization, regardless of the amplitude of $B_x$ and $B_y$, it can be shown that the magnitude of the $B_1^+$ component will be maximum when the two cartesian components are in the correct phase quadrature. For $B_x = Ae^{i\phi_x}$ and $B_y = Be^{i\phi_y}$, we derive the magnitude square of the $B_1^+$ component (Eq. (2)).

$$\begin{aligned} |B_1^+|^2 &= \frac{1}{4} |Ae^{i\phi_x} + iBe^{i\phi_y}|^2 = \frac{1}{4} |Ae^{i\phi_x} + Be^{i\left(\phi_y + \frac{\pi}{2}\right)}|^2 \\ &= \frac{1}{4} \left( A^2 + B^2 + 2 \operatorname{Re} \left( ABe^{i\left(\phi_x - \phi_y - \frac{\pi}{2}\right)} \right) \right) \\ &= \frac{1}{4} \left( A^2 + B^2 + 2AB \cos\left(\phi_x - \phi_y - \frac{\pi}{2}\right) \right) \end{aligned} \qquad (2)$$

We observe that the magnitude of $B_1^+$ is maximum for $\phi_x - \phi_y = \frac{\pi}{2} + 2k\pi$ with $k \in \mathbb{N}$ for A and B strictly positive real numbers.

We focus our study on two specific excitations, the vertical linearly polarized (LP) excitation corresponds to a vector $[B_x, B_y]_{LP} = [0, 1]$ while the circularly polarized (CP) excitation is $[B_x, B_y]_{CP} = \frac{1}{\sqrt{2}}[i, 1]$ with $i = \sqrt{-1}$ and normalized magnitude. Assuming that the presence of a metasolenoid placed vertically in the birdcage coil only affects the $B_y$ component by multiplying the input $B_y$ with a complex coefficient $\alpha e^{i\phi_m}$ leaving the $B_x$ component unchanged. We obtained the following modified vectors $[B_x, B_y]_{LP} = [0, \alpha e^{i\phi_m}]$ and $[B_x, B_y]_{CP} = \frac{1}{\sqrt{2}}[i, \alpha e^{i\phi_m}]$ after adding the metasolenoid contribution in each case. Consequently, the rotating $B_1^+$ component for each excitation gives:

$$[B_1^+]_{LP} = \frac{\alpha}{2} e^{i\phi_m} \quad \text{and} \quad [B_1^+]_{CP} = \frac{i}{2\sqrt{2}} (1 + \alpha e^{i\phi_m}) \qquad (3)$$

We observe that for LP vertical excitation the $B_1^+$ magnitude will equal $\frac{\alpha}{2}$, half of the enhancement provided by the metasolenoid regardless of the added phase term $\phi_m$.

However, in the case of CP excitation, the interaction with the metasolenoid will modify the initial circular polarization to an elliptical polarization with $B_1^+$ magnitude depending on $\phi_m$. As shown in (Eq2), the resulting elliptical polarization will give a maximum $B_1^+$ magnitude if the two cartesian components modified by the interaction with the metasolenoid are in phase quadrature. By adjusting the phase delay of the CP excitation as $[B_x, B_y]_{CP} = \frac{1}{\sqrt{2}}[ie^{i\phi_m}, 1]$, we obtain after interaction with the metasolenoid $[B_x, B_y]_{CP} = \frac{1}{\sqrt{2}}[ie^{i\phi_m}, \alpha e^{i\phi_m}]$ and $[B_1^+]_{CP} = \frac{ie^{i\phi_m}}{2\sqrt{2}}(1 + \alpha)$. This modified CP excitation allows to compensate for the metasolenoid added phase term and maximise the resulting $B_1^+$ magnitude to $\frac{1+\alpha}{2\sqrt{2}}$. Comparing the $B_1^+$ magnitude obtained with the metasolenoid for each excitation, we observe that the LP vertical excitation will give higher $B_1^+$ magnitude than the phase corrected CP excitation if the enhancement factor from the metasolenoid is higher than $\alpha_0 = \sqrt{2} + 1 \approx 2.4$. This result can be generalized to all linear mode resonators enhancing a single RF magnetic field component. Such resonators targeting breast MRI have been described in Jandaliyeva et al. [19] with enhancement factor much higher than $\alpha_0$ suggesting that LP excitation could lead to higher $B_1^+$ efficiency improvement than CP excitation.

## 3. Materials and methods

### 3.1. Electromagnetic numerical simulations

Design, geometry and capacitor configuration were calculated using electromagnetic finite element software (CST Microwave Studio 2019). A model of an empty birdcage was used to optimize the metasolenoid resonant frequency. Fig. 1 illustrates the design of a birdcage and metasolenoid used in simulations. Fig. 1A shows the numerical setup which was used for tuning a metasolenoid for a 3 T frequency. The 16-leg birdcage model had a 35 cm bore radius, a length of 75 cm and legs width of 5 cm. The birdcage model included a cylindrical shield with 40 cm radius and 85 cm length. The metasolenoid was placed in the centre of the tuned birdcage. Fig. 1B contains a schematic representation of the PCB plate of the metasolenoid with sizes 164 (L) x 118 (H) mm². Copper
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** The metasolenoid inside the MRI birdcage. (A) The metasolenoid consists of 10 split-loop resonators (SLRs), each constructed by etching copper strips onto four rectangular FR-4 panels with a thickness of 2 mm. The metasolenoid is assembled by joining the edges of the four panels to form a closed box-like structure. (B) The design of a single FR-4 panel, with length L = 164 mm and height H = 118 mm, includes capacitors placed in the gaps (g = 3 mm) between the copper strips. Each copper strip has a width w = 3 mm and is spaced by a distance d = 8 mm from adjacent strips. (C) The equivalent circuit of an individual SLR. Each SLR forms a square loop with capacitors positioned at the midpoints of each side, effectively creating an LC circuit with the value of the capacitors to tune the resonance frequency of the structure. After calibration was chosen combination of three capacitors C₁ and one capacitance C₂.</figcaption></figure>

strips with sizes 164 (L) × 3 (w) mm² and gap's width g = 3 mm in the middle of the strip located equally at d = 11 mm between each other. The proposed structure consisted in 10 SLRs. Fig. 1C depicts the equivalent electric circuit of an individual SLR. For SAR simulations, the birdcage coil model was loaded with a multi-tissue voxel model of a female model, specifically the Ella model from the Virtual Family model package [26], which was also modified by the breast model from the UWCEM Numerical Breast Phantom repository [27]. Numerical results were obtained for 1 W stimulated power with the matching circuits of each birdcage port retuned after introduction of the metasolenoid.

### 3.2. Sample fabrication and testing

The metasolenoid was constructed of 4 etched FR-4 plates (164 × 118 × 2 mm³) soldered as a rectangular box shape with additional 0.5-mm-thick copper strips in each corner. Each plate contained 10 metallic strips (164 × 3 mm²) evenly spaced by 11 mm between adjacent strips. Each strip presented a 3-mm-gap to solder ceramic capacitors (Johanson Technology and Knowles). Capacitance value was fixed to tune the

resonant frequency of the resonator to 123.2 MHz, the proton Larmor frequency in our MRI scanner (MAGNETOM Vida 3T, Siemens Healthineers). Measurements of the resonator frequency were performed in the MRI scanner by coupling an 80-mm-diameter non-resonant pick-up loop coil placed 10 mm above the metasolenoid and measuring the reflection coefficient (S11 parameter) with a vector network analyser (Anritsu MS2026C). Prior to the measurement the VNA was calibrated with the coaxial cable and the pick-up loop used as a short circuit. For phantom measurements, a cylindrical phantom with 125-mm-diameter and 210-mm-height was used. The material properties were set equal to the real phantom that was subsequently used in the experimental measurements: ε = 61.9 and σ = 0.80 S/m (MVG, body liquid 300).

### 3.3. Phantom experiments

Phantom measurements were performed using the 3 T MRI scanner with the whole-body birdcage coil (BC) used for both transmission and reception of signals. Flip angle (FA) maps were acquired in sagittal orientation using a preconditioning RF pulse with TFL readout sequence [28], TR/TE = 7030/1.86 ms, FA = 8°, Sinc pulse = 80°, FOV = 289 × 289 mm, matrix size = 128 × 128, and 13 slices of 10 mm thickness with 20 mm interslice spacing. Signal acquisitions were performed with a 3D Gradient echo sequence without acceleration, TR/TE = 5/2.3 ms, FA = 5°, FOV = 280 × 280 × 240 mm, matrix size = 128 × 128 × 48. The same gradient echo sequence was performed without RF input to acquire noise samples and compute signal to noise ratio. A series of GRE images were performed with a flexible 18-channel receive coil array wrapped around the phantom. In this case the phantom was rotated and placed along the z-axis.

Our 3 T scanner was equipped with a dual transmit channel that allowed us to control the amplitude and phase of the signal for the two birdcage ports. Such a system was leveraged to investigate how the birdcage coil efficiency was influenced by the presence of the metasolenoid resonator with different input polarizations. In every measurement configuration, the reference voltage, used by the scanner for RF power calibration, was modified in order to reach a target value of 80° in the centre of the phantom. This allowed us to maintain the accuracy of the FA mapping method [28]. The resonant interaction of the vertical component of the RF magnetic field was expected to amplify the input amplitude and add a phase term. As discussed in the theoretical section, such a phase term did not affect the LP excitation result but could affect the CP excitation result. We corrected for this added phase term by setting each port amplitude to be equal and modifying the phase delay between the ports until a maximum of B₁ efficiency was found.

## 4. Results

Fig. 2 presents the reflection coefficient measured at the pickup loop in presence of metasolenoid with and without our phantom. Without phantom, we observed a sharp resonance just above our Larmor frequency. Once the phantom was set, the resonance was damped and tuned to 123.2 MHz.

As discussed in the theoretical part and in the experimental methods, our first test after installing the metasolenoid in the birdcage was to correct for a potential phase delay introduced by the metasolenoid resonance. Fig. 3 presents the B₁ efficiency defined as the FA averaged in the ROI (see Fig. 4) normalized by the reference voltage used to perform the FA map sequence as a function of the phase delay between the two ports (90° being the initial quadrature). We observe a maximum of B₁ efficiency when the phase delay reaches 170° indicating that the magnetic fields are in quadrature inside the metasolenoid. This phase delay was used for following CP excitation experiments.

Experimental measurements of FA maps were performed and presented in Fig. 4. Three distinct cases are shown with different birdcage excitation polarization depicted by pictograms in the lower right corner
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** S₁₁ measurements of a non-resonant loop antenna with diameter D = 80 mm placed on top of the metasolenoid. The blue solid line shows the S₁₁ measurement without phantom loading. The red solid line shows the S₁₁ when the metasolenoid is loaded with the phantom. The vertical black line denotes the working frequency of the 3 T MRI scanner at 123.2 MHz. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Experimentally measured B₁ efficiency (FA averaged in the ROI divided by reference voltage) in function of the phase difference between the two birdcage ports while the input power is distributed equally between the ports.</figcaption></figure>

of each FA map. As discussed in the methods section, the reference voltage, used by the scanner for RF power calibration, was modified so as to reach a target value of 80° in the centre of the phantom. Each panel presents the ratio between the FA and the reference voltage as a measure of the B₁ efficiency of the coil. First, the conventional circular polarization (CP) excitation was performed for the birdcage alone (Fig. 4B) and in presence of the metasolenoid (Fig. 4E). Second, the vertical linear polarization excitation was tested with the birdcage alone (Fig. 4C) and in presence of the metasolenoid (Fig. 4F). As explained in the previous paragraph, instead of a quadrature phase delay between the birdcage port, a phase difference of 170° was used to maximise the CP excitation response when the metasolenoid was present. Finally, the horizontal linear polarization (LP) was tested with the birdcage alone (Fig. 4D) and in presence of the metasolenoid (Fig. 4G). We observed that the presence

of the metasolenoid allowed us to significantly reduce the reference voltage required to reach the target angle in CP and vertical LP excitations. The opposite effect was observed in the horizontal LP excitation where the average FA in the ROI was reduced in presence of the resonator for a given reference voltage. The B₁ efficiency was compared between cases with the birdcage alone and in presence of the metasolenoid. B₁ efficiency was increased by a factor of 3.2 and 5.8 for the CP and vertical LP excitation, respectively. On the contrary, for the horizontal LP excitation, the presence of the metasolenoid led to a 32 % B₁ efficiency reduction. Interestingly, in presence of the metasolenoid, the maximum B₁ efficiency was obtained with vertical LP excitation with 0.88 deg./V (+26 %) compared to 0.70 deg./V in CP excitation.

The experimental SNR maps are presented in Fig. 5. We performed a GRE image using the birdcage as transmit coil, calibrated for CP excitation, and a conventional 18-channel receiver array wrapped around the phantom in horizontal position (Fig. 5A). In a second setup, we placed the metasolenoid around the phantom in vertical position and used the birdcage coil as a transmit and receive coil for CP excitation (Fig. 5B) and vertical LP excitation (Fig. 5C). In the three cases, the reference voltage was adjusted to obtain the same FA in the ROI. The SNR maps were obtained by dividing GRE signal intensities by the standard deviation computed in the noise image with Rician noise correction. The 18-channel coil achieved the highest average SNR (560.4 a.u.) in the ROI. The metasolenoid demonstrated competitive performance, with average SNR values of 416.1 a.u. for circular polarization and 421.3 a.u. for linear polarization.

We performed electromagnetic full wave simulation to investigate the influence of the metasolenoid with respect to specific absorption rate (SAR). A female human voxel model was added in our birdcage coil model. Akin to experiment methodology, we performed a reference case with the birdcage transmit coil for CP and vertical LP excitations. In a second step, the metasolenoid was placed around one of the breasts of the voxel model. Three positions of the metasolenoid with slight lateral adjustments (10, 9 and 8 cm away from the median vertical line) were shown. All results are obtained for 1 W stimulated power after retuning the birdcage ports. Additional information regarding the matching circuit influence has been included in supplemental materials. The results for CP and LP excitation are presented in Fig. 6 for the reference case and three metasolenoid positions. B₁⁺ amplitude was significantly increased in the breast ROI when the metasolenoid was included. In terms of B₁ efficiency, for a 10 cm lateral offset, it increased 5.6-fold (from 0.19 in Fig. 6A to 1.05 µT.W⁻⁰.⁵ Fig. 6C) in case of CP excitation and 22.6-fold in case of vertical LP excitation (from 0.06 in Fig. 6B to 1.24 µT.W⁻⁰.⁵ in Fig. 6D). The adjustments of the metasolenoid position showed more impact on the max local SAR₁₀g with a minimal value for both CP and LP excitation at 9 cm lateral offset in in Fig. 6e and in Fig. 6f. For every position, we observed higher B₁⁺ amplitudes (between +19 % to +17 %) for the metasolenoid driven by LP excitation compared to traditional CP excitation (comparing Fig. 6C–6E–6G with Fig. 6D–6F–6H respectively).

Table 1 summarizes all four configurations presented in Fig. 6. It shows the transmit efficiency calculated as the ratio between B₁⁺ amplitude averaged in the ROI, with the square root of max local SAR₁₀g. Comparing to the reference birdcage results, the best transmit efficiency increase was obtained for the 9 cm offset position with 5.4-fold (from 0.46 to 2.49 µT.W⁻⁰.⁵.kg⁰.⁵) and 20-fold (from 0.13 to 2.59 µT.W⁻⁰.⁵.kg⁰.⁵) for CP and LP excitations, respectively.

## 5. Discussion

Our results illustrated that B₁ efficiency and transmit efficiency could be improved at 3 T using a dedicated metamaterial inspired structure. Both numerical and experimental results showed that the proposed structure was able to enhance the RF magnetic field within the metasolenoid. Experimental results with a tissue-mimicking phantom indicated a significant enhancement in the B₁ efficiency, allowing to reduce the input power required to achieve a given FA in the ROI. This
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Experimental setup and flip-angle maps normalized by reference voltage. The experimental setup in the 3 T MRI scanner MAGTETOM Vida shows the placement and orientation of the metasolenoid and phantom relative to the MRI bore. The top row (B, C, D) shows flip angle maps normalized by the reference voltage ($B_1$ efficiency) with the birdcage in circular (B), vertical linear (C) and horizontal linear (D) polarized excitation without metasolenoid. The bottom row (E, F, G) displays the same for cases with the metasolenoid installed around the phantom. The blue rectangle indicates the region of interest to calculate the average flip angle normalized by reference voltage ((FA/V)), and green dashed lines mark the position of the metasolenoid. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Experimental measurements of SNR maps. Panels A-C show SNR map with the birdcage using an 18-channel receive coil (A), panels (B) and (C) represent SNR maps for circular and vertical linear polarized excitation modes with presence of metasolenoid. The blue rectangle indicates the region of interest to calculate the average SNR value, and green dashed lines mark the position of the metasolenoid. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

improvement was observed in both quadrature (CP) and vertical linear (LP) excitation of the birdcage coil. However, the horizontal linear excitation did not contribute to this enhancement because it remained orthogonal with the RF magnetic field resonant mode. As a matter of fact, the presence of the metasolenoid had reduced the efficiency of the horizontal linear excitation, indicating that the metasolenoid could not be considered transparent to the orthogonal component of the RF magnetic field.

Interestingly, we observed that, in presence of the metasolenoid, the maximum $B_1$ efficiency was obtained with vertical LP excitation with 0.89 deg./V (+26 %) compared to 0.70 deg./V in CP excitation with corrected phase delay. This aspect was not covered by previous work dedicated to metasolenoid structures. In fact, we demonstrated theoretically that a metasolenoid resonator can be more efficient in LP as compared to CP excitation. Results from Fig. 4 showed how the presence of the metasolenoid affects both vertical and horizontal LP excitations. We can apply these to modify the input vector from the birdcage port while assuming that the phase delay due to the metasolenoid was compensated. We obtained the modified vertical LP input [$B_x;B_y$] = [0;5.81] and the modified CP input [$B_x;B_y$] = $1/\sqrt{2}$ [*i*\*0.68;5.81]. When comparing the $B_1^+$ amplitude obtained with these two input vectors using

Eq. (1), it can be demonstrated that the modified vertical LP excitation gives a 27 % larger $B_1^+$ amplitude as compared to the modified CP. This result is very close to the valued (+26 %) experimentally measured which confirms our understanding of the resonant enhancement of the RF magnetic field in the metasolenoid.

Although the SNR gain was not the main objective of this study, we intended to make sure that the presence of the metasolenoid was not increasing RF noise or creating artefacts in conventional GRE images. We observed that the SNR in the phantom ROI was only reduced by 25 % when compared to the image obtained with an 18-channel receive array. Further studies on volunteers are needed to confirm the translation of these promising results towards clinical practice. One important aspect to focus on would be the investigation of bilateral breast imaging with two metasolenoids placed side by side. Close distance tends to reduce the efficiency of the resonators due to mutual coupling. Recent work proposes a solution to mitigate this effect by adding decoupling transformers between two passive Helmholtz resonators at 1.5 T [29].

Further numerical simulations with human voxel model were performed to validate the methodology in a more realistic scenario. We observed a higher $B_1$ efficiency gain as compared to the experimental data. This apparent disagreement can mostly be attributed to tissue
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** B<sub>1</sub><sup>+</sup> and local SAR simulation results. Numerically calculated B<sub>1</sub><sup>+</sup> and local SAR<sub>10g</sub> maps normalized to 1 W of stimulated power in 3 T birdcage simulation (each port re-tuned after introduction of the metasolenoid). Green dotted line shows region-of-interest (ROI) in the breast used for averaging. Each column presents the B<sub>1</sub><sup>+</sup> amplitude maps for CP and LP excitations (uppercase letters) and corresponding SAR<sub>10g</sub> for CP and LP excitations (lowercase letters). From left to right, the reference case with the birdcage and three cases in presence of the metasolenoid with decreasing lateral offset (10, 9 and 8* cm) from the median vertical line. To be noted, the 8 cm offset would not be compatible with bilateral breast imaging as a second structure should be placed symmetrically around the second breast. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

loading differences given that breast tissue has a high fat content which is expected to reduce the conductivity and the relative permittivity. Of interest, the B₁ efficiency gain we reported for CP excitation i.e. 5.6 is close to the 5.1 value reported in simulations with a similar metasolenoid setup [19]. Here again we observed that the metasolenoid yielded a higher B₁ efficiency (+18 %) when using LP excitation compared to CP excitation. However, the transmit efficiency calculations in Table 1 showed that the LP excitation was penalized with higher maximum local SAR leading to similar value for the SAR efficiencies in CP and LP excitations (2.38 and 2.39 µT.W.kg⁻¹) for 10 cm lateral offset. We performed additional simulations with the metasolenoid slightly shifted to avoid local SAR hotspot and found that a 1-cm displacement allowed to reduce SAR and obtain higher transmit efficiency, 2.49 µT. W⁻⁰.⁵.kg⁰.⁵ and 2.59 µT.W⁻⁰.⁵.kg⁰.⁵ for CP and LP excitation,

respectively. However, the absolute value of transmit efficiency reached by LP excitation compared to CP excitation remained similar, ranging from -0.5 % to +8 % when adjusting the position of the metasolenoid. While we allowed a 2 cm lateral shift in the single metasolenoid case, it is important to note that this configuration will not be compatible with a bilateral configuration that requires two resonators to be placed symmetrically around each breast. Additional numerical studies with different voxel model configurations could be useful to support the generalisation of our results.

## 6. Conclusion

A 3 T metasolenoid passive resonator has been designed, built and the corresponding performance has been assessed using phantom
**Table 1**

Numerical results of the amplitude of $B_1^+$ averaged in the ROI, max local SAR$_{10g}$ and transmit efficiency ($B_1^+$ normalized by square root of max local SAR). Four configurations are shown, the reference case without metasolenoid and in presence of the metasolenoid and three cases in presence of the metasolenoid with decreasing lateral offset (10, 9 and 8* cm) from the median vertical line. All configurations are shown for both CP and LP excitations. To be noted, the 8 cm offset would not be compatible with bilateral breast imaging as a second structure should be placed symmetrically around the second breast.

<table><thead><tr><th></th><th>Excitation polarisation</th><th>B<sub>1</sub><sup>+</sup> avg in ROI, μT</th><th>Max SAR<sub>10g</sub>, W, kg<sup>-1</sup></th><th>B<sub>1</sub><sup>+</sup>/√maxSAR<sub>10g</sub>, μT.W<sup>-0.5</sup>,kg<sup>0.5</sup></th></tr></thead><tbody><tr><td rowspan="2">Reference</td><td>CP</td><td>0.19</td><td>0.17</td><td>0.46</td></tr><tr><td>LP</td><td>0.05</td><td>0.19</td><td>0.13</td></tr><tr><td rowspan="2">Metasolenoid Offset 10 cm</td><td>CP</td><td>1.04</td><td>0.19</td><td>2.39</td></tr><tr><td>LP</td><td>1.24</td><td>0.27</td><td>2.38</td></tr><tr><td rowspan="2">Metasolenoid Offset 9 cm</td><td>CP</td><td>1.02</td><td>0.17</td><td>2.49</td></tr><tr><td>LP</td><td>1.21</td><td>0.22</td><td>2.59</td></tr><tr><td rowspan="2">Metasolenoid Offset 8 cm</td><td>CP</td><td>0.98</td><td>0.20</td><td>2.19</td></tr><tr><td>LP</td><td>1.15</td><td>0.24</td><td>2.36</td></tr></tbody></table>

measurements. $B_1$ efficiency was improved by a factor of 3.2 and 5.8 times for CP and LP excitation in phantom experiments, respectively. To our knowledge, this type of resonator has never been studied with different birdcage input polarizations. We confirmed that the metasolenoid specifically amplified the vertical component of the RF magnetic field produced by the birdcage. Consequently, and according to analytical calculations, we demonstrated that the metasolenoid together with LP excitation had a 26 % higher efficiency than with CP excitation. However, numerical simulations of the metasolenoid with a human voxel model showed that such a gain may be balanced by a corresponding increase in local SAR leading to a similar SAR efficiency performance with respect to the birdcage polarization. Besides, we confirmed, *in silico*, that the presence of the metasolenoid increased the SAR efficiency by 5 times when compared to conventional birdcage used as a transmit coil which agrees with the simulations previously reported in the literature. Finally, SNR measurements in a phantom confirmed that the presence of the solenoid yielded SNR values within the same order of magnitude as a conventional 18-channel receive array. Further studies on volunteers are needed to confirm the translation of these promising results towards clinical practice, especially investigating the performance of the metasolenoid for bilateral breast imaging.

**CRediT authorship contribution statement**

**Dmitrii Tikhonenko:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis. **Kai-zad Rustomji:** Writing – review & editing, Project administration, Conceptualization. **Christophe Vilmen:** Resources. **Arnaud Durand:** Resources. **Georges Nouari:** Resources. **Stefan Enoch:** Funding acquisition, Conceptualization. **David Bendahan:** Writing – review & editing, Project administration, Conceptualization. **Redha Abdeddaim:** Writing – review & editing, Project administration, Funding acquisition, Conceptualization. **Marc Dubois:** Writing – review & editing, Supervision, Methodology, Investigation, Formal analysis, Conceptualization.

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Acknowledgements**

This work received support from the French Government under the France 2030 Investment Plan and French "Investissements d'Avenir"

programme, as part of the Initiative d'Excellence d'Aix-Marseille Université, AMIDEX: AMX-19-IET-002, AMX-23-CI-01, AMX-23-CPJ-10 and AMX-23-EQ-FO-009.

**Appendix A. Supplementary data**

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jmr.2025.107992.

**Data availability**

Data will be made available on request.

**References**

[1] Magalie Viallon, et al., State-of-the-art MRI techniques in neuroradiology: principles, pitfalls, and clinical applications, Neuroradiology 57 (2015) 441–467.

[2] Patrick W. Stroman, et al., The current state-of-the-art of spinal cord imaging: methods, Neuroimage 84 (2014) 1070–1081.

[3] Gerwin P. Schmidt, Maximilian F. Reiser, Andrea Baur-Melnky, Whole-body imaging of the musculoskeletal system: the value of MR imaging, Skeletal Radiol. 36 (2007) 1109–1119.

[4] Elmar M. Merkle, Brian M. Dale, Erik K. Paulson, Abdominal MR imaging at 3T, Magn. Reson. Imaging Clin. 14 (1) (2006) 17–26.

[5] Peter B. Roemer, et al., The NMR phased array, Magn. Reson. Med. 16 (2) (1990) 192–225.

[6] Wingchi E. Kwok, Basic principles of and practical guide to clinical MRI radiofrequency coils, RadioGraphics 42 (3) (2022) 898–918.

[7] Norman B. Konyer, et al., Comparison of MR imaging breast coils, Radiology 222 (3) (2002) 830–834.

[8] Hou-Tong Chen, Antoinette J. Taylor, Yu. Nanfang, A review of metasurfaces: physics and applications, Rep. Prog. Phys. 79 (7) (2016) 076401.

[9] Andrew Webb, et al., Novel materials in magnetic resonance imaging: high permittivity ceramics, metamaterials, metasurfaces and artificial dielectrics, MAGMA 35 (6) (2022) 875–894.

[10] Zhiwei Guo, et al., Metamaterial-enhanced magnetic resonance imaging: a review, Adv. Photon. Nexus 3 (5) (2024), 054001–054001.

[11] Dean Darnell, Trong-Kha Truong, Allen W. Song, Recent advances in radio-frequency coil technologies: flexible, wireless, and integrated coil arrays, J. Magn. Reson. Imaging 55 (4) (2022) 1026–1042.

[12] Tomohiro Sahara, et al., Development of inductively coupled wireless radio frequency coil for magnetic resonance scanners, in: 2007 Inaugural IEEE-IES Digital EcoSystems and Technologies Conference, IEEE, 2007.

[13] Akbar Alipour, et al., Improvement of magnetic resonance imaging using a wireless radiofrequency resonator array, Sci. Rep. 11 (1) (2021) 23034.

[14] Alena V. Shchelokova, et al., Volumetric wireless coil based on periodically coupled split-loop resonators for clinical wrist imaging, Magn. Reson. Med. 80 (4) (2018) 1726–1737.

[15] Ekaterina Brui, et al., Volumetric wireless coil for wrist MRI at 1.5 T as a practical alternative to Tx/Rx extremity coil: a comparative study, J. Magn. Reson. 339 (2022) 107209.

[16] Stanislav I. Maslovski, et al., Artificial magnetic materials based on the new magnetic particle: metasolenoid, Prog. Electromagn. Res. 54 (2005) 61–81.

[17] Xiangzheng Kong, et al., Compact metamaterial-inspired RF coil based on split-ring resonators to enhance the magnetic field for 1.5 T magnetic resonance imaging, Opt. Commun. 556 (2024) 130280.

[18] Alena Shchelokova, et al., Ceramic resonators for targeted clinical magnetic resonance imaging of the breast, Nat. Commun. 11 (1) (2020) 3840.

[19] Aigerim Jandaliyeva, Viktor Puchnin, Alena Shchelokova, Volumetric wireless coils for breast MRI: a comparative analysis of metamaterial-inspired coil, Helmholtz coil, ceramic coil, and solenoid, J. Magn. Reson. 359 (2024) 107627.

[20] Viktor Puchnin, et al., Metamaterial inspired wireless coil for clinical breast imaging, J. Magn. Reson. 322 (2021) 106877.

[21] Aigerim Jandaliyeva, et al., Control of the near magnetic field pattern uniformity inside metamaterial-inspired volumetric resonators, Photon Nanostruct. Fundam. Appl. 48 (2022) 100989.

[22] Viktor Puchnin, et al., Quadrature transceive wireless coil: design concept and application for bilateral breast MRI at 1.5 T, Magn. Reson. Med. 89 (3) (2023) 1251–1264.

[23] Manushka V. Vaidya, et al., Dependence of and field patterns of surface coils on the electrical properties of the sample and the MR operating frequency, Concepts Magn. Reson. Part B: Magn. Reson. Eng. 46 (1) (2016) 25–40.

[24] Cecil E. Hayes, et al., An efficient, highly homogeneous radiofrequency coil for whole-body NMR imaging at 1.5 T, J. Magn. Reson. 63 (3) (1985) 622–628.

[25] G.H. Glover, et al., Comparison of linear and circular polarization for magnetic resonance imaging, J. Magn. Reson. 64 (2) (1985) 255–270.

[26] Andreas Christ, et al., The Virtual Family—development of surface-based anatomical models of two adults and two children for dosimetric simulations, Phys. Med. Biol. 55 (2) (2009) N23.
[27] Matthew J. Burfeindt, et al., MRI-derived 3-D-printed breast phantom for microwave breast imaging validation, IEEE Antennas Wireless Propag. Lett. 11 (2012) 1610–1613.

[28] Sohae Chung, et al., Rapid B1+ mapping using a preconditioning RF pulse with TurboFLASH readout, Magn. Reson. Med. 64 (2) (2010) 439–446.

[29] Pavel Tikhonov, et al., A wireless bilateral transceiver coil based on volume decoupled resonators for a clinical MR mammography, J. Magn. Reson. 380 (2025) 107941.