Introduction

Magnetic resonance imaging (MRI) is an indispensable diagnostic method in modern medicine, characterized by its intrinsic advantages of no ionizing radiation, excellent tissue contrast, and the capacity for multiparametric imaging1,2. Nonetheless, MRI often faces the challenge of extended acquisition times compared to other radiological imaging modalities, such as X-rays and computed tomography (CT), thereby limiting the application of MRI3,4. Over the past decades, substantial efforts have been devoted to accelerating MRI acquisitions, resulting in various techniques, including fast pulse sequences5, parallel imaging6, simultaneous multislice imaging7, compressed sensing8, and machine learning approaches9, among others. Among these techniques, parallel MRI (pMRI) has emerged as a practical method, capable of achieving two- to three-fold acceleration in clinical settings, thus facilitating its successful integration into clinical practice10. In a typical pMRI sequence, a multi-channel surface coil array is employed to simultaneously acquire signals while reducing the number of phase encodings, thereby shortening image acquisition time. Subsequently, advanced algorithms represented by sensitivity encoding (SENSE)11 and generalized auto-calibrating partially parallel acquisitions (GRAPPA)12 are employed to reconstruct the full image based on data received from each coil. However, the reduction in encoding steps inevitably leads to a decline in the signal-to-noise ratio (SNR), a pivotal metric for assessing MRI image quality. Sufficient SNR serves as the foundation for acquiring high-resolution images with superior tissue contrast13,14. Therefore, enhancing the SNR in pMRI constitutes a significant pursuit within the field of MRI research.

The emergence of metamaterials provides an effective approach for enhancing the SNR in MRI15,16. Metamaterials are a class of artificially engineered materials characterized by periodic sub-wavelength structures, and demonstrate unique properties that are not inherent in natural substances. The sub-wavelength periodic structures are commonly named by unit cells or meta-atoms. As a result of the intricate interactions between electromagnetic waves and meta-atoms, metamaterials precise control over wave propagation and the distribution of electromagnetic fields. These capabilities enable the utilization of metamaterials in various applications related to the manipulation of electromagnetic waves, including but not limited to super lenses17, invisibility cloaks18, and perfect absorbers19, among others. Moreover, metamaterials exhibit near-field confinement and electromagnetic field enhancement effects. These characteristics are particularly advantageous in the context of MRI, as they enable effective interaction with and amplification of the radiofrequency (RF) magnetic field, ultimately leading to improvements in SNR20,21,22,23,24.

Early-generation metamaterials for MRI improvement are with a linear response and enhance the RF field during both transmission and reception stages. In an MRI sequence, the RF transmission field (B1+) operates at a power level of approximately 1 kW and necessitates a high level of homogeneity, while the RF reception field (B1−) operates at a significantly lower power level, approximately 1 μW25. Enhancing the reception field is of paramount importance for improving the SNR. Conversely, enhancing the transmission field may cause problems such as inhomogeneous excitation and safety concerns due to increased eddy current heating26. To selectively enhance the reception field without affecting the RF transmission field, nonlinear metamaterials (NLMMs) for MRI have been proposed27,28,29,30,31. The nonlinearity of NLMMs enables them to sense the excitation field strength and adjust their resonance properties accordingly. Consequently, NLMMs can be designed to selectively enhance either the RF transmission or reception field. In most cases, they specifically enhance the RF reception field exclusively, resulting in more effective SNR improvement without interfering with the RF transmission field.

Metamaterial-based magnetoinductive lenses could improve the performance of pMRI32,33; however, these metamaterials are linearly responsive, and the SNR improvement has not been characterized in a quantitative manner. Most NLMMs for MRI reported so far were employed in conjunction with the body coil or single-loop surface coil, with limited exploration into metamaterial structures suitable for integration within existing surface coil arrays34. The augmentation of surface coil arrays through the incorporation of NLMMs has merit, particularly as surface coils and pMRI are gaining significance in clinical practice. By optimizing the coupling strength between NLMMs and surface coil arrays, it is possible to achieve enhanced SNR in surface coil imaging without causing the coil to deviate excessively from its normal operational state, thus further elevating the image quality of pMRI. The integration of clinically compatible NLMMs with pMRI techniques holds the promise of increasing SNR while maintaining shorter acquisition time.

Herein, we present the design and validation of an NLMM for pMRI using a commercial receiving-only surface coil array. The NLMM consists of a closely packed array of nonlinear meta-atoms (NMAs). We devise an electromagnetic (EM) field-circuit joint simulation process for numerically modeling the nonlinear response. Through experimental pMRI evaluation, we demonstrate that the NLMM, in combination with the surface coil array, can achieve 3-fold SNR enhancement in the phantom and in-vivo imaging. These findings underscore the SNR improvements enabled by NLMMs in pMRI and their consequent boosting on image quality.

Results

Design of nonlinear meta-atoms and metamaterials

The NLMM consists of the NMA, which is an inductive solenoid coupled with a varactor-loaded split ring resonator (VLSRR), as shown in Fig. 1a, b. The NMA supports multiple resonant modes with varying current distributions within the solenoid and VLSRR components. The fundamental mode, as illustrated in Fig. 1c, is the in-phase mode, in which the induced currents in the solenoid and VLSRR are in the same direction. In this mode, the induced currents generate a magnetic field along the axial direction of the NMA, leading to an enhancement of the RF magnetic field. The nonlinear response of the NMA to RF incident power originates from the varactor within the VLSRR. The varactor’s junction capacitance is intrinsically linked to the depletion layer thickness, which is controlled by the bias voltage across the junction35. In the VLSRR, which can be conceptualized as an LC resonator, the bias voltage across the varactor is generated through electromagnetic induction and rectification effects when an incident wave is present. Consequently, the resonant frequency of the VLSRR becomes related to the incident power, and the resonant frequency of the NMA and the NLMM also shifts with the incident power changing36.

Fig. 1: Structure design and theoretical response of the NLMM.
figure 1

a Schematic diagram illustrating the arrangement of NMAs to form the NLMM. b Composition of the NMA within the NLMM. c Magnetic field distribution of the in-phase resonance mode. d Theoretical oscillation amplitude of the VLSRR section and the solenoid section, along with their respective reflection spectra in (e). f Theoretical oscillation amplitude of the NMA, which is formed by coupling the VLSRR and solenoid, and the corresponding reflection spectra in (g).

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By strategically designing the resonance frequency of the NLMM at low incident power to align with the MRI operating frequency, the NLMM can enhance the B1− field to improve the SNR during the RF reception stage. In the transmission stage, the NLMM automatically detunes to avoid interfering with the B1+ field and affecting sample excitation.

We modeled the NMA as a coupled linear resonator and nonlinear resonator and described the model using coupled-mode theory (CMT)37,38. The CMT model is of the single-port and two-resonance type, which can be expressed as39:

$$\left[ \, \begin{array}{c}j\omega {a}_{1}\\ j\omega {a}_{2}\end{array}\right]=\left[\begin{array}{cc}j(1-\lambda |{a}_{1}|){{{{\rm{\omega }}}}}_{1,0}-({\gamma }_{e1}+{\gamma }_{o1}) & jk-\sqrt{{\gamma }_{e1}{\gamma }_{e2}}{e}^{-j{\theta }_{2}}\\ jk-\sqrt{{\gamma }_{e1}{\gamma }_{e2}}{e}^{j{\theta }_{2}} & j{\omega }_{2,0}-({\gamma }_{e2}+{\gamma }_{o2})\end{array}\right]\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\end{array}\right]+\left[\begin{array}{c}\sqrt{2{\gamma }_{e1}}\\ \sqrt{2{\gamma }_{e2}}{e}^{j{\theta }_{2}}\end{array}\right]|{{{{\rm{s}}}}}_{+}|$$
(1)

where ai is the mode amplitude, ωi,0 is the original resonance frequency, γei and γoi are the decay rates of radiation and intrinsic loss, k and θ2 are the coupling coefficient and phase difference between the two resonators, respectively, λ is the nonlinear coefficient of the VLSRR, s+ is the excitation wave, and the subscripts i represent the VLSRR (i = 1) and the solenoid (i = 2). The derivation and solution of Eq. (1) can be found in Supplementary Section 3, and the parameter values for the CMT model of the NMA are listed in Supplementary Table 2.

We investigated the response of the individual VLSRR and solenoid. When the non-diagonal elements of the coefficient matrix are zero, Eq. (1) describes the individual response of these two resonators. As shown in Fig. 1d, the maximum mode amplitude occurs at the original resonant frequencies of the resonators at the low excitation amplitude. As the excitation amplitude increases, the response of the solenoid remains constant, while the resonance frequency of the VLSRR redshifts and the relative mode amplitude decreases. Once the excitation amplitude reaches a threshold ( | s+|> 0.014), a bi-stable state transition occurs and the amplitude-frequency response spectrum of the VLSRR exhibits an abrupt change27.

In the NMA, mutual capacitive and inductive interactions between the VLSRR and the solenoid results in coupling between them. We considered the coupling coefficients in Eq. (1) to determine the resonance response of the NMA. The calculated results (Fig. 1f) show that the NMA exhibits two resonance modes corresponding to in-phase and anti-phase induced currents in the solenoid and VLSRR. The first-order resonance mode is the desired mode with improved field enhancement capabilities. The NMA also exhibits a nonlinear response to the excitation power due to the coupling between the VLSRR and the solenoid. As the excitation amplitude increases, the resonant frequencies corresponding to both modes redshift, and the peak amplitude of the oscillations decreases sharply, indicating a weakening of the magnetic field enhancement effect. Simultaneously, the abrupt change in the spectrum curve remains because of the bi-stable state transition in the nonlinear system.

We further calculated the spectra of the reflection coefficient r from the mode amplitude results39:

$$r=\frac{{s}_{-}}{{s}_{+}}=-1+\sqrt{2{\gamma }_{e1}}\frac{{a}_{1}}{\left|{s}_{+}\right|}+\sqrt{2{\gamma }_{e2}}{e}^{-j{\theta }_{2}}\frac{{a}_{2}}{\left|{s}_{+}\right|}$$
(2)

where s- is the reflection wave leaving the resonator. In Figs. 1e and 1g, the dips in the reflection coefficient spectra signify the resonance frequencies. The first-order resonant frequency at the low excitation amplitude matches the RF operating frequency of MRI, and the first-order dip redshifts and increases as the excitation amplitude rises. This observation indicates that, under the power level of the MRI RF transmission field (about 1 kW), the NLMM exhibits virtually no resonance response at the RF operating frequency of MRI. Consequently, the NLMM functions as an RF field enhancer exclusively at low incident power levels, automatically detuning itself and remaining inactive at higher power levels.

Our current NMA-based metamaterials design provides additional advantages over other NLMM schemes. In previous designs, a single large VLSRR equivalent in size to the metamaterial array was used for detecting incident power27,29. The large VLSRR, with dimensions similar to those of a surface coil element, may pose challenges such as strong coupling and coil impedance mismatch. In contrast, smaller VLSRRs exhibit reduced coupling with the coil and allow each NMA to display nonlinear responses equally and independently, making them more compatible with surface coils. Details of the coupling effect analysis between the VLSRR and coil element, as well as the quality factor estimation of the NMA, can be found in Supplementary Section 1 and Supplementary Fig. 1.

Numerical modeling of the nonlinear response and field distribution

Simulating nonlinear resonators directly through computational electromagnetic methods has long presented a formidable challenge. In previous investigations of voltage-controlled40,41 and power-controlled27,42,43 metamaterials with varactors, conventional approaches often modeled the varactor as a constant capacitor. The simulation tool then conducted a parameter sweep of the capacitance value, indirectly reflecting variations in junction capacitance caused by bias voltage or incident power changes. Advancements have enabled the direct simulation of voltage-controlled metamaterials by importing varactor S-parameters obtained from circuit simulations into lumped elements of electromagnetic simulations44,45. However, approaches for directly modeling the power-controlled response of metamaterials have not been fully investigated.

To improve the understanding of the nonlinear response of metamaterials in temporal and frequency domains, we propose an EM field-circuit joint simulation process for our NLMM with varactors, which numerically simulates the nonlinear response at various incident powers. The process began with the creation of a comprehensive model for the NLMM, incorporating a varactor (SMV1235-079LF, Skyworks) component as a lumped element according to the SPICE model specification46. The model was then excited by a sinusoidal signal with a specified frequency and amplitude, and its temporal evolution was computed using the time-domain solver. We subjected the results from the solver to a post-processing procedure, primarily employing the Fast Fourier Transform (FFT) algorithm on both the input and output signals of the excitation port. The reflection coefficient at the prescribed frequency was determined from the ratio of transformed signals in the frequency domain, and by varying the frequency and amplitude of the excitation signal, we derived reflection spectra corresponding to different incident power levels. Additionally, we examined the spatial field distribution under various excitation conditions. This EM field-circuit joint simulation approach allows us to understand and analyze the behaviors of the NLMM across spectra and spatial field distributions. Details about the simulation process are provided in Supplementary Section 4 and Supplementary Fig. 3. Moreover, we observed the establishment of bias voltage links to electric field distribution across the varactor under varying incident power, as shown in Supplementary Fig. 4.

Figures 2a, b illustrate the resonance response from the simulation and experimental measurement at low (−20 dBm) and high (5 dBm) incident power. The experimental reflection coefficient was obtained by measuring the S11 parameter of the NLMM using a vector network analyzer (VNA). At low incident power, the simulation and experiment results agree well, showing a strong resonant response at the first-order resonant frequency and a weak resonant response at the second-order. At high incident power, the simulation result shows a redshift of the resonant frequency of the NLMM and a corresponding increase in the reflection coefficient. The trend and magnitude of these changes approach those observed in the experimental measurements, though the simulated spectrum does not exhibit the abrupt change observed experimentally. This discrepancy may arise from the fact that the simulation time step length is much smaller than the excitation signal period, leading the solver to treat the varactor’s characteristic changes as a slow and continuous process. Nevertheless, the EM field-circuit joint simulation provides a reasonably accurate representation of the nonlinear response of the NLMM.

Fig. 2: Demonstration of the NLMM’s resonance response and magnetic field distribution obtained through EM field-circuit joint simulation.
figure 2

a, b Comparison of frequency responses obtained through simulation and experiment at low and high power levels, respectively. c Slice location and coordinate diagram illustrating the two-dimensional magnetic field distribution simulation results. d, e Distribution of simulated magnetic field enhancement ratios at different incident power levels, with Pin = −20 dBm and Pin = 60 dBm. f Magnetic field enhancement ratio as a function of distance along the dashed line in (d).

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Figures 2d, e show the spatial distribution of the RF magnetic field enhanced by the NLMM located on the profile in Fig. 2c under different incident powers. When the incident power is low (−20 dBm), the magnetic field near the NLMM experiences a maximum 34-fold enhancement attributed to the resonance of the NLMM. In contrast, at high incident power (60 dBm), there is minimal enhancement in magnetic field strength as the NLMM exhibits weakened resonance at the original frequency. The simulation results verify that the NLMM automatically enhances the field during the MRI RF reception phase while remaining inactive during the RF transmission phase.

We used Faraday’s Law and the Biot-Savart Law to calculate the theoretical RF magnetic field enhancement ratio at low incident power to verify the simulation results. According to Faraday’s Law, an alternating RF magnetic field generates an induced electromotive force and induces current in the solenoid and VLSRR. The current flowing through a micro-section ds of a wire generates a magnetic field at a point at a distance r. The Biot-Savart Law provides the strength of the induced magnetic field as follows47:

$$d{{{\boldsymbol{B}}}}=\frac{{\mu }_{0}}{4\pi }I\frac{d{{{\boldsymbol{s}}}}\times {{{\boldsymbol{r}}}}}{{\left|{{{\boldsymbol{r}}}}\right|}^{3}}$$
(3)

where I is the current in the wire, ds is the positional vector of the segment of the wire, and r is the vector from the wire segment to the point where the magnetic field is calculated.

To determine the magnetic field at an arbitrary point induced by the current in the solenoid and VLSRR, we calculated the cumulative magnetic induction strength produced by each wire segment using Eq. (3). Details of the calculation are provided in Supplementary Section 5 and Supplementary Fig. 5. Figure 2f shows the theoretical calculation of the magnetic field enhancement ratio along the central axis of the NMA in the NLMM, demonstrating good agreement with the simulation results. The maximum SNR enhancement ratio of approximately 40 times is observed in the NLMM region. In the sample region, the maximum magnetic field enhancement is about 34 times at the upper edge of the metamaterial, with the enhancement ratio gradually approaching 1 at a distance of approximately 55 mm from the NLMM. Furthermore, we noted that the magnetic field enhancement of the metamaterial is driven by both the induced magnetic field of the solenoid and the VLSRR. Theoretical calculations reveal that the magnetic field enhancement from the solenoid contributes about 2/3 of the total magnetic field enhancement, while the VLSRR contributes about 1/3.

MRI validation of NLMM using a phantom

We conducted experiments on a Siemens 3 T Prisma MRI system to assess the effectiveness of the NLMM in improving the SNR of imaging using a surface coil array. The surface coil array used was the Body 18 (Siemens Healthcare), an 18-channel receive-only surface coil array designed for torso imaging. The experimental setup for phantom imaging is shown in Fig. 3a. The NLMM was positioned on top of the coil Body 18, and the gap between them was mechanically adjusted using an acrylic plate holder to control the coupling effect strength. The phantom was situated approximately 4 mm above the top of the NLMM. More details of the experimental MRI validation are provided in Supplementary Section 7.

Fig. 3: 3 T MRI of a phantom using the NLMM enhanced surface coil array.
figure 3

a Illustration of the experimental setup. b Reference image with the phantom positioned directly on top of the coil Body 18, and illustration of the region of interest (ROI) and the region of standard deviation of noise (σnoise) calculated. c–e Images obtained with the NLMM at different gaps between the NLMM and coil Body 18. f SNR comparison between the reference and NLMM-enhanced images at the optimal gap. Insets: σnoise and maximum SNR enhancement ratio (max RSNR) at different gaps.

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Initially, we scanned the reference image using the coil Body 18 in the absence of the NLMM (Fig. 3b) using the fast low angle shot (FLASH) sequence. In the acquisition process, the phantom was placed close to the upper surface of the coil Body 18, which is the optimal working condition for the surface coil. This reference image served as a baseline for assessing the enhancement of image SNR. Subsequently, we positioned the NLMM between the coil Body 18 and the phantom. The separation gap between the NLMM and the coil affects the coupling strength; thus, we determined the optimal gap value by conducting MRI tests at various gaps using the same pulse sequence settings. The acquired images for varied gaps are shown in Fig. 3c–e. In the presence of the NLMM, there was an increase in MRI image signal strength at various gap values. As the gaps increased, the enhanced region decreased, and the highest signal amplitude diminished. We then evaluated the SNR in the region of interest (ROI) by calculating the ratio of the mean signal value (μsignal) to the standard deviation of the noise (σnoise), i.e., SNR = μsignal /σnoise. The σnoise was determined by measuring the standard deviation of background regions located in the upper left and upper right corners of the image, following the standard SNR calculation approach48. As the gaps between the coil Body 18 and the NLMM increased, the σnoise decreased (Fig. 3f insets).

The measured results can be attributed to the coupling effect between the NLMM and the surface coil array, which leads to both enhanced coil sensitivity and increased coil losses. Coil sensitivity, defined as the magnetic field strength generated by a unit current in the coil, is improved due to the magnetic field enhancement achieved by the NLMM. The improvement in coil sensitivity diminishes as the gap between the NLMM and coils increases. The additional losses introduced by the NLMM can be considered from two perspectives. First, the NLMM introduces additional impedance to coils, with the real part of the impedance increasing as the gap between the coils and metamaterials decreases at the MRI operating frequency49. Second, the surface coil array is designed with individual coils decoupled from each other50. The introduction of the NLMM results in indirect coupling between initially decoupled coils, introducing additional noise that increases as the coils and metamaterials are closer.

Considering that the μsignal and the σnoise decrease at different rates as the gap increases, there is an optimal gap where the enhanced SNR reaches its maximum. We determined that the optimal gap value was approximately 20 mm (Fig. 3f insets). Figure 3f shows the comparison of the SNR along the phantom’s height direction between the reference image and the image with optimal metamaterial enhancement. The maximum SNR enhancement achieved was 3.2 times, with the enhancement depth reaching 55 mm. The SNR enhancement ratio was lower than the RF magnetic field enhancement ratio, since the SNR is inversely proportional to the square root of the resistance of coils and the resistance derived from power dissipation in the sample51. Additionally, the NLMM introduces extra resistance as a load on the coils and additional losses.

To further demonstrate that the NLMM has minimal impact on the B1+ field, we measured the relationship between the image signal and the flip angle (FA), and assessed the FA map using the dual-angle method52. The FA is a parameter represent the intensity and duration of the RF transmission pulse. The measurement results showed that the trend of signal variation with flip angle remained consistent after applying the NLMM, with an average local FA increase of 2.1°. This confirms that the NLMM had minimal effect on the B1+ field and enhanced the SNR by improving the B1- field. Additionally, using the EM field-circuit joint simulation we proposed, we evaluated the specific absorption rate (SAR) and the safety of the NLMM. Simulation results indicated that the local average SAR value increased by a factor of 1.12, suggesting that the SAR increase remained within safety standards53. Details of the measurement and simulation methods and results are provided in Supplementary Section 8 and Supplementary Fig. 8.

After validating that the NLMM improved the SNR of MRI with the surface coil array, we proceeded to conduct pMRI experiments using the GRAPPA reconstruction algorithm while maintaining the optimal gap between the NLMM and the coil Body 18. In pMRI, the SNR is linked to the SNR in fully acquired imaging through the following relationship54:

$${SN}{R}_{{pMRI}}=\frac{{SN}{R}_{{full}}}{g\sqrt{R}}$$
(4)

where g is the local geometry factor (g-factor) which measures the geometry-related SNR loss during parallel imaging with the surface coil array. R is the acceleration factor in pMRI, and the image acquisition time is inversely proportional to R.

In the experimental pMRI validation, we set the acceleration factor to 2, 3, and 4. With the increase of the acceleration factor, there was an increase in aliasing artifacts in the images (Fig. 4a). When comparing images obtained with and without the NLMM at the same acceleration factor, we observed an enhancement in the image signal at the bottom regions near the NLMM. Since noise introduced by parallel imaging is spatially inhomogeneous, and to follow the method in55 while remaining consistent with previous non-parallel imaging practices, we used the background regions in the upper-left and upper-right corners of the image to calculate the σnoise. Figure 4b shows the SNR enhancement ratio at the same acceleration factor with the NLMM. The results demonstrated an SNR enhancement ratio of up to approximately 3 times, with an enhancement depth of around 55 mm.

Fig. 4: 3 T parallel MRI of a phantom and an apple using the NLMM enhanced surface coil array.
figure 4

a Comparison of phantom images acquired by surface coil array with or without NLMM at different acceleration factors. b SNR enhancement ratio as a function of distance. c Comparison of apple images obtained with or without NLMM enhancement at different acceleration factors, highlighting the improvement in imaging quality at 4-fold acceleration.

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In addition to the experimental validation on the phantom, we extended our imaging experiments to samples with anatomical structures to demonstrate the performance of the NLMM. As an example, we performed pMRI on an apple using the widely adopted T2-weighted turbo spin echo (TSE) sequence. Since the NLMM remains inactive during the 90° and 180° pulse transmissions and only enhances the echoes, standard imaging parameter settings could be used directly. The MRI images of the apple are presented in Fig. 4c. With the integration of the NLMM, the SNR of the lower portion of the apple, near the metamaterial, was enhanced across acceleration ratios of 2, 3, and 4. In particular, at a 4-fold acceleration, the structure between the core and stem of the apple was difficult to distinguish without the NLMM, resulting in a loss of SNR. However, by incorporating the NLMM into the setup, the details of the structure in the core and stem became distinguishable once again. The pMRI experiment with the apple highlights that the use of NLMM can significantly reduce imaging time while maintaining image quality.

MRI validation of NLMM using in-vivo lower leg imaging

To further demonstrate the potential of the NLMM in practical and clinical settings, we conducted MRI scans on lower legs of healthy volunteers. As depicted in Fig. 5a, the coil Body 18 was employed for in-vivo human imaging. The NLMM was placed on the surface coil array with a separation distance of 20 mm, the optimal distance determined from the phantom tests. The lower leg was secured on the NLMM with the help of a leg holder. For reference, the lower leg contacted the surface coil to achieve optimal SNR and image quality using only the surface coil. We utilized a T2-weighted turbo spin echo (T2w TSE) sequence for the MRI scanning, with detailed sequence parameters provided in Supplementary Section 7.

Fig. 5: 3 T parallel MRI of the human lower leg using the NLMM-enhanced surface coil array.
figure 5

a Illustrations of the experimental setup illustrations and side views with and without NLMM. b–d Comparisons of images acquired with NLMM and without NLMM at different acceleration factors. Insets show enlarged images highlighting tissue detail distinctions improved by the NLMM. e–g SNR quantification for different acceleration factors at varying discrete distances, as labeled in (b).

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The presence of the NLMM enhanced MRI signals, as evidenced by images acquired within the region of interest (ROI) shown in Fig. 5b. We quantitatively compared the SNR at different distances within the ROI. The results, presented in Fig. 5e, demonstrated that the NLMM achieved a maximum SNR improvement of 3.3 times, with the enhancement effect observable within a 40 mm range. The slight discrepancies between images with and without the NLMM could be attributed to the weight of legs. When the NLMM was applied, the leg holder supported the lower leg, allowing the surrounding tissue to freely drop. In contrast, in the reference case, the full weight of the leg rested on the contact surface between the leg and the surface coil array.

Furthermore, we validated the SNR improvement in the in-vivo pMRI. Figures 5c, d illustrate images obtained using the GRAPPA algorithm with 3-fold and 4-fold acceleration. We zoomed in on the superficial regions of the lower leg, where the NLMM enhanced the visibility of the separation of connective tissue between fat lobules. This visibility was maintained even at 3-fold and 4-fold acceleration factors. Conversely, without the NLMM, image details appeared blurred and indistinct due to the accelerated acquisition. Quantitative SNR results are presented in Figs. 5f, g. While SNR values were lower in parallel imaging, the SNR enhancement ratio attributed to the NLMM remained comparable to that observed in non-parallel imaging. At 4-fold acceleration, the enhancement depth slightly decreased to 35 mm due to increased aliasing noise. This pMRI study on the human leg, employing metamaterial-enhanced surface coils, underscores the potential of the NLMM in improving SNR and imaging speed in clinical MRI applications.

It should be noted that our NLMM currently exhibits limited penetration depth for SNR enhancement, with a depth of 55 mm in phantom imaging and approximately 40 mm in the in-vivo imaging. This limitation could potentially affect the contrast-to-noise ratio (CNR). Since metamaterials enhance SNR locally and adipose tissue is often located near the body surface, the signal amplification is more pronounced in these regions, which may spoil the CNR in certain situations. However, the CNR variation might be mitigated by developing customized image post-processing algorithms. Despite the localized SNR improvement, our results still demonstrate the practical significance of the NLMM, particularly for medical imaging and diagnosis of superficial anatomical regions such as the thyroid56, carotid artery57, temporo-mandibular joint58, and cerebral cortex59, among others. There is considerable room for improving the magnetic field enhancement ratio and the penetration depth of the NLMM. Future research could focus on utilizing low-loss materials for NLMM fabrication and further optimizing the coupling efficiency between the NLMM and the surface coil.

Discussion

In this study, we propose and validate an NLMM composed of an array of solenoidal NMAs, aimed at enhancing the SNR of MRI when using a surface coil array and in pMRI. Significantly, the NLMM has minimal impact on the RF transmission field, with an average local FA increase of 2.1°, while the primary SNR improvement stems from enhancing the RF reception field. Consequently, the NLMM can be seamlessly integrated into MRI systems using receive-only surface coil arrays, thereby improving the performance of pMRI. We have achieved a maximum SNR enhancement of 3-fold in the phantom and in-vivo imaging, for non-accelerated and 2- to 4-fold accelerated pMRI. The NLMM provide an SNR enhancement effect with a penetration depth of 55 mm in phantom imaging. In the in-vivo imaging, the penetration depth decreases to 40 mm, primarily due to dielectric loss.

The approach we present for designing, modeling, and simulating the NLMM offers a method for analyzing future structures with power-dependent responses. Unlike previous works on nonlinear metamaterials that relied on conventional static simulations, we adopt an EM field-circuit joint simulation process to directly simulate the nonlinear response. The nonlinear response characteristic enables the NLMM to excel in enhancing the SNR of pMRI using a surface coil array without requiring modifications to the pulse sequence or the coil. Our MRI experiments have verified that the NLMM can help maintain image quality even when employing higher acceleration factors. The achieved 3-fold SNR enhancement is equivalent to approximately doubling the static magnetic field strength or the average of nine times signal acquisition60, showing the potential of metamaterials in improving pMRI.

Parallel imaging using surface coil arrays has significantly influenced clinical imaging by meeting the demands for improved sensitivity and reduced scan time in clinical MRI. The construction of parallel MRI surface coil arrays with increased channel numbers has been demonstrated to enhance SNR and optimize the g-factor61. Compared to the approach of integrating more electronic components and cables into the limited space of coils to increase channel numbers, metamaterials offer a passive and easily deployable method to enhance SNR in parallel imaging. Metamaterials hold promise for facilitating high-sensitivity, rapidly acquired MRI in the future. This study validates the integration and benefits of combining clinical imaging coils with artificially structured materials, thereby advancing MRI technology.

Methods

Fabrication of the metamaterial

Each individual NMA within the NLMM was fabricated using fused deposition modeling (FDM) 3D printing. We then wound a copper wire along the grooves of the scaffold to form both a solenoid and a split ring. To create the varactor-loaded split ring resonator (VLSRR), we soldered a varactor (SMV1235-079LF, Skyworks) into the gap of the split ring. Detailed description of the fabrication process is provided in Supplementary Section 2 and Supplementary Fig. 2, while the geometric parameter values of the NLMM are listed in Supplementary Table 1.

Characterization of the metamaterial

The characterization of the NLMM was conducted using a vector network analyzer (VNA, N5230C, Agilent) equipped with an inductive loop antenna. This setup enabled the excitation of resonance modes of the NLMM and measurement of the S11 parameter to obtain the reflection spectrum of the metamaterial. In the reflection spectrum, the dips corresponded to the resonance modes. During characterization, the excitation power was swept over a range from −30 dBm to 5 dBm. We investigated the variation in the resonant response of the NLMM caused by loading phantoms with different filling materials, including water solution and edible oil. The material properties of the phantoms had a slight impact on the resonant frequency and nonlinear response of the NLMM. The measurement method and results are detailed in Supplementary Section 6 and Supplementary Fig. 6.

Numerical simulation of the nonlinear response

Numerical simulations were carried out using CST Studio Suite 2023. The simulation model replicated the dimensions of the fabricated sample. The scaffold material within the NLMM was modeled as a dielectric with a permittivity of 2 + 0.06i. The varactor was modeled as a lumped element by importing its SPICE model file, and a sinusoidal-type signal was employed as the excitation signal. For simulating the reflection spectrum, excitation was applied using a loop antenna and discrete port, mimicking the VNA characterization setup. For the magnetic field distribution simulation, waveguide ports were used for excitation. Further details regarding the numerical simulation are provided in Supplementary Section 4.

MRI validation

MRI experiments were conducted on a MAGNETOM Prisma 3 T MRI system (Siemens Healthcare). In the surface coil array imaging and pMRI experiments, a Body 18 receive-only surface coil array (Siemens Healthcare) was employed for RF reception, while the body coil was used for RF transmission. The surface coil array was configured to default settings, with all coil elements activated. We conducted a preliminary experiment to locate the NLMM on the coil Body 18, as detailed in Supplementary Section 7 and Supplementary Fig. 7.

The experimental setup required placing the coil Body 18 flat on the patient bed and positioning an NLMM composed of 2 × 2 NMAs on top of the surface coil. An acrylic holder was used to mechanically adjust the gap between the NLMM and the coil Body 18, allowing for regulation of the coupling effect strength to determine the optimal coupling.

For phantom imaging, a FLASH sequence was utilized with the following parameters: TE = 4.62 ms, TR = 100 ms, flip angle = 70 degrees, slice thickness = 2 mm, matrix size = 256 × 256 mm, and pixel size = 1 × 1 mm (as shown in Supplementary Table 3). In the case of sample validation using an apple, a T2-weighted TSE sequence was employed with the following parameters: TE = 89 ms, TR = 7970 ms, flip angle = 150 degrees, slice thickness = 3 mm, matrix size = 180 × 180 mm, and pixel size = 0.47 × 0.58 mm. The turbo factor in TSE was set to 17, with a 30% phase oversampling (as shown in Supplementary Table 4). In pMRI, the GRAPPA algorithm was used with acceleration factors of 2, 3, and 4, respectively.

The experimental validation involving human subjects was approved by the Tsinghua University Science and Technology Ethics Committee (Medicine) (Project No: THU01−20230202), and informed consent was obtained from the volunteer. The volunteer was a 24-year-old male, randomly recruited from graduate students at the university. No sex or gender analysis was performed. For in vivo validation of the human lower leg, a T2-weighted TSE sequence was employed with the following parameters: TE = 67 ms, TR = 4000 ms, flip angle = 150 degrees, slice thickness = 3 mm, matrix size = 140 × 140 mm, and pixel size = 0.3 × 0.3 mm. The turbo factor in TSE was set to 14, with no phase oversampling (as shown in Supplementary Table 5).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.