Metamaterial structure design and analysis

The magnetic metamaterial reported herein is comprises of an array of metallic helices (Fig. 1), the coupling of which leads to a synergy and bulk material property. The synergy of the metamaterial array yields the bulk property of a resonant mode in which the direction of the electric current induced by an applied RF field is identical in each unit cell. Upon excitation by an external RF field, the thusly-induced currents in the resonant mode lead to a dramatic enhancement of the RF field. With respect to MRI, when the frequency of the resonant mode approximates the resonance frequency of the MRI system, marked gains in both transmit and receive local RF magnetic fields (B 1 + and B 1 −, respectively) are achieved, ultimately leading to gains in SNR.

Fig. 1 Schematic of the magnetic metamaterial. The metamaterial array composes unit cells featuring metallic helices, which are made of copper wiring with central polymeric scaffolding. Scale bar is 3 cm. (Dimensions included for scale and employed in subsequent analyses, see Methods) Full size image

In order to elucidate the determinants of the resonant mode of the metamaterial, electromagnetic theory can firstly be employed to define the equivalent self-inductance and mutual inductance of discrete unit cells as29:

$$L_{ij} = \frac{\mu_{0}}{4\pi | I_{i}I_{j} |} {\int \hskip -7pt \int} {\mathrm{dr}}_{i}{\mathrm{dr}}_{j}\frac{J ( r_{i} ) \cdot J ( r_{j} )}{| r_{i} - r_{j} |},$$ (1)

where r i and r j are the integration elements along the path of the helix, I i and I j are the equivalent current in the helix, and J(r i ) and J(r j ) are the current densities (vectors) at r i and r j . Eq. 1 represents the self-inductance when i is equal to j and mutual inductance when i and j are different. At the resonant state, the amplitude of the current follows a sinusoidal profile along the path of the wire of the helix with a maximum (I 0 ) at the center and zero at the ends of the wire29. Thusly, \(I_{i,j} = I_0/\sqrt 2\). The self-capacitance and mutual capacitance can be calculated by finding the inverse of the coefficients of the potential matrix30 and, by employing electromagnetic theory, be expressed as:

$$P_{ij} = \frac{1}{4 \varepsilon_{0} \varepsilon_{\mathrm{r}} | Q_{i}Q_{j} |} {\int \hskip -7pt \int} {\mathrm{d}}r_{i} {\mathrm{d}}r_{j} \frac{\rho (r_{i} ) \rho ( r_{j} )}{ | r_{i} - r_{j} | }.$$ (2)

$$\left[ {\begin{array}{*{20}{c}} {C_{11}} & \cdots & {C_{1m}} \\ \vdots & \ddots & \vdots \\ {C_{m1}} & \cdots & {C_{mm}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {P_{11}} & \cdots & {P_{1m}} \\ \vdots & \ddots & \vdots \\ {P_{m1}} & \cdots & {P_{mm}} \end{array}} \right]^{ - 1},$$ (3)

in which Q i and Q j are the equivalent charge amounts in the helix and ρ(r i ) and ρ(r j ) are the charge densities (scalars) at r i and r j . The phase difference between the current and charge distribution is π/2 and, therefore, the maximal charge density is at the two ends of the helical wire (q 0 ) and zero at the middle of the wire. Taking into consideration the charge distribution, \(Q_{i,j} = q_{0} l_{\mathrm{t}} \pi / \sqrt{2}\) where l t is the total length of the wire wound to form the helix. The diagonal elements of the capacitance matrix yield the self-capacitance, while the remaining elements yield the corresponding mutual capacitance.

Upon implementation of the discrete unit cells into a metamaterial array, the effect of coupling must be considered, given its ultimate effect upon the resonant mode of the array. As the geometry of each unit cell is identical, the coupling coefficient of two adjacent helices may be expressed as k = C m /C + L m /L, in which C m and L m represent the mutual capacitance and mutual inductance, respectively, and C and L represent the self-capacitance and self-inductance, respectively. The total coupling coefficient (k), as well as the contributions from capacitance coupling (k c ) and inductance coupling (k l ) are plotted as a function of unit cell separation distance in Fig. 2. k c and k l are both negative and k c is approximately 3-fold larger than k l , demonstrating that a much larger contribution to the total coupling stems from the capacitance coupling between unit cells. In order to validate these theoretical results, numerical analyses were also conducted to derive the coupling coefficient and plotted in Fig. 2 (please see Supplementary Note 1 for further details).

Fig. 2 Coupling coefficients between adjacent unit cells. Coupling coefficients as a function of unit cell separation distance derived from both theoretical and numerical analyses Full size image

Finally, due to the coupling between the discrete unit cells of the array, the resonant modes of the metamaterial may be derived by employing the coupled mode theory and solving the following equation system31,32:

$$\frac{{{\mathrm{d}}a_n\left( t \right)}}{{{\mathrm{d}}t}} = - \left( {{\mathrm{j}}\omega _0 + {\mathrm{\Gamma }}_n} \right)a_n\left( t \right) + {\mathrm{j}}\mathop {\sum}

olimits_{k = 1}^{m,k

e n} {K_{kn}a_k\left( t \right),\,n = 1, \ldots ,m},$$ (4)

in which \(\omega _0 = 1/\sqrt {LC}\) represents the resonant angular frequency of a single helix, Γ n is the intrinsic decay introduced by the material and radiation losses, K kn is the coupling factor among unit cells and is related to the coupling coefficient k as K kn = kω 0 /2, and m is the total number of unit cells. The loss of each unit cell is omitted and only the coupling between adjacent unit cells is considered for simplicity to derive the resonant modes of the metamaterial (please see Supplementary Note 2 for further details). The eigenvalue represents the resonant frequency of the resonant mode and the corresponding eigenvector indicates the resonant strength and direction of each unit cell of the mode. By analyzing the resultant matrix, one can identify that the current direction in each unit cell is identical for the resonant mode with the highest resonant frequency, which will be referred to as the working mode.

The working mode of a metamaterial ordered as a 4 × 4 array as a function of unit cell separation distance was calculated, compared with simulation results, and illustrated in Fig. 3a. The degree of difference between the simulation results and the coupled mode theory increases as the coupling increases with decreasing separation distance, which is due to the weak coupling assumption of the coupled mode theory. However, as the separation distance increases, the theoretical results demonstrate a high degree of agreement with the simulation results. The working modes for metamaterials composed of different numbers of unit cells are also illustrated in Fig. 3b with identical spacing between unit cells. The colors correspond to the resonant strength of each unit cell derived from the coupled mode theory. The unit cells at the center are found to resonate with the greatest strength, while the resonant strength decreases for the unit cells positioned along the periphery, thereby leading to larger degrees of magnetic field enhancement at the center of the metamaterials. For purposes of comparison, the relative magnetic field strength at the middle cross-section of a 4 × 4 metamaterial array is also simulated in CST Studio Suite 2017 and plotted as the inset at the bottom left corner in Fig. 3a, demonstrating a similar resonant strength pattern to that derived from the coupled mode theory.

Fig. 3 Resonant mode of the metamaterials. a The frequency of the working mode of the metamaterial as a function of separation distance of unit cells. Inset on the bottom left plots the simulated magnetic field at the middle cross-section of the metamaterials. Inset on the top right depicts the model used for simulation. b The resonant strength derived from the coupled mode theory for metamaterials with 4 × 4, 6 × 6, 8 × 8, and 10 × 10 arrays. The resonant strength is normalized to the maximum value Full size image

Magnetic field distribution and field enhancement

Upon deployment of the metamaterials inside the bore of an MRI, by matching the frequency of the working mode to the working frequency of the MRI, the working mode can be excited by the RF magnetic field generated by the MRI (both transmit and receive RF magnetic fields; B 1 + and B 1 −, respectively). The RF magnetic field leads to the induction of an AC voltage across the helix as V = nS (dB 1 +)/dt. The impedance of the helix can be expressed as Z i = jωL + 1/jωC + R, where L and C are the equivalent inductance and capacitance of the unit cell in the array, respectively, and R is the equivalent resistance, which is composed of the ohmic loss of the wire, as well as the dielectric loss of the scaffolding material. Therefore, the current inside the metallic helix can be calculated by I = V/Z i . The induced magnetic field contains two components, along the axial direction and the radial direction. Only the magnetic field component along the axial direction of the helix contributes to the magnetic field enhancement and can be calculated as33:

$${B_{Z} = \frac{\mu_{0}nI_{0}a}{4 \pi l}{\kern 1pt} {\int}_{\!\!\!\!- l/2}^{l/2} {\mathrm{cos}} \left( \frac{\pi z\prime }{l} \right){\kern 1pt} {\int}_{\!\!\!0}^{2 \pi } \frac{a - \rho {\mathrm{cos}}(\varphi \prime )}{[\rho ^{2} - 2a \rho{\kern 1pt} {\mathrm{cos}}(\varphi \prime ) + a^{2} + (z - z\prime )^{2}]^{3/2}}{\mathrm{d}}\varphi \prime {\mathrm{d}}z \prime ,}$$ (5)

in which I 0 is the amplitude of the current at the middle of the helix, n is the number of turns of the helix, a is the radius of the helix, l is the height of the helix, ρ and z are the polar coordinates of the target point, and cos(πz’/l) reflects the current distribution along the helix (please see Supplementary Note 3 for further details). The magnetic field enhancement ratio can be calculated by the summation of the magnetic field components (including the magnetic field generated by each unit cell and B 1 +) and normalizing to B 1 +, with the current strength in each unit cell indicated by the eigenvector of the coupled mode theory.

The magnetic field enhancement ratio at the mid-portion of the metamaterials along the axial direction is calculated and plotted as a function of distance from the metamaterial array in Fig. 4a. The magnetic field enhancement analysis is divided into three regions. Region A (shaded orange in Fig. 4a) is within the metamaterial and, although extremely high magnetic field enhancement is realized, this region is impractical for application to MRI imaging, as is considered herein. Region B (shaded yellow in Fig. 4a) extends from surface of the metamaterial to a distance where the direction of the magnetic field changes to the same direction as B 1+ and the peak enhancement ratio outside the metamaterial is realized. Region C begins at and extends beyond the peak enhancement ratio distance and may be considered the optimal region for placement of a sample to be imaged in MRI (‘sample region’). The inset illustrates an extended analysis of the enhancement ratio of the sample region (Region C), indicating maximum magnetic field enhancement ratios of 5.84 and 6.01 for calculation and simulation results, respectively. Figure 4b depicts the simulated magnetic field distribution at the highlighted cross-section of the inset. The field enhancement ratio along the white dashed line is plotted in Fig. 4a, demonstrating a high degree of agreement with the theoretical results. In addition to the decay of the magnetic field enhancement, the direction of the magnetic field in the vicinity of the metamaterials also varies at different locations, as shown in the Supplementary Fig. 3. The magnetic field strength decay and direction variation lead to the gradient in the SNR enhancement.

Fig. 4 Magnetic field enhancement ratio of the metamaterials. a Magnetic field enhancement ratio as a function of distance from the center of the metamaterial array (along the white dashed line in b) with simulation and calculation results in blue and red. Regions A, B, and C correspond to the regions within the metamaterial array, the region extending from the surface of the metamaterial to the peak enhancement ratio, and the ‘sample region’ beyond the peak enhancement, respectively. The inset illustrates an extended plot of Region C (‘sample region’). b The magnetic field strength along the axial direction distributed at the cross-section depicted as the blue rectangle in the inset. c The theoretical magnetic field enhancement ratio of metamaterials with unit cells with different diameters Full size image

Finally, the magnetic field enhancement ratios of metamaterials composed of unit cells with different diameters were also theoretically modeled. As plotted in Fig. 4c, with the increase in the diameter, the peak enhancement ratio decreases accordingly and exhibits a relatively slower decay rate as a function of distance from the metamaterial array.

Noise analysis and SNR improvement

The expression for SNR in MRI, originally formulated by Hoult and Lauterbur34,35, is:

$${\mathrm{SNR}} \propto \frac{{\omega ^2B_{\mathrm{c}}}}{{\sqrt {r_{{\mathrm{coil}}} + r_{{\mathrm{sample}}}} }},$$ (6)

where B c is the magnetic field strength generated with unit current in the receiving coil and r coil is the resistance of the receiving coil. r sample is the equivalent resistance of the sample in the receiving coil, which can generate the same amount of heat with unit current as the heat generated by the induced current in the sample by the RF magnetic field36,37, as detailed in Supplementary Note 4. Through the application of the magnetic metamaterial, B c may be enhanced due to the RF magnetic field enhancement detailed above, thereby increasing the signal. However, the introduction of the metamaterial also leads to an increase in the noise by increasing the total power dissipation, with a nearly 1.5-fold increase in the noise derived for our design (please see Supplementary Note 5 for further details).

If B c is constant and r sample is much larger than r coil , the substitution of Eq. S18 in Supplementary Note 5 into Eq. 6 would seem to indicate that both signal and noise scale to an identical degree, thereby resulting in an absence of a net gain in SNR. Importantly, however, the introduction of the metamaterial results in the generation of a magnetic field gradient (see Fig. 4a). Therefore, as the degree of RF magnetic field enhancement along the sample in proximity to the metamaterial exceeds the increase in noise, significant gains in SNR are realized in Region C of Fig. 4a. Also notable is the capacity to tune the RF magnetic field enhancement gradients of the metamaterial through precise array design (such as unit cells with different diameters, see Fig. 4c), enabling the tailored application of metamaterials in MRI as a function of the tissue depth of interest, which is demonstrated below. Furthermore, it is noted that r sample is proportional to the 4th order of the radius of the sample (please see Supplementary Note 5 for further details). Thus, when imaging small parts (knees, wrists, etc.), r coil dominates the noise and the degree of RF magnetic field enhancement approximates the ultimate gains in SNR. The effect of the metamaterial on the electric field in MRI has also been investigated, demonstrating that the metamaterial has little effect on the electric field in the region of interest, thereby avoiding an increase in specific absorption rate (SAR) resulted from the application of the metamaterial during the imaging process (please see Supplementary Note 6 for further details).

Experimental MRI validation

Helices with identical geometries to the theoretical models employed in the analyses above were fabricated and employed for subsequent experimental validation. We employed 3D printing to fabricate hollow cylinders with grooves along their outer surface to serve as scaffolding (ABS P430). Subsequently, copper wires were wound along the 3D printed grooves in order to form the helices. Finally, the individual unit cells were aligned in an array in order to construct the complete magnetic metamaterial. Metamaterial arrays of two distinct geometries (4 × 4 array with unit cell diameter of 3 cm, 3 × 3 arrays with unit cell diameter of 5 cm) were fabricated and designed with a resonance frequency of 63.8 MHz, the operating frequency of the 1.5 T MRI employed herein. The two geometries of the metamaterials served to experimentally illustrate the capacity for tuning the RF magnetic field enhancement gradients (see Fig. 4c) such that the degree of SNR enhancement imaging may be optimized for more superficial (spine, for example) or deeper structures (liver, for example).

MRI experiments were performed in a clinical 1.5 T MRI and the birdcage body RF coil, located within the bore of the MRI system, was used for both RF transmission and reception (B 1 + and B 1 −, respectively). The resonant frequency detuning of the body RF coil due to the presence of the metamaterials is trivial based on our simulation (from 64.14 to 64.62 MHz) and may be readily compensated by the MRI system. Phantom MRI samples were prepared for imaging and composed of 2% agar gel in a 3D printed cubic mold with side length of 10 cm. Gradient echo imaging was performed both with and without the presence of the two distinct metamaterial arrays placed along the undersurface of the phantoms (top surface of the metamaterial array placed 2 cm from the bottom surface of the agar phantom; see Fig. 5a) (please see Methods for further details regarding MRI experiments). Given that the metamaterials led to enhancement of both transmission and reception RF energy (5.84 as shown in Fig. 4a), the transmission RF energy during the metamaterial-enhanced experiments was decreased such that, upon enhancement of B 1 +, would approximate the transmission RF energy of control experiments employed for comparison of SNR. MRI images of the phantom with and without the presence of the 4 × 4 metamaterial array with unit cell diameter of 3 cm are shown in Fig. 5b, c, respectively, demonstrating a 4.2-fold increase in SNR. In addition, the degree of SNR enhancement achieved experimentally is compared to simulation and theoretical results in Fig. 5d, demonstrating a high degree of agreement.

Fig. 5 Experimental results with and without metamaterials. a Experimental setup of metamaterial array beneath 3D printed magnetic resonance imaging (MRI) phantom containing 2% agar gel. b MRI image in presence of 4 × 4 metamaterial array with unit cell diameter of 3 cm (transmission radio frequency (RF) power of MRI experiment reduced, see Methods; the dashed line used for comparison of signal-to-noise ratio (SNR) in d). c Control MRI image in the absence of metamaterial array (transmission RF energy not reduced, see Methods). Scale bars correspond to 2 cm in b, c. d Comparison of SNR enhancement ratios of experimental, simulation, and calculation results. The bars for experimental results show the standard deviation of the measured SNR enhancement ratio Full size image

Subsequently, a similar experimental approach was also conducted with metamaterials featuring a unit cell diameter of 5 cm in a 3 × 3 array. The MRI image in the presence of this metamaterial array is shown in Fig. 6a and a comparison of the experimental SNR enhancement ratios of the two distinct array geometries is shown in Fig. 6b. Notably, a lower peak SNR enhancement and slower decay rate as a function of distance from the metamaterial array are demonstrated with the larger unit cell diameter, similar to Fig. 4c.

Fig. 6 Experimental results with larger unit cells. a Magnetic resonance imaging (MRI) image in presence of 3 × 3 metamaterial array with a diameter of 5 cm (transmission radio frequency (RF) power of MRI experiment reduced, see Methods; the dashed line used for comparison of signal-to-noise ratio (SNR) in b). The scale bar is 2 cm. b Comparison of SNR enhancement ratios of metamaterial arrays with distinct unit cell diameters. The larger unit cell (blue line) exhibits lower peak SNR enhancement and slower decay rate as a function of distance from the top surface of the metamaterial array. The bars for experimental results show the standard deviation of the measured SNR enhancement ratio Full size image

Finally, MRI experiments were performed using a chicken leg to demonstrate the SNR enhancement capabilities of the magnetic metamaterial arrays. The experimental setup depicted in Fig. 5a was employed, along with identical MRI pulse sequences, but having replaced the phantom with a chicken leg. Images were acquired in the plane parallel to the top surface of the metamaterial array at distances of 2.25 cm (for Fig. 7a, b) and 2.50 cm (for Fig. 7c, d) from the array. In this case, when compared to images acquired in the absence of the metamaterial array (Fig. 7b, d), SNR enhancement of up to 3.6-fold was achieved (Fig. 7a, c) in the presence of the metamaterial. As a point of further consideration, currently, an alternative approach to increasing SNR commonly employed in clinical MRI is the use of signal averaging. In this case, all other experimental conditions remaining fixed, a 13.9-fold increase in scan time would be required to achieve a similar increase without the presented metamaterial in SNR through signal averaging, highlighting the potential utility of metamaterial-enhanced MRI.