Metasurfaces of Capacitively Loaded Metallic Rings for Magnetic Resonance Imaging Surface Coils

1 Vol.:(0123456789) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports Metasurfaces of capacitively loaded metallic rings for magnetic resonance imaging surface coils Manuel J. Freire This work investigates the use of a metasurface made up of a two-dimensional array of capacitively loaded metallic rings to enhance the signal-to-noise ratio of magnetic resonance imaging surface coils and to tailor the magnetic near-field radio frequency pattern of the coils. It is found that the signal- to-noise ratio is increased if the coupling between the capacitively loaded metallic rings in the array is increased. The input resistance and the radiofrequency magnetic field of the metasurface loaded coil are numerically analyzed by means of an efficient algorithm termed the discrete model to determine the signal-to-noise ratio. Standing surface waves or magnetoinductive waves supported by the metasurface introduce resonances in the frequency dependence of the input resistance. The signal- to-noise ratio is found to be optimal at the frequency corresponding to a local minimum existing between these resonances.The discrete model is used in an optimization procedure to fit the structural parameters of a metasurface to enhance the signal-to-noise ratio at the frequency corresponding to this local minimum in the input resistance. It is found that the signal-to-noise ratio can be greatly improved if the mutual coupling between the capacitively loaded metallic rings of the array is made stronger by bringing them closer or by using rings of squared shape instead of circular. These conclusions derived from the numerical results provided by the discrete model are double-checked by means of numerical simulations provided by the commercial electromagnetic solver Simulia CST and by experimental results. Numerical results provided by CST are also shown to demonstrate that the surface impedance of the array of elements can be adjusted to provide a more homogeneous magnetic near-field radio frequency pattern that ultimately leads to a more uniform magnetic resonance image at a desired slice. This is achieved by preventing the reflection of propagating magnetoinductive waves at the edges of the array by matching the elements arranged at the edges of the array with capacitors of suitable value. Metamaterials (MM) are artificial periodic structures made up of sub-wavelength resonant elements that have emerged in the past two decades as a new class of electromagnetic materials. MM can be engineered to achieve exotic electromagnetic properties1. Thus, for example, one of the most striking properties of MM is the ability of a slab with negative permittivity/permeability to behave as a lens with sub-wavelength resolution for the electric/magnetic field1–3. MM applications range from microwave and radiofrequency (RF) bands to optical frequencies4, with new functionalities in imaging technology recently explored in the optical range5–7. MM exhibit an inherent narrowband response due to the resonant nature of its constituent elements. This narrowband response is usually considered the main drawback of MM in different fields of application. However, this is not the case for an application with an inherent narrow band of operation such as magnetic resonance imaging (MRI), which involves the detection of RF signals ranging from tens to hundreds of MHz with bandwidths of tens of kHz8. Bulk MM made up of three-dimensional (3D) arrays of elements have been explored for their use in MRI in the form of RF magnetic flux guides and lenses9–22. Compared to bulk MM, metasurfaces (MS) made up of two-dimensional (2D) arrays are a desirable alternative, as they are thin and flexible structures that are easier to manufacture and can provide the same performance as bulk structures in certain applications23. In the literature, MS have been proposed for application in MRI to provide local enhancement of the signal-to-noise ratio (SNR) of MRI coils24–28(SNR is the main figure of merit that assesses the quality of MRI). All of these MS reported are based on arrays of conducting wires. In the present work, the application in MRI of MS made up of elements consisting of split metallic rings loaded with lumped capacitors, or capacitively loaded rings (CLR), is investigated. In previous works, the author has explored the use of bulk MM made up of 3D arrays of CLR to OPEN Department of Electronics and Electromagnetism, Facultad de Física, University of Seville, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain. email: [email protected] 2 Vol:.(1234567890) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ locally increase the SNR of surface coils14–22. These bulk structures were designed to behave as lenses of negative permeability with sub-wavelength resolution14–21 or as slabs that exhibit zero22 or high permeability22 for the RF magnetic field. In any case, these bulk structures were designed following a homogenization procedure to model a 3D array of CLR as a medium with an effective parameter (permeability)1,15. In the present work, a different viewpoint is required for the analysis of MS made up of 2D arrays of CLR. An MS is characterized by a surface impedance rather than an effective parameter. In addition, as will be discussed later, an MS made up of CLR supports propagating surface waves. In the present work, the ability of the MS of CLR to provide local enhancement of the SNR of surface coils is investigated, and a key point of this investigation is the role that the excitation of standing surface waves plays in the frequency dependence of the SNR. Moreover, the possibility of tailoring the near-field of an MS-loaded coil by modulating the impedance of the CLR is also investigated, as well as the role played once again by the excitation of surface waves. In particular, it will be shown that, by adding a suitable terminal impedance in all the CLR arranged along the edges of the array, the reflection of surface waves at the edges of the array is prevented and this leads to a more homogeneous field pattern at certain distance. This ultimately allows one to obtain a more uniform MRI image in a desired slice, which is of great interest in MRI. Figure1.a shows a picture of the configuration used in this work to carry out the investigation mentioned above: a squared surface coil loaded with a 2D array of CLR and a conducting sample that resembles human tissue. The picture corresponds to an implementation in the commercial electromagnetic solver Simulia CST, which is used in this work, as described later. The square coil is split and capacitors are inserted into the gaps to homogenize the current, as usual in the designs of MRI coils29. Figure1.b shows a schematic of the CLR array together with geometric parameters. In general, an MRI coil is electrically modeled as a circuit consisting of a series inductance with resistance. This resistance is the input resistance, Rin , introduced into the coil by the ohmic losses of the sample or tissue8. In general, the SNR provided by the coil is proportional to the ratio between the sensitivity, or the RF magnetic field per unit current produced by the coil ,B1 , as specifically noted in MRI, and the square root of Rin 8: This magnetic field produced by a single coil is linearly polarized. To obtain circular polarization, two coils are required, arranged with their axes perpendicular to each other, and fed in phase quadrature8. For the configuration shown in Fig.1.a, B1 corresponds to the total field produced by the coil and the CLR array. The CLR array can enhance the field of the coil but also introduces additional losses in the coil. Thus, Rin for the configuration of Fig.1.a corresponds to the losses introduced in the coil by both the sample and the CLR array. The contribution to Rin of ohmic losses due to coil metallization can be neglected in MRI systems for operation frequencies corresponding to high field values (1 T to 7 T) and ultra-high ( ≥7 T)8. In the present work, the study is carried out for a configuration operating at the typical frequency of a 1.5 T MRI system, which is 63.6 MHz, and the ohmic losses associated with the copper coil can be neglected. The SNR provided by the configuration coil in Fig.1.a can then be enhanced by the CLR array if the total field is maximized and, at the same time, the additional input resistance introduced by the CLR array is minimized. In the present work, the structural parameters of the CLR array are optimized to achieve both goals. The use of small parasitic loops magnetically coupled to a receiving surface coil to locally improve coil sensitivity has been explored since the early days of MRI30. More recently, the use of arrays of passive loops magnetically coupled to a volume coil has also been explored in high-field systems to locally enhance the transmit field of a birdcage coil and thus to homogenize the total transmit field31. In any case, since parasitic or passive loops are magnetically coupled with the active coil (receiver or transmitter coil) through an axial configuration, the mutual inductance between them is positive. Thus, the currents induced in the passive loops by Faraday’s law are in phase with the current in the active coil if the reactance of the passive loops is negative or capacitive. This happens at frequencies below the frequency of resonance of the loops. Therefore, below the frequency of resonance, the fields of the passive loops and the coil are added, and the sensitivity is enhanced. In the present work, the CLR in the configuration under study are also designed to work below their resonance frequency to (1) SNR ∝ B 1 √Rin Figure1. (a) Picture of the configuration to be analyzed in this work: a squared surface of length L loaded with a 2D CLR array and a cubic conducting sample resembling human tissue. (b) Schematic of the array of CLRs with periodicity a, external radius of the ring r, strip width w and capacitance of the lumped capacitor C. 3 Vol.:(0123456789) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ behave capacitively. The CLRs are also magnetically coupled between them, and therefore the array supports the waves due to voltages induced by time-varying magnetic fields or magnetoinductive (MI) waves32,33. Since the CLR of the array in Fig.1 are arranged in a coplanar configuration, the supported MI waves are backward32,33. For the study shown in this work, it is worth noting that, due to the finite size of the array, standing MI waves can be excited with wavelengths that are proportional to the size of the array. In the present work, the study of the configuration shown in Fig.1.a is carried out using a dedicated algorithm developed for the analysis of coils in the presence of CLR arrays and a half-space of conductance similar to that of human tissue19. This algorithm was developed for the numerical analysis of MM, taking into account the discrete and finite nature of the structure. From now on, this algorithm will be termed a discrete model in the text. With this discrete model, the value of Rin and the spatial dependence of B1 along the coil axis are calculated for the configuration shown in Fig.1.a with different structural parameters. Since the SNR is proportional to the ratio B1/√Rin , as stated in expression (1), and it is of interest for the analysis to compare the SNR obtained for different structures, for the present analysis, the SNR will be considered not proportional but equal to this ratio. Calculating B1 and √Rin with the discrete model is very quick compared to the use of commercial electromagnetic solvers. This allows us to study the dependence of the SNR along the coil axis with the structural parameters of the CLR array in an efficient optimization procedure. In the discrete model, a array equation for unknown currents in the CLR and coil is solved19. The diagonal elements of the impedance array correspond to the selfimpedances of the CLR and the off-diagonal elements correspond to the mutual inductances between the CLR. The study carried out with the analysis of different structures with the discrete model shows that the value of Rin introduced by the CLR array exhibits resonances at frequencies corresponding to the excitation of standing MI waves. Furthermore, the optimal SNR is obtained at the frequency that corresponds to a local minimum of Rin between these resonances. A significant outcome of the study is the fact that as the mutual coupling between the CLR increases, the value of this local minimum of Rin is reduced, and consequently the SNR increases. In practice, this is achieved by reducing the periodicity of the array. Thus, one of the main conclusions of the present work is the finding that the SNR of a surface MRI coil can be enhanced by means of MS of strongly coupled elements rather than weakly. Once an optimal structure is determined from the point of view of the SNR along the coil of the axis with the discrete model, the 3D magnetic field pattern is obtained by using the commercial electromagnetic solver Simulia CST and compared with the field produced by the coil in the absence of the array for comparison purposes. A square coil and a CLR array are manufactured with the optimal structural parameters and experimental results are obtained to validate the comparison mentioned above. These experimental results are obtained by measuring with a vector network analyzer (VNA) the transmission coefficient S21 between the squared coil and a small probe immersed inside a saline solution phantom, both in the presence and absence of the intermediate CLR array placed between the coil and the phantom. The coil is matched to 50  in each case (see the Appendix for a demonstration of the relation between the transmission coefficient S21 for a matched coil and the SNR). As has been pointed out, the main conclusion of the study is that increasing the mutual coupling between CLR results in an increase of the SNR and that this can be achieved in practice by bringing the CLR closer, that is, by reducing the periodicity. Moreover, it is also shown that once the periodicity cannot be more reduced for a given size of the CLR, a stronger coupling can still be obtained and, therefore, a higher increase in the SNR can be achieved by using squared rings instead of circular rings. Finally, numerical results provided by Simulia CST are also shown to demonstrate that the RF magnetic field pattern produced by a CLR array can be tailored, for example, to produce a homogeneous field pattern at a plane at a desired distance and, therefore, to obtain a homogeneous MR image at a desired slice. This is achieved by considering a remedy used to avoid reflections of MI waves in a 1D array of resonators of finite length and that consists of adding a terminal impedance at the ends of the 1D array32,33. This remedy is extended to the present case of a 2D CLR array. In the present work, the reflections at the edges of the 2D CLR array of the MI waves propagating in the structure are suppressed by matching the CLR arranged along the edges of the array with a suitable capacitance value. The paper is structured as follows. In Sect. “MI waves and the input impedance of a coil loaded with a CLR array”, it is analyzed the role that MI waves excited by a coil in finite-size CLR arrays play in the frequency dependence of Rin . The standing MI waves supported by the array are shown to lead to resonances in Rin , wit h a local minimum for Rin between these resonances. In Sect. “MI waves and the SNR of a coil loaded with a CLR array”, the dependence of the SNR value with the frequency dependence of Rin is analyzed. In Sect. “A stronger CLR coupling reduces Rin and enhances SNR”, a result of very practical interest is demonstrated, which is the fact that the SNR can be enhanced by increasing the mutual coupling between CLR in the array, and that this can be achieved by bringing the CLR closer to each other or by using squared elements rather than circular. In Sect. “Tailoring the near field”, an example of a CLR array design is shown which provides a more homogeneous field pattern. This is achieved by suppressing the reflection of MI waves at the edges of the array. Conclusions are drawn in Sect. “Conclusion”. In Appendix “Linear relation between SNR and S 21 ”, the relation between SNR as defined in expression (1) and the transmission coefficient or scattering parameter S21 measured with a VNA is demonstrated. MI waves and the input impedance of a coil loaded with a CLR array For the analysis, it is helpful to previously discuss the frequency dependence of the input impedance of a coil loaded with a homogeneous slab of MM with negative permeability ( µ<0 ), and then establish an analogy with the frequency dependence of the input impedance of a similar coil loaded with a CLR array. Figure2.a shows the frequency dependence of the real and imaginary parts of µ for a homogeneous slab of MM implemented in Simulia CST for its simulation. The figure shows in the inset the screenshot of the simulation structure implemented that corresponds to a slab 15 ×15 ×1.8 cm3 and a squared coil of 12 cm in length and 1 cm in width of the strip. In the simulation, three 154 pF capacitors are evenly distributed in three gaps along the strip to homogenize the 4 Vol:.(1234567890) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ current in the coil, and a port is placed in the fourth gap. The dispersion parameters of µ are chosen to obtain µ=−1−j0.2 at 63.6 MHz, that is, the operating frequency of a 1.5 T MRI system. This complex value of µ corresponds to that of a MM lens previously analyzed by the author14–16. Figure2.b shows the results provided by Simulia CST for the frequency dependence of the input resistance Rin and the input reactance Xin introduced by the MM slab in the coil. Rin shows two maximums at frequencies that correspond to the excitation of magnetostatic waves on the slab. Between these two maximums, at the frequency that corresponds to the real part of µ equal to −1 (63.6 MHz), Rin shows a minimum and Xin cancels. In view of the above, it is now interesting to discuss the frequency dependence of the input impedance for a similar coil loaded with a CLR array. For simplicity, the discussion will focus on the analysis of only the values of Rin . To do this, a numerical analysis is performed that makes use of the discrete model mentioned in the Introduction Section19. For the sake of simplicity, follow-up analysis in this section will be carried out in the absence of the conducting sample. The discrete model has already been validated previously for the calculation of Rin 19. However, at this point, a comparison is made between the calculation with the discrete model and the measurement of Rin for a particular structure, to consider this comparison as a reference to the accuracy of the present study. Thus, Fig.3 shows the value of Rin measured for a 12 cm long squared coil, in the absence of a conducting sample or phantom, loaded with an array of 6×6 circular CLR fabricated with the following structural parameters: r= 6 mm, w = 2 mm, a = 15 mm and capacitors by Passive Plus with C= 470 pF and tolerance ±1% . The distance between the coil and the array is 6 mm. Figure3 also shows the numerical results provided by the discrete model for Rin , which are totally in agreement with the experimental results. The numerical results shown in Fig.3 are obtained after fitting the following parameters in the discrete model to these values: a=15.6 mm, C=473 pF. While the peaks shown in Fig.2 in the frequency dependence of Rin for the homogeneous MM slab correspond to the excitation of magnetostatic waves, the peaks shown in Fig.3 for the CLR array correspond to the excitation of standing MI waves whose wavelength fits the finite size of the array. Once the precision of the discrete model has been established by the agreement shown in Fig.3, the discrete model is used to study the frequency dependence of the position of the peaks or resonances in Rin with the size of the array, that is, with the number of elements along the length of the array. Figure4 shows the dispersion relation calculated32 for the array analyzed in Fig.3 for a=15 mm and C=470 pF, in the approximation of the interactions of the first neighbors and in the approximation of the interactions of the first and second neighbors. As mentioned above, the peaks or resonances in Rin for a CLR array correspond to the excitation of standing MI waves associated with different wavelengths in the dispersion relation of MI waves. Note that, since the Figure2. (a) Frequency dependence of the real and imaginary parts of the complex permeability µ of a homogeneous MM slab with the dispersive parameters of a MM lens previously analyzed by the authors14–16. The dispersion parameters of µ are chosen to obtain µ=−1−j0.2 at 63.6 MHz. Inset: picture of a 12-cm squared coil loaded with the MM slab of dimensions 15 ×15 ×1.8 cm3. (b) Frequency dependence of the input resistance Rin and the input reactance Xin of the coil in the presence of the MM slab obtained with Simulia CST. 5 Vol.:(0123456789) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ RF magnetic field pattern produced by the coil that excites the array is maximum at the center of the array, the standing MI waves that can be excited by the coil and that lead to the peaks in Rin must correspond to standing MI waves with an odd number of half-wavelengths throughout the length of the array. From the dispersion relation shown in Fig.4, in the approximation of the interactions between the first and second neighbors, Table1 shows the values of the frequencies in MHz that correspond to the normalized wavenumber kπ/a such that a half-wavelength or three half-wavelengths fits the length of an array of 6×6 unit cells or 8×8 unit cells. Figure5 shows the calculation provided by the discrete model for the frequency dependence of Rin in the 12 cm long square coil loaded with an array of 6×6 unit cells or 8×8 unit cells with the same structural parameters as in the dispersion relation shown in Fig.4. Thus, Table1 also shows the frequency values in MHz for the position of the peaks or resonances in Rin in the curves from Fig.5. The agreement shown in Table1 between the frequency values for the peaks or resonances provided by the discrete model in the calculation of Rin and the frequency values provided by the dispersion relation confirms that the resonances in Rin correspond to the excitation of standing MI waves such that half-wavelength or three half-wavelengths of the standing wave fit the length of an array of 6×6 unit cells or 8×8 unit cells. Note that this agreement requires one to consider both first- and second-neighbor interactions. Figure3. Rin measured and calculated for a 12 cm long square coil loaded with an array of 6×6 CLR. Structural parameters in the experiment: r = 6 mm, w = 2 mm, a = 15 mm, C = 470 pF ±1% . Str uctural parameter in the discrete model: r = 6 mm, w = 2 mm, a = 15.6 mm, C = 473 pF. The array is placed 6 mm from the coil. Figure4. Dispersion relation calculated for the array analyzed in Fig.3 with r=6 mm, w=2 mm, a=15 mm and C=470 pF, in the approximation of the interactions of the first neighbors and in the approximation of the interactions of the first and second neighbors. Table 1. Frequency values in MHz for the corresponding wavenumbers in the dispersion relation of Fig.4 in the approximation of both the first and second neighbors such that a half-wavelength or three half-wavelengths fits the length of an array of 6×6 unit cells or 8×8 unit cells, and frequency values in MHz for the peaks or resonances in Rin of Fig.5. 6×6 units cells Dispersion relation Resonance in R in /2 = 6a =⇒ ka/π = 1/6 = 0.166 66.6 MHz 66.5 MHz 3/2 = 6a =⇒ ka/π = 3/6 = 0.500 64.6 MHz 64.6 MHz 8×8 units cells Dispersion relation Resonance in R in /2 = 8a =⇒ ka/π = 1/8 = 0.125 66.8 MHz 66.9 MHz 3/2 = 8a =⇒ ka/π = 3/8 = 0.375 65.3 MHz 65.3 MHz 6 Vol:.(1234567890) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ Furthermore, it is worth noting also that Fig.5 shows that as the size of the array increases from 6×6 to 8×8 unit cells, the peaks move to higher frequencies. This is due to the backward characteristic of the dispersion relation, which is clearly shown in Fig.4. As the size of the array increases from 6×6 to 8×8 , the standing MI waves that can resonate within the size of the array have larger wavelengths, so smaller wavenumbers, and this corresponds to higher frequencies in the backward dispersion relation. MI waves and the SNR of a coil loaded with a CLR array Once it has been demonstrated that the peaks in Rin already correspond to standing MI waves with half-wave- length or three half-wavelengths resonating throughout the array, it is next checked that the SNR provided by the coil in the presence of the array is optimal at the frequency corresponding to the local minimum of Rin between the peaks. To carry out this study, the discrete model is used to calculate from the expression (1) the SNR along the axis of the 12 cm long square coil loaded with an array of 8×8 elements similar to that previously analyzed in Fig.5, and in the presence of a half-space with conductivity σ=0.7 S/m and dielectric constant ε=70 . For comparison purposes, this calculation is performed for values of Rin that correspond to the peaks and to the local minimum between the peaks, and, for practical interest, by fixing the calculation frequency so that the value of interest of Rin is always fixed for each case at 63.6 MHz, which corresponds to the working frequency of a 1.5T MRI. This is achieved by tuning the value of C for each calculation. Figure6 shows the calculated frequency response of Rin for several values of C. A vertical dashed line highlights for each case the frequency of 63.6 MHz, and points out the position of a peak in Rin for C=495 and C=518 pF, and the minimum between peaks for C=504 pF. For comparison purposes, two additional values of C=484 pF and C=536 pF are also chosen to perform SNR calculations beyond the peaks. Figure7 shows the results of the SNR calculation provided by the discrete model along the coil axis at a frequency of 63.6 MHz for the different capacitance values of Fig.6. It must be noted that in Fig.7, the curve for C=504 pF is above all for almost the whole distance range along the coil axis, indicating that the SNR is optimal for this case, which corresponds to the situation depicted in Fig.6.c, that is, with the coil operating at 63.6 MHz at a minimum of Rin between the peaks. A stronger CLR coupling reduces Rin and enhances SNR In previous sections, it has been demonstrated that a CLR array introduces resonances in the input resistance, Rin , of a coil, and that the SNR provided by the coil in the presence of the array is optimal at the frequency corresponding to the minimum value that Rin takes between these resonances. Once this has been established, an optimization procedure is followed based on the structural parameters of the array to reduce the magnitude of the minimum value of Rin and therefore increase the SNR. The discrete model is used to calculate the frequency dependence of Rin for a 12 cm square coil loaded with an array, starting from the structure of 8×8 elements analyzed in Figs.6 and 7, and tuning the values of a and C to fix the minimum value of Rin at a frequency of 63.6 MHz, and at the same time to reduce the magnitude of this minimum. Figure8 shows a comparison between the frequency dependence of Rin for the starting structure with a=15 mm and C=504 pF and the final structure obtained after the optimization procedure with a=12.5 mm and C=554 pF. It must be noted that, although this optimized structure has a reduced periodicity (from 15 mm to 12.5 mm), the CLR are of the same size, and despite this reduction of the periodicity, the CLR in the optimized array are not in touch. The curves in Fig.8 show that the minimum value of Rin at the frequency of 63.6 MHz between the peaks (marked with × in the curves of the figure) is considerably reduced for the optimized structure. In particular, the value of Rin is reduced from 34  to 8  . As it was explained in the Introduction Section, the enhancement of the SNR comes from the reduction in the value of Rin and from the addition of the fields produced by the coil and the CLR. This last condition is fulfilled if the CLR operate at a frequency below the frequency of resonance, as was also pointed out. In the array analyzed above with a=12.5 mm, for the nearest neighbors in the same row, the discrete model provides a value for the mutual inductance of Mr=−0.48 nH and for the nearest neighbors on the same diagonal, a value of Md=−0.13 nH. These values are negative since they correspond to the mutual inductances of CLR arranged in coplanar configuration. The self-inductance of an isolated CLR is 13.45 nH, calculated from the values of the external radius and the width of the strip using Figure5. Calculations provided by the discrete model for the values of Rin in a 12 cm square coil loaded with CLR arrays with r=6 mm, w=2 mm, a=15 mm, C=470 pF and different number of elements: 6×6 and 8×8 unit cells. Each array is placed at 6 mm from the coil. 7 Vol.:(0123456789) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ Figure6. Calculations provided by the discrete model for the values of Rin of a 12-cm square coil in length loaded with 8×8 CLR arrays with r=6 mm, w=2 mm, a=15 mm and different values of C: (a) C=484 pF, (b) C=495 pF, (c) C=504 pF, (d) C=518 pF, (e) C=536 pF. The dashed vertical lines indicate the frequency of 63.6 MHz that corresponds to the operating frequency of a 1.5T MRI system. Each array is placed 6 mm from the coil. 8 Vol:.(1234567890) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ a known model34. Thus, mutual inductance with all nearest neighbors reduces the self-inductance to a value of 11.01 nH, and with C=554 pF this provides a resonance frequency of 64.4 MHz. Therefore, the CLR actually behave capacitively since the operation frequency of 63.6 MHz is below the resonance frequency of 64.4 MHz. Taking this into account, and the significant reduction in Rin from 34  to 8  shown by the results in Fig.8, the reduction of the periodicity from a=15 mm to a=12.5 mm should lead to a clear enhancement of the SNR as established by expression (1). To check this, Fig.9 shows the calculation carried out with the discrete model for the SNR along the coil axis at 63.6 MHz for each of the two arrays analyzed in Fig.8, in the presence of a conducting half-space with σ=0.7 S/m and ε=70 . The results shown in Fig.9 demonstrate that the axial SNR is significantly enhanced for the CLR array with reduced periodicity a=12.5 mm. Because reducing the periodicity while maintaining the size of the CLRs must lead to an increase in the mutual inductance between neighboring CLRs, it can be concluded that strengthening the coupling between elements in the array leads to an enhancement of the SNR. This can be considered one of the main conclusions of the present work. As a double check, the results shown in Fig.9 have been verified using the numerical results obtained with Simulia CST. As mentioned, a commercial electromagnetic solver such as Simulia CST is very time consuming compared to the discrete model, if we are interested in computing only Rin and the SNR along the coil axis as part of an optimization procedure. However, once an optimal structure has been found with the discrete model, it is of interest to use software such as Simulia CST to obtain the 3D field pattern of the RF magnetic field produced by the optimal configuration. In MRI, the average effective RF magnetic field generated by a coil is noted as B+ 1 , and corresponds to the contribution of the longitudinal (on the axis of the coil) and transverse components of the RF field produced by the coil that are perpendicular to the static magnetic field, B0 , of the MRI system8. In MRI, B0 is oriented along the z axis in a Cartesian coordinate system, and the longitudinal and transverse components of the RF field coil that are perpendicular to B0 are noted as B1x and B 1 y , respectively (the sketch Figure7. Calculations provided by the discrete model for the SNR along the axis of a 12 cm long square coil loaded with arrays of 8×8 CLR with a=15 mm, r=6 mm, w=2 mm and different values of the C that correspond to the different values indicated in Fig.6: 484 pF, 495 pF, 504 pF, 518 pF, 536 pF. All curves are calculated for the same frequency of 63.6 MHz that corresponds to the operating frequency of a 1.5 T MRI system. Each array is placed at 6 mm from the coil and the array is 6 mm far from a conducting half-space with σ=0.7 S/m and ε=70 . Figure8. Calculations provided by the discrete model for the value of Rin for a 12 cm square coil in length loaded with two different arrays of 8×8 CLR with the same size ( r= 6 mm, w=2 mm) but different values of a and C: a=15 mm, C=504 pF (solid line); a=12.5 mm, C=554 pF (dashed line). The crosses in the curves point to the frequency of 63.6 MHz. Each array is placed 6 mm from the coil. 9 Vol.:(0123456789) Scientific Reports | (2023) 13:2998 | https://doi.org/10.1038/s41598-023-30185-y www.nature.com/scientificreports/ depicted in Fig.1 is in accordance with this). B+ 1 is a circularly-polarized field that rotates in the same direction as nuclear precession and is expressed as B+ 1= B1 x+ jB1 y . Thus, Fig.10 shows screenshots of the configurations under analysis implemented in CST together with the maps obtained in the simulations for the root-mean-square (RMS) value of B+ 1 . For comparison purposes, these configurations correspond to the coil of 12 cm in length in the absence (Fig.10a) and in the presence (Fig.10b) of the optimized array of 8×8 elements with a=12.5 mm. In both cases there is a cubic conducting phantom that resembles human tissue with conductivity σ=0.7 S/m and a dielectric constant ε=70 . The separation distance between the coil, the array and the phantom is 6 mm. Also, in both cases the coil is matched to 50  through the corresponding circuit matching network. To conform to reality, the metallic pads of the real capacitors have been implemented in the CST model. This leads to an increase in the inductance of the total metallic path of the CLR. For this reason, to maintain the results at the same frequency of 63.6 MHz, the value of C=554 pF used with the discrete model must be reduced to C=525 pF in the CST simulation. The results show that the field is highly enhanced by the presence of the array with a=12.5 mm up to penetration depths of 3 cm inside the conducting sample. For ease of comparison, Fig.11 shows the profiles of B+ 1 along the coil axis for the maps shown in Fig.10, which correspond to an array with a=12.5 mm, and also for the values obtained for the array with a=15 mm. Figure9. Calculations provided by the discrete model at 63.6 MHz for the SNR along the axis of a 12 cm long squared coil (dotted line) loaded with each of the 8×8 CLR arrays analyzed in Fig.8: a=15 mm and C=504 pF (dashed line); a=12.5 mm and C=554 pF (solid line). In both cases r=6 mm and w=2 mm. Each array is placed at 6 mm from the coil and the array is at 6 mm from a conducting half-space with σ=0.7 S/m and ε=70 . The results are normalized to the value obtained at x=0 in the absence of arrays (dotted line). Figure10. Screenshots of a perspective view and a front view of the configurations under analysis together with the RMS B + 1 maps provided by Simulia CST for (a) a squared coil of 12 cm in length placed at 6 mm from a cubic sample resembling human tissue with σ=0.7 S/m and ε=70 and (b) the same coil loaded with an array of 8×8 CLR with parameters r= 6 mm, w=2 mm, a=12.5 mm, C=525 pF, the array being placed placed at 6 mm from the coil and the cubic sample at 6 mm from the array. The coil is matched to 50 Ohms in each case trough the corresponding circuit matching network.