## 2 Materials and Methods
### Theoretical Model  
As previously introduced, magnetic metasurfaces are specially engineered structures capable of exhibiting unconventional properties, such as enhanced or even negative magnetic permeability. Typically, they consist of arrays of resonant unit cells, whose dimensions are determined by the operating frequency and critically influence the resulting magnetic field distribution. When appropriately designed, the entire metastructure exhibits resonant behavior, allowing the effective magnetic permeability to be tailored over specific frequency  
Figure 1: Flow chart of the proposed methodology. bands and enabling the enhancement of the magnetic field strength, in accordance with [PERSON]'s second equation. \[\
abla\times B=\mu_{0}\mu_{r}(J+i\omega e_{o}E), \tag{1}\]  
where \(B(T)\) and \(E(V/m)\) are, respectively, the magnetic induction and the electric fields; \(\mu_{0}(H/m)\) and \(e_{o}(F/m)\) are, instead, the vacuum magnetic permeability and dielectric permittivity; \(\mu_{r}\) is the relative magnetic permeability; \(\omega=2\pi f\), where \(f\) is the operative frequency. Finally, \(J\left(A/m^{2}\right)\) is defined as the current density. Therefore, according to (1), it is possible to enhance the magnetic field strength by increasing both the conductance and the displacement currents, acting on the engineered relative magnetic permeability realized by the metasurface. One of the most important MRI RF coil parameters, fundamental for imaging quality, is the homogeneity of the generated magnetic field. It is important to note that electromagnetic metasurfaces are typically designed under the assumption of infinite extent, which inherently ensures a homogeneous response. In such cases, each unit cell shares identical boundary conditions--being surrounded by an infinite array of neighbors--and consequently carries the same current distribution. However, at relatively low operating frequencies, such as those used in MRI, the physical size of each unit cell becomes large enough that only a limited number can be used to construct the metasurface. Under these conditions, where the metasurface is finite, edge or truncation effects can become significant, potentially degrading the performance in terms of magnetic field homogeneity. It is easy to understand that only the central unit-cells are surrounded by a large number of elements, while, on the external borders, an asymmetry can be spotted. For this reason, to restore the ideal metasurface behavior, an analytic algorithm is employed to control each unit-cell current [32], and thus, the overall magnetic field distribution. Specifically, by accounting for both the self-impedance of each unit cell and the mutual coupling between them, it becomes possible to compensate for the truncation effects associated with the finite size of the metasurface. If the active RF coil used to excite the metasurface is denoted as element 1, and the metasurface unit cells are indexed from 2 to \(N\), the overall system can be described using an impedance matrix as follows:  
\[ \begin{pmatrix}Z_{11}&Z_{12}&Z_{13}&\ldots&Z_{1N}\\ Z_{21}&Z_{22}&Z_{23}&\ldots&Z_{2N}\\ Z_{31}&Z_{32}&Z_{33}&\ldots&Z_{3N}\\ \vdots&\vdots&\vdots&\ldots&\vdots\\ Z_{N1}&Z_{N2}&Z_{N3}&\ldots&Z_{NN}\\ \end{pmatrix}\begin{pmatrix}I_{1}\\ z_{1}\\ c_{2}I_{x}\\ c_{3}I_{x}\\ \vdots\\ c_{N}I_{x}\\ \end{pmatrix} = \begin{pmatrix}V_{1}\\ 0\\ \vdots\\ 0\\ 0\\ \end{pmatrix} , \tag{2}\]  
where the current flowing in the \(i\)-th cell can be written as follows:  
\[I_{i}=c_{i}I_{x}\qquad with\ i=2,3,\ldots,N, \tag{3}\]  
in which \(c_{i}\) is the \(i\)-th complex current coefficient and \(I_{x}\) is an arbitrary reference value.