## 2 Materials and Methods  
### MP-SAWS Principle  
The substrate of the MP-SAWS is made of the piezoelectric material LiNbO\({}_{3}\), integrating interdigitated electrode transducers (IDTs) in the middle layer and metal reflector networks on both sides. The parameters of various piezoelectric materials are listed in Table 1. LiNbO\({}_{3}\) has a higher electromechanical coupling coefficient compared to other common piezoelectric materials [46]. The portion between the interdigital electrodes and the reflector network is used to detect the resonance frequency of the surface acoustic wave (SAW). The schematic diagram is shown in Figure 1.  
The IDT serves dual roles as both the transmitter and the signal-conversion element of the MP-SAWS. It is deposited onto the piezoelectric substrate. When an AC electric signal is applied to the IDT, vibrations occur in the piezoelectric material caused by the reverse piezoelectric effect, thus exciting the MP-SAWS, which oscillates at an identical frequency to the AC signal. The wave travels across the surface of the substrate. After the mechanical wave reaches the reflector network, it returns to the IDT via the reflector grids. Following the completion of wave transmission, the IDT converts the mechanical wave into an electrical signal using the positive piezoelectric effect and transmits it to the receiver. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **ZnO** & **AIN** & **ST-Cut Quartz** & **128\({}^{\circ}\) Y-X LiNbO\({}_{3}\)** & **36\({}^{\circ}\) Y-X LiTaO\({}_{3}\)** \\ \hline Moulus (GPa) & 110–140 & 300–350 & 71.7 & 130–170 & 205 \\ Poisson’s ratio & 0.36 & 0.22–0.29 & 0.17–0.2 & 0.24–0.28 & 0.17–0.2 \\ Piezo-constant d33 & 12 & 4.5, 6.4 & 2.3 (d11) & 12 & 12 \\ (PC/N) & 1.5–1.7 & 3.1–8 & 0.1–0.2 & 5–11.3 & 5–6.6 \\ \hline Acoustic velocity of longitudinal (transverse) waves (m/s) & 6336 & 10,150–11,050 (5800) & 5000–5960 (3159) & 3680–3980 & 4160–4220 \\ (m/s) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Common piezoelectric material parameters [46]. The structure of MP-SAWS allows for the addition of a test area to change the physical field of the measured object. Consequently, the measured information is reflected through the variation in the SAW's velocity or frequency [2]. In the simulation software, the constitutive equation describing the piezoelectric phenomenon is represented as follows [47]:  
Stress-Charge Form:  
\[\mathrm{T}=c_{E}S-e^{T}E \tag{1}\]  
\[\mathrm{D}=\mathrm{e}\mathrm{S}+\varepsilon_{S}E \tag{2}\]  
Strain-Charge Form:  
\[\mathrm{S}=s_{E}T+d^{T}E \tag{3}\]  
\[\mathrm{D}=\mathrm{d}\mathrm{T}+\varepsilon_{T}E \tag{4}\]  
The following equations are also provided:  
\[c_{E}=s_{E}^{-1} \tag{5}\]  
\[e=ds_{E}^{-1} \tag{6}\]  
\[\varepsilon_{S}=\varepsilon_{T}-ds_{E}^{-1}d^{T} \tag{7}\]  
In the equations, S represents the strain tensor, \(s_{E}\) represents the compliance matrix, T represents the stress tensor, d represents the piezoelectric constant, E represents the external electric field, D represents the electric displacement, and \(\varepsilon_{T}\) represents the dielectric constant. To enable magnetic field measurements and their application to the human body, magnetostrictive Terfenol-D, and biocompatible, flexible polydimethylsiloxane (PDMS) materials were introduced [48; 49; 50; 51]. When subjected to an external magnetic field, magnetostrictive materials undergo changes in their magnetization state, resulting in alterations in length and volume. Based on magnetostrictive properties, these materials can be effectively coupled with MP-SAWS. The device is encapsulated with a PDMS shell material on its outer side. PDMS is biocompatible, flexible, thermally stable, and non-magnetic. These characteristics allow PDMS to enhance the biocompatibility of MP-SAWS as a casing material without compromising the sensor's ability to accurately detect cardiac parameters [52].