Mechanics Modeling of Electrodes for Wireless and Bioresorbable Capacitive Pressure Sensors

Bio-implantable pressure sensors are attractive for many life-threatening clinical scenarios that require real-time monitoring of the internal pressure in the human body, such as the intracranial pressure after traumatic brain injury, the pulmonary hypertension associated with serious heart and lung diseases, and the intracompartmental pressure [1–3]. For the need of temporary implantation, bioresorbable devices are attractive as they can minimize inflammatory responses and eliminate the need for a second surgery [4,5]. Several bioresorbable pressure sensors reported recently were typically based on resistive systems, detecting the resistance change caused by the pressure-induced deformation of the thin film resistor [6–8]. However, such resistance pressure sensors rely on percutaneous wires to collect data, which poses a risk of infections and limits the patient’s natural movements [6–8].

Lu et al. [9] developed a wireless and bioresorbable capacitive pressure sensor to overcome the above-mentioned drawbacks of the resistance system. As shown in Fig. 1, the wireless capacitive pressure sensor [9] was based on an induction-capacitor (LC) resonance circuit (left part in Fig. 1(a)), which was implanted in the body for sensing the pressure, and an external antenna circuit (right part in Fig. 1(a)), which was out of the body for wireless readout. The key structure of the pressure sensor was a parallel-plate capacitor in the LC-resonance circuit that was mainly composed of a top electrode, a bottom electrode, and a sealed air cavity in between (Fig. 1(b)), to convert the pressure in the body applied on the implanted capacitor into a capacitance variation that can be measured electrically. The signal process was that the applied pressure led to the downward deflection of the top electrode, and thereby the change in the electrode separation and associated change in the capacitance such that the resonance frequency in the same LC circuit changed relatedly and was read out wirelessly via the external antenna circuit.

Two devices [9] were fabricated with different capacitors: capacitor A with a conventional air cavity and capacitor B with an expanded air cavity. Figure 2a shows the schematic of capacitor A. The top electrode was a three-layer structure with a zinc (Zn) foil in the middle of two poly (lactic-co-glycolic acid) (PLGA) plates, which was named the PLGA–Zn–PLGA composite plate, with the thickness of 5–2–5 µm. The bottom electrode was made of magnesium (Mg) and parallel to the top electrode with an initial separation (h) of 50 µm. There was a sealed air cavity in the shape of a cylinder with a depth of 50 µm and the same radius as the electrodes (R = 1.5 mm). As PLGA, zinc, and magnesium are bioresorbable materials such that the entire capacitor was bioresorbable and implantable. Young’s modulus and the Poisson’s ratio of materials are 1.6 GPa and 0.25 for PLGA; 100 GPa and 0.34 for zinc; and 45 GPa and 0.34 for magnesium. Figure 2b shows the schematic of capacitor B. A magnesium plate with a central hole with a diameter of R and a circular profile with the equivalent radius of R_out² was added to expand the air cavity. The zinc foil (in the top electrode) and the bottom electrode remained the same circular shape (R = 1.5 mm) as capacitor A. Because of the expanded cylindrical profile (radius R_out = 2.7 mm, depth 50 µm) capacitor B has a much larger initial volume of 1.15 mm³ of air cavity than 0.35 mm³ from capacitor A. The deformable plate over the air cavity is a composite plate in the following, with the cross section of PLGA–Zn–PLGA for the radius 0 < r < R, where R = 1.5 mm is the same as capacitor A, and the cross section of PLGA-Mg (thickness: 10 µm and 50 µm, respectively) for the radius R < r < R_out, where R_out = 2.7 mm. Mg has a much larger elastic modulus (45 GPa) and thickness (50 µm) than PLGA (1.6 GPa and 10 µm) such that its bending stiffness is ∼3,500 times larger. Therefore, the outer part of the electrode is modeled as a single Mg layer.

Experimental results showed that, with all parameters (e.g., materials properties, thickness of the electrode, plate area, and initial separation between two electrodes) fixed, the sensor sensitivity based on capacitor B was around 5 times better than that of capacitor A due to expanding air cavity inside the capacitors. As the applied pressure increases, the top electrode deflects downward, thereby reducing the volume of the sealed air cavity. Consequently, the pressure inside the sealed air cavity increases following the ideal gas law. For capacitor A, this pressure inside the air cavity increases quickly as the air cavity is small (Fig. 2c), therefore providing substantial resistance to deflection of the electrode. The volume of the air cavity is much larger for capacitor B because of its expanding air cavity (Fig. 2d); therefore, the pressure inside increases slowly, leading to a larger deflection of the electrode and improved sensitivity.

This paper establishes an analytic model for deformation of the electrodes of the wireless and bioresorbable capacitive pressure sensor, which shows analytically the sensitivity dependence on the sensor geometry and material properties, leading to the optimized design of wireless, implantable, bioresorbable, and high-sensitivity capacitive pressure sensor.

The sensitivity of the capacitive pressure sensor results mainly from the sensitivity of the capacitance change to the applied pressure. For parallel-plate capacitors (e.g., capacitors A and B in Fig. 2), the capacitance is defined as C = ɛS/d, where d and S are the separation and overlapping area between two parallel conducting electrodes, respectively, and ɛ is the dielectric permittivity of air. As the bottom electrode was fixed in the experiments, the capacitance change ΔC results from the downward deflection of the top electrode under uniform applied pressure and is given by $ΔC/C0=w¯/(h−w¯)$ [9], where C₀ is the zero-pressure capacitance, h is the initial separation between two parallel electrodes, and $w¯=2/R2∫0Rwrdr$ is the average of deflection w over the area of the top electrode. For $w¯≪h$⁠, the capacitance change ΔC becomes linearly proportional to $w¯$⁠. In the mechanics models below “the electrode” is used, instead of “the top electrode” for simplicity, since only the top electrodes deform in capacitors A and B.

Let p denote the applied pressure on top of the electrode (outside the capacitor) and Δp the increase of air pressure below the electrode (inside the sealed air cavity), where Δp is related to the average deflection $w¯$ via the ideal gas law. The deflection w is then obtained analytically in terms of the pressure difference p − Δp via the elastic plate theory [10] for capacitor A in Sec. 2.1 and capacitor B in Sec. 2.2.

For capacitor B, let D denote the bending stiffness of the symmetric, three-layer composite plate (PLGA–Zn–PLGA, same as capacitor A) of the central part (0 < r < R) of electrode, and D_out the bending stiffness of Mg plate for the outer part (R < r < R_out) of the electrode (D and D_out are given in Appendix  A).

The sensitivity dependence on the geometry and material properties is studied in this section based on the analytical model in Sec. 2. The baselines of geometry and material properties for capacitors A and B are [9]: the radius of the electrode R = 1.5 mm for capacitor A; the central and outer radii of the electrode R = 1.5 mm and R_out = 2.7 mm, and initial volume of the air cavity V₀ = 1.15 mm³ for capacitor B; and (for both capacitors A and B) initial separation h = 50 µm; thickness t_PLGA = 5 µm, t_Zn = 2 µm, and t_Mg = 50 µm; elastic modulus E_PLGA = 1.6 GPa, E_Zn = 100 GPa, and E_Mg = 45 GPa; and Poisson’s ratio v_PLGA = 0.34, v_Zn = 0.25, and v_Mg = 0.28.

For capacitor A, as shown in Fig. 3(a), the average deflection of the electrode $w¯$ given by the exact solution (Eq. (5)) is very close (∼4% difference) to the linear $w¯$∼p relation (Eq. (6)) for p < 30 mmHg, which covers the pressure range required for in vivo measurements, such as the intracranial pressure < 10 mmHg [2], pulmonary artery pressure < 25 mmHg [11], and intracompartmental pressure < 30 mmHg [12]. The deformation resistance in Eq. (6) due to bending stiffness of the electrode 192D/R⁴ = 0.013 N/mm³ is negligible as compared to that due to pressure increase in the air cavity p₀/h = 2.026 N/mm³. For a wider range of radius R (1–3 mm) and thickness h (25–100 µm) reported for bio-implantable pressure sensors [6–8], the gas resistance term p₀/h always dominates (>15 times of) the bending resistance 192D/R⁴ (Fig. 3(b)).

Therefore, the traditional strategy to increase the sensitivity by reducing initial separation h (consequently reducing the bending resistance 192D/R⁴) does not work for the bio-implantable wireless pressure sensors discussed here, as reducing h actually increases the gas resistance term p₀/h, leading to the slightly reduced sensitivity. As to be shown in the next section for capacitor B, one strategy to increase the sensitivity is to reduce the gas resistance by increasing the initial volume of the air cavity V₀.

Figure 5(a) shows that the average deflection of the electrode $w¯$ is essentially linear with respect to the pressure p for capacitor B; therefore, the linear $w¯∼p$ relation in Eq. (19) holds. For the baseline values given at the start of Sec. 3, the deformation resistance in Eq. (19) due to bending stiffness of the electrode 192D/λR⁴ = 0.039 N/mm³ is ∼3 times that of capacitor A but is still negligible as compared to that due to the pressure increase in the air cavity $p0/V0/πRout2$ = 2.016 N/mm³. The slopes of the lines in Fig. 5(a) increase with V₀ because the gas resistance term $p0/V0/πRout2$ decreases. Different from capacitor A, the sensitivity of capacitor B is directly related to $w¯center$ in Eq. (21), instead of the average deflection of the electrode $w¯$⁠. Figures 5(a) and 5(b) show that the average deflection of the central electrode $w¯center$ is also essentially linear with respect to the pressure p. As $w¯center$ is linearly proportional to $w¯$ (Eq. (21)), increasing V₀ reduces the gas resistance term $p0/V0/πRout2$⁠, therefore improving sensitivity. For a given pressure p, the average deflection of the entire electrode $w¯$ is independent of D_out/D (consistent with Eq. (18)). However, the average deflection of the central electrode $w¯center$ increases with D_out/D, yielding improved sensitivity of capacitor B. The baselines give D_out/D = 1539. For the limit of D_out/D → ∞, $w¯center$ is the same as Eq. (6) for capacitor A by replacing h with a larger value V₀/πR², i.e.,

Analytical models of electrode deformation for wireless and bioresorbable capacitive pressure sensors have been developed to improve the sensitivity of these pressure sensors. Two capacitors, one with a conventional air cavity and the other with an expanded air cavity, are studied via the theory of elastic composite plate. It is shown, due to the compact design, the increase of air pressure in the sealed air cavity overwhelms the bending stiffness of the electrode to become the main resistance to deflection, therefore dominating the sensitivity of the capacitors. The traditional strategy of increasing sensitivity by reducing the initial separation between electrodes is not suitable for these wireless and bioresorbable capacitive pressure sensors. Instead, increasing the initial volume of the air cavity provides an effective way to increase the device sensitivity.

This work was supported by the National Natural Science Foundation of China (Grant No. 52005331).

There are no conflicts of interest.

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Continuity of shear force Q_r at r = R gives C = 0. The other continuity conditions at r = R give the constants A, B, and F in Eqs. (11)–(13).