## Abstract

Bio-implantable pressure sensors are of great significance for many life-threatening clinical applications that require real-time monitoring of the internal pressure of the human body. Wireless and bioresorbable capacitive pressure sensors overcome the shortcomings of the traditional, resistance-based pressure sensors (e.g., leading to infections and restrictions of natural body motion movements), but they have low sensitivity. One effective way to improve the sensitivity is to increase the volume of the dielectric (air) cavity. Analytic models are established in this paper for the deformation of the electrodes in the wireless and bioresorbable capacitive pressure sensor, and the models show explicitly the sensitivity dependence on the sensor geometry and material properties. The models show that the pressure increase in the air cavity overwhelms the bending stiffness of the electrodes, therefore dominating the deflection of the electrode in capacitors. The traditional strategy to reduce the initial separation between electrodes is not suitable. Instead, increasing the initial volume of the air cavity provides an effective strategy to improve the sensitivity of the bio-implantable wireless pressure sensors.

## 1 Introduction

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 [13]. 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 [68]. 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 [68].

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.

Fig. 1
Fig. 1
Close modal

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 Rout2 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 Rout = 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 < Rout, where Rout = 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.

Fig. 2
Fig. 2
Close modal

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.

## 2 Analytical Model

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.

### 2.1 Capacitor A.

The air cavity has the initial pressure p0 and the initial volume of πR2h (Fig. 2a). For the electrode deformed by $w¯$ due to the applied pressure, the volume of the air cavity decreases to $πR2(h−w¯)$ (Fig. 2c), and the internal pressure increases to p0 + Δp. The ideal gas law at a constant temperature requires $p0πR2h=(p0+Δpp)πR2(h−w¯)$, which gives the increase of the air pressure in the cavity
$Δp=p0w¯h−w¯$
(1)
The electrode is modeled as an elastic thin plate, and its axisymmetric deformation w satisfies [10]
$D(d2dr2+1rddr)(d2dr2+1rddr)w=p−Δp$
(2)
in the polar coordinate r, where D is the bending stiffness of the electrode, and is given in Appendix  A for a three-layer composite plate. The boundary conditions are w = 0 and dw/dr = 0 at the clamped edge r = R, where R is the radius of the electrode. The solution to the above equation is
$w=p−Δp64D(R2−r2)2$
(3)
Its average is
$w¯=2R2∫0Rwrdr=R4192D(p−Δp)$
which, together with Δp in Eq. (1), gives the equation for $w¯$
$w¯=R4192D(p−p0w¯h−w¯)$
(4)
Its solution is
$w¯=2p192DR4+p+p0h+(192DR4+p+p0h)2−4192DR4ph$
(5)
This nonlinear dependence of $w¯$ on p degenerates to the following linear relation at the small applied pressure
$w¯=p192DR4+p0h$
(6)
where 192D/R4 and p0/h represent the deformation resistance due to the bending stiffness of the electrode and the pressure increase in the air cavity, respectively.
The normalized capacitance change ΔC/C0 is related to the average deflection of the electrode of capacitor A by
$ΔCC0=w¯h−w¯=1192DhpR4+p0p−1$
(7)

### 2.2 Capacitor B.

Let p0 and V0 denote the initial pressure and volume of the air cavity in capacitor B, respectively, and $w¯=2/Rout2∫0Routwrdr$ the average deflection over the entire area 0 < r < Rout of the top electrode. For the electrode deformed by $w¯$ due to the applied pressure, the volume of the air cavity decreases to $V0−πRout2w¯$. For the internal pressure increasing to p0 + Δp, ideal gas law at a constant temperature requires $p0V0=(p0+Δp)(V0−πRout2w¯)$, which gives the increase in the air pressure in the cavity
$Δp=p0w¯V0πRout2−w¯$
(8)

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 Dout the bending stiffness of Mg plate for the outer part (R < r < Rout) of the electrode (D and Dout are given in Appendix  A).

The solution of the equilibrium equation Eq. (2) for 0 < r < R is
$w=p−Δp64Dr4+14Ar2+B$
(9)
where A and B are constants to be determined. For R < r < Rout, Eq. (2) is modified by replacing the bending stiffness D with Dout, and its solution satisfying the boundary conditions w = 0 and dw/dr = 0 at the radius r = Rout is
$w=p−Δp64Dout(r4−Rout4−4Rout4lnrRout)+F4(r2−Rout2−2Rout2lnrRout)+C4(r2lnr−Rout2lnRout−r2+Rout2−2Rout2lnRoutlnrRout+Rout2lnrRout)$
(10)
where F and C are constants to be determined. Continuity at r = R of the deflection w|r=R−0 = w|r=R+0, slope $dwdr|r=R−0=dwdr|r=R+0$, bending moment Mrr|r=R−0 = Mrr|r=R+0 and shear force Qr|r=R−0 = Qr|r=R+0 gives A, B, C and F as (Appendix  B)
$A=−[DDout(1+ν¯+2f−1)−νout](1−f)+(1+f)[DDout(1+ν¯)−νout](1−f)+(1+f)p−Δp8DR2$
(11)
$B=p−Δp64DR4+p−Δp64DoutRout4(1−f2−2lnf)−F4Rout2lnf$
(12)
$F=−[DDout(1+ν¯)−νout](1−f2)+(1+f2)[DDout(1+ν¯)−νout](1−f)+(1+f)p−Δp8DoutRout2$
(13)
and C = 0, where $f=R2/Rout2$, and $ν¯$ and νout are the effective Poisson’s ratios of the central part (given in Appendix  B) and the outer part (Mg), respectively.
The average deflection over the entire electrode 0 < r < Rout is
$w¯=2Rout2∫0Routwrdr=2Rout2(∫0Rwrdr+∫RRoutwrdr)$
(14)
Substitution of Eqs. (9) and (10) in the equation earlier, together with Eqs. (11)(13), give
$w¯=p−Δp192DλR4$
(15)
where
$λ=f+(1−f)Df2Dout{4+f+f2−3[DDout(1+ν¯)−νout](1−f2)+(1+f2)[DDout(1+ν¯)−νout](1−f)+(1+f)}$
(16)
Substitution of Eqs. (8) into (15) yields the equation for $w¯$ as
$w¯=λR4192D(p−p0w¯V0πRout2−w¯)$
(17)
Its solution is
$w¯=2p192DλR4+p+p0V0πRout2+(192DλR4+p+p0V0πRout2)2−4192DλR4pV0πRout2$
(18)
This nonlinear dependence of $w¯$ on p degenerates to the following linear relation at the small applied pressure
$w¯=p192DλR4+p0V0πRout2$
(19)
The normalized capacitance change of capacitor B shown in the Sec. 3 is related to the average deflection $w¯center=2/R2∫0Rwrdr$ of the central part 0 < r < R of the electrode, by
$ΔCC0=w¯centerh−w¯center$
(20)
and $w¯center$ is obtained as
$w¯center=w¯λ+w¯λ3Df2Dout{[DDout(1+ν¯)−νout](1−f2)+(1+f2)[DDout(1+ν¯)−νout](1−f)+(1+f)(1−f+2lnf)−2lnf}$
(21)
For small applied pressure p, it is linearly proportional to p
$w¯center=p192DR4+λp0V0πRout21+3Df2Dout{[DDout(1+ν¯)−νout](1−f2)+(1+f2)[DDout(1+ν¯)−νout](1−f)+(1+f)(1−f+2lnf)−2lnf}$
(22)

## 3 Results and Discussion

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 Rout = 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 tPLGA = 5 µm, tZn = 2 µm, and tMg = 50 µm; elastic modulus EPLGA = 1.6 GPa, EZn = 100 GPa, and EMg = 45 GPa; and Poisson’s ratio vPLGA = 0.34, vZn = 0.25, and vMg = 0.28.

### 3.1 Capacitor A.

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 [68], the gas resistance term p0/h always dominates (>15 times of) the bending resistance 192D/R4 (Fig. 3(b)).

Fig. 3
Fig. 3
Close modal
For capacitor A, Fig. 4 shows that the normalized capacitance change ΔC/C0 (Eq. (7)) is approximately linear with respect to pressure p (<30 mmHg), but is relatively insensitive to the initial separation h for a large range from 50 µm to 1000 µm such that Eq. (7) becomes
$ΔCC0≈pp0$
(23)
Fig. 4
Fig. 4
Close modal

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.

### 3.2 Capacitor B.

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 Dout/D (consistent with Eq. (18)). However, the average deflection of the central electrode $w¯center$ increases with Dout/D, yielding improved sensitivity of capacitor B. The baselines give Dout/D = 1539. For the limit of Dout/D → ∞, $w¯center$ is the same as Eq. (6) for capacitor A by replacing h with a larger value V0/πR2, i.e.,

$w¯center=p192DR4+p0V0πR2$
(24)
which yields the maximum sensitivity. The normalized capacitance change ΔC/C0 then becomes
$ΔCC0=w¯centerh−w¯center=1192DhpR4+p0phV0πR2−1$
(25)
Fig. 5
Fig. 5
Close modal
For capacitor B, ΔC/C0 is also linearly proportional to p, and the capacitor sensitivity increases ∼3.5 times as compared to that of capacitor A (Fig. 6), as the initial volume of the air cavity increases ∼3.3 times. Equation (23) then becomes
$ΔCC0≈pp0V0πR2h$
(26)
when 192D/R4p0πR2/V0. It is clear that the sensitivity (slope of $ΔC/C0∼p$) is linear with V0; therefore, increasing V0 is an effective strategy to improve sensitivity. However, the sensitivity saturates as V0 increases such that p0πR2/V0 becomes negligible as compared to 192D/R4
Fig. 6
Fig. 6
Close modal

## 4 Conclusions

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.

## Footnote

2

The actual shape of the air cavity in the plane was rectangular with the size of 3.9 mm × 5.9 mm and it is modeled as a circle of radius for simplicity.

## Acknowledgment

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

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

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

### Appendix A: Bending Stiffness of the Electrode

For a three-layer symmetric composite plate with Zn sandwiched by two PLGA layers, the bending moment Mrr is related to stress σrr by
$Mrr=∫−tPLGA−12tZntPLGA+12tZnσrrzdz=2∫0tPLGA+12tZnσrrzdz$
(A1)
where the integration is along the thickness t of each layer, and tPLGA is the thickness of each PLGA layer. Substitution of stress in terms of bending curvatures κrr and κθθ gives
$Mrr=2∫012tZnEZn1−νZn2(κrr+νZnκθθ)z2dz+2∫12tZntPLGA+12tZnEPLGA1−νPLGA2(κrr+νPLGAκθθ)z2dz$
(A2)
where E and ν are the elastic modulus and Possion’s ratio of each layer, respectively. The integration above yields
$Mrr=D(κrr+ν¯κθθ)$
(A3)
where the effective bending stiffness is
$D=EZntZn312(1−νZn2)+EPLGA12(1−νPLGA2)[(2tPLGA+tZn)3−tZn3]$
(A4)
and effective Poisson’s ratio is
$ν¯=νZnEZntZn312(1−νZn2)+νPLGAEPLGA12(1−νPLGA2)[(2tPLGA+tZn)3−tZn3]EZntZn312(1−νZn2)+EPLGA12(1−νPLGA2)[(2tPLGA+tZn)3−tZn3]$
(A5)
The bending stiffness of the outer electrode is dominated by a single layer of Mg and is given by
$Dout=EMgtMg312(1−νMg2)$
(A6)

### Appendix B: Deflection of Electrodes for Capacitor B

For the radius 0 < r < R, the moment Mrr is
$Mrr=Dκrr+ν¯Dκθθ=D(p−ΔpD316r2+A2)+ν¯D(p−Δp16Dr2+A2)$
(B1)
The shear force Qr is
$Qr=Dddr(d2dr2+1rddr)w=(p−Δp)r2$
(B2)
For the radius R < r < Rout, the moment Mrr is
$Mrr=Doutκrr+νoutDoutκθθ=Dout[p−Δp16Dout(3r2+Rout4r2)+C4(2lnr+1+2Rout2lnRoutr2−Rout2r2)+F2(1+Rout2r2)]+νoutDout[p−Δp16Dout(r2−Rout4r2)+C4(2lnr−2Rout2lnRoutr2−1+Rout2r2)+12F(1−Rout2r2)]$
(B3)
The shear force Qr is
$Qr=Doutddr(d2dr2+1rddr)w=Dout(p−Δp2Doutr+Cr)$
(B4)

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

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