Path loss modeling and verification of implantable human-body communication based on a point source field

Abstract

BACKGROUND:

Deep brain stimulation is a method of nerve regulation that uses human body conductance characteristics; signals are diffused and transmitted through human tissues to regulate diseases.

OBJECTIVE:

We analyzed the distribution and the transmission mechanism of implantable electrical signals from a point source field in the human body using the point source field implanting channel model.

METHODS:

The model was established using a mathematical modeling method, with reasonable boundary conditions and assumptions. Further, we established an equivalent numerical solution model to verify its accuracy and compared the model to published experimental results to evaluate its consistency.

RESULTS:

The analysis results of the two models revealed that both had errors of $<$ 1%. A comparison of the experimental results to published experimental data revealed that the error between the two was $<$ 4 dB, and the model displayed excellent consistency.

CONCLUSIONS:

Our proposed model is accurate and exhibits good consistency with published experimental results. We therefore conclude that the proposed model is reliable.

Keywords

Deep brain stimulation body conduction communication implantable point source field

1. Introduction

With the development of integrated circuits and the microelectronics industry, implanted sensors are increasingly being used in the human body. According to their functions, these electronic devices can be divided into two categories: (1) devices that monitor or record the physiological or biochemical parameters of normal or abnormal living organisms [22], such as implantable blood glucose meters and capsule gastroscopes; (2) auxiliary devices that transmit signals to organisms or the human body to control or improve the function of abnormal biological tissues [2]. Examples of this second category include nerve stimulators [3, 4] and implantable cochleae [5, 6]. These devices ensure continuous real-time measurement and control of various physiological and biochemical parameters under natural physiological conditions.

Using an implantable measuring device means that information about the human body can be measured without contacting the skin. This, in turn, causes interference factors to be greatly reduced, allowing a higher signal-to-noise ratio (SNR) for the obtained signal [7]. Furthermore, when used in the regulation of organs and tissues, this technique can obtain ideal stimulation and control responses, which is conducive to monitoring the recovery of injuries, and in disease control [8]. Therefore, the development of implantable electronic devices will greatly aid advances in modern medicine, especially nerve stimulation.

To understand the functions of an implantable medical electronic device, it must be capable of communicating with the outside world. There are two common communication methods; namely, wired communication [9] and wireless communication. In wired communication, an implanted device is connected to external monitors with communication cables to transmit signals. The communication cable is required to penetrate the biological tissue; this can easily lead to tissue infections and complications. When the organism or human body in question move, the communication connection wire generates noise, thereby reducing the SNR. Wireless communication commonly uses electromagnetic coupling communication [22], radio frequency communication [10, 11], and intra-body communication [12]. Electromagnetic coupling and radio frequency communication both have a high communication frequency owing to their wide bandwidth, and thus have high communication rates. However, in a high-frequency environment, human tissues have a strong ohmic loss effect on the signal, thus resulting in very large signal attenuation. Strong high-frequency fields can also cause damage to organisms and/or human tissues, and this type of communication also requires a larger space to implant communication coils and antennas than galvanic coupling human-body communication. The application of these two methods has therefore been greatly limited, especially in implantable brain devices, such as those used for deep nerve regulation.

In galvanic coupling intra-body communication, human tissues are used as signal transmission media [13]. This avoids any interference caused by complex connecting lines. Furthermore, as the signal is directly coupled with the human body, no space is needed to accommodate the communication coil or the communication antenna. The frequency of galvanic coupling intra-body communication is also below 1 MHz, meaning that its electric field is very low and its impact on the organism is minimal. This means that this method has few restrictions in terms of the placement of the implanted device.

In this study, the transmission mechanism of a point source field signal in the human body channel was investigated using an implantable signal transmission channel model. The model, which is based on a point source field, was established through mathematical modeling. The accuracy of the model solution was tested by solving its results (two model calculation results) under the same boundary conditions. Previously published experimental data were also used to evaluate its validity [14].

2. Methods

According to the volume conductor theory, first, we simplified the arms and legs of the study area into a multi-layer cylinder with a length of $h$ and a maximum radius of $r_{N}$ . A pair of spot electrodes were placed inside the cylinder to act as an implantable device. Their coordinates were $({r_{0},\theta_{0},z_{0}})$ (positive electrode position) and $({r_{0},\theta_{0}+\pi,z_{0}})$ (negative electrode position). The radii of the cylinder’s different layers from inside to outside were $({r_{1},r_{2},r_{3},\cdots,r_{N-1},r_{N}})$ , where $r_{1}$ represents the innermost radius and $r_{N}$ represents the outermost radius (Fig. 1). The electrical characteristics of the research object’s tissues were considered to be isotropic, and the conductivity and relative permittivity in the corresponding layers were $({\sigma_{1},\sigma_{2},\sigma_{3},\cdots,\sigma_{N-1},\sigma_{N}})$ and $({\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\cdots,\varepsilon_{N-1},% \varepsilon_{N}})$ , respectively.

Figure 1.

Simplified implantable multi-layer volume conductor model.

According to the quasi-static electromagnetic field theory [15], the electric field vector, $\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}\over{% E}}$ , can be expressed as the gradient of a scalar potential, as shown in Eq. (1). According to Ohm’s law, the current distribution, $\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}\over{% J}}$ , can be obtained, and thus, Eq. (2) can be derived:

$\displaystyle\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{E}}=-\nabla\varphi(R)$ (1) $\displaystyle\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{J}}_{0}=\tilde{\sigma}\mathord{\buildrel\lower 3.0pt% \hbox{$\scriptscriptstyle\rightharpoonup$}\over{E}}$ (2)

where $\tilde{\sigma}$ denotes the complex conductivity of the medium. In a quasi-static electromagnetic field, the field’s current distribution is a superposition of the induced current and the applied current, such that Eq. (3) can be obtained:

$\displaystyle\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{J}}=\mathord{\buildrel\lower 3.0pt\hbox{$% \scriptscriptstyle\rightharpoonup$}\over{J}}_{0}+\mathord{\buildrel\lower 3.0% pt\hbox{$\scriptscriptstyle\rightharpoonup$}\over{J}}_{\textit{impressed}}$ (3)

In a quasi-static electric field, $\nabla\cdot\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{J}}=0$ must be satisfied; thus, Eq. (4) can be derived:

$\displaystyle\nabla\cdot({\nabla\varphi(R)})=\frac{\mathord{\buildrel\lower 3.% 0pt\hbox{$\scriptscriptstyle\rightharpoonup$}\over{J}}_{\textit{impressed}}}{\sigma}$ (4)

where $\nabla\cdot\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{J}}_{\textit{impressed}}=\frac{I}{4\pi}$ .

According to Green’s function of the vector field, the following equation can be derived in isotropic uniform volume conductors:

$\displaystyle\frac{1}{r}\frac{\partial}{\partial}\left({r\frac{\partial\varphi% }{\partial r}}\right)+\frac{1}{r^{2}}\frac{\partial^{2}\varphi}{\partial\theta% ^{2}}+\frac{\partial^{2}\varphi}{\partial z^{2}}=-\frac{I}{4\pi\tilde{\sigma}}% \delta({r-r_{0}})\delta({\theta-\theta_{0}})\delta({z-z_{0}})$ (5)

According to the Fourier series expansion of the $\delta$ function, the following equations can also be derived:

$\displaystyle\left\{{\begin{array}[]{l}\frac{1}{r}\frac{\partial}{\partial}% \left({r\frac{\partial\aleph_{sn}}{\partial r}}\right)-\left({\frac{n^{2}}{r^{% 2}}+\frac{m\pi^{2}}{h^{2}}}\right)\aleph_{sn}=-\frac{1}{\tilde{\sigma}}\delta(% {r-r_{0}})\\ \\ \delta({\theta-\theta_{0}})=\frac{1}{2\pi}\sum\nolimits_{n=1}^{\infty}{e^{jn({% \theta-\theta_{0}})}}\\ \\ \delta({z-z_{0}})=\frac{2}{h}\sum\nolimits_{n=1}^{\infty}{\sin\left(\frac{m\pi z% }{h}\right)\sin\left(\frac{m\pi z_{0}}{h}\right)}\\ \end{array}}\right.$ (6)

where $\aleph_{sn}$ is the radial function and $h$ is the volume conductor length ( $h\approx 2$ ).

2.1 Boundary conditions

On the basis of the volume conductor theory, in order to solve the model, the boundary conditions of a quasi-static volume conductor need to be satisfied [16, 17]:

$\displaystyle\left\{{\begin{array}[]{l}\nabla^{2}\aleph_{sn}({r|{r_{0}}})=0\\ \\ \frac{\partial\varphi({r,\theta,z})}{\partial r}|_{r=r_{N}}=0\\ \\ \varphi_{s}({r_{s}^{+},z})=\varphi_{s}({r_{s}^{-},z})\\ \\ \mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}\over{% J}}_{s}({r_{s}^{+},z})=\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle% \rightharpoonup$}\over{J}}_{s}({r_{s}^{-},z})\\ \end{array}}\right.$ (7)

In solving the model, it was considered that the potential at both ends of a volume conductor at infinite distance approach zero. We assumed the following:

$\displaystyle\left\{{\begin{array}[]{l}\varphi\left({r,\theta,z}\right)|_{z=0}% =0\\ \varphi\left({r,\theta,z}\right)|_{z=h}=0\\ \end{array}}\right.$ (8)

2.2 Potential distribution model

According to the governing equation and the boundary conditions of the point source field shown in Eqs (4)–(8), we can obtain the potential distribution model of galvanic coupling intra-body communication, based on the point source field, using a mathematical solution method:

$\displaystyle\varphi_{s}({r,\theta,z})=\frac{I}{2\pi^{2}h}\sum\nolimits_{n=1}^% {\infty}\sum\nolimits_{s=1}^{N}\sum\nolimits_{m=1}^{\infty}[{e^{jn({\theta-% \theta_{0}})}}]\left[{\sin\left(\frac{m\pi z}{h}\right)\sin\left(\frac{m\pi z_% {0}}{h}\right)}\right]\left[A_{sn}I_{n}\left({\frac{m\pi r_{s}}{h}}\right)+B_{% sn}K_{n}\left({\frac{m\pi r_{s}}{h}}\right)+\frac{1}{\tilde{\sigma}_{\textit{% source}}}\Im\left(\frac{m\pi r_{s}}{h}\right)\right]$ (9)

Where

$\displaystyle\Im\left({\frac{m\pi r_{s}}{h}}\right)=\left\{{\begin{array}[]{ll% }K_{n}\left({\frac{m\pi r_{0}}{h}}\right)I_{n}\left({\frac{m\pi r_{s}}{h}}% \right)&0<r_{s}\leqslant r_{0}\\ \\ I_{n}\left({\frac{m\pi r_{0}}{h}}\right)K_{n}\left({\frac{m\pi r_{s}}{h}}% \right)&r_{0}<r_{s}\leqslant r_{N}\\ \end{array}}\right.$ (10)

where $I$ denotes the externally applied current intensity ( $A$ ).

2.3 Channel gain model

We analyzed the attenuation characteristics of the channel through channel gain. This, in turn, enabled us to analyze the transmission characteristics of signals in the channel:

$\displaystyle G({r,\theta,z})|_{dB}=20\log_{10}\left({\frac{\varphi_{RX}({r,% \theta,z})}{\varphi_{TX}({r,\theta,z})}}\right)$ (11)

where $\varphi_{RX}({r,\theta,z})$ represents the coupling potential at the receiving end, and $\varphi_{TX}({r,\theta,z})$ represents the coupling potential at the transmitting end.

2.4 Model verification and performance

We designed a numerical solution to verify the accuracy of the analytical solution and used data found in previous studies to verify the performance of the model.

To verify the accuracy of the model solution, we used numerical simulation software (COMSOL Multiphysics 5.3) to establish a numerical solution verification model. The tissue’s electrical parameters were selected from human tissue (skeleton, muscle, fat, and skin) at 100 kHz [18, 19, 20]. The geometric parameters are listed in Table 1. Both models were assumed to have the same boundary conditions. We compared the channel attenuation in the frequency domain across different distances.

Table 1
Geometric parameters of the model

Subject	Source (m)	Bone (m)	Muscle (m)	Fat (m)	Skin (m)
r	24e ${}^{-3}$	20e ${}^{-3}$	26e ${}^{-3}$	29e ${}^{-3}$	32e ${}^{-3}$

To verify the consistency of the models, we selected the data given in [14] for verification. We obtained the parameters of the two samples in the paper through proportional scaling estimation, as shown in Table 2.

Table 2

Sample parameter estimation

Sample	Tissue	r (m)
		$d=$ 40 mm	$d=$ 50 mm	$d=$ 60 mm
1	Bone	26.63e-3	26.63e-3	26.63e-3
	Muscle	41.85e-3	40.85e-3	39.03e-3
	Fat	46.89e-3	45.41e-3	43.65e-3
	Skin	49.10e-3	47.43e-3	45.76e-3
	Source	35.80e-3	34.80e-3	33.83e-3
2	Bone	26.73e-3	26.73e-3	26.73e-3
	Muscle	42.455e-3	41.25e-3	40.05e-3
	Fat	46.18e-3	44.87e-3	43.51e-3
	Skin	48.38e-3	46.95e-3	45.51e-3
	Source	36.00e-3	35.00e-3	34.00e-3

3. Experimental verification and analysis of results

3.1 Preliminary verification

The results of the preliminary verification are shown in Fig. 2.

Figure 2.

Verification analysis of model solution.

Figure 3.

In vitro verification experiment [15].

Figure 4.

Results at distances of (a) 40 mm, (b) 50 mm, and (c) 60 mm.

Figure 2 shows that the solutions of the two models exhibited good consistency, and the model errors were both below 1%. Both solutions of the models can be considered accurate.

3.2 In vitro verification

In vitro experiment, a network analyzer (4395A, Agilent Technologies, Santa Clara, CA, USA) and a differential probe (1141A, Agilent Technologies, Santa Clara, CA, USA; 1142A, Agilent Technologies, Santa Clara, CA, USA) were used. The network analyzer was used to measure the signal attenuation from the point source to the surface electrodes and the differential probe was used to break the common ground loop between the transmitter and the receiver of the network analyzer. The experimental layout is shown in Fig. 3.

In the in vitro experiment, two pig legs (from pigs that died within five hours of the experiment) are randomly chosen as experimental samples. Because of the inter-individual difference, the geometric parameters of the samples changed accordingly; at the time of measurement, we recorded the perimeter of the signal source (samples) and the positions of all detection points [14]. Prior to the experiment, the bristles on the legs’ surfaces were removed. The legs were then scrubbed with cotton soaked in alcohol multiple times to remove dust and dead skin, to improve the electrodes’ adhesion to the skin surface.

To avoid the effect of leg joints on signal transmission, the experiment was performed between the first joint and the second joint (the distance was approximately 140 mm as shown in Fig. 3). At a point 10 mm away from the first joint, we inserted silver electrodes with an exposed needle into the pig leg as the transmitting electrodes (TX). This pair of electrodes were symmetrical in the cylinder formed by the skeleton. The exposed portion of the electrode needle had a maximum radius of 1 mm and a length of 1 mm; its volume was 1/3 $\times$ 3.14 $\times$ 0.5 $\times$ 0.5 $\times$ 1 $=$ 0.262 mm ${}^{3}$ . The implantation depth was $k$ ( $k=$ 40 mm), and the distance from the transmitting electrode (TX) was $d$ ( $d=$ 60, 50, and 40 mm). The receiving electrodes (RX) were a pair of 4 $\times$ 4 cm AgCl/AgCI electrodes; they were symmetrical in the cylinder along the skin surface. In the experiments, the receiving electrode (RX) was implemented using a fixed surface communication distance; the implantable electrode’s transmission was changed according to the depth of the electrode. Changes in the fixed electrode’s implantable depth allow the electrode’s distance to be received at the surface. This was done to analyze the attenuation of the signal in the channel. Before measurement, the instruments were connected according to the layout shown in Fig. 3.

The model calculation results and the experimental results are shown in Fig. 4a–c.

The model’s results were highly consistent with the experimental results. In the frequency domain, the error between the modeled and the experimental results was below 4 dB, implying that the model is feasible.

4. Discussion

At present, the verification of the model is based on published pig feet data, but there would likely be differences between this medium and the human body. However, considering that much of the technology is still in the research stage, according to the medical research criteria, it is reasonable and feasible to use animals for preliminary verification. Future research should consider the design of non-invasive implantable intra-body communication systems for experimental verification.

In building the model, we assumed that human tissues were isotropic for simplification. Considering that there are some differences in the transverse and tangential directions of human tissues due to fiber growth, we will continue to consider the tissue characteristics during future research.

5. Conclusions

In this study, a mathematical modeling method was used to analyze the distribution and transmission mechanisms of electrical signals based on the point source field in the human body. This enabled the creation of an implantable channel model based on the point source field. An equivalent numerical solution model showed that the interpretation of our model was accurate, and comparison with previously published experimental data further verified the model’s accuracy. The model’s solutions and experimental results had excellent consistency; thus, the proposed model can be considered feasible.

Footnotes

Acknowledgments

This work was supported in part by the Leading Talent Training Project of Neijiang Normal University under Grant 2017 [Liu Yi-He]; in part by the Innovative Team Program of the Neijiang Normal University under Grant 17TD03; in part by the Foundation of Ph. D. Scientific Research of the Neijiang Normal University under Grant RSC201704; in part by the Sichuan Science and Technology Program under Grant 2019YJ0181; in part by the Sichuan Province Academic and Technical Leader Training Funded projects under Grant 13XSJS002; in part by the National key research and development program of China under Grant 2016YFC0100800 and Grant 2016YFC0100802; and in part by the Foundation of Ph. D. Scientific Research of Neijiang Normal University under Grant 2019 [Zhang Shuang] and Grant 2019 [Wang Jiujiang].

Conflict of interest

None to report.

Appendix

In the cylindrical volume conductor, when the electrode center $r_{0}\neq r_{N}$ , meaning that the electrode is not located on the conductor surface, the cylindrical conductor is zoned by the electrode based on $r=r_{0}$ . Therefore, the radial function $\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})$ satisfies the following homogeneous equation:

(A1) $\displaystyle\nabla^{2}\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})=0$

The general solution of the modified Bessel function in equation (A1) may be expressed as

(A2) $\displaystyle\nabla^{2}\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})=\left% \{{\begin{array}[]{ll}A_{sn}^{(+)}I_{n}({kr^{+}})+B_{sn}^{(+)}K_{n}({kr^{+}}),% &0<r\leqslant r_{0}\\ A_{sn}^{(-)}I_{n}({kr^{-}})+B_{sn}^{(-)}K_{n}({kr^{-}}),&r_{0}<r\leqslant r_{N% }\\ \end{array}}\right.$

where $k=\frac{m\pi}{h}$ and $A_{sn}^{(+)}$ , $B_{sn}^{(+)}$ , $A_{sn}^{(-)}$ , $B_{sn}^{(-)}$ denote the coefficients of the zones $I_{n}({kr})$ and $K_{n}({kr})$ divided by $r=r_{0}$ , respectively.

In the source points in which the electrodes are located, the potential and current around the electrodes meet the boundary conditions of source points presented in [22]; therefore, the radial function satisfies

(A3) $\displaystyle\left\{{\begin{array}[]{l}\aleph_{sn}({r,\theta,z|r_{0},\theta_{0% },z_{0}})|_{r=r_{0}^{+}}=\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})|_{r=% r_{0}^{-}}\\ \frac{\partial\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})}{\partial r}|_{% r=r_{0}^{+}}-\frac{\partial\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})}{% \partial r}|_{r=r_{0}^{-}}=-\frac{1}{r_{0}\tilde{\sigma}_{\textit{source}}}\\ \end{array}}\right.$

Hence, Eq. (A3) can be derived:

(A4) $\displaystyle\left\{{\begin{array}[]{l}A_{sn}^{(+)}I_{n}({kr^{+}})+B_{sn}^{(+)% }K_{n}({kr^{+}})=A_{sn}^{(-)}I_{n}({kr^{-}})+B_{sn}^{(-)}K_{n}({kr^{-}})\\ \left[A_{sn}^{(+)}I^{\prime}_{n}(kr^{+})+B_{sn}^{(+)}K^{\prime}_{n}(kr^{+})% \right]-\left[A_{sn}^{(-)}I^{\prime}_{n}({kr^{-}})+B_{sn}^{(-)}K^{\prime}_{n}(% {kr^{-}})\right]=-\frac{1}{kr_{0}\tilde{\sigma}_{\textit{source}}}\\ \end{array}}\right.$

Meanwhile, when $r_{0}=r_{s^{\ast}}$ , $s^{\ast}\in s=1,2,\cdots,N-1,N$ shows that the electrode is located at the function between the $k$ -th layer and the $k$ -th layer of the $k+1$ -th volume conductor. Because $K_{n}(x){I}^{\prime}_{n}(x)-{K}^{\prime}_{n}(x)I_{n}(x)=\frac{1}{x}$ , the following can be derived:

(A5) $\displaystyle\aleph_{sn}({r,\theta,z|r_{0},\theta_{0},z_{0}})=A_{sn}I_{n}\left% ({\frac{m\pi r_{s}}{h}}\right)+B_{sn}K_{n}\left({\frac{m\pi r_{s}}{h}}\right){% }+\frac{1}{\tilde{\sigma}_{\textit{source}}}\left\{{\begin{array}[]{ll}K_{n}% \left({\frac{m\pi r_{0}}{h}}\right)I_{n}\left({\frac{m\pi r_{s}}{h}}\right)&0<% r_{s}\leqslant r_{0}\\ I_{n}\left({\frac{m\pi r_{0}}{h}}\right)K_{n}\left({\frac{m\pi r_{s}}{h}}% \right)&r_{0}<r_{s}\leqslant r_{N}\\ \end{array}}\right.$

where $k=\frac{m\pi}{h}$ , and $A_{sn}$ and $B_{sn}$ denote the coefficients of the zones $I_{n}({kr_{s}})$ and $K_{n}({kr_{s}})$ , respectively, divided by $r=r_{0}$ .

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Path loss modeling and verification of implantable human-body communication based on a point source field

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSIONS:

Keywords

1. Introduction

2. Methods

Table 1 Geometric parameters of the model

3.1 Preliminary verification

4. Discussion

5. Conclusions

Footnotes

Acknowledgments

Conflict of interest

Appendix

References

Table 1
Geometric parameters of the model