# Wireless Powering Efficiency Assessment for Deep-Body Implantable Devices

Icaro V. Soares, , Mingxiang Gao, Zvonimir Sipus, Anja K. Skrivervik,
and Denys Nikolayev This paper is an expanded version from the 2021 IEEE
Wireless Power Transfer Conference. This work was supported in part by
l’Agence Nationale de la Recherche (ANR, project ANR-21-CE19-0045-01) and in
part by the French region of Brittany through the SAD “EM-NEURO” project.
(Corresponding author: D. Nikolayev, denys.nikolayev@deniq.com)I. V. Soares
and D. Nikolayev are with the Univ Rennes, CNRS, IETR – UMR 6164, FR-35000
Rennes, France.M. Gao and A. K. Skrivervik are with the Microwave and Antenna
Group, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne,
Switzerland.Z. Sipus is with the Faculty of Electrical Engineering and
Computing, University of Zagreb, 10000 Zagreb, Croatia.

###### Abstract

Several frequency-dependent mechanisms restrict the maximum achievable
efficiency for wireless powering implantable bioelectric devices. Similarly,
many mathematical formulations have been proposed to evaluate the effect of
these mechanisms as well as predict this maximum efficiency and the
corresponding optimum frequency. However, most of these methods consider a
simplified model, and they cannot tackle some realistic aspects of implantable
wireless power transfer. Therefore, this paper proposed a novel approach that
can analyze the efficiency in anatomical models and provide insightful
information on achieving this optimum operation. First, this approach is
validated with a theoretical spherical wave expansion analysis, and the
results for a simplified spherical model and a bidimensional human pectoral
model are compared. Results have shown that even though a magnetic receiver
outperforms an electric one for near-field operation and both sources could be
equally employed in far-field range, it is in mid-field that the maximum
efficiency is achieved, with an optimum frequency between 1–5 GHz, depending
on the implantation depth. In addition, the receiver orientation is another
factor that affects the efficiency, with a maximum difference between the best
and worst-case scenarios around five times for an electric source and over 13
times for the magnetic one. Finally, this approach is used to analyze the case
of a wirelessly powered deep-implanted pacemaker by an on-body transmitter and
to establish the parameters that lead to the maximum achievable efficiency.

###### Index Terms:

Biomedical electronics, implants, pacemakers, radiation efficiency, wireless
power transfer.

##  I Introduction

Wireless Power Transfer (WPT) has been the subject of many works and
publications mainly due to its technological and commercial appeal. It was
founded in the works of Nikola Tesla, who in the first place proposed the use
of electromagnetic waves to wirelessly power electric devices. Even though the
theoretical aspects have been studied throughout the years and are well
consolidated, the physical realization and the proposition of different
techniques for its application have gained a stronger impulse in the last 20
years [1]. Overall, one way to classify the different WPT techniques that use
electromagnetic waves as the energy carries is by the radiation region that
comprises the distance between the transmitter and receiver. For instance, the
near-field techniques are based on the near-field coupling between antennas
that can be either inductive or capacitive [2].

Consequently, due to the short distance between antennas, these techniques can
achieve high efficiency, and they are able to deliver high power levels [3].
This efficiency can be even further increased by electromagnetic resonance
[4]. On the other side, the radiative or far-field techniques can cover a wide
distance range with the drawback of lower efficiency. Therefore, they usually
employ highly directive antennas to reduce the power dispersed in free-space
and thus, increase the received power [5]. A new mid-field wireless power
transfer technique has been proposed to achieve higher efficiency than the
far-field techniques and larger distance ranges than their near-field
counterparts. In mid-field WPT, the distance between transmitter and receiver
is in the order of a wavelength, and, in this case, the electromagnetic fields
are composed of reactive and propagating components [6, 7].

In the context of biomedical engineering, the application of WPT for
implantable bioelectronic and biosensor devices is envisaged to monitor
physiological processes [8], drug delivery [9], and organ stimulation through
electric signals [10], to name a few. Therefore, the different WPT techniques
have already been applied for wireless powering bioelectronic implants. For
instance, near-field WPT is employed for subcutaneous pacemakers [11], [12],
neural and spinal cord stimulators [13], cochlear [14], and ocular implants
[15], among other. All these applications share the same characteristic of
having a small implantation depth. Conversely, far-field techniques are used
for ultra-low-power implants usually applied for biotelemetry [8]. However,
for powering deep-implanted devices, the mid-field WPT is mostly used. For
example, it has been applied for cardiac implants and pacemakers [16, 17, 18],
gastrointestinal endoscopy capsules [19], [20], and neural implants [21].

Several physical, technical, and legal aspects impose limitations to the in-
body WPT application. First, the body tissues are composed of dynamic, highly
heterogeneous, dispersive, and lossy media that deteriorate the efficiency of
WPT techniques which in other applications presents a reasonable efficiency.
The two prominent factors that reduce WPT efficiency are the electromagnetic
wave attenuation in these lossy tissues, which is proportional to the
operating frequency, and the reflection in the interface between different
media [22]. Moreover, the miniaturized implantable receivers are usually both
physically and electrically small, further reducing the maximum achievable
efficiencies [23], not to mention other sources of losses such as impedance
mismatching, misalignments, and internal resonances in the body organs. Apart
from that, another major concern is the exposure to the electromagnetic fields
once some WPT techniques can lead to hazardous exposure levels [24].
Therefore, the WPT system parameters -– such as operation frequency, nature of
the source, and implantation depth -– must be judiciously chosen in order to
achieve the highest efficiency possible, taking into account the exposure
level regulations.

Several models have been proposed to mathematically assess the maximum
radiation efficiency and optimal frequency for implantable bioelectronic
devices. The planar stratified heterogeneous model [25, 26] is a simple
approach that allows an analytical formulation that includes the wave
reflection between different layers. In this model, the equivalence principle
replaces a volumetric surface current distribution for an in-plane electric
current density source that produces the same electromagnetic fields. Although
this model is mathematically more straightforward and provides a physical
inside of the problem under analysis, it considers the human body as planar
and infinite media. On the other hand, the spherical wave expansion [27, 28,
29] is a semi-analytical approach that represents the human body as a
spherical model composed of a homogenous or concentric multilayered medium. In
this case, the source is represented by an electric or magnetic dipole
encapsulated by a small sphere positioned anywhere inside the spherical
phantom. Then, the electric and magnetic fields are decomposed in vector
spherical harmonics, and, from orthogonality relationships, the problem is
mathematically solved by a linear system. This approach is computationally
simpler than a full-wave analysis and can lead to insightful results once the
level of achievable power density just outside the body is mainly influenced
by the source position than the shape of the phantom itself. However, this
approach is difficult to implement for more complex sources and realistic body
models. For this intend, a full-wave computation needs to be carried out to
solve the inhomogeneous wave equation numerically given an arbitrary finite-
sized source [23].

This work aims to investigate the electromagnetic energy exchange between an
optimal on-body transmitter and an implantable receiver, as well as to analyze
the influence of the system’s parameters on the wireless power transfer
efficiency. For this intend, a general 2D-axisymmetric full-wave approach is
developed, providing the best-case scenario for the energy transfer efficiency
as a function of the frequency, the implantation depth, and the nature of the
receiver, i.e., electric or magnetic. The basis for this problem formulation
and some results for a simplified spherical phantom model have been presented
[30]. However, the proposed approach is further extended and validated with a
theoretical model based on spherical wave expansion in this paper. Then, both
methods are used to evaluate the frequency-dependent radiation efficiency for
several implantation depths considering electric and magnetic receivers, and
these results surpass the efficiency levels previously presented in the
literature with other approaches.

Previous studies on modeling electromagnetic wave propagation in the human
body mainly focused on wireless communications with implantable antennas [28,
31] medical treatments based on electromagnetic waves such as hyperthermia
[32, 33, 34, 35], or dosimetry [36]. However, the investigation of wave
propagation for WPT applications [16, 25] involves a specific focus on the
electromagnetic interaction between an on-body transmitter and an in-body
receiver. Therefore, the optimal operating conditions identified for in-body
biotelemetry and hyperthermia applications might differ from the WPT ones.
Specifically, for the in-body biotelemetry, the goal is essentially to
maximize the omnidirectional radiation performance through the high-contrast
tissue–air interface [22]. For hyperthermia, the main objective is to control
the local and non-linear [i.e., $\hat{\varepsilon}(\textbf{E})$)] absorption
of energy close to the source [33]. As a result, most deep-body implantable
devices operate in the MedRadio band (401–457 MHz) [37]; the optimal band for
hyperthermia is between 130 MHz and 500 MHz [38], whereas for WPT, the maximum
efficiency is achieved around 1–3 GHz [25, 26, 30, 39]. This work studies the
electromagnetic exchange in an in-/on-body WPT system, derives, and
demonstrates the optimal operating conditions in terms of efficiency and
exposure as a function of implantation depth and nature of the power source
antenna.

Moreover, this study applies for the first time the 2D-axisymmetric full-wave
formulation to assess the wireless power transfer efficiency in an anatomical
scenario considering a deep-implanted mm-sized pacemaker in a transverse cut
model for the human pectoral. In this way, it is possible to analyze the real-
life application of WPT for powering a cardiac stimulator such as presented in
[16]. In addition, different configurations for electric and magnetic on-body
sources are investigated, and the results for the anatomical model are
compared with the simplified spherical phantom. Finally, the origin for the
losses is discussed from these results, and ways to mitigate them and achieve
maximum efficiency are investigated from parametric analysis of on-body
transmitter that wirelessly powers the implanted pacemaker.

##  II Efficiency Assessment Using The Simplified Spherical Model

The problem under analysis consists of a wireless power transfer system
composed of an implantable bioelectronic device that acts as a receiver and a
transmitter radiating structure conformal to the body. In order to determine
the achievable wireless power transfer efficiency, the human body is modeled
as a dispersive homogeneous spherical phantom of radius $R_{p}$ with complex
permittivity $\hat{\varepsilon}_{r}(f)$ equivalent to the human muscle tissue
[40], as is shown in Fig. 1a. Inside this phantom, the implantable device is
represented as a lossless $\varepsilon_{r}(f)$ capsule-shaped, perfectly
matched to the wave impedance in the surrounding tissue, length $L$, and
radius $R_{c}$. This spherical model allows comparing the results obtained
with the proposed approach to semi-analytical solutions based on spherical
wave expansion. In addition, this simplified model disregards the shape
complexities and heterogeneities leading to more general results that are
better suitable for an initial analysis of implantable wireless power transfer
systems.

Figure 1: Geometrical representation of the problem under analysis: (a-b)
simplified spherical model (not to scale). A muscle-equivalent spherical
phantom with radius $R_{p}$ and dispersive complex permittivity
$\hat{\varepsilon}_{r}(f)$), surrounded by a lossless matched medium with
${\varepsilon}_{r}(f)$, encloses the implant. Both (a) and (b) share the same
axis and are surrounded by the same matched medium. For the: (a)
2D-axisymmetric model, this implant is insulated by a capsule-shaped volume
with ${\varepsilon}_{r}(f)$ and the receiving electric or magnetic antenna is
modeled as their corresponding current distribution $\textbf{J}_{s}$ on the
cylindrical aperture, whereas for the (b) spherical wave expansion model the
source is a lossless bubble with ${\varepsilon}_{r}(f)$ and radius a, and the
receiver are elementary electric and magnetic dipoles. (c) Anatomical 2D model
of a human pectoral region comprising nine dispersive tissues. An on-body
source represents the transmitter, and the receiver is a deep-tissue pacemaker
implanted inside the heart at a depth of 37 mm and filled by a lossless
dielectric with ${\varepsilon}_{r}(f)$, matched to the wave-impedance in the
myocardium.

To eliminate losses due to reflection on the interface between the body and
the external medium and thereby ensure the maximum WPT efficiency, the
permittivity of the region around the phantom is considered as the real part
of $\hat{{\varepsilon}}_{r}(f)$, i.e., the external medium is lossless
${{\varepsilon}}_{r}(f)$ and matched to the wave impedance in the muscle. In
this way, the on-body transmitter is well matched to the tissue, which can be
physically realized using a high dielectric constant layer. In addition, this
model disregards any impedance mismatching in the antennas and any system-
level losses in electronic circuitry.

To mathematically formulate the problem under analysis, an equivalent problem
can be proposed based on reciprocity [41]. For this intent, the radiation
source is modeled as the electric current density distribution
$\textbf{J}_{s}$ inside the phantom, as indicated in Fig. 1a. The reciprocity
can be applied in this case once the medium is linear and there is no
polarization mismatch. Consequently, the radiated fields induce a power flow
in the lossy phantom and, from energy conservation, the transmitted power
$P_{t}$ is given by:

$P_{t}=P_{r}+P_{d}+i2\omega(\overline{W_{m}}-\overline{W_{e}}),$ (1)

where $P_{r}$ and $P_{d}$ are respectively the received and dissipated powers,
whereas $\overline{W_{m}}$ and $\overline{W_{e}}$ are the time-average
magnetic and electric energies, respectively.

From the time-harmonic electric E and magnetic H fields, as well as the source
electric $\textbf{J}_{s}$ and magnetic $\textbf{M}_{s}$ current densities,
each one of the total power components can be calculated as follow:

$P_{t}=\oint_{\Omega_{s}}\left(\frac{1}{2}\textbf{E}\times\textbf{H}^{*}\right)\cdot\textup{d}\mathbf{s},$
(2a)
$P_{r}=\oint_{\Omega_{p}}\left(\frac{1}{2}\textbf{E}\times\textbf{H}^{*}\right)\cdot\textup{d}\mathbf{s},$
(2b)
$P_{d}=\frac{1}{2}\int_{\Omega_{p}}\sigma\left|\textbf{E}\right|^{2}\;\textup{d}v,$
(2c)

where $\sigma$ is the phantom’s electric conductivity conductivity which is
related to the imaginary part of the phantom’s complex permittivity
$\Im\left[\hat{\varepsilon}_{r}(f)\right]$ through
$\sigma(f)=2\pi\Im\left[\hat{\varepsilon}_{r}(f)\right]$. The integration
domains $\Omega_{s}$ and $\Omega_{p}$ indicate that the integral is performed
on the surface of the source and the phantom, respectively. Besides, the
integrals (2a-2c) can be considerably simplified due to the problem’s axial
symmetry. Finally, the wireless power transfer efficiency $\eta$ is defined
as:

$\eta\equiv\frac{\Re(P_{r})}{\Re(P_{t})}=1-\frac{\Re(P_{d})}{\Re(P_{t})},$ (3)

with $\Re$ being the real part of the complex variable under brackets.

Hereafter, the implantable bioelectric device is referred to as the receiver
antenna, whereas the transmitter antenna is the radiating on-body structure.

###  II-A Full-wave 2D-Axisymmetric Analysis

In order to numerically evaluate the wireless power transfer efficiency using
the proposed approach, this model was implemented using the full-wave
2D-axisymmetric formulation available in COMSOL Multiphysics (it is assumed
that there is no variation of electromagnetic fields in the $\varphi$
direction). Electric and magnetic sources, with size $L=1\;\textup{cm}$ and
$R_{c}=L/3$ are considered in this computation. Each source is represented by
the electric $\textbf{J}_{s}^{E}(r,\varphi,z)$ and magnetic
$\textbf{J}_{s}^{H}(r,\varphi,z)$ surface current densities given by [23]:

$\textbf{J}_{s}^{E}=\left[0,0,\cos\left(\frac{\pi z}{L}\right)\right],$ (4a)
$\textbf{J}_{s}^{H}=\left[0,1,0\right].$ (4b)

Then, the wireless power transfer efficiency in (3) can be calculated from the
evaluated integrals (2a-2c).

###  II-B Spherical Wave Expansion Analysis

The results from a semi-analytical approach based on spherical-wave modal
expansion are used as a reference to validate the proposed model. In this
model depicted in Fig. 1b, the electromagnetic fields E and H in a spherical
structure free of charges can be expressed using vector spherical harmonics M
and N [41]:

$\begin{Bmatrix}\textbf{E}\\\
-i\eta\textbf{H}\end{Bmatrix}=\sum_{n,m}a_{mn}\begin{Bmatrix}\textbf{M}_{mn}\\\
\textbf{N}_{mn}\end{Bmatrix}+b_{mn}\begin{Bmatrix}\textbf{N}_{mn}\\\
\textbf{M}_{mn}\end{Bmatrix},$ (5)

with

$\textbf{M}_{mn}=\mathbf{\nabla}\times\textbf{r}\psi_{mn},$ (6a)
$\textbf{N}_{mn}=\frac{1}{k}\mathbf{\nabla}\times\mathbf{\nabla}\times\textbf{r}\psi_{mn},$
(6b)

where $k$ denotes the wave number of the considered media, r is the unit
position vector in the $r$-direction, and $\psi_{mn}$ is the solution to the
Helmholtz differential equation:

$\psi_{mn}=\frac{1}{kr}Z_{n}(kr)P_{n}^{m}(\cos\theta)e^{im\varphi}.$ (7)

The Schelkunoff type spherical Bessel or Hankel functions are indicated by
$Z_{n}$, whereas $P_{n}^{m}$ are the associated Legendre functions of the
first kind [42]. From the orthogonality relationships for spherical harmonics,
the equation (5) becomes a linear system whose solution provides the modal
representation for the electromagnetic fields. More details about this
implementation can be found in [28].

The integrals (2a-2c) can be evaluated from the spherical model coefficients
and vectors and, thus, the efficiency can be obtained through (3). It is
important to notice that this theoretical approach can also be applied to off-
centered sources [27, 28].

###  II-C Results and Discussion

The wireless power transfer efficiency for the electric $\eta_{E}$ and
magnetic $\eta_{H}$ sources were evaluated from the electromagnetic fields
calculated through the 2D-Axisymmetric model (2DA) and the Spherical Wave
Expansion (SWE) model. In Fig. 2, the efficiency for both sources is shown as
a function of the frequency for different phantom radius, which is
asymptotically equivalent to the variation in the implantation depth. Besides,
the chosen range comprises most of the applications for wireless powered
implants, from subcutaneous to deeply implanted devices.

First of all, the results obtained from the two approaches previously
described are compared. As it can be seen, the efficiency evaluated with the
2D-Axisymmetric model agrees with the one calculated with Spherical Wave
Expansion, especially at the vicinity of the optimum frequency in which the
error between them is 0.64% for the electric receiver and 2.07% for the
magnetic one. However, for lower frequencies, an offset between them is
noticeable mainly due to the difference in the source formulation.

Figure 2: Wireless power transfer efficiency for an (a) electric $\eta_{E}$
and (b) magnetic $\eta_{H}$ receiver as a function of the frequency and the
phantom radius $R_{p}$ (asymptotically equivalent to the implantation depth)
evaluated through the 2D-Axisymmetric (2DA) and the Spherical Wave Expansion
(SWE) models.

Regarding the efficiency behavior over the frequency, for near-field
operations, i.e., at lower frequencies, the magnetic source outperforms the
electric one for all evaluated implantation depths. It becomes clear from
Table I once the optimum frequency for the magnetic receiver is below the far-
field region, roughly starting at 1 GHz. In addition, at the near-field
region, $\eta_{E}$ sharply increases until reaching its maximum, whereas
$\eta_{H}$ initially increases then reaches a plateau at the maximum
efficiency. Conversely, both receivers present similar decaying behavior in
the far-field region, and $\eta_{H}$ asymptotically approaches $\eta_{H}$ at
frequencies above the optimum one for the electric receiver.

As the reflection losses can be disregarded, this overall behavior for both
receivers can be explained by the fact that in low frequencies, the near-field
losses are higher, whereas, in the far-field region, the attenuation losses
become more significant [27]. Thus, the optimum frequency corresponds to the
trade-off point between both loss mechanisms. Moreover, once the near-field
losses affect more electric than magnetic sources, the efficiency at low
frequencies is significantly lower for this receiver.

TABLE I: Maximum Efficiency and Optimum Frequency for the 2d-Axisymmetric Model as a Function of the Phantom’s Radium (Asymptotic to the Implantation Depth) $R_{p}$ (mm) | Electric Receiver | Magnetic Receiver
---|---|---
20 | 35.44% at 1.26 GHz | 44.92% at 794.3 MHz
37 | 13.43% at 1.26 GHz | 19.01% at 398.1 MHz
50 | 6.83% at 1 GHz | 10.44% at 631 MHz
100 | 0.49% at 1 GHz | 2.036% at 100 MHz

The analysis carried out in this paper considers fixed dimensions for the
receiver. However, a further study on the peak efficiency and optimum
frequency dependence on the receiver dimensions is presented in [23]. It has
shown that the receiver length $L$ is the parameter that most impacts the
efficiency; for instance, the larger the electric source, the higher the
efficiency will be. On the other hand, for a magnetic source, the peak
efficiency is reached within the range $1.5Rc<L<3Rc$. In addition, the
fundamental limit on the electrically small antennas efficiency states that a
reduction in the receiver size will also lead to a reduction in its radiation
efficiency. Therefore, the receiver size considered in the previous analysis
not only is in the range that leads to maximum efficiency but also corresponds
to the size of practical implementation of implantable and ingestible antennas
[37].

##  III Efficiency Assessment Using the Anatomical Pectoral Model

A 2D heterogeneous model of the human pectoral region presented in Fig. 1c
with nine dispersive media with complex permittivity equivalent to those in
the different human tissues [40] was implemented in COMSOL Multiphysics in
order to analyze the wireless powering efficiency in a more realistic
scenario. Based on the wirelessly powered pacemaker proposed in [16], the
receiver is represented by a circular region with a radius of 1 mm inside the
heart with implantation depth $d$ equal to 37 mm.

Similarly as described for the spherical model in Section II, the electric
permittivity of the region external to the body is the same as a lossless skin
${\varepsilon}_{skin}(f)$, whereas the also lossless medium inside the
pacemaker ${\varepsilon}_{muscle}(f)$ is perfectly matched to the wave
impedance of the myocardium. In this way, the reflection losses can be
disregarded.

For the analysis of this problem as a 2D model, it is assumed that the
implantable receiver and the on-body transmitter are perfectly aligned, and
therefore the $z$-axis invariance is valid, i.e.,
$\textbf{E}(x,y,z)=\textbf{E}(x,y)e^{ik_{z}z}$, where $k_{z}$ is the out-of-
plane propagation constant. Moreover, due to the high tissue dispersivity, the
radiated power is attenuated at short distances from the transmitter.
Therefore, the computational cost can be reduced by analyzing only the cross-
section of the body region in which the receiver is implanted.

The same approach described in Section II was applied to the 2D pectoral model
to calculate the maximum wireless power transfer efficiency in a realistic
scenario and compare these results with those obtained with the simplified
spherical model. The reciprocity theorem is again evoked for calculation
purposes, and the radiation source is considered inside the heart. This
premise is still valid once the different media are linear and there is no
polarization mismatch. Consequently, the obtained efficiency calculated with
(3) is valid for the original problem of powering the implantable device.

However, differently from the 2D-axisymmetrical case, different receiver
configurations and orientations can be used in the realistic planar model, as
shown in Fig. 3.

Figure 3: Different receiver configurations are used for the realistic model
analysis. An electric receiver can be defined through (a) a surface electric
current distribution $\mathbf{J}_{s}^{E}$, (b) an electric point dipole
$\mathbf{n}_{p}^{E}$, and (c) an out-of-plane magnetic current $I_{M}$.
Alternatively, a magnetic receiver can be modeled as (d) a surface magnetic
current distribution $\mathbf{J}_{s}^{H}$, (e) a magnetic point dipole
$\mathbf{n}_{p}^{H}$, and (f) an out-of-plane line current $I_{E}$. The
contour lines depict the electric $\mathbf{E}$ and magnetic $\mathbf{H}$ field
distribution in arbitrary units.

The first possibility is to assign a Surface Current Distribution (SCD) over
the line of length $L$ (in the 2D simulation) for the electric
$\textbf{J}_{s}^{E}(x,y,z)$ and magnetic $\textbf{J}_{s}^{H}(x,y,z)$ receivers
according to:

$\textbf{J}_{s}^{E}=\begin{cases}\cos\left(\frac{\pi
y}{L}\right)\;\textup{A/m}\;\mathbf{\hat{y}},\text{ if vertically oriented}\\\
\cos\left(\frac{\pi x}{L}\right)\;\textup{A/m}\;\mathbf{\hat{x}},\text{ if
horizontally oriented,}\end{cases}$ (8a)
$\textbf{J}_{s}^{H}=\begin{cases}1\;\textup{V/m}\;\mathbf{\hat{y}},\text{ if
vertically oriented}\\\ 1\;\textup{V/m}\;\mathbf{\hat{x}},\text{ if
horizontally oriented.}\end{cases}$ (8b)

Henceforward, vertically oriented means that the receiver is oriented in the
$y$-axis direction ($\mathbf{\hat{y}}$) whereas horizontally in the $x$-axis
direction ($\mathbf{\hat{x}}$), according to the coordinate system depicted in
Fig. 3.

Subsequently, the next alternative is to define the receiver as an Electric
Point Dipole (EPD) or Magnetic Point Dipole (MPD). In each one, the electric
$\textbf{n}_{p}^{E}$ or magnetic $\textbf{n}_{p}^{H}$ dipole moment vector is
assigned as follows:

$\mathbf{n}_{p}^{E}=\begin{cases}1\;\textup{A$\cdot$m}\;\mathbf{\hat{y}},\text{
if vertically oriented}\\\ 1\;\textup{A$\cdot$m}\;\mathbf{\hat{x}},\text{ if
horizontally oriented,}\end{cases}$ (9a)
$\mathbf{n}_{p}^{H}=\begin{cases}1\;\textup{A$\cdot$m}^{2}\;\mathbf{\hat{y}},\text{
if vertically oriented}\\\ 1\;\textup{A$\cdot$m}^{2}\;\mathbf{\hat{x}},\text{
if horizontally oriented.}\end{cases}$ (9b)

Finally, it is also possible to define an out-of-plane Magnetic Current (MC)
for modeling the electric receiver or an out-of-plane Line Current (LC) for
the magnetic one. Respectively for each case, the electric or magnetic fields
are orthogonal to the current distribution direction and the position vector.
Therefore, the out-of-plane magnetic $I_{M}$ = 1 V and electric $I_{E}$ = 1 A
currents must be assigned following the direction and orientation shown in
Fig. 3c and Fig. 3f, respectively.

By taking into account the three source types and the vertical and horizontal
orientations, it leads to six possible configurations for each electric and
magnetic receiver. Finally, the wireless power transfer efficiency was
evaluated for all the described scenarios, and the results are presented in
Fig. 4. It is possible to see that the efficiencies evaluated with different
sources agree between them, which demonstrates the flexibility of the proposed
model for dealing with different source configurations.

Figure 4: Efficiency was evaluated with the realistic pectoral model for an
(a) electric $\eta_{E}$ and (b) magnetic $\eta_{H}$ receiver implanted in the
heart. Different receiver orientations (vertical and horizontal) and
configurations {Surface Current Distribution (SCD), Electric (EPD) and
Magnetic Point Dipole, out-of-plane Magnetic Current (MC) and Line Current
(LC)} are compared.

The source orientation is the factor that mainly affects the radiation
efficiency. In the best-case scenario, the sources are horizontally-oriented,
whereas the vertical orientation corresponds to the worst position for the
receiver. Since the receiver is vertically oriented, the radiation beam
propagates towards the direction parallel to the chest boundary, which is the
closest to the receiver and, therefore, the region that contributes the most
for the integral (2b). This behavior was also pointed out in [22] but, from
the results shown in Fig. 4, it is possible to evaluate the difference between
best and worst-case scenarios quantitatively. Namely, for the best
positioning, the electric source presents an optimum frequency 590 MHz higher
with a maximum $\eta_{E}$ approximately five times the one verified with a
vertically oriented source. A similar offset in efficiency magnitude can be
verified between a horizontal and a vertical magnetic source. For lower
frequencies, the best positioning leads to an efficiency around 2.5 times
higher; however, the maximum difference between them surpasses 13 times in the
far-field region.

It is also noticeable that the radiation efficiency behavior for an electric
or magnetic receiver obtained with the realistic pectoral model agrees with
those obtained with the simplified spherical model. Namely, for the
horizontally oriented electric sources, $\eta_{E}$ increases in the near-field
range with a peak of 0.77% at 1.74 GHz, close to the optimum frequency shown
in Table I for the same implantation depth of 37 mm, then decreasing for
higher frequencies. In contrast, $\eta_{H}$ reaches its maximum at lower
frequencies, remaining almost constant around 10% in the near-field, before
decreasing for frequencies above 1 GHz. The sharp reduction in radiation
efficiency by a magnetic source in the far-field region is explained by the
fact that this kind of source achieves its optimal performance as an
electrically small antenna [43, 44], i.e., when $L<\lambda/10$ which is
satisfied only in the near-field region.

As expected, there is a significant reduction in the maximum efficiencies
obtained with the realistic model compared with the simplified spherical
model. First of all, the latter considers a homogeneous medium equivalent to
the human muscle tissue, whereas in the pectoral model, several organs with
different electromagnetic properties are taken into account. For this reason,
two predominant loss sources are responsible for this efficiency
deterioration: the increased attenuation, mainly in the fat and muscle layers
where the power is dissipated before reaching the receiver in the heart; and
the reflection losses due to the wave-impedance contrast between the medium
that composes the tissues. In this last case, with the multiple reflections,
most of the reflected power is dissipated in the tissue reducing the amount of
power that reaches the receiver encapsulation. Therefore, this loss is
accounted on (1) also through $P_{d}$. Moreover, in a realistic scenario, the
shape of the organs may lead to resonance modes responsible for local peaks
and valleys in efficiency for some frequencies, such as 3.63 GHz for the
electric source and around 1.74 GHz for the magnetic source.

##  IV WPT Efficiency as a Function of the Transmitter Parameters

The case study of a wirelessly powered pacemaker is analyzed to establish the
optimum parameters that lead to the maximum efficiency in a physically
realizable application. In this case, the human pectoral model in Fig. 1a is
used. However, the transmitter is now localized in the chest, as shown in Fig.
5a. This transmitter comprises a substrate with the same dielectric properties
as the medium external to the body and a lossless superstrate (or buffer) with
thickness $d$ and relative electric permittivity $\varepsilon_{r}^{b}$.

The electric transmitter in Fig. 5b is defined by assigning a surface current
distribution $\textbf{J}_{s}^{E}(x,y,z)$ given by:

$\textbf{J}_{s}^{E}=\left[\cos\left(\frac{\pi x}{L}\right),0,0\right],$ (10)

in which $L$ is the transmitter length. Concurrently, the magnetic transmitter
is modeled as a unitary out-of-plane electric current $I_{E}$ = 1 A on the
edge of the source with opposite phases, as illustrated in Fig. 5c. As seen
from the field distribution for the electric and magnetic transmitters, they
are, for instance, equivalent to a dipole and loop antenna, respectively.

In this configuration, the two main transmitter parameters that can be changed
to achieve the maximum wireless power transfer efficiency are the buffer
thickness $d$ and its permittivity $\varepsilon_{r}^{b}$. Therefore, a
parametric analysis was carried out for both parameters within the ranges:
$1\leq d\leq 50$ mm and $1\leq\varepsilon_{r}^{b}\leq 80$. Furthermore, the
lower and upper bounds for $\varepsilon_{r}^{b}$ were chosen so that this
range comprises the permittivities of all nine tissues in the frequency spam
from 100 MHz to 5 GHz. Consequently, the effective permittivity for this
pectoral region is also in the chosen interval.

The operating frequency must be chosen and fixed to carry out this parametric
study. Based on the results shown in Fig. 4, the maximum theoretical WPT
efficiency for a deep-implanted pacemaker charged by an electric source
transmitter is achieved at the frequency of 1.74 GHz. Considering the
frequency bands allocated for Industrial, Scientific, and Medical (ISM) usage,
the closest band to the optimal frequency is centered at 2.45 GHz.
Henceforward, this is the frequency chosen for all following analysis, and the
transmitter length $L$ was set as $\lambda/2$ at this frequency in free space.

In order to evaluate the wireless power transfer system for this
configuration, (3) can be still be applied with the results from integrals
(2a-2c); however, the integration domains need to be reformulated. In this
case, $\Omega_{s}$ indicates that the integral is performed on the boundary of
the pacemaker, whereas $\Omega_{p}$ represents the transmitter’s boundary.

Figure 5: Analysis setup for the parametric analysis of the efficiency in a
wireless powered pacemaker application: (a) close-up view of the on-body
transmitter with length $L$ composed of a radiation source positioned apart
from the skin by a lossless superstrate (buffer) with thickness d and
permittivity $\varepsilon_{r}^{b}$ and with a substrate matched to the medium
external to the body. (b) Electric source modeled as a surface current density
$\mathbf{J}_{s}^{E}$ with the electric field distribution indicated by the
arrows and the magnetic field by the isolines. (c) Magnetic source modeled as
an out-of-plane current $\mathbf{I}_{E}$ with the magnetic field distribution
indicated by the arrows and the electric field by the isolines.

The contour levels for the efficiency as a function of the buffer thickness
and permittivity are shown in Fig. 6a for electric and Fig. 6b for magnetic
sources. As it can be seen, the buffer parameters that lead to the maximum
efficiency are $d=$ 7 mm and $\varepsilon_{r}^{b}=$ 80\. The maximum
efficiency is obtained with the highest permittivity can be explained once the
wave-impedance mismatching at the transition buffer-skin is practically
eliminated. In addition, taking into account this permittivity, the maximum
occurs exactly at $\lambda/2$. As proved in [45] by considering the
superstrate as a transmission line, it corresponds to the resonance condition,
once the impedance seen in the transition transmitter-skin is equal to the
impedance seen in the source, which leads to a high voltage at the source
position, thus increasing the radiation efficiency. Moreover, it can be seen
that as the buffer’s thickness increases, so decrease the optimum
$\varepsilon_{r}^{b}$ in such a manner that a $\lambda/2$ periodicity is
obeyed, as it also happens with a lossless transmission line.

On the other side, the maximum efficiency obtained with a magnetic source is
approximately an order of magnitude lower when compared with the electric one.
It means that, at the frequency of 2.45 GHz, the magnetic source is sub-
optimal, as it was also predicted in the graph of Fig. 4b, once it does not
satisfy the electrically small antenna condition at this frequency. However,
the maximum efficiency is achieved with $d=$ 6 mm and $\varepsilon_{r}^{b}=$
46.5. It is important to mention that, in this case, this maximum occurs in a
significantly narrow distance range, which in a practical realization would
require precise positioning of the source.

In order to classify and quantify the losses, the optimal scenario in which
the transmitter is composed of an electric source and a buffer with $d=$ 7 mm
and $\varepsilon_{r}^{b}=$ 80 is considered. Three main loss mechanisms can be
identified: reflection losses due to the wave impedance contrast between the
on-body transmitter and the body tissue, antenna near-field losses, and the
attenuation losses in the tissues. The two first losses can be mitigated by
adequately designing the WPT system, whereas the latter cause is unavoidable.
For instance, approximately 7% of the transmitted power is lost due to
reflection in this optimum configuration, whereas 93% is due to attenuation
and near-field losses combined. However, the optimal buffer parameters justify
the low reflection losses and reduce the near-field region and its associated
losses.

This fact raises a concern about the electromagnetic exposure levels for in-
body WPT applications. The surface profiles for the local specific absorption
rate (SAR) normalized by the transmitted power are shown in Fig. 7 for the
optimal electric and magnetic sources. First of all, the SAR profiles follow
the field distribution shown in Fig. 5b and Fig. 5c, being higher in the
center for the electric source, whereas it reaches its maximum around the
source edges for the magnetic. In addition, in both cases, the peak SAR is
located in the skin layer, and no significant SAR values are observed in the
vital organs. However, by comparing both SAR levels, it is noticed that the
magnetic source leads to a larger SAR, almost twice the value for the electric
source, even with much lower efficiency levels. Since the SAR is related to
the attenuation in the tissues that cannot be physically mitigated, it is
possible to conclude that the efficiency is maximized by properly choosing the
transmitter parameters, and the electromagnetic exposure is minimized.

Figure 6: Contour plots for the efficiency as a function of the buffer
thickness $d$ and permittivity $\varepsilon_{r}^{b}$ for an (a) electric
source $\eta_{E}$ and a (b) magnetic source $\eta_{H}$. The star indicated the
global maximum position, and the dotted lines indicate the position
corresponding to $\lambda/2$ its integer multiples as a function of
$\varepsilon_{r}^{b}$.

Finally, the fact that the efficiency obtained in this case is significantly
lower when compared with the optimum efficiency presented in Fig. 4 shall be
addressed. It can be explained by the fact that the transmitter is considered
to be a conformal structure covering the entire body surface in the ideal
scenario described in Section III. In contrast, the current analysis takes
into account a mono-antenna transmitter, leading to a much smaller integration
domain $\Omega_{p}$ thus reducing the received power. In practical terms, it
means that the single antenna mid-field WPT solutions currently in literature
are sub-optimal by many orders of magnitude. Therefore, to physically achieve
the maximum efficiency theoretically obtained, further developments in mid-
field WPT are required based on conformal antenna arrays such as in [46] and
techniques for electromagnetic focusing in lossy media that have already been
investigated for other biomedical applications [47].

Figure 7: Surface plot of the local specific absorption rate (SAR) in the
region between the on-body transmitter and the deep-implanted receiver
normalized by the transmitted power considering an optimal (a) electric and
(b) magnetic source.

##  V Conclusion

In order to wireless power implantable bioelectronic devices, several
techniques can be employed depending upon the application characteristics.
However, the combination of different physical mechanisms imposes a limit to
maximum achievable wireless powering efficiency. Among them, the principal
mechanism is the electromagnetic attenuation in the dispersive tissues, the
reflection losses, and the fundamental efficiency limitation of electrically
small antennas. In addition to it, there are system-level losses and resonant
modes in the body cavities that also contribute to efficiency deterioration.
Therefore, this work focuses on evaluating the wireless powering efficiency of
deep-body implanted devices and studying how the system’s parameters could be
tuned to reach the highest efficiency possible in a practical application.

For this intend, a mathematical formulation for this problem was proposed
based on reciprocity and T-symmetry principles as well as considering an on-
body conformal source, perfectly matched to the human tissue. Such hypotheses
lead to optimum energy focusing and, thus, to the highest possible efficiency.
In order to validate this approach, a semi-analytical Spherical Wave Expansion
analysis was carried out, and the results for a simplified spherical
representation for the human body were compared.

This first analysis indicates that a magnetic receiver outperforms the
electric one at the near-field frequency range, whereas, in the far-field,
both receivers show similar performance. Therefore, the optimum frequency is
in the mid-field region and depends upon the source type and implantation
depth, leading to the highest efficiency.

Once the proposed approach was validated, it was applied to an anatomical
model focused on a deep-implanted wireless powered pacemaker application [16].
This analysis verified that the overall behavior for both sources agrees with
the one predicted by the simplified spherical model, including the optimum
operation in the mid-field range. Furthermore, this optimum frequency also
agrees with the results presented in literature obtained through different
approaches [22, 23, 25, 26, 27, 28, 29]. Even though there is a significant
efficiency reduction in the realistic model results, it is because now the
wave impedance contrast between the different tissues imposes reflection
losses that were disregarded in the simplified model. Moreover, with this
model, it was verified that the receiver orientation is another practical
aspect that can reduce efficiency. Considering the source positioning, the
maximum difference in efficiency between the best and worst-case scenarios is
around five times for an electric receiver, whereas, for the magnetic one,
this difference can be above 13 times. The effect of the source orientations
is justified once in the best case, i.e., for a horizontally oriented
receiver, the power flows towards the chest line, which is the closest point
to the receiver, contributing more for the overall received power. It is
important to mention that the flexibility in analyzing different source types
is another contribution of the proposed approach compared to other methods of
analysis.

Finally, a wireless power transfer system considering a deep-implanted
pacemaker as the receiver and a multilayered (substrate-source-buffer)
transmitter positioned in the chest was considered to analyze the impact of
the transmitter’s parameters on the powering efficiency as well as to provide
information on how these parameters could be tuned in order to achieve the
maximum efficiency. This parametric study has shown that at the 2.45 GHz ISM
band, the electric transmitter leads to efficiency over a degree of magnitude
higher than a magnetic one. In addition, the maximum efficiency can be
achieved with a lossless high-permittivity buffer, matched to the skin
permittivity, and thickness equal to $\lambda/2$ in this medium. Furthermore,
other local maxima can be obtained for thicker and lower permittivity
superstrates, respecting a $\lambda/2$ periodicity.

To summarize, the approach proposed in this paper is able to predict the
maximum achievable powering efficiency considering simplified or anatomical
body models as well as to provide information on the parameters that can be
optimized in order to achieve this maximum efficiency. In addition, even
though the case study of a deep-implanted pacemaker was analyzed in this
paper, the same approach could also be used to evaluate the powering
efficiency for other deep-body implantable devices.

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