Volume Conduction for Powering Deeply Implanted Networks of Wireless Injectable Medical Devices: A Numerical Parametric Analysis

The use of networks of wireless active implantable medical devices (AIMDs) could revolutionize the way that numerous severe illnesses are treated. However, the development of sub-mm AIMDs is hindered by the bulkiness and the transmission range that consolidated wireless power transfer (WPT) methods exhibit. The aim of this work is to numerically study and illustrate the potential of an innovative WPT technique based on volume conduction at high frequencies for powering AIMDs. In this technique, high frequency currents are coupled into the tissues through external electrodes, producing an electric field that can be partially picked-up by thin, flexible, and elongated implants. In the present study, the system formed by the external electrodes, the tissues and the implants was modeled as a two-port impedance network. The parameters of this model were obtained using a numerical solver based on the finite element method (fem). The model was used to determine the power delivered to the implants’ load (PDL) and the power transmission efficiency (PTE) of the system. The results allow the identification of the main features that influence the PDL and the PTE in a volume conduction scenario and demonstrate that volume conduction at high frequencies can be the basis for a non-focalized WPT method that can transfer powers above milliwatts to multiple mm-sized implants (< 10 mm3) placed several centimeters (>3 cm) inside the tissues.


I. INTRODUCTION
The use of active implantable medical devices (AIMDs) has the potential to revolutionize the health industry for years to come [1], [2]. In particular, the development of dense networks of miniaturized (mm-sized or less), minimally invasive and distributed AIMDs that allow for efficient interaction with the nervous system could drastically change the way that several major diseases are approached (e.g., tetraplegia, tremor suppression) [3]- [5], and facilitate the development of next generation wearable robots [6].
The associate editor coordinating the review of this manuscript and approving it for publication was Masood Ur-Rehman .
However, miniaturization, and thus the development of these envisioned swarms of networked AIMDs, has been hampered by the way implantable electronic devices are powered. On the one hand, current batteries have limited power density for allowing the implementation of mm-sized devices that can operate for long periods of time. On the other hand, several wireless power transfer (WPT) methods have been developed for supplying power to battery-less AIMDs but, to the best of our knowledge, in all cases require relatively bulky components within the implants for receiving the energy transferred by the remote transmitter. Inductive and ultrasonic coupling are the two most widely used WPT methods [7]. Both methods are useful to transfer powers in the order of mW to a single AIMD [8]- [13], but they present several limitations for distributed networks of AIMDs.
In the case of inductive coupling, the power delivered to the implant load (PDL) and power transmission efficiency (PTE) notably decrease as the separation between the transmitter and the receiver coils increases, as well as with misalignments between them [14]. Recently, to minimize these losses, some authors have proposed the use of a third coil located between the transmitter and the receiver [9], [15]. In particular, this approach has been used for powering a distributed network of mm-sized implants for brain-computer interface applications [16]. However, the cm-sized intermediate coil was also implanted, thus creating a more invasive system. It is difficult to conceive the use of this approach in soft tissue regions such as the limbs or the abdomen, as this additional coil would be excessively stiff and bulky.
Ultrasonic WPT can deliver powers in the order of mW to implants located several centimeters inside the tissues when the ultrasonic waves are focused on a specific region [17]. However, there is a trade-off between the received power and the spatial range; high power will be obtained by focusing the beam, while low power will be obtained if the beam is spread. This reduces the applicability of ultrasounds for powering networks of distributed AIMDs.
In the last years, we have explored the use of volume conduction to power networks of implants [18]- [21]. The WPT technique proposed does not require embedding bulky components inside the implants to electrically power them. The technique consists in the delivery of bursts of high-frequency currents through pairs of external textile electrodes and the development of thread-like devices, which can be deeply implanted by minimally invasive procedures. The electric currents, due to the conductive properties of the tissues, produce a voltage gradient that can be picked up by thin (<1 mm) and elongated (∼3 cm) implants. The implants draw power from the high-frequency current bursts through electronic rectification. The rectifier, together with the supplied electronics can be integrated into a microcircuit or an application-specific integrated circuit (ASIC). This implies that the implants can be shaped as thin and flexible devices that can be deployed by injection or catheterization [22]. Fig. 1a illustrates an envisioned scenario, where multiple thread-like AIMDs are placed in the muscle of an upper arm, and the electric field is applied by two external electrodes made of conductive fabric.
In [18] we demonstrated that injectable stimulators based on off-the-shelf components could be powered and addressed from an external unit. In [19], we proposed a two-port network to model the whole system for circuit simulation. Recently, we have developed an analytical model able to determine the maximum power that an implant can locally obtain considering an infinite homogeneous medium [20]. We have also conducted an experimental study in humans demonstrating the feasibility of this WPT technique (preprint [21]). However, to the best of our knowledge, the complete transmission link (i.e., the combination of FIGURE 1. a) Envisioned scenario where a swarm of thread-like AIMDs are powered using volume conduction. b) Simplified model of an upper arm that consists in a multilayered concentric cylinder (skin, fat, muscle, cortical bone, and cancellous bone), two external electrodes and two cylindrical electrodes that emulate an implant. c) Two-port network model of the coupling between the external electronic system and the implant electronics. external electrodes, multilayered tissue, and implants) for the described technique has not been studied in detail yet. For this reason, the present work aims to numerically study the volume conduction transmission link and its potential for powering networks of deeply implanted AIMDs in human limbs. In particular, this work is a parametric analysis in which the dependence on several geometrical and anatomical parameters is analyzed. Unlike the local model presented in [20], here it is considered a more realistic scenario that includes the whole transmission link (Fig. 1b). Additionally, the system is modeled using a two-port network (Fig. 1c) instead of using an analytical approach, since an interpretable closed analytical expression is not feasible because the medium is finite and includes multiple tissue layers. The parameters VOLUME 9, 2021 of the two-port network were obtained using a finite element method (fem) solver. In order to parametrically analyze how different geometrical and anatomical parameters influence the PDL and the PTE, the limb was simplified using a concentric cylindrical approach as it has been done in previous studies in which volume conduction has been analyzed for communications [23]- [25].

II. BACKGROUND: VOLUME CONDUCTION AND ELECTRICAL SAFETY
International safety standards [26], [27] identify two main sources of risk with respect to human exposure to radio frequency electromagnetic fields: neuromuscular stimulation and tissue heating. The risk of stimulation is directly related to the electric field, E (V rms /m), and the frequency applied, being especially relevant for frequencies below 100 kHz. In particular, the IEEE safety standard [26] considers that for continuous sinusoidal waveforms at frequencies above 5 MHz the risk of electrostimulation is negligible. Regarding tissue heating due to Joule's effect, the standards define a maximum specific absorption rate (SAR) to avoid tissue overheating. The SAR is expressed in W/kg and can be calculated at any point as where σ is the tissue conductivity (S/m), and ρ is the tissue density (kg/m 3 ). The SAR is averaged over 10 g of tissue for 6 minutes. The safety standards establish a limit of 20 W/kg for human extremities in controlled environments [26], [27]. In this work, the electrostimulation risk is not addressed as we propose the use of currents with a frequency of 6.78 MHz, which is above the 5 MHz limit. The 6.78 MHz corresponds to the central frequency of an ISM radio band, thus the possibility of interfering radiocommunications is minimized. We advise against the use of higher ISM radio bands (above 10 MHz) to avoid losses due to skin effect, as at those frequencies the skin effect may be very substantial [28], especially in cases where the AIMDs are implanted several centimeters into the tissues.

III. MODELING
The volume conduction links formed by the external electrodes, the upper arm tissues and the implants' electrodes were modeled using multi-port networks. The impedance parameters of these networks were obtained using a fem solver.

A. GEOMETRY
The upper arm geometry was simplified as a five-layer concentric cylinder, as shown in Fig. 2a-b, where each layer emulates a tissue: dry skin, fat, muscle, cortical bone, and cancellous bone (Fig. 2). Table 1 reports the thicknesses used for each layer. Similar models have been broadly used to model the upper arm in galvanic coupling communication models [23]- [25]. Since all the tissues can   be considered electrically isotropic for high frequencies [29], and the quasi-electrostatic condition is considered, each layer was simply electrically characterized with two scalars: conductivity (σ ) and permittivity (ε). The electrical values of these parameters, for a frequency of 6.78 MHz, were taken from [30], while the tissue density was obtained from [31]. These parameters are reported in Table 2.
Although we have proposed and demonstrated the use of external electrodes made of conductive fabric that could be embedded in garments [18], in this simplified model the external electrodes consisted in a pair of cylindrical rings embracing the skin with dielectric properties and density equivalent to those of stainless steel [32]. Their dimensions are also reported in Table 1. Unless otherwise indicated, the separation distance between the edges of both electrodes was 100 mm and their length was 30 mm.
Regarding the elongated implant, for the sake of simplicity, it was modeled as an aligned pair of stainless-steel cylindrical electrodes separated at a given distance. Their dimensions are reported in Table 1. Unless otherwise stated, the implant electrodes were placed at a depth of 30 mm. They were longitudinally aligned with the multilayered cylinder, placing one electrode at the longitude of 95 mm and the other one at 135 mm. For both electrodes, the transverse position was set to zero (see Fig. 2a-b).

B. MULTI-PORT NETWORK MODEL
The system composed of the external electrodes, the tissues, and the implants' electrodes, is linear and reciprocal for low current densities [33]. For this reason, it can be modeled with a multi-port network. For the case of a single implant, the system can be modeled as a two-port network [34] where V ex (V rms ) and I ex (A rms ) are the transmitter voltage and current phasors, while V im (V rms ) and I im (A rms ) are the implant voltage and current phasors, and the impedances are defined as where Z ex corresponds to the impedance between the two external electrodes, Z im is the impedance between the implant electrodes, and Z exim is the transimpedance between the external electrodes and the implant electrodes. For the studied frequency, the tissue impedance seen by the implant electrodes is mostly resistive due to the biological properties of the tissues (see Table 2 ). Assuming a purely resistive Z im , the maximum power is transferred to the load when the implant load (R L ) matches Z im . For this reason, R L was set to be equal to the real part of Z im . Consequently, the implant current was defined as For the cases in which two implants were considered, the two-port model (2) was expanded to a multiport model where

C. EXTRACTION OF PARAMETERS FOR THE MULTI-PORT NETWORK
For the described scenario, the quasi-static electric field approach can be assumed, since for a frequency of 6.78 MHz, the electric field wavelength is much larger than the model dimensions.
The model geometry described in Fig. 2 was implemented in COMSOL Multiphysics 5.3 (from COMSOL, Inc., Burlington, MA, US) to calculate the electric field distributions and the parameters of the multi-port network model using the ''Electric Currents (ec)'' physics from the ''AC/DC module''. This physics mode considers the quasistatic approach and solves the following equations where J is the current density (A/m 2 ) and ρ c is the charge being ω the angular frequency of the field (rad/s), and D the electric displacement field (C/m 2 ), And The used mesh consisted of two tetrahedral meshes. The first one was applied in the regions close to the implant electrodes (<5 mm from their centers), imposing a maximum edge length of 0.1 mm. This refined mesh was used to obtain a proper resolution and therefore accurate impedance values for the implant electrodes. The second one comprises the rest of the geometry and was automatically generated. The total number of elements was comprised between 1,000,000 and 1,200,000.
The multi-port impedance parameters were extracted simulating the delivery of a known current (f = 6.78 MHz) through the selected electrodes and measuring the received voltage at the electrodes of interest, see (3). To calculate the averaged SAR, the normalized values of the simulated E rms and σ were stored in a regular 3-D grid (steps of 1 mm) for the whole geometry. Then, using MATLAB R2019a (from MathWorks, Inc., Natick, MA, US), the local SAR (1) was averaged over cubes with a volume of ∼10 cm 3 (edge equal to 21 mm and an approximately mass of 10 g) for each of the grid points. The peripheral points that could not be the center of a cube, as the cube included points outside the tissues, were assigned to the highest spatial-average in which they were enclosed as indicated by the IEC/IEEE 62704-1 standard [35].

D. POWER DELIVERED TO THE IMPLANT AND POWER TRANSFER EFFICIENCY
The PDL can be calculated as The active power (P in ) coupled to the tissues by the external electrodes is calculated as where ϕ is the phase (rad) between V ex and I ex . As most of the power is dissipated at the tissues and only a tiny portion can be dissipated at the implants, the applied power (12) can be approximated as Therefore, the PTE is calculated as where N is the total number of implants.

A. ELECTRIC FIELD INSIDE THE TISSUES
The amplitude of the external high frequency current that can be coupled to the tissues is limited by the SAR restrictions. For the nominal model of Table 1, it was found that the applied current must be below 0.46 A rms (at 6.78 MHz), since above this value, the local averaged SAR will exceed the 20 W/kg limit. If not otherwise indicated, this will be the applied current for the following cases. Fig. 3 shows the averaged SAR distribution when the external electrodes apply 0.46 A rms at 6.78 MHz. The results highlight that the averaged SAR is focused close to the external electrodes and has a maximum value of 20 W/kg. Fig. 4a shows the electric field distribution along four longitudinal lines at the depths of 5 mm, 20 mm, 35 mm, and 50 mm (see Fig. 2a). The maximum electric field is located close to both external electrodes (see LL1, which runs through the middle of the fat layer). However, it can be observed how the electric field tends to be uniform a few centimeters away from the external electrodes. Considering the most central segment of the model (i.e., longitudes between 95 mm and 125 mm) the electric field can be considered depth-independent (e.g., at a depth of 50 mm the electric field is just 1.5% lower than at a depth of 5 mm, for the entire range). The electric field at the midpoint between the external electrodes (i.e., longitude position equal to 110 mm) is about 155 V rms /m. Considering an implant with a length of 30 mm and aligned parallel to the applied electric field, the voltage that this implant can pick up is 4.65 V rms (see Fig.4b). It is worth noting that, although this will be the maximum rms amplitude because of the SAR limitation, it will be possible to obtain much higher peak amplitudes by delivering the ac currents in the form of short bursts [20], [21].

B. IMPLANTATION DEPTH
To study the PDL and PTE as a function of the implantation depth, the longitudinal position of the implant electrodes was set to 95 mm and 125 mm respectively while the transverse position was zero for both electrodes. The electrodes' depth  was swept from the peripheral fat layer (depth = 3 mm) to the cancellous bone (depth = 50 mm), see Fig. 5.
The results show that the maximum PDL is obtained when the implant electrodes are surrounded by muscle tissue (PDL = 11.8 mW, PTE = 0.07%, and depth = 22 mm). The lowest PDL is obtained when the implant is surrounded by fat tissue (PDL = 0.4 mW, PTE = 0.004%, and depth = 7 mm). This difference in the PDL between different layers is mostly related to the tissue impedance seen by the implant electrodes. At a depth of 5 mm (i.e., fat tissue) the tissue impedance seen by the implant electrodes (Z im ) is 8612-1262i , while for a depth of 25 mm (i.e., muscle tissue) is 464-50i .
As the maximum power is transferred to the load approximately when R L matches the real part of Z im , Z im directly affects the obtained PDL (11). Nevertheless, the obtained PDL as a function of the depth is hardly affected by the maximum voltage that an implant can pick up since, for the studied area, the maximum voltage that  an implant could receive is not related to its implantation depth (Fig. 2b). Fig. 6 illustrates the influence of the separation distance between the inner edges of the external electrodes. In this figure the separation distance between the external electrodes was symmetrically increased from 30 mm to 120 mm while the implant electrodes were kept at the default position (i.e., an implantation depth of 30 mm, a transverse position of 0 mm, and a longitudinal position of 95 mm and 135 mm for the first and second electrode respectively). The amplitude of the applied current was adjusted to obtain a SAR of 20 W/kg, resulting in a current amplitude of 0.46 A rms for the entire range studied. The results show that the PDL is reduced when the external electrodes are aligned over the implant electrodes (i.e., a separation distance of 30 mm regarding the depth axis), as the electric field E is focused on the more superficial layers (see Fig. 4a). In this case, the implant picked up 2.8 mW with a PTE of 0.03%. PDL increases with the separation distance of the external electrodes. The maximum power transferred is for the case where the external electrodes were separated 120 mm. For this case, the PDL is 11.6 mW and the PTE is 0.068%. PTE has its maximum (0.072%) for a separation of 90 mm.

D. EXTERNAL ELECTRODES WIDTH
To determine the influence of the width of the external electrodes, their size was swept from 10 mm to 35 mm (Fig. 7) while the separation distance between the edges of the electrodes was kept constant at 100 mm. The current applied for each band size was set to obtain a maximum averaged SAR of 20 W/kg (Fig. 7a) ranging from 0.345 A rms to 0.49 A rms . The results show that as the band electrodes get wider, the PDL and PTE obtained are also higher (Fig. 7b).

E. FAT THICKNESS
One of the anatomical parameters of the model that varies more between individuals is the fat thickness [36]. To study how the fat thickness can influence in the PDL and the PTE, the thickness of this layer was swept from 1 mm to 25 mm, while the thicknesses of the dry skin, cancellous bone and cortical bone were kept constant, and the muscle thickness was modified to obtain a total radius of 50 mm (i.e., the limb radius was kept constant to allow the comparison between cases). Again, the external current I ex was set to obtain a maximum averaged SAR of 20 W/kg (Fig. 8a) ranging from 0.34 A rms to 0.69 A rms . The results indicate that to avoid tissue over-heating, the applied I ex must be reduced as the fat thickness is increased (Fig. 8a). Interestingly, the results show that as the fat layer gets thicker, the power that an implant placed in the muscle can obtain is higher (Fig. 8b).  The minimum PDL value corresponds to a 5 mm fat layer (PDL = 10.6 mW, and PTE = 0.067%).

F. INTERACTION OF MULTIPLE IMPLANTS
To determine how neighboring implants may influence the PDL of each individual implant, a parametric sub-analysis was performed varying the separation distance of two parallel implants along two different axes independently. In this sub-analysis it was assumed a minimum separation distance of 3 mm between both parallel devices due to implantation limitations. In this sub-analysis it was also assumed that the load of both implants corresponds to the real part of Z im .
In Fig. 9a a longitudinal displacement was done while fixing the length of both implants to 30 mm. The position of Implant 1 was kept constant, while Implant 2 was longitudinally displaced along an axis 3 mm more superficial. The longitudinal position of the left electrode of Implant 2 was swept from 45 mm to 145 mm. During this displacement, the implantation depth of both electrodes of Implant 2 was kept constant at 27 mm and their transverse position was 0 mm (see inlet Fig. 9a or ''S9a.gif'').
In the second case, the displacement was done with respect to the implantation depth (Fig. 9b). Here, the location of Implant 1 was kept constant at a depth of 30 mm, while the depth position of Implant 2 was swept from 3 mm to 27 mm (i.e. 3 mm away from Implant 1), keeping the transverse position equal to zero, and both electrodes aligned parallel to the electrodes of Implant 1 regarding the longitudinal axis (see inlet in Fig. 9b or ''S9b.gif'').   9 shows that the PDL moderately decreases for both implants as the separation distance between electrodes is reduced. For instance, the PDL obtained by Implant 1 when the electrodes of both implants are placed parallel at a depth separation of 3 mm is 9.5 mW (i.e., a reduction of 19.5% compared to the single implant case, see Fig. 5b). However, this power reduction becomes less than 10% for depth separations above 10.5 mm. When two electrodes of different implants are placed consecutively on the longitudinal axis (e.g., a longitudinal separation of ± 30 mm regarding the left electrodes, and a depth separation of 3 mm, see Fig. 9a) the obtained power is also reduced.
In this case, the PDL is reduced 11% compared to the single implant case (Fig. 5b). The substantial reduction in PDL observed in Fig. 9a for large longitudinal separations and in Fig. 9b for large depth separations is not due to the interaction between the implants, but to the non-uniformity of the electric field and to the change of tissue type respectively.

G. FIBROUS CAPSULE AROUND THE IMPLANT
The presence of an artificial implant within the human body will trigger physiological reactions that will end up producing a fibrous tissue capsule around the device, even if the implanted device is considered to be biocompatible [37]. In the systems considered here, this encapsulation will modify the tissue impedance seen by the implants, and therefore the PDL. The two main factors that will contribute to the impedance change will be the capsule conductivity and the capsule thickness. A sub-analysis was conducted to study the potential impact of the encapsulation. The conductivity of the fibrous tissue can be considered mostly frequency independent, with a conductivity of 0.16 ± 0.1 S/m for a fibrotic capsule formed around silicone rubber [38]. Thus, here a fibrotic conductivity of 0.16 S/m was assumed. The implant electrodes were positioned at the default location of Table 1, within muscle tissue. The modeled thicknesses of the fibrotic capsule ranged from 0.1 to 1 mm. Fig. 10 shows that, for the described scenario, a capsule of just 0.1 mm substantially reduces the maximum power that the implant could pick up; the PDL drops from 11.6 mW (Fig.  5) to 6.6 mW. Interestingly, the rate of PDL drop decreases with the thickness and for a tenfold thickness, 1 mm, the PDL is still considerably high (3.7 mW).

H. IMPLANT ELECTRODE SIZE
The size of the implant electrodes has a direct effect on the impedance across them (i.e., Z im ) [20]. To study their contribution to the PDL and PTE, the length of the implant electrodes was changed from 1 mm to 3.5 mm for two different electrode diameters (0.5 mm and 0.25 mm), while the separation distance between electrodes was kept constant. The results show that for the studied range, the PTE depends linearly on the PDL, and confirm that as in [20] the power is maximized by increasing the size of the implant electrodes (Fig. 11).

I. RADIUS OF THE MODELED LIMB
The radius of the limbs varies within the population [39]. In Fig. 12 the radius of the multilayered concentric cylinder was swept from 40 mm to 60 mm, while the thicknesses of the layers were scaled to preserve the proportions of Table 1. The implant was placed at a depth equivalent to half of the cylinder's radius, the transverse position was zero, the longitude of the implant was 30 mm and the longitudinal position of its electrodes was 95 mm and 135 mm (i.e., centered regarding the inter-electrode region). The applied   Table 1. The limb radius increment is illustrated in the supplementary material (''S12.gif'').
current was limited to obtain a maximum SAR of 20 W/kg ranging from 0.34 A rms to 0.58 A rms (Fig. 12a).
The results show, that both PDL and PTE decrease as the multilayered cylinder radius increases (Fig.12b). The PDL for a radius of 60 mm is 8.3 mW (i.e., a reduction of 48% compared to the 40 mm case), and the PTE for the 60 mm case is 0.046% (i.e., a reduction of 69% compared to the 40 mm case).

V. DISCUSSION
The numerical results obtained here indicate that the proposed WPT technique based on volume conduction can be a proper method to power networks of deeply implanted AIMDs at the limbs. This claim is supported by three pieces of evidence presented here. First, powers above 10 mW can VOLUME 9, 2021 be obtained with thread-like thin and elongated implants (e.g., diameter equal to 500 µm and length 30 mm), while the externally applied currents comply with safety limitations (i.e., SAR ≤ 20 W/kg for extremities, and a frequency above 5 MHz). This obtained power is orders of magnitude higher than the power required for most existing AIMDs [10]. Second, the generated electric field is coarsely uniform within the region encompassed by the external electrodes. Thus, the maximum voltage that the implants can receive is almost independent on their implantation depth; deeply implanted devices can be powered. Third, the presence of multiple implants placed just a few millimeters apart has a minor impact on their PDL, while it increases the PTE of the system.
An unanticipated and counterintuitive finding is that as the fat layer becomes thicker, the power picked up by the implant placed at the muscle tissue becomes higher. This is attributable to two main reasons. First, fat tissue is more insulating than the underlying muscle tissue and this enables safe delivery of higher currents: a thicker fat layer around the external electrodes (i.e., lower conductivity than the muscle layer) reduces the edge effect of the external electrodes, making it possible to increase the applied external current. Second, the increase in thickness of the fat layer at the expense of the muscle layer causes that more current will flow through the muscle, increasing the electric field within this layer. Therefore, the PDL of the implants increases.
In this study, for the sake of simplicity and because the waveform of the applied current does not influence the PDL (for the case where the implant load is a purely resistive load [20]) the applied external current was considered purely sinusoidal. However, by applying the external currents in the form of short bursts, the peak voltage that an implant can obtain using volume conduction can be increased. This is relevant for most electronic implants, and particularly for digital implants, as these devices will typically require a minimum voltage to operate. By delivering the ac current in the form of bursts rather than continuously, the received peak voltage can be increased without incrementing the complexity of the implant's electronics (i.e., in most of the scenarios, the implant circuit will not require bulky boost converters). Regarding the safety limitations, if the current bursts are applied with a repetition frequency F (Hz) of tens of hertz, and the burst has a duration B (s), (1) becomes However, it must be considered that, by delivering the current in the form of bursts rather than continuously, the field is no longer purely sinusoidal. It must be verified that the low frequency components of the modulation harmonics cannot produce undesired stimulation [40].
Another interesting result is that, even considering a capsule formation with a thickness of 1 mm around the implant electrodes (i.e., twice the diameter of the considered electrode), the implant would be able to harvest powers above 3.7 mW for the studied geometry. However, it must be mentioned that although in this study it was used a fibrotic tissue conductivity of 0.16 S/m, which is the most used in the literature [38], the electrical properties of the fibrotic tissue are not clearly determined [41], [42]. Therefore, the electrical properties and growth of the fibrotic capsule are key features to study in future long term in vivo studies.
In this work, the size of the implant was intentionally kept constant. This is due to the fact that the separation distance between the implant electrodes and the dimensions of the electrodes were deeply studied in [20]. Considering that the implant electrodes are part of a flexible tubing that includes the implant electronics, if the diameter of this tubing coincides with the electrodes' diameter (500 µm), the volume of the AIMDs would be 6.5 mm 3 . The small cross section of the implants combined with their thread-like shape will allow their percutaneous injection. Previous in vivo studies emphasized the relevance of inserting the whole implant into the target muscle [43]. Muscles experience longitudinal displacements with respect to neighboring muscles, therefore, if an implant crosses an interface between them, it will be subjected to a great mechanical stress. Additionally, due to the corrosive body environment, stress can induce chemical changes and lower the strength of the materials. To avoid these damages we are currently developing and mechanically validating new robust and resilient [44] implant encapsulations using testbeds for accelerated validation of AIMDs [45]. Table 3 compares the obtained results against stateof-the-art AIMDs deployed more than 3 cm deep. This comparison highlights the potential of the proposed technique to obtain high power densities (i.e., PDL per implant volume).
Regarding the external electrodes, the results show that to power deeply implanted devices (e.g., depth above 10 mm) it is preferable a separation distance considerably larger than the length of the implant assuming that the implant is placed in the central region between both external electrodes. However, if this separation distance is too large, the PTE is reduced because, as the separation distance between the external electrodes increases, more power is dissipated through the tissues, while the PDL remains almost constant. Another noteworthy observation related to the external electrodes is that wider external electrodes allow the application of higher external electric currents, thereby increasing the PDL. The results show that for a wider electrode, Z ex is reduced, and consequently, the PTE improves. Nevertheless, the width of the band electrodes is obviously limited by the available space on the limbs.

VI. CONCLUSION
In this study it is numerically demonstrated that the proposed WPT technique based on volume conduction can be a safe and effective method for powering electronic devices deeply implanted within human limbs. Powers above 10 mW can be obtained by thin (diameter < 1 mm) thread-like implants, which can be easily implanted using minimally invasive procedures such as percutaneous injections.
The results show that the maximum PDL and PTE are mostly related to the tissue impedance as measured across the implant electrodes (Z im ). The main feature that affects this impedance, apart from the implant electrode size [20], is the kind of tissue that surrounds the implant: the PDL ranges from 10.4 mW in muscle tissue to 0.4 mW in fat tissue. Other features that affect PDL and PTE are the size and location of the external electrodes. Larger external electrodes allow the application of higher external currents, increasing the PDL. Additionally, to maximize the PTE in deeply implanted devices, the external electrodes must have a separation distance moderately larger than the implant's length. The geometrical dimensions and the tissue anatomy of the limbs also contribute to the PDL and PTE. For example, having a fat layer above 10 mm significantly increases the PDL for an implant placed inside muscle tissue. Furthermore, as the diameter of the modeled limb increases, the PDL and PTE are decreased. Nevertheless, it is worth noting that transferred powers of several mW were obtained in all the cases reported here.
It must be acknowledged that although tens of milliwatts can be safety transferred using the proposed WPT technique, the PTE of this method is considerably lower compared to the PTE of focalized WPT methods (e.g., inductive coupling). However, it also must be acknowledged that the proposed WPT technique is a non-focalized method and that the implants do not interfere substantially in terms of power extraction if they are spaced few millimeters away. Therefore, the proposed WPT technique can be particularly suitable to power distributed networks of minimally invasive implants.
JESUS MINGUILLON was born in Granada, Spain, in 1990. He received the M.Sc. degree in telecommunication engineering, the M.Sc. degree in electrical and electronic engineering, and the Ph.D. degree in information and communication technologies from the University of Granada, Granada, in 2014, 2015, and 2018, respectively. He worked on the design and development of non-invasive brain-computer interfaces and other biosignal-based systems. Moreover, he investigated the use of these systems in daily life contexts and published his main results in international high-impact journals and conferences. He is currently a Postdoctoral Researcher with the Biomedical Electronics Research Group (BERG), Pompeu Fabra University, Barcelona, Spain. His main research focuses on the design and development of power supply and bidirectional communication systems (hardware-software) for volumeconduction-based implants, in the framework of two European projects, namely Electronic AXONs (wireless microstimulators based on electronic rectification of epidermically applied currents) and EXTEND (bidirectional hyper-connected neural systems). , he was with the Biomedical Applications Group, Centre Nacional de Microelectrònica, Bellaterra, Spain. He is the author or coauthor of 50 publications in peer-reviewed journals, four book chapters, and more than 35 conference contributions. He is the inventor or co-inventor of 11 families of patent applications. His research is focused on bioelectrical phenomena and, in particular, on exploring the use of these phenomena for implementing new methods and devices for biomedical applications. Specifically, his main research interests include electroporation, particularly for tissue ablation, electrical bioimpedance for diagnostic purposes, and injectable electronics for neuroprosthetics. VOLUME 9, 2021