Power Transfer by Volume Conduction: In Vitro Validated Analytical Models Predict DC Powers Above 1 mW in Injectable Implants

Galvanic coupling, or more precisely volume conduction, has been recently studied by different research groups as a method for intrabody communications. However, only in a very few occasions its use for powering implants has been proposed and proper analyses of such capability are still lacking. We present the development and the in vitro validation of a set of analytical expressions able to estimate the maximum ac and dc powers attainable in elongated implants powered by volume conduction. In particular, the expressions do not describe the complete power transfer channel but the behavior of the implants when the presence of an electric field is assumed. The expressions and the in vitro models indicate that time-averaged powers above 1 mW can be readily obtained in very thin (diameter < 1 mm) and short (length < 15 mm) implants when ac fields that comply with safety standards are present in the tissues where the implants are located. The expressions and the in vitro models also indicate that the obtained dc power is maximized by delivering the ac field in the form of short bursts rather than continuously. The study results support the use of volume conduction as a safe option to power implants.


I. INTRODUCTION 1
In the field of active implantable medical devices (AIMDs), miniaturization is hampered because of the use of batteries and inductive coupling as energy sources [1]- [4]. Modern battery technologies offer too low energy densities for submillimetric miniaturization [5] and power transfer by inductive coupling, although it offers the additional benefit of allowing communications [6]- [11], requires embedding wide and rigid coils within the implants.
The associate editor coordinating the review of this manuscript and approving it for publication was Masood Ur-Rehman . 1 A small portion of this study (the analysis on absolute maximum attainable power) was presented in a conference paper [48].
In addition, research efforts are being carried out for developing the so-called energy harvesters, also known as energy scavengers, which may provide electrical energy to AIMDs from energy sources available within the human body; typically from movements and temperature gradients [20], [21]. However, all these methods require bulky and rigid parts (e.g., piezoelectric crystals or photodiodes) within the implant that are typically much larger than the electronics they feed. To the best of our understanding, for power consumptions in the order of 1 mW, none of these methods is capable of allowing the development of AIMDs shaped as thin rods with diameters below 1 mm, which is a conformation highly beneficial in terms of minimal invasiveness as it allows percutaneous deployment (i.e., injection instead of surgery) [22]- [24].
We have recently shown in vivo that galvanic coupling, or more precisely volume conduction, can be an effective power transfer method for injectable AIMDs [25].
Remarkably, although galvanic coupling for intrabody communications has been studied lately by different research groups [26], it appears that recently only in a very few occasions, besides in our own publications, its use for powering implants has been explicitly proposed [27]- [29]. It must be noted, however, that in a few additional studies power transfer by volume conduction is also proposed but referred to as capacitive coupling [30]- [32].
To the best of our knowledge, we are pioneering the development of thread-like implants to be deployed deeply within tissues and to be powered by volume conduction. Other authors on power transfer by volume conduction have favored planar configurations to be deployed subcutaneously as these configurations obviously exhibit higher power transfer efficiency. However, we deem those planar configurations to be of little practical interest. They require much more complex implantation procedures than those required for tubular geometries which can be deployed by injection or catheterization [23], [33], [34].
The present study is intended to support the exploration of volume conduction as a power transfer method for allowing the development of very thin and flexible AIMDs suitable for deployment through injection or catheterization. We present the development, and the in vitro validation, of a set of analytical expressions able to estimate the maximum ac and dc powers attainable in elongated AIMDs which are powered by volume conduction. In particular, the expressions estimate the maximum ac and dc powers that the implants can obtain (or ''harvest'') when ac fields that comply with safety standards are present in the tissues where the implants are located.

II. METHODS: MODELS A. GENERAL FEATURES OF THE MODELS
The expressions developed in this study model the presence of an elongated implant within a tissue where an electric field exists due to external delivery of current. The modeled implant consists of two pick-up electrodes at its opposite ends and an electrical load (i.e., the electronics to be powered). This setup would represent hypothetical scenarios of use for the proposed power transfer method such as the scenario depicted in Fig. 1.
The electric field is modeled as homogeneous at the location of the implant and it consists in a sinusoidal waveform of frequency f and amplitude E peak that can be applied continuously or in the form of bursts with duration B and repetition frequency F (Fig. 2). Unless otherwise stated, f = 5 MHz. The angle formed by the field and the implant (i.e., direction defined by the two electrodes of the implant) is α. Unless otherwise stated, α = 0.
In order to develop concise analytical expressions, the following assumptions and simplifications were made (most of them illustrated in Fig. 3): 1) The implant is modeled as two conductive spheres, representing its pick-up electrodes (diameter D and inter-center separation L) and a resistance R Load across FIGURE 1. Hypothetical scenario of use for the proposed power transfer method. An ac electric field is generated in the forearm tissues across a pair of external electrodes. The implants draw electrical power using two electrodes located at their opposite ends. The devices could be used, for instance, as electrical stimulators or sensors.

FIGURE 2.
The delivered electric field consists in a sinusoidal waveform of frequency f and amplitude E peak that can be applied continuously or in the form of bursts with duration B and repetition frequency F . these two electrodes that accounts for the electronics of the device (Fig. 3a).
2) The volume of homogeneous medium surrounding the implant is infinite relative to the dimensions of the implant.
3) The medium surrounding the implant (i.e. body tissue) is resistive. That is, rather than modeling the tissue surrounding the implant with, for instance, a Cole impedance model as it would be appropriate in a multi-frequency study, the tissue is modeled with a single resistance. This assumption is supported by the fact that at a single sinusoidal frequency the passive electrical properties of any material are appropriately modeled by a RC circuit (see Fig. 3b) and, at the frequencies considered here (1 MHz < f < 20 MHz), soft tissues are predominately resistive and the capacitive component can be neglected [35]. In particular, the phase of the admittivity of muscle tissue is only 11.5 • at 1 MHz, 8.7 • at 10 MHz and 10.8 • at 20 MHz [36]. 4) The passive electrical properties of the medium (i.e., body tissue) surrounding the implant are isotropic. This assumption is supported by the fact that when high-frequency current is applied through the tissues, the capacitive membranes of the cells are virtually short-circuited, thus making the tissue isotropic [37]. Even in the case of muscle tissue, which is highly anisotropic at low frequencies, it can be considered isotropic for frequencies above 1 MHz [38]. 5) The skin effect is considered negligible. The skin effect causes ac current density to concentrate near the surface of the conductors and hinders electric field penetration. However, in human tissues, because their relatively poor conductivity, the skin effect only becomes significant at frequencies well above 10 MHz [40]. As latter discussed, for power transfer by volume conduction, we recommend the use of field frequencies below 20 MHz because of the skin effect. 6) The electrode-electrolyte interface impedance is considered negligible. This assumption is supported by the fact that the interface impedance of metal electrodes is roughly equivalent to a capacitance above 0.1 F/m 2 [39] that, for the electrode dimensions and the frequencies considered here, translates into an impedance magnitude in the order of tenths of ohm or a few ohms (e.g., 0.4 for a spherical electrode with a diameter D of 0.5 mm and at a frequency f of 5 MHz). Therefore, the electrode-electrolyte impedance is negligible to the equivalent resistance across the implant electrodes (see Section II.C).

B. LIMITATIONS IMPOSED BY SAFETY STANDARDS
The safety standards developed by IEEE [41] and the International Commission on Non-Ionizing Radiation Protection (ICNIRP) [42], [43] implicitly limit the maximum value of E peak . These safety standards for human exposure to electromagnetic fields identify two general sources of risk regarding passage of radiofrequency (RF) currents through the body: 1) risk of thermal damage due to the Joule heating and 2) risks caused by unsought electrical stimulation of excitatory tissues with harmful effects ranging from mild perception to death by ventricular fibrillation. The limitations specified by the standards regarding heating are indicated as a limitation to the so-called Specific Absorption Rate (SAR). It can be calculated at any point of the tissue with the following expression where σ is the electrical conductivity of the tissue (S/m), ρ is the mass density of the tissue (kg/m 3 ) and E rms is the root mean square value of the electric field in the tissue (V/m). Both the IEEE standard and the ICNIRP guidelines indicate the same SAR limitations averaged over 6 minutes and 10 g mass of tissue for localized exposure in the frequency range from 100 kHz to 3 GHz. For general public and at any human body location the SAR limit is 2 W/kg. For occupational exposure or persons in controlled environments -as would be the case considered here -this limit is 10 W/kg and it is further increased up to 20 W/kg if the localized exposure is at extremities. The SAR limit entails the following limitation for continuous sinusoidal fields and for fields applied in the form of sinusoidal bursts Unless otherwise stated, here the SAR limit of 10 W/kg averaged over 10 g of tissue has been considered. At 5 MHz and in muscle tissue (σ = 0.59 S/m [36], ρ ≈ 1000 kg/m 3 [44]), this field amplitude limit is roughly 180 V/m for FB = 1 and 580 V/m for FB = 0.1.
The standards indicate that above a specified frequency the risks caused by unsought electrical stimulation are negligible and only the SAR limitation applies. This frequency is 5 MHz in the case of the IEEE standard, which we deem more precisely defined than the ICNIRP standard regarding the specification of the limits to avoid the risks of unsought stimulation. This eliminates the risk of unsought electrical stimulation for continuous sinusoidal fields. However, for sinusoidal fields applied in the form of bursts, the IEEE standard rightfully indicates that the low frequency harmonics must be checked and it specifies 5MHz 0 E n MPE n ≤ 1 (4) where E n is the magnitude of the n th Fourier component of the exposure E, and MPE n is the maximum permissible exposure for n frequency. For persons in controlled environments and exposed tissues that are neither the brain nor the heart, the standard determines a MPE of 2.1 √ 2 V/m for frequencies below 3350 Hz. For frequencies above, the MPE is calculated as This condition establishes another limit to E peak that can be lower than the threshold specified by the SAR limitation (3) for low duty cycles (see appendix A). For muscle tissue at 5 MHz (σ = 0.59 S/m [36]), if FB ≥ 0.1 then this second threshold is higher than that imposed by the SAR limitation and hence it can be neglected.

C. ABSOLUTE MAXIMUM ATTAINABLE POWER
The implant is seen as a load by the tissues surrounding it. Therefore, since the passive electrical properties of tissues are linear, a Thévenin equivalent circuit can be implemented (Fig. 3b).
This Thévenin model (circuit outside dashed square/box in Fig. 3b composed of v Th (t) and R Th ) locally models the tissues surrounding the implant, the implant electrodes and the presence of the electric field. a) The implant is modeled as two conductive spheres, representing its pick-up electrodes, and a resistance across these two electrodes that accounts for the electronics of the device. b) The tissue surrounding the implant, the implant electrodes and the presence of an electric field are modeled with a Thévenin equivalent circuit. To obtain closed analytical expressions, the combined impedance of the tissue and the electrodes is simply modeled with a resistance (see II.A). This approximation can be understood as a three-step process: first, the accepted Cole impedance model for living tissues is simplified to an RC circuit considering that the frequency of the field is constant. Second, as stated in the text, because of the high operating frequencies, the tissue is considered predominantly resistive and C T is neglected. Third, since the impedance of the electrodes is much smaller than the impedance across them, the impedance of the electrodes is neglected.
The Thévenin voltage is the open-circuit voltage. That is, the voltage across the implant electrodes when R Load = ∞. If the electric field present at the location of the implant is sinusoidal, the Thévenin voltage (v Th (t)) is also sinusoidal with amplitude [45] V Th peak = E peak L cos (α) (6) or, in general, for any field waveform The power drawn by the load is For the circuit described in Fig. 3b, P Load is maximized when R Load matches R Th . Hence the maximum power that the implant can draw is The Thévenin resistance, R Th , is the equivalent resistance across the implant electrodes as seen from the implant (R Load = ∞). In an infinite volume, the resistance across two conductive spheres with a separation distance much larger than their diameter (L D) can be approximated by [46] Hence the absolute maximum power that the implant can draw is

D. MAXIMUM UNREGULATED DC POWER
Most AIMDs will require dc power for functioning. In Fig. 4 it is represented the most likely circuit topology to be employed in AIMDs for extracting a dc power: a diode bridge full-wave rectifier combined with a smoothing capacitor. For analyzing the behavior of this system, a procedure similar to that described in [47] is followed. Two assumptions are made. First, it is assumed that the ripple factor of the rectifier is almost zero. That is, the load voltage, v Load (t), is assumed to be constant (i.e., v Load (t) = V Load ) once the circuit has reached steady-state.
Second, the diodes of the full-wave rectifier are modeled by the series combination of an ideal diode, an ideal voltage source (V Diode ) which accounts for the forward voltage and a resistance (R Diode ). Initially, it is also assumed that the electric field at the location of the implant is a continuous sinusoidal field. Fig. 5 illustrates the main waveforms of the circuit under the previous assumptions.
The constant load voltage, V Load , can be expressed as being equal to the value of the Thévenin voltage at specific time points where k is an integer, T r is the period of the rectified signal, ω is the angular frequency of the field and ϕ is a phase angle VOLUME 8, 2020 that relates both voltages. In particular, for k = 0 On the other hand, i Load (t) can be expressed as Since And since the average of i C (t) must be zero The diode bridge only conduces when |v Th (t)| > v Load (t) + 2V Diode . Therefore (17), as shown at the bottom of this page.
As indicated in Fig. 5, i Implant flows between kT r − t 2 and kT r + t 2 . These time points can be related to the phase of v Th (t) defining a new phase angle Then I Load can be calculated using (16) and (17) as follows From (13) and (19), the power obtained at the load is It can be numerically verified that for a given V Th peak and a given V Diode there is an optimum value of V Load that maximizes the obtained dc power. The absolute maximum possible dc power will be obtained in the case of no loses, that is, in the case of ideal diodes (i.e., V Diode = 0 and R Diode = 0). In this case (18) can be rearranged as Comparing (13) with (21), we can assert that, for this case, θ is equal to ϕ. Thus (20) becomes Th peak π R Th cos ϕ (sin ϕ − ϕ cos ϕ) . (22) And it can be numerically determined that P Load dc ideal is maximum when ϕ ≈ 1.166 = ϕ opt (V Load ≈ 0.394 V Th peak and R Load ≈ 1.35R Th ). Substituting (6) into (22), and noticing that E peak has to fulfill (2), the maximum dc power that the implant can draw is max (P Load dc ideal ) = 4 max (SAR) ρL 2 π R Th σ cos ϕ opt sin ϕ opt −ϕ opt cos ϕ opt . (23) For conductive spherical electrodes R Th is calculated as (10), and in this case (23) becomes max (P Load dc ideal ) = 4 max (SAR) ρDL 2 cos ϕ opt sin ϕ opt − ϕ opt cos ϕ opt .
The ratio between the maximum dc power that the implants can draw (24) and the absolute maximum attainable power (11) is The above expressions were found under the assumption of a continuous sinusoidal field. For fields in the form of 37812 VOLUME 8, 2020 sinusoidal bursts with duration B and repetition frequency F, where the product FB is the duty cycle of the bursts and it is comprised between FB min and 1, where FB min is the minimum duty cycle to prevent unsought stimulation. Hence the obtained power is If V Th peak 2V Diode the phase angle θ defined in (18) will be equal to ϕ, see (13). Therefore, replacing θ by ϕ, and substituting (13) into (27) lead to

E. MAXIMUM VOLTAGE-REGULATED DC CURRENT
In most practical cases the dc voltage will have to be regulated at a specific magnitude (e.g., +3.3 V) for powering the circuitry of the AIMD. Although switching regulators may be an option, because of size limitations, low-dropout (LDO) linear regulators will probably be preferred.
The circuit in Fig. 6 represents such scenario. Linear regulators provide a regulated dc voltage at their output (V Load ) if the dc voltage at their input satisfies V Reg ≥ V Load + V Drop , where V Drop is the so-called dropout voltage, and they waste a small portion (I Q ) of the input current (I Reg ) so that I Load = I Reg − I Q .
In this case, since V Load is fixed, the objective is not to determine which is the maximum dc power at the load (V Load I Load ) but the maximum current that can be obtained.
The load seen by the diode-bridge and the smoothing capacitor (C 1 ) is the combination of the actual load (R Load ) and the load of the LDO regulator. Hence the power expressions obtained above must be equaled to The maximum regulated dc current that can be obtained (I Load ) is that one that ensures that the regulator works properly, that is, that for V Reg = V Load + V Drop .

III. METHODS: SETUP FOR IN VITRO VALIDATION OF THE MODELS
An in vitro experimental setup that replicates the assumptions made to develop the previous expressions was implemented (Fig. 7). The ac electric field was delivered by two 86.5 mm×30 mm parallel aluminum plates acting as the external electrodes.  These two electrodes were attached to opposite internal sides of a polycarbonate container (inner dimensions: 84 mm×86.5 mm×30 mm) filled with a NaCl solution whose conductivity was measured with a conductivity tester (HI 98312 by Hanna). Sinusoidal voltages, either continuous or in the form of bursts, were generated across the electrodes by the combination of a function generator (4064 by BK Precision) and a custom-made high-frequency (HF) high-voltage (HV) amplifier.
In all instances, the applied voltage was measured using a digital oscilloscope (TPS2014 by Tektronix). Unless otherwise stated, the amplitude of the applied voltage (A) was adjusted to obtain a SAR of 10 W/kg (in particular, for the described experimental setup, this corresponds to 2.2 W). That is, where d is the distance between the plate electrodes (84 mm). The dimensions of this setup were chosen to be large for maximizing the uniformity of the electric field at the location of the pick-up electrodes while allowing the use of the custom-made high-frequency high-voltage amplifier (limited to 180 V, 1 A and 5 MHz).
The pick-up electrodes of the implants were represented by stainless steel (SAE 317) spherical electrodes with three different diameters: 0.5 mm, 1 mm, and 1.5 mm. Each electrode was laser welded to a 5 cm piece of 32 AWG enameled copper VOLUME 8, 2020 wire. The ends of the copper wires were connected to the validation circuits described below.

A. VALIDATION CIRCUIT FOR ABSOLUTE MAXIMUM ATTAINABLE POWER
To validate (11), the pick-up electrodes were connected to a high-precision potentiometer acting as a variable resistor (see validation circuit A. in Fig. 7). Before connection, the resistance of the potentiometer (R Load ) was adjusted to match R Th as provided by (10). Such adjustment was performed using a multimeter (38XR-A by Amprobe). After connection, the load voltage, v Load (t), was recorded using the digital oscilloscope and max (P Load ) was calculated as the square of the rms recorded voltage divided by the value of R Load , considering a time interval of 5 µs.

B. VALIDATION CIRCUIT FOR MAXIMUM UNREGULATED DC POWER
To validate (20) and (27), it was implemented a PCB with the implant circuit represented in Fig. 4. The diodes of the diode-bridge were Schottky diodes (RB521ZS-30 by Rohm Semiconductor). The capacitor C had a capacitance of 10 µF. R Load was again a high-precision potentiometer acting as a variable resistor.
The diode model parameters used in the analytical expressions were: V Diode = 0.2 V and R Diode = 14 .
In the experiments reported below in which the value of R Load was not fixed, the expression for P Loaddc was numerically maximized for obtaining the optimum ϕ value and hence the optimum R Load was found for maximum power. When the value of R Load was fixed, the expression for P Load dc FB was numerically maximized for obtaining the optimum FB value.

C. VALIDATION CIRCUIT FOR MAXIMUM VOLTAGE-REGULATED DC CURRENT
A LDO regulator (ADP7112 by Analog Devices) adjusted to produce +3.3 V was mounted on the referred PCB to implement the implant circuit in Fig. 6. To study its analytical behavior, its quiescent current (I Q ) was set to 80 µA and its dropout voltage (V Drop ) to 30 mV. Both values are given by the manufacturer for I Load = 10 mA. The capacitor C had a capacitance of 10 µF.

A. DEPENDENCY ON INTER-ELECTRODE DISTANCE AND ON ELECTRODES DIAMETER
At f = 5 MHz, for a SAR of 10 W/kg and for the properties of a saline solution that resembles the properties of muscle tissue, Fig. 8 displays a set of results obtained by applying the analytical expressions together with the corresponding validation results using the in vitro setup. In particular, the absolute maximum attainable power, max (P Load ), in Fig. 8a, and the maximum obtained dc (unregulated)  (27). Circles: experimental results; a) using Fig. 7.A, b) and c) the circuit represented in Fig. 4. a) Absolute maximum attainable power. b) maximum dc (unregulated) attainable power for a continuous sinusoidal field. c) maximum dc (unregulated) attainable power for a sinusoidal field applied in the form of bursts (F = 10 kHz, B = 50 µs).
power, max (P LoadDC ) and max (P LoadDC FB ), in Fig. 8b and in Fig. 8c, are represented as a function of the inter-electrode distance (L) and the electrodes diameter (D). Besides confirming the validity of the models, these results indicate that dc powers above 1 mW can be obtained in very thin (diameter < 1 mm) and short (length < 15 mm) implants. Harvested dc power is larger when the sinusoidal electric field is delivered in the form of bursts (Fig. 8c), than when it is delivered continuously (Fig. 8b). However, even when the field is delivered in the form of bursts, the obtained dc power is still considerably smaller than the absolute maximum attainable power (Fig. 8a). This fall in performance becomes especially noticeable for short inter-electrode distances. As further stressed in the following sub-sections, fall in performance is more severe for short implants because the voltage picked up by the implant electrodes approaches the magnitude of the forward voltage of the rectifier diodes and hence the relative energy losses are higher.

B. DEPENDENCY ON THE CONDUCTIVITY OF THE MEDIUM
As (11) indicates, and as we in vitro demonstrated in [48], the absolute maximum attainable power is independent of the conductivity of the medium.
Therefore, here only the dependency of max (P LoadDC ) on the conductivity of the medium is inspected. In Fig. 9, harvested dc power is surveyed for three different NaCl solutions with conductivities representative of those found in human tissues. In the case of a continuous field (Fig. 9a) it can be observed a notable dependency on the conductivity of the medium.
For instance, for L = 3 cm, the obtained dc power for σ = 0.2 S/m nearly doubles that for σ = 1.52 S/m. The main reason for this dependency is that E peak , and hence V Thpeak , must be reduced at higher conductivities for keeping the SAR value (2) and that causes bigger relative losses at the diodes of the full-wave rectifier.
On the other hand, if the field is applied in the form of bursts (Fig. 9b), since E peak , and hence V Thpeak , can be higher for the same SAR, the obtained dc power is not only higher but also less dependent on the conductivity of the medium. For instance, for L = 3 cm, the dc power for σ = 0.2 S/m is only about 20% higher than that for σ = 1.52 S/m.

C. DEPENDENCY ON LOAD RESISTANCE AND IMPEDANCE MATCHING
In the two previous sub-sections it was applied the optimum R Load to maximize the harvested power. Since, in most scenarios, R Load will be a non-adjustable parameter, it is worth inspecting how the harvested power depends on it.
In Fig. 10 it is displayed the dependency of the maximum attainable power on the value of R Load for the range from 200 to 200 k in the case of a modeled implant with D = 1 mm and L = 30 mm in a medium with σ = 0.58 S/m. For these parameters R Th ≈ 550 . If the voltage is not rectified, that is, if the resistive load is directly connected to the implant electrodes, the dissipated power at the load (P Load ac ) is maximum when R Load = R Th . That is the point at which the absolute maximum attainable power is reached. Similarly, a (lower) maximum harvested dc power is found at a R Load = R Load Opt dc if the sinusoidal electric field is applied continuously.
In both cases the obtained power decreases substantially when R Load is lower or higher than the optimum one. However, if the electric field is applied in the form of bursts, the drop in harvested dc power can be mitigated, and actually reverted, for loads above R LoadOptdc . The use of bursts intro-   (11). Blue trace and markings: maximum dc power for an ac field applied in the form of bursts (duty cycle optimized) (27). Solid lines: results from the analytical expressions. Markings: experimental results using the circuits represented in Fig. 4 (dc) and Fig. 7.A (ac). b) Optimum duty cycle, FB, for obtaining the results in subFig. a).
duces an additional parameter for allowing power optimization: the duty cycle, FB.
By reducing FB the loss in performance due to the increase of R Load can be compensated. Moreover, since by reducing FB the field amplitude, E peak , can be increased for the same SAR value, the losses associated with the diodes are reduced and hence the power obtained at the load is higher VOLUME 8, 2020 than that obtained for a continuous electric field at R Load = R Load Opt dc . As it can be observed in Fig 10, the obtained dc power reaches values (∼ 6 mW) close to the absolute maximum attainable power (∼ 7 mW). Nevertheless, FB cannot be reduced without limit.
As explained in Section II.B and in the appendix, FB must be limited to prevent unsought stimulation. In the case illustrated in Fig. 10, FB was limited accordingly and that causes the drop of power observable when R Load is larger than 57 k .

D. MAXIMUM VOLTAGE-REGULATED DC CURRENT
For a range of inter-electrode distances (L ∈[0.5 cm, 5.5 cm]), Fig. 11 shows the maximum dc current attainable at the output of the linear regulator (Fig. 6) when the ac electric field is applied in three different ways: continuously, in bursts with a constant duty cycle (FB = 0.2) and in bursts with a duty cycle optimized for the load seen by the diode-bridge.
For short implants it can be clearly noticed the benefit of using ac fields in the form of bursts. If a continuous ac field is employed, voltage regulator operation is impossible for implants of 2 cm or shorter whereas, for that same length, if the field is applied in the form of bursts, operation is possible and currents of 500 µA can be readily drawn by the load.
In addition, the results displayed in Fig. 11 also illustrate the importance of optimizing the duty cycle: to obtain a current of 500 µA the minimum implant length is 1.65 cm in the case of optimum FB whereas such minimum length has to be increased to 2.56 cm if the constant duty cycle is used.

V. DISCUSSION
As stated in the introduction, only in a very few occasions the use of volume conduction for powering AIMDs has been proposed and studied. And, to the best of our knowledge, no commercial AIMDs use it. The reasons why other teams are reluctant -or did not conceive -to use volume conduction for power thin and elongated implants can only be guessed.
We conjecture that the reluctance mainly arises from not recognizing two facts about volume conduction. First, large magnitude high frequency (> 1MHz) currents can safely flow through the human body if applied as short bursts. Second, to obtain a sufficient voltage drop across its two intake (pickup) electrodes, the implant can be shaped as a thin and flexible elongated body (Fig. 1) which is a configuration highly suitable for minimally invasive deployment through injection or catheterization.
One of the few studies in which volume conduction has been studied for powering elongated implantable devices is [29]. Although that study reinforces the main notion of the present study (i.e., volume conduction can be an effective method for powering AIMDs), it must be noted that some of the results and conclusions of that study are implausibly benign according to the assumptions and results of the present study. For instance, that study reports that with an interelectrode distance of only 1 mm the implant can harvest a dc voltage of 0.44 V when the SAR is below 1 W/kg (f = 13.9 MHz, FB = 1). We deem this result is implausible because a harvested dc voltage of 0.44 V by an implant placed 5 cm inside the tissues would imply V Th ≥ 0.44 V which in turn would imply E Th ≥ 440 V/m in its surrounding tissues. Furthermore, since volume conduction is a non-focalized wireless power transfer method, the electric field is expected to be higher in more superficial locations. Although we do not have enough information to compute the exact value of the SAR averaged over 10 g of tissue, the magnitude of the electric field in the implant location together with the distance between external electrodes (from 2 to 5 cm), leads us to be skeptical about the feasibility of achieving the powers reported in [29] without exceeding the limits established by international safety standards. Another result of that study that we deem implausible is the absence of a significant impact on the obtained power by the relative angle between the external electrodes and the implant electrodes. Since the electric field direction depends on the orientation of the external electrodes, such result would imply that the obtained power is independent of α. And this, as illustrated in Fig. 12, is far from what our models and previous results [25] indicate: if the electric field is perpendicular to the implant then the harvested power is null.
Powering AIMDs by volume conduction requires the delivery of currents through external electrodes so that electric fields appear in the tissues where the AIMDs are located. For the sake of brevity and simplicity, the aspects related to current delivery are intentionally neglected in the present study; the presence of a homogeneous electric field is assumed. We will treat these aspects in subsequent studies. However, there is an aspect worth briefly discussing now: the impact of the skin effect. In particular, at 10 MHz the current density 10 cm deep within muscles, J 10 cm , is approximately half 37816 VOLUME 8, 2020 results from the analytical model (11). Circles: experimental results (the two-spheres arrangement for the implant electrodes in Fig. 7 was rotated with respect to the plate electrodes) using the circuit represented in Fig. 7.A.
of the current density at the surface, J 0 ; and at 100 MHz J 10cm = 0.1J 0 . Therefore, it appears sensible not to use frequencies significantly above 10 MHz if it is intended to power deep seated implants. That is why we recommend the use of frequencies below 20 MHz (but above 5 MHz to prevent unsought stimulation).
Another limitation of the present study is that, again for the sake of simplicity, the shape of the pick-up electrodes was restricted to spheres. This limits their surface area and hence hinders reducing R Th . It is worth noting that, for a given electrode diameter (e.g., the maximum allowed by the lumen of a needle), larger powers than those reported here will be harvested if the electrodes are shaped as cylinders rather than as spheres since they will have a lower R Th [46]. Recalling from (8), this will imply higher power harvested.
The results obtained here show minor differences between the model and the data obtained in vitro. These differences are most likely due to geometrical tolerances in the in vitro setup, and limitations in terms of resolution and accuracy imposed by the measurement equipment.
In terms of power transfer efficiency, the specificity of the setup used in this study limits the direct comparison of the results with those obtained with other wireless power transfer technologies such as inductive coupling and ultrasonic power transfer. Therefore, this comparison falls out of the scope of this paper. However, it is worth noting that we envision systems with the power transfer efficiencies (PTE) ranging from 0.01% to 1%. For instance, we performed a numerical study in which it was simulated power transfer by volume conduction to stimulation implants in an anatomically realistic leg model and we obtained a PTE of about 0.05% with an input power requirement of less than 4 W, which is low enough to grant the use of small portable rechargeable batteries [49]. Additionally, it must be noted that the results obtained here demonstrate that the maximum attainable power can be well above 1 mW and that such power is enough to supply the electronics of most AIMDs. For instance, power consumption of the electronics of a recently commercialized injectable glucose sensor (Eversense by Senseonics Inc.) is in the order of 250 µW [50].

VI. CONCLUSION
This study supports the use of volume conduction as a safe option to power very thin and flexible AIMDs. It provides a theoretical framework for optimizing both the design of the implants and the applied electric fields used to power them.
The developed analytical expressions, which are in vitro validated in the present study, indicate that dc powers above 1 mW can be readily obtained in very thin (diameter < 1 mm) and short (length < 15 mm) implants when ac electric fields that comply with safety standards are present in the tissues where the implants are located. Other main conclusions are: 1) The obtained dc power is maximized by delivering the ac field in the form of short bursts rather than continuously.
2) The maximum attainable dc power depends on the conductivity of the medium. Such dependency is minimized by delivering the ac field in the form of short bursts rather than continuously. 3) Both the absolute maximum attainable power and the maximum attainable dc power exhibit a distinctive maximum for a specific load resistance (i.e., optimum load resistance). If the ac field is delivered in the form of bursts rather than continuously, it is possible to set the value of the optimum load resistance by adjusting the duty cycle of the bursts.
The use of volume conduction for powering AIMDs could overcome some of the limitations imposed by other technologies in terms of size and invasiveness. This would improve the deployment and usability of AIMDs in several applications such as bioelectronic medicines [51], neuro-prostheses [52] and implantable sensors [53].

APPENDIX
Appendix A. Maximum field amplitude limited by unsought simulation in the case of sinusoidal electric fields applied in the form of bursts.
The waveform obtained for this case considering a single period can be mathematically expressed as where E peak must fulfill (3)  the E n magnitude of the n th Fourier component of the exposure field can be determined. Finally, the obtained set of E n has to fulfill (4) to avoid possible stimulation. In this studied case, (4) limits the maximum E peak when FB tends to be much lower than 1. This fact is illustrated in Fig. 13. Considering a SAR = 10 W/kg, f = 5 MHz, σ = 0.59 S/m, and ρ = 1000 kg/m 3 it can be noticed that for FB> 0.05 and F > 100 Hz the maximum E peak is limited due to the SAR restriction (3). On the other hand, when FB is decreased (4) has to be considered. Remarkably, considering the same FB, for higher F, higher E peak amplitudes can be safely applied.
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, Spain, 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 volume-conduction-based implants, in the framework of two European projects: Electronic AXONs (wireless microstimulators based on electronic rectification of epidermically applied currents) and EXTEND (Bidirectional Hyper-Connected Neural Systems).
ANTONI IVORRA received the Ph.D. degree in electronics engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 2005. He is currently an Associate Professor with Universitat Pompeu Fabra (UPF), Barcelona, where he leads the Biomedical Electronics Research Group. Prior to joining the UPF, in 2010, he enjoyed a four-year postdoctoral stay at the University of California at Berkeley, from 2005 to 2009, followed by an eight-month Postdoctoral position at the CNRS-Institut Gustave Roussy, Villejuif, France. From 1998 to 2005, 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 topics are electroporation, particularly for tissue ablation, electrical bioimpedance for diagnostic purposes, and injectable electronics for neuroprosthetics.