Impedance detection of the electrical resistivity of the wound tissue around deep brain stimulation electrodes permits registration of the encapsulation process in a rat model

Microelectrodes were custom-made from round Pt/Ir alloy (Pt90/Ir10) wires, which were insulated with polyesterimide and bared at the tips (Figure 1). The microelectrodes were purchased from Polyfil, Zug, Switzerland (unipolar electrodes) and FHC, Bowdoin, ME, USA (bipolar electrodes). Their distal ends were connected to cables with biocompatible insulation. To avoid excessive heating by soldering, the cables were connected with conductive silver glue, which was covered by biocompatible heat-shrink tubing and sealed with biocompatible silicon glue (NuSil Technology, Carpinteria, CA, USA). The unipolar electrodes were driven and measured against a gold-wire counter electrode (length 30 mm, diameter 200 μm). The bipolar electrodes did not require an additional counter electrode.

Figure 1

Control and animal measurements of the impedance were conducted using a Sciospec ISX3-spectrometer (Sciospec Scientific Instruments, Pausitz, Germany) with a HP16047D-test fixture (Agilent Technologies Deutschland GmbH, Böblingen, Germany). The DBS and counter-electrode wires were connected to the test fixture. The Sciospec measuring software was programmed to average over five measuring cycles at a measuring voltage of 12.5 mV_PP in a frequency range from 100 Hz to 10 MHz. The real (Real(Z*)) and imaginary (Imag(Z*)) components of 401 frequency points were logged with a personal computer. The measuring frequencies were equidistant at a logarithmic frequency scale.

Prior to each measurement, the impedance spectrometer was open, short and load calibrated. Each measurement was repeated three times to improve the statistical validity. Finally, the logged data were transferred to Matlab (The MathWorks™, Version 7.9.0.529, Natick, MA, USA) for further processing. The impedance was measured within the frequency range from 100 Hz to 10 MHz for two reasons: i) the steep slopes of the needle-shaped stimulation pulse of the DBS signal are rich in high harmonic frequencies [26] and ii) complete impedance spectra were required for the extraction of the tissue resistivities (c.p. also to [33]).

Control measurements of the impedance were conducted in beakers with a calibration solution that had a conductivity of σ_Cal=0.1307 Sm^-1 (HI77100C, HANNA Instruments Deutschland GmbH, Vöhringen, Germany). For measurements, the electrodes were immersed to approx. 90% of their shaft lengths. The gold-wire counter electrode was oriented approximately perpendicular to the shaft of the unipolar electrode (cf. Figure 3). Figure 2 provides a schematic summary and introduces the terms used in the description of the electrode impedance.

Figure 2

Figure 3

In the complex plane, the transition frequency from a linear (“constant phase”) to a semicircle shape indicates the cessation of electrode polarization processes. Their contributions to the overall impedance vanish for increasing frequencies, which renders the impedance to pure resistance and capacitance contributions of the wiring and bulk medium (or of the brain tissue) that surrounds the electrode [26]. For ideal electrodes and pure electrolytes, the parallel circuit of R_Bulk and C_Bulk would result in a perfect semicircle with its center at the abscissa. The transition frequency between the linear and semicircle branches depends on the electric medium and electrode-surface properties. The constant phase element (CPE) provides a phenomenological description of the frequency-dependent impedance contributions of the electrode processes and the double layer properties at the electrolytic interfaces (see equivalent circuit in Figure 6; [42, 43]). Electrotechnically speaking, the CPE may be considered a serial cascade of parallel resistor-capacitor pairs with consecutively increasing time constants [44].

When impedance data were recorded with zero bias voltage, special elements for the charge-transfer resistance and the double-layer capacitance may be neglected [26]. In this case, the CPE may be mathematically described by [45, 46, 47]:

where ω indicates the angular frequency 2πf and j=−1.$j=\sqrt{-1}.$Q is a constant with a dimension of Fs^(α-1). The exponent α, with 0 < α < 1, describes the slope of the linear branch in the complex plot. The limiting cases α=0 and α=1 describe a purely resistive or capacitive impedance behavior,

respectively. The straight lines in Figure 2A are described by equation (1), and the semicircles reflect the impedance properties of the parallel resistor-capacitor pair.

For a cubic measuring volume with bulk properties, confined by two plane-parallel electrodes of area a² and distance a, R_Bulk and C_Bulk are:

where ε₀ = 8.85 10 ^-12 F/m and γ = a²/a = a indicating the permittivity of vacuum and the cell constant (electrode-geometry factor), respectively. These parameters determine the so-called Maxwell-Wagner frequency, f_MW, which is an inherent property of each aqueous medium:

For the calibration solution with a conductivity of σ_Cal = 0.1307 Sm^-1 and a relative permittivity of ε_rel = 80, the Maxwell-Wagner frequency is approx. 29 MHz. This frequency would also be obtained from R_Bulk , C_Bulk and the generalized cell constant γ. Nevertheless, from the semicircles of the bipolar and unipolar electrodes in Figure 2A, characteristic frequencies of 825 kHz and 1.79 MHz were identified. The explanation of these experimental deviations requires higher capacitive contributions than predicted by equation (3) (see subsequent discussion).

Experimental determination of the cell constants Because of the uncertainties in the capacitive contributions, we experimentally determined the cell constants γ in [m] of the electrodes from their resistances. The resistances were derived from the impedance spectra of four unipolar and four bipolar electrodes in a calibration solution. The specific conductivity σ_Cal = 0.1307 Sm^-1 of the calibration solution corresponds to ρ_Cal = 7.6511 Ωm at 21.3 °C. It was cross-checked with a temperature-compensated conductometer (Seven2Go, Mettler Toledo GmbH, Gießen, Germany). In the calibration solution, the cell constant γ may be obtained from the resistance R_Cal and the specific conductivity σ_Cal [48]:

To determine R_Cal, the CPE and bulk contributions must be separated in the impedance spectra. Thus, we used a phenomenological interpretation of the impedance spectra (Figure 2A). While the low frequency CPE part of the measuring data enabled a linear regression, the high frequency (bulk) part could be fitted by a semicircle [26, 49]. Matlab was used for fitting. The overall impedance of the electrode results from the series connection of the two contributions that can be obtained from equations (1) and (2) (Figure 2C). Accordingly, extrapolations of the straight line towards high frequencies and the semicircle towards low frequencies provide the same point on the abscissa, which corresponds to the electrode resistance R_Cal (Figure 2A).

With the cell constants, the R_Bulk data of the animal models may be recalculated into effective specific resistivities:

ρ_Bulk represents integrative tissue parameters with timedependent contributions of the free interstitial fluid, adventitia, and neuronal tissue. Of the two methods for the extrapolation of R_Bulk from the complex plot, we prefer fitting of the straight branch (Figure 2A). We believe that this approach ensures a better comparability of the calibration with the in vivo tissue data. The main reason is the complicated frequency dependence of the in vivo impedance in the semicircle range. This is caused by the dielectric structure of the brain tissue, which introduces additional dispersion processes and a deformation of the semicircle (cf. subsequent experimental spectra).

Male Wistar Han rats (240 - 260 g) were obtained from Charles River Laboratory, Sulzfeld, Germany and housed under temperature-controlled conditions in a 12 h lightdark cycle with conventional rodent chow and water provided ad libitum. The study was conducted in accordance with the European Community Council directive 86/609/EEC for the care of laboratory animals and was approved by the local Animal Care Committee (LALLF M-V/TSEM/7221.3-1.2-019/10).

For electrode implantation, naive rats were anesthetized by intraperitoneal application of ketamine hydrochloride (10 mg per 100 g body weight, Ketanest S^®, Pfizer, Karlsruhe, Germany) and xylazine (0.5 mg per 100 g body weight, Rompun^®, Pfizer). The eyes were protected from dehydration using Vidisic^® (Bausch and Lomb, Berlin, Germany). The surgical procedure was performed using a stereotactic frame (Stoelting, Wood Dale, IL, USA). The skull was opened by a dental rose-head bur (Kaniedenta, Herford, Germany) prior to electrode implantation into the right hemisphere. The stimulating tips of the electrodes were localized in the subthalamic nucleus (STN), which comprises one of the most important target regions for the treatment of PD in humans. The tip coordinates relative to bregma were: anterior-posterior (AP) = -3.5 mm, medial-lateral (ML) = 2.4 mm and dorsalventral (DV) = -7.6 mm (Paxinos and Watson, 2007). The electrodes were fixed to the skull by an adhesive-glue bridge of dental acrylic (Pontiform automix 10:1, Müller & Weygandt GmbH, Büdingen, Germany), including an anchor screw that was tightened to the skull on the left hemisphere. Figure 4 illustrates the unipolar electrode model.

Figure 4

Following electrode implantation, the cables of the stimulating and counter electrode contacts were subcutaneously implanted with a central dorsal outlet port. After surgery, the wound was sutured, and the rats received 0.1 ml novaminsulfone (Ratiopharm, Ulm, Germany) and 4 ml saline subcutaneously. To prevent hypothermia, the rats were exposed to red light until the normalization of vital functions.

For chronic instrumentation, a plug connector was crimped to the cables of the DBS electrodes one week after electrode implantation. The connector ensured flexibility in the use of the commercial rat jackets (Lomir Biomedical, Quebec, Canada), which contained the stimulators and batteries in a custom-made textile backpack. The miniaturized DBS stimulators had a PMMA housing and a 12-V battery for the constant current source of the stimulator. For details, see [22].

The rats received a unipolar electrode with a gold-wire counter electrode or a bipolar electrode with parallel shifted tips (Table 1). The same stimulators were used for both electrode types. Impedance measurements were conducted daily for approx. 12 minutes. For measurements, the rats were anesthetized with ketamine/xylazine.

For an ideal electrode, the same cell constant would apply for the R_Bulk and C_Bulk values obtained from the conductivity and permittivity of the immersing medium (equation (2)). Accordingly, the R_Bulk−C_Bulk time constant in Figure 2 with R_Bulk = R_Cal and C_Bulk = C_Cal should correspond to the Maxwell-Wagner frequency of the calibration solution according to equation (3) (refer to Table 2 for parameters). Nevertheless, to explain the time constants of the semicircles in Figure 2A, higher electrode resistances or additional electrode capacitances (C_Add) that were significantly increased compared with the bared electrode tips (Table 2) had to be assumed for both electrode types. (C_Add) were approx. 10.5 pF and 10.9 pF for the unipolar and bipolar electrodes, respectively. In tests with water, alcohol and air, these capacitances were changed consistently with the different permittivities of the media (results not shown).

Clearly, the largest contribution to (C_Add) stemmed from the immersed parts of the electrode shafts in aqueous solution (Figure 2C). Reasonable estimates for the shaft capacitances were obtained from the effective areas of the electrode insulations. These areas were estimated with the assumptions of open cylinders located in the central plains of the insulations with diameters of 225 (unipolar) and 150 μm (bipolar) (cf. Figure 1). In the estimation of the shaft capacitances, we assumed mean immersion depths of the electrodes of 90%, an insulation thickness of 25 μm, and a relative permittivity of 3 for the polyesterimide insulation [51].

For the unipolar electrode, a shaft capacitance of approx. 8.7 pF was obtained. This is slightly below the required (C_Add) of approx. 10.9 pF likely because the stray capacitances of the measuring beaker and the external wiring were not taken into account. For the bipolar electrode, geometrical capacitance estimations are not easy. Applying the previously described estimations to the two parallel wires, capacitances of approx. 5.7 pF were obtained for each wire insulation. Nevertheless, the capacitances cannot simply be summed because they are switched in series with respect to their metal cores. We suggest that stray capacitances across the measuring beaker, between the individual wires and the shielding of the impedance device, play a role for each wire. This finding, together with the external wiring likely led to the obtained (C_Add) of approx. 10.5 pF.

Other sources of experimental deviations include differences in the immersion depth of the electrodes, as well as uncertainties in the wiring of the measuring setup, the uneven thickness of the insulation layers and the areas of the bared electrode tips. We suggest that a general understanding of the capacitive electrode properties is important for a correct picture. Thus, it should be noted that the capacitive electrode environment and the wiring conditions are different in the animal experiments. In this manuscript, an exact quantitative interpretation of the capacitive electrode properties is not required for parameter interpretation (see discussion).

We demonstrated that R_Bulk=R_Cal correctly reflects the cell constant of the bared electrode tips and the resistivity of the calibration solution. The reason is that resistive currents require galvanic contact between the electrolytic solution and the electrode metal, which is present only at the bared electrode tips. Table 2 summarizes the experimental and numerical results. The good agreement between the experimental and numerical cell constants demonstrates the applicability of our approach.

Figure 5 illustrates simulated distributions of the electric field around both electrode types. Please note that the values were calculated with average mesh sizes of approx. 16 μm (unipolar electrode) and 10 μm (bipolar electrode). This size is of the order of the effective cell-soma size of approx. 10 μm.

Figure 5

Figure 6

Field strength peaks were identified at the electrode tips and the edges of the passivations. For the two electrode types, maximum peaks of 33.9 kV/m (unipolar) and 78.2 kV/m (bipolar) appeared at the electrode front edges (pointing arrows in Figure 5). The factor of 2.3 between the field strength peaks corresponds to the quotients of the cell constants (cf. Table 2; quotient of experimental results: 2.28; quotient of numerical results: 2.37).

Figure 6 presents impedance measurements with the bipolar electrodes in dependence on the day after implantation. Each of the plotted Real(Z*) spectra was averaged over four electrodes of different animals. Despite the individual differences, the scatter is also a result of the low signal-to-noise-ratio caused by the low measuring voltage (12.5 mV_PP). It was used to minimize current effects on the brain tissue.

In the low frequency range of the spectra, the characteristic increase in the Real(Z*) values for decreasing frequencies suggests the influence of electrode processes, which makes the extraction of tissue properties difficult [26, 52]. Below 14 kHz, a characteristic decrease in the Real(Z*) spectra occurred at day two after implantation, followed by a systematic increase up to day seven. DBS was initiated at day eight and resulted in another impedance decrease at day 13.

At first glance, the relations were less clear when the raw data of the brain impedance were plotted in the complex plane. Figure 7 presents examples. Similar to the calibration solution, the high frequency branches reflect the bulk properties of the medium that surrounds the electrode. Nevertheless, the additional structural dispersions in the encapsulation tissue deformed the plots and flattened the semicircle branches [29]. Please compare Figure 2A (electrodes measured in calibration solution) with Figures 7A1 and 7B1 (electrodes measured in the brain).

Figure 7

In general, semicircles that span a larger abscissa range indicate an increased resistance of the encapsulation tissue. The slopes of the straight (“constant phase”) branches of the unipolar electrodes were reduced with time after implantation, whereas the slopes of the bipolar electrodes were rather stable (compare Figures 7A and B).

The interpretation of the daily measurements required a rigorous refinement of the large data numbers. In our straightforward approach, we derived the R_Bulk values by fitting the CPE branches (cf. Figure 2). The effective specific resistivities ρ_Bulk were calculated with equation (5) using the experimental cell constants of Table 2. In the calculations, the SEMs of the cell constants and the laws of error propagation were taken into account. Our data interpretation discriminated against fluctuations of single impedance values.

Figure 8 summarizes the resistivities of the bipolar and unipolar electrodes for two weeks after implantation. The resistivities of the brain tissue around the tips were similar for both electrode types and exhibited characteristic time-courses. Nevertheless, the mean resistivities of the bipolar electrodes changed within a substantially larger range (between approx. 2.2 and 28 Ωm) compared with the unipolar electrodes (between approx. 4.6 and 9.2 Ωm). Typically, the resistivities decreased at the second day after implantation, followed by an increase. The initiation of the DBS stimulation at day eight resulted in another resistivity decrease.

Figure 8

Figure 9 indicates the resistivity behavior of each of the bipolar electrodes summarized in Figure 8. The differences in the four individual DBS electrodes suggest that the ingrowth process may be individually traced.

Figure 9

Custom-made unipolar and bipolar Pt/Ir electrodes with polyesterimide insulated shafts enabled long-term experiments for more than six weeks [22]. The analysis of their physicochemical properties and impedance spectra led to the idea of registering changes in the effective specific resistivity of the medium around the bared electrode tips to trace the ingrowth process in the rat brain.

In previous experiments with the unipolar setup, we tested different counter electrodes, which were pierced

into the neck skin of rats [34]. The counter electrodes comprised dental wire rings composed of biocompatible, nickel-free steel alloy or arrays of small suture clips. The arrays were also used to detect “pure” counter electrode effects between pairs of clips and enabled the impedance changes to be monitored during the ingrowth process. We determined that the tested counter electrodes, their position, shape or material did not influence the impedance of the unipolar electrode (results not published). In the experiments, we also tested the goldwire electrode used in the current study. Its advantages included the chemical stability, biocompatibility and ease of handling because it did not require additional surgical measures (compare with Figure 4C).

At first glance, the complex plots of the impedance data in the calibration solution appeared typical for bare metal electrodes (Figure 2). The constant phase behavior reflected the electrode polarization processes at low frequencies, whereas the semicircles indicated the resistor-capacitor (RC) behavior of the bulk medium at frequencies beyond the cessation of electrode polarization. The R_Bulk resistances (Figure 2) reflect ionic currents in the bulk medium, which is in galvanic contact with the bared electrode tips. In the calibration solution, the resistances R_Cal = R_Bulk are given by equation (2) or (5) using the conductivity and cell constants (Table 2). The corresponding capacitances (C_Bulk) obtained from the medium permittivity and the cell constants (equation (2), Table 2) were superseded by C_Add in the measuring setup as demonstrated by the high capacitances required to explain the characteristic frequencies of the semicircles (Figure 2A). These frequencies were substantially lower than the Maxwell-Wagner dispersion frequency of the calibration solution of approx. 29 MHz. Nevertheless, the diameters and shapes of the semicircles, as well as the DC and low frequency behavior of the electrodes are not altered by C_Add. In the calibration solution, geometric fits of the two branches by a straight line and a semicircle merge at the real axis, which provides the same resistances R_Cal=R_Bulk.

This behavior is inherent to our circuit model, which effectively consists of the CPE in series with a parallel RC pair (Figure 2C). Beyond the CPE dispersion at high frequencies (equation (1)), the extrapolated merging point at the real axis is defined by R_Bulk. For the behavior of the circuit, it is unimportant which of the CPE poles C_Add is connected to because significant capacitive current across the electrode insulation flows only at frequencies where potential electrolytic processes at the insulation surface have previously dispersed by capacitive CPE bridging.

For the experimental and numerical determinations of the cell constants, the specific conductivity of 0.1307 S/m of the calibration solution was used. This conductivity corresponds with human grey matter at 65 kHz. At the basic frequency of the DBS signal of 130 Hz, its conductivity is 0.0915 S/m and increases with frequency [53]. Sixty-five kHz corresponds to the 500^th harmonic frequency of the 130-Hz DBS pulse. Taking into account all harmonic frequencies up to 65 kHz ensures a good fidelity of the pulse shape [26]. This frequency is sufficiently high to prevent electrode processes from influencing the experiments (Figure 2A), whereas wiring inductances and capacitances may be neglected in the experimental determination of the cell constants (Figure 2B), as well as in animal experiments.

For the results of the finite-element simulations, the domain geometries around the electrodes were important. For the bipolar electrode, the domain geometries did not impose a problem because the current source and current drain were both located in the center of the calculation space with vanishing current contributions in the periphery. In contrast, the results for the unipolar electrode sensitively depended on the location and geometry of the counter electrode. Experimental resistances or cell constants were useful parameters to test the applicability of the geometry chosen for the numerical model.

For example, the numerically calculated resistance for the unipolar electrode with a spherical counter electrode at an infinite distance was only 82.8% of the resistance obtained with the cylindrical counter-electrode (dashed in Figure 3A). Using the infinitely distant spherical counter electrode as a reference for a hypothetical spherical “equivalent electrode”, a diameter of 188.9 μm was analytically calculated [48]. Neglecting electrode processes, this equivalent electrode has the resistance of the unipolar electrode and the same voltage-current behavior. At larger distances, it generates the same field distribution as the actual electrode. Equivalent electrodes enable intuitive considerations of all relevant parameters. We used these considerations to confirm that our stimulators could provide sufficient voltage for the adjusted current at the given tissue impedances.

In the impedance measurements, the same voltages were used for all electrodes. In the calibration solution, the CPE slope-exponents were α=0.73 and α=0.78 for the fresh bipolar and unipolar electrodes, respectively (Figure 2). Exponents closer to unity reflect increased capacitive currents and reduced conductive (Ohmic) currents, which are mediated by electric double layer processes. The slightly increased conductive contributions for the bipolar electrode are likely caused by the increased local current densities at electrode sites with increased curvature. An overall increased curvature results from the increased edge-to-plain-surface ratio of the bipolar electrode (cf., for example, to the characteristic V-I curves of the electrode in [26]). The increased current densities in the immediate vicinity of the metal contacts result in increased contributions to the overall resistivity compared with the unipolar electrode. This is one reason for the increased sensitivity of the bipolar electrode for the surrounding medium properties.

For the unipolar electrode, the contribution of the counter electrode to the impedance may be neglected throughout the frequency-measuring range. The reason is its very large surface area, which results in high capacitive and low conductive current densities. Accordingly, the CPE impedance of the unipolar electrode (Figure 2) is solely generated by currents that pass through the double layer at a single tip, whereas the CPE impedance of the bipolar electrode is generated at two tips switched in series.

Calculations with equation (1) demonstrated that two CPEs of equal electrical properties in series exhibit the same slope in the complex plain as the single CPE. The halved effective capacitance shifts the location of the frequency points at the CPE plot upwards (Figure 2A). As a result, the CPE effects are pronounced in a wider frequency range of the spectrum. This is another reason for the increased sensitivity of the bipolar electrode for the surrounding medium properties. Moreover, in our experiments, CPE fitting was facilitated because more measuring points were located further away from the frequency range of the interference with the structural dispersions.

The medium capacitances at the bared electrode tips in the control measurements were superimposed with additional shaft and wiring capacitances (Table 2). In the animal model, these additional capacitances are likely increased because of the complete “immersion” of the electrodes and the subcutaneous wiring. An exact analysis was hindered by the structural dispersions of the brain tissue, which deformed the semicircle branch in the animal experiments.

There are two additional things that must be considered: i) C_Add introduce low-pass filter characteristics for the electrode wiring. This will not lead to distortions of the DBS needle pulse-shape as long as the cut-off frequency is greater than 1 MHz, i.e., the highest significant harmonic frequencies of the signal (cf. to [26] for the Fourier content of the DBS signal). For our setup, this condition was roughly fulfilled. ii) The electrical double layer effects at the insulation-medium interface should not contribute to the CPE properties. This condition was fulfilled because significant currents across this interface require the capacitive bridging of the insulation. At the high frequencies required for bridging, electrical double layer effects may be neglected (compare area specific capacitances of the electric double layer and the insulation layer).

In the animal model, the deformation and flattening of the semicircle by structural dispersions of the brain tissue in the frequency range of the β-dispersion resulted in a vertical shift of its center [29]. As a result, fitting the intercepts of the semicircles with the real axis became impossible (compare with Figure 7). The fits of the linear CPE branches provided reasonable results, which we interpret as the effective resistivities of the brain tissue that surrounds the electrode tip (Figure 8). To discriminate against the influence of the structural dispersions, we neglected measuring points in the transition ranges between the linear and semicircle branches in the fits.

The detected resistivity changes were qualitatively similar for both electrode types; however, they were substantially more pronounced for the bipolar electrodes (Figure 8). A characteristic decrease at the second day was followed by a successive increase in the electrode resistivity at an individual pace from the third day on (Figure 9). These findings are consistent with the findings of [33]. These authors ascribed the impedance increase of a unipolar electrode in the brain of a rhesus macaque to the foreign body reaction, which was accompanied by the formation of the adventitia. The authors left the impedance decrease one day after implantation, which is visible in their figure 4B, uncommented. We suggest that these drops are characteristic and caused by the influx of wound fluid into the electrode-tissue interface [34]. [33] also described a stimulation-on-induced decrease in the electrode impedance, which stabilized in their experiments approximately five weeks post implantation. In our experiments, starting the continuous stimulation after eight days induced a resistivity decrease, followed by stabilization over the following days of stimulation.

The thickness of the adventitia that ultimately encapsulates the DBS electrodes is correlated with the intensity of the foreign body reaction [54]. Without a stimulating signal, the intensity of the reaction depends on the electrode material [38]. Adventitia formation is also influenced by the presence of the stimulating signal [41] likely because electrochemical reactions at the surface of stimulated electrodes alter their effective properties as “foreign bodies”.

In the model, the edges of the electrodes were assumed to form perfect 90°-angles between the cylindrical flanks and the bottoms of the electrode tips (Figure 3).

Microscopic investigations of the electrodes indicated rounded edges (Figure 1). Rounding leads to lower field strengths near the edges compared with our simulation results. For future setups, it is desirable to have electrodes with edges of defined curvatures. This approach will enable the improvement of model precision and the generation of better-defined fields in the brain.

The homogeneous electrical tissue properties assumed in the simulations correspond to a random orientation of the neuronal cells, which is abstracted from their dielectric structure. These structures introduce the complex frequency dependent properties of tissues and cell suspensions [55], which were ignored for simplicity. Our parameters avoided field strengths at sites of high electrode curvature that may have caused membrane poration and cell destruction. The relations are discussed in [5].

Because of the smaller electrode geometries, in general, increased current densities occur in animal experiments compared with patients. In the experiments, we used the same current magnitude for both electrode types. Given their different cell constants, this approach led to stimulation voltages that were approximately twice as high for the bipolar electrode compared with the unipolar electrode (Figure 5). The increased electrode voltage resulted in increased field strengths and current densities at the surface of the bipolar electrode because these parameters are proportional to the electrode voltage [48]. Nevertheless, the field decays over shorter distances from the smaller tips of this electrode (Figure 5). For the bipolar electrodes, the surface voltage of the unipolar electrodes would be reached at constant currents of approx. 100 μA because of the factor-of-two higher resistance. Although this would reduce the power consumption for the stimulator, it would reduce the reach inside the brain even further [5].

Despite the electrode tip geometries, models that are more realistic must consider the rat skull geometry, counter-electrode location, and inhomogeneous and frequency-dependent tissue properties to improve the simulation of the V-I dependence, the field distributions and the DBS effects in the rat brain.