In Silico Study of Local Electrical Impedance Measurements in the Atria - Towards Understanding and Quantifying Dependencies in Human

<italic>Background:</italic> Electrical impedance measurements have become an accepted tool for monitoring intracardiac radio frequency ablation. Recently, the long-established generator impedance was joined by novel local impedance measurement capabilities with all electrical circuit terminals being accommodated within the catheter. <italic>Objective:</italic> This work aims at <italic>in silico</italic> quantification of distinct influencing factors that have remained challenges due to the lack of ground truth knowledge and the superposition of effects in clinical settings. <italic>Methods:</italic> We introduced a highly detailed <italic>in silico</italic> model of two local impedance enabled catheters, namely IntellaNav MiFi OI and IntellaNav Stablepoint, embedded in a series of clinically relevant environments. Assigning material and frequency specific conductivities and subsequently calculating the spread of the electrical field with the finite element method yielded <italic>in silico</italic> local impedances. The <italic>in silico</italic> model was validated by comparison to <italic>in vitro</italic> measurements of standardized sodium chloride solutions. We then investigated the effect of the withdrawal of the catheter into the transseptal sheath, catheter-tissue interaction, insertion of the catheter into pulmonary veins, and catheter irrigation. <italic>Results:</italic> All simulated setups were in line with <italic>in vitro</italic> experiments and <italic>in human</italic> measurements and gave detailed insight into determinants of local impedance changes as well as the relation between values measured with two different devices. <italic>Conclusion:</italic> The <italic>in silico</italic> environment proved to be capable of resembling clinical scenarios and quantifying local impedance changes. Significance: The tool can assists the interpretation of measurements in humans and has the potential to support future catheter development.


I. INTRODUCTION
E LECTRICAL impedance measurements have a long history in the medical and biomedical field. Historical studies have shown that different kinds of biological tissues are characterized by different conductivity spectra [1] attributed to the microscopic composition of the materials [2]. Besides the composition of the material and the measurement frequency, electrode arrangement and temperature are major determinants of the measured impedance.
During invasive cardiac electrophysiological studies, generator impedance measurements have been an established method to monitor the delivery of radio frequency energy during ablation since decades [3], [4]. The transthoracic impedance of the radio frequency energy delivery pathway between an intracardiac and a cutaneous dispersive electrode assists differentiation of tissue contact during ablation. However, the bulk impedance of the torso blurs measurements [5], [6] and impedes detailed assessment of tissue characteristics in the region of interest next to the catheter. Recently, two novel catheters have been introduced to the market that aim at a more locally focused impedance assessment in the vicinity of the catheter with all injecting and measuring electrodes being built into the intracardiac catheter itself [4]. The radio frequency ablation catheters IntellaNav MiFi OI [7] and IntellaNav Stablepoint [8] (Boston Scientific, Malborough, MA, USA) come with a four-electrode and a three-electrode impedance measurement circuit implemented within the catheter, respectively. During ablation, the so-called DirectSense technology measures the magnitude of the local impedance (LI). An LI drop resulting from a combination of resistive tissue heating and subsequent myocardial destruction and lesion formation is used as a surrogate for lesion quality and durability [7], [9]. Compared to the transthoracic generator impedance, the LI emphasizes local changes in impedance while being less susceptible to far field artifacts [9]- [12]. Despite an increased influence of the local surroundings on the measurement compared to the generator impedance, the LI is still sensitive to the three-dimensional arrangement of materials and their properties surrounding the catheter [13]. LI may therefore not be mistaken for lumped impedance measurements, which condense all influencing properties to an infinitesimal element. Besides the monitoring of ablation lesion formation, LI has also shown potential to characterize cardiac tissue and differentiate between healthy myocardium and fibrotic or scar tissue [10]- [12]. Atrial fibrillation as the most common sustained cardiac arrhythmia poses a major burden for both patients and global health care systems. Since current treatment approaches result in unsatisfactory success rates, novel methods of tissue characterization such as the LI need further exploration.
A major challenge in the expansion of the diagnostic value of intracardiac LI measurements are confounding factors. Not only different tissue compositions but also the distance and angle between the catheter and the tissue, the surrounding tissue geometry, an overlap of catheter and transseptal sheath, and sodium chloride (NaCl) solution irrigation influence the measurement, amongst others. Many of these effects can be observed in in human studies but lack quantification due to the superposition of multiple effects and an unknown ground truth. Therefore, the differentiation between the target measure and confounding factors has remained uncertain. In vitro and ex vivo experiments can help to shed light on different scenarios but are costly and depending on the experimental setup, the underlying ground truth still remains under-determined.
In this work, we present for the first time a highly detailed in silico framework that models the IntellaNav MiFi OI catheter and the IntellaNav Stablepoint catheter in combination with different clinically relevant surroundings. After validation of the framework by in vitro measurements in standardized setups, clinically relevant scenarios such as the effect of the distance and angle between catheter and tissue, scar tissue, the insertion of the catheter into a pulmonary vein or a transseptal sheath, and NaCl solution irrigation were investigated and compared to in vitro and in human measurements.
With a highly detailed comparison between different catheter geometries and the investigation of isolated scenarios to quantify various clinically relevant effects, this work paves the way for an inexpensive enhancement of the understanding of intracardiac LI measurements and future catheter development.

A. In Silico -Geometrical Setup
Both clinically available LI enabled radio frequency ablation catheters were modeled in high detail as depicted in Fig. 1(a) to (d). Measures were taken from product specification sheets [14], [15] as well as calibrated photographs yielding a resolution below 100 μm.
The IntellaNav MiFi OI comes with a 4.5 mm tip electrode, three ring electrodes of 1.3 mm width and 2.5 mm spacing, three evenly distributed mini electrodes of 0.8 mm diameter embedded in the tip electrode, six irrigation holes, and a cooling chamber filled with NaCl solution. The interior of the catheter is electrically isolated from the electrodes and accommodates thin electrical and mechanical steering wires. While neglecting the latter, the interior of the catheter shaft was filled with insulating material in the model.
The IntellaNav Stablepoint is similarly composed of a 4 mm tip electrode, three ring electrodes of 1.3 mm width and 4.0 mm | 2.5 mm | 2.5 mm spacing, six irrigation holes, and a cooling chamber filled with NaCl solution. The tip does not embed any mini electrodes. Proximal to the tip, the diameter expands conically to the shaft diameter. The existence of the force sensing spring between the tip electrode and the distal ring electrode in the interior of the catheter [8] was assumed to be negligible with respect to the spread of the electrical field outside the catheter. Therefore, the interior of the catheter shaft was filled with insulating material as well.
A transseptal sheath was implemented on the model of the 8.5 F Agilis NxT steerable introducer (Abbott, Chicago, IL, USA) as depicted in Fig. 1(e).
Detailed measures of the implemented catheter and sheath geometries are shown in Fig. 1.
The respective catheter was embedded in a 140 mm × 140 mm × 140 mm box filled with either blood or NaCl solution as displayed in Fig. 2(a) for all simulation setups. Geometry definition and tetrahedral meshing was done with Gmsh (version 4.5.6) [16]. Mesh resolution was adapted to the size of local structures being the highest at the IntellaNav MiFi OI mini electrodes and the lowest at the outer boundary of the surrounding box. The meshes were comprised of 2.5 million to 5 million tetrahedral elements. 1) Standardized NaCl Solutions: Either catheter was placed in the surrounding box filled with NaCl solution of eight different molar concentrations starting from 0.02 mol l in steps of 0.01 mol l up to 0.09 mol l . 2) Transseptal Steerable Sheath: Within a surrounding box filled with blood, either catheter was withdrawn into the transseptal sheath model with the distance d Sh describing the distance between the catheter tip and the distal edge of the sheath (compare Fig. 2(d)). Negative distances describe the withdrawal of the catheter into the sheath. d Sh was varied from −2 mm in steps of 0.5 mm up to 19.5 mm.
3) Tissue: Either catheter was placed in the surrounding box filled with blood. A square patch of tissue measuring 110 mm × 110 mm × 2.5 mm was placed below the catheter resembling a piece of atrial myocardial tissue of typical wall thickness [17] (compare Fig. 2(a)). The distance d T between the catheter and the tissue was varied from −2 mm to 10 mm in steps of 0.5 mm (compare Fig. 2(b)). Negative distances represented an immersion of the catheter into the tissue. Mechanical interaction was not modeled. Instead, the catheter simply displaced the tissue.
In a second step, the angle α T between the catheter and the tissue was varied from 0 • to 180 • in steps of 15 • . For 90 • < α T ≤ 180 • , one of the mini electrodes pointed directly towards the tissue. For 0 • ≤ α T < 90 • , the two remaining electrodes were pointed towards -but not directly towards -the tissue (compare Fig. 2(b)). The pivot was located at the intersection of the catheter's distal plane and the outer wall of the catheter shaft at the left and the right, respectively. The experiment was conducted for five different distances between catheter and tissue d T ∈ {0.0 mm, 0.5 mm, 1.0 mm, 2.0 mm, 4.0 mm}.

4) Transmural Lesion:
The general tissue setup as described above was complemented by a central line of scar tissue of width w Sc ∈ {3 mm, 6 mm} (compare Fig. 2(a)) representing ablated tissue from a previous procedure or natively developed myocardial scar. For two different distances between catheter and tissue (d T = 0 mm and d T = 1 mm), either catheter was moved perpendicularly to the line of scar starting at a distance to the center of the line of scar of d Sc = −10 mm, crossing the line of scar for d Sc = 0 mm up to a distance of d Sc = 10 mm at a default step size of 1 mm and a decreased step size of 0.5 mm for |d Sc | < 2 mm.

5) Pulmonary Vein:
The insertion of either catheter into a pulmonary vein (PV) was simulated by extending the general tissue setup by a perpendicular tube filled with blood (compare Fig. 2(c)). For a PV wall thickness th PV = 2 mm, four different inner PV radii r PV ∈ {2 mm, 3 mm, 4 mm, 6 mm} were implemented. For r PV = 6 mm, additional PV wall thicknesses of th PV ∈ {1 mm, 3 mm, 4 mm} were modeled. Either catheter was inserted into the PV quantified by the distance d T to the surface of the tissue. Negative distances represent states with the respective catheter being inside the PV while positive distances represent states of catheter elevation above the tissue. d T was varied from −20 mm (full immersion) to 10 mm (full extraction) in steps of 1 mm. 6) Irrigation: Catheter irrigation was modeled by placing a sphere of physiological NaCl solution at each center of the irrigation holes displacing all encircled blood elements (compare Fig. 2(e)). The radius r NaCl of the NaCl spheres was varied from 0 mm to 2 mm in steps of 0.05 mm.

B. In Silico -Material Properties
The tetrahedral elements of the geometrical meshes were assigned conductivity values characteristic for the respective material at 14.5 kHz as summarized in Table I. Due to the significant dependency of conductivities on the temperature, the latter had to be regarded for.
The in vitro setups with NaCl solutions of different concentrations were conducted at different temperatures and compared to in silico experiments based on conductivities published by Gabriel et al. [1], which lack an explicit statement about temperature. Comparing to the conductivity of 0.5 % NaCl solution given for 20 • C [18] suggests that Gabriel et al. measured at a slightly higher temperature T Gab (compare Table I). With a temperature coefficient of approximately 2.1 % • [19] and the reference values from [18], Gabriel et al. most likely measured NaCl solutions significantly below body temperature as opposed to their measurements of biological tissue. Since the data set [1] was consistent in itself, the exact temperature was deemed insignificant for the validation setups with NaCl solutions of different concentrations.
All other in silico experiments were parameterized with conductivities given for blood, myocardial tissue, and scar tissue at body temperature (BT) as well as physiological 0.9 % NaCl solution for catheter irrigation at an approximate lab temperature of 20 • C as listed in Table I. Due to the lack of an explicit reference for the conductivity of physiological 0.9 % NaCl solution, the latter was linearly interpolated from the conductivities of 0.5 % and 1 % NaCl solution at 20 • C [18] as listed in Table I.

C. In Silico -Impedance Forward Simulation
The spread of the electrical field was simulated with the software EIDORS v3.10 [23] and MATLAB R2021a (The Math-Works, Inc., Natick, MA, USA). In short, EIDORS solves the Poisson equation with a finite element model F . The injection currents are given as boundary conditions. The current density and the potential field are the solution. The voltage v between two electrodes is extracted as the potential difference and is dependent on the given conductivities σ at the elements of the model and the stimulation pattern q of the electrode model with v = F (σ, q) [23].
Stimulation and measurement circuits were defined according to the clinical system: A four-terminal circuit with current injection between the distal tip electrode and the proximal ring electrode was combined with measurements between the mini electrodes and the distal ring electrode for the Intel-laNav MiFi OI [7]. The three voltage measurements resulting from either mini electrode to the distal ring electrode were reduced to their maximum value following the clinical system. The IntellaNav Stablepoint was set up as a three-terminal circuit with current injection between the distal tip electrode and the proximal ring electrode and voltage measurement between the distal tip electrode and the distal ring electrode.
An alternating current of 5 μA peak-to-peak amplitude at 14.5 kHz was modeled. The complete electrode model was used [24]. The resulting voltage amplitude |v| was then divided by the amplitude of the injected current to obtain LI as the magnitude of the impedance.

D. In Vitro Setup
All measurements were conducted with the Rhythmia HDx system (Boston Scientific, Malborough, MA, USA), the Intel-laNav MiFi OI, and the IntellaNav Stablepoint. To validate the simulation framework, NaCl solutions of different concentrations and known conductivity σ were prepared. The molar mass as given in [1] starting from 0.02 mol l up to 0.09 mol l in steps of 0.01 mol l was converted to weight percentages. The respective amount of NaCl was weighed out with a scale of 10 −3 g resolution and 10 −3 g precision and dissolved in 250 ml of de-ionized water. For all concentrations, the NaCl dissolved completely and formed an aqueous solution. A thermometer of 0.1 • C resolution was used to keep track of the solution's temperature. The LI was measured with both catheters in each solution at 7 to 13 different temperatures between 18.2 • C and 38.8 • C. For comparability, the LI at three different temperatures -namely 21 • C, 25 • C, and 36 • C -was interpolated and compared to the simulated results for the respective NaCl solutions.
Additionally, the behavior of LI with tissue contact was measured in vitro. A tissue phantom composed of 100 ml de-ionized water, 3 g agar-agar, and 0.0499 g NaCl [25] was prepared. The expected conductivity of 0.16 S m at 25 • C matched the conductivity of cardiac tissue at 14.5 kHz well. Since in vitro measurements were taken at 20.5 • C in this work, the actual conductivity might have deviated slightly due to the difference in temperature. Typical temperature coefficients reported for similar materials justified to neglect deviations caused by the described change in temperature [19], [26]. Additionally, a piece of smooth left atrial porcine tissue was used. The tissue phantom and the tissue sample were mounted at an elevated ring in order not to disturb measurements by the mount in 0.35 % NaCl solution. Either catheter was positioned at the tissue phantom and the tissue sample in orthogonal and parallel orientation.
The effect of catheter irrigation with physiological NaCl solution on LI was investigated by increasing the flow rate of the HAT 500 irrigation pump (Osypka AG, Rheinfelden, Germany) from 0 ml min to 2 ml min and 17 ml min in a 250 ml bath of 0.35 % NaCl solution. A flow rate of 2 ml min is clinically applied in standby mode while the flow rate is typically adjusted to 17 ml min during ablation. The bath model did not include circulation.

E. In Human Setup
Clinical measurements complemented the in silico analysis of catheter sheath interaction and its effect on the LI. Either catheter was located in the left atrium passing the inter-atrial septum via the transseptal sheath, namely the Agilis NxT steerable introducer. An X-ray scan verified that the proximal ring electrode was outside of the sheath. Starting from a central position in the left atrial bloodpool without endocardial contact, the catheter was gradually pulled back into the sheath at constant speed while recording the LI. Clinical LI was represented by its moving average calculated with a sliding window of 1.5 s width as provided by the electroanatomical mapping system. All in human measurements were approved by the local ethics committee and were conducted in accordance with the Declaration of Helsinki. Written informed consent was obtained from all patients. Fig. 3 presents LI values measured in vitro in aqueous NaCl solutions prepared according to Table I   The simulated bloodpool LI for a blood conductivity σ = 0.7 S m [22] as given in Table I  Starting at a simulated bloodpool of 87 Ω and 139 Ω, the in silico LI measured with the IntellaNav MiFi OI and the IntellaNav Stablepoint first increased by more than 2 Ω for the distal edge of the sheath being located between the proximal and the 2 nd to proximal ring electrode. The steep increase of LI began upon the coverage of the distal ring electrode by the sheath. For full sheath coverage, the LI increased up to 1353 Ω and 2200 Ω for the IntellaNav MiFi OI and the IntellaNav Stablepoint, respectively. For the distal edge of the sheath being located between the distal ring electrode and the tip electrode, an interim decrease in steepness formed a plateau especially pronounced for the IntellaNav Stablepoint.

A. Aqueous NaCl Solutions
Both simulated traces compared well with the clinically measured traces.

1) Catheter Distance and Orientation:
Again starting from a bloodpool LI of 87 Ω and 139 Ω for the Intel-laNav MiFi OI and the IntellaNav Stablepoint, respectively, the LI increased with decreasing distance to the tissue surface for perpendicular catheter positions (α T = 90 • ) as shown in Fig. 5. At a distance d T = 3.5 mm and d T = 2.5 mm, the LI exceeded the bloodpool LI by more than 2 % for the Intel-laNav MiFi OI and IntellaNav Stablepoint, respectively. At a distance d T = 0 mm, the LI exceeded the bloodpool LI by 16.0 % and 14.9 % for the IntellaNav MiFi OI and IntellaNav Stablepoint, respectively. The closer the catheter approached the tissue, the steeper the LI increased. For negative distances d T , i.e. the catheter entering the tissue, the increase in LI per distance was approximately constant. For the IntellaNav MiFi OI, a small plateau in LI formed between d T = −1.0 mm and d T = −1.5 mm. Fig. 6 presents the simulated LI values for changing angles α T between the catheter and the tissue for selected distances. For both catheters and all distances, the traces were w-shaped. Starting from a perpendicular position and approaching a parallel position, LI first dropped and then increased again. The LI for parallel catheter orientation at a distance d T = 0 mm exceeded the LI for perpendicular catheter positions by 14.0 Ω (α T = 0 • ) and 12.9 Ω (α T = 180 • ) for the IntellaNav MiFi OI and by 9.4 Ω for the IntellaNav Stablepoint. While the traces were symmetric to α T = 90 • for the IntellaNav Stablepoint, the LI depended on the orientation of the mini electrodes for the IntellaNav MiFi OI as indicated by the mirrored trace in Fig. 6. Catheter orientations with one of the measuring mini electrodes being directed to the tissue ((90 • < α T ≤ 180 • )) exceeded those LI values of the same distance and angle for which none of the mini electrodes pointed directly towards the tissue (0 • ≤ α T < 90 • ).
In vitro measurements with the respective catheter touching a tissue phantom or a tissue sample perpendicularly and in parallel yielded comparable differences between the parallel and orthogonal position. The LI for the parallel position exceeded the LI of the perpendicular position by approximately 11 Ω and 10 Ω for the IntellaNav MiFi OI and IntellaNav Stablepoint, respectively.
2) Transmural Lesion: Due to the higher conductivity of connective tissue compared to healthy myocardium, the LI typically drops in the vicinity of myocardial lesions. In these setups, the dependency of the LI on the extent of the scar and the relative position of the catheter was investigated. Fig. 7 shows LI  traces for the virtual catheter passing linear lesions of 3 mm and 6 mm width. For direct tissue contact (d T = 0 mm), the absolute drop was larger for the IntellaNav Stablepoint due to the higher baseline LI for either lesion width. The percentage drop based on the LI at maximum distance to the lesion, however, was similar with 3.8 % and 6.0 % for lesion widths of 3 mm and 6 mm for the IntellaNav MiFi OI and a percentage drop of 3.9 % and 5.7 % for the IntellaNav Stablepoint. Increasing the vertical distance d T between the catheter and the myocardial tissue by 1 mm caused a larger drop in the baseline LI than either of the scars for both catheters. Fig. 8 shows the potential field (a) and the current density (b) for the setup with a scar width w Sc = 3 mm and direct tissue contact. With a similarity to an electrical dipole field, the current spreads between the injecting electrodes. The current density in   Fig. 9 shows characteristic LI traces for progressive introduction of an ablation catheter into the PV. Fig. 9(a) displays simulated LI values for the IntellaNav MiFi OI. The LI increased from 87 Ω in the simulated bloodpool up to peak values between 93 Ω and 176 Ω depending on the radius r PV and the thickness th PV of the PV. According LI traces for the IntellaNav Stablepoint are presented in Fig. 9(b). Starting from a simulated bloodpool LI of 139 Ω, the LI increased up to 145 Ω to 240 Ω depending on r PV and th PV .

D. Insertion Into a Pulmonary Vein
The radius r PV was found to be a strong determinant of the maximum LI reached upon insertion of the catheter into the PV. While the narrowest simulated PV with r PV = 2 mm yielded a maximum LI of 176 Ω with the IntellaNav MiFi OI and 240 Ω with the IntellaNav Stablepoint, an increase of the radius by 1 mm resulted in a maximum LI of only 127 Ω and 186 Ω, respectively.
Peak values for all parameterizations of the PV corresponded with the insertion of the tip of the ablation catheter into the PV in the in silico experiments. The slight decrease for deeper insertions was related to the passing of the surrounding tissue plate that additionally elevated the LI at its maximum.
The thickness of the vein tissue took additional influence on the absolute LI value, as shown for a vein radius r PV = 6 mm in Fig. 9.

E. NaCl Solution Irrigation
Fig. 10 displays simulated LI values for flushing of the catheters with physiological NaCl solution exiting the cooling lumen at the irrigation holes. Varying the bubble radius r NaCl from 0 to 2 mm mimicked changing the irrigation flow rate. The LI remained indifferent to NaCl bubbles up to a radius r NaCl = 0.7 mm and r NaCl = 0.55 mm with less than 1 % change compared to the in silico measurement in plain bloodpool of 87 Ω and 139 Ω for the IntellaNav MiFi OI and the the Intel-laNav Stablepoint, respectively. For the IntellaNav MiFi OI, the LI then slightly increased reaching a maximum elevation of 1.4 Ω above the bloodpool for r NaCl = 0.8 mm when the NaCl bubbles barely reached the mini electrodes' distal edges. Afterwards, LI values decreased with increasing r NaCl down to 76.4 Ω for r NaCl = 2 mm. The LI decreased monotonously for the IntellaNav Stablepoint down to 116.6 Ω for r NaCl = 2 mm. Fig. 10(a) and (b) display in vitro traces of LI for onset and offset of irrigation at different flow rates. In vitro measurements at a flow rate of 2 ml min revealed an instantaneous drop of 0.6 Ω and 1 Ω and oscillations in LI of 0.6 Ω and 1 Ω peak-to-peak amplitude tracing back to the cylinders of the irrigation wheel compressing the irrigation tube for the IntellaNav MiFi OI and the IntellaNav Stablepoint, respectively. For the IntellaNav Stablepoint, LI dropped abruptly by 2.5 Ω upon the onset of irrigation at 17 ml min . The gradual decrease of LI is the result of a small bath volume mixing with the irrigation fluid of higher conductivity.

IV. DISCUSSION
In summary, we presented an in silico environment that resembled in human and in vitro LI measurements to a high degree of detail and allowed for quantification of distinct influences on the measurement with known ground truth.

A. Aqueous NaCl Solutions
Model validation with standardized aqueous NaCl solutions of known conductivity was successful and proved the suitability of the simulation environment. NaCl solutions can be assumed to be of mostly resistive character at a measurement frequency of 14.5 kHz. Thus, the hyperbolic-like relationship between conductivity and LI can be explained as impedance reduces to resistance in this setup and resistance is reciprocally related to conductivity.
For in vitro experiments, aqueous NaCl solutions at 0.35 % to 0.4 % mass concentration at 21 • C were shown to serve well as dielectric equivalent of human blood at body temperature for a measurement frequency of 14.5 kHz.
A perfectly linear relationship between LI measurements with the IntellaNav MiFi OI and the IntellaNav Stablepoint as described by equation (1) is of great clinical value. Translation of findings and reference values between both catheters can extrapolate clinical trials to the respective other device and reduce efforts. Measurements with different instances of the catheters resulted in minor deviations of the linear coefficients and could potentially be caused by slight manufacturing differences or by the fact that all in vitro catheters had been used for radio frequency ablation before.

B. Transseptal Steerable Sheath
In silico experiments revealed that LI started to increase notably as soon as the sheath passed the proximal ring electrode. LI measurements for both, substrate and lesion characterization in clinical practice, should therefore always assure full withdrawal of the catheter out of the sheath in order to prevent confounding influences on the measured LI.

C. Catheter Tissue Interaction
The elevation of LI in tissue contact above the bloodpool LI ranged from 14 Ω for 0 mm distance to the tissue, i.e. 0 g so-called "contact force," to 33 Ω for −2 mm distance to the tissue and compared well to clinically observed mean ranges between 16 Ω and 20 Ω [27] for the IntellaNav MiFi OI. The simulated upper bound for an immersion depth of 2 mm thus likely overestimates the LI for clinical mean contact force applications due to the disregard of realistic tissue deformation.
Sulkin et al. had performed detailed in vitro experiments on catheter tissue interaction with the IntellaNav MiFi OI and found a nonlinear monotonic increase of LI as the catheter approximated the tissue at an angle of 90 • [7]. The in silico results generated in this work matched the shape of the curve very well but yielded scaled absolute values and slopes presumably due to differences in the underlying conductivity of tissue and blood. The right ventricular tissue used by Sulkin et al. was presumably thicker than the atrial tissue modeled with a thickness of only 2.5 mm in this work and could explain the higher absolute values and slopes in their study. Additionally, the natural variability of the conductivity of tissue samples causes a spread of measured LIs [7] that could account for the scaled results. The specific conductivities chosen in this work are only one sample of the natural spread of human myocardial conductivity.
Changing the angle between catheter and tissue resulted in higher LIs for more parallel compared to orthogonal catheter orientation for distances d T > −2 mm both in the work by Sulkin et al. and the in silico experiments in this work. Garrott et al. [8] observed a mean LI difference of 13 Ω between perpendicular and parallel catheter orientation of the IntellaNav StPt which is well in line with the in silico experiments presented here.
In silico experiments with the IntellaNav MiFi OI presented a small plateau for an immersion into atrial tissue by 1.0 mm to 1.5 mm as well as an abrupt decrease in LI for an angle α T = 180 • that were not in line with the trend of the respective adjacent distances and angles. Presumably, the close interaction between the measuring mini electrode and the tissue caused both observations.
Clinical studies report different ranges of LI values for healthy and scar tissue, e.g. 109 Ω ± 15 Ω and 104 Ω ± 12 Ω [10], 111 Ω ± 14 Ω and 92 Ω ± 16 Ω [11], and 132 Ω ± 12 Ω [12], respectively, for the IntellaNav MiFi OI. The variability in range may be explained by different operators and differences in typically applied contact force which remains uncontrolled for the IntellaNav MiFi OI. In line with previously published clinical observations, scar tissue presented lower LI compared to healthy myocardium due to the increase in extracellular space and the resulting increase in conductivity in the in silico model as well. Slightly lower values for both healthy and scar tissue in the in silico study as depicted in Fig. 7 in comparison with the clinical observations [10]- [12] could either be caused by the choice of conductivies in the in silico model or from a lower contact force. While the in silico model operates at an equivalent of 0 g so-called "contact force" for the experiments on scar tissue, typical clinical values range from 5 g to 20 g. The larger the lesion area within the footprint of the catheter, the lower the LI dropped. The results presented in Fig. 7 emphasize the importance of direct tissue contact and controlled contact force for quantitative applications of LI measurements. Drops in baseline LI caused by only 1 mm distance to the endocardial surface exceeded LI drops caused by transmural lesions. Since the exact values depend on the scar and tissue conductivity provided to the model and scar conductivity was approximated by the conductivity of connective tissue, a validation of the conductivity of atrial scar tissue would strengthen the finding but was out of the scope of this work.
Myocardial tissue was modeled as homogeneous, isotropic block. The effect of fiber direction and three-dimensional atrial structures remains unlit within the scope of this work. Future studies will have to shed light on more detailed models of the myocardium.

D. Insertion Into a Pulmonary Vein
In silico experiments demonstrated the strong dependency of the LI measured inside a PV on the radius of the vein. Vein tissue was modeled indifferently from myocardial tissue for simplicity although the substrates clearly differ histologically and can be assumed to further alter the LI measured in human PVs.

E. NaCl Solution Irrigation
Both IntellaNav MiFi OI and IntellaNav Stablepoint come with an open irrigated tip with the purpose of cooling the electrode during ablation delivery. Typically, catheters are flushed with 0.9 % so-called physiological NaCl solution at lab temperature. However, 0.9 % NaCl solution deviates by a factor of approximately 2 from human blood in terms of conductivity (compare Table I). Earlier studies have shown that the irrigation fluid during radio frequency ablation delivery takes influence on lesion formation. Highly conductive irrigation fluids such as physiological NaCl solution attract current flow and thus reduce the current flowing through the target tissue resulting in reduced energy deposition and smaller lesions compared to irrigation with less conductive fluids such as 0.45 % NaCl solution or dextrose water [28]- [31]. Similarly, awareness should be drawn to irrigation fluids for LI measurements during ablation delivery and substrate characterization. In particular, two cases have to be distinguished: the effect of constant irrigation flow rates > 0 ml min and the effect of changing flow rates. Constant flow rates mainly relate to the application of LI substrate mapping while a change of flow rate alludes to the use case of radio frequency ablation delivery. In either case, clinical LI is mostly interpreted in differential manner comparing to the bloodpool reference or the LI at the start of the ablation as opposed to absolute values.
With the typical increase of the irrigation flow rate from a default flow of 2 ml min to 17 ml min or 30 ml min during radio frequency power delivery, the amount of irrigation fluid surrounding the catheter tip presumably increases and causes an LI drop by default that is not related to tissue heating as commonly attributed to LI drops during ablation. With the results presented in Fig. 10 and the assumption that the irrigation fluid is quickly flushed by circulatory blood flow, the LI drop caused by changes of the irrigation flow rate seem to be mostly negligible seen in the context of typically required minimum LI drops of 12 Ω to 16 Ω during radio frequency power delivery with the IntellaNav MiFi OI [9].
For the use case of LI substrate mapping, LI differences of few ohms become of importance. However, a constantly low flow rate of 2 ml min limits the potential for flawing the measurement. Interpreting LI only in differential manner, irrigation will impact the result if the distribution of NaCl close to the catheter tip changes, e.g. due to blood flow. Additionally, the higher conductivity of NaCl solution compared to tissue and blood causes less current to flow through the target of interest.
The in silico investigations in this work are clearly limited to the oversimplified spherical geometries of NaCl irrigation fluid at the catheter tip as well as the lack of a clear correlation between spherical radius in the model and clinical irrigation flow rates. Similarly, the in vitro setup lacks a model of circulatory blood flow. Including a fluid dynamics model could bring more detailed insights into the influence of irrigation and irrigation changes on the measured LI.

F. Sensitivity
Slight variations in the catheter dimensions resulted in notable changes of LI especially for the respective measuring electrodes.
For quantitative analyses, a detailed geometrical model of the catheter under investigation is therefore of high importance.
In silico experiments in this work demonstrated that selected phenomena of interest such as the presence of scar tissue result in minute changes in LI while recording conditions such as the loss of optimal wall contact cause changes in the same or even higher order of magnitude. In a clinical environment under the presence of measurement noise, the detectable range of changes in LI will further decrease, which emphasizes the necessity of establishing ideal wall contact, amongst other recording conditions under control of the operator.
In clinical setups, the inflation and deflation of the lungs is an additional confounding factor with evident impact on the LI measurement [27] due to the close proximity of the lungs to selected parts of the cardiac chambers. While the conductivity of inflated lungs is reported to be 0.0954 S m at 14.5 kHz, the conductivity increases to 0.247 S m in deflated state [21]. Since the respiratory state of the patient is a known parameter, respiratory oscillations in LI traces could be compensated for.
An estimate of the relative contribution of sample volumes in vicinity to the catheter to the measured LI would be of high interest in order to assess the suitability of catheters and electrode arrangements for impedance measurements. The close proximity of the catheter will take significantly more influence on the measured impedance for LI measurements as compared to generator impedance measurements. Specific examples such as varying the distance between catheter and tissue, scar, and sheath as well as varying the volume of NaCl irrigation fluid were presented in this work. However, a systematic analysis does not only require the variation of the sample volume position but also of its size and conductivity. Future studies should systematically shed light on this aspect in order to further optimize catheter and electrode arrangement for LI measurements.

V. CONCLUSION
With this work, we introduced and validated an in silico model including highly detailed catheter and sheath geometries in combination with a simplified myocardial geometry to study local electrical impedance measurements with intra-atrial catheters. Clinically relevant scenarios such as catheter-tissue interaction in terms of angle, distance, and substrate, the insertion of the catheter into a PV, the withdrawal into the transseptal sheath, and catheter irrigation were reflected in the model. Forward simulations of the electrical field gave insights in the quantitative effects of isolated and combined changes in parameters on the LI. The presented environment proved to be a highly valuable tool that provides deeper insight into the clinical interpretation of LI and has the potential to support future catheter development.