In silico and in vitro conductivity models of the left heart ventricle

An elastic material with conductive electrical properties is required in order to build a realistic model of a human heart. In this section, the method to customize the electrical properties of silicone for the in vitro model is described. We selected silicone, because the elastic properties of this material are similar to those of heart muscle, it is more durable, than for example, agar-agar, and easy to process. The most important advantage is that various additives can be combined with silicone to mimic the electrical properties of heart muscle tissue.

We selected silicone with a shore scale of A00 (Silikonfabrik.de, Ahrensburg, Germany), which is in the range of the elastic properties of heart tissue, for the in vitro ventricle phantom. Depending on the point of time in the cardiac cycle, heart muscle tissue has an elasticity modulus between 20 kPa and 280 kPa [8]. Silicone with a shore scale of A00 has an elasticity modulus of about 100 kPA according to Gent et al. [9]. The electrical conductivity of pure silicone is about σs=10−10Sm \sigma _s = 10^{ - 10} {{\rm{S}} \over {\rm{m}}} . Carbon black powder (VULCANR XC72, Cabot Corporation, Boston, USA) together with carbon cutlets with an average length of 3mm (R&G Faserverbundwerkstoffe GmbH, Waldenbuch, Germany) were mixed into the silicone to manipulate the electrical properties. A concentration variation to reach the percolation limit (the threshold above a strong increase in conductivity can be measured) was conducted. Therefore, a mixture of both materials, carbon powder and carbon cutlets, was added to the silicone by simultaneously guaranteed proper compounding. Different samples with a concentration of 1% of carbon black powder and a variation of carbon cutlet concentration (between 0.7 % and 1%) were investigated. All samples were cast into a standardized cylindrical form with a diameter of 28mm, and subsequently degassed (100 kPa) and cured to ensure reproducibility. A weight of 35.6 g was placed on all samples to ensure the same contact pressure, since the contact area influences the impedance measured. An LCR-meter E4980A from Keysight (Santa Rosa, USA) was applied for the analysis of samples. All measurement results are described in detail in [10]. To summarize the results from our previous work: all samples had pure ohmic behavior. Difficulties in homogenization of samples were experienced when using only carbon cutlets. Instead, a mixture of both carbon cutlets and powder showed homogeneous samples and reproducible results. A conductivity of about σC=0.306 Sm \sigma _C = 0.306\,{{\rm{S}} \over {\rm{m}}} within the range of physiological heart tissue (0.05 Sm−0.3Sm) (0.05\,{{\rm{S}} \over {\rm{m}}} - 0.3{{\rm{S}} \over {\rm{m}}}) was reached with a concentration of 1% of carbon black powder and 0.7% of carbon cutlets.

Three dimensional (3D)-printed low-cost molds that replicate the anatomical shape of a left ventricle were developed to cast the in vitro silicone phantoms. Therefore, data from embodi3D [11] of a left ventricle was taken from a computer tomographic image series consisting of 300 slices with a slice thickness of 0.4mm each. The model was simplified by removing the left atrium, pulmonary veins and the coronary arteries. Additionally, the ventricle walls were smoothed. These simplifications allowed for 3D printing of molds and the replication of the inner blood cavity. The modification of the ventricle was done with Fusion 360 (AUTODESK Corporation, San Rafael, USA), a computer-aided design (CAD) software. The simplified ventricle shape was used to construct the 3D casting molds by cutting the volume of the ventricle out of a cuboid. The volume of the cuboid was reduced to a minimum without affecting the stability of the casting molds to save printing material and time. The casting molds were printed with polylactide using a wall thickness of 1.75mm and a layer thickness of 0.2mm. The CAD model and the 3D-printed casting molds are shown in Fig. 1a) and 1b), respectively.

Fig. 1

The manufacturing of hollow ventricle phantoms was necessary to emulate the inner blood cavity. Therefore, we tested two methods to obtain a soluble core. Paraffin wax was poured into a negative core mold and then boiled out of the silicone phantom. This method required additional molds but was highly cost-effective. As an alternative, 3D-printed polyvinylalcohol cores were used and separated from the silicone by dissolving in warm water. Here, the reproducibility of the phantom became easier, as fine structures can be reproduced consistently. Furthermore, the 3D-printing of the blood cavity allows for a rapid customization of the model. The development of the core (red core in Fig. 1a)) in CAD was done by scaling the ventricle used for the casting molds to an end-diastolic volume of about 120 ml. This operating point is typical for measurement of PV-Loop related parameters (e.g. end diastolic volume). Smaller ventricle sizes can be simulated by applying external pressure or internal suction displacing the walls according to the volume shift. In addition, the aorta was replaced by a cylindrical shape such that a hose can be poured directly into the silicone phantom.

Two ventricle phantoms were cast for comparison. One made of pure silicone (see Fig. 1c)) and the other made of silicone with a carbon concentration (see Fig. 1d)) as described in the previous section Customized silicone. A wall thickness of 8mm − 10mm was fabricated with a blood volume of about 140 ml, including the cylindrical part of the aorta, which is within the physiological range of a man's heart having a wall thickness between 4mm − 10mm and a diastolic volume of 67ml − 155 ml [12]. Both ventricles were connected to a water tank with a polyvinylchlorid (PVC) hose. A saline solution with a conductivity of σsaline=0.658Sm \sigma _{{\rm{saline}}} = 0.658{{\rm{S}} \over {\rm{m}}} close to the conductivity of blood σblood=0.701Sm \sigma _{{\rm{blood}}} = 0.701{{\rm{S}} \over {\rm{m}}} was used to fill the hollow ventricles. The conductivity of the solution was validated with the conductivity meter HI 8733 (HANNA Instruments, Woonsocket, Rhode Island, USA).

The impedance measurement system for continuous monitoring inside the in vitro model is based on the arrangement defined by Baan et al. [5]. Ten electrodes are attached to a catheter to replicate a beneficial electrode configuration for catheter-based VADs. This allows for the future application of our device in animal experiments as well. A miniaturized and low-cost measurement system was developed, which records the impedance in real-time while simultaneously allowing for changes of measurement parameters, such as measurement frequency, sampling rate and signal pre-amplification. The impedance measurements were performed using a 6–French ten-electrode electrophysiology catheter (VANGUARD AG, Berlin, Germany). The electrodes are made from platinum and evenly distributed on the catheter at a distance of 5mm.

We created a modular hardware configuration to allow for rapid modifications in the prototype phase. The schematic arrangement of the printed circuit boards (PCBs) designed is shown in Fig. 2. It consists of a controller area network (CAN) communication board, the Launchpad MSP430G2553 (Texas Instruments Incorporated, Dallas, USA), a power supply PCB and a measurement PCB.

Fig. 2

The power supply PCB provides 3.3 V and 5V by two low-dropout regulators. Additionally, a charge pump is integrated which generates −3.3V for multiplexer and amplifier circuit.

The centerpiece of the measuring unit is the microcontroller MSP430G2553 integrated on the launchpad, which is used for communication via serial peripheral interface with the integrated circuits on the CAN-PCB and the measurement PCB.

The analog front-end AFE4300 from Texas Instruments Incorporated (Dallas, USA), which is a cost-effective and highly integrated circuit used in body fat scales, featuring bioelectrical impedance analysis, is mounted on the measurement PCB. The bioelectrical impedance analysis function can be used to perform reliable four-point impedance measurements between 1 kHz and 100 kHz obtaining magnitude and phase for various frequencies. It should be noted that an impedance range of 1Ω − 2.8kΩ is covered by using the AFE4300 front-end for body fat scales. In our measurement setup, the expected impedance range is approximately between 0Ω and 100Ω, resulting in low utilization of the 16 Bit resolution range of the analog-to-digital converter (ADC), which is embedded in the AFE4300. However, this problem was solved by an optional pre-amplification of the voltage signal integrated onto the measurement PCB. Additionally, the PCB includes the multiplexer DG4051EEG-T1-GE3 (Vishay Intertechnology, Inc., USA, Malvern) to switch between different sensing electrodes of the catheter. The current injection is permanently attached to the outer electrodes (1 and 10) of the electrophysiology catheter. The ADC of the AFE4300 can convert the voltage difference of two electrodes simultaneously. We applied external multiplexing to enable faster sensing at all electrodes (2 − 9), since the internal multiplexing of the AFE4300 is too slow to perform heart cycle synchronous measurements. The maximal conversion rate for one measurement is 95 Hz, which results from the maximal sampling rate of the ADC, the time for multiplexing of all channels and communication delays. A conversion rate of 4 Hz was selected for this study. The AFE4300 is calibrated with known resistors to convert the values recorded by the ADC into impedance values. While the calibration with two reference values can be described by a straight-line, the use of four resistors requires a linear multi-point interpolation, so that the four values measured are used to calculate three straight-line equations. Therefore, a calibration network consisting of four resistances is integrated on the measurement PCB. The calibration resistances selected for this study can be taken from Table 1. The AFE4300 has two different operation modes for further processing of the measurement signal. Both phase and the magnitude are calculated in the I/Q-demodulation mode. In this mode, two signals are generated which are proportional to the real and imaginary part of the impedance and must be sampled individually.

We use the full-wave rectifier, however, in a different mode where only the magnitude of the impedance signal is determined. Therefore, the voltage drop at the impedance is calculated by injecting a sinusoidal current using (1) u(t)=A|Z|sin (ω0t+θ). u(t) = A\left| Z \right|\sin \left( {\omega _0 t + \theta } \right). Here, Z is the magnitude and θ is the phase of the impedance at given frequency ω₀. The resulting DC value, which is proportional to the magnitude, is rectified by the AFE4300 as described in (2) DC=2T∫T2A|Z|sin (ω0t+θ)dt=2A|Z|π=K|Z|. DC = {2 \over T}\int_{{T \over 2}} {A\left| Z \right|\sin \left( {\omega _0 t + \theta } \right){\rm{d}}t = {{2A\left| Z \right|} \over \pi }} = K\left| Z \right|. The factor K is then determined with the calibration resistances. The full-wave rectification is much faster and the calibration needs to be performed at only one measurement frequency. Please refer to the data sheet [13] for a full description of features of the AFE4300 and the implementation of the serial peripheral interface communication.

A resistance of 1 kΩ to limit the current amplitude for cardiac current injection and a capacitance of 1 μF for the suppression of DC voltages were inserted into the injecting path. This leads to an effective injected current amplitude of about 254 μA.

The in vitro models and the measurement system developed are going to be integrated into the heart volume test bench presented in an earlier work [14]. Therefore, a CAN bus was implemented to guarantee corresponding communication standards. The CAN-PCB includes a CAN controller and a transceiver that communicate via serial peripheral interface with the microcontroller. All signals on the CAN bus are collected by the MicroAutoBox II (dSPACE GmbH, Paderborn, Germany), which runs in real-time. The MicroAutoBox II is used as an overall system to adapt all parameters in real-time and record all quantities measured. The user interface developed displays the impedances of all segments of the catheter measured individually. In addition, the measurement frequency, the sampling rate of the AFE4300, and the selection of the calibration resistances are visualized and can be changed during the measurement.

The conducted research is not related to either human or animals use.

The validation of the impedance measurement system presented was carried out by comparing its results to metal film resistors with an accuracy of ±1%. The conversion rate of the AFE4300 was set to 4 Hz and the measurement frequency for the injected current was set to 64 kHz. Two networks of seven serial connected resistances net1 (seven resistances of 12 Ω) and net2 (seven resistances of 22 Ω) were connected to the measurement system, so that each of the seven channels was tested. One measurement of all channels in less than 250 ms reached an accuracy of 99.34 % for each channel tested with both resistance networks (net1 and net2).

In addition, we performed a sensitivity analysis of the measurement system at the measurement frequency of 64 kHz to investigate the influence of varying resistance values of one channel on the others. The analysis identifies the limitations of our system and allows the correct interpretation of measurement results. For this, one resistor of net2 is substituted with a different resistor (12.1 Ω, 33.3 Ω, 46.8 Ω or 98.4 Ω) and the conversion rate was varied between 4 Hz to 98 Hz. As a result, the overall error increases up to 10 %. This error was observed as an increase in impedance for all channels, at the highest possible conversion rate of 95 Hz and the substitution of a resistance of 98.4 Ω. This may be due to a transient phenomenon in the AFE4300, where the stationary state seems to be reached faster when similar resistances are monitored.

As we do not expect resistances with large ohmic differences between the catheter segments, it is possible for future applications to measure the cardiac cycle synchronously with high conversion rates. Since we only consider static measurements in this study, we suggest that all measurements presented have an accuracy in the range of 99.34 %.

The measured resistances of both ventricle phantoms are illustrated for each electrode segment in Figure 4 on the left side. The insulating ventricle is shown in blue and the conductive ventricle in red. We observed that resistances close to the apex are larger than those measured in the middle of the phantom. This confirms measurement results expected, as the volume of the blood cavity, the most conductive material, is larger in the middle than in the apex region. The minimum of the resistance measured is reached in the segment between electrodes 6 and 7. The measurements from the conductive ventricle phantom differ from the insulating ventricle measurements at an average of −3.55Ω ± 0.198Ω. This offset is proportional to the increase in the cross-sectional area, directly referring to the thickness of the muscle wall of 8 mm – 10 mm. In the case of the insulating material, the current remains in the blood chamber, while in conductive material, parts of the injecting current flows into the ventricle myocardium. Therefore, the cross-sectional area through which all the injected current flows is smaller, leading to an overall increase in resistance in the case of the insulating phantom. The resistances measured in both ventricles show the same shape across the different segments, thus, we can assume that the catheter is similarly placed in both ventricles and that the geometry of both cast phantoms correlates well.

Fig. 4

A two-dimensional map of the relative current density of the simulations with insulating and conductive material is depicted in Figure 5: the current pathways of a ventricle with insulating properties are shown in the left side and the current pathways of the ventricle with conductive properties are shown on the right side. It is clearly noticeable that in the case of insulating material, the current remains in the blood chamber (Fig. 5, left), while in the case of conductive material, a significant part of the current flows into the ventricle myocardium (Fig. 5, right). All materials have been defined as discussed and the catheter was placed in a position in the middle of the ventricle.

Fig. 5

The estimated resistances for all segments between adjacent electrodes are presented in Figure 4 on the right side. The insulating ventricle is illustrated in blue and the conductive ventricle in red. It is noticeable that higher resistances appear in the region close to the apex, whereas the lowest resistances can be observed at the segments in the middle of the catheter (4 − 5). As discussed, this may be due to the lower volume of blood in the apex region. Comparing the insulating and the conductive ventricle, conductive muscle walls reduce impedance. On average, the difference of the resistance between both settings is −1.612Ω ± 0.334Ω. Both curves converge at the segment close to the aortic valve (2−3). As these electrodes are close to the blood filled aorta, the muscle walls show a reduced impact on the current distribution at the catheter.

A similar catheter position in both scenarios had to be found for the comparison of measurements obtained at the in vitro models and the simulation results from the in silico model. The catheter in the FE model can be shifted and rotated as indicated by the arrows on the right side in Figure 3. One of the difficulties in this study was the placement of the electrode catheter in the silicone phantoms, since the phantoms are not transparent. To solve this problem, we set a fixed insertion distance of 9 cm from electrode 1 to the beginning of the PVC hose. With this setup the assumed catheter position is central in the ventricle without touching the phantom walls. In the FEM simulation, a distance of 9 cm from electrode 1 to the end of the ventricle, where only the PVC hose remains, was set. The results shown in Figure 5 and 4 are taken with this catheter location.

Looking at the results from Figure 4 and comparing measurements (left) and simulations (right), it can be observed that the shapes of the curves are quantitatively similar. The assumption from section Impedance measurement system that resistances measured for different segments are within a narrow ohmic range (±4Ω) is validated with the simulations, suggesting that the measurement system presented allows for highly accurate impedance monitoring. Comparing measurement and simulation results, it can be noticed that a shift of resistance minima appears from position 4 − 5 in the simulation to 6 − 7 in the in vitro models. This may be due to the fact that the real catheter is flexible and, therefore, the determination of its position is not very accurate.

Additionally, the difference of insulating to conductive material is higher in the measurements than in the simulations. A possible explanation is the difficult placement of the core inside the casting molds that may lead to a shift of wall thickness in the in vitro model. We suggest only using polyvinylalcohol cores for future phantom casting. Nevertheless, the resistance measurements in the area of the apex are reproduced in the simulation, concluding that this area was manufactured as planned. The simulation and measurement results differ the most closest to the connecting PVC hose. Furthermore, the difference between the conductive and the insulating ventricle is more significant in the measurements compared to the simulation. Our assumption is that the shape of the ventricle phantoms in the aortic valve area differs from the CAD model, and therefore, from the model in CST Studio. Silicone was used to glue the phantom to the PVC hose while ensuring waterproofness, which resulted in more material. Small differences in the conductivity and wall thickness of the modeled myocardium can lead to differences in absolute values observed.

Furthermore, there are some limitations of the presented study. The background material of the models was kept at a fixed conductivity of water. However, the conductivity of the surrounding tissue changes for instance by respiration (inflated: 0.104Sm 0.104{{\rm{S}} \over {\rm{m}}} ; deflated: 0.265Sm @ 64kHz 0.265{{\rm{S}} \over {\rm{m}}}@64{\rm{kHz}} . This phenomenon must be taken into account for dynamic operation. In addition, the deformation of the lung has an effect on the anatomy of the heart. Therefore, the development of differently shaped hearts would be advantageous to cover a wide range of anatomic differences. Furthermore, changes in conductivity due to changes in wall thickness, as they occur in the contracting heart, should also be investigated for the developed models.