Time-variant left ventricle models for intracardiac impedance analysis

The cylinder model represents a simple approximation of the left ventricle, maintaining a uniform diameter throughout its height and thus disregarding the ventricle’s conical geometry. Nevertheless, the foundational equations for left ventricular volume estimation via impedance have been derived using this cylindrical assumption. Baan et al. formulated a conductance-to-volume conversion equation predicated on the cylindrical segments that collectively approximate the LV [5]. Furthermore, Wei et al. conducted an in-vitro assessment of the conductance catheter measurement system employing cylindrical vessels and numerical finite element models to emulate conductance catheter readings [11, 6]. Thus, the cylinder model is appropriate for the evaluation of these algorithms. The LVV of a cylinder of length l and radius r is given by 1 LVVcylinder=πr2l. \[\text{LV}{{\text{V}}_{\text{cylinder}}}=\pi {{r}^{2}}l.\]

The dominant factor contributing to volume changes of the ventricle is the alteration of its diameter. Prior research [12, 13] has shown that variations in ventricular length exert only a small influence. Hence, this change in length is neglected here and this model is kept at a fixed length of 90 mm. Thus, simulating the cardiac cycle with volumes ranging from 45 mL to 140 mL yields a diameter varying between 26 mm and 44mm.

In its basic form, as described in literature, this model lacks the aorta and myocardium. To enhance the model, a cylinder representing the aorta with a diameter of 30 mm is added and the ventricle itself is modeled by myocardium of constant thickness. Although myocardial thickness fluctuates with contraction and relaxation during the cardiac cycle, these variations are locationspecific, and comprehensive data reflecting myocardial thickness changes throughout the ventricle during the cardiac cycle are not available. Hence, the myocardium is represented with a consistent thickness of 12 mm.

The electrode catheter utilized for the simulated impedance measurements is positioned in the center of the model. A 7 F catheter (outer diameter of 2 mm) equipped with 10 electrodes, each 1 mm wide and spaced at 10 mm intervals from each other, is incorporated into the model. To prevent current injection into the myocardium, the bottom electrode is situated 5 mm from the apex, while the top electrode is placed just above the aortic valve. The first column of Figure 1 depicts the final model in both the end-systolic and end-diastolic phases.

Figure 1:

The four ventricle models. The blood is displayed in red and the ventricle is semitransparent red. The surrounding lung tissue is not shown.

The bullet model offers a more accurate representation of the LV’s morphology. This model is frequently utilized for LVV calculations in 2D-echocardiography methods [14, 15, 16]. It integrates a cylindrical upper section with an ellipsoidal lower segment, mirroring the tapering apex of the LV, with each segment accounting for half of the LV’s total length. Consequently, the ellipsoidal segment has two minor axes with the same length as the radius r and a major axis measuring half the total length l. The LVV for the bullet model can be expressed by 2 LVVbullet=56πr2l. \[\text{LV}{{\text{V}}_{\text{bullet}}}=\frac{5}{6}\pi {{r}^{2}}l.\]

The simulation parameters mirror those of the cylinder model. The LV is modeled with a constant length of 90 mm, while the diameter, derived from eq. (2), varies between 28 mm and 48 mm over the cardiac cycle. The myocardium enveloping the model has a thickness of 12 mm. Additionally, the model incorporates a cylindrical aorta, 30 mm in diameter. The catheter is positioned partially within the aorta, aligned with the longitudinal axis. Given the model’s fixed length, the uppermost electrode is consistently situated above the aortic valve for the duration of the cardiac cycle. The resultant model, captured at both end-systole and end-diastole, is illustrated in the second column of Figure 1.

The truncated prolate spherical model (TPS) introduces a variation in ventricle length. Initially proposed by Domingues et al. [17] as a mathematical model for determining the surface area of the human LV, this model has been applied to compute LVV from cardiac imagery. It was subsequently validated in bovine subjects for predicting LVV for volumes reaching 164 mL [18].

The application of the TPS model for intracardiac impedance simulations, as in this study, is not documented in existing literature. Schiller et al. [19] have delineated methods for gauging LV dimensions using two-dimensional echocardiography. They suggest dividing the TPS model’s long axis into segments measuring 2/3 l and 1/3 l. With a short axis length of r, the TPS model’s LVV is given by 3 LVVTPS=3πr3. \[\text{LV}{{\text{V}}_{\text{TPS}}}=3\pi {{r}^{3}}.\]

The resulting model at end-systole and end-diastole phases is shown in the third column of Figure 1. In this model, the LV length varies between 67 mm and 98 mm, and the diameter ranges from 34 mm and 50 mm throughout the cardiac cycle. The length, thus, exceeds the typical physiological range of 84 mm to 93 mm [13]. Despite this, the TPS model is advantageous in evaluating the impact of catheter positioning shifts along the longitudinal axis. The myocardium encasing the model is again portrayed with a uniform thickness of 12 mm. The catheter is anchored at the apex, akin to patient measurements, which results in electrode displacement relative to the aortic valve as volumes vary, potentially extending into the aorta. The aorta is once more represented as a cylinder with a 30 mm diameter.

The physiological model is based on image data of a human heart. It has been created from images at 5 distinct time points and thus is only available at discrete time steps of the cardiac cycle. Besides the changes in volume, the model incorporates changes of ventricle length, inner diameter, and myocardium thickness. Thus, this model is the most realistic, but also more complex. In the following the model creation process and the resulting model parameters are presented.

The physiological model has been created from CT images of a heart. Data were taken from the opensource 4D statistical shape model of the whole heart published by Unberath et al. [9]. This model is one of few open-source models that includes data at different time points of the cardiac cycle and was originally developed as a numerical phantom for cone-beam CT simulation. It contains surface meshes from ten-phase angiography measurements of the aorta, atria, ventricles, and left ventricular myocardium at ten phases during the cardiac cycle derived from data of 20 patients. The model is implemented in the author’s open-source simulation and reconstruction framework CONRAD. A metric volume phantom was created using the multi-material phantom renderer and exported in TIF format. The image processing package Fiji, a distribution of ImageJ, was used to generate the surface meshes. Segmentation of the blood inside the four chambers, left ventricular myocardium, and the aorta was done by thresholding. This approach extracts objects from the background by classifying pixels into a group whose intensities are larger than a specific value. To avoid holes in the final model, individual thresholds were used for each compartment. Autodesk Meshmixer© and the software Open Flipper were used to remove holes in the structures and for surface smoothing. The final surface meshes of the models were imported into the FEM simulation tool CST Studio as STL files and inserted with the following procedure to remove overlapping portions. First, the blood of the LV, left atrium, and aorta was added, so that these structures form one solid. Afterwards, the created left heart blood was inserted into the myocardium. The resulting structure was inserted into the right heart, which consists of the right atrium and ventricle. The resulting model is depicted in the fourth column of Figure 1.

The model embodies both left and right ventricles, which are filled with blood, and the ascending aorta all depicted in red. The LV is surrounded by the semitranslucent red left ventricular myocardium. The catheter is placed with the tip at the apex and the top extending into the aorta into the LV. This results in an off-center position close to the ventricular wall.

Throughout the cardiac cycle, the LVV varies between 73 mL and 146 mL, resulting in a stroke volume of 73 mL. Myocardial thickness ranges between 8 mm and 16 mm. The left ventricular inner diameter fluctuates between 40 mm and 50 mm, while the length varies from 81mm to 90 mm.

For the simulation of the LV geometry during the cardiac cycle a reference volume is needed. To ensure the results’ comparability across all models, a uniform volume cycle was adopted. Since the physiological model is only available at fixed steps, the volume cycle of this model is used as the reference volume.

The image data of the physiological model are available for ten discrete time points during the cardiac cycle. However, due to the extensive manual labor involved in modeling and simulation, only half of these time points are utilized for the simulation of the physiological model. For the volume reference, the LVV at all 10 time steps has been evaluated and a shape-preserving piecewise cubic interpolation was applied. The resulting volume cycle is shown in Figure 2, with the values at each of the ten available phases marked by a circle. The analytical models are continuously available and can be simulated at any volume of the reference curve.

Figure 2:

LVV during one cardiac cycle. The first three models are simulated for volume increments of 5 mL, while the physiological model is only available for the 5 indicated volumes.

The simulation of the electrical fields requires the electrical properties of each model component. The simulation models comprise of four materials: blood, myocardial tissue, the surrounding tissue, and the catheter. Both, the left ventricle and the aorta contain blood, the ventricle is modelled by myocardium. The surrounding matter is modeled by lung tissue, reflecting the heart’s primary anatomical enclosure. Each material is presumed to be homogenous and anisotropic.

The simulated impedance measurements were conducted at 50 kHz. The electrical properties of blood, myocardium and background tissue at this frequency are derived from Gabriel et al. [20]. The electrical conductivities σ for blood, myocardium, and lung tissue are then respectively 0.7 S/m, 0.19 S/m, and 0.1 S/m. The associated relative permittivities ϵ_r are given by 5197, 16982, and 4272, correspondingly.

Additionally, the catheter is modelled with perfectly conducting electrodes connected by polyimide sections with conductivity σ = 2 × 10^-8 S/m and permittivity ϵ_r = 3.52.

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