FEM-based volume estimation using electrode catheter measurements

Admittance is measured using a 4-electrode configuration to minimize the influence of lead resistance R_l and electrode polarization effects [12]. Fig. 1 illustrates the measurement concept, where two electrodes are used to inject an electrical current and two other electrodes are used to measure a potential difference. The (transfer) admittance Y is then defined as the ratio of injected current I to the measured voltage V between the voltage-sensing electrodes: Y = I/V . We use a 10-electrode intracardiac catheter, so that in principle 5040 distinct 4-electrode current–voltage configurations are possible if all permutations of injection and measurement electrode pairs are considered.

Figure 1:

Close-up of a section of the catheter, with a conceptual drawing of a single 4 electrode measurement. All electrode combinations with both measurement electrodes inside of the injection electrodes are considered.

For real-time monitoring, only a limited number of measurements can be measured. To obtain a feasible and informative measurement set, we systematically reduce the number of configurations based on physical considerations and expected signal quality. First, the reciprocity principle implies that exchanging current injection and voltage measurement pairs does not provide additional information, allowing us to discard reciprocal duplicates. Second, we exclude configurations in which the current-injecting electrodes lie between the voltage-sensing electrodes along the catheter, as these tend to exhibit a low signal-to-noise ratio (SNR). Applying these criteria yields a reduced set of 210 4-electrode configurations.

For algorithm development and evaluation, we use this set of 210 measurements as a reference and investigate further reductions motivated by sensitivity analysis. Let Y ∈ ℂ^m denote the vector of simulated admittances for all m selected 4-electrode configurations at a given operating point, and let p_i be a scalar model parameter of interest (e.g., LVV, catheter offset, or blood conductivity σ_bl). The relative sensitivity of the measurements with respect to parameter p_i is defined as (1) Spi=1Y⊙∂Y∂pi, {\bf{S}}_{p_i } = {1 \over \bf{Y}}\odot {{\partial \bf{Y}} \over {\partial p_i }}, where division is performed element-wise and ⊙ denotes the Hadamard (element-wise) product.

We compute sensitivities for three parameters: LVV, horizontal catheter offset o, and blood conductivity σ_bl, using the cylindrical left ventricle model with a LVV of 70 mL and the catheter in the center. The model is fully described in the upcoming simulation section. To enable comparison across the parameters with different scales, each sensitivity vector is normalized to 1.

The resulting sensitivity values for all 210 configurations are shown in Fig. 2, where the measurements are sorted three times according to the magnitude of S_LVV, S_o, and S_σbl, respectively. When sorted by LVV sensitivity, the sensitivities with respect to catheter offset and blood conductivity vary without strong correlation, indicating that some measurements are predominantly sensitive to LVV. In contrast, sorting by catheter offset or blood conductivity reveals a strong correlation between S_o and S_σbl, suggesting that these two parameters are more difficult to distinguish based solely on the admittance data.

Figure 2:

Normalized sensitivity of the measurements. Sorted by three different parameters: LVV (left), blood conductivity (center), catheter offset (right).

This analysis shows that different 4-electrode configurations emphasize different parameters and that a partial separation of LVV from catheter offset and blood conductivity is feasible, while offset and σ_bl are more tightly coupled. We thus exclude the conductivity of blood as an estimation parameter and only estimate LVV alone, and LVV and catheter offset. Based on the sensitivity ranking, we construct a reduced measurement set by taking the union of the ten most sensitive configurations for each parameter, resulting in 19 distinct 4-electrode measurements due to overlap between the corresponding subsets. In the following, we therefore evaluate the estimation algorithm using both the full set of 210 measurements and this reduced set of 19 sensitivity-optimized configurations to explore the trade-off between measurement effort and estimation performance.

The parameter estimation problem is formulated as the adjustment of a finite-element forward model such that the simulated 4-electrode admittance measurements match the measured data as closely as possible. Let the parameter vector p ∈ ℝⁿ contain the quantities to be estimated (e.g., LVV and catheter offset), and let Y_sim(p) ∈ ℂ^m denote the corresponding vector of simulated admittances for all selected measurement configurations. For a given measurement vector Y_meas ∈ ℂ^m, the residual is defined as (2) r(p)=Ysim(p)-Ymeas. {\bf{r}}(\bf{p}) = {\bf{Y}}_{{\rm{sim}}} ({\bf{p}}) - {\bf{Y}}_{\rm meas}.

The inverse problem is then the non-linear least-squares minimization of the residual norm. We solve this problem by an iterative Gauss–Newton scheme. An overview of the algorithm is shown in Fig. 3.

Figure 3:

High-level overview of the FEM-based estimation algorithm using intracardiac admittance measurements.

To apply this gradient-based optimization scheme, we compute the Jacobian matrix (3) J(p)∈ℂm×n, Jij(p)=∂ri(p)∂pj , {\bf{J}}({\bf{p})} \in {\mathbb{C}}^{m \times n} ,\,\,\,\,\,\,\,\,\,J_{ij} ({\bf{p}}) = {{\partial r_i ({\bf{p}})} \over {\partial p_j }}, with i ∈ {1, . . . ,m} indexing the measurements and j ∈ {1, . . ., n} indexing the parameters. Since the measurements Y_meas are independent of the parameters, the Jacobian can be expressed directly in terms of the simulated admittances as (4) Jij(p)=∂Ysim,i(p)∂pj . J_{ij} ({\bf{p}}) = {{\partial Y_{{\rm sim},i} (\bf{p})} \over {\partial p_j }}.

The derivatives are evaluated numerically using finite differences: central differences are employed in the interior of the admissible simulation region, and one-sided differences are used near the bounds of the simulation region. Step sizes for the finite differences are chosen small enough to approximate the derivatives accurately while avoiding excessive numerical noise.

At iteration k, the residual r_k = r(p_k) and Jacobian J_k = J(p_k) are evaluated at the current parameter estimate p_k. Linearizing r(p) around p_k yields (5) r(pk+Δp)≈rk+JkΔp, {\bf{r}}({\bf{p}}_k + \Delta {\bf{p}}) \approx {\bf{r}}_k + {\bf{J}}_k \Delta {\bf{p}}, and the Gauss–Newton step Δp_k is obtained by solving the normal equations (6) (J kHJk+λI)Δpk=-J kHrk, ({\bf{J}}_k^{\rm{H}} {\bf{J}}_k + \lambda {\bf{I}})\Delta {\bf{p}}_k = - {\bf{J}}_k^{\rm{H}} {\bf{r}}_k , where (·)^H denotes the conjugate transpose, I is the identity matrix, and λ > 0 is a small Tikhonov regularization parameter [13] introduced purely for numerical stability. The regularization does not aim to promote sparsity, as all parameters are expected to be non-zero, but to mitigate ill-conditioning in J kHJk {\bf{J}}_k^{\rm{H}} {\bf{J}}_k . Equation (6) is solved by an LU factorization of the system matrix.

The tentative update is applied as (7) pk+1=pk+αRe(Δpk), {\bf{p}}_{k + 1} ={ \bf{p}}_k + \alpha {\mathop{\rm Re}\nolimits} (\Delta {\bf{p}}_k ), where α ∈ (0, 1] is a scalar step size. We performed backtracking line search to determine the optimal step size α [14]. During the majority of iterations α was close to 0.4. For that reason, we then used a constant α = 0.4 instead of determining the optimal step size during each iteration to increase the performance of the algorithm. Finally, the new parameters are constrained to a plausible range covering physiological and pathological values by clipping each parameter to its allowed range: LVV is constrained between 0 mL and 200 mL and the catheter offset between 0 and the maximum physically possible value depending on the current LVV.

The iterative procedure is terminated either after a maximum of 20 iterations or earlier if the update becomes sufficiently small in all estimated parameters. Specifically, the iteration stops if the step size satisfies (8) α|Δoffset|<0.4 mm, α|ΔLVV| <1 ml \alpha |\Delta _{{\rm{offset}}} |< 0.4\,{\rm{mm}},\,\,\,\,\,\,\,\alpha |\Delta _{{\rm{LVV}}} |\,< 1\,{\rm{ml}} for the parameters included in the current estimation scenario. For cases where only LVV is estimated, only the corresponding condition is enforced, whereas for joint LVV and offset estimation both criteria must be satisfied.

The simulation model does not completely replicate a real measurement. The deviations are due to a different model geometry, differences in the electrical properties, deviations due to numerical issues, as well as, measurement errors. To compensate for these deviations we performed a calibration of the measurements. We used an independent measurement of the LVV at minimal and maximal volume, similar to the established practice for conductance/admittance catheter approaches [8, 9]. For each 4-electrode configuration, we then assumed an affine mapping between measured and corrected admittance, (9) Ycorr=aYmeas+b, Y_{corr} = aY_{meas} + b, with a and b chosen accordingly to remove the deviation from the independent measurement.

The algorithm was evaluated in two settings: A simulation, covered in this section and an experiment described in the following section.

Although the left ventricle has a conical shape, analytical simulations have shown that for admittance measurements a simple cylindrical ventricle shape with fixed length closely resembles a full anatomical model. For this reason, most analytical models [15, 9, 16] have been developed using this simplification. Here, we used our cylindrical model, developed in our previous work [17] with a varying diameter, fixed ventricle length of 90 mm and myocardium thickness of 12 mm and surrounding tissue.

The electrical conductivity σ and relative permittivity ε for each material were chosen for a measurement at 50 kHz based on physiological data [18]. We selected for blood σ_bl = 0.7 S/m and ε_bl = 5197; for myocardium σ_myo = 0.19 S/m and ε_myo = 16982; and for the background tissue σ_bgrd = 0.1 S/m and ε_bgrd = 4272.

Finite-element simulations were performed using EIDORS, an open-source MATLAB toolkit for low frequency/quasi-static electrical simulations [19]. Geometries, electrode placements, and material properties were defined in EIDORS and discretized by Netgen using conforming tetrahedral meshes. The resulting sparse linear system from the FEM discretization was assembled and solved with EIDORS’ forward solver to obtain electrode voltages and resulting transfer admittances. Mesh density was selected via hyperparameter tuning to ensure numerical convergence and stability.

Figure 4:

Experimental setup; CAD drawing of phantom model phantom (top); Real phantom submerged in NaCl solution (bottom).

To perform measurements in a controlled setup, we have developed a 3D printed left ventricle phantom that allows for changes in volume (Fig. 4). Ventricle contraction was emulated by approximating the cylindrical geometry by a 6-sided aperture-like design, with each segment sliding in a fixture to achieve the change in diameter. The phantom is driven by a microcontroller-controlled motor to allow for automatic changes in volume. This allows for a volume range from 5 mL to 210 mL, while also enabling dynamical measurements that emulate the cardiac motion during the heart cycle.

Since conducting filament with a conductivity similar to myocardium is not available, we have approximated the myocardium as non-conducting. The phantom was printed with PLA and was fully submerged in an NaCl solution with an electrical conductivity of 0.7 S/m, modeling the blood in the ventricle. The impedance measurements were performed at 50 kHz using a voltage injection of 10 mV resulting in currents below 6 μA, thus below the legal limit of 10 μA for a cardiac floating patient auxiliary current. The 10-electrode catheter (Inquiry, Abbott) used in the measurement was fixated in the center of the ventricle (not shown in the figure) and connected to an impedance analyzer (ISX-3, Sciospec).

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

In a first step, we evaluated the proposed estimation algorithm using the cylindrical FEM ventricle model with physiological electrical properties described in the simulation subsection of the Method section. All simulations were performed at 50 kHz using the 10-electrode catheter and either the full set of 210 4-electrode configurations or the reduced set of 19 sensitivity-optimized configurations.

We first considered the case where only volume is estimated while the catheter is assumed to be centered and all other parameters are fixed at their nominal values. For a true volume of 70 mL, we computed the 2-norm of the residual, eq. (2), as a function of volume for three scenarios: (i) using all 210 measurements without noise, (ii) using the reduced set of 19 measurements without noise, and (iii) using the reduced set with added measurement noise. The resulting curves are shown in Fig. 5.

Figure 5:

Residual norm as a function of volume for the physiological simulation with a true volume of 70 mL: comparison of all 210 measurements, the reduced set of 19 measurements, and the reduced set with added noise.

For both noise-free cases, the residual exhibits a clear minimum at the true volume, indicating good identifiability of volume from the admittance data. The residual curve obtained from the reduced measurement set closely matches that of the full set, demonstrating that the sensitivity-based selection retains most of the volume information while substantially reducing the number of measurements. In the noisy scenario, the residual values are overall similar to the noise-free case, however, the minimum becomes slightly broader. Nevertheless, the location of the minimum remains near the true volume, suggesting that the volume estimate is robust with respect to the considered level of measurement noise.

Quantitatively, the estimation always yields an absolute error below 1 mL for all volumes, the small deviation is just below the stopping criterion. This error is expected due to the different meshes.

In a second simulation study, we jointly estimated volume and horizontal catheter offset to assess whether the algorithm can separate these two parameters despite their coupled influence on the admittance measurements. Using the physiological FEM model with a true volume of 70 mL and a 4 mm catheter offset, we computed the residual norm on a grid in the two-dimensional parameter space and visualized the resulting residual map for the reduced set of 19 measurements. Fig. 6 shows the residual map together with the Gauss–Newton iteration paths for four different initial guesses.

Figure 6:

Residual map for the simulation using the reduced set of 19 measurements; the true parameter combination is marked in red, and four Gauss–Newton iteration trajectories from different initial guesses are overlaid.

The residual map exhibits a well-defined basin of attraction with a single pronounced minimum located near the true volume–offset combination, indicating local convexity around the solution and good joint identifiability of both parameters for the chosen electrode configurations. For all four initialization points, the Gauss–Newton iterations converge toward the same region in parameter space within the prescribed maximum number of iterations, illustrating that the algorithm is robust with respect to the initial guess. These findings support the use of the reduced, sensitivity-optimized measurement set for simultaneous volume and offset estimation in the physiological simulation setting. We repeated this analysis for 50 different targets and achieved a mean estimation error for the volume of 2.4% regardless of the chosen initial conditions.

In a second step, we validated the approach using the dynamic 3D-printed ventricle phantom submerged in a NaCl solution and equipped with the 10-electrode catheter as described in the method section. In contrast to the physiological simulation, the phantom walls are nonconductive, and no background tissue is present, which has been adjusted in the estimation algorithm accordingly.

For the experiments, the phantom was operated in a volume range from 25 mL to 200 mL in increments of 5 mL. Intracardiac admittance measurements were acquired at each step. The volume was then estimated using both the full set of 210 measurements and the reduced set of 19 measurements. Fig. 7 shows the reconstructed volumes.

Figure 7:

Estimated volume over actual volume of the phantom experiment for the full set of 210 measurements and the reduced set of 19 measurements.

Both reconstructions closely follow the identity line, indicating that the algorithm can reliably estimate the volume in this controlled setting. The volume curves obtained from the full and reduced measurement sets are in good agreement, with a mean error of 3.2% and 2.2 %, which suggests that the sensitivity-based measurement reduction remains effective in the presence of model–phantom discrepancies and measurement noise.

Finally, we investigated joint estimation of volume and catheter offset using data from the phantom experiment. Volume and offset were estimated using both the full set of 210 measurements and the reduced set of 19 measurements. Fig. 8 shows the reconstructed volumes.

Figure 8:

Estimated volume for the joint volume and offset estimation in the phantom. Full measurement set of 210 measurements and the reduced set of 19 measurements.

The experimental residual map exhibits a distinct minimum region, although less symmetric and more elongated than in the purely simulated case, reflecting the combined effects of phantom geometry, non-conductive walls, and measurement noise. Nonetheless, the Gauss–Newton trajectories converge toward the same region in parameter space, indicating that the algorithm can still reconcile volume and catheter offset from the experimental admittance data after calibration. These results demonstrate that the proposed FEM-based estimation approach is transferable from numerical simulations to a physical test bench and can cope with moderate model mismatch and experimental imperfections.