Spatial-dependent regularization to solve the inverse problem in electromyometrial imaging

Abstract

Recently, electromyometrial imaging (EMMI) was developed to non-invasively image uterine contractions in three dimensions. EMMI collects body surface electromyography (EMG) measurements and uses patient-specific body-uterus geometry generated from magnetic resonance images to reconstruct uterine electrical activity. Currently, EMMI uses the zero-order Tikhonov method with mean composite residual and smoothing operator (CRESO) to stabilize the underlying ill-posed inverse computation. However, this method is empirical and implements a global regularization parameter over all uterine sites, which is sub-optimal for EMMI given the severe eccentricity of body-uterus geometry. To address this limitation, we developed a spatial-dependent (SP) regularization method that considers both body-uterus eccentricity and EMG noise. We used electrical signals simulated with spherical and realistic geometry models to compare the reconstruction accuracy of the SP method to those of the CRESO and the L-Curve methods. The SP method reconstructed electrograms and potential maps more accurately than the other methods, especially in cases of high eccentricity and noise contamination. Thus, the SP method should facilitate clinical use of EMMI and can be used to improve the accuracy of other electrical imaging modalities, such as Electrocardiographic Imaging. Graphical abstract The spatial-dependent regularization (SP) technique was designed to improve the accuracy of Electromyometrial Imaging (EMMI). The top panel shows the eccentricity of body-uterus geometry and four representative body surface electrograms. The bottom panel shows boxplots of correlation coefficients and relative errors for the electrograms reconstructed with SP and two conventional methods, the L-Curve and mean CRESO methods.

Keywords: Electromyometrial imaging; Inverse problem; Regularization.

Fig. 1.. Schematic of EMMI system.

Top left, a patient-specific body-uterus geometry is obtained and segmented from an MRI scan while the patient is wearing up to 256 MRI-compatible markers. Lower left, body surface potentials are recorded from up to 256 pin-type unipolar electrode patches placed in positions corresponding to the MRI-compatible markers. Middle, EMMI software combines the two data sets to reconstruct uterine surface (top right) electrograms (electrical waveforms over time at each uterine site) and (middle right) potential maps (electrical activity across the uterus at a single time point). (Lower right) Activation times can be derived from the electrograms to construct isochrone maps. EMMI software is an in-house developed MATLAB package able to solve the inverse problem.

Fig.2.. Flowchart of the derivation of λ_SP.

Body-uterus geometry (G) and body surface potentials (ϕ_B) are shown in the blue boxes at the top left. Transfer matrix A and its singular value decompositions are shown in the grey boxes at the top right. The eccentricity (e) and signal-to-noise-ratio (SNR) are defined and shown in the yellow boxes. M, N, m, n, and Z are defined in the key. The procedure to calculate λ_SP by minimizing a cost function is shown in the orange boxes. The level threshold function p(a) as a function of unknown variable a is described in equation (a). j_k(a) represents the maximum index t of singular basis corresponding to variable a, where ${\sum }_{\mathrm{v}=1}^{\mathrm{t}}{\mathrm{X}}_{\text{kv}}\le p_k$ in equation (b). g(a) is derived by applying a moving average filter to j, and this process is denoted by S(j) in equation (c). The under-curve ratio r(a) is defined as the ratio between the area under the curve of g(a) and the total area (= NZ) of cumulative X in equation (d). r = q [e, SNR, 1]. By minimizing the difference between r(a) and r, an optimized a can be computed, which can be used compute p, g, λ_k, and λ_SP.

Fig. 3.. Geometry and physics settings of bioelectric simulation.

a, three-layer spherical geometry. Arrows represent the three orthogonal dipole directions: normal direction (green arrow), and two horizontal directions (red and blue). b, three-layer RPI geometry. c, schematic of bio-electricity simulation setting with three volume conductors, Ω_UB, Ω_AU, and Ω_A. O_B and O_U represent the centers of the body surface and uterine surface, respectively. R_B, R_U, and R_A represent the radius of the three spherical geometries. A dipole was placed on the intrauterine surface, Γ_A. The body surface point C is nearest to uterine point A. All images are right lateral views.

Fig. 4.. Reconstruction accuracy of electrograms and potential maps for spherical geometry.

a, a representative body surface potential map, where warm colors indicate positive potential and cool colors indicate negative potential. Black mesh inside the color map is the uterine surface geometry. b, Distribution of the optimal site-specific λ. Blue dots are discrete points on the simulated body surface. a and b both show right lateral views. c, reconstructed electrogram correlation coefficient (CC). d, reconstructed potential map CC. e, reconstructed electrogram relative error (RE). f, reconstructed potential map RE. In c, d, e, and f, red indicates optimal λ, blue indicates mean CRESO λ, and green indicates L-Curve λ. Uterine sites with index of 1~300 are located at the posterior, and sites with index of 601~900 are located at the anterior. Potential maps with index of 1~900 are correlated with posterior current dipole source, and index 1801~2700 are correlated with anterior current dipole source. The average peak-to-peak white Gaussian noise is 44 microvolts, which is 15% of the signal.

Fig. 5.. Reconstruction accuracy of electrograms and potential maps using different regularization methods for multiple spherical geometry settings and noise levels.

a – d, CC and RE values of electrograms and potential maps generated by the mean CRESO method subtracted from the CC and RE values of electrograms and potential maps generated by the SP method. e - h, CC and RE values of electrograms and potential maps generated by the L-Curve method subtracted from the CC and RE values of electrograms and potential maps generated by the SP method. In all graphs, warm colors represent positive differences and cold colors represent negative differences. X-axis, index of datasets; Y-axis, index of uterine site or potential map; Z-axis, difference in values. The datasets were sorted by geometry eccentricity and noise level, and a large index means large eccentricity and high noise level (See data file S2).

Fig. 6.. Comparison of reconstruction accuracy of three regularization methods under RPI Geometry with single current dipole.

a. reconstructed electrogram accuracy of λ_SP, mean CRESO λ, and L-Curve λ. b, reconstructed potential map accuracy of λ_SP, mean CRESO λ, and L-Curve λ. Cross boxes indicate CC values and striped boxes indicate RE values. Blue, orange, and pink represent mean CRESO method, SP method, and L-Curve method, respectively. n.s., not significant, * P < 0.05, ** P < 10⁻², *** P < 10⁻³, **** P < 10⁻⁴, compared to SP method.

Fig.7.. Comparison of reconstructed uterine surface electrograms using three regularization methods.

Simulated and reconstructed electrograms (0 to 10 s) from the indicated sites A to D. CC and RE values are labeled on the top of each electrogram panel. The average noise level of the BSPMs used is 15%.

Fig.8.. Evaluation of reconstruction accuracy of three regularization methods under RPI Geometry with multiple current dipoles.

a, a body surface potential map with RPI geometry setting shown in right and anterior views. Black mesh represents RPI uterine surface. Black arrows represent dipole locations and directions. b, Reconstructed electrogram accuracy of λ_SP, mean CRESO λ, and L-Curve λ. The electrogram accuracy of L-Curve λ was too small to show. c, Reconstructed potential map accuracy of λ_SP, mean CRESO λ, and L-Curve λ. Cross-hatched boxes indicate CC values, and striped boxes indicate RE values. Blue, orange, and pink represent mean CRESO method, SP method, and L-Curve method, respectively, n.s., not significant, * P < 0.05, ** P < 10⁻², *** P < 10⁻³, **** P < 10⁻⁴, compared to SP method.

Figure 9.. Reconstruction accuracy of SP method with 1400 vs. 256 body-surface sites.

a, RPI body-uterus geometry. The blue dots represent the 256 body surface sites. b, reconstruction accuracies with 1400 (black) vs. 256 (red) body surface sites. The median, first quartile, and third quartile of CC and RE values are shown by error bars.