Automated Fetal Magnetocardiography Reconstruction Using a Biophysical Forward Model

1 Introduction

• An abdominal sensor array measures fields amid noise and stronger maternal activity; reduced shielding and greater environmental-noise sensitivity raise reconstruction demands.Source: §1 · p. 2, Figure 1 · p. 2

Related work

• ICA fragments sources and requires error-prone manual annotation, while template-based filters need manual selection or many-beat averages and neither approach models the source-field relationship.Source: §1 · p. 2

2.3 Forward model

• The approach models each cardiac source as a time-varying three-dimensional dipole and relates its moment and position to measured fields through a forward model.Source: §1 · p. 2, §2.3 · p. 5

1 Introduction

• Synthetic data recover ground-truth dipole parameters accurately, while real recordings agree with manual spatial-filtering processing and reported normative physiological values.Source: §1 · p. 3

Related work

• Prior fMCG dipole models tracked fetal position and movement from reconstructed signals; this work instead uses dipoles for signal reconstruction.Source: §1 · p. 2, Related work · p. 3

2.4 Signal reconstruction with regularized least-squares

• The pipeline preprocesses measurements and estimates source parameters with regularized least squares, using whitened field observations and gradient-based optimization.Source: §2 · p. 3, §2.2.4 · p. 4, §2.4 · p. 6

2.4.1 Regularization

• To address fetal–maternal crosstalk, the regularizer minimizes cross-correlation between their estimated dipole moments across multiple time lags.Source: §2.4.1 · p. 6

2.5 Selective Averaging

• Optional selective averaging of similar-rate heartbeats improves waveform detail for cardiac interval assessment, trading beat-to-beat variability for signal quality.Source: §2.5 · p. 7

3 Results

• Synthetic recordings show low-error recovery, while real-data reconstructions agree with spatial filtering and yield cardiac intervals within reported healthy-fetus ranges.On real recordings, interval differences were below typical inter-annotator variability.Source: §3 · p. 7, §3.3 · p. 10

3.1 Validation on synthetic data

• Field-domain errors are low for all methods; the forward model and spatial filtering achieve the smallest errors, while only the forward model estimates dipole moments and positions.Synthetic sources were static, testing parameter identifiability rather than the fixed-position assumption; supplementary movable-dipole tests indicate the static model is a reliable trade-off when positions vary slowly.Source: §3.1 · pp. 7–8, Figure 3 · p. 8

3.2 Reconstructing cardiac dipole activity

• Real-data reconstructions yield physiologically plausible fetal waveforms comparable to splined ICS and spatial filtering, while uniquely providing source-domain signals and a full set of 3S field channels.The reconstructed dipole time series spans the recording; the fetal signal is two orders of magnitude weaker than the maternal signal.Source: §3.2 · pp. 8–9

3.3 Fetal cardiac time intervals

• All interval ICC values exceed 0.9, and mean absolute differences of 1.85–6.36 ms are below the 7–19 ms inter-annotator variability range.Measured intervals for both methods fall within the reported 95% prediction intervals for healthy fetuses.Source: §3.3 · pp. 10–11

3.4 Spatiotemporal cardiac representation

• The three-dimensional dipole vector forms a cardiac-cycle loop, and its axis-dependent projections produce different waveform morphologies across x, y, and z.In the illustrated example, the P-wave is prominent on x and z but weak on y; the T-wave is nearly absent, with only slight visibility on z.Source: §3.4 · p. 11, Figure 6 · p. 11

3.5 Automated cardiac delineation based on vectorcardiographic representation

• ECG-trained delineation transfers zero-shot to reconstructed fMCG vector signals, with near-perfect QRS detection, high P-wave detection despite larger timing errors, and more challenging T-wave detection.The authors attribute T-wave difficulty to the larger ECG–MCG domain shift and note that augmentation or fMCG fine-tuning could improve performance.Source: §3.5 · p. 12

3.6 Reconstructed dipole positions

• Estimated maternal dipole positions are more dispersed, while remaining separated from the tightly clustered fetal dipoles near the sensor array.Position estimates provide spatial information that can distinguish fetal from maternal sources when heart rate or signal amplitude is ambiguous.Source: §3.6 · pp. 12–13

4 Discussion

• Real fMCG evaluation can establish agreement with reference methods rather than absolute accuracy, and performance on pathological morphologies remains unestablished.The cohort contains only healthy fetuses.Source: §4 · p. 13
• The method uses a single equivalent dipole; multipole or distributed-source models could capture additional structure but bring severe ill-posedness and computational complexity.Multi-dipole and time-varying extensions would require strong fetal cardiac and movement priors that do not currently exist.Source: §4 · pp. 13–14
• Spatial localization adds information for distinguishing fetal and maternal activity when heart rate alone is insufficient, and estimated source positions separate the two sources.Position and orientation may also provide anatomical insights, but validating that possibility requires simultaneous reference measurements, which are currently unavailable.Source: §4 · p. 13, Figure 8 · p. 14
• Automated reconstruction supports potential clinical adoption, but diagnosis still requires experts and fMCG is not yet routine clinical practice.The reconstructed vectors may also support standardization across subjects and gestational age.Source: §4 · pp. 13–14

5 Conclusion

• This work presents the first forward model-based reconstruction approach for fMCG, estimating dipole source signals in a physically motivated representation.Source: §5 · p. 14
• Synthetic experiments recover underlying source activity, while real-data validation shows strong agreement with manual spatial filtering-based processing (ICC > 0.9) and anatomically consistent dipole positions.Source: §5 · p. 14
• The framework automates fMCG reconstruction and returns source signals in a physiologically interpretable cardiac vector domain, supporting potential translation into prenatal care.Source: §5 · p. 14

S1.1 Removing faulty sensor channels

• Channels with signal ranges below 1 pT or with more than half their samples outside the global variability range are excluded as faulty or outlying.The channel checks target insufficient variability and abnormal temporal variability relative to valid channels.Source: S1.1 Removing faulty sensor channels · p. 18

S1.2 Removing temporal artifacts

• A 7-level stationary Haar wavelet transform and robust segment-wise thresholds identify temporal artifacts, after which only artifact-free segments of at least 10 seconds are retained.Periods with substantial motion artifacts or magnetic interference are excluded.Source: S1.2 Removing temporal artifacts · p. 18

S2 ICA annotation limitations

• ICA often fragments cardiac sources across components, making annotation of subtle P- and T-waves laborious and sometimes impractical, while overlapping subspaces and nonstationary signals can impair separation.Figure S1 illustrates mixed and noisy components alongside more readily identified R-peak series.Source: S2 ICA annotation limitations · pp. 18–19, Figure S1 · p. 19

S3 Regularization

• Cross-correlation regularization reduces overlap between reconstructed fetal and maternal source components.Figure S2 contrasts reconstructions without and with the penalty.Source: S3 Regularization · p. 19, Figure S2 · p. 20

S3.1 Regularization strength

• The regularization-weight analysis used 31 real recordings and manually curated R-peak positions as its reference.Source: S3.1 Regularization strength · p. 20
• Maternal leakage in the weaker fetal reconstruction decreases monotonically as regularization increases, while fetal leakage in maternal components also declines but less strongly.For this cohort, λ values within [0.1, 0.5] provide a good balance; smaller values tend to leave more crosstalk, while larger values risk reducing amplitude and detection performance.Source: S3.1 Regularization strength · pp. 20–21
• Fetal beat-detection F1 improves from 0.906 without regularization to 0.999 around λ = 0.3, then drops for λ ≳1; maternal performance remains near perfect across values.Figure S3 summarizes crosstalk, amplitude change, and beat-detection performance across regularization weights; fetal amplitude eventually decreases at larger weights.Source: Figure S3 · p. 21, S3.1 Regularization strength · p. 21

S4.1 Annotation procedure

• Intervals were manually marked on averaged fetal waveforms in the field domain, with P, PR, QRS, QT, and RR defined by their wave onsets, offsets, or successive QRS peaks.QTc was calculated using Bazett’s formula.Source: S4.1 Annotation procedure · pp. 21–22, Figure S4 · p. 22

S4.2 Interval agreement across gestational age

• Agreement was assessed using one recording per patient (N = 22); the nine repeated recordings were excluded from the independent analysis, and QTc was excluded because its paired difference follows QT.The comparison used ordinary least-squares trends against gestational age; spatial filtering is a reference method, not ground truth, so absolute accuracy cannot be established.Source: S4.2 Interval agreement across… · p. 22
• Gestational age explained less than 2% of the variance in RR, P, PR, and QT differences; QRS showed R2 = 0.14, a −1.11 ms trend below annotator variability.Repeated recordings fell within 95% prediction intervals for all intervals except QT, whose opposite-sign exceptions indicate greater spread rather than a trend.Source: S4.2 Interval agreement across… · p. 22

S5 Relaxing the fixed dipole position constraint

• Time-varying dipole positions extend the framework to longer recordings or fetal movement, but add parameters and require regularization for stable reconstruction.The fixed-position model remains a sufficient, interpretable baseline for the main-text cardiac interval analysis.Source: S5 Relaxing the fixed dipole position… · pp. 22–23, S5.4 Clinical output robustness under… · p. 26

S5.1 Piecewise constant parametrization and Total Variation regularization

• Piecewise-constant position segments control the degrees of freedom, while a Total Variation penalty discourages rapid changes between consecutive segment positions.The TV term is normalized by the number of consecutive differences, making it independent of recording length and the number of optimized parameters.Source: S5.1 Piecewise constant… · p. 23

S5.2 Parameter sweep

• At λTV = 100, moving-dipole reconstruction matched static SDNN within 0.4 ms per patient using 1-second segments, with comparable performance from 0.5 to 2 seconds.Without sufficient regularization, performance degraded as shorter segments increased the degrees of freedom.Source: S5.2 Parameter sweep · p. 24, Figure S6 · p. 24

S5.3 Reconstructed position trajectories

• Reconstructed trajectories were smooth and spatially bounded, but their dependence on segment length leaves it unclear whether they represent physical motion or compensate for model mismatch.Validation would require independent fetal movement tracking, and movement priors remain future work.Source: S5.3 Reconstructed position trajectories · p. 24, Figure S7 · p. 25

S5.4 Clinical output robustness under positional ambiguity

• At λTV = 100, fetal moment error was 6.4% for TV regularized versus 7.5% for the static baseline, while SDNN error was 0.5% versus 0.2%.With moderate TV weights, moment and SDNN errors could be minimized simultaneously at levels comparable to the static dipole baseline.Source: S5.4 Clinical output robustness under… · p. 26, Figure S8 · p. 26

• Dipole positions were mapped into bounded, differentiable ranges based on device dimensions and physiological constraints, while moments were initialized with a robust pseudo-inverse estimate.The pseudo-inverse discarded singular values below a threshold to support stable initialization.Source: S6 Optimization details · pp. 26–27
• Group-wise gradient normalization can compensate for differing magnitudes across fetal and maternal moments and spatial coordinates; parameters were rescaled to comparable scales before optimization.Source: S6 Optimization details · pp. 26–27
• Reasonable hyperparameter variations yielded qualitatively similar results, while longer optimization primarily improved data fitting without substantially changing reconstruction quality.Experiments used 500 iterations, with learning rates reduced when loss plateaued and a 50-step warmup.Source: S6 Optimization details · p. 27

S7.1 Data

• The delineator was trained on LUDB (200 recordings) and ISP (475 recordings), with LUDB fiducial annotations converted to dense segmentation masks.Source: S7.1 Data · p. 27
• Kors transformation converted eight ECG leads into three orthogonal vectorcardiographic components while preserving temporal annotations, creating input analogous to the dipole source domain.Source: S7.1 Data · p. 27, S7 Automated cardiac delineation via… · p. 27

S7.2 Model architecture

• A 1D U-Net with spatial and temporal attention took three-channel vectorcardiographic input and produced four classes: background, P-wave, QRS, and T-wave.Source: S7.2 Model architecture · p. 28

S7.3 Data preprocessing & augmentation

• ECG preprocessing included resampling, notch and bandpass filtering, normalization, and two-second windows; training augmentation varied timing, amplitude, orientation, baseline wander, and artifacts.Source: S7.3 Data preprocessing & augmentation · p. 28

S7.4 Training

• Training used focal loss with class weights, and the 1,373 training and 295 validation windows were split while preserving proportions from both ECG datasets.The model was trained for 100 epochs using AdamW, with learning-rate reduction on validation plateaus and early stopping.Source: S7.4 Training · p. 28

S7.5 Evaluation

• Evaluation matched predicted and reference fiducial points within 150 ms, counting unmatched predictions and annotations as false positives and false negatives, respectively.The reported metrics covered detection performance and fiducial timing precision.Source: S7.5 Evaluation · pp. 28–29

S7.6 Performance on ECG

• On 295 held-out ECG windows, all wave boundaries except the T-wave offset were detected perfectly; the T-wave offset achieved an F1-score of 99.92%.Timing-error variability was within CSE recommendations, and the delineator learned vectorcardiographic features from ECG data.Source: S7.6 Performance on ECG · p. 29, Table S1 · p. 29
• The ECG-trained delineator transferred by zero-shot inference to 31 fMCG recordings, supporting feasibility of transferring learned cardiac features to the dipole domain.Source: S7 Automated cardiac delineation via… · p. 27, S7.6 Performance on ECG · p. 29