Efficient Fetal-Maternal ECG Signal Separation from Two Channel Maternal Abdominal ECG via Diffusion-Based Channel Selection

There is a need for affordable, widely deployable maternal-fetal ECG monitors to improve maternal and fetal health during pregnancy and delivery. Based on the diffusion-based channel selection, here we present the mathematical formalism and clinical validation of an algorithm capable of accurate separation of maternal and fetal ECG from a two channel signal acquired over maternal abdomen. The proposed algorithm is the first algorithm, to the best of the authors' knowledge, focusing on the fetal ECG analysis based on two channel maternal abdominal ECG signal, and we apply it to two publicly available databases, the PhysioNet non-invasive fECG database (adfecgdb) and the 2013 PhysioNet/Computing in Cardiology Challenge (CinC2013), to validate the algorithm. The state-of-the-art results are achieved when compared with other available algorithms. Particularly, the F1 score for the R peak detection achieves 99.3% for the adfecgdb and 87.93% for the CinC2013, and the mean absolute error for the estimated R peak locations is 4.53 ms for the adfecgdb and 6.21 ms for the CinC2013. The method has the potential to be applied to other fetal cardiogenic signals, including cardiac doppler signals.

Before describing the linear combination idea, recall the well-know vectocardiogram (VCG) and its relationship with the ECG signals. It has been well known that the ECG signal, denoted as a continuous time series E : [0, T ] → R, where T > 0 is the observation time, is the projection of the representative dipole current of the electrophysiological cardiac activity on a predesigned direction [7]. Denote the dipole current as a three dimensional continuous time series d : [0, T ] → R 3 . If we could record d(t), it is called the VCG signal. Physiologically, for a normal subject, d(t) is oscillatory with the period τ > 0, which is about 1 second, in the sense that d(t) ∼ d(t + τ ) for all t ∈ [0, T − τ ]. Suppose t l , l = 1, . . . , m, where m is the number of cardiac cycles over the period [0, T ], is the timestamp corresponding to the maximal amplitude point of the l-th cardiac cycle. We call the vector the cardiac axis. For a given ECG signal, there is an associated projection direction v ∈ R 3 so that E is 13 the projection of d(t) on v; that is, E(t) = v T d(t). It has been well known that depending on v, we could 14 acquire different aspects of the cardiac information. We mention that in general, v changes according to 15 time due to the cardiac axis deviation caused by the respiratory activity and other physical movements. To 16 simplify the discussion, we do not take these facts into account.
Denote d m to be the mother's VCG and d f to be the fetus' VCG. Denote c m to be the mother's cardiac axis and c f to be the fetus' cardiac axis. Fix two abdominal lead placements and record two aECG signals, denoted as x 1 and x 2 . Denote v m,i ∈ R 3 and v f,i ∈ R 3 to be the projection directions of the mother's VCG and fetus' VCG corresponding to x i , where i = 1, 2. Obviously, we have where i = 1, 2, and it is possible that the fetal cardiac activity is relatively weak in both x 1 and x 2 . To resolve this problem, we consider the following linear combination scheme. Take a linear combination of x 1 and x 2 by and v m,1 = v m,2 , and hence the set antipodal points of the unit circle, The lag map is a well-known method widely applied to study a given time series, and its theoretical 31 foundation has been well established in [3,8,9]. In brief, it allows us to reconstruct the structure underlying conditions, Ψ f ,L could recover the manifold up to a diffeomorphism. Since the cardiac activity is periodic, 38 the corresponding "underlying manifold" is a one-dimensional circle representing the cardiac dynamics 39 that is diffeomorphic to the unit circle S 1 , and the lag map of the cardiac activity time series leads to a 40 point cloud supported on another one-dimensional simple closed curve.

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The above-mentioned important property of the lag map allows up to examine the quality of the 42 reconstructed fECG. If f ∈ R N is the true fECG signal, or a good estimation of the fECG signal, 43 we obtain an one-dimensional simple closed curve by the point cloud X f ,L : On the other hand, if the tempted fECG estimator f ∈ R N fails to be a good estimator of the fECG signal, where d is the degree function defined on the vertex set as shown in [6, 10] that if X is sampled from a low dimensional Riemannian manifold, when α = 1 and of the Laplace-Beltrami operator of the Riemannian manifold. In general, this allows us to reconstruct the manifold by applying the diffusion geometry and the spectral embedding theory, which is commonly 80 known as the DM algorithm [6]. The robustness of the GL and DM has been studied in [12,11].

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In our problem, due to the periodic oscillation intrinsic to the fECG we have interest, the α-normalized 82 graph Laplacian associated with X f ,L gives us the Laplace-Beltrami operator over a simple closed curve.

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It follows that asymptotically, the first two non-trivial eigenvectors are the sine and cosine functions. We 84 could thus take this fact into account and design the signal quality index for the channel selection purpose.