A New Approach for Extracting and Characterizing Fetal Electrocardiogram

2.1 Theoretical background

2.1.1 Wavelet

A wavelet Figure 2 is a function φ(t) which satisfies certain mathematical criteria [11-13].

1.2.png

Figure 2. Morlet wavelet

-Wavelet must have finite energy: 

$€=\int_{-\infty}^{+\infty}|\psi(\mathrm{t})|^{2}$      (5)

- Admissibility constant:

$\mathcal{Q}=\int_{0}^{+\infty} \frac{|\widehat{\psi(\mathrm{f})}|^{2}}{\mathrm{f}} \mathrm{df}$     (6)

-The Fourier transform of ψ (t)

$\widehat{\Psi(\mathrm{f})}=\int_{-\infty}^{+\infty} \psi(t) e^{-i 2 \pi f t} d t$     (7)

2.1.2 Continuous wavelet transform CWT

The continuous wavelet transform (CWT) is a very effective time-scale method for characterizing complex signals. A common application of the (CWT) is in the detection of edges and singularities of signals using multi-resolution analysis. The (CWT) of a signal x(t) with an analyzing wavelet ψ(t) can be defined as:   

$T(a, b)=\frac{1}{\sqrt{a}} \int_{-\infty}^{+\infty} x(t) \psi^{*}\left(\frac{t-b}{a}\right) d t$     (8)

where, $\mathrm{a}, \mathrm{b} \in \mathbb{R}$ and $\mathrm{a} \neq 0, \psi^{*}$ is a complex conjugate parameter, respectively [14, 15]. The normalized wavelet is often written more compactly as     

$\psi_{\mathrm{a}, \mathrm{b}}=\frac{1}{\sqrt{\mathrm{a}}} \psi\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right)$     (9)

At a specific scale a and location b the relative contribution of the signal energy is given by the two-dimensional wavelet energy density function:

$E(a, b)=|T(a, b)|^{2}$     (10)

A plot of E(a,b) is known as a scalogram or Times- Scales Image (TSI).

2.2 Algorithm

The approach presented in this article is detailed in the algorithm below dissolved in Figure 3.

1.3.png

Figure 3. Algorithm

2.3 Description of algorithm

We consider the following linear instantaneous mixture (Eq. (1)) of two real valued sources. So, the CWT of x₁ and x₂ can be written as:

$\mathbf{T}_{1}(\mathbf{a}, \mathbf{b})=\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{x}_{1}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}$

$=\mathrm{a}_{11}\left[\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{1}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}\right]+$

$\mathrm{a}_{12}\left[\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{2}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}\right]$

$\mathbf{T}_{2}(\mathbf{a}, \mathbf{b})=\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{x}_{2}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}=$

$\mathrm{a}_{21}\left[\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{1}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}\right]+$

$\mathrm{a}_{22}\left[\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{2}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}\right]$

Thus:

$\left\{\begin{array}{l}\mathrm{T}_{1}(\mathrm{a}, \mathrm{b})=\mathrm{a}_{11} \mathrm{T}_{11}+\mathrm{a}_{12} \mathrm{T}_{22} \\ \mathrm{T}_{2}(\mathrm{a}, \mathrm{b})=\mathrm{a}_{21} \mathrm{T}_{11}+\mathrm{a}_{22} \mathrm{T}_{22}\end{array}\right.$     (11)

where:

$\mathrm{T}_{11}=\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{1}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}$ is the $\mathrm{CWT}$ of $\mathrm{s}_{1}$

And 

$\mathrm{T}_{22}=\frac{1}{\sqrt{\mathrm{a}}} \int_{-\infty}^{+\infty} \mathrm{s}_{2}(\mathrm{t}) \psi^{*}\left(\frac{\mathrm{t}-\mathrm{b}}{\mathrm{a}}\right) \mathrm{dt}$ is the $\mathrm{CWT}$ of $\mathrm{s}_{2}$

At a specific scale a and location b the relative contribution of the signal energy is given by the two-dimensional wavelet energy density function:

$\left\{\begin{array}{l}\mathbf{E}_{1}=\left|\mathbf{T}_{1}(\mathbf{a}, \mathbf{b})\right|^{2} \approx \mathbf{a}_{11}^{2}\left|\mathbf{T}_{11}(\mathbf{a}, \mathbf{b})\right|^{2}+\mathbf{a}_{12}^{2}\left|\mathbf{T}_{22}(\mathbf{a}, \mathbf{b})\right|^{2} \\ \mathbf{E}_{2}=\left|\mathbf{T}_{2}(\mathbf{a}, \mathbf{b})\right|^{2} \approx \mathbf{a}_{21}^{2}\left|\mathbf{T}_{11}(\mathbf{a}, \mathbf{b})\right|^{2}+\mathbf{a}_{22}^{2}\left|\mathbf{T}_{22}(\mathbf{a}, \mathbf{b})\right|^{2}\end{array}\right.$     (12)

So the ratio Γ can be written as:

$\Gamma=\frac{E_{1}}{E_{2}}=\frac{a_{11}^{2}\left|T_{11}(a, b)\right|^{2}+a_{12}^{2}\left|T_{22}(a, b)\right|^{2}}{a_{21}^{2}\left|T_{11}(a, b)\right|^{2}+a_{22}^{2}\left|T_{22}(a, b)\right|^{2}}$     (13)

Therefore, our goal is to determine the coefficients a₁₁, a₁₂, a₂₁ and a₂₂ in order to solving Eq. (1). Also, the heart rate of the fetus is very high compared to that of her mother. Then, there are at least two pairs (ai, bi) and (aj, bj) in the time-scale space such as the continuous wavelet transform of s₁ or s₂ is equal to zero i.e.

$\mathrm{T}_{11}=0$ and $\mathrm{T}_{12} \neq 0$ or $\mathrm{T}_{11} \neq 0$ and $\mathrm{T}_{12}=0$

This implies that:

if $\mathrm{T}_{11}=0$ and $\mathrm{T}_{22} \neq 0$

So:

$\Gamma=\frac{E_{1}}{E_{2}}=\frac{a_{11}^{2}\left|T_{11}(a, b)\right|^{2}}{a_{21}^{2}\left|T_{11}(a, b)\right|^{2}}=\frac{a_{11}^{2}}{a_{21}^{2}}$

$\frac{a_{11}}{a_{21}}=\sqrt{\Gamma}=\Gamma_{1}$      (14)

if T₁₁≠0 and T₂₂=0

So:

$\Gamma=\frac{E_{1}}{E_{2}}=\frac{a_{12}^{2}\left|T_{22}(a, b)\right|^{2}}{a_{22}^{2}\left|T_{22}(a, b)\right|^{2}}=\frac{a_{12}^{2}}{a_{22}^{2}}$

$\frac{a_{12}}{a_{22}}=\sqrt{\Gamma}=\Gamma_{2}$     (15)

So by replacing Eq. (14) and Eq. (15) in Eq. (1) we get:

$\left\{\begin{array}{l}x_{1}=a_{11} s_{1}+a_{12} s_{2} \\ x_{2}=a_{21} s_{1}+a_{22} s_{2}\end{array}\right.$

$\left\{\begin{array}{l}\mathrm{x}_{1}=\Gamma_{1} \cdot \mathrm{a}_{21} \mathrm{s}_{1}+\Gamma_{2} \cdot \mathrm{a}_{22} \mathrm{s}_{2} &(a)\\ \mathrm{x}_{2}=\mathrm{a}_{21} \mathrm{s}_{1}+\mathrm{a}_{22} \mathrm{s}_{2}&(b)\end{array}\right.$

$s_{1}=\frac{x_{1}-\Gamma_{2} \cdot x_{2}}{a_{21}\left(\Gamma_{1}-\Gamma_{2}\right)}$     (16)

$-\Gamma_{2} \cdot(b)$ implies: s₁ presents the Fetal ECG: FECG

It is interesting to note that the calculation of Γ₁ and Γ₂ allows us to determine the desired signal. What is particularly striking in this work is the extreme ease for determining the ratios Γ₁ and Γ₂. The continuous wavelet transforms (CWT) as a matter of fact is a mathematical microscope which allows the detection of the frequency content of a signal and especially the detection of all the irregularities contained in an electrocardiogram. As we know, the most important wave in an electrocardiogram is the QRS complex which provides all medical parameters of the electrical activity of the heart. The highlight point of this approach is based on the fact that the signal recorded by the electrode placed on the mother's abdomen contains both, the QRS complexes corresponding to the fetus and to its mother, although the amplitude of the fetal electrocardiogram  is much lower, the fetus' heart rate is higher than that of its mother, so the QRS complexes will be located clearly in the time-scale domain or  scalogram and therefore we can easily locate all points (a_i , b_i) where one of the sources occurs alone i.e. (T₁₁=0 and T₂₂≠0) and (T₁₁≠0 and T₂₂=0) as shown in Figure 4 and Figure 5. Therefore, it is enough to determine the ratio Eq. (13) at each point (a_i, b_i) and by determining the period of repetition of these points we will make the average of all the founded values Eq. (14), Eq. (15).

1.4.png

Figure 4. Process of locating points (a_i, b_i)

1.5.png

Figure 5. Data mixtures x₁ and x₂

The question which then arises is how to extract the fetal ECG from a mixture a fetal and maternal ECG. The proposed algorithm this paper is based on the continuous wavelet transform as an advanced and very rich mathematical tool which makes it possible to locate in time and in precise chronological order the frequencies contained in a stationary or no stationary signal and it is this property which allowed us to separate the foetal and maternal contributions in the time-scale domain. As a result, we were able to extract the fetal electrocardiogram. The simulation of the algorithm was done on Python and Matlab and it is applied to a real signal from the DAISY database, the results found are in perfect agreement with the results of the database. I really want to emphasize the simplicity of the algorithm and its notable reliability which allowed us to make a deep characterization leading to a successful early detection. What is more important here is the possibility of implementation of the algorithm in real time through the use of embedded processor and this allows regular monitoring of pregnant women to reassure themselves about the cardiac state of their foetuses. Currently, pediatric cardiologist uses ultrasound, which unfortunately remains a limited and non-rich tool and which can’t completely show the foetal morphology malformations a statement that what is expected, as stated in the introduction Prenatal screening for structural congenital heart disease allows the preparation of postnatal intervention in most instances; in a minority of cases (mainly critical left-heart lesions), fetal cardiac intervention can be considered. Finally, we can’t deny that this paper presents an important tool for detecting cardiac malformations for fetuses. Future works will be the concretization of this approach by an important industrial product.