Open access peer-reviewed chapter

# Mathematical Morphology and the Heart Signals

Written By

Taouli Sidi Ahmed

Reviewed: 02 March 2022 Published: 20 April 2022

DOI: 10.5772/intechopen.104113

From the Edited Volume

## Biosignal Processing

Edited by Vahid Asadpour and Selcan Karakuş

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## Abstract

Nowadays, signal processing is integrated into most systems for analyzing and interpreting ECG and PCG signals. Its objectives are multiple and mainly include compensating for the addition of artifacts to the signals of interest, and extracting information that is not visible by direct visual analysis. Considering that useful clinical information is found in the characteristic waves of the ECG, in particular, the P wave, the QRS complex, the T wave and the heart sounds of the PCG signal. These signals provide important indicators for the diagnosis of heart disease because they reflect physiological processes. These are non-stationary signals that are very sensitive to noise, hence the need to have optimal conditions to record them. It is necessary to use appropriate signal processing tools for noise suppression and wave detection of the ECG signal. Our method uses Morphological filtering, multi-scale morphological and the other by top-hat transform, which are based on mathematical morphology. The latter is based on mathematical operators called opening and closing morphology operators. These methods are also tested, with the aim of removing the noise and detection of the waves of the ECG signal and of the pathological sounds of the PCG signal.

### Keywords

• ECG signal
• PCG signal
• morphological filtering
• multi-scale morphological
• top-hat transform

## 1. Introduction

In the Western world, the leading cause of death is cardiovascular diseases. Even if the knowledge acquired in cardiology is great, the heart has not yet revealed all its secrets. However, doctors have many ways to study and verify its proper functioning. In particular, they use cardiac signals, the electrocardiogram (ECG) and the phonocardiogram (PCG) which are an important tools in the diagnosis of cardiac pathologies.

The ECG is the signal reflecting the recording of the bioelectrical activities of the cardiac system. It is rich in information on the functional aspects of the heart and the cardiovascular system.

The electrocardiogram includes three important waves called the P wave, complex QRS and T wave which translate respectively the atrial activity, the ventricular activity and ventricular repolarization. From these waves are determined intervals known by the PR interval which defines the atrioventricular conduction time, the ST segment which corresponds to the ventricular repolarization phase, a phase during which the ventricular cells are all depolarized, and the RR interval which indicates the cardiac period, i.e. the time between two successive beats. Inverting it, we get then the heart rate commonly expressed in beats per minute.

The QT interval reflects all ventricular activity, i.e. phases of depolarization and repolarization. The time intervals between these different ECG waves provide important indicators for the diagnosis of heart disease because they reflect physiological processes of the heart and autonomic nervous system [1, 2, 3].

The analysis of these different intervals often involves the study of their variability.

At the level of the PCG signal, this signal produces two noises (S1 and S2) during the opening and closing of the valves under normal conditions. Two other noises (S3 and S4) with significantly lower amplitudes than the first two sometimes appear at the level of the cardiac cycle due to the effect of pathology or age [4].

The S1 sound occurs just after the onset of systole and is preferentially due to the closure of the atrioventricular valves. The S2 sound is produced at the beginning of diastole and is due to the closure of the aortic and pulmonary valves. However, doctors use heart auscultation to hear two normal sounds S1 and S2 using a medical instrument called a stethoscope. Also, cardiac synchronization can simultaneously gives the two physiological signals ECG and PCG, such that the S1 noise appears at the end of the R peak and the S2 noise at the end of the ECG segment, as for the S3 and S4 noises, they originate respectively at the end of the P wave and in the middle of the diastolic phase of the electrocardiogram (Figure 1).

Generally, the recording conditions of the ECG and PCG make that the signals are necessarily noisy by processes other than cardiac. These artifacts can be of physiological origin (skin, muscle, breathing…) or environmental (mains current, electromagnetic artifacts, placement of the electrode …).

The practitioner who analyses the ECG can then be discomfort by the presence of noise: in the case where, for example, he looks for the existence of a normal sinus rhythm and he is looking for the presence of the P wave preceding the R wave, the P wave, which is of low amplitude, can be drowned in noise. In the same way, a strong variation of the base line can prevent discerning an anomaly of the over- or under-shift type of the S-T segment, for example. To be able effectively segment heartbeats without altering clinical information, a certain number of pre-treatments are necessary. The purpose of this step is attenuate, or at by eliminates noises present in the raw ECG signal such as baseline variations or sector the interference at 50 Hz. Also, the doctor finds it difficult to listen to the S1 and S2 heart sounds due to the presence of heart murmurs and this prevents the doctor from diagnosing the patient.

These heart murmurs are an additional noise; it was produced by a turbulent circulation of blood toward the heart. In fact the objective is to filter the non-stationary ECG and PCG signals and the extraction useful clinical information is found in the time intervals defined by the ECG waves characteristic, including the P wave, QRS complex, T wave, PR interval, ST segment, and interval QT, the defined time intervals between two characteristics ECG waves provide important indicators for the diagnosis of heart disease because they reflect physiological processes.

However, it is clear that to achieve this study, it is essential to perform a pre-processing of the ECG and PCG signals in order to then detect the different waves of the ECG signal.

In this article the morphological transform is used to remove noise and to detect the characteristics of the ECG and PCG signal.

This transformation uses mathematical morphology to realize, in particular the morphological filtering and the top hat transform. Mathematical morphology, based on set operations, provides an approach to developing nonlinear signal processing methods in which the shape of a signal’s information is incorporated [5]. In these operations, the result of one set of data transformed by another set depends on the shapes of the two sets involved. A structuring element must be designed according to the shape characteristics of the signal that must be extracted. There are two basic morphological operators: erosion and dilation. Opening and closing are derived operators defined in terms of erosion and dilation [6].

Dilation reduces the peaks in a signal and to widen the valleys, erosion fills in the valleys and thickens the peaks in the signal; opening removes the peaks but preserves the valleys, and closing fills in the valleys, removes the wells (or valleys). The “closing” and “opening” operators behave like filters; we will speak of “Morphological Filter” [7].

An important number of researches work using different tools and methods of noise filtering have been presented in the literature. The methods often based on classical linear high-pass filtering, low-pass or band-pass [8, 9, 10], linear adaptive filtering [11], filtering based on neural networks [12, 13, 14, 15], have been proposed to eliminate noise affecting the line of basis of the ECG signal. The major disadvantage of these methods is the distortion of the signal due to the overlapping of the spectra of the ECG, PCG and their noises. On the other hand, a large number of methods have been proposed for the detection of ECG signal waves [12, 13, 14, 15, 16, 17, 18, 19, 20]. The majority of these methods are based on adaptive filtering or thresholding, which shows the limitation of the application. The emergence of treatment method in the non-stationary case has helped the researchers to develop new tools better suited to filtering. Techniques based on the wavelet theory have already proved their worth for the filtering of noise from the ECG signal. Donoho and Johnston proposed a denoising method by thresholding of wavelet [21, 22]. The denoising method by wavelet thresholding was treated the wavelet coefficients by a threshold which must be chosen in advance. Approaches for estimate the value of this threshold can be found in [23, 24].

In this chapter, the pre-processing is realized in two stages, a stage of the correction of the line base by Morphological filtering and the top-hat transform for remove noises that are based on mathematical operators called derivation operators opening and closing morphology. Followed by the second step which is consisted of detecting the waves of the ECG signal and the cardiac murmurs of the PCG signal by morphology operators and a multi-scale structuring element.

## 2. Theory of mathematical morphology

### 2.1 The basic principle

The morphological transform is very widespread in the domain of signal processing and image processing because of its robustness and its simple and fast calculation [25, 26].

Mathematical morphology, based on set operations, provides an approach to the development of nonlinear methods of signal processing, in which the form of a signal’s information is incorporated [27]. In these operations, the result of a data set transformed by another set depends on the shapes of both sets involved. A structuring element must be designed according to the characteristics of shape of the signal to be extracted.

There are two basic morphological operators: erosion and dilation .

Opening and closing are derived operators defined in terms of erosion and dilation [28].

These operators are described in detail below with the corresponding mathematical expressions. Throughout this document denotes the discrete ECG signal of point size and the symmetrical structuring element of M points.

a. Erosion.

To obtain the eroded function of f(x), we attribute to f(x) its minimum value in the domain of the structuring element B and this, at each new displacement of B (Figure 2). The following formula illustrates the erosion of the function f(x) (original signal) by a structuring element B plane:

εBf=fBn=minm=0,M1fnBmE1

b. Dilation.

To obtain the dilated function of f(x), we attribute to f(x) its maximum value in the field of the structuring element B and this, with each new displacement of B (Figure 2). The following formula illustrates the dilation of the function f(x) (original signal) by a structuring element B plane:

δBf=fBn=maxm=0,M1fn+BmE2

Erosion shrinks peaks and the crest lines. The peaks narrower than the element structuring disappear. At the same time, it widens the valleys and the minima [29].

The dilation produces the opposite effects (fills in the valleys and thickens the peaks).

c. Opening.

As in mathematical morphology, the opening consists of the erosion followed by dilation. The opening of f(x) by the structuring element B plane has the following consequences on the initial function (Figure 3):

The opening removes the peaks but preserves the valleys [30], according to the equation:

γBf=foB=fBBE3

d. Closing.

As in mathematical morphology, closing consists of dilation followed by erosion (Figure 3). The closing of f(x) by the structuring element B plane, for its part, has the following consequences on the starting function [31]:

The closing fills the valleys [32] as follows:

φBf=fB=fBBE4

The “closing” and “opening” operators behave like filters; these are in the same time morphological Filters [33].

The opening and closing by adjunction create a simpler function than the initial function, by softening the latter in a nonlinear manner.

Opening (closing) eliminates positive (negative) peaks respectively that are narrower than the structuring element.

The opening (the closing) is located below (above) the initial function.

e. Structuring element.

After the selection of the morphology operator, the structuring element (SE) is the next component of the morphological analysis to be defined. Generally, only when the shape of the signal matches those of the structuring element that the signal can be preserved. Therefore, the shape, length (domain) and size (amplitude) of the structuring element should be chosen according to the signal to be analyzed. The shapes of the structuring element can vary regularly or irregularly, such as a triangle, line (flat), or a semicircle.

### 2.2 Filter construction

#### 2.2.1 Morphological filter

In signal processing, the term “filter” is not very precise, it depends on the context in which it is used. It sometimes implies convolution, sometimes also any operation that produces a new function. On the contrary, mathematical morphology defines it very precisely [34, 35, 36]:

Any increasing and idempotent transformation on a trellis defines a morphological filter.

• Increase:

This hypothesis is the most fundamental. It ensures that the basic structure of the trellis, i.e. the order relation, is preserved during a morphological filtering.

This property causes the filter to generally lose information.

• Idempotence:

By definition, an idempotent transformation transforms the signal into an invariant.

This property often appears, but implicitly, in the descriptions. We say of an optical filter that it is red, or of an amp that its bandwidth is 50 kHz. Here, we will pose it as an axiom.

Idempotence is reached either after a single pass or as a limit by iteration. More generally, a sequence of operations, taken as a whole, can be idempotent.

Finally, note that when linear filters are idempotent, then they do not admit an inverse: they lose information, which brings them near to morphological filters.

#### 2.2.2 Average filter

In practice, operators morphologic are based on different application scenarios in signal processing [37, 38, 39, 40, 41, 42]. Sometimes it is difficult to obtain prior knowledge of the positive (positive peak) or negative (negative peak) pulse characteristics of the signal, especially when both positive and negative pulses are used. We can make by the combination of the four operators an average filter presented by the following formula:

AVGf=fg+fog/2E5

Figure 4 illustrates the results obtained not used to flatten the positive and negative peaks.

#### 2.2.3 Top hat transform

The concept of Top-Hat, due to F. Meyer, is a residue intended to eliminate the slow variations of the signal, or to amplify the contrasts. It therefore applies mainly to functions (Figure 5).

We call Top-Hat the residual between the identity and an invariant opening under vertical translation [43]:

ρf=ffoBE6

We define in the same way a dual top-hat, residue between a closure and the identity:

ρf=fBfE7

#### 2.2.4 Multi-scale morphology

f and s, represent respectively, a discrete signal and the structuring element (SE) for a morphological analysis. The morphology operator R, based on multi-scale analysis [44, 45], can be defined as a set Tλλ>0λN, where Rλf=λRf/λ.

Multi-scale erosion and dilation are defined by:

fsλ=λf/λs=fλsE8
fsλ=λf/λg=fλsE9

and λg=ss.sλ1times.

The original purpose of multi-scale morphology analysis is based on the morphology composition of the structuring element g is to improve the speed of morphology analysis by a large scale of structuring element and to expand the domains of application in signal processing.

The original purpose of multi-scale morphology analysis is based on the morphology composition of the structuring element g is to improve the speed of morphology analysis by a large scale of structuring element and to expand the domains of application in signal processing.

## 3. Application of morphological filtering to physiological signals

### 3.1 ECG signal filtering algorithm

The opening and closing operators proposed in this work for ECG signal processing, is shown in Figure 6 below.

It consists of different phases. A first phase which consists in detecting variations in the baseline from the original noisy ECG signal. A second phase which consists in correcting these variations (correction of the baseline), and a third phase which consists in suppressing the remaining noises to finally generate the filtered ECG signal.

These different phases are realized through the development of a sequence of opening and closing operators.

This sequence is based on exploiting the different characteristics of baseline drift and noise contamination in ECG signals, different structuring elements and different morphological operators. For baseline correction, an opening operator followed by a closing operator is defined; as well as noise cancelation. They are described in detail in the following subsections.

#### 3.1.1 Baseline correction algorithm

To correct for baseline variations, the variations of that line (fb signal) detected from the original signal (fo) are subtracted from that signal to generate the corresponding signal fcb (i.e., an isoelectric line).

The signal fb is obtained through a sequence of opening and closing operators with structuring elements Bo and Bf suitably chosen as the one described below.

fb=foBoBfE10
=fbΘBoBoBfΘBf
fcb=ffbE11

#### 3.1.2 Noise suppression algorithm

The approach adopted for noise suppression appeal of Top-hat transformations, which are translated by the equations below.

f=fcbBfcbBE12
=fcbBfcb+fcbfcbB
=fcbB1ΘB2fcb+fcbfcbΘB1B2

fcbBfcb and fcbfcbB are of the type of morphological transformation by Top Hat. This transformation is a high pass filter.

fcbBfcb and is called the black top hat transformation, which is used to extract negative pulses (or negative peaks); is called the white top hat transformation, which is used to extract positive pulses (or positive peaks). Thus the filter can be used to extract positive and negative pulses simultaneously [46].

Figure 7 represents the block diagram describing the structure of the top-hat transformation for noise suppression.

It consists of three steps: The first concerns the acquisition of ECG signals (fcb: ECG signal after baseline correction).

This step is followed by another step that allows noise removal. This step uses the morphology operators defined in Eq. (12).

B1 and B2 are structuring elements for opening and closing. These operators are employed simultaneously on the signal. The next step is the subtraction between the two closing and opening operators to generate a filtered ECG signal.

The proposed algorithm is implemented under MATLABR14 environment. It is tested on a set of ECG signals (noisy) from the MIT-BIH database [47].

### 3.2 PCG signal filtering algorithm

Figure 7 below illustrates the stages of noise suppression and heart murmurs.

The first step is to normalize the PCG input signal. A second step is to remove noises, and a third step is to remove heart murmurs.

#### 3.2.1 PCG signal normalization

The original signal has been normalized before performing any operation. The PCG signal is re-sampled at 8000 Hz with 16 bits of precision and converted to WAVE format and normalized by:

xNort=xtmaxxtE13

The aim of normalization is to reduce the amplitudes of the PCG signal.

#### 3.2.2 Noise suppression algorithm

Noise suppression uses the morphological filter, which is presented by the following equations:

fp=fBBE14
fc=ffpE15

fp represent the detection of the positive and negative peaks by the closing of the element.

B1 followed by the opening of B2 to finally generate the filtered signal fc. So the signal fc is the subtraction of the original signal and the signal fp.

#### 3.2.3 Heart murmur algorithm

The filter applied for the suppression of murmurs is the average filter; it is described in the equation below.

fp=fBBE16
fc=ffpE17

The detection of murmurs is based on two morphological operators, namely closing and opening, to suppress peaks. The average between the closing and opening operator gives the result fp. After the subtraction between the f signal and fp to generate a filtered signal fc.

### 3.3 ECG signal wave detection algorithm

The detection of the RR interval can be accomplished only after a good detection of the R peak. This detection goes through three stages. A step which allows locating the positive peaks R by the opening operation and another step which allows locating the negative peaks R by the application of the closing operation. The final step to arrive at these peaks and valleys is the subtraction of the filtered ECG signal f with the two previous operations.

The following formula illustrates the extraction of R positive peaks:

PeaksRR=ffoBE18

And the following formula illustrates the extraction of valleys or negative peaks R:

ValleysRR=ffBE19

Formula (18) and (19) are equivalent to:

PeaksRR+ValleysRR=ffoBBE20

Structuring element B is used to detect peaks and valleys. In this case it is considered as a geometric shape “horizontal segment of different lengths”. The following formula shows the:

bt=1sit<k0sinonE21

where K is a constant.

One of the problems of the morphological approach is to determine the optimal value of K. If it is too small, the transformation will be sensitive to high frequency noise, and the peaks are incorrectly determined. However, K which must be less than the peak width R. Research has shown excellent results by using values between [55–60] ms [48].

After detecting the RR rhythm, the next step is to detect the T wave. It uses the erosion operator to build maximum thresholding. Knowing that the amplitude of the T wave is enormously greater than the Q wave and the S wave in the normal or abnormal case of the ECG signal.

The following formula illustrates the detection of the T wave:

fBm=maxn=0,.,M1fn+BmnforN>Mandm=M1,..,N1E22

Where the sets f and B are underlined by fn=01.N1 and B=01.M1.

The next phase is the detection of the Q and S waves of the ECG signal. It uses a morphological operator called closing to detect the negative peaks. On the other hand, the morphology of Q and S waves in an ECG signal are of negative amplitudes. For this, we have chosen the most adapted closing operator to detect the negative peaks. Finally, the detection of the P wave is based on the opening operator which is combined with a structuring element of multi-scale, according to the equation:

fp=fB1B2E23

The signal is operated by two successive opening operators with a structuring element of an equal scale 2. This means that the number of opening operations and the number of the structuring element is two scales.

## 4. Results and interpretations

### 4.1 ECG signal filtering

The proposed morphological approach for baseline correction and noise suppression in the ECG signal was tested by the use of the MIT-BIH arrhythmia database.

Figure 8 illustrate respectively the noisy ECG signal, the ECG signal after baseline correction and the resulting filtered ECG signal.

After acquiring the ECG signal, the next step is baseline correction. It consists of the application of morphological operations: “opening and closing”.

The signal is first opened by a structuring element Bo, which means the application of two morphology operations “erosion + dilation”, to remove the peaks and preserve the valleys. This “opening” operation generates a signal consisting of valleys which are suppressed, using the second operation which is the closing “dilation + erosion”, so this operation uses another structuring element Bf. We achieved to the signal that represents the estimate of the derivative of the baseline (or variations of the baseline).

The structuring element BoBf is used for baseline correction. In this case BoBf is chosen as a geometric shape corresponding to an horizontal segment of different lengths. Different lengths of the structuring element Bo and Bf are employed considering that the construction of the structuring element for the correction of the baseline depends on the duration of the characteristic wave and the sampling frequency (Fs Hertz) of the ECG signal. If the length of a characteristic wave is TLsecond, the number of samples of this wave is FsTL, which signify the structuring element Bo should have a length greater than FsTL.

Figure 9 shows the closing operation which uses a structuring element to remove the valleys left by the opening operation. The length of the structuring element Bf must be longer than the length of the Bo.

The ECG signal, the most characteristic waves are the P wave, the T wave and the QRS complex, which are generally less than 0.2 second.

Therefore, Lo, the length of Bo is 0.2Fs and Lf, the length of Bf is typically chosen to be longer than the structuring element Bo, at approximately 1.5Lo.

Since we employ Fs=360Hz as a sampling frequency, we get Lo=0.2Fs=72 and Lc=1.5Lo=108.

The final step to arrive at the baseline correction is the subtraction of the noisy ECG signal fo with fb (signal describing the baseline variations).

After baseline correction, the next step is noise removal. It consists of the application of morphology operators of the morphological transformation by top hat.

In fact, the input signal fcb is processed simultaneously by the operations “close” and “open”, followed by a subtraction, to end up with the filtered ECG signal f.

The morphological transformation by Hat-Top permits to extract the positive and negative impulses simultaneously by the application of the operations of opening and closing.

It should be noted that the shape of the structuring element in the noise suppression is different compared to the baseline correction. Indeed, it can take two different shapes and of equal lengths: a triangular shape B1 for maintain peaks and valleys on a straight shape (segment of zero amplitude) B2.

In our case, the size of the structuring element was fixed at 5 samples, with the value of the structuring element of B1=01510 and B2=00000. This value is set in an empirical way where the initial values of minimums and maximums are set to the optimal values in the noise removal step.

The algorithm is applied respectively to records 209 and 234 of the MIT-BIH Arrhythmia database. Figures 10 and 11 illustrate, that the filtered ECG signal does not present any variation of the base line, also the different waves of the signal are clearly highlighted, and without any deformation.

### 4.2 PCG signal filtering

The proposed morphological approach for noise suppression and detection of heart murmurs from the PCG signal was tested from the Pascal Classifying Heart Sound Challenge [49].

Figure 12, shows a noisy signal that has been filtered by the morphological filter. This filter uses two successive operators including closing and opening.

These operators are combined with a single structuring element B to suppress noise. For this, the selection of the shape and size of B is very important so as not to distort the information of the signal.

In this context, the length L of B is 0.15*Fs and the sampling frequency Fs = 8000 Hz, so we get L = 1200.

It depends on the duration of heart sounds S1 and S2 which are basic frequency (0.1–0.15) second, (70–0.14) ms, respectively.

After noise suppression, the next step is heart murmur suppression. It uses an average filter, which is combined with a structuring element B.

Knowing that heart murmurs are a frequency content above 100 Hz. Starting from this hypothesis, the length L of B = 0.15*f, f = 100 Hz then we obtain that L = 15.

Figure 13 below shows heart murmur suppression.

### 4.3 ECG signal wave detections

The proposed morphological approach for detecting the beginnings and endings of QRS complexes, the T wave and the P wave of the ECG signal were tested from the MIT-BIH arrhythmia database.

For each (filtered) ECG input signal, the following procedures were performed:

(a) pre-processing the ECG signal; (b) detecting the RR heart rate by the morphological filter; (c) detection of the T wave by maxima of the erosion operator; (d) detection of the Q and S waves of the ECG signal by the closure operator; (e) the detection of the P wave is detected by the opening operator with a multi-scale structuring element.

Figures 14 to 16 illustrate the results obtained on three different recording representing different morphologies where the power of the morphological transformation algorithm is clearly shown.

## 5. Conclusion

The characteristic waves of an electrocardiogram (ECG) signal and the heart sounds of the phonocardiogram (PCG) signal provide important indicators for the diagnosis of heart disease; they reflect physiological processes of the heart and autonomic nervous system. The analysis and interpretation of these physiological signals make it possible to highlight new phenomena, which is sometimes possible to explain at the physiological level, and which leads to a better understanding of the overall functioning of the heart.

The automatic analysis of ECG and PCG signals provides the cardiologist with the information needed to diagnose cardiac pathologies. The implementation of reliable algorithms for the processing of ECG and PCG signals, making it possible to detect the useful information carried by the ECG and PCG signals remains a major concern for technicians.

A set of algorithms using morphology transforms has been developed. These algorithms concerning:

• On the one hand the pre-processing of ECG and PCG signals where an algorithm has been developed and implemented: one for the elimination of baseline ripples and the other for noise suppression.

• and on the other hand, the analysis of ECG and PCG signals where an algorithm has been developed for the detection of QRS complexes, the T wave, the P wave of the ECG signal and the murmurs of the PCG signal.

• The algorithms were evaluated by applying them to ECG and PCG signals from the universal database ‘MIT-BIH’ and Dataset from “Pascal Classifying Heart Sound Challenge, respectively.

Morphological filtering which uses two morphological operators opening and closing for the correction of the baseline.

In noise suppression, the Top Hat transformation was used. It combines the subtraction of the closing and opening morphology operators.

For the detection of ECG signal parameters, typically the QRS complex, the T wave, and the P wave, two techniques have been presented which are called multi-scale morphology and morphological operators.

At the PCG signal level, we used a morphological filter for noise suppression and heart murmur detection.

These techniques, which are based on mathematical morphology, are very effective in estimating rapid changes in the morphology of ECG and PCG signals.

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Written By

Taouli Sidi Ahmed

Reviewed: 02 March 2022 Published: 20 April 2022