This paper proposes a combined coarsegrained multifractal method to discriminate between distressed and normal foetuses. The coarsegraining operation was performed by means of a coarsegrained procedure and the multifractal operation was based on a structure function. The proposed method was evaluated by one hundred recordings including eighty normal foetuses and twenty distressed foetuses. We found that it was possible to discriminate between distressed and normal foetuses using the Hurst exponent, singularity, and Holder spectra.
Foetal distress is often the result of reduction in respiratory exchange between the mother and the foetus. In most cases, foetal distress is strongly related to intrauterine growth retardation [
The value of analysis of heart rate variability (HRV) to provide a means of diagnosis and prognosis of heart disease is now well established. HRV time series has now become the elementary basis from which most analyses and processes are operated.
Due to the nonstationary and nonlinear nature of HRV time series, many recent studies have tried to take full advantage of the nonlinear nature of heart rate variability by analysing the complexity of time series. This complexity analysis of the foetal heart rate (FHR) that has its roots in adult HRV research can be conceived of in many ways. However, it was probably the scale invariance properties observed through power law spectral density [
The starting point of the present study was based on two approaches, the first being that of Wang et al. [
The second approach was based on the studies by [
In response to these two kinds of research, we investigated a coarsegrained multifractal analysis of the foetal heart rate in order to discriminate healthy from distressed foetuses.
Although the present study has certain similarities to those proposed by Wang et al. [
Our study aimed to improve the differentiation between normal and distressed foetuses by investigating the time scale dependency of the multifractal features of the FHR in depth. To do so we investigated the multifractal analysis originating from a structure function from a coarsegraining point of view.
To demonstrate the value of our approach, we tested the proposed method on a dataset derived from normal and distressed foetuses.
Our system comprised a personal computer and a Doppler ultrasound unit. The latter device contained three groups of four transducers and a Doppler acquisition board. The transducers exploring the foetal heart were nonfocused and monoelement. The transducers placed on the mother’s abdomen were circular in shape, with a diameter of 13.5 mm and an acoustic power of 1 mW/cm^{2}. Each transducer transmitted a sinusoidal pulse at 2.25 MHz with a pulse repetition frequency of 1 kHz. The wave was propagated through the mother’s abdomen towards the foetal heart.
The backscattered signal was converted into an electrical signal and amplified to compensate for the attenuation of 1 dB/cm/MHz. The signal was then demodulated in phase (I) and quadrature (Q).
The Doppler signals were acquired at CHRU “Bretonneau” Tours, France. The consent of each patient was obtained and the study was approved by the Ethics Committee of the Clinical Investigation Centre for Innovative Technology of Tours (CICIT 806 CHRU of Tours). All patients were over eighteen years of age and pregnancies were single. One hundred examinations (eighty normal foetuses and twenty distressed foetuses) were recorded in this study. Gestational ages of foetuses ranged from 25 to 39 weeks were monitored for 30 minutes. FHR was evaluated as proposed by [
As previously reported, the foetal heart rate was estimated in real time from ultrasound Doppler signals [
Each time series
Figure
Time and spectral representations of a Brownian motion. (a) Original time series superimposed on the coarsegrained time series (
Scheme of different processes used to calculate coarsegrained multifractal descriptors.
Figures
As previously shown by [
For
For
Figure
Effects of coarsegraining on multifractal descriptors for different fBm of Hurst exponents
Due to the nonstationary nature of the coarsegrained time series analysed, a shortterm procedure was performed. This procedure consisted of evaluating multifractal descriptors from subsignals
Among all the existing methods supplying multifractal descriptors, we used the structure function of order
The structure function is by far the simplest method to implement compared to DFA, box counting, and wavelet methods.
Using
The real signals under consideration were not theoretical signals. This means that mathematical demonstrations operating exclusively on theoretical signals are not systematically applicable in practice.
Several multifractal analyses showed that it was more possible to discriminate between normal and distressed subjects for
The structure function that we used in this study is defined [
If
The structure function
Multifractal parameters for a normal foetus (in blue) and a distressed foetus (in green). (a) Structure function
Other multifractal descriptors such as the singularity spectrum
Multifractal parameters for a normal foetus (in blue) and a distressed foetus (in green). (a) Singularity spectrum
FHR of a normal foetus and a distressed foetus.
Several measurements were performed in order to quantify the different trends observed in the multifractal indicators
The relative error
where
The relative error
where
The relative error
where
The relative error
where
From our own dataset composed of one hundred recordings, each time series of 7200 points was coarsegrained for 6 different scales. From each coarsegrained signal, subsignals composed of 720 points and overlapping by 97% were analysed with multifractal tools.
Figure
Boxplot of Hurst exponents versus scale. Normal foetus (in blue) and distressed foetus (in red).
Figure
Boxplot of
Figure
Boxplot of
Figure
Boxplot of
To conclude, Table
Relative errors of different multifractal parameters between the two groups of foetuses for different scales.
Scale  1  2  3  4  5  6 

RE_{1}  0.37 

0.33  0.29  0.26  0.24 
RE_{2}  0.32 


0.37  0.36  0.35 
RE_{3}  0.11 

0.11  0.10  0.09  0.08 
RE_{4}  0.41 

0.41  0.38  0.36  0.33 
Finally, although the present study was quite similar to that presented in [
Furthermore, although a large number of research studies have mainly been based on the use of partition functions (DFA, boxcounting and wavelet approaches) on the pretext that structure functions do not operate for negative orders, we have shown here (i) that the use of such structure functions is fully justified due to the simplicity of implementation and (ii) that structure functions completely fulfil their role in distinguishing between healthy and distressed foetuses.
Note that, as our proposed methodology was that of investigating offline, we plan to evaluate multifractal descriptors one line in the near future.
This study was supported by the “Agence Nationale de la Recherche” (Project ANR07TECSAN023, Surfoetus). Furthermore, the authors would like to thank the Clinical Investigation Centre for Innovative Technology of Tours (CICIT 806 CHRU of Tours) and Professor F. Perrotin’s team in the Obstetric Department for their support in recording the signals.