J Med Syst (2007) 31:529–536 DOI 10.1007/s10916-007-9094-8
ORIGINAL PAPER
Determining Fractal Dimension of Umbilical Artery Doppler Signals Using Hurst Exponent Fatma Latifoğlu & Sadık Kara & Mehmet Güney
Received: 26 June 2007 / Accepted: 21 August 2007 / Published online: 15 September 2007 # Springer Science + Business Media, LLC 2007
Abstract Doppler signals from the umbilical artery of 20 women with normal pregnancy between 18 and 20 weeks of gestation were recorded. The AR spectral analysis method has been used to obtain the Doppler sonograms of umbilical artery belonging to normal pregnant subjects and fractal dimension curves were calculated using Hurst exponent. RI; PI and S/D indexes have been calculated from the maximum frequency envelope of Doppler sonograms and from the fractal dimension curve. Area under the curve from ROC curve for RI, PI and S/D indexes derived from maximum frequency waveform were calculated as 0.931, 0.959, 0.938, respectively and area under the curve for RI, PI and S/D indexes derived from fractal dimension curve were calculated as 0.933, 0.961, and 0.941, respectively. These results show that, the Doppler indexes derived from fractal dimension curve are as sensitive as Doppler indexes derived from maximum velocity curve. Power Spectral Density graphics were derived from Doppler signals and Hurst exponent values calculated to evaluate the blood flow changing during pregnancy. ROC curve for PSDHURST index was calculated as 0.97. According to this result, PSDHURST index is more sensitive to detect the blood flow changing than traditional Doppler indexes.
F. Latifoğlu : S. Kara (*) Department of Electrical-Electronics Engineering, Erciyes University, 38039 Kayseri, Turkey e-mail:
[email protected] M. Güney Faculty of Medicine, Obstetric and Gyneology, Süleyman Demirel University (SDU), Isparta, Turkey
Keywords Umbilical artery . Doppler signals . Fractal dimension . Hurst exponent . Power spectral density
Introduction Doppler ultrasound provides noninvasive means to evaluate vascular perfusion of the pregnant umbilical artery. Doppler umbilical artery blood flow velocity waveform measurements are used in perinatal surveillance for the evaluation of fetal condition [1]. Doppler systems are based on the principle that ultrasound, emitted by an ultrasonic transducer, is returned partially towards the transducer by the moving red blood cells, thereby inducing a shift in frequency proportional to the emitted frequency and the velocity along the ultrasound beam. Spectral analysis is a preferred method of displaying quantitative Doppler information, so that the evaluation in time of the velocity distribution may be followed [2–6]. Since the velocity components are proportional to the frequency shifts, it is possible to track the velocity distribution by obtaining the power spectral density (PSD) estimates. The most usual way of displaying the information resulting from the spectral analysis is the Doppler sonogram. In a sonogram, the horizontal axis (t) represents time, the vertical axis ( f ) frequency and the gray level intensity represents the power level of the corresponding frequency at each point of time. As the color tone of the sonogram goes into black, the power level is increased and as it becomes lighter, the power level diminishes. By monitoring the sonogram, variation of the spectral properties of the Doppler signal and a number of extents related to the blood flow can easily be tracked.
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The umbilical artery waveform in normal pregnancy is characterized by forward velocity levels which remain high in diastole and increase gestational age [7]. Lowered diastolic velocities and occasionally reverse flow, are seen in certain cases of fetal compromise. These waveforms which changes have been associated with raised placental resistance [8]. When the sonogram of the Doppler signals is obtained, the mean and/or the maximum frequency waveform can be estimated. The maximum frequency waveform extracted from sonogram has almost invariably been used o obtain indices used for fetal circulation. A number of indices such as resistance index (RI), pulsality index (PI) and systolic: diastolic ratio (S:D) can then be calculated to describe the shape of the maximum frequency envelope over a cardiac cycle [8–11]. These indices have recently been used obstetrics field [2, 12, 13]. We have employed these indices from maximum velocity waveforms derived using Autoregressive (AR) modelling. As a new technique fractal analysis method has been investigated in the area of medical ultrasound [14]. Fractals of which B. B. Mandelbrot gave a detailed description in his works, has been successfully applied in many areas [15]. Its application in the area of medical ultrasound has been evaluated and its worthiness in this area has been searched [16, 17]. The Hurst exponent is also directly related to the fractal dimension, which gives a measure of the roughness of a surface. In this study, firstly AR spectral analysis method was used to obtain the Doppler sonograms of umbilical artery which belong to normal pregnant subjects and secondly, maximum frequency envelopes extracted from umbilical artery Doppler sonograms were obtained using image processing method. During normal pregnancies, there is a reduction of resistance to flow throughout pregnancy so this affects the Doppler indices. Thirdly, traditional RI; PI and S:D indices were calculated from the maximum frequency envelope to evaluate the variation of the umbilical artery blood flow velocity with the increasing gestational age. Secondly, as a new technique, we have used the fractal concept to accurately quantify the Doppler blood flow signal for the evaluation of the fetal and placental circulation. This paper presents the results of studies in which the fractal analysis method is used to analyze umbilical artery Doppler signals from pregnant women and RI, PI and S:D indices derived from fractal dimension curve. These indices were compared with indices obtained from classic sonogram envelope. Thirdly, PSD graphics were derived from umbilical artery Doppler signals using AR method. Hurst exponent values were calculated from PSD graphics as a new index to evaluate the variation of the umbilical artery blood flow velocity with the increasing gestational age.
J Med Syst (2007) 31:529–536
Materials and methods Hardware and demographic characteristics Umbilical artery Doppler signals were recorded from 20 normal pregnant women when they were both in second trimester (18–24 weeks) and third trimester (32–40 weeks). We used the data with 2 s duration. Maternal smoking, multiple pregnancies, a diagnosed fetal abnormality before recruitment, previous history of preeclampsia, intrauterine growth retardation, abruption placenta or preterm delivery, and history of any preexisting medical condition were reasons for not being included. Doppler signal acquisition was conducted in the Department of Obstetrics and Gynecology in Süleyman Demirel University hospital. The system hardware was composed of Digital Doppler Ultrasound unit that can work in the pulsed mode, linear ultrasound probe, input–output card and a personal computer. A personal computer (PC) was used for storage, displaying and spectral analysis of the acquired Doppler data. Ultrasound Doppler studies were performed with a Medison Sonace 8800 model Doppler ultrasound unit. A combined trans-abdominal real-time and pulsed Doppler system (carrier frequencies 5.0 and 3.75 MHz, respectively) was applied at 18–40 weeks of gestation. Signals reflected from the umbilical artery were recorded to derive out the Doppler shift frequencies. The Doppler recordings were performed by one examiner. The high pass filter was set at 40 Hz to attenuate low frequency components originating from the vessel wall and the sample volume length was set between 0.2 and 0.3 cm to cover the center of the umbilical artery. Flow velocity waveforms from the umbilical artery were obtained from the free-floating loop. In all tests performed on the subjects, the insonation angle and the presetting of the ultrasound were kept constant. The insonation angle was adjusted via electronic steering methods and manually in order to keep a constant value of 60° on a longitudinal view. The amplification gain was carefully set to obtain a clean spectral output with minimized background noise on the spectral display. The audio output of ultrasound unit was sampled at 44,100 Hz and then sent to a PC via an input–output card [18]. MATLAB 7.1® software package was used to calculate Hurst exponent from PSD and RI, PI and S/D indices from maximum frequency envelope of Doppler sonogram and from fractal dimension curve. Analysis of t test showed statistical significance for all indexes performed. Signal processing of umbilical arterial Doppler signals Signal processing to obtain the traditional Doppler indices PI, RI and S:D indexes and Hurst exponent values were
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performed on the umbilical artery Doppler signals in the following order: (1) Doppler signals were denoised using wavelet soft thresholding method. (2) The umbilical artery Doppler sonograms were extracted from the AR method. Acquired Doppler data was grouped in frames of 512 data points and the AR methods were applied on these frames and computed using Hanning window with %50 overlap. (3) Image processing method that was image enhancement, thresholding and edge detection was used to extract maximum frequency envelope from the Doppler sonograms. (4) Standard Doppler indices PI, RI and S:D were calculated from maximum frequency envelope. (5) The Doppler audio signals were divided into short overlapping segments of 10 ms. duration (5 ms. overlap) and fractal dimension were calculated using Hurst exponent. (6) Standard Doppler indices PI, RI and S:D were calculated from fractal dimension curve. (7) Power Spectral Density (PSD) graphics were obtained using the AR method. (8) Hurst exponent (PSDHURST) values were calculated from PSD graphics as a new index to evaluate whether there is variation during gestation. (9) ROC curve analysis were used for the Doppler indexes and for the PSDHURST indexes. Denoising of Doppler signals Denoised umbilical artery Doppler signals using soft thresholding-based denoising algorithm method were performed. The soft thresholding-based denoising algorithm has been proven to have the advantages of optimizing Mean
Fig. 1 Umbilical artery Doppler sonogram developed for 32 week pregnant woman
Fig. 2 Maximum frequency envelope developed for 32 week pregnant woman
Square Error (MSE) and keeping the smoothness of the denoised signal [19].The underlying model for the noisy signal is F ðnÞ ¼ F0 ðnÞ þ σeðnÞ
ð1Þ
Where the time n is equally sampled, F(n) is the noisy signal, F0(n) is the true signal, e(n) is supposed as a Gaussian white noise N(0, 1), and σ is the noise level. AR method for spectral analysis of umbilical arterial Doppler signals Because of the random spatial distributions of red blood cells, the Doppler signal is considered as a time-varying random signal. This signal has a nonstationary character in frequency can be considered stationary on a time interval of duration shorter than 20 ms [6]. Umbilical artery Doppler
Fig. 3 Demonstration of three velocity waveform indices
532 Table 1 Standard Doppler indexes derived from maximum frequency envelope
J Med Syst (2007) 31:529–536 RI
Subject number
Mean
Std. deviation
Minimum
Maximum
2.trimester Hurst exponent 3. trimester Hurst exponent PI 2.trimester Hurst exponent 3. trimester Hurst exponent S/D 2.trimester Hurst exponent 3. trimester Hurst exponent
20 20 Subject number 20 20 Subject number 20 20
0.728 0.568 Mean 1.2520 0.90 Mean 1.93 1.40
0.072 0.071 Std. deviation 0.15 0.13 Std. deviation 0.35 0.18
0.60 0.44 Minimum 0.98 0.72 Minimum 1.32 1.18
0.83 0.70 Maximum 1.44 1.24 Maximum 2.93 1.86
sonograms have been derived from AR method (Fig. 1). AR spectral modelling has been performed to display and analyze the spectral domain of the umbilical artery Doppler signals from pregnant woman. Among several methods of estimation of the AR model parameters (Yule Walker equation, Burg algoritm, least squares algorithm), in this study, the Burg method for estimating the AR parameters is used. Since AR-Burg method is computationally efficient and yields stable estimates, PSD estimates of carotid arterial Doppler signals are obtained by using this method. The AR-Burg method is based on the minimization of the forward and backward prediction errors and estimation of the reflection coefficient [20]. Acquired Doppler data was grouped in frames of 512 data points and we have utilized PSD using AR-Burg method on these frames [20].
was used to optimize the AR model order in this study [21]. Final Prediction Error, which selects the model order p by minimizing the function FPE(p) defined as: FPEð pÞ ¼ s 2 ð pÞ ðN þ p þ 1Þ ðN p 1Þ
ð2Þ
Selection of the model order of the parametric process is an important issue. If it is not chosen the order high enough, it may come with a too smooth spectrum, on the other side, too low. There are a number of criteria to optimize the AR model order. Akaike’s final prediction error (FPE) criterion
Where s2 is the estimate white noise variance (prediction error power) for the pth order AR model, p is the model order and N is the number of the samples of the output sequences. It could be seen that while s2 decreases with increasing p, the term (N+p+1)/(N−p−1) increases. The FPE is an estimate of the prediction error power when the prediction coefficients must be estimated from the data. The term (N+p+1)/(N−p−1) accounts for the increase in variance of the prediction error power estimator due to the inaccuracies in the prediction coefficient estimates. Selection of the model order was based an examination of the consistency of several order determination methods under noise-free conditions for each of the analyzed data length realizations. FPE yielded curves that asymptotically approached a minimum with knee around p=18 [22, 23]. Maximum frequency envelope obtained from Doppler sonogram is seen in Fig. 2. Three waveform indices to be calculated from maximum frequency envelope of Doppler
Fig. 4 Variance Fractal Dimension Curve developed for 32 week pregnant woman
Fig. 5 Power Spectral Density graphic for 32 week pregnant woman
Selection of model order for parametric methods
J Med Syst (2007) 31:529–536 Table 2 Standard Doppler indexes derived from fractal dimension curve
533 RI
Subject number
Mean
Std. deviation
Minimum
Maximum
2.trimester Hurst exponent 3. trimester Hurst exponent PI 2.trimester Hurst exponent 3. trimester Hurst exponent S/D 2.trimester Hurst exponent 3. trimester Hurst exponent
20 20 Subject number 20 20 Subject number 20 20
0.735 0.567 Mean 1.251 0.893 Mean 1.98 1.41
0.074 0.069 Std. deviation 0.15 0.14 Std. deviation 0.35 0.19
0.60 0.44 Minimum 0.96 0.72 Minimum 1.44 1.16
0.88 0.68 Maximum 1.44 1.26 Maximum 2.93 1.88
sonogram RI, PI and S:D indices are illustrated in Fig. 3 and in Table 1, Calculations of these indices are illustrated. PSD graphics as seen in Fig. 4 were obtained using AR method and Hurst exponent values were calculated from PSD curve. Fractal dimension of umbilical arterial Doppler signals using hurst exponent Fractals are objects which possess a form of self-scaling: Parts of the whole can be made to fit the whole in some way or the other in some way by shifting and stretching. Fractals features represent the morphology of the signals in some way or the other. These morphological differences can be picked up and used for several applications. They have already found applications in the field of traffic control, image analysis and compression. Such utility of fractal analysis is a source of motivation to consider it as a useful tool for feature extraction in Doppler signal analysis. Doppler ultrasound flow signals can be regarded as a quasi-wide-sense stationary random process during a short enough time periods. Therefore the fractal dimension of Doppler signal waveforms were analyzed in 10 ms interval. Fractal Dimensions are a measure of the self-similarity of the signals. A lot of dimensions have been defined in this field. In this study, we calculated Hurst exponent of the Doppler time series to characterize their fractal behaviours [15]. The variance fractal dimension (VFD) was used to estimate the fractal dimension. The VFD for a one dimensional time series is calculated via the Hurst exponent as VFD=2−H, where H is defined as: H¼
lim
ðt2 t1 Þ!0
log ½Varðsðt2 t1 ÞÞ 2: log ½t2 t1
Table 3 Hurst Exponent values derived from PSD curve
PSD
ð3Þ
HURST
2.trimester Hurst exponent 3. trimester Hurst exponent
Where, t2 −t1, and the slope of the regression line determines H. Figure 5 shows the fractal dimension curve which resulted from applying fractal analysis to the umbilical artery Doppler signal from 32 week pregnant woman. In Table 2, Calculations of three indexes from fractal dimension curve are illustrated. Trend in which the power of the spectrum is inversely proportional to the frequency f according to a 1=f α power law. This trend characterizes the spectra of most biological signals. When the spectrum is plotted in a log–log scale, the 1/f trend appears as a straight line with slope -α. For fractal time series there is a simple relation between the slope α and the Hurst exponent. H ¼ ðα 1Þ=2
ð4Þ
Fit is seen in Fig. 8 PSD curve and spectral trend of PSD curve. In Table 3 calculations of Hurst exponent from PSD curves are illustrated. ROC curve ROC curves have been used as measures for the accuracy of diagnostic tests in medicine and other fields when the test results are continuous measures. ROC curve is one of the best methods of evaluating the performance of a test and defining appropriate decision threshold. The best choice of threshold will then depend on a number of factors including the consequences of making both types of false classification (false positive and false negative) and the prevalence of disease in the target population. ROC curves display the relationship between sensitivity (truepositive rate) and 1-specificity (false-positive rate) across all possible threshold values that define the positivity of a
Subject number
Mean
Std. deviation
Minimum
Maximum
20 20
0.747 0.886
0.036 0.060
0.69 0.78
0.82 0.98
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J Med Syst (2007) 31:529–536
Fig. 6 ROC curve for RI,PI, S:D indexes derived from Doppler sonogram AUC for RI=0.931, AUC for PI=0.959, AUC for S:D=0.938)
Fig. 8 Power spectrum of umbilical artery Doppler signals from 32 week pregnant plotted in a log–log scale. The 1/f-trend is shown by a straight line
disease or condition [24, 25]. A good test is one for which sensitivity rises rapidly and specificity hardly increases at all until sensitivity becomes high. Hence, the closer the curve is to the upper left corner, the higher the overall accuracy of the test. ROC curve which is seen in Figs. 6, 7 and 9 represents the performance of the test parameter which is RI, PI and S/D indexes derived from Doppler sonogram and from fractal dimension curve and PSDHURST from power spectral density of umbilical artery Doppler signals.
signals in the time domain have not extra information about duration of pregnancy. Therefore these signals were analysed in the frequency domain to examine blood flow. During normal pregnancy, there is a reduction of resistance to flow throughout pregnancy. Fig. 1 shows umbilical artery Doppler sonogram derived from AR spectral analysis method. Also, Fig. 2 shows maximum frequency envelope of umbilical artery Doppler sonogram. Fig. 4 shows the fractal dimension curve derived from umbilical artery Doppler signals using Hurst exponent. It can be seen from Figure 4 that fractal dimension curve and maximum frequency curve are very similar. Calculations of RI, PI and S:D indexes from maximum frequency curve and fractal dimension curve are seen Table 1 and Table 2,
Results and discussion Doppler signals reflected from the umbilical artery were recorded to derive out the Doppler shift frequencies and denoised using wavelet soft thresholding method. Doppler
1.00
1.00
Sensitivity
.75
.50
Source of the Curve S/D
.25
Sensitivity
.75
.50
.25
PI
0.00 0.00
RI
.25
.50
.75
1.00
0.00 0.00
.25
.50
.75
1.00
1 - Specificity Fig. 7 ROC curve for RI,PI, S:D indexes derived from fractal dimension curve (AUC for RI=0.933, AUC for PI=0.961, AUC for S:D=0.941)
1 - Specificity Fig. 9 ROC curve for PSDHURST index derived from Power Spectral Density (AUC for PSDHURST =0.97)
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respectively. During the pregnancy RI mean values derived from maximum frequency curve decrease from 0.73 to 0.57, PI mean values decrease from 1.25 to 0.90 and S/D mean values decrease from 1.93 to 1.40. Also, during the pregnancy RI mean values derived from fractal dimension curve decrease from 0.72 to 0.57, PI mean values decrease from 1.25 to 0.89 and S:D mean values decrease from 1.98 to 1.41. Because of reduction of resistance to blood flow during 18–40 week gestation, It is seen that there is a reduction of standard Doppler indexes derived from maximum frequency waveform and fractal dimension curve. It is seen that in Fig. 6, ROC curves for RI PI and S/D values derived from maximum velocity curve and in Fig. 7, ROC curves for RI PI and S/D values derived from fractal dimension curve. Area under the curve (AUC) for RI, PI and S/D indexes derived from maximum frequency waveform were calculated as 0.931, 0.959, 0.938, respectively, and AUC for RI, PI and S/D indexes derived from fractal dimension curve were calculated as 0.933, 0.961, and 0.941, respectively. These results show that, the Doppler indexes derived from fractal dimension curve are as sensitive as Doppler indexes derived from maximum velocity curve. As wilcoxon t test, three Doppler indices derived both maximum frequency waveform and fractal dimension curve for pregnant subject both second trimester and third trimester is ρ
