Order selection in Doppler blood flow signal spectral analysis using autoregressive modelling

Adaptive spectral doppler estimation

In this paper, 2 adaptive spectral estimation techniques are analyzed for spectral Doppler ultrasound. The purpose is to minimize the observation window needed to estimate the spectrogram to provide a better temporal resolution and gain more flexibility when designing the data acquisition sequence. The methods can also provide better quality of the estimated power spectral density (PSD) of the blood signal. Adaptive spectral estimation techniques are known to provide good spectral resolution and contrast even when the observation window is very short. The 2 adaptive techniques are tested and compared with the averaged periodogram (Welch's method). The blood power spectral capon (BPC) method is based on a standard minimum variance technique adapted to account for both averaging over slow-time and depth. The blood amplitude and phase estimation technique (BAPES) is based on finding a set of matched filters (one for each velocity component of interest) and filtering the blood process over slow-time and averaging over depth to find the PSD. The methods are tested using various experiments and simulations. First, controlled flow-rig experiments with steady laminar flow are carried out. Simulations in Field II for pulsating flow resembling the femoral artery are also analyzed. The simulations are followed by in vivo measurement on the common carotid artery. In all simulations and experiments it was concluded that the adaptive methods display superior performance for short observation windows compared with the averaged periodogram. Computational costs and implementation details are also discussed.

Determining fractal dimension of umbilical artery Doppler signals using Hurst exponent

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 PSD(HURST) index was calculated as 0.97. According to this result, PSD(HURST) index is more sensitive to detect the blood flow changing than traditional Doppler indexes.

Pattern Detection of Atherosclerosis from Carotid Artery Doppler Signals using Fuzzy Weighted Pre-Processing and Least Square Support Vector Machine (LSSVM)

Carotid Artery Doppler Signals were recorded from 114 subjects, 60 of whom had Atherosclerosis disease while the rest were healthy controls. Diagnosis of Atherosclerosis from Carotid Artery Doppler Signals was conducted using Fuzzy weighted pre-processing and Least Square Support Vector Machine (LSSVM). First, in order to determine the LSSVM inputs, spectral analysis of Carotid Artery Doppler Signals was performed via Autoregressive (AR) modeling. Then, fuzzy weighted pre-processing based is proposed expert system, applied to inputs obtained from spectral analysis of Carotid Artery Doppler Signals. LSSVM was used to detect Atherosclerosis from Carotid Artery Doppler Signals. All data set were obtained from Carotid Artery Doppler Signals of healthy subjects and subjects suffering from Atherosclerosis disease. The employed expert system has achieved 100% classification accuracy using a 10-fold Cross Validation (CV) method.

Discontinuous doppler signals simulating respiratory misregistration: Effect on autoregressive frequency spectra

In this study, we have produced discontinuous Doppler signals of carotid artery and internal jugular vein, simulating respiratory misregistration. The aim of the study is to observe the effect of signal discontinuity and its duration on power spectral density vs. frequency graphs obtained by Autoregressive Modeling. The signals were recorded from ten male volunteers. Signal interruption was performed by moving the sampling volume in and out of the vessel bidirectionally. To estimate the effect of on-line recording time and signal discontinuity on frequency spectra, we have worked on a control data of 30s with continuous signal, and three sets of data with artificially interrupted signals of 30, 60 and 90s duration. Maximum power spectral density, area under the power spectral density, and frequency level corresponding to maximum power spectral density were calculated on frequency spectra. The frequency level corresponding to maximum power spectral density provides the most statistically stable finding in our preliminary data. The signal duration of the signal had no significant effect on the statistical stability of the frequency level.

Comparison of fast Fourier transformation and autoregressive modelling as a diagnostic tool in analysis of lower extremity venous signals

In this study, we have compared the efficacy of autoregressive modelling (ARM) and fast Fourier transformation (FFT) of Doppler signals from lower extremity veins of healthy volunteers in various physiologic situations. Compared to FFT, ARM produced smooth spectra and less spectral broadening both in sonograms and power spectra. However, faulty positioning of the peaks along the time axis in FFT-derived power spectral density curves show that FFT is not a suitable method if these graphs are to be used as a diagnostic tool. Analysis of ARM-based venous sonograms and power spectral density graphs revealed that FFT should not be used in signals with high power spectral density levels and low-frequency bandwidth within limited segments of time.

Doppler angle estimation using AR modeling

The transit time spectrum broadening effect has long been explored for Doppler angle estimation. Given acoustic beam geometry, the Doppler angle can be derived based on the mean Doppler frequency and the Doppler bandwidth. Spectral estimators based on the fast Fourier transform (FFT) are typically used. One problem with this approach is that a long data acquisition time is required to achieve adequate spectral resolution, with typically 32-128 flow samples being needed. This makes the method unsuitable for real-time two-dimensional Doppler imaging. This paper proposes using an autoregressive (AR) model to obtain the Doppler spectrum using a small number (e.g., eight) of flow samples. The flow samples are properly selected, then extrapolated to ensure adequate spectral resolution. Because only a small number of samples are used, the data acquisition time is significantly reduced and real-time, two-dimensional Doppler angle estimation becomes feasible. The approach was evaluated using both simulated and experimental data. Flows with various degrees of velocity gradient were simulated, with the Doppler angle ranging from 20 degrees to 75 degrees. The results indicate that the AR method generally provided accurate Doppler bandwidth estimates. In addition, the AR method outperformed the FFT method at smaller Doppler angles. The experimental data for Doppler angles, ranging from 33 degrees to 72 degrees, showed that the AR method using only eight flow samples had an average estimation error of 3.6 degrees, which compares favorably to the average error of 4.7 degrees for the FFT method using 64 flow samples. Because accurate estimates can be obtained using a small number of flow samples, it is concluded that real-time, two-dimensional estimation of the Doppler angle over a wide range of angles is possible using the AR method.

A study on the optimum order of autoregressive models for heart rate variability

Heart rate variability (HRV) has been used as a non-invasive marker of the activity of the autonomic nervous system and its spectrum analysis gives a measure of the sympatho-vagal balance. If short segments are used in an attempt to improve temporal resolution, autoregressive spectral estimation, where the mode] order must be estimated, is preferred. In this paper we compare four criteria for the estimation of the 'optimum' model order for an autoregressive (AR) process applied to short segments of tachograms used for HRV analysis. The criteria used were Akaike's final prediction error, Akaike's information criterion, Parzen's criterion of autoregressive transfer function and Rissanen's minimum description length method, and they were first applied to tachograms to verify (i) the range and distribution of model orders obtained and (ii) if the different techniques suggest the same model order for the same frames. The four techniques were then tested using a true AR process of known order p = 6; this verified the ability of the criteria to estimate the correct order of a true AR process and the effect, on the spectrum, of choosing a wrong model order was also investigated. It was found that all the four criteria underestimate the true AR order; specifying a fixed model order was then looked at and it is recommended that an AR order not less than p = 16, should be used for spectral analysis of short segments of tachograms.

Application of autoregressive methods to multigate spectral analysis

Multigate analysis is known to be capable of detecting accurate blood velocity profiles from human vessels. Experimental systems so far presented in the literature use time-domain frequency estimations and, more recently, the fast Fourier transform (FFT) for real-time analysis of Doppler signals from multiple range cells. This experimental study is aimed at evaluating the application of an autoregressive (AR) method (Burg algorithm) to multigate Doppler analysis. Both in vitro and in vivo results were collected with a commercial Duplex scanner coupled with a prototype multigate unit developed in our laboratory. The same multigate signals are, thus, processed according to both the FFT and the Burg algorithms. The related spectral and maximum frequency profiles are reported and statistically compared. AR, implemented with the Burg algorithm, is demonstrated to be a way to perform multigate spectral analysis with reduced spectral variance, suitable for maximum velocity profile extraction through a simple threshold.

Developments in cardiovascular ultrasound: Part 1: Signal processing and instrumentation

One of the major contributions to the improvement of spectral Doppler and colour flow imaging instruments has been the development of advanced signal-processing techniques made possible by increasing computing power. Model-based or parametric spectral estimators, time-frequency transforms, station-arising algorithms and spectral width correction techniques have been investigated as possible improvements on the FFT-based estimators currently used for real-time spectral estimation of Doppler signals. In colour flow imaging some improvement on velocity estimation accuracy has been achieved by the use of new algorithms but at the expense of increased computational complexity compared with the conventional autocorrelation method. Polynomial filters have been demonstrated to have some advantages over IIR filters for stationary echo cancellation. Several methods of velocity vector estimation to overcome the problem of angle dependence have been studied, including 2D feature tracking, two and three beam approaches and the use of spectral width in addition to mean frequency. 3D data acquisition and display and Doppler power imaging have also been investigated. The use of harmonic imaging, using the second harmonic generated by encapsulated bubble contrast media, seems promising particularly for imaging slow flow. Parallel image data acquisition using non-sequential scanning or broad beam transmission, followed by simultaneous reception along a number of beams, has been studied to speed up 'real-time' imaging.

A study of the spectral broadening of simulated Doppler signals using FFT and AR modelling

Doppler ultrasound is used clinically to detect stenosis in the carotid artery. The presence of stenosis may be identified by disturbed flow patterns distal to the stenosis that cause spectral broadening in the spectrum of the Doppler signal around peak systole. This paper investigates the behaviour of the spectral broadening index (SBI) derived from wide-band spectra obtained using autoregressive modelling (AR), compared with the SBI based on the fast-Fourier transform (FFT) spectra. Simulated Doppler signals were created using white noise and shaped filters to analyse spectra typically found around the systolic peak and to assess the magnitude and variance of AR and FFT-SBI for a range of signal-to-noise ratios. The results of the analysis show a strong correlation between the indices calculated using the FFT and AR algorithms. Despite the qualitative improvement of the AR spectra over the FFT, the estimation of SBI for short data frames is not significantly improved using AR.

Comparison of Doppler signal analysis techniques for velocity waveform, turbulence and vortex measurement: a simulation study

Simulated time-varying Doppler signals incorporating bandwidth, power variation and vortex simulation have been used to compare a number of signal analysis techniques with a view to optimising the accuracy of convective velocity waveform, spectral broadening and vortex signal estimation. The short-time Fourier transform (STFT), the autoregressive (AR) modified covariance estimator, the time-frequency pseudo-Wigner-Ville and Choi-Williams distributions and a partial stationarising algorithm were investigated for a range of some analysis parameters (such as window duration, AR model order). It was found that all methods could estimate the convective velocity waveform well and all the nonclassical methods were an improvement over the STFT for bandwidth estimation with the stationarising method giving the lowest error. For vortex measurement, using parameters that were optimum for mean frequency and bandwidth estimation, the stationarising, modified covariance, pseudo-Wigner-Ville with a 10-ms window and Choi-Williams methods gave improved performances compared with the STFT.

Comparison of FFT- and AR-based sonogram outputs of 20 MHz pulsed Doppler data in real time

Real time sonogram outputs of autoregressive (AR) and Fast Fourier Transform (FFT) spectral analysis of 20 MHz pulsed ultrasonic Doppler blood flowmeter are presented. Data obtained from coronary, renal, iliac, digital and mesenteric arteries were processed using AR- and FFT-based spectral analysis techniques and interpretable sonograms were constructed. In comparison with the FFT-based sonogram outputs. AR-based sonogram outputs for 20 MHz pulsed Doppler data provide better results. Hence, the AR modeling is strongly recommended for small vessels with diameters between 1 and 2 mm.

An adaptive approach to computing the spectrum and mean frequency of Doppler signals

Modern ultrasound Doppler systems are facing the problem of processing increasingly shorter data sets. Spectral analysis of the strongly nonstationary Doppler signal needs to shorten the analysis window while maintaining a low variance and high resolution spectrum. Color flow imaging requires estimation of the Doppler mean frequency from even shorter Doppler data sets to obtain both a high frame rate and high spatial resolution. We reconsider these two estimation problems in light of adaptive methods. A regularized parametric method for spectral analysis as well as an adapted mean frequency estimator are developed. The choice of the adaptive criterion is then addressed and adaptive spectral and mean frequency estimators are developed to minimize the mean square error on estimation in the presence of noise. Two suboptimal spectral and mean-frequency estimators are then derived for real-time applications. Finally, their performance is compared to that of both the FFT based periodogram and the AR parametric spectral analysis for the spectral estimator, and, to both the correlation angle and the Kristoffersen's [8] estimators for the mean frequency estimator using Doppler data recorded in vitro.

Differences in the power structures of Fourier transform and autoregressive spectral estimates of narrow-band Doppler signals

There is considerable interest in the application of autoregressive (AR) spectral analysis to ultrasonic Doppler signals. Sonograms produced using this technique are, however, very different from those produced using classic Fourier transform methods. Simulations have shown that the heights of the peaks in the AR spectra of narrow-band signals are not necessarily proportional to signal power, and should be used with caution in the context of Doppler signal processing.

Extracting instantaneous mean frequency information from Doppler signals using the Wigner distribution function

The Wigner distribution function (WDF) does not require the analysed signals to be stationary within the time segment used for analysis, and instantaneous frequency (IF) information can be obtained from it. Because of the influence of the cross-power of the signal components, however, the interpretation of the IF results is physically clear only for monocomponent signals with infinite data lengths. The IF results for multicomponent signals also suffer from spike problems and are quite unstable even when the signal-to-noise ratio is high. It is suggested that a "pseudo-instantaneous mean frequency," which uses the positive part of the WDF to follow the power distribution changes among frequency components, is used as a simple and rapid way to track frequency changes of Doppler signals. Simulation results show that the pseudo-instantaneous mean frequency does not have the same spike problems and gives stable and relatively accurate information about frequency changes when the sampling frequency is properly chosen.

Cardiac Doppler blood-flow signal analysis. Part 2. Time/frequency representation based on autoregressive modelling

Doppler spectrograms obtained by using autoregressive (AR) modelling based on the Yule-Walker equations were investigated. A complex AR model using the in-phase and the quadrature components of the Doppler signal was used to provide blood-flow directions. The effect of model orders on the spectrogram estimation was studied using cardiac Doppler blood flow signals taken from 20 patients. The 'final prediction error' (FPE) and the 'Akaike's information criterion' (AIC) provided almost identical results in model-order selection. An index, the spectral envelope area (SEA), was used to evaluate the effect of window duration and sampling frequency on AR Doppler spectrogram estimation. The statistical analysis revealed that the SEA obtained from AR modelling was not sensitive to window duration and sampling frequency. This result verified the consistency of the AR Doppler spectrogram. The white-noise characteristics of the AR modelling error signal indicated that the Doppler blood-flow signal can be adequately modelled as a complex AR process. With appropriate model orders, AR modelling provided better Doppler spectrogram estimates than the periodogram.

Application of autoregressive analysis to 20 MHz pulsed Doppler data in real time

The real time application of autoregressive (AR) spectral analysis to a 20-MHz pulsed Doppler blood flowmeter is presented. The system consists of a TMS 320C25 digital signal processor with a 80286 based PC/AT microcomputer and associated interfacing circuitry. The AR method was used for in vivo spectral analysis of the signals obtained from a 20-MHz pulsed Doppler flowmeter in real time. The data obtained from digital and coronary arteries were processed using both AR and FFT spectral analysis methods. Also the data obtained from a stenosis coronary artery under surgical operation were processed using both methods. When the results are compared, it is seen that autoregressive analysis has given better results. Therefore the technique can be used in the examining of small vessels such as renal, iliac, mesenteric, coronary and digital arteries.

A unified approach to modeling the backscattered Doppler ultrasound from blood

A unified approach to modeling the backscattered Doppler ultrasound signal from blood is presented. The approach consists of summing the contributions from elemental acoustic voxels each containing many red blood cells (RBC's). For an insonified region that is large compared to a wavelength, it is shown that the Doppler signal is a Gaussian random process that arises from fluctuation scattering, which implies that the backscattered power is proportional to the variance of local RBC concentrations. As a result, some common misconceptions about the relationship between the backscattering coefficient and hematocrit can be readily resolved. The unified approach was also used to derive a Doppler signal simulation model which shows that, regardless of flow condition, the power in the Doppler frequency spectrum is governed by the exponential distribution. For finite beamwidth and paraxial flow, it is further shown that the digitized Doppler signal can be modeled by a moving average random process whose order is determined by the signal sampling rate as well as the flow velocity profile.

On the usefulness of singular value decomposition-ARMA models in Doppler ultrasound

The singular value decomposition (SVD) autoregressive moving average, (ARMA) procedure is applied to computer-generated synthetic Doppler signals as well as in-vivo Doppler data recorded in the carotid artery. Two essential algorithmic parameters (the initially proposed model order and the number of overdetermined equations used) prove difficult to choose. The resulting spectra are very dependent on these two parameters. For the simulated data models orders of (3, 3) provide good results. However, when applying the SVD-ARMA algorithm to in-vivo Doppler signals no single set of model orders was capable of producing consistent spectral estimates throughout the cardiac cycle. Altering the model orders also necessitates the selection of new algorithmic parameters. Hence, the SVD-ARMA approach cannot be considered suitable as a spectral estimation technique, for real-time Doppler ultrasound systems.

Low-order AR models for mean and maximum frequency estimation in the context of Doppler color flow mapping

Autoregressive (AR) techniques are investigated by developing mean and maximum frequency estimators suitable for use in Doppler color flow mapping systems, where they are most needed. The estimators are based on low-order (for computational efficiency) AR models applied to complex signals whose real and imaginary parts are the in-phase and quadrature components of the analytical Doppler signal, respectively. A large number of simulated data sequences generated by a sinusoidal computer model and having different number of samples, spectral shapes, bandwidths, and signal-to-noise ratios are used to examine the performance (bias and variance) of the estimators in a systematic manner. Comparisons are made with the established autocorrelation technique, whose output is shown to be identical to one of the AR mean frequency estimators described.

Asymmetry of Doppler spectrum in stenosis differentiation

The asymmetry of the spectral distribution of ultrasonic Doppler flow velocity signals, assessed using the coefficient of skewness, is discussed as a criterion of stenosis differentiation. Its performance is compared with that of the index of turbulence intensity for both in vitro and in vivo flow Doppler signals, recorded distal to a stenosis. The power spectral distributions are computed using the direct Fourier transform and maximum likelihood method. The asymmetry of spectral distribution has proved to be a more efficient criterion than the turbulence intensity. The maximum likelihood method ensures better stenosis differentiation than the direct FFT method.

Selection of a spectral analysis method for the assessment of velocity distribution based on the spectral distribution of ultrasonic Doppler signals

The direct Fourier transform method, autoregressive modelling, the maximum likelihood method and the Wigner-Ville distribution were applied to the Doppler signal obtained from a fully insonated laminar model flow. The appreciation of the spectral method was based on the properties of the ratio variance/(fmean)2 (INT) of the spectrum. The basic criterion was the sensitivity of INT to the analysis parameters, especially the data window. The results of spectral analysis, as well as the properties of INT, were strongly affected by the method applied. The maximum likelihood method appeared best suited for the purpose of assessment of velocity distribution and is expected to give the best results in the case of in vivo blood flow. The performances of other discussed methods were inferior, due to their stronger incompatibility with the signal properties.