Optimized order estimation for autoregressive models to predict respiratory motion

Purpose

To successfully ablate moving tumors in robotic radio-surgery, it is necessary to compensate for motion of inner organs caused by respiration. This can be achieved by tracking the body surface and correlating the external movement with the tumor position as it is implemented in the CyberKnife $^\circledR $ Synchrony system. Tracking errors, originating from system immanent time delays, are typically reduced by time series prediction. Many prediction algorithms exploit autoregressive (AR) properties of the signal. Estimating the optimal model order $p$ for these algorithms constitutes a challenge often solved via grid search or prior knowledge about the signal.

Methods

Aiming at a more efficient approach instead, this study evaluates the Akaike information criterion (AIC), the corrected AIC, and the Bayesian information criterion (BIC) on the first minute of the respiratory signal. Exemplarily, we evaluated the approach for a least mean square (LMS) and a wavelet-based LMS (wLMS) predictor.

Results

Analyzing 12 motion traces, orders estimated by AIC had the highest prediction accuracy for both prediction algorithms. Extending the investigations to 304 real motion traces, the prediction error of wLMS using AIC was found to decrease significantly by 85.1 % of the data compared to the original implementation

Conclusions

The overall results suggest that using AIC to estimate the model order $p$ for prediction algorithms based on AR properties is a valid method which avoids intensive grid search and leads to high prediction accuracy.     

A real-time autoregressive spectrum analyzer for Doppler ultrasound signals

A system based on a digital signal processor and a microcomputer has been programmed to estimate the maximum entropy autoregressive (AR) power spectrum of ultrasonic Doppler shift signals and display the results in the form of a sonogram in real-time on a computer screen. The system, which is based on a TMS 320C25 digital signal processor chip, calculates spectra with 128 frequency components from 64 samples of the Doppler signal. The samples are collected at a programmable rate of up to 40.96 kHz, and the computation of each spectrum takes typically 3.2 ms. The feasibility of on-line AR spectral estimation makes this type of analysis an attractive alternative to the more conventional fast Fourier transform approach to the analysis of Doppler ultrasound signals.     

Doppler angle estimation of pulsatile flows using AR modeling

In quantitative ultrasonic flow measurements, the beam-to-flow angle (i.e., Doppler angle) is an important parameter. An autoregressive (AR) spectral analysis technique in combination with the Doppler spectrum broadening effect was previously proposed to estimate the Doppler angle. Since only a limited number of flow samples are used, real-time two-dimensional Doppler angle estimation is possible. The method was validated for laminar flows with constant velocities. In clinical applications, the flow pulsation needs to be considered. For pulsatile flows, the flow velocity is time-varying and the accuracy of Doppler angle estimation may be affected. In this paper, the AR method using only a limited number of flow samples was applied to Doppler angle estimation of pulsatile flows. The flow samples were properly selected to derive the AR coefficients and then more samples were extrapolated based on the AR model. The proposed method was verified by both simulations and in vitro experiments. A wide range of Doppler angles (from 3o degrees to 78 degrees) and different flow rates were considered. The experimental data for the Doppler angle showed that the AR method using eight flow samples had an average estimation error of 3.50 degrees compared to an average error of 7.08 degrees for the Fast Fourier Transform (FFT) method using 64 flow samples. Results indicated that the AR method not only provided accurate Doppler angle estimates, but also outperformed the conventional FFT method in pulsatile flows. This is because the short data acquisition time is less affected by the temporal velocity changes. It is concluded that real-time two-dimensional estimation of the Doppler angle is possible using the AR method in the presence of pulsatile flows. In addition, Doppler angle estimation with turbulent flows is also discussed. Results show that both the AR and FFT methods are not adequate due to the spectral broadening effects from the turbulence.     

Performance of time-frequency representation techniques to measure blood flow turbulence with pulsed-wave Doppler ultrasound

The current processing performed by commercial instruments to obtain the time-frequency representation (TFR) of pulsed-wave Doppler signals may not be adequate to characterize turbulent flow motions. The assessment of the intensity of turbulence is of high clinical importance and measuring high-frequency (small-scale) flow motions, using Doppler ultrasound (US), is a difficult problem that has been studied very little. The objective was to optimize the performance of the spectrogram (SPEC), autoregressive modeling (AR), Choi-Williams distribution (CWD), Choi-Williams reduced interference distribution (CW-RID), Bessel distribution (BD), and matching pursuit method (MP) for mean velocity waveform estimation and turbulence detection. The intensity of turbulence was measured from the fluctuations of the Doppler mean velocity obtained from a simulation model under pulsatile flow. The Kolmogorov spectrum, which is used to determine the frequency of the fluctuations and, thus, the scale of the turbulent motions, was also computed for each method. The best set of parameters for each TFR method was determined by minimizing the error of the absolute frequency fluctuations and Kolmogorov spectral bandwidth measured from the simulated and computed Doppler spectra. The results showed that different parameters must be used for each method to minimize the velocity variance of the estimator, to optimize the detection of the turbulent frequency fluctuations, and to estimate the Kolmogorov spectrum. To minimize the variance and to measure the absolute turbulent frequency fluctuations, four methods provided similar results: SPEC (10-ms sine-cosine windows), AR (10-ms rectangular windows, model order = 8), CWD (w(N) and w(M) = 10-ms rectangular windows, sigma = 0.01), and BD (w(N) = 10-ms rectangular windows, alpha = 16). The velocity variance in the absence of turbulence was on the order of 0.04 m/s (coefficient of variation ranging from 8.0% to 14.5%, depending on the method). With these spectral techniques, the peak of the turbulence intensity was adequately estimated (velocity bias < 0.01 m/s). To track the frequency of turbulence, the best method was BD (w(N) = 2-ms rectangular windows, alpha = 2). The bias in the estimate of the -10 dB bandwidth of the Kolmogorov spectrum was 354 +/- 51 Hz in the absence of turbulence (the true bandwidth should be 0 Hz), and -193 +/- 371 Hz with turbulence (the simulated -10-dB bandwidth was estimated at 1256 Hz instead of 1449 Hz). In conclusion, several TFR methods can be used to measure the magnitude of the turbulent fluctuations. To track eddies ranging from large vortex to small turbulent fluctuations (wide Kolmogorov spectrum), the Bessel distribution with appropriate set of parameters is recommended.     

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.     

A comparison method for mean frequency estimators for Doppler ultrasound.

Mean frequency estimators as used in pulsed Doppler ultrasound equipment should provide an accurate (quality) and consistent (robustness) estimate over a wide range of signal conditions. In a simplified signal model, the main parameters to consider are the noise level, mean frequency, bandwidth and power of both the Doppler signal and the stationary component over a given time window. It may be expected that one estimator for a given parameter combination exhibits a good performance while another estimator for the same parameter combination behaves poorly. To allow direct comparison between different types of frequency estimators, a method is introduced to evaluate the quality and robustness of estimators for a common signal space covering a wide range of realistic parameter combinations. The method is illustrated using three different mean frequency estimators: (1) a first order autoregressive estimator in combination with a stationary echo filter; (2) a second order autoregressive estimator; and (3) a complex linear regression estimator in combination with a stationary echo filter. It is concluded that, for the parameter combination considered, the complex linear regression estimator exhibits the best quality (low variance and bias of the estimate) and robustness (consistent quality for all parameter combinations).     

Noninvasive assessment of left ventricular isovolumic contraction and relaxation with continuous wave Doppler aortic regurgitant velocity signals: an in vivo validation study.

The purpose of this study was to provide fundamental in vivo validation of a method with the use of aortic regurgitant (AR) jet signals recorded with continuous wave (CW) Doppler for assessing left ventricular (LV) isovolumic contraction and relaxation. Preliminary studies have suggested that analysis of CW Doppler AR velocity signals permits the estimation of LV positive and negative dP/dt. We studied 19 hemodynamically different states in 6 sheep with surgically induced chronic aortic regurgitation. CW AR velocity spectra and high-fidelity LV and aortic pressures were recorded simultaneously. Rates of LV pressure rise and fall (RPR and RPF) were calculated by determining the time interval between points at 1 m/s and 2.5 m/s in the deceleration and acceleration slopes of the CW Doppler AR velocity envelope (corresponding to a pressure change of 21 mm Hg). RPR and RPF calculated by CW Doppler analysis for each state were compared with the peak positive dP/dt and negative dP/dt, obtained from the corresponding high-fidelity LV pressure curve, respectively. The LV peak positive and negative dP/dt derived by catheter ranged from 817 to 2625 mm Hg/s and from 917 to 2583 mm Hg/s, respectively. Multiple regression analysis showed that Doppler RPR correlated well with catheter peak positive dP/dt (r = 0.93; mean differences, -413 +/- 250 mm Hg/s). There was also good correlation and agreement between Doppler RPF and the catheter peak negative dP/dt (r = 0.89; mean difference, -279 +/- 239 mm Hg/s). Both Doppler-determined RPR and RPF underestimated their respective LV peak dP/dt. CW Doppler AR spectra can provide a reliable noninvasive estimate of LV dP/dt and could be helpful in the serial assessment of ventricular function in patients with aortic regurgitation.     

Estimation of the intrinsic dimensionality of fMRI data

A new method based on an autoregressive noise model of order 1 is introduced to the problem of detecting the number of components in fMRI data. Unlike current information-theoretic criteria like AIC, MDL, and related PPCA, which do not incorporate autocorrelations in the noise, the new method leads to more consistent estimates of the model order, as illustrated in simulated and real fMRI resting-state data.     

Mathematical modelling of ventilation mechanics

Routine application of rule of thumb parameter sets in clinical practice pushes model visions to the background, including the complete framework of assumptions, simplifications, suppositions and conditions. But: models can be a very strong tool, when applied selectively — that means, with a clear idea of destination, definition, parameter selection and verification.This article discusses universal issues of modelling — based on ventilation mechanics models in intensive care medicine.Abbreviations AIC AKAIKE's information theoretic criterion - ARX auto regressive model with external input - C compliance - E elastance - EX expiratory - FPE final prediction error - IN inspiratory - LMS least mean squares method - R resistance - s² dispersion - P, V, V pressure, volume and flow at airway opening - PCV pressure controlled ventilation     

Simulation of Doppler ultrasound signals for a laminar, pulsatile, nonuniform flow.

A simulation for Doppler ultrasound quadrature signals from pulsatile, nonuniform flow is presented. It is an extension of an earlier simulation presented by Jones and Giddens (1990a) which was valid for laminar, uniform, steady flow and which included the stochastic characteristics introduced by scattering particles which enter and leave the sample volume at random times. Fourier transform and autoregressive spectral analysis techniques are used to compare the simulated signals to Doppler signals collected from an in vitro flow setup. Power spectra, Doppler frequency estimates and standard deviations of these estimates serve as standards of comparison. Results show that the simulation model generates realistic quadrature signals. The study improves the understanding of the physics of the Doppler process and shows that it can be modeled for complex flow conditions. The input parameters of the simulation are the Doppler instrument parameters and flow characteristics. This allows the simulation to be used for transducer design as well as in the study of the applicability of signal analysis techniques to Doppler ultrasound.     

Tutorial on Univariate Autoregressive Spectral Analysis

In the present paper, the theoretical basis of autoregressive (AR) modelling in spectral analysis is explained in simple terms. Spectral analysis gives information about the frequency content and sources of variation in a time series. The AR method is an alternative to discrete Fourier transform, and the method of choice for high-resolution spectral estimation of a short time series. In biomedical engineering, AR modelling is used especially in the spectral analysis of heart rate variability and electroencephalogram tracings. In AR modelling, each value of a time series is regressed on its past values. The number of past values used is called the model order. An AR model or process may be used in either process synthesis or process analysis, each of which can be regarded as a filter. The AR analysis filter divides the time series into two additive components, the predictable time series and the prediction error sequence. When the prediction error sequence has been separated from the modelled time series, the AR model can be inverted, and the prediction error sequence can be regarded as an input and the measured time series as an output to the AR synthesis filter. When a time series passes through a filter, its amplitudes of frequencies are rescaled. The properties of the AR synthesis filter are used to determine the amplitude and frequency of the different components of a time series. Heart rate variability data are here used to illustrate the method of AR spectral analysis. Some basic definitions of discrete-time signals, necessary for understanding of the content of the paper, are also presented. An erratum to this article is available at.     

Multigate doppler signal analysis using 3-D regularized long AR modelling.

Autoregressive (AR) modelling has already been proposed as an alternative to fast Fourier transform to process ultrasound (US) Doppler signals. Previous works introduced long AR models, set up under a regularization framework. The latter may be in 1-D (frequency) or 2-D (frequency and space or time). This study generalizes the spectrum regularization in the three dimensions frequency, space and time. The problem of the penalization function is addressed, and a new convex solution is proposed, taking into account possible nonstationarity of the Doppler signal. The parameter tuning is based on simulations using a standard Doppler signal model. The first results show that this processing improves the spectral estimation, and is well suited to flow interpretation.     

Stimulated acoustic emission: pseudo-Doppler shifts seen during the destruction of nonmoving microbubbles

The purpose of this study was to evaluate the appearance and the characteristics of stimulated acoustic emission (SAE) as an echo contrast-specific color Doppler phenomenon with impact on myocardial contrast echocardiography (MCE). Stationary microbubbles of the new contrast agent SH-U 563A (Schering AG) were embedded within a tissue-mimicking gel material. Harmonic power Doppler imaging (H-PDI), color Doppler and pulse-wave Doppler data were acquired using an HDI-5000 equipped with a phased-array transducer (1.67/3.3 MHz). In color Doppler mode, bubble destruction resulted in random noise like Doppler signals. PW-Doppler revealed short "pseudo-Doppler" shifts with a broadband frequency spectrum. Quantification of SAE events by H-PDI demonstrated an exponential decay of signal intensities over successive frames. A strong linear relationship was found between bubble concentration and the square root of the linearized H-PDI signal for a range of concentrations of more than two orders of magnitude (R = 0.993, p < 0.0001). Intensity of the H-PDI signals correlated well with emission power (R = 0.96, p = 0.0014). SAE results from disintegration of microbubbles and can be demonstrated by all Doppler imaging modalities, including H-PDI. Intensity of SAE signals is influenced by the applied acoustic power and correlates highly with the concentration of microbubbles. Because intensity of SAE signals correlates highly with echo contrast concentrations, analysis of SAE signals might be used for quantitative MCE.     

An autoregressive model-based method for contrast agent detection in ultrasound radiofrequency images

This paper presents a spectral autoregressive method dedicated to the detection of ultrasound contrast agents (USCA) from radiofrequency (rf) data. The method is based on second-order autoregressive (AR) modeling of the rf signal. Contrast agents induce a second harmonic, which may be efficiently detected through the AR spectrum using the magnitude of the second AR spectral peak (SM2). In contrast to multipulse methods that process two or more rf frames, our method processes a single rf frame. The method is tested by numerical simulation and on in vitro data for contrast agent concentrations ranging from 10(3) to 50 x 10(3) bubbles/ml (2 x 10(-6) to 10(-4) volumic concentration) and mechanical index (MI) ranging from 0.1 to 0.36. The results show that the proposed parameter SM2 enables one to detect correctly the contrast agent, in particular at low concentration and MI (the minimum difference in SM2 between tissue and USCA is 10 dB). Furthermore, the in-vitro data demonstrates that an adapted smoothing technique reduces the variability of SM2 and provides accurate and stable segmentation of the contrast agent perfusion region.     

Forecasting Models of Emergency Department Crowding

Objectives:  The authors investigated whether models using time series methods can generate accurate short-term forecasts of emergency department (ED) bed occupancy, using traditional historical averages models as comparison.
Methods:  From July 2005 through June 2006, retrospective hourly ED bed occupancy values were collected from three tertiary care hospitals. Three models of ED bed occupancy were developed for each site: 1) hourly historical average, 2) seasonal autoregressive integrated moving average (ARIMA), and 3) sinusoidal with an autoregression (AR)-structured error term. Goodness of fits were compared using log likelihood and Akaike's Information Criterion (AIC). The accuracies of 4- and 12-hour forecasts were evaluated by comparing model forecasts to actual observed bed occupancy with root mean square (RMS) error. Sensitivity of prediction errors to model training time was evaluated, as well.
Results:  The seasonal ARIMA outperformed the historical average in complexity adjusted goodness of fit (AIC). Both AR-based models had significantly better forecast accuracy for the 4- and the 12-hour forecasts of ED bed occupancy (analysis of variance [ANOVA] p < 0.01), compared to the historical average. The AR-based models did not differ significantly from each other in their performance. Model prediction errors did not show appreciable sensitivity to model training times greater than 7 days.
Conclusions:  Both a sinusoidal model with AR-structured error term and a seasonal ARIMA model were found to robustly forecast ED bed occupancy 4 and 12 hours in advance at three different EDs, without needing data input beyond bed occupancy in the preceding hours.     

Respiratory rate extraction from pulse oximeter and electrocardiographic recordings

We present an algorithm of respiratory rate extraction using particle filter (PF), which is applicable to both photoplethysmogram (PPG) and electrocardiogram (ECG) signals. For the respiratory rate estimation, 1 min data are analyzed with combination of a PF method and an autoregressive model where among the resultant coefficients, the corresponding pole angle with the highest magnitude is searched since this reflects the closest approximation of the true breathing rate. The PPG data were collected from 15 subjects with the metronome breathing rate ranging from 24 to 36 breaths per minute in the supine and upright positions. The ECG data were collected from 11 subjects with spontaneous breathing ranging from 36 to 60 breaths per minute during treadmill exercises. Our method was able to accurately extract respiratory rates for both metronome and spontaneous breathing even during strenuous exercises. More importantly, despite slow increases in breathing rates concomitant with greater exercise vigor with time, our method was able to accurately track these progressive increases in respiratory rates. We quantified the accuracy of our method by using the mean, standard deviation and interquartile range of the error rates which all reflected high accuracy in estimating the true breathing rates. We are not aware of any other algorithms that are able to provide accurate respiratory rates directly from either ECG signals or PPG signals with spontaneous breathing during strenuous exercises. Our method is near real-time realizable because the computational time on 1 min data segment takes only 10 ms on a 2.66 GHz Intel Core2 microprocessor; the data are subsequently shifted every 10 s to obtain near-continuous breathing rates. This is an attractive feature since most other techniques require offline data analyses to estimate breathing rates.     

Optimal design of multi-subject blocked fMRI experiments

The design of a multi-subject fMRI experiment needs specification of the number of subjects and scanning time per subject. For example, for a blocked design with conditions A or B, fixed block length and block order ABN, where N denotes a null block, the optimal number of cycles of ABN and the optimal number of subjects have to be determined. This paper presents a method to determine the optimal number of subjects and optimal number of cycles for a blocked design based on the A-optimality criterion and a linear cost function by which the number of cycles and the number of subjects are restricted. Estimation of individual stimulus effects and estimation of contrasts between stimulus effects are both considered. The mixed-effects model is applied and analytical results for the A-optimal number of subjects and A-optimal number of cycles are obtained under the assumption of uncorrelated errors. For correlated errors with a first-order autoregressive (AR1) error structure, numerical results are presented. Our results show how the optimal number of cycles and subjects depend on the within- to between-subject variance ratio. Our method is a new approach to determine the optimal scanning time and optimal number of subjects for a multi-subject fMRI experiment. In contrast to previous results based on power analyses, the optimal number of cycles and subjects can be described analytically and costs are considered.     

Reduced-order autoregressive modeling for center-frequency estimation

The center frequency of a narrowband, discrete-time random process, such as a reflected ultrasound signal, is estimated from the parameter values of a reduced, second-order autoregressive (AR) model. This approach is proposed as a fast estimator that performs better than the zero-crossing count estimate for determining the center-frequency location. The parameter values are obtained through a linear prediction analysis on the correlated random process, which in this case is identical to the maximum entropy method for spectral estimation. The frequency of the maximum of the second-order model spectrum is determined from these parameters and is used as the center-frequency estimate. This estimate can be computed very efficiently, requiring only the estimates of the first three terms of the process autocorrelation function. The bias and variance properties of this estimator are determined for a random process having a Gaussian-shaped spectrum and compared to those of the ideal FM frequency discriminator, zero-crossing count estimator and a correlation estimator. It is found that the variance values for the reduced-order AR model center-frequency estimator lie between those for the ideal FM frequency discriminator and the zero-crossing count estimator.     

Adaptive,autoregressive spectral estimation for analysis of electrical signals of gastric origin

The electrical activity of the human stomach, which normally shows a frequency of about 0.05 Hz, may be studied non-invasively by either cutaneous electrogastrography (EGG) or surface magnetogastrography (MGG). Detection of changes in frequency with time may be useful to characterize gastric disorders. The fast Fourier transform (FFT) has been the most commonly used method for the automated spectral analysis of the signals obtained from the EGG or the MGG. We have used an autoregressive (AR) parametric spectrum estimator to analyse simulated signals of gastric electrical activity, and to evaluate the results of human studies using EGG and MGG. In comparison with the FFT, our results showed that the AR spectrum estimator provided more detailed qualitative information about frequency variations of short duration simulated signals than the FFT. In the human studies, the AR estimator was as good as the conventional FFT methods in detecting physiological changes in frequency and in identifying abnormal recordings. We conclude that the AR spectral estimator may provide a better qualitative analysis of frequency variations in small portions of the signal, and is as useful as the FFT to analyse human EGG or MGG studies.