Optimal selection of frequencies for estimating parameters fromrespiratory impedance data

The goal of this study was to evaluate whether optimal selection of a reduced number of frequency points would still result in statistically reliable parameter estimates. A direct-search technique is described which optimally places a small number of frequencies so that the volume of the parameter joint confidence region is minimized. The accuracy of the parameters estimated from a full data set (50 evenly spaced points) is compared to that achievable with optimal designs using 20, 10, or 5 frequency points. The techniques were applied to parameters obtained from healthy dogs and humans. Results indicated that with ten optimally chosen frequencies most parameter uncertainties are only slightly higher than that achievable with 50 frequencies while parameter uncertainties increase greatly when only five optimal points are used. This suggests that the technique of forced oscillation permits identification of the distribution of respiratory system properties without the need for extensive data acquisition     

Estimating Respiratory Mechanical Parameters in Parallel Compartment Models

Four iterative parameter estimation algorithms were used to obtain estimates in three parallel compartment models of the respiratory system. The stability of the parameter estimates and the agreement between the forced random noise impedance data and the model's response were evaluated for each algorithm-model combination. The combination of a two-stage simplex algorithm with a five element model provided the most stable parameter estimates and the second best fit to the data.     

Parameter Estipiates in a Five-Element Respiratory Mechanical Model

Using four sets of forced random noise impedance data from each of five normal subjects and five patients with obstructuve lung disease, we computed parameter estimates for a three-element series model and a five-element parallel compartment model. For normal subjects, the five-element model provided no better fit to the impedance data than did the simple series model. Estimates obtained from normal subjects using this three-element model were reasonable and reproducible within 25 percent. For all subjects with lung disease, the five-element model provided a significantly (p 0.05) better fit than the three-element model. Estimates for parameters representing central inertance and resistance, airway compliance, and peripheral resistance were reasonable and reproducible-within 18 percent. However, estimates for the compliance of the lung and chest wall were more variable since measured impedance appeared to be insensitive to this parameter in the frequency range used.     

Confidence bounds on respiratory mechanical properties estimatedfrom transfer versus input impedance in humans versus dogs

Using parameters typical of a dog, we have shown that estimates for the parameters in the six-element model of Dubois et al. would be very unreliable if either input (Z(in)) or transfer (Ztr) data from only 2-32 Hz were fit. It has subsequently been shown that this model is not appropriate for human Z(in) from 2-320 Hz. However, several studies have continued to apply the model to human Ztr data from only 2-32 Hz. In this study a sensitivity analysis is used to determine whether and why the six-element model could be applicable to lower frequency (less than 64 Hz) Ztr data in humans, but not Z(in) data over any frequency range. We first predicted the joint parameter uncertainty bounds assuming a fit to either 2-32 Hz Z(in) or Ztr data created from literature based mean parameter values. Consistent with previous studies, we predicted that the estimates will be very unreliable if obtained from Z(in) data for humans or dogs, or from Ztr data from dogs. Surprisingly, however, the reliability of several parameter estimates from human Ztr data from only 2-32 Hz are reasonable. We next evaluated the variability in 2-64 Hz based Ztr parameter estimates by comparing experimental variability in two healthy human subjects (over 10 and 13 trials) to theoretical and Monte Carlo numerical predictions based on a single trial. Again, the Ztr parameters were reliable. A simulation study was used to describe the reasons for enhanced reliability when using human Ztr data. It is shown that this reliability is largely dependent on alveolar gas compressibility, Cg.(ABSTRACT TRUNCATED AT 250 WORDS)     

A nonstatistical approach to estimating confidence intervals aboutmodel parameters: application to respiratory mechanics

Estimates of parameters obtained by fitting models to physiologic data are of little use unless accompanied by confidence intervals. The standard methods for estimating confidence intervals are statistical, and make the assumption that the fitted model accounts for all the deterministic variation in the data while the residuals between the fitted model and the data reflect only stochastic noise. In practice, this is frequently not the case, as one often finds the residuals to be systematically distributed about zero. In this paper, we develop an approach for assessing confidence in a parameter estimate when the order of the model is clearly less than that of the system being modeled. Our approach does not require a parameter to have a single value located within a region of confidence. Instead, we let the parameter value vary over the data set in such a way as to provide a good fit to the entire data set. We apply our approach to the estimation of the resistance of the respiratory system in which a simple model is fitted to measurements of tracheal pressure and flow by recursive multiple linear regression. The values of resistance required to achieve a good fit are represented as a modified histogram in which the contribution of a particular resistance value to the histogram is weighted by the amount of information used in its determination. Our approach provides parameter frequency distribution functions that convey the degree of confidence one may have in the parameter, while not being based on erroneous statistical assumptions.     

Standard errors on respiratory mechanical parameters obtained by forced random excitation

Equations describing the standard errors of forced random impedance data and derived parameters in terms of various data collection and data processing factors were developed and verified. The equation indicate that to obtain reliable estimates: 1) 16 ensembles are adequate when coherence is greater than 0.9, and that 32 ensembles are adequate when the coherence is between 0.8 and 0.9; 2) the impedance of the bias tube should be at least two times the impedance of the respiratory system in the bandwidth of the applied noise; and 3) the spectrum should include at least 20 frequencies with at least 2 and preferably more below 10 Hz. Fortunately, all of these constraints can be satisfied with most subjects. This analysis also provides a basis for using weighted regression in estimating resistance, inertance, and compliance parameters, and for separating observed parameter variability into methodological and physiological components.     

Improved parameter estimation by noise compensation in the time-scale domain

It was shown recently that parameter estimation can be performed directly in the time-scale domain by isolating regions wherein the prediction error can be attributed to the error of individual dynamic model parameters [1]. Based on these single-parameter equations of the prediction error, individual model parameters error can be estimated for iterative parameter estimation. An added benefit of this parameter estimation method, besides its unique convergence characteristics, is the added capacity for direct noise compensation in the time-scale domain. This paper explores this benefit by introducing a noise compensation method that estimates the distortion by noise of the prediction error in the time-scale domain and incorporates that as a confidence factor to bias the estimation of individual parameters error. This method is shown to improve the precision of the estimated parameters when the confidence factors accurately represent the noise distortion of the prediction error.     

Physiological interpretations based on lumped element models fit torespiratory impedance data: use of forward-inverse modeling

Respiratory impedance (Zrs) data at lower (less than 4 Hz) and higher (greater than 32 Hz) frequencies require more complicated inverse models than the standard series combination of a respiratory resistance, inertance, and compliance. In this paper, a forward-inverse modeling approach was used to provide insight on how the parameters in these more complicated inverse models reflect the true physiological system. Forward models are set up to incorporate explicit physiological and anatomical detail. Simulated forward data are then fit with identifiable inverse models and the parameter estimates related to the known detail in the forward model. It is shown that inverse fitting of low frequency data alone will not allow a distinction between frequency dependence due to airway inhomogeneities and frequency dependence due to tissue viscoelasticity. With higher frequency data, a forward model based on an asymmetric branching airways network was used to simulate Zrs from 0.1-128 Hz with increasing amounts of nonuniform peripheral airway obstruction. Here, inverse modeling is more amenable to sensibly separating estimates of airway and tissue properties. A key result, however, is that changes in the tissue parameters of an inverse model (which provides an excellent fit to Zrs data) will appropriately occur in response to inhomogeneous alterations in airway diameters only. The apparent altered tissue properties reflect the decreased communication of some tissue segments with the airway opening and not an explicit change at the tissue level. These phenomena present a substantial problem for the inverse modeler. Finally, inverse model fitting of low and high frequency Zrs data simultaneously with a single model is not helpful for extracting additional physiological detail. Instead, separate models should be applied to each frequency range.     

Endothelial cell electrical impedance parameter artifacts produced by a gold electrode and phase sensitive detection

Frequency dependent cellular micro-impedance estimates obtained from a gold two-electrode configuration using phase sensitive detection have become increasingly used to evaluate cellular barrier model parameters. The results of this study show that cellular barrier function parameter estimates optimized using measurements obtained from this biosensor are highly susceptible to both time dependent and systematic instrumental artifacts. Based on a power spectral analysis of experimentally measured microelectrode voltages, synchronous, 60 Hz, and white Gaussian noise were identified as the most significant time dependent instrumental artifacts. The reduction of these artifacts using digital filtering produced a corresponding reduction in the optimized model parameter fluctuations. Using a series of instrumental circuit models, this study also shows that electrode impedance voltage divider effects and circuit capacitances can produce systematic deviations in cellular barrier function parameter estimates. Although the implementation of an active current source reduced the voltage divider effects, artifacts produced by coaxial cable and other circuit capacitive elements at frequencies exceeding 1 kHz still remained. Reducing time dependent instrumental fluctuations and systematic errors produced a significant reduction in cellular model barrier parameter errors and improved the model fit to experimental data.     

Measurement of fractional order model parameters of respiratory mechanical impedance in total liquid ventilation

This study presents a methodology for applying the forced-oscillation technique in total liquid ventilation. It mainly consists of applying sinusoidal volumetric excitation to the respiratory system, and determining the transfer function between the delivered flow rate and resulting airway pressure. The investigated frequency range was f ∈ [0.05, 4] Hz at a constant flow amplitude of 7.5 mL/s. The five parameters of a fractional order lung model, the existing "5-parameter constant-phase model," were identified based on measured impedance spectra. The identification method was validated in silico on computer-generated datasets and the overall process was validated in vitro on a simplified single-compartment mechanical lung model. In vivo data on ten newborn lambs suggested the appropriateness of a fractional-order compliance term to the mechanical impedance to describe the low-frequency behavior of the lung, but did not demonstrate the relevance of a fractional-order inertance term. Typical respiratory system frequency response is presented together with statistical data of the measured in vivo impedance model parameters. This information will be useful for both the design of a robust pressure controller for total liquid ventilators and the monitoring of the patient's respiratory parameters during total liquid ventilation treatment.     

Closed-loop identification: application to the estimation of limb impedance in a compliant environment

The force and position data used to construct models of limb impedance are often obtained from closed-loop experiments. If the system is tested in a stiff environment, it is possible to treat the data as if they were obtained in open loop. However, when limb impedance is studied in a compliant environment, the presence of feedback cannot be ignored. While unbiased estimates of a system can be obtained directly using the prediction error method, the same cannot be said when linear regression or correlation analysis is used to fit nonparametric time- or frequency-domain models. We develop a prediction error minimization-based identification method for a nonparametric time-domain model augmented with a parametric noise model. The identification algorithm is tested on a dynamic mass-spring-damper system and returns consistent estimates of the system's properties under both stiff and compliant feedback control. The algorithm is then used to estimate the impedance of a human elbow joint in both stiff and compliant environments.     

Estimating transition-probabilities in a dynamic graphic model withunobservable variables

This paper emphasizes the utility of graphic models in describing partially observed dynamic systems, and establishes a method for estimating the parameters of the model. A dynamic graphic model with an associated graphic structure, which consists of a sequence of chain graphs with two consecutive graphs in the sequence connected by directed links, is described. The chain graphs describe relationships among the contemporaneous variables; the directed links describe the relations between noncontemporary variables. The paper assumes that some of the variables are unobservable when the model is in use, but partial observation of these variables is allowed in an estimation phase, by performing autopsies of the system: stopping the system and observing its state, including destructive observation. A recursive estimation method for the parameters is given and a simulation study evaluates its performance. In conclusion, the parameters of the model can be estimated, the RMS errors of the estimates are larger for those parameters associated with longer time (because of their dependence on previous estimates), and the larger the dependence between unobservable and observable variables, the better the parameter estimates     

Muscle-joint models incorporating activation dynamics,moment-angle, and moment-velocity properties

Muscle input/output models incorporating activation dynamics, moment-angle, and moment-velocity factors are commonly used to predict the moment produced by muscle during nonisometric contractions: the three factors are generally assumed to be independent. The authors examined the ability of models with independent factors, as well as models with coupled factors, to fit input/output data measured during simultaneous modulation of the fraction of muscle stimulated (recruitment) and joint angle inputs. The models were evaluated in stimulated cat soleus muscles producing ankle extension moment, with regard to their potential applications in neuroprostheses with either fixed parameters or parameter adaptation. Both uncoupled and coupled models predicted the output moment well for random angle perturbation sizes ranging from 10° to 30°. For the uncoupled model, the best parameter values depended on the range of perturbations and the mean angle. Introducing coupling between activation and velocity in the model reduced this parameter sensitivity; one set of model parameter values fit the data for all perturbation sizes and also fit the data under isometric or constant stimulation conditions. Thus, the coupled model would be the most appropriate for applications requiring fixed parameter values. In contrast, with continuous parameter adaptation, errors due to changing test conditions decreased more quickly for the uncoupled model, suggesting that it would perform well in adaptive control of neuroprostheses     

Maximum-likelihood versus maximum a posteriori parameter estimation of physiological system models: the C-peptide impulse response case study

Maximum-likelihood (ML), also given its connection to least-squares (LS), is widely adopted in parameter estimation of physiological system models, i.e., assigning numerical values to the unknown model parameters from the experimental data. A more sophisticated but less used approach is maximum a posteriori (MAP) estimation. Conceptually, while ML adopts a Fisherian approach, i.e., only experimental measurements are supplied to the estimator, MAP estimation is a Bayesian approach, i.e., a priori available statistical information on the unknown parameters is also exploited for their estimation. In this paper, after a brief review of the theory behind ML and MAP estimators, we compare their performance in the solution of a case study concerning the determination of the parameters of a sum of exponential model which describes the impulse response of C-peptide (CP), a key substance for reconstructing insulin secretion. The results show that MAP estimation always leads to parameter estimates with a precision (sometimes significantly) higher than that obtained through ML, at the cost of only a slightly worse fit. Thus, a three exponential model can be adopted to describe the CP impulse response model in place of the two exponential model usually identified in the literature by the ML/LS approach. Simulated case studies are also reported to evidence the importance of taking into account a priori information in a data poor situation, e.g., when a few or too noisy measurements are available. In conclusion, our results show that, when a priori information on the unknown model parameters is available, Bayes estimation can be of relevant interest, since it can significantly improve the precision of parameter estimates with respect to Fisher estimation. This may also allow the adoption of more complex models than those determinable by a Fisherian approach.     

Estimating lung mechanics of dogs with unilateral lung injury

Extended least-squares algorithms using transpulmonary pressure and airway flow data from ventilatory waveforms were studied for their ability to track parameters of one- and two-compartment models of lung mechanics. A recursive extended least-squares algorithm with discounted measures estimated parameters of discrete-time models during synchronized intermittent mandatory ventilation. In tests on seven dogs developing oleic acid-induced unilateral hemorrhagic pulmonary edema, the one-compartment estimator responded rapidly and appropriately to changes in mechanics: compliance fell to 0.55 +/- 0.15 of its initial value and resistance rose by a factor of 1.8 +/- 0.5 in 3 h following injection of oleic acid. One-compartment parameter estimates revealed a difference between the airway resistance of inspiration and expiration. Two-compartment estimates were seldom physiologically plausible. The difference between inspiratory and expiratory resistance may have caused the two-compartment estimator to fail when applied to data from the entire respiratory cycle; when only expiratory data were used for estimation, the two-compartment estimates were meaningful. These estimates demonstrated increasing lung inhomogeneity after oleic acid was injected; at the end of 3 h, the ratio of the time constants of the two compartments ranged from 5 to 20 in six of the seven dogs. We conclude that the one- and two-compartment estimates may be combined to provide a meaningful assessment of lung mechanics.     

Joint Estimation of States and Parameters for an Input Nonlinear State-Space System with Colored Noise Using the Filtering Technique

This paper concerns the state and parameter estimation problem for an input nonlinear state-space system with colored noise. By using the data filtering and the over-parameterization technique, we transform the original nonlinear state-space system into two identification models with filtered states: one containing the system parameters and the other containing the noise model’s parameters. A combined state and parameter estimation algorithm is developed for identifying the state-space system. The key is that the estimation of system parameters uses the estimated states, and the estimation of states uses the preceding parameter estimates. A simulation example is provided to show that the proposed algorithm can work well.     

Frequency domain accuracy of identified 2-D causal AR-models

The author studies parametric estimation of 2-D autoregressive models using the least squares method. The analysis is concentrated on the frequency domain accuracy of the estimated models. First results for the accuracy of the parameter estimates are discussed. The estimates are asymptotically Gaussian distributed. The variance of the estimated model evaluated in the frequency domain can be expressed using these results for the parameters. This, however, gives no insight of the dependence on the true transfer function. An illuminating result is obtained if one lets the model order tend to infinity. The limiting results show good correspondence with Monte-Carlo simulations even for small data sets, using low model orders     

An Exponential Damage Model for Strength of Fibrous Composite Materials

When modeling experimental data observed from carefully performed tensile strength tests, statistical distributions are typically used to describe the strength of composite specimens. Recently, cumulative damage models derived for predicting tensile strength have been shown to be superior to other models when used to fit composite strength data. Here, an alternative model is developed which is based on an exponential cumulative damage approach. The model is shown to exhibit a similar structural form to the other models in the literature so that previous theory for cumulative damage models can be utilized to find parameter estimates.     

Frequency-dependent mutual resistance and inductance formulas for coupled IC interconnects on an Si-SiO2 substrate

A highly accurate closed-form approximation of frequency-dependent mutual impedance per unit length of a lossy silicon substrate coplanar-strip IC interconnects is developed. The derivation is based on a quasi-stationary full-wave analysis and Fourier integral transformation. The derivation shows the mathematical approximations which are needed in obtaining the desired expressions. As a result, for the first time, we present a new simple, yet surprisingly accurate closed-form expression which yield accurate estimates of frequency-dependent mutual resistance and inductance per unit length of coupled interconnects for a wide range of geometrical and technological parameters. The developed formulas describe the mutual line impedance behaviour over the whole frequency range ( i.e. also in the transition region between the skin effect, slow wave, and dielectric quasi-TEM modes). The results have been compared with the reported data obtained by the modified quasi-static spectral domain approach and new CAD-oriented equivalent-circuit model procedure.     

一种P3/P4多相码雷达信号参数快速估计方法

针对Wigner-Ville变换、Radon-Wigner变换估计P3/P4多相编码雷达信号参数存在运算量大、估计精度低等问题,提出了基于Radon-Ambiguity变换和分数阶傅立叶变换（FRFT）联合分析的参数快速估计方法。该方法采用Radon-Ambiguity变换估计信号调制斜率,采用分数阶傅立叶变换估计载频、周期等,通过两次一维搜索峰值来估计信号的参数,与Wigner变换、Radon-Wigner变换的二维搜索峰值相比,运算量大大降低,且提高了参数估计精度。仿真结果证明了该方法的有效性。