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.     

A rapidly converging algorithm for estimating respiratory mechanical parameters in a five-element model

A rapidly converging algorithm for computing values for respiratory mechanical parameters from forced random noise independance data was developed and verified. The algorithm, which was based on a five-element Mead-type model, minimized the sum of squared differences between the model's response and experimental data, while imposing a nonnegativity constraint on the parameter values. It yielded parameter values that showed excellent agreement with values obtained previously using standard nonlinear regression analysis, but required much less computer time, 10 s versus 1 h. When this algorithm is coupled with the forced random impedance data collection techniques, it provides a rapid noninvasive method for estimating respiratory inertance, central resistance, peripheral resistance, and airway compliance. The problem of estimating peripheral compliance was not solved by this algorithm.     

Estimation of Mechanical Parameters in Multicompartment Models Applied to Normal and Obstructed Lungs During Tidal Breathing

A technique is presented which allows quantitative assessment of the use of parallel compartment models for characterizing pulmonary mechanical function during tidal breathing. A model consisting of a conducting airway leading to two parallel parenchymal regions is used. Numerical simulation and sensitivity analysis indicated that a) the compliance of the conducting airway was not significant under the experimental conditions of interest and that b) estimates of the distribution of central and peripheral resistances would not be precise. The techniques were demonstrated using measurements of transpulmonary pressure, flow, and volume changes during tidal breathing obtained from a human subject with normal lungs and a human subject with obstructed lungs. Optimal estimates of the parameters were obtained by minimizing the difference between the model output and experimental data combined from two breathing frequencies. In the estimation procedure, the sum of the peripheral compliances was constrained to equal the independently measured static lung compliance. This constraint was critical for correct evaluation of nonuniform mechanical lung function. From the parameter estimates, the ratio of parenchymal time constants was about five in the subject with normal lungs and 60 in the subject with obstructed lungs. These results suggest that a full study with several normal and obstructed lung subjects is warranted.     

Statistical Measures of Parameter Estimates from Models Fit to Respiratory Impedance Data: Emphasis on Joint Vanabilities

To describe respiratory mechanical impedance data, many investigators have proposed electromechanical models and then fit them to data using formal parameter estimation techniques. This approach has resulted in confusion as to how to interpret the resulting estimated values, and hence as to which model is most appropriate. A key cause of this confusion is that most studies rely on the quality of fit between the model and the data as the only measure of model validity rather than performing adequate statistical analysis of the parameter estimates themselves. This paper describes several statistical measures that should be applied to parameter estimates obtained from forced oscillation data. Specifically, we describe standard errors of the parameter estimates, confidence intervals for each parameter estimate, and the joint confidence region for the parameters. Much emphasis is placed on the joint confidence region which, unlike the interval, allows for simultaneous variations in parameters. The measures are applied to an often used six-element model for respiratory impedance data of dogs from 4 to 64 Hz. This application indicated that even when fitting data over this frequency range, parameter estimates are not well defined and the parameter estimated with least accuracy is airway resistance.     

Dynamic Characteristics of the Renal Arterial Microvasculature of the Dog Obtained by Simulation

To develop a measurement technique for microvascular stiffness and resistance, an iterative algorithm (NOLESQ) combining steepest descent and Taylor series (Gauss-Newton) methods was used to estimate model parameters to provide a minimal least squares error at 5-ms intervals of the predicted blood flow to the measured flow. Renal arterial pressure and flow data of high dynamic and static accuracy were used. The data were in blocks of 2500 values each (12.5-s samples). A reasonable fit was obtained with only three parameters: arterial conductance, compliance, and a steady pressure at or just beyond the glomerulae. Other models, incorporating large artery resistance, blood mass, pressure dependent conductance, and pressure dependent compliance were studied. Criteria for acceptance were reasonable values from prior knowledge of biological materials, similarity of parameter values between consecutive samples and between animals, convergence, and error of fit reduced to the noise level of the input data.     

Limitations of Frequency Dependence as a Measure of Airway Obstruction

The purpose of this study is to develop a pulmonary model and determine the frequency response sensitivity of mechanical parameters such as impedance, dynamic compliance, and dynamic resistance as a function of individual airway properties. Computer simulations of a three compartment model of various physiological cases were used to determine lung parameters as a function of frequency, peripheral airway contribution to total airway resistance, and relative percent obstruction of the peripheral airways. Provided our present concepts of the lung are valid and adequately incorporated into the present model, our results indicate the utility of frequency dependence as a measure of airway obstruction.     

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.     

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.     

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.     

A nonlinear model of the arterial system incorporating apressure-dependent compliance

The three-element modified Windkessel model has been widely used to study the characteristics of the systemic arterial system. This model provides most of the features of the systemic input impedance, but does not describe the nonlinear effect of the pressure dependence of arterial compliance. The current investigation examines the hemodynamic consequences of such an inclusion. Simultaneous aortic pressure and flow during control and brief descending aortic occlusion were measured in open chest anesthetized experimental dogs. A numerical procedure was implemented to compute constant compliance linear and nonlinear compliance model-predicted pressure waveforms with flow as the input. Results show that the nonlinear compliance model in general can more accurately predict the measured pressure waveforms during control and during acute pressure loading. The difference between the predicted waveforms is more pronounced when blood pressure is high and when the pulse pressure is large.     

Reduced Models of Arterial Systems

Simple models that manifest input impedances of arterial systems are compared. An improvement upon documented two-, three-, and five-element models is presented. The classical two-element model (the windkessel) accounts for the lowest frequency components, and the three-element model (the modified windkessel) accounts for both low-and high-frequency components of the spectrum of interest. Five-element models, however, by allowing for reflection, can account for principal features over the entire frequency range of interest.     

Forced Oscillatory Parameters of the Canine Respiratory System with Altered Vagal Tone

Total respiratory impedance was obtained by forced oscillations ranging from 0.9 to 16 Hz in six anesthetized intubated dogs during a control period, after bilateral vagotomy, and during bilateral vagal stimulation. Values for forced oscillatory resistance, compliance, and inertance were calculated using regression analysis with a linear model. Mean values ±SD for the control period were 2.49 ±0.18 cmH2O ·L?1 ·s, 0.0254 ±0.0039 L ·cmH2O?1, and 0.0849 ±0.0055 cmH2O ·L?1·s2, respectively. These were similar to previously reported values. Vagotomy produced only small changes in these parameters. Vagal stimulation produced a 33 percent increase in resistance, a 20 percent decrease in compliance, and an 8 percent increase in inertance. Changes in resistance and compliance were consistent with reported effects in the literature. Thus, the transient mechanical changes induced by vagal stimulation can be characterized by this technique.     

Thoracic Impedance Gradient with Respect to Breathing

The transthoracic mutual-impedance responses to lung ventilation were measured for 37 normal subjects with 14 orthogonal lead systems. Spatial intravariability resulted from small errors in electrode placement on the thoracic surface. Temporal intravariability was determined by repeating the measurements for one subject on five consecutive days. Insignificant correlation coefficients were obtained between impedance sensitivity to lung ventilation and either age, height, weight, or chest-to-back distance of the subject. Intra-and intersubject variability were found to be of the same size for a given lead system. While the significant spatial intervariability can be easily correlated with local resistance changes due to lung volume shifts, the intersubject variability did not lend itself to correlation with subject somatotype variables.     

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.     

Impedance characterization and modeling of electrodes for biomedical applications

A low electrode-electrolyte impedance interface is critical in the design of electrodes for biomedical applications. To design low-impedance interfaces a complete understanding of the physical processes contributing to the impedance is required. In this work a model describing these physical processes is validated and extended to quantify the effect of organic coatings and incubation time. Electrochemical impedance spectroscopy has been used to electrically characterize the interface for various electrode materials: platinum, platinum black, and titanium nitride; and varying electrode sizes: 1 cm2, and 900 microm2. An equivalent circuit model comprising an interface capacitance, shunted by a charge transfer resistance, in series with the solution resistance has been fitted to the experimental results. Theoretical equations have been used to calculate the interface capacitance impedance and the solution resistance, yielding results that correspond well with the fitted parameter values, thereby confirming the validity of the equations. The effect of incubation time, and two organic cell-adhesion promoting coatings, poly-L-lysine and laminin, on the interface impedance has been quantified using the model. This demonstrates the benefits of using this model in developing better understanding of the physical processes occurring at the interface in more complex, biomedically relevant situations.     

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)     

Left-ventricular dynamic model based on constant ejection flowperiods

Experiments with constant ejection flow periods on the rabbit left ventricle suggest that left ventricular pressure can be described by a time varying three-element model consisting of elastance Ee(t), resistance R(t), and series-elastance Es(t). Previous experiments demonstrated the existence of a "deactivation effect" after the cessation of a constant ejection flow period, which could be described by a decrease of elastance Ee(t). This paper presents a simulation model based on findings of constant ejection flow experiments, and tested on measured pressure and volume data. The results show that when the model is fitted on one single beat, left ventricular pressure can satisfactorily be described by a three-element model without deactivation. However, the model does not predict isovolumic pressure at end-ejection volume. When isovolumic pressure has to be described by the model as well, introduction of deactivation is necessary. The quality of the model was further tested by fitting it to two beats with different ejection parameters. Deactivation again was necessary for a good fit. Only with a deactivation effect in the model, the component values found are close to the normal range found with CFP experiments in the rabbit left ventricles. From the simulation results it can be concluded that (at least for constant ejection flow periods) elastance, resistance, series-elastance, and deactivation effects all are necessary in describing (and predicting) left ventricular pressure.     

一种新的七元联合稀疏型表示及其应用

为了进一步提高椭圆曲线密码体制中k1P+k2Q的计算效率,该文提出了一种新的七元联合稀疏型。对任一整数对,给出了新七元联合稀疏型的定义和算法,证明了新七元联合稀疏型的唯一性,并证明了新七元联合稀疏型的平均联合Hamming密度约为0.3023。采用新七元联合稀疏型计算k1P+k2Q时,比最优三元联合稀疏型减少了0.1977l次点加运算,比一种五元联合稀疏型减少了0.031l次点加运算,比另一种七元联合稀疏型减少了0.0392l次点加运算。