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.     

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)     

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.     

A Nonlinear Model Combining Pulmonary Mechanics and Gas Concentration Dynamics

To elucidate the various mechanisms by which pulmonary mechanics affect the distribution of gas species throughout the lungs, a multicompartment model relating pressure differences, flows, volumes, and gas species concentrations has been developed. The alveolar regions of the model are nonlinearly elastic and the pressure-flow relation of their associated small airways is volume dependent. Various combinations of parameter values were chosen, including cases in which the model was mechanically uniform (normal) and nonuniform (obstructive). Computer solutions of model equations were obtained for both piecewise-exponential and sinusoidal transpulmonary pressure inputs. Clinical measures of mechanical uniformity and gas concentration homogeneity were evaluated along with unobservable indexes. Results indicate how the distribution of mechanical variables affects the distribution of gas species concentration within the lungs. For the nonuniform (obstructive) model, the gas is distributed more inhomogeneously at higher frequencies and lower lung volumes. The distribution of initial dead space gas to the compartments as well as pendelluft tend to decrease this inhomogeneity. Dynamic compliance for the non-uniform model was frequency dependent at each of the three volume operating points investigated, whereas the semilog nitrogen washout curve was essentially linear for some frequencies and volumes while nonlinear for others. Consequently, inferences about distributions of mechanical parameters and intrapulmonary gas may require that clinical measurements be obtained together at several frequencies and volume operating points.     

How inhomogeneities and airway walls affect frequency dependence and separation of airway and tissue properties

It has been proposed that during mild-to-moderate bronchoconstriction one can partition airway and tissue properties on the basis of input impedance (Zin) acquired from 0.1 to 5 Hz (K.R. Lutchen, B. Suki, Q. Zhang, F. Peták, B. Daróczy, and Z. Hantos. J. Appl. Physiol. 77: 373-385, 1994). The approach is to apply a homogeneous lung model that contains airway resistance and viscoelastic tissue damping and elastance parameters. The tissue parameters account for the frequency dependence in lung resistance (RL) and elastance (EL). We present an anatomically consistent asymmetrically branching airway model to address two key questions: 1) How will lung inhomogeneities, airway wall shunting, and tissue viscoelasticity contribute to increased frequency dependence and levels of RL and EL during lung constriction? and 2) How much can lung inhomogeneities and airway wall shunting contribute to our assessment of airway, tissue, and overall lung properties derived from Zin? The model incorporates nonrigid airway walls and allows for explicit control over the type and degree of inhomogeneous airway constriction or tissue changes. Our results indicate that, from 0.1 to 5 Hz, airway wall shunting does not become important unless the entire lung periphery experiences significant constriction. Mild-to-moderate inhomogeneous peripheral airway constriction produces a relatively minor additional frequency dependence in RL and EL beyond that due to the tissues alone. With more extreme constriction, however, there is a marked frequency-dependent increase in EL. This phenomenon may render it impossible to distinguish from a single frequency measurement whether an increase in EL during bronchoconstriction is a consequence of a true increase in tissue stiffening or simply a consequence of airway phenomena. Finally, Zin from 0.1 to 5 Hz can be used to provide a reasonable separation of airway and tissue properties for mild-to-moderate homogeneous or inhomogeneous lung constriction. However, during more severe disease, inhomogeneities and/or wall shunting will produce substantial overestimation of tissue damping and hysteretic properties. In fact, the only reliable indicator of a real change in the tissues may be a change in the estimate of tissue elastance that is based on data extending to a sufficiently low frequency.     

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     

Importance of low-frequency impedance data for reliably quantifying parallel inhomogeneities of respiratory mechanics

The ability to reliably measure total respiratory input impedance Z/sup rs/ from 0.25 to 4 Hz has only recently been reported and only in healthy subjects. The real part of Z/sup rs/ decreased substantially with frequency. One explanation is provided by the Otis model, which contains parallel resistance-compliance time-constant inhomogeneities. Several investigators have suggested the use of this model at the level of estimating its parameters by fitting the model to data. Such an approach would permit quantification of the functional inhomogeneity of an individual's respiratory system and may be useful diagnostically. In this study, experimental data and a sensitivity analysis are combined to specify the requirements and limitations associated with estimating the parameters. The data acquisition technique was improved to acquire Z/sup rs/ as low as 0.125 Hz in seven healthy subjects. The Otis model provided an excellent fit to the data with reasonably low intra- and intersubject variability.<>     

Assessment of time-domain analyses for estimation of low-frequency respiratory mechanical properties and impedance spectra

Time-domain estimation has been invoked for tracking of respiratory mechanical properties using primarily a simple single-compartment model containing a series resistance (Rrs) and elastance (Ers). However, owing to the viscoelastic properties of respiratory tissues, Rrs and Ers exhibit frequency dependence below 2 Hz. The goal of this study was to investigate the bias and statistical accuracy of various time-domain approaches with respect to model properties, as well as the estimated impedance spectra. Particular emphasis was placed on establishing the tracking capability using a standard step ventilation. A simulation study compared continuous-time versus discrete-time approaches for both the single-compartment and two-compartment models. Data were acquired in four healthy humans and two dogs before and after induced severe pulmonary edema while applying sinusoidal and standard ventilator forcing. Rrs and Ers were estimated either by the standard Fast Fourier Transform (FFT) approach or by a time-domain least square estimation. Results show that the continuous-time model form produced the least bias and smallest parameter uncertainty for a single-compartment analysis and is quite amenable for reliable on-line tracking. The discrete-time approach exhibits large uncertainty and bias, particularly with increasing noise in the flow data. In humans, the time-domain approach produced smooth estimates of Rrs and Ers spectra, but they were statistically unreliable at the lower frequencies. In dogs, both the FFT and time-domain analysis produced reliable and stable estimates for Rrs or Ers spectra for frequencies out to 2 Hz in all conditions. Nevertheless, obtaining stable on-line parameter estimates for the two-compartment viscoelastic models remained difficult. We conclude that time-domain analysis of respiratory mechanics should invoke a continuous-time model form.     

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.