Adam G. Polak

A forward model for maximum expiration

The maximum expiratory flow-volume (MEFV) curve is a sensitive test of respiratory mechanics. Several mathematical models for forced expiration have been developed, but they suffer from various shortcomings. It is impossible to calculate the parts of the MEFV curve beyond the flow limiting conditions and computational algorithms do not allow a direct calculation of maximal flow. In the present work a complex, nonlinear forward model, including exciting signal and static recoil pressure-lung volume descriptions and 132 parameters, has been constructed. Direct determination of maximum flow is achieved by means of successive approximation algorithm. The model enables prediction of flow during forced expiration in the whole range of forced vital capacity.

Preliminary study on the accuracy of respiratory input impedance measurement using the interrupter technique

Respiratory input impedance contains information about the state of pulmonary mechanics in the frequency domain. In this paper the possibility of respiratory impedance measurement by interrupter technique as well as the accuracy of this approach are assessed. Transient states of flow and pressure recorded during expiratory flow interruption are simulated with a complex, linear model for the respiratory system and then used to calculate the impedance, including three states of respiratory mechanics and the influence of the measurement noise. The results of computations are compared to the known, theoretical impedance of the model. At 1 kHz sampling rate, the optimal time window lays between 100 and 200 ms and is centred around the pressure jump caused by the flow interruption. The proposed algorithm yields satisfactory accuracy in the range from 10 to 400 Hz, particularly to 150 Hz. Depending on the simulated respiratory system state, the error of calculated impedance (relative Euclidean distance between the vectors of computed and theoretical values), for the window of 190 ms, varies between 5.0% and 7.1%.

A model-based method for flow limitation analysis in the heterogeneous human lung

Flow limitation in the airways is a fundamental process constituting the maximal expiratory flow-volume curve. Its location is referred to as the choke point. In this work, expressions enabling the calculation of critical flows in the case of wave-speed, turbulent or viscous limitation were derived. Then a computational model for the forced expiration from the heterogeneous lung was used to analyse the regime and degree of flow limitation as well as movement and arrangement of the choke points. The conclusion is that flow limitation begins at similar time in every branch of the bronchial tree developing a parallel arrangement of the choke points. A serial configuration of flow-limiting sites is possible for short time periods in the case of increased airway heterogeneity. The most probable locations of choke points are the regions of airway junctions. The wave-speed mechanism is responsible for flow choking over most of vital capacity and viscous dissipation of pressure for the last part of the test. Turbulent dissipation, however, may play a significant role as a supporting factor in transition between wave-speed and viscous flow limitation.

Analysis of multiple linear regression algorithms used for respiratory mechanics monitoring during artificial ventilation

Many patients undergo long-term artificial ventilation and their respiratory system mechanics should be monitored to detect changes in the patients state and to optimize ventilator settings. In this work the most popular algorithms for tracking variations of respiratory resistance (R(rs)) and elastance (E(rs)) over a ventilatory cycle were analysed in terms of systematic and random errors. Additionally, a new approach was proposed and compared to the previous ones. It takes into account an exact description of flow integration by volume-dependent lung compliance. The results of analyses showed advantages of this new approach and enabled to form several suggestions. Algorithms including R(rs) and E(rs) dependencies on airflow and lung volume can be effectively applied only at low levels of noise present in measurement data, otherwise the use of the simplest model with constant parameters is preferable. Additionally, one should avoid including the resistance dependence on airflow alone, since this considerably destroys the retrieved trace of R(rs). Finally, the estimated cyclic trajectories of R(rs) and E(rs) are more sensitive to noise present in pressure than in the flow signal, and the elastance traces are estimated more accurately than the resistance ones.

An Error-Minimizing Approach to Regularization in Indirect Measurements

Indirect measurements often amount to the estimation of the parameters of a mathematical model that describes the object under investigation, and this process may numerically be ill conditioned. Various regularization techniques are used to solve the problem. This paper shows that popular regularization methods can be depicted as special cases of a generalized approach based on a penalty term in the minimized criterion function and how different kinds of a priori knowledge can be engaged into each of them. A new function, which depends on the estimate bias and variance, is proposed to find a regularization parameter that minimizes the error of estimation, as well as a novel approach for nonlinear estimation that results in the iterative minimization (IM) method. The superiority of IM with respect to the conventional Marquardt procedure is demonstrated. Based on analysis, it also follows that the regularization technique can be used even in the case of numerically well-conditioned indirect measurements, decreasing the total error of estimation.

A metrological model for maximum expiration

A metrological model for maximum expiration formulated by means of complex model reduction and selection of a parameter set for estimation was the purpose of this investigation. Reduction of the model was based on the forward-inverse method. Carrying out regression analysis showed that the properties of the airways were well approximated by exponential functions. The cross-sectional areas, airway lengths and compliances were replaced by these exponential relationships. Some other dependencies between mechanical quantities were also included. This reduced the number of parameters from 132 to 12. The complex and reduced models for maximal expiration were implemented on a PC and compared. The sensitivity index expressing the effect of parameter reduction was 3.2%. The selection of parameters for estimation was based on sensitivity analysis. This analysis showed that the reduced model was insensitive or weakly sensitive to eight parameters. This allowed for the selection of four parameters that influence the shape of the maximal expiration curve more than the others.

Measurement of Respiratory Input Impedance Using the Interrupter Technique – A Model Study

Abstract Respiratory input impedance contains information about the state of pulmonary mechanics in the frequency domain. In this paper the possibility of respiratory impedance measurement by interrupter technique as well as the accuracy of this approach are assessed. Transient states of flow and pressure recorded during expiratory flow interruption are simulated with a complex model for the respiratory system and then used to calculate the impedance. The results of computations are compared to the known impedance of the model, showing that the proposed algorithm yields satisfactory accuracy from 10 to 400 Hz, particularly in the range between 20 and 150 Hz.

Selection of identifiable parameters from the reduced model for forced expiration

Forced expiration is the most common maneuver applied to test lung function. It has been shown that the registered maximum expiratory flow-volume (MEFV) curve is effort-independent and simultaneously sensitive to respiratory disorders. The aforementioned link between the respiratory mechanics and the shape of the MEFV curve suggests a question if one is able to estimate lung parameters from the flow-volume data. Early studies exposed main difficulties following a huge number of parameters of a computational model, and finally failed in solving the problem. Recently, a reduced model for the forced expiration with 16 free parameters has been proposed. The aim of this study is to choose a set of identifiable parameters from this reduced model. We apply a procedure starting from the analysis of the estimate sensitivity to measurement data and then completing the selection of parameters based on the computed sensitivity vectors and taking into account a trade-off between random and systematic errors of estimation. In effect, a mathematical model for the forced expiration with 9 identifiable parameters is proposed. This set includes the parameters of scaling functions used in the reduced model as well as properties of lung tissue. The parameters should be estimated from the descending part of the MEFV curve. We conclude also that the MEFV curve is insensitive to a shift of the lung recoil characteristics, and consequently, residual volume of the lung cannot be deduced from the forced expiratory data.

REDUCED MODEL FOR FORCED EXPIRATION AND ANALYSIS OF ITS SENSITIVITY

Abstract Pathological changes in the lung modify the shape of the flow-volume curve registered during forced expiration. Computational models allowing simulation of the test results are too complex for the estimation of their parameters. In this study a complex model was reduced by introduction of the functions scaling airway properties. Influence of the individual parameters of the reduced model on the flow-volume curve was evaluated by means of the sensitivity analysis. The conclusion is that the parameters of the scaling functions and elastic properties of lung tissue affect the measured data most significantly and that the descending part of the curve should be used to assess them.

Optimization of Recursive Algorithms for Respiratory Mechanics Monitoring during Artificial Ventilation

Many patients undergo long-term artificial ventilation and their respiratory system mechanics should be monitored to detect changes in the patient’s state and to optimize ventilator settings. In this work two recursive algorithms for tracking respiratory resistance and elastance over ventilatory cycles, i.e. the recursive least squares and the Kalman filter, have been analyzed using the forward-inverse modeling approach. The results show that the optimal values of algorithm parameters, minimizing the error of estimation, reveal a bimodal character. In effect the algorithms can be used with similar success to monitor both the long-time and intracyclic variations of respiratory mechanics. The optimal values of these parameter depend however on the noise level present in measured signals of ventilatory flow and pressure.