Analytical solution to Windkessel models using piecewise linear aortic flow waveform.

Objective.Deriving time-domain analytical solutions to two- three- and four-element Windkessel models, which are commonly used in teaching and research to analyse the behaviour of the arterial pressure-flow relationship.Approach.The governing (first-order, non-homogeneous, linear) differential equations are solved analytically, based on a piecewise linear function that can accurately approximate typical aortic flow waveforms.Main results.Closed-form expressions for arterial pressure are obtained both in transient conditions and in steady-state periodic regime.Significance.In most cases Windkessel models are studied in the frequency domain and when studied in the time domain, numerical methods are used. The main advantage of the proposed expressions is that they are an explicit, exact, and easily understood mathematical description of the model behaviour. Moreover, they avoid the use of Fourier analysis or numerical solvers to integrate the differential equations.

A functional mathematical model to simulate the single-breath nitrogen washout.

A nonlinear dynamic model is proposed to reproduce and interpret the influence of pulmonary inhomogeneities on the single-breath nitrogen washout (SBNW) curve. The model is characterized by two parallel zones. In each zone, the upper airways are described by a Rohrer resistor. Intermediate airways are represented as a collapsible segment, the volume of which depends on transmural pressure. Smaller airways are described by a resistance which increases when transpulmonary pressure decreases. The respiratory region is modeled as a Voigt element. Three different conditions were simulated: a reference case, characterized by airway-parameter values for normal conditions, and two pathological states corresponding to different levels of disease. In the reference case, a straight line was a good approximation of SBNW phase III and the last point of departure of the nitrogen trace from this line unambiguously identified the onset of phase IV. The slope of phase III rose with disease severity (from a 1.1% increase in nitrogen concentration per 1000 ml of expired volume in the reference case to 3.6% and 7.7% in the pathological cases) and the distinction between phases III and IV became less evident. The results obtained indicate that the slope of phase III depends primarily on nitrogen-concentration differences between lung zones, as determined by different mechanical properties of the respiratory airways. In spite of the simplified representation of the lungs, the similarity of the simulation results to actual data suggests that the proposed model describes important physiological mechanisms underlying changes observed during SBNW in normal and pathological patients.

An EMG-driven model applied for predicting metabolic energy consumption during movement.

The relationship between mechanical work and metabolic energy cost during movement is not yet clear. Many studies demonstrated the utility of forward-dynamic musculoskeletal models combined with experimental data to address such question. The aim of this study was to evaluate the applicability of a muscle energy expenditure model at whole body level, using an EMG-driven approach. Four participants performed a 5-min squat exercise on unilateral leg press at two different frequencies and two load levels. Data collected were kinematics, EMG, forces and moments under the foot and gas-exchange data. This same task was simulated using a musculoskeletal model, which took EMG and kinematics as inputs and gave muscle forces and muscle energetics as outputs. Model parameters were taken from literature, but maximal isometric muscle force was optimized in order to match predicted joint moments with measured ones. Energy rates predicted by the model were compared with energy consumption measured by the gas-exchange data. Model results on metabolic energy consumption were close to the values obtained through indirect calorimetry. At the higher frequency level, the model underestimated measured energy consumption. This underestimation can be explained with an increase in energy consumption of the non-muscular mass with movement velocity. In conclusion, results obtained in comparing model predictions with experimental data were promising. More research is needed to evaluate this way of computing mechanical and metabolic work.

Automatic detection of maximal oxygen uptake and ventilatory threshold.

Maximal oxygen uptake (VO(2max)) and ventilatory threshold (VT) are the most common measurements in exercise physiology laboratories for the objective characterization of the physiologic state of metabolic and respiratory systems. Several techniques for their identification were proposed in the literature: the aim of the present study was to review them and assess their performance when applied to experimental data. In the present study, the criteria to detect VO(2max) and VT from respiratory gas-exchange data were analysed and automatic procedures for the identification of these parameters were implemented. These procedures were then applied to experimental data in order to assess the verifiability, repeatability and sensitivity to measurement noise of each proposed method. The results suggest plateau- and RISE-105- as the most reliable automatic procedures for determining VO(2max), while respiratory exchange ratio-, ventilatory equivalent for O(2)- and P(ET,O2)-criteria appear to be the most reliable automatic procedures for estimating VT.

Estimation of expiratory resistance: a simulation study.

It is of particular importance to detect and quantify obstructive pathological conditions in mechanically ventilated patients, especially in the presence of expiratory flow limitation (EFL), in order to help the clinicians in the choice of the most appropriate ventilation and pharmacological strategies. Aim of this work is to test by simulation a non invasive procedure for estimating the total apparent expiratory resistance of the respiratory system (Rtae). The proposed procedure is based on a time-varying two-element viscoelastic model characterized by the variable resistance Rtae and by a constant compliance estimated by the end-inspiratory occlusion technique. A non linear, dynamic, morphometric model of respiratory mechanics, based on Weibel's representation of the tracheobronchial tree, was used to simulate normal and obstructive respiratory conditions, leading to EFL, during artificial ventilation. The proposed resistance was computed in all simulated cases when the 50% and the 75% of tidal volume has been exhaled during a normal expiration. Rtae appeared to be dependent on the degree of airway obstruction and could provide useful information on how the airway compression varies during expiration.

Helium-oxygen ventilation in the presence of expiratory flow-limitation: a model study.

A comparison between air and heliox (80% helium-20% oxygen) ventilation was performed using a mathematical, non-linear dynamic, morphometric model of the respiratory system. Different obstructive conditions, all causing expiratory flow limitation (EFL), were simulated during mechanical ventilation to evaluate and interpret the effects of heliox on tidal EFL and dynamic hyperinflation. Relative to air ventilation, intrinsic positive end-expiratory pressure did not change with heliox if the obstruction was limited to the peripheral airways, i.e. beyond the seventh generation. When central airways were also involved, heliox reduced dynamic hyperinflation (DH) if the flow-limiting segment remained in the fourth to seventh airway generation during the whole expiration, but produced only minor effects if, depending on the contribution of peripheral to total apparent airway resistance, the flow-limiting segment moved eventually to the peripheral airways. In no case did heliox abolish EFL occurring with air ventilation, indicating that any increase in driving pressure would be without effect on DH. Hence, to the extent that chronic obstructive pulmonary disease (COPD) affects primarily the peripheral airways, and causes EFL through the same mechanisms operating in the model, heliox administration should not be expected to appreciably reduce DH in the majority of COPD patients who are flow-limited at rest.

A simulation study of expiratory flow limitation in obstructive patients during mechanical ventilation.

Although normal lungs may be represented satisfactorily by symmetrical architecture, pathological conditions generally require accounting for asymmetrical branching of the bronchial tree, since lung heterogeneity may be significant in respiratory diseases. In the present study, a recently proposed symmetrical dynamic morphometric model of the human lung, based on Weibel's regular dichotomy, was adapted to simulate different physiopathological scenarios of lung heterogeneity. The asymmetrical architecture was mimicked by modeling different conductive airway compartments below the main bronchi, each compartment being characterized by regular branching. The respiratory zone and chest wall were described by a Voigt body and a constant elastance, respectively. Simulation results allowed us to investigate the influence of the main mechanisms involved in expiratory flow limitation and dynamic hyperinflation in mechanically ventilated COPD patients. In brief, they showed that convective gas acceleration plays a key role in reproducing a negative relationship between driving pressure and expiratory flow. Moreover, reduced lung elastance due to emphysema resulted in a remarkable increase in dynamic hyperinflation, although it did not significantly modify expiratory flow limitation. Finally, the presence of a normal lung compartment masked pathological behaviors, preventing standard techniques from revealing expiratory flow limitation in affected compartments.

A dynamic morphometric model of the normal lung for studying expiratory flow limitation in mechanical ventilation.

A nonlinear dynamic morphometric model of breathing mechanics during artificial ventilation is described. On the basis of the Weibel symmetrical representation of the tracheo-bronchial tree, the model accurately accounts for the geometrical and mechanical characteristics of the conductive zone and packs the respiratory zone into a viscoelastic Voigt body. The model also accounts for the main mechanisms limiting expiratory flow (wave speed limitation and viscous flow limitation), in order to reproduce satisfactorily, under dynamic conditions, the expiratory flow limitation phenomenon occurring in normal subjects when the difference between alveolar pressure and tracheal pressure (driving pressure) is high. Several expirations characterized by different levels of driving pressure are simulated and expiratory flow limitation is detected by plotting the isovolume pressure-flow curves. The model is used to study the time course of resistance and total cross-sectional area as well as the ratio of fluid velocity to wave speed (speed index), in conductive airway generations. The results highlight that the coupling between dissipative pressure losses and airway compliance leads to onset of expiratory flow limitation in normal lungs when driving pressure is increased significantly by applying a subatmospheric pressure to the outlet of the ventilator expiratory channel; wave speed limitation becomes predominant at still higher driving pressures.

A simulation model of the oxygen alveolo-capillary exchange in normal and pathological conditions.

This paper presents a mathematical model of the oxygen alveolo-capillary exchange to provide the capillary oxygen partial pressure profile in normal and pathological conditions. In fact, a thickening of the blood-gas barrier, heavy exercise or a low oxygen partial pressure (PO2) in the alveolar space can reduce the O2 alveolo-capillary exchange. Since the reversible binding between haemoglobin and oxygen makes it impossible to determine the closed form for the mathematical description of the PO2 profile along the pulmonary capillaries, an approximate analytical solution of the capillary PO2 profile is proposed. Simulation results are compared with the capillary PO2 profile obtained by numerical integration and by a piecewise linear interpolation of the oxyhaemoglobin dissociation curve. Finally, the proposed model is evaluated in a large range of physiopathological diffusive conditions. The good fit to numerical solutions in all experimental conditions seems to represent a substantial improvement with respect to the approach based on a linear approximation of the oxyhaemoglobin dissociation curve, and makes this model a candidate to be incorporated into the integrated descriptions of the entire respiratory system, where the datum of primary interest is the value of end capillary PO2.