Positive End Expiratory Pressure and Expiratory Flow Limitation: A Model Study

Patients suffering from chronic obstructive pulmonary diseases, frequently exhibit expiratory airflow limitation. We propose a mathematical model describing the mechanical behavior of the ventilated respiratory system. This model has to simulate applied positive end-expiratory pressure (PEEP) effects during expiration, a process used by clinicians to improve airflow. The proposed model consists of a nonlinear two-compartment system. One of the compartments represents the collapsible airways and mimics its dynamic compression, the other represents the lung and chest wall compartment. For all clinical conditions tested (n=16), the mathematical model simulates the removal of expiratory airflow limitation at PEEP lower than 70–80% of intrinsic end-expiratory pressure (PEEPi), i.e. the end-expiratory alveolar pressure (PAet) without PEEP. It also shows the presence of an optimal PEEP. The optimal PEEP contributes to decrease PAet from 7.4 ± 0.9 (SD) to 5.4 ± 0.9 hPa (p < 0.0001; mild flow limitation) and from 11.8 ± 1.1 to 7.8 ± 0.7 hPa (p < 0.0001; severe flow limitation). Resistance of the collapsible compartment is decreased from 53 ± 7 to 8.2 ± 5.9 hPa.L^–1.s (p < 0.0001; mild flow limitation) and from 80 ± 11 to 6.9 ± 5.4 hPa.L^–1.s (p < 0.0001; severe flow limitation). This simplistic mathematical model gives a plausible explanation of the expiratory airflow limitation removal with PEEP and a rationale to the practice of PEEP application to airflow limited patients.     

Modeling expiratory flow from excised tracheal tube laws.

Flow limitation during forced exhalation and gas trapping during high-frequency ventilation are affected by upstream viscous losses and by the relationship between transmural pressure (Ptm) and cross-sectional area (A(tr)) of the airways, i.e., tube law (TL). Our objective was to test the validity of a simple lumped-parameter model of expiratory flow limitation, including the measured TL, static pressure recovery, and upstream viscous losses. To accomplish this objective, we assessed the TLs of various excised animal tracheae in controlled conditions of quasi-static (no flow) and steady forced expiratory flow. A(tr) was measured from digitized images of inner tracheal walls delineated by transillumination at an axial location defining the minimal area during forced expiratory flow. Tracheal TLs followed closely the exponential form proposed by Shapiro (A. H. Shapiro. J. Biomech. Eng. 99: 126-147, 1977) for elastic tubes: Ptm = K(p) [(A(tr)/A(tr0))(-n) - 1], where A(tr0) is A(tr) at Ptm = 0 and K(p) is a parametric factor related to the stiffness of the tube wall. Using these TLs, we found that the simple model of expiratory flow limitation described well the experimental data. Independent of upstream resistance, all tracheae with an exponent n < 2 experienced flow limitation, whereas a trachea with n > 2 did not. Upstream viscous losses, as expected, reduced maximal expiratory flow. The TL measured under steady-flow conditions was stiffer than that measured under expiratory no-flow conditions, only if a significant static pressure recovery from the choke point to atmosphere was assumed in the measurement.     

Identification of a Non-Linear Model as a New Method to Detect Expiratory Airflow Limitation in Mechanically Ventilated Patients

Expiratory flow limitation (EFL) can occur in mechanically ventilated patients with chronic obstructive pulmonary disease and other disorders. It leads to dynamic hyperinflation with ensuing deleterious consequences. Detecting EFL is thus clinically relevant. Easily applicable methods however lack this detection being routinely made in intensive care. Using a simple mathematical model, we propose a new method to detect EFL that does not require any intervention or modification of the ongoing therapeutic. The model consists in a monoalveolar representation of the respiratory system, including a collapsible airway that is submitted to periodic changes in pressure at the airway opening: EFL provokes a sharp expiratory increase in the resistance Rc of the collapsible airway. The model parameters were identified via the Levenberg-Marquardt method by fitting simulated data on the airway pressure and the flow signals recorded in 10 mechanically ventilated patients. A sensitivity study demonstrated that only 8/11 parameters needed to be identified, the remaining three being given reasonable physiological values. Flow-volume curves built at different levels of positive expiratory pressure, PEEP, during "PEEP trials" (stepwise increases in positive end-expiratory pressure to optimize ventilator settings) have shown evidence of EFL in three cases. This was concordant with parameter identification (high Rc during expiration for EFL patients). We conclude from these preliminary results that our model is a potential tool for the non-invasive detection of EFL in mechanically ventilated patients.     

Inspiratory flow in the nose: a model coupling flow and vasoerectile tissue distensibility.

We have developed a discrete multisegmental model describing the coupling between inspiratory flow and nasal wall distensibility. This model is composed of 14 individualized compliant elements, each with its own relationship between cross-sectional area and transmural pressure. Conceptually, this model is based on flow limitation induced by the narrowing of duct due to collapsing pressure. For a given inspiratory pressure and for a given compliance distribution, this model predicts the area profile and inspiratory flow. Acoustic rhinometry and posterior rhinomanometry were used to determine the initial geometric area and mechanical characteristics of each element. The proposed model, used under steady-state conditions, is able to simulate the pressure-flow relationship observed in vivo under normal conditions (4 subjects) and under pathological conditions (4 vasomotor rhinitis and 3 valve syndrome subjects). Our results suggest that nasal wall compliance is an essential parameter to understand the nasal inspiratory flow limitation phenomenon and the associated increase of resistance that is well known to physiologists. By predicting the functional pressure-flow relationship, this model could be a useful tool for the clinician to evaluate the potential effects of treatments.     

Flow limitation and wheezes in a constant flow and volume lung preparation

To facilitate the study of respiratory wheezes in an animal lung model, an isovolume, constant-flow excised dog lung preparation was developed. Dog lungs were inflated to 26 +/- 4 cmH2O and coated with layers of epoxy glue and polyester compound. A rigid shell 2 mm thick was obtained around the entire pleural surface and the extra-pulmonary airways. The adhesive forces between the pleura and the shell were strong enough to hold the lung distended after the inflation pressure was removed. Holes 2 mm diam were drilled through the shell over one of the lung lobes in an array, 4 cm across. The holes penetrated the pleural surface, so that constant flow could be maintained in the expiratory direction by activating a suction pump connected to the trachea. Downstream suction pressure and flow rate were measured with a mercury manometer and a rotameter, respectively. Sounds were recorded by a small (0.6 cm OD) microphone inserted into the trachea. When suction pressure was increased, flow initially increased to 31 +/- 3 l/min. Further increase of suction pressure caused only very slight additional increase in flow (i.e., flow limitation). During this plateau of flow, a pure tone was generated with acoustic properties similar to respiratory wheezes. Both the flow plateau and the wheezing sounds could be eliminated by freezing the lungs. It is concluded that wheezing sounds were associated with flow limitation in this preparation. It is suggested that the stable acoustic properties obtained by this preparation may become useful in the analysis of mechanisms of wheezing lung sounds generation.     

A model of the coronary vascular bed and its verification by a physiological experiment

We analysed the relationship between artery pressure (AP) and coronary flow (F) in the canine coronary bed, using an electrical analog model of the coronary circulation. The model contained a capacitance of epicardial vessels, input and terminal resistances, diode, and the number of e.m.f., simulated the intramyocardial pressure and zero-flow pressure. These e.m.f. are assumed to be a linear functions of left ventricular and aortic pressure. The value of coronary blood flow was calculated from experimental curves of AP and LVP and setting parameters. Good agreement was obtained between theoretical and experimental curves of coronary blood flow.     

An asymptotic model of viscous flow limitation in a highly collapsed channel.

A viscous flow through a long two-dimensional channel, one wall of which is formed by a finite-length membrane, experiences flow limitation when the channel is highly collapsed over a narrow region under high external pressure. Simple approximate relations between flow rate and pressure drop are obtained for this configuration by the use of matched asymptotic expansions. Weak inertial effects are also considered.     

2D gasdynamic simulation of the kinetics of an oxygen-iodine laser with electric-discharge generation of singlet oxygen

The kinetic processes occurring in an electric-discharge oxygen-iodine laser are analyzed with the help of a 2D (r, z) gasdynamic model taking into account transport of excited oxygen, singlet oxygen, and radicals from the electric discharge and their mixing with the iodine-containing gas. The main processes affecting the dynamics of the gas temperature and gain are revealed. The simulation results obtained using the 2D model agree well with the experimental data on the mixture gain. A subsonic oxygen-iodine laser in which singlet oxygen is generated by a 350 W transverse RF discharge excited in an oxygen flow at a pressure P = 10 Torr and the discharge tube wall is covered with mercury oxide is simulated. The simulated mixing system is optimized in terms of the flow rate and the degree of preliminary dissociation of the iodine flow. The optimal regime of continuous operation of a subsonic electric-discharge oxygen-iodine laser is found.     

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.     

Computer modeling of the human systemic arterial tree

A model of the human systemic arterial tree has been devised, based on a lumped-parameter-circuit approximate form. This model has been set up and studied on an analog computer. A feature of this simulation is the division of the arterial system into sections whose lengths are inversely proportional (approximately) to their cross-sectional area-or what is termed ‘equal-volume’ modeling.

Great care was exercised in the determination of the model parameters, using expressions for these parameters from a recent paper by Rideout and Dick on fluid flow in distensible tubes, with numerical values based on measurements reported in the medical literature.

The simulated pressure and flow waveforms obtained with the model compare favorably with data recorded from the normal adult human, and exhibit such well-known features as distal delay and peaking of pressure pulses. The aortic input impedance vs. frequency curve checks well against measurements on the human. The model also provides a simple means for determination of cardiac output, cardiac work and cardiac power under various assumed conditions such as variation of heart rate.     

Numerical solutions for steady and unsteady flow in a model of the pulmonary airways

A computational model is presented for unsteady flow through a collapsible tube with variable wall stiffness. The one-dimensional flow equations are solved for inlet, outlet and external conditions that vary with time and for a tube with time-dependent, spatially-distributed local properties. In particular, the effects of nonuniformities and local perturbations in stiffness distribution in the tube are studied. By allowing the flow to evolve in time, asymptotically steady flows are calculated. When simulating a quasi-steady reduction in downstream pressure, the model demonstrates critical transitions, the phenomena of wave-speed limitation and the sites of flow limitation. It also exhibits conditions for which viscous flow limitation occurs. Computations of rapid, unsteady changes of the exit pressure illustrate the phenomena occurring at the onset of a cough, and the generation and propagation of elastic jumps.     

Tracheal stenosis: a flow dynamics study.

Patients referred for treatment of tracheal stenosis typically are asymptomatic until critical narrowing of the airway occurs, which then requires immediate intervention. To understand how tracheal stenosis affects local pressure drops and explore how a dramatic increase in pressure drop could possibly be detected at an early stage, a computational fluid dynamics (CFD) study was undertaken. We assessed flow patterns and pressure drops over tracheal stenoses artificially inserted into a realistic three-dimensional upper airway model derived from multislice computed tomography images obtained in healthy men. Solving the Navier-Stokes equations (with a Yang-shih k-epsilon turbulence model) for different degrees of tracheal constriction located approximately one tracheal diameter below the glottis, the simulated pressure drop over the stenosis (DeltaP) was seen to dramatically increase only when well over 70% of the tracheal lumen was obliterated. At 30 l/min, DeltaP increased from 7 Pa for a 50% stenosis to, respectively, 46 and 235 Pa for 80% and 90% stenosis. The pressure-flow relationship in the entire upper airway model (between mouth and end of trachea) in the flow range 0-60 l/min showed a power law relationship with best-fit flow exponent of 1.77 in the absence of stenosis. The exponent became 1.92 and 2.00 in the case of 60% and 85% constriction, respectively. The present simulations confirm that the overall pressure drop at rest is only affected in case of severe constriction, and the simulated flow dependence of pressure drop suggests a means of detecting stenosis at a precritical stage.     

Effects of near-bed current speeds on the distribution and behaviour of the lobster, Homarus gammarus (L.)

The distribution of the lobster, Homarus gammarus (L.) appears to be limited to areas of the sea bed with rocky outcrops. It seems likely that this limitation operates because lobsters need to be able to avoid currents created by tidal and wave action and that it is size selective. A series of experiments was made with lobsters in a flume tank, to observe the effects of a range of current speeds and substratum types on lobster behaviour. In addition, depth profiles of flow rates were made by Pitot tube measurements to estimate flow rates within the boundary layer immediately above the sea bed. Predictions of the influence of natural water currents on lobster behaviour have been made using available data on near-bed current speeds. The implications of these predictions to fisheries management and reef construction plans are discussed.     

Simulating pH effects in an algal‐growth hydrodynamics model1

Models and numerical simulations are relatively inexpensive tools that can be used to enhance economic competitiveness through operation and system optimization to minimize energy and resource consumption, while maximizing algal oil yield. This work uses modified versions of the U.S. Environmental Protection Agency's Environmental Fluid Dynamics Code (EFDC) in conjunction with the U.S. Army Corp of Engineers' water‐quality code (CE‐QUAL) to simulate flow hydrodynamics coupled to algal growth kinetics. The model allows the flexibility of manipulating a host of variables associated with algal growth such as temperature, light intensity, and nutrient availability. pH of the medium is a newly added operational parameter governing algal growth that affects algal photosynthesis, differential availability of inorganic forms of carbon, enzyme activity in algae cell walls, and oil production rates. A single‐layer algal‐growth/hydrodynamic model without pH limitation was verified by comparing solution curves of algal biomass and phosphorus concentrations to an analytical solution. Media pH, now included in the model as a growth‐limiting factor, can be entered as a measured value or calculated based on CO₂ concentrations. Upon adding the ability to limit growth due to pH, physically reasonable results have been obtained from the model both with and without pH limitation. When the model was used to simulate algal growth from a pond experiment in the greenhouse, a least‐squares fitting technique yielded a maximum algal production (subsequently modulated by limitation factors) of 1.05 d^?1. Overall, the measured and simulated biomass concentrations in the greenhouse pond were in close agreement.     

重力对锥形血管中血流压力的影响——基于考虑体位改变的三维流固耦合数学模型

重力是体位改变过程中最基本的生物力学刺激因素．血流压力是表征心血管功能状态的一个基本指标．目前,体位改变影响心血管系统的确切内部机制尚不清楚．为此,采用在流体和固体方程中分别引入体力项的方法,建立一个基于血流动力学概念的三维流固耦合数学模型,用以研究体位改变,确切量化重力对血流压力的影响．通过数值计算,得到以下结果．水平卧位条件下：a．单一血管中血流压力由无重力影响的轴对称二维分布变为重力影响下的三维不对称分布;b．随着进出口压差由小变大,重力对压力分布和极值的影响由大变小,当压差值分别达到10 665.6 Pa(80 mmHg)和2 666.4 Pa(20 mmHg)时,重力的影响就不再随进出口压差增大而变化;对三维单一流体,重力影响的总体趋势类似．对正、倒直立位,压力均为二维轴对称分布,其重力影响强度约为水平卧位的2倍以上．结果表明：基于血流动力学概念,引入体力项,建立三维流固耦合模型为研究体位改变提供了一种新思路,重力对单一血管中血流压力分布和大小的影响因体位不同而不同,并与进出口压差密切相关,提示,若血管进出口压差较小,忽略重力影响,不考虑体位改变,以二维轴对称模型来研究血管中血流状态,须谨慎解释所得结果．     

A new technique to demonstrate flow limitation in partial expiratory flow-volume curves in infants

Partial expiratory flow-volume (PEFV) curves in infants are generated by applying a compressive pressure over the chest wall with an inflatable jacket. This study addresses two issues: pressure transmission to and across the chest wall and whether flow limitation can be identified. Eleven infants sedated with chloral hydrate were studied. Pressure transmission to the chest wall, measured with neonatal blood pressure cuffs placed on the infant's body surface, was 72 +/- 4% of jacket pressure during compression maneuvers. The pressure transmission to the air spaces, determined by measuring airway pressure during a compression maneuver against an occluded airway, was 56 +/- 6% of jacket pressure. A significant amount of the applied pressure is therefore lost across both the jacket and chest wall. Rapid pressure oscillations (RPO) were superimposed on static jacket pressures while expiratory flow was measured. Absence of associated oscillations of flow measured at the mouth was taken to indicate that flow was independent of driving pressure and therefore limited. Flow limitation was demonstrable with the RPO technique in all infants for jacket pressures greater than 50 cmH2O; however, it was evident at jacket pressures less than 30 cmH2O jacket pressure in four infants with obstructive airway disease. The RPO technique is a useful adjunct to the compression maneuver utilized to generate PEFV curves in infants because it facilitates the recognition of expiratory flow limitation.     

Determinants of long-term facilitation in humans during NREM sleep.

Long-term facilitation (LTF) is a prolonged increase in ventilatory motor output after episodic peripheral chemoreceptor stimulation. We have previously shown that LTF is activated during sleep following repetitive hypoxia in snorers (Babcock MA and Badr MS. Sleep 21: 709-716, 1998). The purpose of this study was 1) to ascertain the relative contribution of inspiratory flow limitation to the development of LTF and 2) to determine the effect of eliminating inspiratory flow limitation by nasal CPAP on LTF. We studied 25 normal subjects during stable non-rapid eye movement sleep. We induced 10 episodes of brief repetitive isocapnic hypoxia (inspired O(2) fraction = 8%; 3 min) followed by 5 min of room air. Measurements were obtained during control and at 20 min of recovery (R(20)). During the episodic hypoxia study, inspiratory minute ventilation (Vi) increased from 6.7 +/- 1.9 l/min during the control period to 8.2 +/- 2.7 l/min at R(20) (122% of control; P < 0.05). Linear regression analysis confirmed that inspiratory flow limitation during control was the only independent determinant of the presence of LTF (P = 0.005). Six subjects were restudied by using nasal continuous positive airway pressure to ascertain the effect of eliminating inspiratory flow limitation on LTF. Vi during the recovery period was 97 +/- 10% (P > 0.05). In conclusion, 1) repetitive hypoxia in sleeping humans is followed by increased Vi in the recovery period, indicative of development of LTF; 2) inspiratory flow limitation is the only independent determinant of posthypoxic LTF in sleeping human; 3) elimination of inspiratory flow limitation abolished the ventilatory manifestations of LTF; and 4) we propose that increased Vi in the recovery period was a result of preferential recruitment of upper airway dilators by repetitive hypoxia.     

Analysis of the collapsibility of the upper airway in a spectrum of sleep-disordered breathing: a modelling approach

Using a simplified model of the upper airways with two independent collapsible elements (nostrils and hypo-pharynx), we calculated the cross-sectional area of these two elements, taking into account pressure drops. We experimentally measured flow and pressure in the fossa and hypo-pharynx in various syndromes. This allowed us to compare the behaviour of the area supplied by our model with the aerodynamic resistance that is often used to analyse upper airway flow limitation events. We showed that nostril and hypo-pharyngeal areas are better correlated than the resistance values and thus concluded that the pressure divided by the square of the flow is a better parameter for analysing flow limitation in upper airways than resistance. Owing to its simplicity, our model is able to supply the area of the collapsible element in real time, which is impossible with more sophisticated models.     

A mathematical model to detect inspiratory flow limitation during sleep.

The physiological significance of inspiratory flow limitation (IFL) has recently been recognized, but methods of detecting IFL can be subjective. We sought to develop a mathematical model of the upper airway pressure-flow relationship that would objectively detect flow limitation. We present a theoretical discussion that predicts that a polynomial function [F(P) = AP(3) + BP(2) + CP + D, where F(P) is flow and P is supraglottic pressure] best characterizes the pressure-flow relationship and allows for the objective detection of IFL. In protocol 1, step 1, we performed curve-fitting of the pressure-flow relationship of 20 breaths to 5 mathematical functions and found that highest correlation coefficients (R(2)) for quadratic (0.88 +/- 0.10) and polynomial (0.91 +/- 0.05; P < 0.05 for both compared with the other functions) functions. In step 2, we performed error-fit calculations on 50 breaths by comparing the quadratic and polynomial functions and found that the error fit was lowest for the polynomial function (3.3 +/- 0.06 vs. 21.1 +/- 19.0%; P < 0.001). In protocol 2, we performed sensitivity/specificity analysis on two sets of breaths (50 and 544 breaths) by comparing the mathematical determination of IFL to manual determination. Mathematical determination of IFL had high sensitivity and specificity and a positive predictive value (>99% for each). We conclude that a polynomial function can be used to predict the relationship between pressure and flow in the upper airway and objectively determine the presence of IFL.     

Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis

Pulse wave propagation in the mature rabbit systemic circulation was simulated using the one-dimensional equations of blood flow in compliant vessels. A corrosion cast of the rabbit circulation was manufactured to obtain arterial lengths and diameters. Pulse wave speeds and inflow and outflow boundary conditions were derived from in vivo data. Numerical results captured the main features of in vivo pressure and velocity pulse waveforms in the aorta, brachiocephalic artery and central ear artery. This model was used to elucidate haemodynamic mechanisms underlying changes in peripheral pulse waveforms observed in vivo after administering drugs that alter nitric oxide synthesis in the endothelial cells lining blood vessels. According to our model, these changes can be explained by single or combined alterations of blood viscosity, peripheral resistance and compliance, and the elasticity of conduit arteries.