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.     

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.     

Mathematical simulation of forced expiration

Flow limitation during forced expiration is simulated by a mathematical model. This model draws on the pressure-area law obtained in the accompanying paper, and the methods of analysis for one-dimensional flow in collapsible tubes developed by Shapiro (Trans. ASME J. Biomech. Eng. 99: 126-147, 1977). These methods represent an improvement over previous models in that 1) the effects of changing lung volume and of parenchymal-bronchial interdependence are simulated; 2) a more realistic representation of collapsed airways is employed; 3) a solution is obtained mouthward of the flow-limiting site by allowing for a smooth transition from sub- to supercritical flow speeds, then matching mouth pressure by imposing an elastic jump (an abrupt transition from super- to subcritical flow speeds) at the appropriate location; and 4) the effects of levels of effort (or vacuum pressure) in excess of those required to produce incipient flow limitation are examined, including the effects of potential physiological limitation.     

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.     

Configuration of maximum expiratory flow-volume curve: model experiments with physiological implications

A two-compartment mechanical model of the lungs was constructed with two parallel peripheral and collapsible bronchi in series with one central and collapsible trachea. Maximal expiratory flow-volume (MEFV) curves similar to those obtained in most dogs and in some humans could be produced: a peak followed by a gently sloping plateau ending in a knee, where flow suddenly fell to a much smaller value approaching zero rather slowly over the last 25 to 50% of the expired vital capacity. It was shown that flow before the knee was limited in the trachea, and after the knee it was limited in the bronchi. Two patterns of changes in the configuration of the MEFV curve could be observed. Pattern of changes affecting the central airway, at a given volume, maximal flow during the first part of the expiration (i.e., before the knee) is decreased; the knee occurs at a lower lung volume; the flow at the beginning of the knee is decreased. This pattern was observed with the following interventions: decreased cross-sectional area of the trachea (partial obstruction); decreased axial tension of the trachea; and, increased frictional loss between the trachea and the bronchi. Pattern of changes affecting the airways in the periphery: the knee occurs at a higher lung volume; at a given volume, flow after the knee becomes smaller; the absolute flow at the start of the knee is almost unchanged. This pattern was observed with the following interventions: decreased cross-sectional area of the peripheral airways (partial obstruction); increased frictional loss upstream to the peripheral airways; and, decreased elastic recoil pressure.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Steady pressure-flow relationship in a cast of the upper and central human airways

Pressure drops across the upper (larynx) and central airways of a human lung cast were measured at steady state inspiratory and expiratory flows. Air, HeO₂ and SF6-O₁ gas mixtures were used at tracheal Reynolds' numbers ranging from 145 to 30 000. The pressure-flow characteristics of the model were analysed using standard pressure-flow diagrams and Moody plots. We found that the asymmetry between inspiratory and expiratory resistances, observed in the central airways (larynx excluded), was markedly reduced in the presence of the larynx. However, static pressure differences were greater across the entire model of the upper and central airways than across the model of the five generations of the tracheo-bronchial tree (without larynx) at the same flow-rates. In addition, our results showed that the presence of the larynx tended to reduce the zone of fully developed laminar flow in the Moody diagram with the higher density gas, while extending the zone of turbulent flow even for the low density gas at low Reynold's numbers.     

Flow of aqueous humor in the canal of Schlemm

A mathematical model is presented for the flow of aqueous humor in Schlemm's canal in the eye. The model introduces a canal segment between two collector channels as a rectangular channel with porous upper wall. Two cases have been considered in the model: (I) the inner porous wall of the canal is rigid; (II) the inner wall is collapsible. Analytical solution of the governing equation in case I is straightforward, whereas the nonlinear equation in case II is solved by an iterative procedure. Aqueous fluid pressure and flow profiles in the proposed model are drawn, and the effects of important parameters on these profiles are brought out and discussed. It is concluded that for case I, resistance to aqueous flow is influenced by the filtration constant of the trabecular and endothelial meshwork and that narrowing of the canal reduces outflow. In case II, an increase in intraocular pressure (IOP) or compliance coefficient of the canal inner wall increases the collapse of the canal, which offers increased resistance to flow resulting in the decreased flow whereas increasing filtration constant facilitates aqueous outflow. These theoretical results suggest that increased IOP or decreased rigidity of the inner wall may contribute to the development of increased resistance as observed in some cases of glaucoma and that increasing values of filtration constant may contribute to the facility of outflow increase.     

Airway stability and heterogeneity in the constricted lung.

The effect of bronchoconstriction on airway resistance is known to be spatially heterogeneous and dependent on tidal volume. We present a model of a single terminal airway that explains these features. The model describes a feedback between flow and airway resistance mediated by parenchymal interdependence and the mechanics of activated smooth muscle. The pressure-tidal volume relationship for a constricted terminal airway is computed and shown to be sigmoidal. Constricted terminal airways are predicted to have two stable states: one effectively open and one nearly closed. We argue that the heterogeneity of whole lung constriction is a consequence of this behavior. Airways are partitioned between the two states to accommodate total flow, and changes in tidal volume and end-expiratory pressure affect the number of airways in each state. Quantitative predictions for whole lung resistance and elastance agree with data from previously published studies on lung impedance.     

Effect of negative expiratory pressure on respiratory system flow resistance in awake snorers and nonsnorers.

In spontaneously breathing subjects, intrathoracic expiratory flow limitation can be detected by applying a negative expiratory pressure (NEP) at the mouth during tidal expiration. To assess whether NEP might increase upper airway resistance per se, the interrupter resistance of the respiratory system (Rint,rs) was computed with and without NEP by using the flow interruption technique in 12 awake healthy subjects, 6 nonsnorers (NS), and 6 nonapneic snorers (S). Expiratory flow (V) and Rint,rs were measured under control conditions with V increased voluntarily and during random application of brief (0.2-s) NEP pulses from -1 to -7 cmH(2)O, in both the seated and supine position. In NS, Rint,rs with spontaneous increase in V and with NEP was similar [3.10 +/- 0.19 and 3.30 +/- 0.18 cmH(2)O x l(-1) x s at spontaneous V of 1.0 +/- 0.01 l/s and at V of 1.1 +/- 0.07 l/s with NEP (-5 cmH(2)O), respectively]. In S, a marked increase in Rint,rs was found at all levels of NEP (P < 0.05). Rint,rs was 3.50 +/- 0.44 and 8.97 +/- 3.16 cmH(2)O x l(-1) x s at spontaneous V of 0.81 +/- 0.02 l/s and at V of 0.80 +/- 0.17 l/s with NEP (-5 cmH(2)O), respectively (P < 0.05). With NEP, Rint,rs was markedly higher in S than in NS both seated (F = 8.77; P < 0.01) and supine (F = 9.43; P < 0.01). In S, V increased much less with NEP than in NS and was sometimes lower than without NEP, especially in the supine position. This study indicates that during wakefulness nonapneic S have more collapsible upper airways than do NS, as reflected by the marked increase in Rint,rs with NEP. The latter leads occasionally to an actual decrease in V such as to invalidate the NEP method for detection of intrathoracic expiratory flow limitation.     

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.     

Numerical simulation of noninvasive blood pressure measurement

In this paper, a simulation model based on the partially pressurized collapsible tube model for reproducing noninvasive blood pressure measurement is presented. The model consists of a collapsible tube, which models the pressurized part of the artery, rigid pipes connected to the collapsible tube, which model proximal and distal region far from the pressurized part, and the Windkessel model, which represents the capacitance and the resistance of the distal part of the circulation. The blood flow is simplified to a one-dimensional system. Collapse and expansion of the tube is represented by the change in the cross-sectional area of the tube considering the force balance acting on the tube membrane in the direction normal to the tube axis. They are solved using the Runge-Kutta method. This simple model can easily reproduce the oscillation of inner fluid and corresponding tube collapse typical for the Korotkoff sounds generated by the cuff pressure. The numerical result is compared with the experiment and shows good agreement.     

Flow through a Collapsible Tube: Experimental Analysis and Mathematical Model

Flow through thin-wall axisymmetric tubes has long been of interest to physiologists. Analysis is complicated by the fact that such tubes will collapse when the transmural pressure (internal minus external pressure) is near zero. Because of the absence of any body of related knowledge in other sciences or engineering, previous workers have directed their efforts towards experimental studies of flow in collapsible tubes. More recently, some attention has been given towards analytical studies. Results of an extensive series of experiments show that the significant system parameter is transmural pressure. The cross-sectional area of the tube depends upon the transmural pressure, and changes in cross-section in turn affect the flow geometry. Based on experimental studies, a lumped parameter system model is proposed for the collapsible tube. The mathematical model is simulated on a hybrid computer. Experimental data were used to define the functional relationship between cross-sectional area and transmural pressure as well as the relation between the energy loss coefficient and cross-sectional area. Computer results confirm the validity of the model for both steady and transient flow conditions.     

Noninvasive determination of upper airway resistance and flow limitation.

We have shown that a polynomial equation, FP = AP3 + BP2 + CP + D, where F is flow and P is pressure, can accurately determine the presence of inspiratory flow limitation (IFL). This equation requires the invasive measurement of supraglottic pressure. We hypothesized that a modification of the equation that substitutes time for pressure would be accurate for the detection of IFL and allow for the noninvasive measurement of upper airway resistance. The modified equation is Ft = At3 + Bt2 + Ct + D, where F is flow and t is time from the onset of inspiration. To test our hypotheses, data analysis was performed as follows on 440 randomly chosen breaths from 18 subjects. First, we performed linear regression and determined that there is a linear relationship between pressure and time in the upper airway (R2 0.96 +/- 0.05, slope 0.96 +/- 0.06), indicating that time can be a surrogate for pressure. Second, we performed curve fitting and found that polynomial equation accurately predicts the relationship between flow and time in the upper airway (R2 0.93 +/- 0.12, error fit 0.02 +/- 0.08). Third, we performed a sensitivity-specificity analysis comparing the mathematical determination of IFL to manual determination using a pressure-flow loop. Mathematical determination had both high sensitivity (96%) and specificity (99%). Fourth, we calculated the upper airway resistance using the polynomial equation and compared the measurement to the manually determined upper airway resistance (also from a pressure-flow loop) using Bland-Altman analysis. Mean difference between calculated and measured upper airway resistance was 0.0 cmH2O x l(-1) x s(-1) (95% confidence interval -0.2, 0.2) with upper and lower limits of agreement of 2.8 cmH2O x l(-1) x s(-1) and -2.8 cmH2O x l(-1) x s(-1). We conclude that a polynomial equation can be used to model the flow-time relationship, allowing for the objective and accurate determination of upper airway resistance and the presence of IFL.     

Location of flow-limiting segments via airway catheters near residual volume in humans

Previous studies have demonstrated sites of flow limitation in the central airways of dogs and humans. At low lung volumes, however, during a forced expiration, it is not clear whether flow-limiting segments (FLS) move into the lung periphery. Using intrabronchial lateral pressure catheters, we located FLS in human subjects at all lung volumes between functional residual capacity (FRC) and residual volume (RV). Three individuals with severe intracranial hemorrhage maintained on ventilators were studied. Partial maximal flow-volume curves were generated from 1 liter above FRC to RV by lowering downstream pressure and using the interrupter technique. Sites of FLS were defined as the most downstream points where lateral pressure did not change with driving pressure. FLS were found in all subjects in the central airways. In one subject, FLS moved from segmental bronchi to the first subsegmental bronchus as RV was approached but not beyond. In the other two subjects, FLS remained fixed in location at all measured lung volumes. At constant volume, multiple FLS were located, all in parallel, e.g., fixed in left upper, left lower, and right middle lobar bronchi. In conclusion, sites of flow limitation remain in the central airways as lung volume approaches RV. FLS may move peripherally within the central airways but not beyond proximal subsegmental bronchi.     

Mechanical output impedance of the lung determined from cardiogenic oscillations.

The beating heart naturally oscillates the lung because of the close juxtaposition between these organs producing cardiogenic oscillations in flow that can be measured at the mouth when the glottis is open. Correspondingly, if the mouth is occluded, the same phenomenon produces cardiogenic pressure oscillations that can be measured just distal to the site of occlusion. The Fourier-domain ratio of these oscillations in pressure and flow constitutes what we call cardiogenic respiratory impedance (Zc). We calculated Zc between about 1.5 and 10 Hz in relaxed normal subjects at functional residual capacity with open glottis. Zc was insensitive to heart rate changes induced by exercise and had an imaginary part close to zero at all frequencies investigated. Its real part was similar to or smaller than resistance determined by the forced oscillation technique. We speculate that Zc measures the flow resistance of the central and upper airways of the lung. Zc may be useful as a means of obtaining information about lung mechanics without the need for an external source of flow perturbations.     

Respiratory mechanics in the normal dog determined by expiratory flow interruption

We recently proposed an eight-parameter model of the respiratory system to account for its mechanical behavior when flow is interrupted during passive expiration. The model consists of two four-parameter submodels representing the lungs and the chest wall, respectively. The lung submodel consists of an airways resistance together with elements embodying the viscoelastic properties of the lung tissues. The chest wall submodel has similar structure. We estimated the parameters of the model from data obtained in four normal, anesthetized, paralyzed, tracheostomized mongrel dogs. This model explains why lung tissue and chest wall resistances should be markedly frequency dependent at low frequencies and also permits a physiological interpretation of resistance measurements provided by the flow interruption method.     

Bronchial mechanical properties and maximal expiratory flows

Flow limitation in a collapsible elastic tube is dependent on the area (A) vs. pressure (P) relationship (the "tube law") for the tube. In this paper, a tube law in which A varies as (1-P)-n1 at negative pressures is assumed. It is shown that wave-speed limitation is possible at negative pressures only if n1 is greater than 0.5. Dissipative limitation is also investigated. Viscous limitation can occur if n1 is greater than 0.5, and turbulent limitation can occur if n1 is not less than 0.4. For values of n1 less than 0.4, flow cannot be limited at negative pressures. Model simulations are used to show that a combination of a value of n1 less than 0.3 together with an area minimum in the bronchial tree produce a minimum (a "hook") in the flow-volume curve. In the vicinity of such hooks, density dependence exceeds the usually accepted theoretical maximum value. Simulations also show that, when n1 is sufficiently small, apparently supramaximal flows appear to be possible.     

Pharyngeal critical pressure in children with mild sleep-disordered breathing.

There is evidence that narrowing or collapse of the pharynx can contribute to obstructive sleep-disordered breathing (SDB) in adults and children. However, studies in children have focused on those with relatively severe SDB who generally were recruited from sleep clinics. It is unclear whether children with mild SDB who primarily have hypopneas, and not frank apnea, also have more collapsible airways. We estimated airway collapsibility in 10 control subjects (9.4 +/- 0.5 yr old; 1.9 +/- 0.2 hypopneas/h) and 7 children with mild SDB (10.6 +/- 0.5 yr old; 11.5 +/- 0.1 hypopneas/h) during stable, non-rapid eye movement sleep. None of the subjects had clinically significant enlargement of the tonsils or adenoids, nor had any undergone previous tonsillectomy or adenoidectomy. Airway collapsibility was measured by brief (2-breath duration) and sudden reductions in pharyngeal pressure by connecting the breathing mask to a negative pressure source. Negative pressure applications ranging from -1 to -20 cmH(2)O were randomly applied in each subject while respiratory airflow and mask pressure were measured. Flow-pressure curves were constructed for each subject, and the x-intercept gave the pressure at zero flow, the so-called critical pressure of the upper airway (Pcrit). Pcrit was significantly higher in children with SDB than in controls (-10.8 +/- 2.8 vs. -15.7 +/- 1.2 cmH(2)O; P < 0.05). There were no significant differences in the slopes of the pressure-flow relations or in baseline airflow resistance. These data support the concept that intrinsic pharyngeal collapsibility contributes to mild SDB in children.     

Properties of steady maximal expiratory flow within excised canine central airways

Using our transistor model of the lung during forced expiration (J. Appl. Physiol. 62: 2013-2025, 1987), we recently predicted that 1) axially arranged choke points can exist simultaneously during forced expiration with sufficient effort, and 2) overall maximal expiratory flow may be relatively insensitive to nonuniform airways obstruction because of flow interdependence between parallel upstream branches. We tested these hypotheses in excised central airways obtained from five canine lungs. Steady expiratory flow was induced by supplying constant upstream pressure (Pupstream = 0-16 cmH2O) to the bronchi of both lungs while lowering pressure at the tracheal airway opening (16 to -140 cmH2O). Intra-airway pressure profiles obtained during steady maximal expiratory flow disclosed a single choke point in the midtrachea when Pupstream was high (2-16 cmH2O). However, when Pupstream was low (0 cmH2O), two choke sites were evident: the tracheal site persisted, but another upstream choke point (main carina or both main bronchi) was added. Flow interdependence was studied by comparing maximal expiratory flow through each lung before and after introduction of a unilateral external resistance upstream of the bronchi of one lung. When this unilateral resistance was added, ipsilateral flow always fell, but changes in flow through the contralateral lung depended on the site of the most upstream choke. When a single choke existed in the trachea, addition of the external resistance increased contralateral flow by 38 +/- 28% (SD, P less than 0.003). In contrast, when the most upstream choke existed at the main carina or in the bronchi, addition of the external resistance had no effect on contralateral maximal expiratory flow.(ABSTRACT TRUNCATED AT 250 WORDS)