J Appl Physiol 100: 384-389, 2006.
First published October 20, 2005; doi:10.1152/japplphysiol.00689.2005
8750-7587/06 $8.00
Viscoelasticity of the trachea and its effects on flow limitation
Nikolai Aljuri,1,2
Jose G. Venegas,1,2 and
Lutz Freitag3
1Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge; 2Department of Anaesthesia, Massachusetts General Hospital, Harvard Medical School, Boston, Massachusetts; and 3Institute for Experimental Pneumology, Lungenklinik Hemer, Germany
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ABSTRACT
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To test the hypothesis that peak expiratory flow is determined by the wave-speed-limiting mechanism, we studied the time dependency of the trachea and its effects on flow limitation. For this purpose, we assessed the relationship between transmural pressure and cross-sectional area [the tube law (TL)] of six excised human tracheae under controlled conditions of static (no flow) and forced expiratory flow. We found that TLs of isolated human tracheae followed quite well the mathematical representation proposed by Shapiro (Shapiro AH. J Biomech Eng 99: 126–147, 1977) for elastic tubes. Furthermore, we found that the TL measured at the onset of forced expiratory flow was significantly stiffer than the static TL. As a result, the stiffer TL measured at the onset of forced expiratory flow predicted theoretical maximal expiratory flows far greater than those predicted by the more compliant static TL, which in all cases studied failed to explain peak expiratory flows measured at the onset of forced expiration. We conclude that the observed viscoelasticity of the tracheal walls can account for the measured differences between maximal and "supramaximal" expiratory flows seen at the onset of forced expiration.
static tube law; dynamic tube law; maximal expiratory flow; forced expiration; supramaximal flow
THEORETICAL AND EXPERIMENTAL STUDIES have shown that forced expiratory flow (
) becomes limited when gas velocity reaches the local wave-speed at a point along the airway: the choke point (4). Such wave-speed is a function of the airway’s transmural pressure (Ptm)-area (Atr) relationship, also called the tube law (TL), and gas density (11). Pedersen et al. (10) tested in healthy and asthmatic human subjects whether peak expiratory (PEF) may be determined by the wave-speed limiting mechanism and concluded that local wave-speed is reached in the central airways at PEF in most subjects (healthy and asthmatic). Likewise, Tantucci et al. (13) found that PEF does not increase in normal subjects by increasing the driving pressure between alveoli and mouth through the negative expiratory pressure application (7), indicating that expiratory limitation occurs even at PEF, supporting the hypothesis that PEF is determined by the wave-speed limiting mechanism. In contrast to Pedersen’s results, Lambert et al. (8) predicted from a fluid mechanical analysis of the maximum expiratory based on airway properties from excised human lungs that the most proximal locations of the -determining site at high lung volumes must be in the main or lobar bronchi; however, Pedersen et al. (10) suggested that invagination of the membranous parts of the airways and axial tension resulting from measurements under dynamic conditions may very well change the measured TLs in a way that is not accounted for in Lambert et al.’s model. Furthermore, D’Angelo et al. (3) showed that during the forced vital capacity maneuver in normal subjects depends on the pattern of the preceding inspiratory maneuver and concluded that the time dependency of the -volume curves is consistent with the presence of viscoelastic elements within the respiratory system. The purpose of the present study was to study the viscoelasticity of the trachea and its effects on limitation. Consequently, we measured the choke point TL under controlled conditions of static and forced expiratory to test whether TL stiffness measured during rapid expiratory transients can account for measured differences in maximal and supramaximal .
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METHODS
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Experimental preparation.
Six human tracheae were excised immediately after the death of adult patients who had died from cancer not involving the central airways. The tracheas were cleaned from surrounding tissues and kept refrigerated at 5°C for <48 h before the beginning of the study. The isolated tracheae were suspended on an experimental apparatus (Fig. 1) that simulated Ptm and imaged cross-sections of the trachea at the choke point during static and dynamic conditions. Tracheae were supported in the vertical position inside a clear Plexiglas cylindrical chamber (6-cm inner diameter) that could be pressurized to simulate pleural pressure (Ppl). The ends of the tracheae were mounted over and secured to thin-walled aluminum (laryngeal and carinal) fittings tailored to the largest possible diameter. The laryngeal fitting was connected to the smaller opening of a conical funnel and then to atmosphere via a 3-cm-inner diameter side branch with an air shutter. A glass window covered the larger opening of the funnel, providing an axial view of the inner walls of the trachea. The carinal fitting was connected to a 16-gallon rigid tank that was fed with air via an adjustable pressure regulator. For measurements under static conditions, the carinal end of the trachea was closed, and the laryngeal end was connected to graded levels of negative pressures. To allow unobstructed view over its entire length, a minimal longitudinal tension was applied to the trachea by adjusting the distance between tracheal fitting just enough to overcome any natural bending. Pressure transducers were connected to the tank and the tracheal supporting chamber to measure the equivalent of alveolar pressure (PA) and Ppl, respectively. A tip pressure transducer catheter (Millar Instruments) was introduced into the trachea from its carinal end and supported by a 3-mm-outer diameter steel tube at a location of minimal cross-sectional area during collapse to measure the choke-point pressure. The alveolar tank and the pleural chamber were interconnected to maintain almost equal pressures, thus Ppl 
PA. was measured at the exit of the funnel’s side branch with a pneumotachograph (Jäger, Würzburg, Germany) and a Valadyne differential pressure transducer. The pressure and signals were recorded digitally in a personal computer.
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Fig. 1. Schematic representation of the experimental setup. , flow; Ppl, pleural pressure; Pin, choke-point pressure; PA, alveolar pressure.
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Experimental procedures.
The inner tracheal walls were recorded through the glass window with a black-and-white video camera equipped with a zoom lens. To delineate the walls of a selected cross section of the trachea, a high-intensity flash ring placed around the supporting chamber was pulsed across a 2-mm-wide circular slit as illustrated in Fig. 1. The axial location of the imaged cross section was selected during forced expiration by advancing the illuminated band along the axis of the trachea to the point showing the lowest Atr, which should be close to the choke point. This location was left unchanged for the rest of the study. Acquired video images were digitized with a frame grabber and stored and processed in a personal computer. Atr was calculated from those images with a computer program that counted the pixels inside the illuminated walls (see Fig. 2). Area measurements were calibrated by imaging under identical conditions a drawing of a 1-cm-side square placed at the same distance of the camera from the lighted section. During static measurements of choke point Atr, the carinal end was kept closed and the pleural chamber was kept opened to atmosphere, while progressive levels of negative Ptm were applied at 
5-min intervals to the inner walls of the trachea, as illustrated in Fig. 2, where 
represents Atr measured at the end of each 5-min interval and normalized by Atr at 0 Ptm (Atr0). As a result, a Ptm- relationship at steady-state conditions was attained and referred to herein as the static TL (STL). During forced expiratory maneuvers, the tank was pressurized to progressive levels of PA while the shutter was closed. At time = 0, the shutter was opened, and as the tank emptied Ptm was determined as the difference between measured choke-point pressure and Ppl, and was determined at different time intervals as illustrated in Figs. 3 and 4, where it can be observed how rapidly (0.1–0.15 s) decreased from unity to a point of minimal value as Ptm rapidly varied and then increased at a much slower rate as Ptm reverses and gradually approaches zero toward the end of expiration. This procedure was repeated for increasing levels of initial PA, as illustrated in Fig. 5, where measured Ptm was plotted against measured together with the measured STL, depicted in circles. The triangles illustrate how, during each expiratory maneuver, Ptm and vary with time as the tank empties, whereas the squares denote the TL measured at PEF at the onset of exhalation, referred to herein as the dynamic TL (DTL).
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Fig. 5. Determination of the dynamic TL (DTL). Repetition of the procedure illustrated in Fig. 4 for increasing levels of PA yields the experimental DTL, depicted in squares, measured at the onset of forced expiration.
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Data analysis.
Measured TLs were assumed to follow the nonlinear relationship proposed by Shapiro (11) for elastic tubes:
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where Kp and n are parametric constants related to the stiffness of the tube wall. Solving for , Eq. 1 can also be written as
 | (2) |
and can therefore be calculated from Eq. 2 using the estimated parameters Kp and n obtained from fitting the measured TLs to Eq. 1. As previously shown by Shapiro (11), the maximal rate (max) that can pass through an airway segment is the product of Atr times the local speed of propagation of long pressure waves (C), which depends on the gas density (
) and the slope of the TL and can be calculated as
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As a result, it is possible to calculate theoretical maximal expiratory flows STL and DTL from the STL and DTL, respectively, simply by derivation of dPtm/d from Eq. 1 and subsequent substitution of from Eq. 2 into Eq. 3 as
 | (4) |
where Kp and n represent the parameters previously obtained from fitting the measured STL and DTL to Eq. 1.
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RESULTS
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Equation 1 provided excellent curve fitting to the measured STL and DTL of all tracheae (Table 1), and we found that DTL was always stiffer than STL, as graphically illustrated for all tracheae in Fig. 6. The time dependency of the trachea and its implication on maximal expiratory are exemplified in Fig. 5. This figure displays how, during forced expiratory maneuvers performed at graded levels of initial PA on trachea 4, initially follows DTL while it decreases from unity to a point of minimal value as Ptm rapidly varies during the start of exhalation and then increases as Ptm reverses and gradually approaches STL as Ptm continues to decrease for the remaining of exhalation. The measured expiratory corresponding to the forced expiratory maneuvers shown in Fig. 5 are plotted against measured PA in Fig. 7 together with the theoretical maximal expiratory flows STL (Fig. 7A) and DTL (Fig. 7B) determined from STL and DTL, respectively, of the same trachea. The symbols on the predicted theoretical maximal expiratory represent the point in time at peak expiratory , demonstrating that measured during an exhalation is limited at its peak by DTL in striking contrast to STL, which predicts significantly smaller maximal expiratory . Moreover, we found that DTL was always higher than STL, and, most significantly, whereas STL was never able to explain observed peak expiratory , DTL always did, as graphically illustrated for all tracheae in Fig. 8. The stars represent measured peak expiratory observed at increasing levels of effort, whereas the squares and circles indicate the theoretical max predicted by the DTL and STL, respectively.
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Fig. 6. Graphical comparison between STL (circles) and DTL (squares) of all tracheae demonstrating that DTL was always stiffer than STL. Symbols represent experimental results, whereas lines denote TLs fitted to Eq. 1.
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DISCUSSION
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We measured TLs of six excised human tracheae under steady-state static conditions and at the onset of forced expiration. For this purpose, we designed a simple experimental apparatus able to generate the required Ptm across the trachea while simultaneously allowing accurate imaging of the choke point Atr and measurement of the resulting expiratory . We found that TLs of isolated human tracheae followed quite well the mathematical representation proposed by Shapiro (11) for elastic tubes. Furthermore, our experimental data agree with the derivations of Shapiro showing that a TL with n < 2.0 is required for limitation to occur, in conformity with a previous study in excised sheep and dog tracheae (1). We found that all but one trachea had n < 2 and exhibited limitation, whereas the one trachea (trachea 1) with n > 2 did not (Table 1, Fig. 8). In retrospect, we cannot rule out for trachea 1 that greater effort would have resulted in expiratory limitation, and this was the reason why we decided to double the effort in all subsequent tracheae.
Most significantly, however, is the fact that in all cases studied DTL was stiffer than STL. As a result, the stiffer DTL predicted theoretical maximal expiratory greater than those predicted by the more compliant STL, which in all cases studied failed to explain the measured maximal expiratory measured at the onset of forced expiration (Fig. 8). Whereas in the past other investigators (6) have reported a time-dependent behavior of the TL, no attempts were made to explain its effects on limitation. To test whether pressure waves in the airways propagate at the speed obtained from maximal expiratory , Suki et al. (12) compared phase velocities of oscillatory pressure waves in excised calf tracheae with wave speeds predicted from measured STLs obtained using the acoustic reflection technique. They found that, even though with decreasing frequency the velocity at which sinusoidal pressure waves propagate in the airways approaches wave speed during maximum expiratory , phase velocity is highly frequency dependent and can be much higher than the wave speed over a wide range of negative Ptm. Furthermore, they concluded that frequency dependency of wall tissue properties was the most important contributor to the observed differences between wave speed and phase velocity. In this study, we test whether the viscoelasticity of the tracheal wall can possibly account for the reported differences in maximal expiratory . How the viscoelasticity of the trachea plays a role on predicted maximal expiratory can be easily appreciated on Figs. 4 and 5. As time passes, conditions change from highly unsteady into steady state, and DTL becomes STL with significant implications on allowable maximal expiratory as graphically demonstrated in Fig. 7, where it can be noted how the different measured expiratory curves share a single path toward the end of expiration while they diverge during the beginning of exhalation when conditions are highly unsteady and peak expiratory are reached at different levels of effort. The symbols on the predicted theoretical maximal expiratory curves represent the point in time at PEF. Noticeably in Fig. 7A, the point at which all curves converge coincides with STL, perhaps signaling the point in time at which expiratory transitions into a quasi-steady state. Even though expiratory continues to increase with increasing effort, once PEF reaches DTL (Fig. 7B), a clear maximum is achieved and PEF remains limited at that maximum independently of applied effort. Even though Pedersen et al. (10) concluded that PEF in general is determined by the wave-speed -limiting mechanism, they believed that PEF would increase with increasing effort because wave-speed is reached at a higher lung volume. Our results, however, are in disagreement with this notion and explain why Tantucci et al. (13) were not able to induce a higher PEF value in normal subjects performing expiratory forced vital capacity maneuvers by increasing the driving pressure between alveoli and mouth through the negative expiratory pressure application (7). In the present study, STL was never able to explain the observed expiratory limitation (Fig. 8), hence the so-called supramaximal are correctly labeled if referred to STL, which is derived from STL, but mislabeled if DTL is considered, which is derived from DTL. Our results demonstrate that supramaximal observed at the onset of expiration is in fact still determined by the wave-speed -limiting mechanism.
In view of these results, we conclude that the viscoelastic choke point properties of the tracheal wall can explain the previously referred to as supramaximal observed at the onset of expiration and offer a plausible explanation for the difference between TL results obtained from measurements under steady-state forced expiratory conditions and at peak expiratory during realistic dynamic conditions (8, 10, 13).
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GRANTS
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This work was sponsored by the Deutsche Forschungs-Gemeinschaft through grant IIB5Fr680 and National Heart, Lung, and Blood Institute Grant HL-38267.
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FOOTNOTES
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Address for reprint requests and other correspondence: N. Aljuri, Massachusetts Institute of Technology, 45 Carleton St., E25-335, Cambridge, MA 02142 (e-mail: nikko{at}mit.edu)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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Copyright © 2006 by the American Physiological Society.
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ABSTRACT
METHODS
, Experimental data. The solid line denotes the fitted tube law (TL).
