INTRODUCTION

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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 (1, 2). Such wave speed is a function of the airway's transmural pressure-area relationship [also referred as tube law (TL)] and gas density (10). At high lung volumes, wave speed is generally reached within the trachea or the main bronchi. On theoretical grounds, it has been proposed that, depending on upstream viscous losses, expiratory flow could become limited at gas velocities lower than the corresponding local wave speeds (7, 10). Furthermore, if the tracheal TL is such that, with increasing effort local, wave speed increases faster than gas velocity, then flow might never become limited by the trachea. Under those conditions, flow might still become limited, but the limiting segment (also called the choke point) would occur at a more compliant peripheral location. Therefore, depending on the morphology of the tracheal TL and the magnitude of the upstream viscous losses, tracheal expiratory flow () at maximal effort could have three different behaviors: limited by wave speed, limited by viscosity, or not limited. Given this dependency, we set out to test the theoretical relationship between choke point TL and expiratory flow limitation using excised animal tracheae of various stiffness and under various degrees of upstream airway resistance (Raw).

To isolate the flow limitation phenomena from unsteady changes in lung recoil and Raw that take place as a result of lung deflation, we used excised tracheae mounted in an apparatus that simulated a constant Raw and allowed quasi-steady levels of alveolar (PA) and pleural (Ppl) pressures.

Contrary to general belief, all the measured expiratory flows obtained during forced expiration were found to be higher than the theoretical maximal expiratory flows derived from the TL measured under static conditions. A possible explanation for this disagreement, as some investigators have pointed out (11, 12), is that increased wall tension caused by downstream migration of the choke point during expiration would effectively result in a choke point TL that would be stiffer than that measured at the same point under the no-flow condition. As a result, higher wave speeds and expiratory flows than those predicted by a TL measured under no-flow (static) conditions could be expected. With this in mind, we imaged the cross section of the trachea at the choke point and estimated the tracheal TL with digital planimetry under static and forced expiratory flow conditions with the assumption that the pressure downstream of the choke point was equal to atmospheric pressure (4, 10). However, we found that most forced expiratory flows measured were still higher than the theoretical maximal flows derived from the TL assessed when static pressure recovery downstream of the point of minimal cross-sectional area (A_tr) was neglected. Therefore, we considered two additional hypotheses: 1) that the area expansion downstream of the choke point was sudden, with boundary layer separation and conservation of momentum and 2) that the area expansion occurred gradually with neglect of viscous losses and conservation of energy.

Glossary

Atr	Tracheal cross-sectional area
Atr0	Atr at Ptr = 0
	Normalized cross-sectional area
C	Wave speed
U	Flow velocity
S	Speed index
	Expiratory flow
	Theoretical frictionless maximal expiratory flow
	Theoretical maximal expiratory flow
Ppl	Pleural pressure
Ptm	Transmural pressure
PA	Alveolar pressure
Pin	Choke point pressure
Kf	Friction factor
	Gas density
TL	Tube law
STL	Static TL
DTL	Dynamic TL
	Pressure recovery term
Raw	Upstream airway resistance
MR	Moderate Raw
HR	High Raw
Kp,n	Parametric factors related to the stiffness of the tube wall

	THEORETICAL CONSIDERATIONS

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Total static pressure loss. Consider the trachea as a collapsible tube and that all upstream pressure losses from the alveolus to the carina can be attributed to a single resistance Raw. For simplicity, let us assume that there is no elastic recoil; thus PA = Ppl. Also, disregard static pressure recovery from the point of minimal cross-sectional area (choke point) to atmosphere; then the static pressure in the choke point (P_in) equals atmospheric pressure = 0. Accordingly, the overall pressure drop or driving pressure is equal to

(1)

where P_up = Raw · ² is assumed to represent the upstream pressure loss, U is the air velocity,  is the density of air, and K_f is a turbulent friction factor that accounts for the local losses during the contraction of flow along the collapsing segment and is a function of tube geometry and Reynolds number. From Eq. 1, we may obtain

(2)

where the total loss factor (L) may be calculated as

(3)

With the aforementioned assumptions, the Ptm difference = P_in  Ppl = PA. A local TL, representing the relationship between Ptm and A_tr normalized by the cross-sectional area at Ptm = 0 (A_tr0), is assumed to follow the relationship proposed by Shapiro (10)

(4)

where  = (A_tr/A_tr0) and K_p and n are parametric constants related to the stiffness of the tube wall.

(5)

(6)

(7)

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Plots of expiratory flow () vs. alveolar pressure (PA) with increasing upstream airway resistance [Raw; subcritical (dashed line) and supercritical (dotted line)]. Solid curve, theoretical maximal expiratory frictionless flow (max = C · Atr, where C is wave speed and Atr is tracheal cross-sectional area) plotted against PA. A: frictionless (Kf = 0). B: including friction (Kf > 0). Dotted line, theoretical maximal expiratory flow ( = C · Atr/). S, speed index; Kf, friction factor.

(8)

View larger version (14K): [in this window] [in a new window]   Fig. 2.   and maximal experimental normalized by vs. parameter n of Eq. 4. Dotted line, theoretical with no Raw; dashed line, with moderate Raw (MR); solid line, with high Raw (HR); ×, experimental with MR; , with HR.

Including static pressure recovery. If the air flowing through a collapsible tube is treated as steady and one dimensional and it is assumed that downstream from the choke point the expansion is sudden, boundary layer separation occurs until the flow gradually returns to a fully developed condition. For this particular flow, conservation of mass requires that

(9)

where U is the air velocity at the choke point and U₀ is the velocity at the point along the tube where the airflow returns to a fully developed condition. If it is further assumed that the average shear stress on the wall can be neglected because the fluid motion near the wall is very sluggish and random, the momentum equation can be written as

(10)

where P_in is the pressure at the choke point and P₀ is the pressure where the flow returns to a fully developed condition. Combination of Eqs. 9 and 10 yields

(11)

If we assume, however, that the expansion downstream from the choke point occurs gradually and disregard any viscous losses, then conservation of energy requires that

(12)

and combination of Eqs. 9 and 12 yields

(13)

Accordingly, depending on the assumption regarding the character of the area downstream of the choke point and assuming further that the upstream viscous losses are represented by a more realistic relationship (8)

(14)

Equation 1 can be written as

(15)

where the parameter  corresponds to the fraction of the stagnation pressure recovered by the flow along the expansion downstream of the choke point:  = 2  2² for a sudden area expansion leading to boundary layer separation, which we will refer to as minimal pressure recovery, and  = 1  ² for total static pressure recovery with no flow separation. Because of this pressure recovery, Ptm no longer equals PA, but

(16)

Finally, solving for in Eq. 15 after substitution of Eq. 16 yields

(17)

(18)

	METHODS

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Experimental preparation. Approval of the Animal Care Committee of the Massachusetts General Hospital was obtained for use of the animals and the protocol used for killing them. Tracheae from three dogs and two sheep were excised after the animals were killed with an intravenous mixture of KCl and pentobarbital sodium. The tracheae were cleaned from surrounding tissues and kept refrigerated for <24 h before the beginning of the study. The tracheae were suspended on an experimental apparatus that simulated upstream viscous losses and Ptm in static and dynamic conditions (Fig. 3). The tracheae were supported in the vertical position inside a clear Plexiglas cylindrical chamber (4 cm ID) that could be pressurized to simulate Ppl. The ends of the trachea 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 2-cm-ID side branch. 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 one or two flow resistors in series made of packed 0.5-mm-ID 10-cm-long glass tubes for moderate resistance (MR) and high resistance (HR), respectively. The free end of the resistors was then connected to a 16-gallon rigid tank that was fed with air via an adjustable pressure regulator. To allow unobstructed view over its entire length, a minimal longitudinal tension was applied to the trachea by adjusting the distance between tracheal fittings just enough to overcome any natural bending.

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Schematic representation of experimental setup. Ppl, pleural pressure.

Assessment of A_tr. The inner tracheal walls were recorded through the glass window with a black-and-white videocamera 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 (Fig. 4). The axial location of the imaged cross section was selected during steady by advancing the illuminated band along the axis of the trachea to the point showing the lowest A_tr, which should be close to the choke point. This location was left unchanged for the rest of the study.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Plots of relationship between transmural pressure (Ptm) and normalized tracheal cross-sectional area ( = Atr/Atr0) measured under static conditions with corresponding images. , Experimental; solid line, fitted to Eq. 4.

View larger version (20K): [in this window] [in a new window]   Fig. 5.   Plots of relationship between normalized transmural pressure (Ptm/Kp) and normalized tracheal cross-sectional area ( = Atr/Atr0) measured under static conditions. ×, dog 1; +, dog 2; , dog 3; *, sheep 1; , sheep 2. Lines represent corresponding curves fitted to Eq. 4: dog 1 (n = 1.0), dog 2 (n = 0.3), dog 3 (n = 1.4), sheep 1 (n = 0.5), and sheep 2 (n = 2.1).

View larger version (18K): [in this window] [in a new window]   Fig. 6.   A: plots of vs. applied PA in trachea of dog 3. Solid line, experimental . Fitted model of with use of experimental Atr and corresponding PA with assumption of no (Kf = 0, ×), minimal (Kf = 0, +), and total static pressure recovery (Kf = 0.27, ) is shown. Fitted model of with use of fitted relationship of Atr and PA with assumption of no (Kf = 0, dotted line), minimal (Kf = 0, dashed-dotted line), and total static pressure recovery (Kf = 0.27, dashed line) is shown. B: logarithmic plots of mean squared errors between fitted model and experimental data of vs. Kf for cases of no (dotted line), minimal (dashed line), and total static pressure recovery (solid line) in trachea of dog 3 with moderate Raw.

	RESULTS

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Static and dynamic TLs. Equation 4 provided excellent curve fitting to the experimental measurements of  (mean r² = 0.99). There was, however, substantial variability in K_p (range 7.1-36) and n (range 0.32-2.1) between the different tracheae. Dynamic TLs (DTLs) were stiffer than static TLs (STLs) for the case of total recovery ( = 1  ²). In partial or no recovery, DTLs were, in some cases, more compliant, as shown by the parameter n in Table 1.

TLs and flow limitation. We found that the fitting of the model with the assumption of minimal (mean r² = 0.81) and no pressure recovery (mean r² = 0.71) was poorer than that with the assumption of total recovery, which yielded excellent fitting (mean r² = 0.996; Fig. 7A). The poor fitting for the minimal or no-pressure recovery assumption was caused by a consistent underestimation of , despite a consistently smaller K_f than that found for the total pressure assumption (Fig. 7B).

View larger version (21K): [in this window] [in a new window]   Fig. 7.   A: logarithmic plots of minima of mean squared errors between fitted model and experimental data of in all tracheae for cases of no, minimal, and total static pressure recovery. , Minima of mean squared errors averaged over all tracheae. B: comparison between fitted model and experimental data of maximal for cases of no (*), minimal (), and total static pressure recovery (+).

View larger version (23K): [in this window] [in a new window]   Fig. 8.   Plots of fitted model and experimental data of vs. PA in sheep 1 (A) and sheep 2 (B). Dotted lines, experimental with MR and HR. Fitted model of with use of experimental Atr and corresponding PA with assumption of total pressure recovery with MR () and HR (*) is shown. Also shown is fitted model of with use of fitted relationship of Atr and PA with assumption of total pressure recovery with MR (solid line) and HR (dashed line). Theoretical maximal frictionless expiratory flow (max = CSTL · Atr) is derived from static tube law (STL; +), and theoretical maximal frictionless expiratory flow (max = CDTL · Atr) is derived from stiffer dynamic tube law (DTL) estimated assuming total pressure recovery with MR () and HR (×).

	DISCUSSION

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We formulated a theoretical model to test the relationship between choke point TL and maximal measured from excised animal tracheae at various degrees of Raw. For this purpose, we developed a simple experimental apparatus able to generate the required Ptm across the trachea while simultaneously allowing accurate imaging of the choke point A_tr. Because of the limited number of tracheae, one cannot characterize the elastic properties of dog or sheep tracheae. However, given the substantial (mechanical) differences among the five tracheae studied (Fig. 5) and the excellent fitting obtained with the model in all conditions, this limited number of studies gives sufficient evidence for the validity of the model relating TLs and flow limitation.

Jones et al. (4) used a similar imaging technique with a fiber-optic bronchoscope to measure A_tr. This technique enabled them to assess the effect of time on airway compliance and to estimate the maximum flow from the measured TL of the compressed airway segment. Their study already suggested that this TL could determine maximum flow and, therefore, pointed out the importance of the experimental assessment of the airway's TL.

The most important findings of our study are as follows: TLs of isolated tracheae followed quite well the mathematical representation proposed by Shapiro (10) for elastic tubes (Eq. 4). The TL assessed under quasi-steady flow conditions (DTL) was stiffer than the TL measured under quasi-steady no-flow conditions (STL) only if a significant pressure recovery was assumed. Only under these conditions could the measured higher than those predicted by a STL be explained. We thus concluded that static pressure recovery from the choke point to atmosphere must have been substantial. Furthermore, we found that upstream viscous losses, represented by Raw, did affect maximal in these isolated tracheae.

Because our imaging technique allows us to assess the TL at a single axial location, we cannot rule out the possibility that the point of maximum A_tr collapse during forced exhalation might have migrated axially, depending on the applied driving pressure. However, because the high-intensity flash ring used to define (delineate) the inner walls of the trachea illuminated a relatively wide (~10-mm) section of the tracheal wall, it is likely that small migrations of the maximal collapse point did not affect our results, since only the minimal cross section was captured by the camera. The choke point TL was assessed only during progressively increasing driving pressure until the highest quasi-steady with our apparatus was obtained. Significant hysteresis was observed during the decreasing phase of the flow, with lower flows for the same driving pressure, an observation described previously by Knudson and Knudson (6).

We found that in all cases studied the TL with A_tr measured during flow conditions and calculated with the assumption of no static pressure recovery was more compliant than the STL. This behavior could not be explained by any effects of longitudinal tension (11, 12), an unstable configuration of the flow-limiting segment during the transient (7), or effects of time on the change in A_tr (4). All these explanations would lead to the opposite result: a stiffer TL measured under flow conditions (DTL). We have to conclude that the assumption of Ptm = PA is erroneous. A fact supporting the hypothesis that a DTL should be stiffer than the STL is that, according to the wave-speed theory, the maximum frictionless flow that can pass through an airway segment is given by C · A_tr. In most of our experiments, exceeded this limit when the STL was used to derive C but never when the stiffer DTL, estimated assuming total pressure recovery, was used.

We also found that the model described by Eq. 17 could not describe the experimental data unless a significant fraction of the static pressure is recovered downstream of the choke point. Even for K_f set to 0, the calculated with the assumption of minimal recovery underestimated the observed (Fig. 7). Only for the case where total static pressure recovery was assumed, positive values of K_f were found and the model predicted well the experimental data. We cannot rule out and it is, in fact, likely that a fraction of the static pressure may have been lost downstream from the choke point. This would imply that our values of K_f were somewhat overestimated but still yield and excellent fit, since the pressure loss term includes the parameters  and K_f with opposite signs (Eq. 18). On the other hand, if one assumes a gradual contraction angle (30-60°), K_f should lie between 0.02 and 0.07 (3), and  can then be calculated by substituting K_f into Eq. 17 and solving for . Figure 9 shows a representative case for this calculated  with these two values of K_f plotted together with the theoretical predicted bounds for the cases of total and minimal static pressure recovery. These results agree with those of Kececioglu et al. (5), who found that  generally fell between the limiting curves of total pressure recovery and minimal pressure recovery for short and intermediate shocks for collapsible penrose tubes. We found, however, that the calculated values of  fell out of the theoretical predicted bounds for  > 0.5. This is probably because friction losses due to tracheal collapse are insignificant compared with the other pressure drops in the system at these low levels of collapse. Thus, because the actual dependence of  on  is not known and may vary from trachea to trachea, it is not possible for this model to predict on the basis of values of K_f estimated from first principles. It seems clear nonetheless that the static pressure from the choke point to atmosphere is not totally lost, as assumed previously (1, 2, 7, 8, 10-12). The stiffness of the tracheal cartilaginous rings and longitudinal tension forces probably prevented a sharp transition of A_tr, allowing significant static pressure recovery.

View larger version (21K): [in this window] [in a new window]   Fig. 9.   Plots of  vs.  in trachea of sheep 1 with MR. Solid line, theoretical predicted bound for case of total static pressure recovery; dashed line, theoretical predicted bound for case of minimal static pressure recovery.  was derived from Eq. 17 for Kf = 0.02 () and Kf = 0.07 (*).

Our experimental data agree qualitatively with the derivations of Shapiro (10) showing that for a very high Reynolds number a TL with n < 2 is required for flow limitation to occur. Lambert (7) also showed that, for wave-speed limitation to occur, m (the reciprocal of n) should be >0.5. We found that all but one trachea had n < 2 in the STL and exhibited limitation with MR and HR. The trachea of sheep 2 (n = 2.1) did not exhibit a plateau at MR or HR (Fig. 8). If an Raw is included in the simplified model proposed by Shapiro, then flow limitation still occurs for TLs with 0 < n < 2 and decreases as Raw is increased (Fig. 2). This theoretical result also agrees with our data showing that, in all cases, measured decreases with increasing Raw. However, measured and theoretical differ quantitatively, because this theoretical result does not include static pressure recovery. Excellent agreement could be achieved only if a substantial degree of static pressure recovery from the choke point to atmosphere took place, as discussed previously (Fig. 7B).

Elliot and Dawson (2) tested the wave-speed theory assuming frictionless and found values that were somewhat lower than C · A_tr. Given that the experimental apparatus used by Elliot and Dawson did not include Raw, it is not surprising that the observed maximal were found close to, although slightly lower than, C · A_tr. Although they attributed this result to possible errors in the measurements of A_tr, the small departure of from C · A_tr could have also been explained if K_f was nonzero (Eq. 6). In our experiments, always limited at values lower than C · A_tr because of the high upstream viscous losses generated by the MR and HR.

We conclude that an exponential form of the TL similar to that proposed by Shapiro (10) for collapsible tubes and Lambert (7) for airways characterized well the compliant properties of excised tracheae measured under static and flow conditions. Experimental data obtained from excised sheep and dog tracheae showed that flow limitation occurred only in tracheae with n < 2, whereas no flow limitation was reached for tracheae with n > 2. We developed a simple lumped-parameter model of limitation that included the effects of upstream and choke point friction pressure losses. The model agrees with wave-speed theory in the limit when Raw and K_f  0 but predicts  < C · A_tr for the more realistic cases where Raw and K_f > 0. This prediction is confirmed by experimental data. These data strongly suggest that a large fraction of the static pressure is recovered downstream of the choke point.