INTRODUCTION

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THE IMPORTANCE OF TRANSPORT and deposition of fine aerosol particles (0.1-1 µm) deep in the lung is becoming increasingly apparent. For example, recent epidemiologic studies have shown that exposure to these particles is strongly associated with mortality and morbidity (6, 12, 25). In addition, the use of small particles for therapeutic drug delivery deep in the lung is undergoing increasing development (13). Nevertheless, the mechanisms governing the transport of particles in this size range from tidal air to residual alveolar gas are not understood. The purpose of this study was to evaluate by means of a physical model the role of "stretch-and-fold" convective mixing in quasi-reversible cyclic ventilation within the pulmonary acinus (2).

Fine particles basically follow the gas flow, because their inertia is not sufficient to effect crossing of streamlines. The characteristics of gas flow in the acinus are therefore critical in determining particle motion. The low-Reynolds-number (Re) gas flow deep in the lung is mainly governed by acinar wall motion, and the principal mode of lung expansion approximately satisfies geometric similarity (1, 7, 8, 15, 23). To the extent that this is true, acinar flow is kinematically reversible (22). Consequently, it has been long assumed that there is little flow-induced mixing in the acinus; it follows that, for fine inhaled particles, most will be exhaled without being mixed with residual alveolar gas and will show negligible deposition within the acinus (5, 21, 24). However, recent experimental studies using bolus techniques (10) or performed in a microgravity environment (4) suggest that fine particles can mix substantially with the residual alveolar gas and deposit deep in the lung, despite their low diffusivity and sedimentation rate. These observations suggest that flow-induced mixing (irreversibility) must occur in the acinar region, which is then responsible for enhanced acinar deposition.

It is important to distinguish between the kinematic irreversibility of the flow field and the irreversible nature of diffusive mixing. Kinematic irreversibility, as used throughout this study, explicitly means that at the end of one breathing cycle the trajectories of tracer particles do not return the particles to their initial configuration. We shall see that this can arise from boundary hysteresis, but, in this context, kinematic irreversibility does not imply that a subsequent cycle of strictly time-reversed boundary motion would not return all trajectories to their initial positions (like a motion picture played backward). This is different from the irreversibility of diffusive origin, insofar as no fluid flow pattern can unmix a diffusively smeared tracer pattern.

Direct experimental evidence of convective mixing of fluids in the lung periphery has recently been presented (19). In ventilated excised rat lungs filled with polymerizable silicone fluids of two colors, we demonstrated that the acinar flow is indeed irreversible and exhibits highly convoluted stretch-and-fold streak patterns. These experimental observations were followed by a theoretical analysis demonstrating the potential significance of stretch-and-fold convective flow in acinar aerosol mixing when it is coupled with diffusion (2).

The origin of stretch-and-fold mixing pattern in the acinus is not clear. We have suggested that chaotic mixing may occur as a result of the presence of saddle points in the acinar flow field (20). Two factors of acinar geometry are key in this mechanism: structural alveolation and the rhythmical expansion and contraction of the alveolar walls. Another possibility for the origin of this type of mixing is a small departure or perturbation in boundary motion from perfect reversibility. For instance, Taylor (18) gave a classic demonstration of perfectly reversible low-Re flow, which could be made completely irreversible merely by the introduction of a very small disturbance in the flow field (see also Ref. 3). Recently, Miki et al. (15) reported a small but consistent geometric hysteresis (i.e., temporal asynchrony) in lung expansion during tidal ventilation. [Note that this is geometric hysteresis, rather than the more common usage of hysteresis associated with the pressure-volume (P-V) loop. Although P-V hysteresis almost certainly contributes to the origin of geometric hysteresis, the consequences of the latter are the focus of this study.] Following Taylor's demonstrations, we reasoned that geometric hysteresis might also induce a substantial acinar flow irreversibility, even though lung geometry during ventilation appears approximately reversible. In turn, such irreversibility may be an important factor in the mixing and ultimate deposition of fine particles.

We tested the hypothesis that even a small amount of geometric hysteresis is sufficient to induce flow irreversibility and evaluated its physiological potency as a mechanism of convective mixing for fine particles deep in the lung. We used a large-scale T-shaped (TEE) junction as a model of an alveolar duct with alveoli (16), created the same degree of boundary hysteresis observed by Miki et al. (15) in live rabbits ventilated tidally, and performed flow visualization studies. The results showed that, with continuing cyclic ventilation, even a small degree of asynchrony could induce a largely irreversible flow with increasingly complex wavy patterns superposed on a net circulation. The evolving tracer displayed a progressively finer lateral distance scale between adjacent tracer streaks. The coupling of this convective pattern with diffusion resulted in a relatively narrow time interval over which a sharp increase in mixing occurred. This confirmed the idea of stretch-and-fold convective mixing proposed by Butler and Tsuda (2) and suggests its importance for the transport of fine particles deep in the lung.

	METHODS

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Apparatus and experimental setup. Acinar fluid dynamics in humans were simulated in a large-scale TEE junction model similar to that used in our previous work (16). The model consisted of a section of square duct (2 cm side), with pistons at each end, connected to a vertical column (2-cm square duct) in the middle (Fig. 1). The system was filled with silicone oil [kinematic viscosity () = 100 cm²/s, diffusivity (D) = 1.5 × 10⁸ cm²/s]. Low-Re cyclic flow was generated in the system by the two pistons, the position and motion of which were independently driven by a combination of an automatic linear stage (model LTS-100X, Sigma Koki) and stepping motor assembly (model PH556-Q, Oriental Motor). The actual motion of the pistons was precisely controlled by a personal computer (model PC9801VX, NEC).

View larger version (21K): [in this window] [in a new window]   Fig. 1.   T-shaped (TEE) junction-acinar duct model: experimental setup. PC, personal computer, , kinematic viscosity.

Wall motion asynchrony. To mimic the geometric hysteresis observed in the lungs of live rabbits by Miki et al. (15), we drove the left and right pistons asynchronously. The degree of hysteresis in data of Miki et al. and the degree of hysteresis in our TEE junction experiments were matched as follows (Fig. 2). Consider a Lissajous representation of two variables. (In the more common hysteresis literature, these are typically P-V loops or force-length loops. In our case, they are two geometric degrees of freedom; we are not trying to match P-V hysteresis.) Hildebrandt (11) and Peslin and Fredberg (17) defined an effective phase angle () for hysteresis loops as follows where A_loop is the area of the loop generated in the Lissajous figure and A_box is the area of the circumscribed rectangle. (Note that because this is a dimensionless ratio, the same equation can be applied to any pair of variables describing a closed loop.) This definition has the desirable property that it reduces to the true phase difference of two sinusoidal variables, where the loop is an ellipse, and therefore can be used in the more general setting of nonelliptical loops. Mindful of the fact that Miki et al. only made measurements at the VT extremes and near the volume midpoints (Fig. 2A) and that, therefore, the actual shape of the loop remains unknown, we chose to induce hysteresis in the pistons' motion by describing a similar diamond-shaped Lissajous figure. As shown in Fig. 2B, the two variables are the displacements of the right and left piston. We quantified the pistons' hysteresis by the acute angle at either extreme vertex. It may be shown that an ellipse with this phase angle, inscribed in the box, will match the diamond's width (i.e., pass through the obtuse vertices). In this sense the diamond's acute angle is an "equivalent" phase to the ellipse and, therefore, to the nonelliptical data of Miki et al. Data of Miki et al. were analyzed after the fashion of Hildebrandt described above; we found that the degree of geometric hysteresis in normal rabbits during tidal ventilation was ~10°. For the piston hysteresis, we therefore chose acute diamond angles of 5, 10, and 20° to span observations of Miki et al. on both sides.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Lissajous representation of geometric hysteresis. A: geometric hysteresis in live rabbit lungs during tidal ventilation observed by Miki et al. (15). S/V, surface area-to-volume ratio; FRC, functional residual capacity; VT, tidal volume; insp, inspiration; exp, expiration. B: geometric hysteresis in TEE junction. Degree of hysteresis in TEE junction was matched to degree of hysteresis in data of Miki et al. by equating effective phase angle () of hysteresis loops.

View larger version (14K): [in this window] [in a new window]   Fig. 3.   Displacement of pistons (XL and XR) plotted as a function of time (t). Motion of each piston was plotted, such that inspiratory displacements are positive for both. Period of cycle was set at 16 s.

Data processing. To quantify the kinematics of the induced flow, we used a colored tracer. The two pistons were initially 8 cm from the center of the vertical duct, and silicone oil colored by a pigment (Thermoplast Red, BASF Japan) was introduced manually with a syringe at the middle of the TEE junction, in a pattern described below. The tracer pattern was photographed at given intervals, typically at the end of each successive cycle, and was also monitored by a videocamera (model IK-636, Toshiba) for later analysis. The patterns were processed by image analysis software (Photoshop, Adove); the evolving tracer lengths were measured by tracing their profiles on a computer screen with a mouse and associated software.

	RESULTS

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In all studies the tracer was introduced as three lines at the boundary of the three ducts with the central core. Figures 4, 5, 7, and 9 show representative photographs of the evolving tracer pattern for several cases of interest, including part of the control studies and representative experiments with mild asynchrony and different VT. The most important characteristic common to all the patterns (with the exception of the control experiments; see below) is that the induced flow is irreversible. This is shown through two features. 1) With increasing cycle number (N), there is a generally increasing complexity in the swirls of tracer apparent in the joining regions (i.e., those within the 3 separate duct branches). 2) There is a net clockwise circulation of tracer in the core region. We chose to quantify the evolving irreversible characteristic by the length of tracer (L) as function of N. Note further that as L increases, the lateral separation of tracer trajectory profiles systematically decreases; this will form part of the basis for our conclusions about the contribution of low levels of boundary hysteresis to mixing and transport.

View larger version (111K): [in this window] [in a new window]   Fig. 4.   Tracer patterns in control experiment (VT/Vduct = 2, where Vduct is duct volume;  = 0). Cycle-by-cycle tracer motion was essentially reversible, even after 10 cycles (L/L0 = 1.05 at N = 10, where L is tracer length and N is number of cycles; this is a negligible lengthening of tracer compared with that observed when   0).

View larger version (82K): [in this window] [in a new window]   Fig. 5.   Tracer patterns at low VT (VT/Vduct = 2) for 3 levels of hysteresis (). Patterns were observed at end of indicated N. Complexity of circulating tracer patterns increased with increasing N and increasing . Despite presence of geometric hysteresis in boundary motion, there was no rotation per se in boundary and, therefore, no clear relationship to fluid circulation. Surprisingly, tracer patterns were essentially the same at common values of N, implying that pattern was a single-valued function of cumulative hysteresis of boundary motion.

Flow irreversibility can arise not only from the controlled level of asynchrony but also from various artifacts associated with the physical model, such as inaccuracy in positioning of the pistons, mechanical vibration, fluid inertia, or small transient effects. To assess this, we performed control measurements of the evolving tracer length with synchronous ventilation, i.e.,  = 0. Figure 4 shows the tracer pattern after N = 0 (initial), 1, 5, and 10. The observation that the pattern returns essentially to its initial configuration even after N = 10 suggests negligible lengthening of the tracer from artifactual sources.

For the low-VT case (VT/V_duct = 2.0), the results of tracer elongation for three different levels of asynchrony ( = 5, 10, and 20°) and at a different N are shown in Fig. 5. Inspection of the patterns reveals a surprising and unexpected finding. Although the patterns increase in complexity with increasing N and increasing , they are essentially the same, even including fine details, for (N,) = (8,5°), (4,10°), and (2,20°) (Fig. 5, middle). Similarly, for (N,) = (20,5°), (10,10°), and (5,20°), the patterns shown in Fig. 5, right, are strikingly similar. This implies that the nature of the increasing complexity of the swirling and circulating tracer pattern is not independently a function of N and level of asynchrony. Rather, it shows that the pattern and, a fortiori, the length (L) are a function dependent almost entirely on the product N, i.e., the cumulative asynchrony presented by the boundary motion. The data of L were therefore plotted against N and are shown in Fig. 6, which shows not only the collapse of the data to a single function of N but also the interesting feature that it is approximately linear. This observation in turn implies that the tracer stretching is an additive function of cycle-by-cycle asynchrony. Note that this is different from the subset of the stretch-and-fold convective mixing treated by Butler and Tsuda (2), where the lengthening is exponential. However, this difference, although substantive, makes only a qualitative difference in the potentiation of transport through convective mixing (see DISCUSSION).

View larger version (25K): [in this window] [in a new window]   Fig. 6.   L normalized by original length (L0) plotted against cumulative hysteresis for low VT (VT/Vduct = 2). L/L0 collapsed as a function of N, i.e., net cumulative boundary hysteresis.

For the high-VT case (VT/V_duct = 4.0), the results of the tracer patterns for two levels of asynchrony ( = 5 and 20°) and at different N are shown in Fig. 7. As in the low-VT case, the complexity of the wavy pattern increased with increasing N and increasing , but, in contrast to the low-VT case, the patterns did not collapse into a single function of N. This is also seen in Fig. 8, which shows L/L₀ as a function of N. The increasing complexity with N still shows an approximately linear dependence on N, but the increasing slope of L vs. N with increasing  implies that L is no longer linearly related to  itself. Thus the general feature of additivity with N remains, related presumably to the general circulatory pattern induced by the boundary motion. However, the surprising observation in the low-VT case that the cumulative circulation was a single-valued function of the cumulative boundary hysteresis does not hold for the high-VT case.

View larger version (54K): [in this window] [in a new window]   Fig. 7.   Tracer patterns at large VT (VT/Vduct = 4) for 2 levels of hysteresis () shown at end of indicated N. Similar to low-VT case, complexity of swirling pattern increased with increasing N and increasing , but patterns did not collapse into a single function of N.

View larger version (26K): [in this window] [in a new window]   Fig. 8.   L normalized by L0 plotted against cumulative hysteresis for large VT (VT/Vduct = 4). Data did not collapse but showed a nonlinear dependence on .

One of the predictions made by Butler and Tsuda (2) is that when convective stretching is coupled with diffusion, there is an evolution of tracer that retains its separate identity (and thus shows little true mixing) up to a time when the shrinking lateral separation of trajectories matches the slowly growing diffusive length scale. At this time, there is a sudden "entropy burst" with subsequent rapid mixing. To test this prediction, we cycled the TEE model for many cycles and attempted to quantify the N near which we could no longer discern clear tracer profiles. Figure 9 (left) shows the highly complex but still distinguishable pattern of tracer trajectory at N = 400. Figure 9 (right) shows the pattern at N = 700, where it is clear that essentially complete mixing has already taken place. We conclude from this 1) that little actual mixing has taken place by N = 400 and 2) that the time for the burst of mixing occurred sometime between N = 400 and N = 700. This observation is compared with the quantitative predictions of Butler and Tsuda in the DISCUSSION.

View larger version (54K): [in this window] [in a new window]   Fig. 9.   Tracer patterns (VT/Vduct = 2,  = 10°) after 400 and 700 cycles. Delineated tracer lines can be clearly seen even after N = 400, but tracer appears to have been truly mixed by N = 700, suggesting that mixing occurred somewhere between N = 400 and N = 700.

	DISCUSSION

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Kinematics of the induced-flow field. The most important finding of these studies, with particular relevance to aerosol mixing and transport deep in the lung, is that even a small amount of geometric or boundary hysteresis is sufficient to induce flow irreversibility. We emphasize that this phenomenon is not directly associated with P-V hysteresis, even though inhomogeneities in the local P-V characteristics will cause geometric or boundary hysteresis. The flow irreversibility we observed occurred even with Re and Womersley parameter significantly <1 and with purely cyclic boundary motion, but with a slight asynchrony between the two moving walls quantitatively comparable to the small geometric hysteresis observed in the lungs of live rabbits by Miki et al. (15).

Mixing. In any cyclic fluid motion where there is a systematic lengthening of tracer along the direction of the local velocity vectors, conservation of mass and incompressibility demand that there be a concomitant narrowing of the distances between adjacent fluid layers. This in turn can be a potent mechanism for mixing and is at the heart of the proposal of Butler and Tsuda (2). In general, the characteristic distance over which diffusive mixing will occur, with or without convection, increases like to , where D is the particle diffusion coefficient and t is time. In our approximately two-dimensional TEE model, the length of the tracer lines increases roughly linearly, i.e., L = L₀(1 + t), and so the separation distance (s) between tracer lines will decrease roughly hyperbolically, i.e., as s = s₀/(1 + t). At any time t, we therefore have two distinct lengths to consider. The increasing diffusive length is the scale over which true mixing takes place, in the sense that the entropy has significantly increased and the effects are strictly irreversible. The convectively determined decreasing separation lengths are kinematic in origin and do not represent true mixing, in the sense that a perfectly time-reversed flow field would restore the original tracer pattern. At a time when these two lengths are comparable, different portions of the fluid that are brought into proximity given by the narrowing convective scale, in fact, become mixed by diffusion. In other words, diffusive mixing, which, in the absence of convection, would take an extremely long time (because of the low diffusivity of the aerosol), can become significant if convection is continually resupplying other trajectories separated by distances over which diffusion is appreciable. This is essentially the argument described more fully by Butler and Tsuda (2).

View larger version (31K): [in this window] [in a new window]   Fig. 10.   Mixing length scales plotted against time. When narrowing separation distance between adjacent tracer lines (descending solid line; data from VT/Vduct = 2,  = 10°; oscillation period = 16 s) becomes comparable to growing diffusive length of tracer (ascending solid line; Ldiff = , where D is diffusivity and t is time and D = 1.5 × 108 cm2/s), a substantial mixing occurs in TEE junction-acinar model (Fig. 9). Scaling these results observed in TEE junction model to aerosols with particle diameters (Dp) in 0.1- to 1.0-µm range (ascending lines) and to acinar fluid flow [descending (with convection) and constant (without convection) dashed lines], we see that time at which significant mixing occurs (intersection of diffusive and convective length curves) is substantially reduced. This implies enhanced aerosol transport in human pulmonary acinus and shows potential importance of "stretch-and-fold" convective mixing mechanism.

Physiological implications. These same results can now be scaled to the physical properties of real aerosols in terms of their diffusivities in air and to the physical properties of real acini in terms of their geometric size. The essence of this scaling argument is that the functional forms for the growth of diffusive lengths and the decrease of trajectory separation distances are independent of particle size and acinar or model channel size; only the constants of proportionality are different. In particular, the diffusivity of aerosol particles is approximately given by the Einstein relation D = k_BTC/3dµ, where k_B is Boltzmann's constant, T is absolute temperature, C is the Cunningham correction factor, d is particle diameter, and µ is the fluid viscosity. The analogous lines for the growth of the diffusive length scales for 0.1- and 1.0-µm aerosol particles are shown in Fig. 10. Similarly, if the flow in the acinus is similar to the flow in the model, then the acinar convective length scales will be equal to those of the TEE multiplied by the ratio of the characteristic size of the acinus and the TEE. We have shown such a plot for an acinus with a typical characteristic diameter of 200 µm.

Conclusion and summary. Using an in vitro TEE junction model of the alveolated pulmonary acinus, we have shown that even small amounts of geometric hysteresis, quantitatively similar to that observed in live rabbits by Miki et al. (15), can induce highly complex flow fields and patterns. In turn, these patterns enjoy the stretching and folding properties postulated by Butler and Tsuda (2) to account for substantial increases in aerosol mixing, far beyond that which would be predicted on the basis of the low diffusivity and mass of fine particles. The flow patterns in vivo are certainly much more complex than those in the TEE junction, implying that their effects are commensurably more pronounced. We conclude that convective mixing exists in the pulmonary acinus in vivo, secondary to the presence of geometric hysteresis, and that this phenomenon may be the origin of substantial deposition of fine particles deep in the lung.