Abstract
Current theories describe aerosol transport in the lung as a dispersive (diffusionlike) process, characterized by an effective diffusion coefficient in the context of reversible alveolar flow. Our recent experimental data, however, question the validity of these basic assumptions. In this study, we describe the behavior of fluid particles (or bolus) in a realistic, numerical, alveolated duct model with rhythmically expanding walls. We found acinar flow exhibiting multiple saddle points, characteristic of chaotic flow, resulting in substantial flow irreversibility. Computations of axial variance of bolus spreading indicate that the growth of the variance with respect to time is faster than linear, a finding inconsistent with dispersion theory. Lateral behavior of the bolus shows finescale, stretchandfold striations, exhibiting fractallike patterns with a fractal dimension of 1.2, which compares well with the fractal dimension of 1.1 observed in our experimental studies performed with rat lungs. We conclude that kinematic irreversibility of acinar flow due to chaotic flow may be the dominant mechanism of aerosol transport deep in the lungs.
 lung
 deposition
 chaos
 fractal
 particulate pollution
convection and diffusion are the two major mechanisms of mass transport for gas molecules and submicrometer aerosols in the pulmonary acinus. For gas transport, diffusion dominates at distances comparable to acinar size and over times comparable to breathing frequencies, and, therefore, theories based on diffusion are probably adequate. By contrast, the particle diffusivity of submicrometersized aerosols is very small, and, therefore, acinar convection, even though it is in a quasiStokes viscous flow regime (26), is correspondingly more important and may dominate aerosol transport. However, current theories describe aerosol transport as a dispersion (diffusionlike) process (e.g., Refs. 7, 10, 11, 19, 34). These theories are based on the following two key assumptions: 1) acinar flow is basically kinematically reversible (i.e., during expiration each fluid particle retraces the path taken during inspiration) (9, 45), and 2) all processes (including the coupling of Brownian diffusivity with the convective flow field and any kinematic irreversibility that may be present) that contribute to irreversible aerosol bolus spreading can be characterized as axial mixing with an effective longitudinal diffusivity (D _{eff}). The first assumption is based on classical fluid mechanics (36), and the second assumption is substantially equivalent to Taylor dispersion (35). As most aerosol studies are currently interpreted in the framework of these dispersion theories, experimental data are often reduced and analyzed through the use of some D _{eff} (e.g., Ref. 30), and many of the recent theoretical research efforts are focused on refining D _{eff} for better fit to experimental data, through which new insights into acinar transport mechanisms are sought (e.g., Ref. 6).
Our laboratory's recent findings (3741), as well as those of others (8, 18), however, have questioned the validity of the basic assumptions that formed the basis of the dispersive theories. We have demonstrated that, because of the peculiar geometry of the alveolated duct and its timedependent motion associated with tidal breathing, under certain conditions, alveolar flow can be chaotic (16, 40). As a consequence, acinar flow can be kinematically irreversible, even though it is a lowReynolds number viscous flow, and lung expansion and contraction are approximately selfsimilar and reversible (1, 13, 14, 24,46). These findings are supported by a discovery in fluid mechanics that chaotic mixing can occur even in a viscous flow (2, 25). We have also developed a theoretical analysis (5) to show that chaotic mixing is radically different from diffusive mixing. In chaotic acinar flow, a tracer bolus undergoes cyclic stretchandfold deformation, resulting in the induction of finer and finer scales in tracer profile with repeated breaths. This process soon reaches a critical moment at which the lateral distance between adjacent tracer striations becomes comparable to the diffusion distance. A burst of mixing occurs at this moment, and mixing is quickly completed.
The objectives of the studies reported here are, through numerical simulations of bolus experiments, to provide key data showing that the behavior of particles in a rhythmically expanding, multiply alveolated duct flow, with a saddle point and associated vortexes in each air pocket, does not satisfy the fundamental assumptions of any dispersion theory and that the shape of a tracer bolus evolves to a stretchandfold fractallike pattern, similar to those found in flow visualization experiments in rat lungs (Tsuda A, Butler JP, and Rogers RA, unpublished observations). The results suggest that1) kinematic irreversibility is the origin of aerosol transport, 2) axial transport cannot be characterizable by an effective diffusivity, and 3) fractal trajectories can occur in most of the alveoli in the acinar tree. The alternative mechanism of aerosol transport that we propose here may, in fact, be the dominant mechanism determining deposition of submicrometer particles deep in the lung.
METHODS
In a previous investigation (40), we used the singlealveolus model to explore the basic physics operating in a viscous flow subjected to cyclic alveolar wall motion. The alveolus model used in the present investigation is also axisymmetric but comprises a central circular channel around which are placed nine tori, equispaced in the axial direction (Fig.1
A). Details of a typical cell are given in Fig. 2. The duct and alveolar walls move in a perfectly kinematically reversible, simple sinusoidal manner with a specific volume excursion C of 25% [C = (V_{max} − V_{min})/V_{min}, where V_{max} and V_{min} are the maximum and minimum volumes of the model, respectively] and a cycle period T of 3 s. These correspond roughly to typical tidal ventilation and respiratory period in human. Any length scale L of the model changes asL(t) = L̄[1 +K sin(nt)], where L̄ is the meanL value, t is time, n = 2π/T, and K = (φ − 1)/(φ + 1), where
The flow field is defined by the full, incompressible NavierStokes equations, which are solved numerically on a multiblock, bodyfitted moving grid using the finite volume code CFX4 (CFDS, AEA Technology, Harwell, UK). This generalpurpose, pressurecorrection code offers a variety of discretization schemes and solution techniques. In these calculations, central differencing is used to model the convection terms, and the implicit backward Euler method is used to advance the solution in time. A combination of the SIMPLEC pressurecorrection method (44) and the RhieChow (28) algorithm to eliminate pressure oscillations on the collocated gird (12) is used in the formulation of the discrete equations. Stone's method (33) is used to solve the discrete velocity equations, and the method of preconditioned conjugate gradients (see, for example, Ref. 22) is used to solve the discrete pressure correction equation. At the inlet, a constantpressure boundary is defined. The value of the pressure on this boundary is set arbitrarily to zero, and, as for incompressible flow, only the pressure gradient is of importance. The noslip condition is enforced on all solid surfaces, which, in the case of moving walls, means that the fluid matches the wall velocity at the fluidwall interface. Tests were carried out to ensure that the solutions are grid independent and converged. For example, increasing the number of grid cells by 30% above that which was eventually used produces an increase of only 0.16% in the predicted maximum velocity. The final 73block grid had a total of 38,349 active cells. With the use of a measure of error due to using backward Euler time stepping, suggested by Roache (29), a time step of T/240 is found to give sufficiently accurate results in that further reductions in the size of the time step used does not produce any significant reduction in the error. These simulations are computationally intensive, with one breathing cycle taking ∼350 h on a Sun Ultra 10.
Particle trajectories are calculated in all three models using a specialpurpose tracking routine. This routine reads a full cycle of flow field and grid data produced by CFX4 and uses this data repeatedly, cycle after cycle, and a predictor and corrector method to track individual particles (fluid elements) over as many cycles as required. The time step used in the particle tracking routine is set independently of that for the flowfield solution. Before the particle track is advanced to the next time, the time step is recalculated, using local flow conditions, to ensure that the particle does not step out of the solution domain. At each particletrack time, a grid and flow field are created from the CFX4 data using bicubic interpolation in space (27) and linear interpolation in time. For numerical efficiency, the routine solves the particle tracks in a stationary computational space, and, at each time step, the particle position is mapped back to physical space before it is written to an output file. The maximum error in the predicted particle position is estimated to be 0.2% of the distance traveled by the particle, and, in most cases, the error is considerably smaller than this.
RESULTS
Flow patterns.
Solving the velocity field of the carrier gas on a moving grid over the physiologically relevant range of flow parameters (Re_{RMS} < 1) in a rhythmically expanding and contracting, multiply alveolated duct, we often detected the presence of slowly rotating recirculation in each alveolus (see Fig.3, A and B). The size of the recirculation flow depends on the ratio between the alveolar flow (Q˙_{A}) (i.e., flow produced by the volume change of the alveolus) and the volumetric ductal flow (Q˙_{D}) (Q˙_{A}/Q˙_{D}). Similar to our previous study (40), we found that the smaller the Q˙_{A}/Q˙_{D}, the larger the alveolar recirculation. When Q˙_{A}/Q˙_{D}is larger than ∼0.1, however, the alveolar flow is largely radial without recirculation (Fig. 3 C). Because, under normal breathing conditions or during moderate exercise, the value ofQ˙_{A}/Q˙_{D} is usually <0.05 in the majority of alveoli along the acinar tree (from the respiratory bronchioles to the last few generations) (37), we expect that most of the alveoli are likely to possess recirculation in their flow field. Importantly, the presence of alveolar recirculation in the cyclically expanding alveoli is topologically associated with the existence of a stagnation saddle point in each alveolar flow field (40), implying that chaotic mixing could originate in most of the alveoli (see discussion).
Interalveolar kinematic mixing.
To demonstrate the effects of a series of saddle points on fluid flow irreversibility, the motion of massless particles was tracked over one ventilation cycle in a ninecell alveolar model for three different ranges of Q˙_{A}/Q˙_{D} (0.0050 <Q˙_{A}/Q˙_{D} < 0.0053, 0.040 <Q˙_{A}/Q˙_{D} < 0.069, and 0.081 <Q˙_{A}/Q˙_{D} < 0.577 shown in Fig.4, A, B, andC, respectively). The particles were initially placed on radial lines across the duct midway between alveoli at three different initial axial locations (shown as three different colors, brown, green, and pink, in Fig. 4). As soon as inspiration begins, particles near the center line convect distally, proportionally to the bulk mean velocities (see t/T = 0.25 in Fig. 4). The lines of particles quickly approach each other and comigrate along the central channel, particularly in the cases of smallerQ˙_{A}/Q˙_{D} (Fig. 4, A orB). The particles that were located initially near the alveolar opening enter the alveolus (Fig. 4). The depth of particle penetration into the alveolus during inspiration depends on the size of alveolar recirculation. When the alveolar recirculation is large (Fig.4 A), the particles penetrate deep into the alveoli, and the particles from the three different lines appear to be mixed during inspiration (see t/T = 0.25 and 0.5 in Fig.4 A).
As expiration starts, the particles start to move in a proximal direction. The reversibility of the particle motion is largely influenced by both the presence and the size of alveolar recirculation. When the alveolar recirculation is large (Fig. 4 A), the particles exhibited three types of behavior. For type I, some particles, which were deep inside the alveolus at the end of inspiration, do not exit and remained trapped inside after expiration. For type II, particles that were inside the alveolus at the end of inspiration exit during expiration but do not come back to their original starting positions. For type III, particles near the center line that remained in the central channel during inspiration virtually retrace their inspiratory paths during expiration and thus arrive back very close to their original starting position. Comparing these three types of particle behavior, we notice that particles that were associated with alveolar recirculation (even for a short period of time) followed irreversible trajectories. When the alveolar recirculation is present but small (Fig. 4 B), no particles are trapped in the alveolus, suggesting that the effects of small alveolar recirculation are insufficient to cause particle trapping. However, particles initially located near the side walls (which eventually became associated with alveolar recirculation) still display irreversible motion (type II). Particles that start near the channel center line come back to their original starting position (type III). In the complete absence of alveolar recirculation (Fig. 4 C), the motion of all particles, even those that traveled inside the alveolus, is reversible.
For the purpose of comparison, the following two additional cases were also conducted: massless particles were tracked in a singlecell model and in an expandable straighttube model (data not shown). Whereas the particles in the singlealveolus model, similar to the ninecell alveolar model, exhibit some irreversibility, especially near the walls (type II), the particles in the expandable straight tube are reversible as predicted (42).
Motion of interface between inhaled tracer fluid and the host alveolar residual fluid.
Regarding massless tracer particles as marked fluid elements, the interface between one fluid and another can be approximated by chords connecting initially adjacent particles (Fig.5 A). In our studies, this piecewise linear approximation to the interface can represent the front of incoming tidal gas or a tracer bolus facing the host alveolar residual gas. As we have demonstrated in Fig. 4, A and B, the kinematically irreversible deformation of the tracer interface seems to be due to its association with alveolar recirculation flow over the respiratory cycle. Thus characteristics (e.g., size and strength) of the alveolar recirculation may be major determinants in these processes. Our studies described here focus on the situation that presumably occurs deep in the acinus, where the alveolar flow exhibits a medium to smallsized recirculation (Fig. 3 B). Larger alveolar recirculation flow (Fig.3 A) is likely to occur near the entrance of acinus (e.g., respiratory bronchioles), and that case is discussed elsewhere (38).
To understand the temporal evolution of interface deformation (approximated as described above) in the case of flows with mediumsized alveolar recirculation, we monitored the motion of a tracer bolus for three cycles. In these simulations, we systematically increased the number of massless particles P(P = 2^{j}; where j= 3, 4, 5,…,13). For each P, the initial radial distribution of particles was adjusted in such a way that each particle represented equal crosssectional annular area. Over each cycle, the interface progressively deforms into “fingerlike” protrusions located especially near the duct walls (Fig. 5 B). Each protrusion consists of complex stretchedandfolded patterns (Fig.5 B, inset 1), and, moreover, as the scale becomes finer, similar and finer stretchedandfolded patterns are revealed (Fig. 5 B, inset 2). This suggests that the observed patterns might be qualitatively selfsimilar over a wide range of length scales. Such selfsimilarity is characteristic of fractal geometry observed in many chaotic systems (32).
To quantify the extent of stretchandfold flow irreversibility and especially to test whether axial spreading could be described by an effective diffusivity, we analyzed the shape of the tracer in axial and lateral directions separately. To characterize the axial phenomena, the axial variance (ς^{2}) of distribution of particles was computed and plotted vs. cycle number (N) as a family in theP employed (Fig. 6). The results show that ς^{2} is independent of P and grows exponentially with increasing N[ς^{2} = 0.0474(e ^{0.457N} − 1)]. It is important to note that ς^{2} does not increase linearly withN; this observation is fundamentally inconsistent with the predictions of the classical dispersion theory (discussed below).
To characterize the lateral phenomena, we examined the extent of tracer stretching [ΔL/L
_{o} = (L− L
_{o})/L
_{o}, whereL
_{o} (= Σ
DISCUSSION
The principal findings of this study are that 1) chaotic fluid motion occurring in a rhythmically expanding and contracting, multiply alveolated duct induces substantial kinematic irreversibility in the acinus, even under lowReynolds number flow conditions, and 2) mixing due to this kinematic irreversibility is fundamentally different from the mixing described in dispersive processes.1 A tracer (bolus) subjected to the chaotic flow field is deformed both axially and laterally. The ς^{2} of particle distribution increases exponentially, rather than linearly, with increasing cycle time; thus axial bolus spreading does not obey the basic rules described in classical diffusive transport theories (35). The cyclebycycle evolution of lateral particle distribution is even more complex. The tracer forms fingerlike protrusions, even after one cycle. The tracer length exponentially increases as Nincreased, forming characteristic stretchandfold fractallike patterns.
Chaotic mixing in the pulmonary acinus.
Viscous flow has been considered kinematically reversible if the boundary motion is reversible (36, 45). In the mid1980s, however, there was a breakthrough discovery in fluid mechanics, namely, that even Stokes flow can be kinematically irreversible if the structure of the flow is chaotic (2, 25). In the last several years, applying this new concept to respiratory fluid mechanics, our laboratory has been studying the role of chaotic flow phenomena in the experimentally observed, yet theoretically unexplained, convective mixing occurring in the lung periphery (5, 16, 37, 40). In our laboratory's previous numerical study (16, 40), we reported that chaotic flow and chaotic mixing can occur in the alveolated duct because of its peculiar geometry and timedependent motion associated with tidal breathing. We found that acinar flow was often slowly rotating in the alveolar air pocket, and the velocity field near the alveolar opening was complex with a stagnation saddle point typical of chaotic flow structure. Performing Lagrangian fluid particle tracking, we further demonstrated that, in such a flow structure, the motion of fluid,x(t), could be highly complex, irreversible, and unpredictable even though it was governed by simple deterministic equations [x(t) = ∫v(x,t)dt,x(0) = x _{o}, wherev(x,t) denotes Eulerian velocity field].
Our initial, numerical investigations performed in a simplified system with an isolated, single alveolus were aimed at discovering and understanding the basic physics operating in a rhythmically expanding alveolar flow (16, 40). These studies, however, did not address the cumulative effects of multiple alveoli (i.e., a series of saddle points) on the fate of inhaled aerosols. There are roughly 300 million alveoli in the human lung (∼10,000 alveoli in each of ∼30,000 acini) (17). This means that, in each acinus, the incoming tidal air may sample roughly 200 alveoli along a longitudinal pathway, from the entrance of an acinus to the terminal alveolar sac (here, we assume that alveoli are uniformly spaced in nine intraacinar airway generations). As demonstrated in Figs. 3 and 4, a series of saddle points and associated vortexes in each air pocket, generated in a cyclically expanding, multiply alveolated duct, make the supposedly reversible lowReynolds number acinar flow highly irreversible and can cause substantial interalveolar convective mixing.
In the context of flow irreversibilities associated with multiple saddle points, it is important to recognize that the estimates that we obtain in this work may significantly underestimate the importance of convective mixing. In particular, we have explicitly assumed and, therefore, constrained the flow field to be axisymmetric, which is to say that all azimuthal velocity components (i.e., the component in the direction perpendicular to the rz plane) are zero. This would imply that the presence of a saddle point in the longitudinal section would be associated with a saddle “line” or “circle” around the ductal axis. Such a flow structure is not only highly unlikely but also would be expected to be unstable and to break into a separate sequence of saddle points. Any such failure to preserve axial symmetry would thus enhance whatever convective mixing is already associated with the axisymmetric alveolated geometry.
Fingerlike protrusion.
In this study, a line of massless particles was introduced in the alveolated duct to represent a fluidfluid interface (Figs. 4 and5 A). The behavior of this interface, such as its reversibility and irreversibility, changes in its shape and size and contains crucial information for understanding the mechanism of mixing between inhaled particles and alveolar residual gas. By tracking the motion of the tracer particles, we have followed the motion of this interface over several cycles. Because the boundary of this line, shown as the point Q in Fig. 5 A, is stationary on the wall because of the noslip condition, the line expands when the alveolar walls expand during inspiration, and the line also tends to contract when the walls contract during expiration. However, the reversibility of this process depended on the nature of the flow fields sampled during expansion and contraction. The segments of the line near the channel center line are enormously stretched axially (see Fig. 4) and sample mostly reversible Poiseuillelike ductal flow fields (Fig. 3). Consequently, the interfacial line near the center line also shows approximately reversible behavior (Fig. 5 B). By contrast, the segments of the lines near the walls (e.g., segments s1,s2, and s3 in Fig. 5 A) sample a series of irreversible alveolar flow fields (e.g., Alv1, Alv2, and Alv3, respectively, in Fig. 5 A) and, consequently, do not return to their original positions (Fig. 5 B). Each of these line segments, separated by points that sample approximately reversible flow fields between alveoli (e.g., R1, R2, and R3) basically form one finger after a cycle (Fig. 5 B). The number of “fingers,” therefore, roughly matches the number of alveoli distal to the initial position. Although, in the present computational model, the number of alveoli that the tracer samples is limited to six (due to computational constraints), in a real acinus, the inhaled bolus is expected to encounter a larger number of alveoli (∼200) along the acinar longitudinal pathway. This implies that the acinar airways are likely to be filled with many longitudinal fingers after a few breathing cycles. This prediction has been confirmed experimentally in our laboratory's recent flow visualization studies performed in rat acini (39).
The size of each finger depended on its resident time in alveolar recirculation. The line segments that were closer to the side walls (e.g., s1 and s2) spent more time in recirculation (Alv1 and Alv2, respectively) and produced longer fingers. As the Nincreased, the tracer repeatedly encountered alveolar recirculation; the number of fingers rapidly increased, and they propagated toward the channel center line. This global evolution of the tracer pattern (i.e., rapid increase in the number of fingers, cyclebycycle lateral propagation), together with progressively finer scale tracer striations (discussed below in detail), are important observations because they indicate a substantial net enhancement of lateral particle transport for deposition.
Axial phenomena–a departure from the conventional dispersion theory.
In current theories, aerosol transport in the pulmonary acinus is described as a dispersion (diffusionlike) process, and the mixing phenomena are couched in the language of a D
_{eff}. The most important feature of this approach is that, in any process that is dispersive in the sense that it can be characterized by an effective diffusivity, the variance of a bolus asymptotically increases linearly in time (or N). Phrased differently, the variance is an additive function over time. The reduction of experimental data through the use of some D
_{eff} has thus been the framework by which many aerosol studies have been conducted (e.g., Ref.30). For instance, bolus spreading in the tracheobronchial tree is commonly characterized by the difference between the inhaled and exhaled variances (proportional to H
^{2}, whereH is the bolus width at half height), given by
In sharp contrast to these ideas, the result of our numerical experiment does not obey this basic rule (Fig. 6). The ς^{2}of bolus particle distribution grows faster than linearly in time, in contrast to the linear growth predicted by all theories that are characterized by aD _{eff}.2It is important to emphasize here that mixing of kinematic origin (e.g., chaotic flow in the acinus) has a fundamentally different nature from mixing described in dispersive mechanisms; thus it cannot be described by any D _{eff}.
Lateral phenomena (fractal patterns).
The results of our study show that complex flow phenomena can occur in the lateral direction. Because of alveolar flow irreversibility, the tracer (i.e., a series of line segments, s1, s2, etc., in Fig.5 A) deforms at every cycle, forming fine characteristic stretchandfold striation patterns (see Fig. 5 B, insets), on top of the global fingerlike protrusion (Fig. 5 B). Consequently, an enormously large lateral diffusion surface (which is represented as a stretched tracer line in our study) evolves over every cycle, suggesting a substantial enhancement of lateral particle transport and subsequent deposition on the acinar walls. Interestingly, we found that exact estimation of tracer length L was not possible because L was strongly dependent on the number of particles forming the tracer (Fig. 7). At sufficiently fine scales (i.e., when the number of test particles used is sufficiently large), the apparent L of the tracer increases exponentially with increasing N.
The analysis of potentially fractal patterns by the wellknown method of box counting (32) was employed. The analysis shows that the distribution of tracer particles evolve in a fractallike manner, with D≈ 1.2. The fact that the tracer pattern exhibits fractal characteristics is not entirely surprising, because the origin of particle irreversibility is due to chaotic alveolar flow, and deterministic chaos often manifests a fractal geometry (23,32). On the other hand, it is important and remarkable that the fingerlike protrusions (i.e., global pattern) found in these simulations are strikingly similar to those found in the longitudinal airway section of our experiments performed with rat lungs (39). Moreover, our initial finding that D≈ 1.2 obtained in finer scale stretchandfold striations in the present numerical study is close to D≈ 1.1 found in those rat experiments (unpublished observations) is encouraging. Although further detailed analyses will be necessary to determine the dependence of the fractal dimension (or indeed if the pattern remains fractal) on model parameters, these similarities suggest that our numerical simulation, although based on highly idealized assumptions, does, in fact, capture the essential features of the underlying mixing mechanism, namely, that lowReynolds number chaotic flow in the acinus determines particle transport in the lung.
Physiological origins of mixing.
Our laboratory has recently proposed two possible origins of “stretchandfold” kinematics in the pulmonary acinus: first, that induced by a small departure (asynchrony) from kinematically reversible motion of alveolar walls (16, 41), and, second, that due to the presence of saddle points associated with alveolar recirculation flows in the acinar flow field (40). With respect to the first mechanism, Miki et al. (24) reported the presence of a small but consistent geometric hysteresis (i.e., temporal asynchrony) in lung expansion during normal tidal ventilation in live rabbits. By matching this degree of geometric hysteresis, we have generated physiologically realistic asynchrony in physical models (41) and computational models (16), which demonstrated that geometrical hysteresis, even if small, can produce stretchandfold patterns and, consequently, induce substantial acinar flow irreversibility. In a related work, Smaldone et al. (31), using gravitational sedimentation of aerosol particles to estimate mean linear intercepts, showed significant geometric hysteresis in excised lungs with largevolume excursions from minimal volume and interpreted their data in terms of respiratory unit recruitment and derecruitment. Those studies, however, being restricted to singlevolume histories, did not address the issue of mixing during cyclic ventilation and so are not strictly comparable to the present work. Finally, flow/volume hysteresis (different flow magnitudes at isovolume points on inspiration and expiration) can lead to differences in the velocity profiles in the central airways; this, in turn, can cause a difference in aerosol deposition between inspiration and expiration (4).
In contrast to these studies, which focus primarily on differing geometric features between inspiration and expiration, in this paper we investigate the second mechanism mentioned above. We believe that the presence of saddle points may be an equally fundamental mixing mechanism responsible for convective mixing in the acinus. Note that such saddle point singularities do not exist in rigid wall models of acinar flow but, through their association with alveolar recirculation, are necessarily linked to the cyclic motion of the alveolar walls. To distinguish clearly this mechanism from the former one (caused by geometric hysteresis), we used only kinematically reversible wall motion in these studies. The present paper extends our previous work (40), which was restricted to a saddle point in a single alveolar space, by considering the effect of multiple saddle points in a multiply alveolated channel. In practice, we believe that both mechanisms, asynchrony and saddle points, coexist in the pulmonary acinus, mutually enhancing each other in producing stretchandfold mixing.
Significance and conclusions.
It is well established that exposure to environmental aerosol pollutants is associated with health risks, ranging from mild to life threatening (47). The exposure and pathophysiological consequences are clearly linked by two independent and separate causal pathways: the exposuredose relationship (i.e., given an aerosol concentration in the ambient air, what is the actual dose delivered to the lung) and the doseresponse relationship (i.e., for a given deposited dose or burden, what is the biological consequence). Despite the large literature on exposure assessment methodologies as well as on the pathophysiological consequences of short and longterm exposures, there is much less known about the mechanisms contributing to the first of these links, and much deposition and mixing data are inconsistent with previous theories of mixing deep in the lung. It is important, therefore, to identify new potential mechanisms that may dominate aerosol transport, even when boundary motion is approximately reversible. We argue in this paper that chaotic mixing is such a candidate. We have shown, through realistic numerical simulation of the lowReynolds number alveolated duct flow (and by comparison with experimental results in rat lungs; unpublished observations), that the peculiar geometry of the alveolated duct structure within the pulmonary acinus and its cyclic motion during breathing can give rise to1) a chaotic type of mixing associated with the presence of saddle points, 2) slow recirculatory flow within the alveoli, and 3) stretching and folding of stream surfaces. These, in turn, can significantly increase mixing, especially laterally, and will also contribute to an increasing ς^{2}(which increases faster than linearly with breath number, an observation that is inconsistent with any dispersal mechanism that can be characterized by an effective axial diffusivity). We suggest that chaotic mixing may be the dominant mechanism of aerosol transport and deposition deep in the lung.
Acknowledgments
This study was supported by National Heart, Lung, and Blood Institute Grants HL47428 and HL54885 and, in part, by Environmental Protection Agency Research Award R827353.
Footnotes

Address for reprint requests and other correspondence: A. Tsuda, Physiology Program, Harvard School of Public Health, Huntington Ave., Boston, MA 02115 (Email: atsuda{at}hsph.harvard.edu).

↵1 It is unlikely that the results on kinematic irreversibility in this paper are important to ventilationperfusion matching, but a quantitative assessment of this remains open.

↵2 In a recent theoretical study in fluid mechanics, Jones and Young (21) showed that the axial variance of tracer fluid elements in lowReynolds number chaotic flow in a twisted pipe grows faster than linearly with time. They also pointed out that the resulting distribution of the tracer particles exhibited a fractal pattern (20). Experimentally, whether variances are in fact additive or not in real lungs, there do not appear to be any data on the growth of bolus variance as a function of breath number. [This point should not be confused with the observation that there is an essentially linear relationship between bolus width and volume of penetration in the bronchial tree (18)].

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10.1152/japplphysiol.00385.2001
 Copyright © 2002 the American Physiological Society