Abstract
Butler, J. P., and A. Tsuda. Effect of convective stretching and folding on aerosol mixing deep in the lung, assessed by approximate entropy. J. Appl. Physiol. 83(3): 800–809, 1997.—There is a surprisingly substantial amount of aerosol mixing and deposition deep in the lung, which cannot be explained by classic transport mechanisms such as streamline crossing, inertial impaction, or gravitational sedimentation with reversible acinar flow. Mixing associated with “stretch and fold” convective flow patterns can, however, be a potent source of transport. We show such patterns in experimental preparations using rat lungs and in the theoretical Baker Transform. In both cases, mixing is associated with the temporal evolution of two length scales. The first is the slowly increasing diffusive length scale. The second is the rapidly decreasing lateral length scale, due to “stretching and folding,” over which diffusion must take place. This interaction leads to aerosol mixing in much shorter times than previously appreciated. Finally, we propose a new method by which to quantify the state of mixing, using an approximation to the entropy of the aerosol concentration distribution. The results of the analysis suggest that stretching and folding may be a key feature underlying peripheral aerosol transport.
 aerosol deposition
 convection
 diffusion
 chaos
 chaotic mixing
 acinar flow
it has long been thought that convective flow patterns in the pulmonary acinus are essentially reversible kinematically (14) and that, consequently, there should be little contribution of convective mixing to the mixing and transport of aerosols (e.g., Ref.3). However, the important work of Heyder et al. (7) has challenged this conventional understanding. They found that there is appreciable aerosol mixing deep in the lung that cannot be accounted for by any known transport mechanisms such as streamline crossing, inertial impaction, or gravitational sedimentation within the context of reversible acinar flow. More recently, we observed a highly complex pattern of convective movement deep in the acinar regions of rat lungs (12) when we used flowvisualization techniques. These findings contradict the hypothesis of simple and reversible kinematics. The origin of these experimental findings is important to our understanding of the pathogenesis of aerosolrelated lung injury, which depends on the dose and site of deposition, and to the search for effective means of delivery of aerosolborne drug therapies.
In sharp contrast to the classic approaches, there has been a recent surge in interest in certain types of flow fields characterized by a “stretch and fold” pattern (9). These form part of the basis of the study of chaotic mixing and are potentially of great importance in the study of aerosol mixing and deposition in the lung because this mechanism may be governing more of the transport dynamics (e.g., Ref.13) than has been hitherto appreciated. The stretch and fold pattern of convection is qualitatively much different from laminar or even turbulent steady flow in tubes, but it is easily visualized. It is evident in common everyday occurrences: mixing of different colors of paint in a can, cream in coffee, and mixing of cinnamon and sugar in a bowl. These all display the striking feature that, over a certain number of cycles, there is an increasingly detailed fine structure in the pattern, which is followed by a threshold number of cycles where the mixing characteristics qualitatively change from one with a fine but welldefined structure to one which is, in effect, completely mixed. This phenomenon is characteristic of the stretch and fold convective kinematics. During such repeated stretching and folding, two important convective characteristics emerge. First, the area over which diffusion can take place grows exponentially in time or cycle number and, second, the lateral or transverse distance scales from one fold to its neighbor decrease exponentially in time or cycle number. These two features together imply that the classic theories [e.g., for steady laminar pipe flow (2, 11) and cyclic flow (15)] are inadequate to explain the interaction of convection with diffusion and the resulting mixing in these types of flow fields.
This paper presents an approach to the quantification of mixing in stretch and fold flow fields by using an approximation to the evolving entropy of the flow field. We discuss several idealized examples of this, in addition to applying these ideas to some experimental data obtained with white and blue silicone polymers used in ventilating rat lungs. We indicate the reasons why classic theories of mixing may fail to appreciate the role of convective mixing and conclude that stretch and fold kinematics is a potent new mechanism that may underlie the mixing and ultimate deposition of fine aerosol particles deep in the lung.
THEORY
Preliminary Considerations
Before developing our theory in detail, we first clarify the concept of “mixing” itself. One of the key features of a completely mixed state of particles is that the particles themselves are indistinguishable. This has several implications. First, there is a complete loss of any information regarding the history of each particle; there is no dependence of any physical property of the mixture that respects the particles’ individual trajectories. Second, this same property implies that there exist length scales over which concepts such as concentration can be usefully defined. That is, a small volume with a given concentration is in all sensible ways equivalent to any other equal volume of the same concentration, and the size of this sample volume defines a length scale over which the mixed sample makes sense. We note that the lower limits of these scales occur at the fine scale of molecular or particle mean free paths, and the upper limits are those lengths over which significant concentration gradients exist.
The determination of these length scales is important to appreciate the evolution of any given concentration distribution toward its mixed state. First, simple diffusion defines the scale ultimately responsible for mixing; any time interval (Δt) is associated with a length (ΔL), which is proportional to
Quantitative Characterization of Mixing; Use of Entropy
The evolving spatial spread, or variance, of a tracer bolus is commonly used to characterize the mixing process when diffusion and convection interact. In a wide class of simple systems, one can show that asymptotically the variance will increase linearly in time, and the constant of proportionality can be taken as an effective diffusivity. However, such considerations only apply in circumstances of steady or quasisteady flow. In contrast, when the convective flow field involves continuous stretching and folding, the mixing cannot be described by a linearly increasing bolus variance. A different measure of mixing is needed.
Consider an initial distribution sharply peaked at some location. A tracer bolus of paint of one color in a can of differentcolored paint or a bolus of aerosol particles of low diffusivity deep in a lung are simple examples. The fact that the particles are concentrated in one particular region, rather than spread out, implies that this is a state of very low probability. Now stir the paint or ventilate the lung. As time evolves, convection and diffusion will change the distribution of particles toward the uniform distribution, a disordered state of maximum probability. This notion is the origin of the concept of entropy. Thus the evolving entropy of the distribution is an appropriate measure of the progressive transition from an initial lowprobability state to a final highprobability state. Apart from multiplicative constants and the scale for measuring concentrations, the entropy density of the particle distribution is classically given by
(see, e.g., Ref. 10), where ρ is the aerosol concentration. The total entropy of any closed region (R) is then given by the volume integral of the entropy density
It is important to appreciate what entropy is measuring and what the role of convection is, as this will be part of our motivation for the approximate entropy calculations to be presented below. First, note that convection alone given by the velocity field of the carrier (u⃗) does not lead to a change in entropy regardless of whether it is a stretching and folding pattern or not. This can be seen by an elementary argument as follows (here and in what follows, we always take the carrier flow to be incompressible, i.e., ∇ ⋅ u⃗ = 0). Local conservation of mass implies that the material derivative of the density vanishes, i.e., Dρ/Dt = 0, which, in turn, implies that along any streamline ρ is constant. Therefore, any function of ρ, in particular the entropy density s and the total entropyS, are constant. In other words, pure convection does not affect mixing, in the precise sense of an increased entropy.
By contrast, the presence of diffusion, with or without convection, necessarily effects mixing, which is manifested in a globally increasing entropy. This can be shown as follows. From the definition of entropy given above, its global rate of change is given by the sum of a volume and a surface integral
where ∂R is the boundary of R,ν⃗ is the outward pointing unit normal on ∂R; note that the fluid velocity u⃗ at any point on ∂R by the noslip condition is equal to the velocity of the boundary at that point (R is still closed, but the boundary may be moving). The first term in dS/dt arises from the time dependence of the entropy density within R; the second term comes from the time dependence of ∂R. Performing the indicated time derivative leads to a volume integral involving ∂ρ/∂t. This may be converted into spatial derivatives by using the local convection diffusion equation ∂ρ/∂t =D∇^{2}ρ −u⃗ ⋅ ∇ρ. The integrand of the volume integral can be written as a pure divergence −∇ ⋅ [(1 + ln ρ)(D∇ρ − ρu⃗) + ρu⃗] plus a nonnegative term (D/ρ)‖∇ρ‖^{2}. By Gauss’ theorem, the volume integral of the divergence term can be written as a surface integral, which can then be combined with the second term in dS/dt above. The only noncanceling term in the surface integral is proportional toν⃗ ⋅ (−D∇⃗ρ), which is identically zero, since there is no diffusive flux across ∂R. The final result is then
A striking feature of this expression is that the fluid velocity u⃗ does not appear explicitly; nevertheless, convection can play a highly significant role through its influence on ∇ρ. Although the role of pure convection, then, does not directly change the entropy, it does modify the length scales over which there are significant variations in concentration and, therefore, contributes to increasing entropy and mixing. This role of convection is particularly important for aerosol mixing, since the diffusivityD for aerosol particles is typically very low. Below, we will focus on the specific case where the lateral length scales associated with stretching and folding will markedly affect the ultimately diffusive origin of the growing entropy.
Approximating the Effect of Diffusion on Entropy
The evolution of the entropy S(t) is exactly soluble in only a handful of highly simplified cases. On the other hand, for a given distribution ρ(r⃗, t) (which may have arisen, for example, from pure convection), we may calculate an approximation to S by “smearing” ρ(r⃗, t) over an appropriate length scale δ. We take as our smearing function a simple Gaussian, and define the approximate entropy S
_{δ} by the volume integral of the approximate entropy density s
_{δ} = −ρ_{δ} ln ρ_{δ}, where ρ_{δ}is given by the convolution
(We assume here that the size of the system is sufficiently large compared with the smearing length δ so that errors associated with using the form for the Gaussian in an infinite medium are negligible.)
A Representative Stretch and Fold Pattern
We now introduce the stretch and fold convective pattern known as the Baker Transform [taken from the recipe for phyllo dough (9)]. There are a number of ways of describing this physically, all of which are essentially equivalent. The simplest is perhaps to consider a twodimensional object in the shape of a square with coordinatesx and y. The sample is “stretched” in they direction to twice its initial length and simultaneously shrunk in the x direction to onehalf its initial length. (In two dimensions, this is a measure preserving deformation.) It is then cut in the x direction halfway along its new y length, and the two rectangular pieces are translated and rejoined side by side in the x direction to make a new square sample. Strictly speaking, this is not a stretch followed by a true “fold,” but it does represent an equivalence to the real phenomenon known commonly as stretching and folding. This is a kinematical description, and it translates into a sequentially evolving concentration distribution as follows. For unit length in the x direction, we define an initial distribution independent of y and equal to 1 onx ∈ [0, 1/2) and 0 on x ∈ [1/2, 1). Over each stretch and fold cycle, this pattern is shrunk to onehalf itsx size and copied to fill the unit x interval. This sequence is shown in Fig. 1. For later convenience in avoiding nonessential questions about boundary conditions, we then extend this pattern by making it periodic inx. That is, let the initial concentration distribution (I
_{0}) be given by
for all integers k. One cycle of stretching and folding transform applied to this function results simply in the function
and in general, after n cycles, we have
Interaction of Convection and Diffusion in the Baker Transform
The Baker Transform provides a useful model of the stretch and fold pattern of convection. Because it is discrete, there are several ways in which to combine diffusion with this transform. Below we describe two such methods. These methods are useful insofar as they give similar results and that they also represent approximations to the solution of general convection/diffusion equations. Furthermore, one of these methods is particularly amenable to post hoc applications involving experimental data on real lungs. Both of the methods described below can be considered a kind of approximate “factorization,” in the sense that diffusion and convection are computed separately. The first is the solution to the discretized (by cycle period T ) convection/diffusion equation. That is, the initial distribution is folded and diffused over one cycle, and this process is then repeated through n cycles. We denote this method by the operator (DF)^{n}, where D andF denote diffusion and folding through a single cycle. (More precisely, the D and F operators are respectively defined by Df ≡ G _{δ} ∗ f andFf (x) ≡ f (2x), and all operator products are interpreted from right to left.) The superscriptn denotes the repetition through n cycles. The second method is to fold the initial distribution through n cycles and then to diffuse over a time nT. This is denoted by the operatorD^{n}F^{n} . This procedure is the one that is useful with real experimental data.
The case (DF)^{n}.
For n = 1, the solution is simply
where the argument of the indicator function is displayed explicitly for the sake of clarity. Similarly, the next iteration yields explicitly
where we have used only the scaling transformations associated with folding and again the additivity of variances for convolved Gaussians. It is then a simple matter to show by induction that the final result can be expressed as
where the variance of the Gaussian satisfies the recurrence relation
The case D^{n}F^{n}.
The solution here is simpler. We already know that the folded distribution after n cycles is given byI
_{n}(x) =I
_{0}(2^{n}
x). Furthermore, the variances of repeated convolutions of Gaussians are additive, yielding a smearing function given by
The variance corresponding to Eq. 12
is, for this case, given by
Diffusive length scale.
The length scale over which diffusive smearing takes place is given by ς_{n}, which is explicitly a function of cycle number n. The behavior of this function for large ndiffers in the two cases considered above. In the
RESULTS
Baker Transform and the Evolving Entropy of Mixing
The results of the approximate computations of the smeared distributions of the periodic Baker Transform are shown in thetop half of Fig. 2. Note that for visual clarity the abscissa shows the interval (0, 2^{−n}); with this doubling of the xscale magnification for each cycle, the sequence of concentration profiles can be easily compared. The initial distribution of aerosol tracer is square; it has concentration unity over onehalf the spatial domain (0 ≤ x < 0.5), is zero for the other half (0.5 ≤ x < 1.0), and is spatially periodic with unit period. For comparison purposes, Fig. 2 A shows the effect on the concentration profile of diffusive smearing with δ = 10^{−2} for cycles 1 through 10. Note the characteristic pattern of a gentle “smoothing” of the initial square wave. Figure 2, B and C, shows the combined effects of the Baker Transform coupled with diffusive smearing on the concentration profile. Figure 2, B and C, shows theD^{n}F^{n} and (DF)^{n} approximations given above. Note the strong similarity of Fig. 2, B and C, and the striking difference of these compared with the effect of pure diffusion shown in Fig. 2 A. In particular, whereas there is an initial smearing over the first few cycles similar to the diffusive case in Fig. 2 A, there is a sharp acceleration in the degree of smoothness associated with an almost violent transition of the concentration profile to being essentially uniform (i.e., complete mixing) by n = 5 or n = 6 in the two approximations shown.
Figure 2, bottom, shows the degree of mixing in these same cases, expressed in terms of the approximate entropy. The formulas developed above, in Quantitative Characterization of Mixing… and in Approximating the Effect of Diffusion on Entropy, are appropriate when the tracer concentration is dilute, i.e., ρ ≪ 1. For the specific examples of coupled diffusion and Baker Transforms, the tracer is not dilute; indeed, the initial conditions include both ρ = 0 and ρ = 1. In this case, the entropy density of the carrier −(1 − ρ) ln (1 − ρ) must also be included. Thus, the total entropy is given by
for ρ_{n} given respectively by Eqs.11
and
13. S_{n}
was computed analytically. Here one sees a particularly graphic demonstration of the difference between simple diffusive mixing and the mixing associated with the coupling of diffusion with stretch and fold patterns. The general characteristic of diffusive mixing is shown by the very gradual rise in entropy over time; this reflects the fact that the variance of bolus distributions increases only linearly with time or cycle number, or the diffusive length scales increasing like
It is clear that the sequential type of factorization (DF)^{n} is a closer approximation to the actual solution of simultaneous diffusion and convection. On the other hand, the experimental results of the stretch and fold pattern at fixed time points or cycle number in real lungs only show the net result of the cyclic sequence. This would represent an inverse problem for which the underlying flow profiles are unknown, and this cannot be solved. By contrast, the factorization given byD^{n}F^{n} represents a combination ofn convective steps followed by n diffusive steps. The former (F^{n} ) can be directly assessed from actual experimental data by using immiscible fluids with negligible diffusion, and the latter (D^{n} ) represents a numerically estimated simulation of the subsequent effect of diffusion over n cycles. In comparingD^{n}F^{n} with (DF)^{n}, we see in Fig. 2 that the experimentally usable method D^{n}F^{n} is quantitatively similar to the (DF)^{n}approximation to the full convection/diffusion equation solution.
Preliminary Observations in Lungs
We performed flow visualization experiments in excised rat lungs (see Fig. 4 legend for detailed methods). Briefly, after filling the lungs sequentially with differentcolored ultralowviscosity polymerizable silicone fluids (blue followed by white), we mechanically ventilated them (1 ml/min) for a few cycles at physiological tidal volumes (∼1 ml) and then waited for the silicone to polymerize. The blue and white volumes were chosen to be roughly equal to assure deep penetration of the bluewhite boundary. Subsequent slices of the lung showed the convective stirring pattern in various regions. The principal findings were as follows. First, mixing of the two colors occurred surprisingly rapidly. In many acinar regions, a uniformly bluishwhite color (representing essentially complete mixing) was achieved within as few as three to five mechanical oscillations. Second, where the interfaces of blue and white compounds were clearly delineated, they exhibited extremely complex stirred convection patterns. A representative example is shown in Figure 4, displaying a typical pattern of the two colors after five cycles, observed on a cross section of a small airway. This airway is ∼1 mm in diameter and is, therefore, near generation 8 or 9 in the rat (17). Highly convoluted stretched and folded mixing patterns can be observed both inside the blue “blob” as well as in the rest of the cross section. The two distinctive regions may represent the presence of a viscous tongue in the convective pattern, superposed on which are the finescale stretching and folding patterns.
To demonstrate how we quantify the extent of mixing of this highly complex system by applying the idea of entropy developed above, we have chosen two representative onedimensional subsets of the cross section, shown as line AB and line CD in Fig. 4. The bold line in Fig. 5, top left, shows the concentration distribution of blue and white colors on line AB.Note the presence of highfrequency fluctuations with a separation distance (lateral folding) of ∼50 μm. Similarly, the bold line in Fig. 5, top right, shows the color distribution corresponding to line CD. Here one sees, in contrast to the generally homogeneous character found on line AB, a separation of the color pattern into two quite distinct regions (sectionsC–E and F–D), separated by ∼400 μm. The middle part (E–F) especially shows highfrequency fluctuations very similar to those seen online AB. We assessed the degree of mixing on each line by determining the diffusive smearing length scale [δ (normalized to unit field length) or δ* = Lδ (the dimensional diffusive length scale, where L is the field length)] required to effect a transition from an unmixed state to a fully mixed state. In Fig. 5, bottom, the extent of mixing as measured by the approximate entropy was plotted as a function of δ*. Similar to the entropy shown in the smeared Baker Transforms (Fig. 2), both sets of experimental data show the entropy increasing with increasing δ* only slowly at first, followed by a relatively sharp transition to an asymptotic state of complete mixing. The convective folding length scale clearly has a great influence on this relationship. For line AB (folding distance ∼50 μm), approximately onehalf of the mixing is completed at a smearing distance of δ* = 10 μm, and at δ* = 100 μm the mixing is nearly complete. On the other hand, line CD exhibits multiple (in this case two) length scales, one ∼50 μm (similar to line AB), and the other a length scale of ∼400 μm, representing the separation of the colors into clearly distinct regimes. At a smearing length δ* of 10 μm, there is little mixing, and even at δ* = 100 μm the mixing is only ∼70% completed.
DISCUSSION
Coupling of Stretch and Fold Patterns With Diffusion
The coupling of the stretch and fold convective patterns with diffusion is a new mechanism that may be important in aerosol mixing and transport deep in the lung. One way of appreciating this phenomenon is to consider the length scales appropriate to convective movement and to diffusive transport over a given time interval. In the absence of convection, diffusive transport occurs over length scales that increase very slowly with time, in particular like
Use of Approximate Entropy as a Tool to Quantify Mixing
The complexity in the interaction of the highly convoluted stretch and fold convection patterns with diffusion cannot be easily or adequately characterized by simple measures such as effective diffusivities (see below). In this paper, we suggest that the entropy of the system is a natural candidate for the quantification of the state of mixing including cases where the interaction of flow and diffusion is complex. As noted in Interaction of Convection and Diffusion in the Baker Transform, one would ideally solve the exact convection/diffusion equation with both mechanisms operating simultaneously, but this is intractable in all but the simplest cases. Furthermore, the experimental data do not show the flow profiles per se but rather the cumulative resultant pattern of the spatial evolution of a tracer (such as the blue and white interface in our experiments). It follows that a post hoc method is necessary to analyze actual experimental observations, and it is for this reason that we propose the approximate entropy computations described in Theory and suggestively validated in Results. In actual practice, therefore, we proceed as follows. For a given test line on a cross section of the air space (now filled with a blue and white silicone mixture) the concentration profile can be smeared with a Gaussian of nondimensional width δ (or dimensional width δ*), from which the approximate entropy S _{δ} can be calculated. A graph ofS _{δ} vs. δ will show a unique value of δ at which the mixing, by this entropic measure, is, say, half completed. This defines a characteristic mixing length δ or δ* for that preparation.
We note that, as the entropy is defined here, there remain two degrees of freedom, namely, the scale of how concentration is measured and the length scale of the region of interest. These correspond to freedom in the scale and zero offset of the computed entropy. To make our proposed method in this paper specific, we have chosen, therefore, to normalize all concentrations to range from 0 to 1, and such that the measure of the interval over which the entropy density is integrated (e.g., line length for line sampling as in Fig. 5) is also unity. With these scales, it follows that the minimum entropy is zero and that the maximum entropy (corresponding to complete mixing) is given by
Relationship of Bolus Dispersion and Mixing
In the classic approach to analyzing the interaction of convection and diffusion, one typically computes the axial spread of a tracer bolus in a convecting coordinate system moving with the mean velocity of the carrier. (This is equivalent to Taylor’s original ideas relating the average flux to the mean concentration gradient.) For flow patterns that are relatively smoothly varying across the cross section, the concentration profile becomes asymptotically Gaussian, with the variance increasing linearly in time. The constant of proportionality relating variance to time is, apart from numerical constants, the effective diffusivity. Indeed, the property of a linearly growing variance is taken as the signature of a diffusive or dispersive process, even though the constants may show significant enhancement over molecular diffusivities due to the coupling of axial convection and transverse diffusion.
The validity of this approach depends on at least two assumptions that may not be appropriate to aerosol transport in real lungs. First, the diffusive limit requires sufficient time for initial distributions to become essentially Gaussian. In vivo, breathing is cyclic, and it is not clear that the longtime steadyflow approximations apply. Furthermore, even with periodic flow conditions, a diffusive limit only obtains when tidal volumes are sufficiently small to be continuously contained within the airways (here the Gaussian profile is a consequence of the central limit theorem on repeated convolutions). However, tidal volumes are more than sufficient to convect from the outside air to the alveolar region, and this implies that such arguments are not valid.
Second, the cyclic flow pattern must be approximately reversible and spatially smooth, in the sense that finer and finer scales are not induced in any tracer profile with repeated breaths. This is the case treated by Watson (15), who analyzed the timeaveraged dispersion of a bolus in a carrier with periodic flow. On the other hand, for convective flow patterns such as the stretch and fold patterns described here, both the simple theoretical example of the Baker Transform as well as the experimental data shown in Fig. 4, the assumption of a smoothly varying and approximately reversible flow pattern is clearly not valid, and a diffusive characterization of mixing in these cases is not appropriate.
The Analogy With the Baker Transform
This particular transform possesses, in a very simple manner, the essential features of the stretch and fold pattern of convection. It is deficient in one respect, which, however, does not change our conclusions about the potential importance of the stretch and fold mechanism leading to acinar mixing. In particular, Bakertype analyses, common in the study of chaos, are usually restricted to steady flow fields and resulting tracer trajectories. Gas flow in the lung, on the other hand, is cyclic, being driven by boundary motion that is close to being kinematically reversible (1, 5, 6, 8, 16). This observation suggests that both the carrier flow field and the resulting trajectories in the acinus would be very simple and smooth. Nevertheless, the blue and white tracer experiments show a clear stretch and fold pattern with substantial fine structure and complexity in the tracer trajectories; this evolving fine structure is precisely the characteristic feature mimicked by the Baker Transform. The origins of this phenomenon in real lungs may include the presence of flow irreversibility associated with the small motion of saddle points in the carrier flow (13) or, perhaps, the presence of small but systematic geometric hysteresis (8).
Our theoretical work here is based on the cycle number n, and the evolving entropy is characterized by its dependence on n. By contrast, the data of Heyder et al. (7) represent bolus dispersion after a single breath. On the other hand, the nature of the branching airway tree, with multiple pathways, path lengths, and path asynchrony implies that each breath is, in effect, equivalent to a stretch and fold sequence of n > 1 or even n ≫ 1. The experimental determination of the effective folding number per breath in real lungs is, therefore, a critical point and it awaits future quantification.
Conclusion
Taken together, the experimental findings of Heyder et al. (7) on aerosol transport and our experimental work in flow visualization (Ref.12 and this paper) strongly suggest a substantive role that convective mixing may play in the ultimate fate of inhaled aerosol particles. The theoretical analysis presented here represents a first step toward capturing the essence of the mechanism by which such convective patterns can strongly influence particle mixing and transport. Two conclusions emerge. First, from direct observations in rat lungs, there does exist a stretch and fold pattern of convection deep in the lung. Second, from our theoretical analysis of the interaction of this type of convection with diffusion, we find this interaction to be a potent candidate for the observed extent of mixing of aerosols in the lung periphery, where the low diffusivity and approximate flow reversibility would have led one to predict little mixing at all.
Acknowledgments
We thank Dr. J. Godleski for help in the experimental preparation and K. Okabe for assistance in morphometry.
Footnotes

Address for reprint requests: J. P. Butler, Physiology Program, Harvard School of Public Health, 665 Huntington Ave., Boston, MA 02115.

This work was supported by National Heart, Lung, and Blood Institute Grants HL47428 and HL54885.
 Copyright © 1997 the American Physiological Society