Abstract
Pulmonary vascular perfusion has been shown to follow a fractal distribution down to a resolution of 0.5 cm^{3} (5E11 μm^{3}). We wanted to know whether this distribution continued down to tissue volumes equivalent to that of an alveolus (2E5 μm^{3}). To investigate this, we used confocal microscopy to analyze the spatial distribution of 4μmdiameter fluorescent latex particles trapped within rat lung microvessels. Particle distributions were analyzed in tissue volumes that ranged from 1.7E2 to 2.8E8 μm^{3}. The analysis resulted in fractal plots that consisted of two slopes. The left slope, encompassing tissue volumes less than 7E5 μm^{3}, had a fractal dimension of 1.50 ± 0.03 (random distribution). The right slope, encompassing tissue volumes greater than 7E5 μm^{3}, had a fractal dimension of 1.29 ± 0.04 (nonrandom distribution). The break point at 7E5 μm^{3} corresponds closely to a tissue volume equivalent to that of one alveolus. We conclude that perfusion distribution is random at tissue volumes less than that of an alveolus and nonrandom at tissue volumes greater than that of an alveolus.
 alveolar perfusion
 fluorescent microspheres
 fractal analysis
 pulmonary blood flow
fractal analysis has been used extensively to analyze the distribution of perfusion throughout the lung (3). The method is to infuse 15μmdiameter fluorescent latex particles into the pulmonary circulation and then count the number of trapped particles in blocks of frozen or airdried lungs (15). The blocks, which range in size from 0.5 mm^{3} to 2 cm^{3}, are grouped mathematically to allow particle counts to be obtained in successive volumes of tissue that range from single tissue blocks up to groups of blocks on the scale of the whole lung (4). For groups of each volume, the mean and SD of the particle counts are calculated for the blocks that compose the group. A fractal plot is assembled by plotting the log of the coefficient of variation (CV; SD/mean) of the group count against the log of the group tissue volume. When this is done for groups of all volumes, it produces a linear plot that shows that the CV of the particle count increases as the group tissue volume decreases. The linearity of such plots is considered to be evidence of a fractal system, which means that perfusion distribution within the lung follows a fractal pattern. Fractals are complex geometric shapes that exhibit a scaleindependent property of selfsimilarity such that the smallest part, no matter how highly magnified, resembles the whole (11). Using this approach, Glenny and colleagues (4) showed that pulmonary perfusion distribution remained fractal down to a tissue volume of 0.5 cm^{3} (5E11 μm^{3}).
It is interesting to speculate whether or not this pattern continues down to the level of the alveolus. An alveolus with a diameter of 75 μm has a volume of 2.2E5 μm^{3}, but present methods do not allow fractal analysis to be conducted at that level of resolution.
To address this, we developed a method utilizing confocal microscopy in which fractal analysis could be conducted to a level of resolution of 1.7E2 μm^{3}. Our approach was to infuse 4μmdiameter fluorescent latex particles into anesthetized, spontaneously breathing rats. We removed and air dried the lungs and then used confocal microscopy to obtain digital images of the particles within lung slices. We divided these images into tissue volume increments that ranged from 1.7E2 to 2.8E8 μm^{3} and counted the number of particles within each increment. The variation in the particle counts among the increments allowed us to establish fractal dimensions for the particle distribution throughout the range of tissue volumes. We found that the distribution was fractal (fractal dimension = 1.29 ± 0.04) within the volume range 7E5 to 2.8E8 μm^{3}. However, in the volume range from 2E2 to 7E5 μm^{3} we measured a fractal dimension of 1.50 ± 0.03, which is the value associated with a random or Poisson distribution. The tissue volume at which the inflection point occurred (7E5 μm^{3}) is approximately equal to that of one alveolus. Our results suggest that flow distribution is nonrandom at tissue volumes greater than that of one alveolus but at subalveolar volumes flow distribution appears to be random.
METHODS
The experimental protocols employed in these studies were approved by the institutional animal care committee of the University of WisconsinMadison.
Animal Preparation
We anesthetized retired male breeder rats (410 ± 26 g; n = 6) with intraperitoneal ketamine (40 mg/kg), xylazine (6 mg/kg), and acepromazine (1 mg/kg) and tied them supine. We placed an indwelling cannula (PE190) into a femoral vein, infused heparin (500 U/kg), and then turned the animals so that they were lying prone. A syringe pump containing the latex particle suspension was connected to the femoral cannula, and the particle solution (2.0 ml) was infused over 5 min. After the infusion, we turned the animals supine and then transected a femoral artery to allow them to exsanguinate. Once spontaneous breathing had stopped, we tied a cannula into the trachea, set the tracheal pressure to 5 cmH_{2}O with air, and opened the chest using a sternumsplitting incision. We removed the lungs from the thorax, raised the airinflation pressure to 20–25 cmH_{2}O, and maintained this pressure for 2 days to allow the lungs to dehydrate.
Using a specimen knife (Pathco), we collected slices (2–3 mm) from eight specific lung regions of each dried lung (Fig.1). Our goal was to determine whether particle distribution differed among regions. We analyzed particle distribution in each slice using the methods described below (Analysis of Particle Distribution).
Particle Infusion
The particle infusion solution was prepared as follows. A suspension of 4μmdiameter rhodaminelabeled latex particles in water (20 or 400 μl; Molecular Probes) was added to enough 3% albumin in water to make a final volume of 2.0 ml. The exact particle suspension volume added was determined from measurements of the particle concentration that we made using a fluorescenceactivated cell sorter (FACScan, BectonDickinson). The albumin coated the particles, neutralized their surface charge, and prevented them from clumping in the presence of ions. We next added salts to the solution (NaCl, NaH_{2}PO_{4}, Na_{2}HPO_{4}) to yield PBS. This infusion solution was gently sonicated to eliminate any chance that clumps had formed. The absence of clumps was confirmed by using the FACScan and by inspecting particles in smears of the solution via a fluorescence microscope. Solutions of two particle counts were prepared and infused: 1E7 (3 rats) or 2E8 (3 rats).
Microvascular Trapping
To confirm that 4μmdiameter particles were trapped within the pulmonary microcirculation, we measured the fraction of infused particles trapped in an additional set of isolated lungs and in the lungs of an additional set of intact rats. Isolated lungs (n = 4) were harvested by using the methods described above, ventilated with air (inspiratory and expiratory pressure: 15 and 5 cmH_{2}O, respectively; 50 breaths/min), and perfused with 3% albumin in PBS from a reservoir set 15 cm above the base of the lung (pulmonary arterial and left atrial pressure: 15 and 0 cmH_{2}O, respectively; flow, 12 ml/min). Latex particles were infused into the arterial inflow line by using a syringe pump (2 ml, 5 min). During and after particle infusion, the venous outflow was collected in a series of tubes, and the number of particles in each tube was measured by using the FACScan.
We measured particle trapping in intact, spontaneously breathing rats (n = 2) by collecting simultaneous lung inflow and outflow blood samples (5 pairs, 1 ml) during particle infusion into a femoral vein. The outflow sample was collected from the abdominal aorta, and the inflow sample was collected from the inferior vena cava (IVC). The IVC sample did not represent all blood entering the pulmonary artery. However, we assumed that IVC flow represented 75% of pulmonary artery flow; we, therefore, reduced the IVC particle counts to 75% of those measured to estimate particle concentrations in the pulmonary artery (9). Particle concentrations in blood samples were measured by using the FACScan.
Analysis of Particle Distribution
Fractal analysis was based on statistical analysis of particle distributions within slices of dried lung obtained from 8 lung regions (Fig. 1). The analyses were conducted by use of digital maps of the particles within each field. The maps were made and analyzed as follows.
Particle maps.
The maps were made by using a laserscanning confocal fluorescence microscope (BioRad MRC 1024ES). Maps were made in each of the eight regions per lung (Fig. 1). The areas mapped were randomly chosen from fields that lacked major vessels or airways (>0.5 mm). Maps of each lung region consisted of two adjacent lowpower maps encompassing an area of 6,732 × 3,366 × 100 μm, plus a highpower map that encompassed an area of 672 × 672 × 100 μm. Each of the two lowpower maps was a composite of 20 sequential confocal images (3,366 × 3,366 μm) that were acquired stepwise into the tissue through a depth of 100 μm by use of the ×4 objective of the microscope. The 20 images were digitally stacked to form a single image that encompassed a thickness of 100 μm. Thus the overall tissue volumes were those encompassed by this stack. The highpower map was prepared similarly by use of the ×20 objective of the microscope. An example confocal image acquired via the ×4 objective is shown in Fig.2.
Each map was a digital image that consisted only of particles and was acquired by using a digital array (512 × 512 pixels) (Fig.3). Thex,y address of each particle within each map was determined by using public domain software (NIH Image, version 1.62). The pixel grayscale value of each map was set to ≥100 (range 1–256) before x,y addresses were determined. This value was chosen because it maximized the number of pixels per particle yet minimized the frequency with which adjacent particles merged into particle clusters. We return to this point in thediscussion.
Particle distribution analysis.
We used multiscale statistical analysis to evaluate particle dispersion within the maps. The foundations of this method are described in thediscussion. To conduct this analysis, each pair of lowpower maps was subdivided stepwise. An example is shown in Fig. 3, which is onehalf of a lowpower map that has been subdivided into a 2 × 2 array that consists of four subdivisions, each of which measures 1,680 × 1,680 × 100 μm (2.8E3 μm^{3}). We counted the number of particles in each of the four subdivisions, as well as in the four subdivisions of the other half of the map, and determined the mean and SD for the counts in all eight subdivisions. We then plotted the CV of this count (SD/mean) against the subdivision tissue volume (2.8E3 μm^{3}). We next subdivided the two images into an 8 × 4 array, counted the number of particles in each of the 32 subdivisions, and again plotted the log of the CV of the particle count against the log of the subdivision tissue volume (step 2). We repeated this process through a total of nine steps, as well as through an additional three steps using the highpower maps. The number of subdivisions and the volume of each are shown in Table 1. To assemble a fractal plot, we plotted the CV of the particle count (vertical axis, log scale) vs. the subdivision tissue volume (horizontal axis, log scale) for all 12 steps and fit a regression line through the data. We assembled separate plots for animals that received 1E7 particles and for those that received 2E8 particles.
The slope of each regression line is a measure of the particle dispersion. In the pulmonary circulation literature, these slopes are described in terms of fractal dimensions because a fractal system can be described by using multiscale analysis (7). The fractal dimension is equal to one minus the slope of the regression line (5). Thus a slope of −0.5 has a fractal dimension of 1.5, which corresponds to a condition in spatial statistics known as complete spatial randomness (CSR; Ref. 8). We will return to this point in the discussion.
Statistics
Results are expressed as means ± SD.
We found that the fractal plots consisted of two slopes: one that corresponded to tissue volumes less than 7.4E5 μm^{3} and one that corresponded to tissue volumes greater than this value. We used a repeatedmeasures analysis of variance to determine whether these two slopes differed among lung regions and between lungs that received 1E7 or 2E8 particles. Pairwise comparisons of factor interactions were made by use of Fisher's protected least significant difference test. Differences were considered to be significant atP ≤ 0.05. All analyses were performed by using SAS statistical software (SAS Institute, Cary, NC).
RESULTS
Particle Trapping
In isolated lungs, venous outflow particle concentrations reached a cumulative maximum of 4.3 ± 1.5% of the particles infused. This meant that 95.7% of the particles were trapped within the lung. In intact animals, abdominal aorta samples concentrations averaged 6.4 ± 3.8% of those in IVC samples. If we assume that blood flow through the IVC represented 75% of the blood flowing into the PA, then the aortic concentrations would have averaged 8.5 ± 5.1% of those in the PA (9). This meant that 91.5% of the particles were trapped within the lungs of the intact animals.
Particle Counts Per Map
The number of particles per lowpower map averaged 91 ± 26 in maps prepared from lungs infused with 1E7 particles. There was an average of 3.9 ± 2.7 particles in each highpower map. In maps prepared from lungs infused with 2E8 particles, there was an average of 2,313 ± 617 particles per lowpower map and 107 ± 33 particles per highpower map. We found no significant differences in the number of particles per map among lung regions (Figs. 4 and5).
Fractal Dimensions
Pooled plots of log CV vs. log subdivision volume (fractal dimension) for all data from all eight lung regions are shown graphically in Fig. 6. Fractal dimensions for individual lungs by lung region are shown in Table2. Fractal dimensions were not significantly different among lung regions in lungs infused with either 1E7 (P = 0.9779) or 2E8 (9 = 0.9935) particles. This is the reason that the data were pooled for graphic display (Fig.6). Upper fractal dimensions were significantly different from lower fractal dimensions within both groups (1E7: upper fractal dimension, 1.52 ± 0.06; lower fractal dimension, 1.47 ± 0.03;P = 0.0003; 2E8: upper fractal dimension, 1.50 ± 0.03; lower fractal dimension, 1.29 ± 0.04; P = 0.0001). Lower fractal dimensions in the 1E7 data were significantly different from those in the 2E8 data (P = 0.0007); upper fractal dimensions between the two groups were not significantly different (P = 0.2684).
DISCUSSION
Our main finding is the noticeable inflection in the fractal plot, especially in the 2E8 data, at a tissue volume of 7.4E5 μm^{3}. To the left of this inflection the data have a fractal dimension of 1.50, which is a value that corresponds to CSR (see Particle Distribution Analysis, below). To the right of the inflection, the fractal dimension is 1.29, which is a value that corresponds to a clustered particle distribution. The tissue volume at the inflection point (7.4E5 μm^{3}) corresponds to a sphere with a diameter of 112 μm, which is approximately that of an alveolus. Thus this change in the fractal dimension occurs at tissue volumes approximately equal to that of one alveolus (Fig. 6).
The interpretation suggested by this is that pulmonary microvascular perfusion is distributed randomly at tissue volumes smaller than the alveolus, and at larger tissue volumes perfusion distribution is clustered. The cause of this clustering is unknown but may be the result of vasoactive control. We consider below (Effect of Parenchyma on Particle Distribution) the possibility that parenchymal distribution may be responsible for this particle clustering.
Particle Distribution Analysis
We used fractal analysis to evaluate latex particle distribution within the confocal particle maps because fractal analysis has been used extensively to describe perfusion distribution throughout whole lungs (6). However, fractal analysis is a specific example of more general methods of spatial pattern analysis. These more general methods use a specific pattern known as CSR as a point of reference. The definition of CSR is based on the notion of spatial independence: if points in space are spatially independent, then knowing the location of a point provides no information about the probability of other points nearby (8). Thus, if one knows that a point falls at coordinates x,y in a CSR pattern, such knowledge makes it no more or less likely that another point falls in some neighborhood around x,y. The only statistical model that can generate a CSR pattern is a homogenous planar Poisson process (HPPP) (2).
If an HPPP was used to generate a figure like that of one of our particle maps (Fig. 3), and if that figure was partitioned into a grid of subdivisions, as we did to conduct fractal analysis, then an HPPP has the important property that the mean number of particles per subdivision (μ) would equal the variance of the particles per subdivision (ς^{2}). Thus, regardless of the subdivision size, the variancetomean ratio, ς^{2}/μ, would be unity for an HPPP. This ratio is known as the dispersion index (DI) and is widely used as a measure of spatial pattern. Under CSR, DI = 1, which means that it is impossible to improve predictions of the location of a point by knowing the location of any other point. However, if DI > 1, it means that the points are clustered more than would be expected under CSR. The points are no longer independent, and knowing that a point is located atx,y increases the probability that another point is in the near neighborhood ofx,y. In contrast, if DI < 1, the points are more regularly spaced than expected under CSR. In the extreme case of DI = 0, all subdivisions would have exactly the same number of particles, as in the distribution of squares on a checker board. If D < 1, and if we know that a point is located at x,y, such knowledge now decreases the probability that another point is in the neighborhood ofx,y and increases the probability that an adjacent point is at some greater distance fromx,y.
These values are incorporated in the GriegSmith method, which is one of the oldest and most widely used methods of analyzing the scale of spatial patterns and which is based on the fact that DI equals unity if the spatial pattern of points was generated by an HPPP (8). This is so regardless of the fineness or coarseness of the grid of subdivisions. In this method, a surface is successively partitioned into finer and finer grids, and at each level the DI or log(DI) is plotted against the subdivision size. Conspicuous kinks in the resulting plot are interpreted as indicating the scale at which important changes in pattern occur. Fractal analysis is an example of this method.
It is rare to find CSR patterns in biological systems. Clustering or aggregation is the norm. In fact, it is quite common for there to be a hierarchy of clustering, or clustering of clusters. As the spatial scale changes, this clustering of clusters often leads to a smooth change in the DI for various spatial biological phenomena. It is well known that this smooth change obeys power laws of the form ς^{2} = αμ^{β} = α(vλ)^{β}, where λ is the mean count per grid at the smallest grid unit; for any subdivision consisting of v units, μ = vλ. In ecology, this model is known as Taylor's power law, and various stochastic models can generate spatial patterns with such structure (10). The power law includes CSR as a special case, for which α = 1 and β = 2.
A fractal plot is a loglog plot of Taylor's power law. In fractal analysis, a nonlinear loglog plot, such as that we found (Fig. 6), is said to be indicative of a nonfractal pattern (5). Describing a relationship between CV and subdivision volume as fractal is equivalent to saying that the mean and variance obey a power law relationship. Such power law relationships are to be expected in random branching processes such as that of the pulmonary circulation.
Although we expressed our data in terms of fractal analysis, it would be more accurate to say that we were interested in knowing whether Taylor's power law held at lung volumes less than 0.5 mm^{3}. Glenny and Robertson considered this question and predicted how the CV of blood flow should behave as the tissue subdivision volume approached what they described as “the anatomic limit of the capillary” (5). At such a volume, they predicted that blood flow would become uniform among lung units, producing a fractal plot that was horizontal (fractal dimension 1.0) as it extended leftward through smaller and smaller tissue subdivision volumes. At larger subdivisions, the plot would be fractal, producing a linear plot with a fractal dimension of 1.16.
Glenny and Robertson's requirement for a conspicuous flattening of the fractal plot might be too strong a condition for observance of uniformity of flow (5). By definition, latex particle distribution is more uniform than CSR whenever DI < 1, which, in terms of a log CV plot (fractal plot), translates into the condition log CV < −1/2log v − 1/2logλ. That is, particle distribution is uniform whenever the fractal plot falls below the line described by −1/2log v − 1/2logλ. The distance from this line measures the departure from CSR. Specifically, the distance δ below the line is measured as δ = logς_{v}−1/2log(vλ). The more negative the value of δ, the more uniform the distribution. Interestingly, δ = 1/2log(DI), which means that the distance of interest is 1/2 the log of the DI. A substantial departure from CSR toward uniformity would not necessarily result in a conspicuous departure from the CSR reference line. For example, a DI of 0.5 (ς^{2}/μ = 0.5μ), which suggests a substantial trend toward uniformity, would only depart from the CSR reference line by 1/2log (0.5) = −0.15. Thus uniformity of flow need not result in curvature as extreme as that proposed by Glenny and Robertson and may include plots that are concave upward. This is what we found in the fractal plot of our data (Fig. 6).
Because of these relationships, plots of log(DI) are easier to interpret than plots of log(CV). If a process is fractal, i.e., linear with respect to log(CV), it is also linear with respect to log(DI). Furthermore, if a process consists of CSR at all scales, the plot of log(DI) has a slope and intercept of zero. We replotted our data as a DI plot to illustrate this point (Fig.7). Visually, it is easier to detect departures from this pattern than from the reference line of −1/2log v − 1/2logλ as in log(CV) plots (fractal plots). The fractal dimension (D _{s}) can be obtained from the regression line of log(DI) as D _{s} = 3/2 − β/2, where β is the regression slope of log(DI).
Effect of Parenchyma on Particle Distribution
Because much of the lung is air space, the distribution of particles within subdivisions of each particle map (Fig. 3) might be a function of the distribution of tissue within the subdivisions. This may be particularly so at the smallest subdivision volumes. For example, if the air space is ignored and perfusion distribution fits a CSR pattern, is particle distribution truly random or is it uniform within the tissue, which itself is CSR? This might result in a pattern of particle distribution that spuriously resembles CSR.
We show in the how we can correct for this by using only that portion of each particle map that includes tissue. We used this approach to examine the distribution of particles in tissue alone, without the confounding influence of air space. Our method was to collect separate but coincident confocal images of parenchyma and latex particles. This was possible because parenchyma exhibited autofluorescence in the green while the particles fluoresced in the red. Thus images of each could be collected simultaneously from the same tissue area by using two of the confocal microscope's photodetectors. This allowed us to use the pixel coordinates of the tissue as the space in which the particle distribution analysis was conducted. We used five pairs of parenchymaparticle images for this analysis. These images were made using only the ×4 objective of the microscope. Thus they represent subdivision volumes ranging from 3.5E3 to 2.8E8 μm^{3}.
A confounding aspect of this approach was to determine the appropriate grayscale value for the parenchymal pixels. Parenchymal fluorescence spanned the 8bit grayscale range of the photodetectors, which meant that fluorescence could range from values of 1 to 256. For analysis, we selected threshold values of >20, >50, and >100. We expressed the results as DI plots, which are shown in Fig.8.
DI plots resulting from this approach (Fig. 8) show deviations from CSR at tissue volumes of between 1E5 and 1E6 μm^{3} at all three grayscale threshold values. The magnitude of the deviation from CSR is less at threshold value >100 than it is at >20, demonstrating that the magnitude of the particle clustering is inversely proportional to the space in which the pattern analysis is conducted. However, the most important finding of this analysis is that the particle distribution deviation from CSR occurs at the same subdivision volume as that seen in Fig. 7, where parenchymal effects were not accounted for. This suggests that the deviation of the perfusion distribution from CSR at tissue volumes equivalent to that of an alveolus is not due exclusively to the distribution of the lung tissue itself, but is due to other factors. This further suggests that perfusion distribution may be controlled down to the alveolar level.
Perfusion distribution throughout the lung is assumed to be controlled by arterioles, which may have diameters as small as 30 μm (12). This is based on the observation that these are the smallest vessels around which a partial circumferential smooth muscle layer may occasionally be seen. If variations in the tone of these vessels were responsible for the particle clustering patterns we observed, we would expect deviations from CSR to first appear among tissue volumes equal to that supplied by individual 30μmdiameter arterioles. We can estimate the number of alveoli supplied by each vessel of this size by dividing the total number of alveoli in the lung by the total number of 30μm arterioles. Data for this calculation are available for dog lungs, for which we calculate a ratio of 750 alveoli per 30μm arteriole (13, 14). We assume a similar ratio for rat lungs. Thus, if perfusion distribution were controlled by 30μmdiameter arterioles, we would expect deviations from CSR to begin among tissue volumes consisting of 750 alveoli each. However, our data show that deviations from CSR begin in tissue volumes equal approximately to that of one alveolus (Fig. 7). Thus we found deviations from CSR in far fewer alveoli than those that would be expected to be supplied by individual arterioles. This again suggests that control of pulmonary microvascular perfusion may exist within much smaller vessels than those previously thought, through some mechanism not yet recognized.
Our results support those of Wagner and colleagues (16), who examined perfusion patterns in individual capillaries of adjacent alveoli. They measured patterns of red cell movement within capillaries of individual alveoli and found that these patterns were repetitive over time and, therefore, fractal. However, they also found that the patterns recorded in one alveolus bore virtually no relationship to those recorded in an adjacent alveolus. Thus temporal patterns of perfusion differed between adjacent alveoli. Our results suggest that clustering of perfusion, based on latex particle distribution, can be seen in tissue volumes equal to that of only one or perhaps a few alveoli. Thus our data and those of Wagner and colleagues demonstrate variations in perfusion patterns among individual or small groups of adjacent alveoli. Our data further suggest that these variations increase with the number of alveoli in the adjacent groups.
Glenny and colleagues (4) recently reported data on the fractal distribution of perfusion in rat lungs down to subdivision volumes of 0.53 mm^{3} (5.3E8 μm^{3}). They utilized an imaging cryomicrotome to analyze the distribution of 15μmdiameter latex particles throughout rat lungs fixed by freezing. A fractal plot through their data had an average value of 1.12. Their results suggest somewhat more clustering of perfusion [greater deviation from CSR (fractal dimension 1.5)] than we found in our 2E8 data, which have a fractal value of 1.26 (Fig. 6). The smallest unit of tissue volume they examined (5.3E8 μm^{3}) is approximately twofold larger than our largest subdivision volume (2.8E8 μm^{3}). Thus our studies extend those of Glenny and colleagues down to subalveolar tissue volumes. Differences in the fractal dimensions between our studies and theirs may be due to differences in the size of the particles used (4 vs. 15 μm) and differences in the methods used to measure particle positions (confocal microscopy vs. cryomicrotomy).
We did not utilize the vertical (z) axis resolution of the confocal microscope to determine the location of each particle within the vertical plane (100μm thickness). Our interest was in determining the distribution of the particles among alveoli. Assuming that a rat alveolus has an average diameter of 75 μm, the lowpower map dimensions (Fig. 1) measured ∼90 × 45 × 1.3 alveoli (14). This is appropriate for evaluating interalveolar perfusion distribution. It is unlikely that distribution in thezaxis differed from that in the x andyaxes.
We used 4μmdiameter particles for our studies because we were interested in perfusion distribution among alveoli, not in flow per gram of tissue. We showed previously that these particles are trapped mainly in interalveolar corner vessels (1). Furthermore, our trapping data suggest that more than 91% of the infused particles were trapped within the pulmonary microcirculation. Thus these particles appear to be well suited for evaluating the distribution of perfusion among alveoli.
One goal of our studies was to determine the optimum number of particles to infuse. The optimum provides the greatest number of particles per map, while minimizing particle clustering. Greater numbers of particles increase pattern analysis resolution. The data (Figs. 6 and 7) show that deviations from CSR were greater for 2E8 particles than for 1E7, which demonstrates the effect of particle numbers on resolution. To determine the fraction of clustered particles in particle maps, we counted the number of pixels occupied by every particle in five randomly selected 2E8 maps. We assumed that single particles consisted of ≤10 pixels. We determined this value from particle maps in lungs infused with 1E7 particles where we assumed that the particles were widely dispersed enough so that only singlets were present; 10 was the largest number of pixels per particle in those images. In the 2E8 maps, 96.5% of the particles consisted of ≤10 pixels. The remaining 3.5% of particle images consisted of 19–33 pixels. We found no particle images that consisted of 11–18 pixels. This supports our belief that single particles consisted of ≤10 pixels and suggests that larger groups of pixels consisted of clusters of three or possibly four particles. However, the fact that these pixel groups accounted for only 3.5% of the total number of particles present suggests that particle clusters had an insignificant effect on the pattern analysis. Thus 2E8 appears to be an ideal number of particles for these studies.
Our goal was to evaluate flow distribution among alveoli, and we used statistical analysis of the particle distribution patterns as a way to accomplish that. We believe that this method provides a way to quantify the distribution of pulmonary microvascular perfusion at the scale of one to a few hundred alveoli. We hope this approach will provide new insights into mechanisms by which the distribution of pulmonary microvascular perfusion is controlled.
In conclusion, our results suggest that flow distribution among alveoli becomes nonfractal at the alveolar level and corresponds to a statistical state of CSR. However, at tissue volumes larger than that of one alveolus, we found that the flow distribution is clustered in a way that can be expressed quantitatively. The cause of this clustering is unknown, but it may be evidence that perfusion is controlled down to the alveolar level.
Acknowledgments
This research was supported by a grant from the Department of Veterans Affairs.
Footnotes

Address for reprint requests and other correspondence: R. L. Conhaim, Univ. of Wisconsin Medical School, Dept. of Surgery, H5/301 Clinical Science Center, 600 Highland Ave., Madison, WI 537927375 (Email:rconhaim{at}facstaff.wisc.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

First published October 11, 2002;10.1152/japplphysiol.00700.2002
 Copyright © 2003 the American Physiological Society
Appendix
The following proof was used to conduct particle DI analysis within the subset of pixels consisting of tissue fluorescence. Results are shown in Fig. 8.
Theorem: Let x_{i}
be the particle counts in theith sample. Let p_{i} be the proportion of the ith sample that is tissue. Define