Abstract
The spatial distribution of pulmonary blood flow is increasingly heterogeneous as progressively smaller lung regions are examined. To determine the extent of perfusion heterogeneity at the level of gas exchange, we studied blood flow distributions in rat lungs by using an imaging cryomicrotome. Approximately 150,000 fluorescent 15μmdiameter microspheres were injected into tail veins of five awake rats. The rats were heavily anesthetized; the lungs were removed, filled with an optimal cutting tissue compound, and frozen; and the spatial location of every microsphere was determined. The data were mathematically dissected with the use of an unbiased random sampling method. The coefficients of variation of microsphere distributions were determined at varying sampling volumes. Perfusion heterogeneity increased linearly on a loglog plot of coefficient of variation vs. volume, down to the smallest sampling size of 0.53 mm^{3}. The average fractal dimension, a scaleindependent measure of perfusion distribution, was 1.2. This value is similar to that of other larger species such as dogs, pigs, and horses. Pulmonary perfusion heterogeneity increases continuously and remains fractal down to the acinar level. Despite the large degree of perfusion heterogeneity at the acinar level, gases are efficiently exchanged.
 pulmonary perfusion
 heterogeneity
 spatial distribution
 rats
with improved spatial resolution of blood flow measurements, pulmonary perfusion has been shown to be quite heterogeneous. The magnitude of this heterogeneity is dependent on the scale of resolution used to examine local blood flow. As progressively smaller lung regions are examined, the observed variability in perfusion increases. Over the range of observations that have been made, this scaledependent heterogeneity has been well characterized by fractal mathematics. Fractal analyses have demonstrated that, as the volume of observation decreases, the measured perfusion heterogeneity increases in a linear fashion on a loglog plot (3, 9). This fractal relationship remains linear down to lung pieces as small as ∼2 cm^{3}.
Higher resolution methods for measuring regional perfusion may demonstrate one of three possibilities. Either perfusion heterogeneity will continue to increase linearly down to the alveolar capillary level, it will continue to increase but in a nonfractal manner, or it will become uniform within regions (2). At a piece size of ∼2 cm^{3}, perfusion heterogeneity, as measured by the coefficient of variation (CV), is 0.45 in dogs (9). If the fractal relationship is extrapolated to an acinar volume of ∼1 mm^{3} (20, 23), the observed perfusion heterogeneity is three times greater. This large variability in blood flow has important implications for gas exchange and the mechanisms matching ventilation and perfusion. The hypothetical volume at which perfusion may become uniform has been designated the “unit of perfusion” (29).
To determine whether blood flow remains fractal down to the level of gas exchange or whether there is a larger unit of homogeneous perfusion, we used an imaging cryomicrotome to measure regional blood flow at the subacinar level in rats. A secondary goal of our study was to determine whether blood flow in the rat is fractal, and, if so, whether the fractal dimension is similar to that in larger laboratory animals.
METHODS
The University of Washington Animal Care Committee approved these animal studies. Five awake male SpragueDawley rats (B and K Universal, Kent, WA) weighing between 262 and 276 g were physically restrained while fluorescent microspheres were injected via a tail vein. Approximately 75,000 (15μm diameter) microspheres (Molecular Probes, Eugene, OR) of two different colors (total 150,000) were injected simultaneously. Two different fluorescent colors were used to reduce the spatial density of microspheres counted within images (see below). Tightly grouped microspheres cannot be resolved as individual points and are undercounted in highflow regions if their spatial density is too great. By using two different colors, the spatial density of any one color is not too great for accurate counting. Before injection, the microspheres were sonicated, vortexed, and mixed in the same syringe. The microspheres were suspended in 0.24 ml of saline and injected over 15 s.
After the microsphere injection, the rats were deeply anesthetized with an intraperitoneal injection of pentobarbital sodium (50–100 mg/kg), their chests were widely opened, and they were exsanguinated. The lungs were removed from the chest, filled via the trachea with optical cutting tissue (Sakura Finetek, Torrence, CA) media until they appeared inflated to total lung capacity, and frozen. We chose this volume to ensure that all lung regions were uniformly expanded so that regional blood flow was measured relative to regional alveolar volume. Optical cutting tissue is a clear liquid containing polyvinyl alcohol and polyethylene glycol that is easily sectioned with a microtome when frozen. The lung blocks were mounted in the imaging cryomicrotome, and the spatial locations of every microsphere were determined by using 30μmthick sections.
The imaging cryomicrotome (Barlow Scientific, Olympia, WA) is a device that determines the spatial distribution of fluorescent microspheres at the microscopic level. Details of the instrument configuration have been previously reported (18). The instrument consists of a Kodak Megaplus 4.2 chargecoupled device video camera (Eastman Kodak, San Diego, CA), a computer (Dell Computer, Round Rock, TX), metal halide lamp (HTI 403W/24, Osram, Sylvania), excitation filterchanger wheel, emission filterchanger wheel, and a cryostatic microtome. Fluorescence images were acquired with the Kodak digital camera (2,000 × 2,000 pixel array) with a 200mm Nikkor lens (Nikon, Tokyo, Japan) by using a macrofocusing baffle. Two motorized filter wheels containing excitation and emission filters were controlled by the computer through stepper motors and microsensors. A customdesigned microtome serially sections frozen organs. Computer control of the microtome motor, emission filter wheel, and image capture and display was accomplished through a virtual instrument written in LabVIEW 5.0.1 (National Instruments).
The cryomicrotome serially sections through the frozen lung with a slice thickness of 30 μm. Digital images of the remaining tissue surface were acquired with appropriate excitation and emission filters to isolate each fluorescent color (Fig.1). Images at each fluorescence excitation/emission wavelength pair were collected and processed so that x, y, and z (slice) locations of each microsphere were determined and saved in a text file. The spatial resolution of the system depends on the magnification used to image the lung but is typically 10 μm in the x and ydirections and 30 μm in the z direction. A threshold algorithm was applied to images of lung cross sections to produce a threedimensional binary map defining the spatial location of parenchyma. Airways and large vessels were identified in the images and designated as nonparenchyma, because microspheres cannot lodge within these structures. This map determines the threedimensional organ space to be sampled and the space for the random microsphere distributions (see below).
Fractal algorithm.
The fractal algorithm is written in C running on an Ultra 10 Workstation (Sun Microsystems, Palo Alto, CA). The general approach is to randomly sample the organ space with spherical volumes, determine the number of microspheres present in each volume, and calculate a CV (CV = SD/mean) for the sampled distribution. The random sampling was performed multiple times to obtain an accurate estimate of the true CV and the SE of the mean CV. The radius of the sampling spheres was incremented by 250 μm, and a new CV was determined at this new sampling volume. The CV was corrected for Poisson noise at each step as described below.
The starting radius in these experiments was 500 μm, resulting in a smallest sampling volume of ∼0.53 mm^{3}. Random sampling without replacement was performed by choosing x,y, and z coordinates from a pseudorandom number generator (Unix, Berkeley, CA). A method known as Marsaglia shuffling (21) was used to ensure a uniform distribution of numbers. Each random spatial point is located in the binary map of the lung. If more than 95% of the sampling sphere (defined by its center point and selected radius) lies within parenchyma, the spherical region is considered adequate to sample. If not, another point is randomly chosen, and the volume around this point is examined. Sampled volumes cannot overlap any portion of a previously sampled region. This sampling process continued until no other spherical regions could be found within the organ. The sampling process was repeated 10 times at each given sampling volume. With the use of this approach, only a fraction of the organ was sampled, but it was done so in an unbiased manner without sampling overlap. A CV of the sampled distribution was calculated each time the lung was sampled. The true CV of the flow distribution was then estimated from the average of the 10 CVs. The radius of the sampling volume was then incremented by 250 μm, and the process was repeated. The sampling volume was repeatedly incremented until fewer than 10 samples could be obtained from the organ space. Figure 2 shows the spatial distribution of the sampling spheres and the number of microspheres counted in each sampled region at four different sizes of sampling spheres.
Because microspheres are discrete particles, Poisson counting noise was superimposed on the observed distribution of microspheres (4, 23). Smaller sampling regions have fewer microspheres and hence more counting noise. This noise can be mathematically removed from the observed CV with the following formula
The fractal dimension of perfusion is determined by
Random distributions.
Microsphere distributions are randomly created for each rat lung as a test of the null hypothesis that blood flow is distributed by a random process. The test distributions are modeled by choosing x,y, and z coordinates from a pseudorandom number generator (Unix, Berkeley, CA) by using Marsaglia shuffling (18) to ensure a uniform distribution of numbers. Only spatial points located within the defined organ boundary for each lung set are used. Hence the microsphere locations that are randomly generated are constrained to the same spatial boundaries as the experimental lung. Although these spatial constraints impose a spatial pattern on the test microspheres, they are randomly distributed within the lung parenchyma. A random distribution is created for each rat lung. The same numbers of random microspheres are created as are counted in each of the rat lungs. The fractal algorithm described above is used to analyze the random distributions.
RESULTS
All animals tolerated the injections well. Table1 presents the total number of microspheres counted in each rat lung, the number of microspheres in the smallest sampling volume, and the number of samples obtained at the smallest sampling volume. At the smallest sampling size, all rats had >26,000 microspheres counted at each iteration. Hence the corrected CV should faithfully represent the true CV in each animal (23).
Figure 3 shows a plot of regional flow data for one pair of rat lungs together with a plot of a random distribution. There are three important points to be noted in Fig. 3. First, the heterogeneity of perfusion continues to increase past the acinar level. Second, the CVs are well determined at each sampling size. Third, the CVs at larger sampling volumes fall below the observed fractal relationship. The fractal dimensions for each rat are presented in Table 2. On average, the fractal dimensions of perfusion were 1.12 across all animals. All data were fit very well by the fractal relationship, with weightedr ^{2} values ≥0.99 in all animals.
Figure 3 also presents the plot of randomly distributed spatial points. Note that the fractal dimension of this data set is close to the expected value of 1.5 (11). The fractal dimensions of perfusion were all significantly less than the fractal dimensions of the randomly distributed points (Table 2).
DISCUSSION
The important findings of this study are that pulmonary blood flow heterogeneity increases past the volume of a ventilatory unit in the rat and that the spatial distribution of pulmonary perfusion is similar between the rat and other larger species.
Although gas exchange occurs throughout the bronchial and pulmonary trees, authors have attempted to define a unit of ventilation in which gas composition is uniform. Using gas washout and modeling methods, Paiva and Engel (21) have proposed that the acinus functions as the gasexchanging unit. Mercer and Crapo (20) measured acinar volumes in rats using threedimensional reconstruction methods. They determined that the volume surrounding an acinus is ∼10 mm^{3}. The volume of the smallest sampling unit in this study is 0.52 mm^{3}. We, therefore, believe that we are sampling blood flow distributions at the level of gas exchange. Determining perfusion heterogeneity below this level is not practical because the structure of the alveolar walls and capillaries determine the distribution of blood flow. In fact, the spatial distribution of microspheres at our smallest level of sampling is likely influenced by parenchymal structures, such as small airways, that cannot be resolved in our system.
One other study has examined the variability in pulmonary blood flow in rats. Kuwahira and colleagues (16) used microspheres to measure the spatial distribution of pulmonary perfusion in 10 awake and 5 anesthetized rats. They dissected the lungs from each animal into 28 samples, determined blood flow to each piece relative to the mean within each animal, and reported a CV of 0.102 across all 420 samples. Our best estimate is that the sample volumes obtained by Kuwahria et al. are equivalent to our sampling volumes of ∼200 mm^{3}. At this sampling volume, we observed an average CV of 0.113. Hence our studies demonstrate similar perfusion heterogeneities at similar scales of resolution.
Despite the much smaller scale of observation, blood flow heterogeneity in these rats is similar to that previously observed in larger awake laboratory animals. Using piece sizes of ∼2 cm^{3} (4,000 times larger than in the present study), CVs of 0.259, 0.295, and 0.356 have been observed in awake sheep (32), dogs (8), and horses (12), respectively. If the fractal relationship remains linear in these larger species, the CV of perfusion will approach 1.15 at the smallest sampling volume in this study. Although this degree of variability in regional perfusion will strain the mechanisms matching ventilation and perfusion, they must manage, given the efficient gas exchange seen in these animals. Larger ventilatory units will compensate somewhat for the greater perfusion heterogeneity in larger animals. It is also possible that a unit of perfusion exists in larger animals and blood flow may become uniform within a given volume of the lung in these larger species. Studies using higher resolution methods in larger animals are needed to answer this question.
Using a new highresolution method, we find that blood flow heterogeneity continues to increase down to the ventilatory unit. There is no evidence of a plateau in the plots, suggesting that perfusion does not become uniform within the range of lung volumes explored. Wagner and coworkers (31) recently demonstrated that blood flow varies over time at the capillary in a fractal pattern. They also demonstrated that flow patterns within neighboring alveolar regions vary independently of each other. Their study suggests that, whereas highflow regions are near other highflow regions, blood flow at the capillary level varies independently within alveoli. One potential explanation is that the geometry of larger feeder vessels determines the spatial correlation of blood flow, whereas other mechanisms determine temporal fluctuations in blood flow at the alveolar level.
Figure 3 demonstrates an important concept about fractal analysis. At the smallest sampling volume, the microsphere determined and the random distributions have the same perfusion heterogeneity. Although the CVs are similar, their spatial patterns are different, as shown by the slope of the fractal line. To illustrate this point, consider a distribution of flows at a given sampling volume that has a spatial pattern in which neighboring volumes have similar flows. In this example, highflow regions are near other highflow regions, and lowflow areas neighbor other lowflow areas. When neighboring pieces are grouped together to form larger sampling volumes, the CV will decrease, but the change (slope) will be relatively small because the combined pieces have similar values. Now consider shuffling the sampled flows in space so that there is no longer any spatial pattern but rather a random distribution. The CV of this distribution will be the same as the spatially ordered distribution because it contains the same flow values. However, when neighboring pieces are now grouped together to form larger sampling volumes, the CV will change significantly more because neighboring pieces do not have similar values. Hence the spatial information in fractal analyses is derived from the slope of the CV as a function of piece size.
We have previously shown that flows to neighboring lung regions are highly correlated (7). Because of this correlation, the numbers of microspheres counted in neighboring regions are not independent of each other. This dependence will yield SDs (and CVs) that may underestimate the true variability in regional perfusion. This problem confounds all measures of heterogeneity within a system that exhibits spatial correlation. Presently there is no solution to this problem, and the results of our study must be viewed as exploratory. This caveat should be kept in mind when our results are assessed.
Despite the large variability in regional perfusion seen in lung regions of ∼2 cm^{3}, Robertson et al. (25) and Melsom et al. (19) have shown that local ventilation and perfusion are tightly matched. Robertson (24) has proposed that the unit of gas exchange is defined by the volume of lung in which gases readily mix, smoothing out local smallscale perfusion inequalities. Work by Tsuda and colleagues (27) suggests that a considerable amount of convective gas mixing may occur within the acinus because of interactions between tidal convective gas movement and alveolar duct anatomy. Overall efficient gas exchange, therefore, likely occurs through matching of ventilation and perfusion on a macroscopic level and convectivediffusive interactions within the acinar spaces at the microscopic level.
With increasingly greater spatial and temporal resolution of physiological measurements, it is becoming evident that biological systems do not have smooth and continuous characteristics. They do, however, maintain a degree of spatial and temporal correlation. Fractal analysis permits the characterization of these processes, or structures, that are not easily represented by traditional analytic tools (11). The construction of selfsimilar structures reveals a potential advantage in coding for growth and development of the necessarily complex vascular networks. Because a complete genetic description for every alveolus and capillary is not possible, it seems logical to propose that elementary recursive rules exist to guide their construction. Although these operational rules are speculative, a coding for selfsimilar structures is clearly the most efficient and appropriate algorithm to explain both the order and complexity of ontogeny. Iversen and Nicolaysen (14) studied the heterogeneity of perfusion in rabbit heart and skeletal muscle. Although the fractal dimensions varied among animals, they found a tight correlation between the fractal dimensions of heart and skeletal muscle within the same animal. Whereas the tight correlation within animals may be induced by shared experimental conditions, the observation suggests the potential for an underlying mechanism that determines perfusion distributions in different muscles in the same animal. Efficiency of the finalized structures is also important to the organism. Lefevre (17) has shown that a selfsimilar branching model of the pulmonary vascular system optimizes the costfunction (energymaterials) relationship while closely approximating physiological and morphometric data. Tsonis and Tsonis (28) have related fractal patterning to minimal energy consumption as well. Recently, West and coworkers (33) proposed an evolutionary advantage to fractal vascular trees that explains the conservation of these structures across many species and phyla. Fractal analysis, therefore, provides a unifying mechanism for development and function of the pulmonary vascular tree.
The fractal dimensions in this study were ∼1.12. This value is similar to those for larger species. Other researchers have reported fractal dimensions of 1.13, 1.14, 1.18, 1.15, and 1.22 in dogs (22), sheep (5), rabbits (6), pigs (1), and horses (26), respectively. Hence it appears that the spatial distribution of blood flow in rat lungs is similar to that in larger laboratory animals.
Pulmonary perfusion distribution is not fractal over the entire range of observations. At larger piece sizes, the observed heterogeneity is always less than that predicted by the fractal relationship. This has been previously explained by statistical uncertainty at the larger piece size (30). However, our data is well defined because of multiple repetitions at larger sampling volumes and continues to show this deviation from the fractal relationship. We believe that this deviation from the fractal relationship must occur because of conservation of blood flow within a confined space. Regions of relatively higher blood flow can only exist at the expense of blood flow to lowflow regions. The fractal branching vascular tree is likely responsible for this steal phenomenon, producing local regions of both high and low flow. Neighboring regions have similar magnitudes of flow, whereas regions distant from each other have negatively correlated flows. At large piece sizes, the fractal algorithm begins to combine pieces that are farther away from each other. Because blood flows to these pieces may be negatively correlated, when averaged together, high and lowflow pieces cancel each other and flows to larger combined pieces tend to be more uniform. This observation deviates from the theoretical fractal model in which there are no boundary conditions (7) but is predicted by fractal models of blood flow distribution to organs with defined spatial boundaries (10, 30). It is also possible that the reduction in the CV could be an artifact of the dependence among samples, as fewer of them are used to calculate the CV at larger piece sizes. This possibility must be explored more thoroughly.
In conclusion, we believe that we are able to measure the distribution of blood flow at the level of gas exchange in rat lungs. These observations show that the observed heterogeneity of blood flow continues to increase down to the acinar level in a fractal manner. In the rat, there is no evidence of a unit of perfusion in which blood flow becomes uniform.
Footnotes

Address for reprint requests and other correspondence: R. Glenny, Karolinska Hospital and Institute, Dept. of Anesthesia and Intensive Care, Bldg. F2, 00, SE171 76 Stockholm, Sweden (Email:glenny{at}u.washington.edu).

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 Copyright © 2000 the American Physiological Society