Abstract
We determined the changes in fractal dimensions and spatial correlations of regional pulmonary blood flow with increasing exercise in race horses (n= 4) by using 15μm fluorescent microspheres. Fluorescence was measured to quantitate regional blood to 1.3cm^{3} samples (n = 1,621–2,503). Perfusion distributions were characterized with fractal dimensions (a measure of spatial variability) and spatial correlations. On average, the fractal dimension decreased with exercise (trot 1.216 to gallop 1.173; P < 0.05) despite a variable fractal dimension at rest. Spatial correlation of flow to neighboring pieces increased with exercise (trot 0.57 ± 0.074 to gallop 0.73 ± 0.051) and was inversely correlated with fractal dimension, indicating better spatial correlation as blood flow distribution becomes more uniform. This is the first study to document a change in fractal dimension as a result of increasing pulmonary blood flow. Spatial differences in response to vasoregulatory mediators may play a role in this phenomenon.
 fluorescent microspheres
 cardiac output
the development of highresolution microsphere technology has advanced the study of regional blood flow distribution within organs (1, 7). The distribution of pulmonary blood flow, for example, once believed to be primarily influenced by gravitational forces, when examined with these new techniques is found to be quite heterogeneous within isogravitational planes (11). The branching pulmonary vascular tree is fractal by nature, and, therefore, fractal methods have emerged as an effective way to describe regional blood flow distribution.
Glenny and Robertson (11) showed that regional distribution of pulmonary blood flow in dogs is fractal and can be characterized by the fractal dimension. Described first by Mandelbrot (23), fractal dimension can be determined from a loglog plot (Fig. 1) of the coefficient of variation (CV) as a function of increasing piece size. If a system is fractal, this plot will have a constant slope. Fractal dimension is equal to 1 − slope of this plot, and, therefore, the more random the relationship among adjacent pieces, the higher the fractal dimension value. With this approach to data analysis, fractal dimension approaches 1 as regional perfusion becomes more spatially correlated (i.e., neighboring piece will have more similar flows), whereas a fractal dimension of 1.5 defines a totally random system. Unlike the CV, the fractal dimension does not change with changes in spatial resolution (i.e., fractal dimension is a scaleindependent measure of scaledependent heterogeneity). Values of fractal dimension have been determined for pulmonary blood flow distribution in several species (dogs 1.09 ± 0.02; sheep 1.14 ± 0.09; Refs. 4, 11) and were noted to remain constant with increasing pulmonary blood flow in an in situ, pumpperfused sheep model, despite a decrease in CV at higher flows (4). Additionally, no change in fractal dimension was observed from rest to exercise in dogs (27). We hypothesize that fractal dimension, because it is determined by a fixed vascular branching pattern, should remain constant regardless of changes in total pulmonary blood flow.
In this study we examine the effect of extremely large changes in total pulmonary blood flow on the fractal dimension, as well as local spatial correlation, of pulmonary blood flow distribution in an in vivo, unanesthetized horse model. This model is important and unique in that it provides incremental and very large increases in pulmonary blood flow via graduated exercise, allowing data to be gathered over an enormous range of pulmonary blood flow while maintaining the natural vascular milieu with its local and organwide vascular autoregulatory mechanisms intact.
METHODS
All methods and animals were approved by the University of Washington (Seattle) and Kansas State University (Manhattan) Animal Care Committees and were in accordance with the American Association for Accreditation of Laboratory Care Animals statement on the care of animals. Before the study, four Thoroughbred geldings (422–500 kg) were trained on a highspeed equine treadmill for 4 wk, and their maximal oxygen consumption (V˙o _{2 max}) was determined. Each horse was prepared for study and monitored as previously described in detail (2, 14). Briefly, a left transverse facial artery catheter and a left internal jugular artery catheter were placed for bloodgas sampling and microsphere injection, respectively. A pulmonary artery catheter was inserted via a right internal jugular vein introducer for withdrawal of reference samples. Two additional catheters, one with a pressure transducer, the other with a thermistor, were placed in the right internal jugular vein and advanced into the pulmonary artery to measure pulmonary artery pressure and core temperature, respectively. Surface electrodes were applied for heart rate determination, and each horse was fitted with a safety harness.
Exercise protocol.
Measurements were made at rest (standing quietly at a 3° incline) and at three progressively more strenuous exercise levels based on previously determinedV˙o _{2 max}. The treadmill speed was increased after resting measurements were obtained to produce a V˙o _{2} ∼34% ofV˙o _{2 max}(trot). Measurements were repeated after a steadystate oxygen consumption (V˙o _{2}) was achieved. Subsequent measurements were made after the treadmill speed was increased to produce a steadystateV˙o _{2} ∼59% ofV˙o _{2 max} (canter) and aV˙o _{2} ∼90% ofV˙o _{2 max}(gallop), in which the final set of measurements was recorded.
Physiological measurements.
Complete details of the physiological measurements were outlined previously (14). Briefly,V˙o _{2} and carbon dioxide production (V˙co _{2}) were measured by using a biasedflow technique through a loosefitting mask strapped to the horse's head. O_{2} and CO_{2}concentrations of expired gas were measured with a mass spectrometer (model 1100, PerkinElmer, Pomona, CA) after the gas passed through a mixing chamber. Exhaled airflow was measured via a pneumotachometer (model MP45, Validyne Engineering, Northridge, CA). Arterial bloodgas determinations were made with a Nova bloodgas analyzer calibrated with standard gases and buffers and corrected to the measured temperature in the pulmonary artery with horsespecific correction factors. Oxygen content was measured from anaerobically obtained arterial and mixed venous blood samples with a LexO_{2}Con (model TL, Hospex, Chestnut Hill, MA). Surface electrodes monitored heart rate, and a transducertipped catheter measured pulmonary arterial pressure. Cardiac outputs were determined by the Fick equation. Hematocrit measurements were made by using the microhematocrit method, and lactate concentration of mixed venous blood was measured with a lactate analyzer (model 23L, Yellow Springs Instruments).
Pulmonary blood flow measurements.
Measurements of pulmonary blood flow distribution were made at each exercise level with 40–65 million 15μm fluorescent microspheres (bluegreen, yellowgreen, orange, red, or crimson; Molecular Probes, Eugene, OR) injected via the left internal jugular vein over ∼15 s. A single, differentcolor microsphere was used at each exercise level, and the order of colors used was picked at random for each animal. Injections occurred after a steadystateV˙o _{2} was achieved, and reference blood was simultaneously drawn during each injection from the pulmonary artery catheter at a rate of 60 ml/min, allowing blood flow calculations in milliliters per minute. A simultaneous injection during a single exercise level in each horse with a secondcolor microsphere (two differentcolored microsphere aliquots mixed in a single syringe) was performed to estimate methodological error.
Lung processing.
At the end of the experimental protocol, each horse was deeply anesthetized, and a tracheotomy was performed before exsanguination. After death, the pulmonary vasculature was flushed with normal saline, and the lungs were removed from the chest and inflated to 35 cmH_{2}O with warmed air and dried for 14 days. After drying, the lungs were placed in a box and surrounded with urethan foam such that in vivo isogravitational planes were parallel to the relevant box surface. Lungs were then sliced into 2.2cmthick coronal sections (16–21 per horse), and each section was systematically sampled by using a rigid XY grid. The grid was placed over each section, and core samples (1.3 cm^{3}) were obtained at 2.3cm intervals from the center of adjacent cores over the entirety of each lung section (Fig. 2). This produced from 1,621 to 2,503 pieces per horse after pieces consisting of >25% airway by volume (88–251 pieces per horse) were discarded. Spatial coordinates were noted for each piece, samples were soaked for 2 days in cellosolve acetate, and extracted fluorescence was measured with a fluorimeter (model LS50B, PerkinElmer).
Pulmonary blood flow was calculated for each lung piece of 1.3 cm^{3} unit volume in milliliters per minute as previously described (2). Relative flow to each piece was determined by dividing the flow to each piece at a given exercise level by the mean flow to all pieces at the same exercise level. The uniform volume of each core sample precluded the need for weight normalization of relative flows.
Fractal analysis.
Fractal plots of pulmonary blood flow data for each exercise state were constructed by iteratively calculating the CV of blood flow for progressively larger piece sizes and plotting the logarithm of CV vs. the logarithm of piece size (Fig. 1). For each animal, we calculated the CV of the relative flow per piece, and then combined neighboring pieces into aggregates of size n _{I} and recalculated CV_{I} for 16 different aggregate sizes. For the larger aggregate sizes, CV_{I}exhibits higher variability in that the CV changes depending on which pieces are combined to form the aggregates. To reduce the variability of the CV of larger piece sizes, we estimated the CV repeatedly via a randomized algorithm to form aggregates. An aggregate center piece is chosen at random from all available lung pieces. Neighboring pieces are then combined to form an aggregate of desired size n _{i} around this center piece. Another center piece is then randomly chosen, and an aggregate around that center is formed in similar fashion. A graphic representation of this is shown in Fig.3. This process is repeated until no more aggregates of the size n _{i} can be formed, and the CV of each aggregate is calculated. By repeating this algorithmL _{I} times, we obtainedL _{I} CV measurements, which were then averaged to produce the mean CV_{I} used in the fractal dimension slope estimation.
Because each CV_{I} is based on different amounts of data, a weighted analysis is required to ensure that the more accurate estimates are given greater weight and to stabilize variance in the regression ANOVA. Through empirical observations on prior data sets, we ascertained that, for aggregate size n
_{I}, good weights are given by the equation
The fractal dimension statistic is related to the degree of local correlation of flow among lung pieces. A large fractal dimension value indicates a low degree of local spatial correlation; i.e., flows to neighboring pieces are relatively dissimilar. Conversely, a small fractal dimension value indicates a high degree of local scaleindependent correlation; i.e., flows to neighboring pieces are relatively similar, regardless of the sample size chosen. An attractive feature of fractal dimension is that it is independent of piece size, whereas the usual correlation coefficient depends on a specific piece size. Fractal dimension supplements and provides different information than does CV, which depends on a specific piece size. Empirically, it is possible to have a large or small value of CV with a large or small fractal dimension value. Thus a useful pair of statistics within an experiment is the CV at piece size = 1 as a measure of heterogeneity and fractal dimension as a measure of scaleindependent spatial correlation. Fractal dimension was calculated by adding the negative of the slope of the linear portion of the fractal plot to 1.
Spatial correlation.
The spatial correlation as a function of distance over a threedimensional space [ρ_{xyz}(d)] was calculated within each lung, as previously described (6), by using Eq.2
Statistical analysis.
Fractal dimensions were analyzed for each horse at each exercise state via a twoway ANOVA; the two conditions of interest were exercise state and subject number. Statistical significance was set at P < 0.05 for all comparisons. A similar ANOVA model was used to compare the spatial correlations as characterized by the actual ρ value at the smallest realizable distance between lung pieces measured (2.3 cm) fromEq. 2 for each animal and exercise state.
RESULTS
Physiological measurements.
All horses demonstrated comparable changes in response to exercise (2). Cardiac output increased over 10fold with exercise, from a mean of 31 l/min at rest to a mean of 348 l/min at maximal exercise. This was accompanied by a marked increase in mean pulmonary arterial pressure (35 to 107 cmH_{2}O) andV˙o _{2} (4.0 to 133.9 ml ⋅ min^{−} ^{1} ⋅ kg^{−} ^{1}). These and the other physiological measurements recorded are summarized in Table 1. Notably, the pulmonary artery pressure at rest did not significantly differ at the end of the experiments, after all four microsphere injections, from that seen before any microsphere administration (data not shown).
Fractal dimension.
The blood flow data fit the fractal model well with an averageR ^{2} value of 0.95 ± 0.032. A set of fractal plots from one horse is shown in Fig. 1. Fractal dimension values for each horse at each exercise state are shown in Fig.4. Twoway ANOVA using subject number (i.e., identity of each horse) and exercise state as conditions of interest was conducted for the 20 fractal dimension values generated and is summarized in Table 2. Fractal dimension values varied significantly (P < 0.05) between horses and with exercise within a given horse. The interaction term (horse + exercise state) is also significant, indicating that the change in fractal dimension with exercise is different from horse to horse. This is largely due to the extreme intersubject variability of the fractal dimension at rest. These variations due to exercise and animaltoanimal comparisons are much larger than those seen in the simultaneous, duplicate microsphere injections within the same horse (Fig. 4).
Fig. 4 shows the fractal dimensions for each horse and the degree to which it changes with increasing exercise. The largest variation in fractal dimension between horses was seen in the resting state, and the differences diminished during exercise. Therefore, a second ANOVA, excluding the fractal dimensions calculated at rest, was done and is summarized in Table 3. The interaction term is no longer significant (P = 0.059), indicating that the change in fractal dimension across exercise states changes in a similar fashion in all the horses studied, a decrease in fractal dimension from trot to canter and a lesser decline or plateau from canter to gallop. Again significant differences are seen in fractal dimensions across exercise states (P = 0.0014) as well as between the individual horses (P = 0.00023). The difference seen between exercise states when the resting fractal dimensions are excluded from the analysis confirms that these changes in fractal dimension with exercise are indeed significant.
Spatial correlation.
The spatial correlation of flow as a function of distance (d) was determined for each horse at each exercise state. A representative plot of the spatial correlation in one horse is shown in Fig. 5, where the line represents the weighted fit of ρ(d) to Eq. 3 . The values for the spatial correlation of flow between adjacent pieces, ρ(2.3 cm), for each horse and exercise state are presented in Table4.
Regional pulmonary blood flow was spatially correlated as previously seen in dogs (6), wherein regional blood flow was most similar for adjacent lung pieces and diverged as a function of increasing distance between lung pieces, resulting in negative correlation at extremes of distance (Fig. 5). This pattern was evident in all horses and all exercise states. To determine whether exercise influenced the extent of spatial correlation, a twoway ANOVA using subject (i.e., horse identity) and exercise state was performed to compare the correlation of flow for adjacent pieces, ρ(2.3 cm). Results of this analysis are summarized in Table 5. The value for ρ(2.3 cm) steadily increases with exercise in horses 1,3, and 4, indicating better correlation of the flow magnitude of adjacent pieces as total pulmonary blood flow increases.Horse 2 shows an initial decrease and then a steady spatial correlation from trot to gallop. Statistically significant differences are seen between different horses and between exercise states within a given horse.
Comparison of fractal dimension and spatial correlation.
With increasing exercise intensity, a decrease in fractal dimension occurs, indicating more spatially organized flow distribution (fractal dimension → 1.0). Conversely, correlation of flow to adjacent pieces increases under these same circumstances [ρ(2.3 cm) → 1.0]. This inverse relationship is illustrated in Fig. 6 and supports the intuitive assumption that regional blood flow will be more closely correlated as its distribution becomes more spatially organized.
DISCUSSION
The important findings from this study are 1) pulmonary blood flow distribution is fractal in exercising horses, 2) fractal dimension for pulmonary blood flow decreases when exercise intensity increases (i.e., increasing pulmonary blood flow), 3) regional pulmonary blood flow is spatially correlated, 4) the spatial correlation of flow differs among horses and exercise states, and5) correlation of flow for adjacent pieces increases with increased pulmonary blood flow.
Microsphere method.
The microsphere method for measuring regional organ blood flow distribution has been previously validated in dogs (7), and the justifications for its use in these experiment have been discussed in prior reports (1, 13, 24). Flow to each lung piece was normalized to the mean flow to all pieces in a given animal. The coring method used to sample the lungs in this study yielded pieces of virtually identical volume; therefore, we did not report blood flow normalized to piece weight, as has been the case for previous studies from this laboratory (8, 10, 11). Weight normalization has been used as a surrogate for the painstaking process of ensuring that each lung piece sampled is of equal volume. The core samples in this study were weighed, and, when weightnormalized relative flows were analyzed, CVs were routinely increased by 2 to 3%, suggesting that additional noise is added by weight normalization.
The lungs were dried at total lung capacity to approximate the alveolar volume distribution in the prone animal. The transpulmonary pressure (alveolar pressure − pleural pressure) determines alveolar volume distribution. Recently, studies in ponies (26) and other species (22,25) have confirmed that the vertical pleural pressure gradient seen in the supine posture is abolished when the animal is turned prone. A uniform distribution of alveolar size has also been documented in the prone posture with the use of various imaging techniques (15, 16, 18).
Drying at total lung capacity (i.e., lungs suspended from the trachea and inflated to 35–40 cmH_{2}O) will, however, alter the lung size and configuration somewhat compared with the in vivo lung. The absence of a confining thoracic cage and diaphragm will result in a slightly larger lung volume. Additionally, the absence of the chest wall, diaphragm, and mediastinal structures and the altered gravity vector when the lung is suspended by the trachea during drying will distort the lung's shape slightly. Neither of these factors is expected to have a significant impact on the major findings of the present study.
In one horse, total volume of dried lung tissue was estimated by calculating the volume of each 2.2cmthick lung slice and summing these volumes for the entire lung. By using this technique, the estimated lung tissue volume was 33.6 liters for horse 3. The number of core tissue samples taken of 1.3 cm^{3} volume can then be used to determine the percentage of total lung sampled in this horse (∼8%). It has been held that ∼400 microspheres per sample is the minimum number required to adequately estimate regional blood flow (3). This “rule” is only important if the goal is to determine the flow to an individual organ piece within a 95% confidence limit. For more general analysis, such as CV (and by association, fractal dimension), many fewer microspheres per piece will yield accurate results (29). From the fluorescent yield of known numbers of each microsphere color, we are able to estimate the number of microspheres per core lung sample. These calculations show that <0.7% of all lung core samples had <400 microspheres present, except in horse 2, in which 11.95 and 2.62% of core samples had <400 yellowgreen and red microspheres present, respectively. This horse had an unusual number of very lowflow pieces and differed from the other horses studied as discussed below.
Because accurate estimates of lung size were not available before the experiments, the microsphere dose was estimated on the basis of each animal's body weight. The number of microspheres used was large compared with studies in other smaller animals, yet we do not think that this changed flow characteristics. Microspheres of 15μm diameter lodge in capillaries, not small arterioles (12). There are no published estimates of the number of pulmonary capillaries present in the horse. Horsfield made casts of the human arterial tree and was able to count arterial vessels down to 10 to 15μm diameters, the number of which they determined to be ∼73 million, with each of these giving rise to hundreds of pulmonary capillaries (17). Given that the horse lung is five to sixfold larger than that of humans, the percentage of vessels occluded will be very small and is not likely to effect flow characteristics. Figure 7 shows a fluorophotomicrograph of a lung sample from this study. It clearly shows microspheres in the alveolar capillaries as well as the paucity of microspheres relative to the number of alveoli. Glenny et al. (9) have observed that multiple microsphere injections over a 5day period do not significantly alter pulmonary blood flow characteristics in dogs. In fact, between days 1 and 5, after a total of 12 million 15μm spheres were injected, relative flow per piece remained highly correlated (r = 0.96)
Horse 2.
Significant differences were seen in the results obtained from analysis of the blood flow data from horse 2 compared with the other animals in this study. As we have previously shown, this horse had an unusual bimodal distribution of pulmonary blood flow at rest, which markedly differed from the unimodal distribution seen in the other horses (14). A significant number of lowflow pieces were seen in the dorsocaudal lung regions of this horse at rest, and, on removal of the lungs, reddened areas were seen in these same regions, compatible with exerciseinduced pulmonary hemorrhage. Although no diagnostic procedure was performed to confirm this, horse 2 is clearly an outlier in the Thoroughbred population, but these differences only slightly impacted the findings of this study when the rest data were included in the analysis of fractal dimension.
Fractal dimension.
We found statistically significant decreases in fractal dimension with increasing total pulmonary blood flow induced by exercise. This finding conflicts with prior results reported in open chest, pumpperfused sheep in which relative distribution decreased but fractal dimension remained constant with increases in pulmonary blood flow from 1.5 to 5.0 l/m (4). Parker and colleagues found no significant change in fractal dimension from rest (1.132 ± 0.006) to exercise (1.149 ± 0.01) in dogs, despite a two to fourfold increase in cardiac output (27). Differences in study design may account for some of these discrepancies. We employed an awake, unanesthetized horse model, which not only provided a much larger (10fold) increase in total pulmonary blood flow but did so in the intact animal in which normal autoregulatory mechanisms continued to influence regional vascular tone. It has recently been shown that, in horses, vasoactive substances evoke different responses as a function of the vessel's spatial location (28). Specifically, 6mm arteries from the dorsocaudal lung relaxed when exposed to methacholine, whereas cranioventral arteries constricted. This phenomenon was endothelium dependent and nitric oxide mediated, a mechanism similar to that which is thought to mediate flowdependent vasodilation (20, 21). It was postulated that similar mechanisms might account for the preferential perfusion of dorsalcaudal lung regions seen when using microsphere techniques (8,14). Indeed, in our prior paper (2), we reported an increase in dorsal distribution of pulmonary blood flow with increasing exercise. Similar alterations in regional vascular tone may be responsible for the change in fractal dimension seen in this study. Regional variation in vascular tone will change the distribution of flow to the lung distal to the affected vessel(s) as well as the proportion of flow to its sister vessel(s) in a dichotomously branching system. Although only described to date in 6mm OD vessels, if a flowmediated vasodilatory signal affects larger, more proximal parent vessels, the proportion of flow to these areas could be significantly increased, thereby altering the effective “geometry” of the vascular tree and the fractal dimension that describes it.
In our prior report (2), CV for pulmonary blood flow in horses did not change significantly with increasing total pulmonary blood flow. Although it is a scaledependent measure of heterogeneity, a stable CV indicates little change in the global heterogeneity of pulmonary blood flow distribution with exercise. The scaleindependent fractal dimension, however, is influenced largely by local correlation of flow magnitude. A decreasing fractal dimension, as was observed in this study, indicates more homogeneous dispersion of flow among spatially proximate pieces (i.e., improved piecetopiece correlation), while global heterogeneity remains relatively unchanged. This observation can be explained intuitively if the lung pieces were shuffled from their actual spatial positions in a random fashion. If this random system were then analyzed, there would be a decrease in spatial correlation and a rise in fractal dimension to 1.5, yet the CV of blood flow would be preserved.
Our observations of pulmonary blood flow during rest and exercise in horses (2, 14) were made with the highresolution fluorescent microsphere technique. These findings are inconsistent with predictions of the traditional fourzone model of pulmonary blood flow distribution (19). The data show a considerable isogravitational heterogeneity that is spatially ordered and nonrandom. Because the zone model cannot be used to characterize these highresolution findings, other models for pulmonary blood flow distribution should be considered. Our findings show that fractal analysis may provide an effective framework for characterizing pulmonary blood flow distribution.
Spatial correlation.
Glenny (6) has previously found that regional pulmonary perfusion is highly correlated over large spatial distances in dogs. The present study found a similar strong correlation of blood flow but across even larger distances (up to ∼60 cm) and further supports the theory that a dichotomously branching vascular tree is the primary factor influencing the distribution of regional pulmonary blood flow. As has been previously observed, correlation was highest among pieces in the closest spatial proximity and lower as distance between pieces increased, becoming negative at extremes of distance. This can be explained by the concept of conservation of flow, in which, given a fixed amount of blood flow to a vascular tree, high flow in one region can only occur at the expense of flow to another region.
Closer correlation was observed in the present study between neighboring pieces [ρ(2.3 cm)] with increasing exercise. This is not surprising given our previous findings that lowflow pieces tended to remain low flow and highflow pieces tended to remain high flow with increasing exercise (2). This finding also supports the fractal dimension data. As previously mentioned, fractal dimension is primarily influenced by local flow correlations, and therefore it is expected that ρ(2.3 cm) will increase (approach 1.0) as fractal dimension decreases. This relationship is graphically illustrated in Fig. 6.
Summary.
We have shown that pulmonary blood flow in horses at rest and at three levels of exercise is fractal in nature and spatially correlated. Additionally, this is the first study to document a change in fractal dimension as a result of increasing pulmonary blood flow. A recent study from this institution has also documented a reduction in fractal dimension with isovolemic hemodilution in rabbits (5). This phenomenon may be at least partially explained by spatial differences in response to vasoregulatory mediators, but further study is needed to better elucidate the mechanism(s) responsible.
Acknowledgments
We thank H. Thomas Robertson for editorial advice and M. Roger Fedde, Howard H. Erickson, and Randall J. Basaraba for expertise during these experiments.
Footnotes

Address for reprint requests and other correspondence: S. E. Sinclair, Div. of Pulmonary and Critical Care Medicine, Univ. of Washington, BB1253 HSB Box 356522, Seattle, WA 98195–6522 (Email:scottes{at}u.washington.edu).

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 Copyright © 2000 the American Physiological Society