Abstract
Highresolution measurements of pulmonary perfusion reveal substantial spatial heterogeneity that is fractally distributed. This observation led to the hypothesis that the vascular tree is the principal determinant of regional blood flow. Recent studies using aerosol deposition show similar ventilation heterogeneity that is closely correlated with perfusion. We hypothesize that ventilation has fractal characteristics similar to blood flow. We measured regional ventilation and perfusion with aerosolized and injected fluorescent microspheres in six anesthetized, mechanically ventilated pigs in both prone and supine postures. Adjacent regions were clustered into progressively larger groups. Coefficients of variation were calculated for each cluster size to determine fractal dimensions. At the smallest size lung piece, local ventilation and perfusion are highly correlated, with no significant difference between ventilation and perfusion heterogeneity. On average, the fractal dimension of ventilation is 1.16 in the prone posture and 1.09 in the supine posture. Ventilation has fractal properties similar to perfusion. Efficient gas exchange is preserved, despite ventilation and perfusion heterogeneity, through close correlation. One potential explanation is the similar geometry of bronchial and vascular structures.
 gas exchange
 lung airway
 lung mechanics
the principal function of the lung is to exchange oxygen and carbon dioxide between blood and inspired air and is dependent on local matching of regional ventilationtoperfusion ratio (V˙a/Q˙). Traditional theory, developed from measurements using radioactive gases and chest wall scintillation counters, proposed thatV˙a/Q˙ matching is accomplished by gravitymediated gradients of both ventilation and perfusion (19, 34, 35).
Experiments using higher resolution techniques in nonprimate species have demonstrated significantly greater heterogeneity of pulmonary perfusion than can be explained by gravitational mechanism alone (10,14, 20, 25). Glenny et al. (10) estimated the maximal contribution of gravity to overall perfusion heterogeneity at ∼7% in dogs. Recently, highresolution measurements of regional perfusion in baboons (9) and indirect measurements of perfusion in humans under sustained microgravity conditions (24) suggested that nongravitational mechanisms of pulmonary perfusion distribution are also important in humans.
If gases are efficiently exchanged, regional perfusion heterogeneity has significant implications for regional ventilation. Wilson and Beck (36) mathematically demonstrated that, for a given heterogeneity of regional perfusion, as regional ventilation heterogeneity increases, the correlation between ventilation and perfusion must improve to preserve a narrow distribution ofV˙a/Q˙ values. Recent highresolution measurements using either aerosolized 0.005μm^{99m}Tclabeled carbon particles (18) or aerosolized 1μm fluorescent microspheres (1, 26) demonstrated that regional ventilation is heterogeneous and highly correlated with local perfusion. Although mechanisms of ventilation heterogeneity have not been well studied, nitrogen washout experiments during space shuttle flights demonstrated persistent ventilation heterogeneity during microgravity, confirming the importance of nongravitational mechanisms on regional ventilation distribution (13, 23).
Close correlation between regional ventilation and perfusion suggests that regional ventilation has spatial characteristics similar to regional perfusion. One method used to characterize regional ventilation and perfusion distributions is fractal analysis. The incentive for applying fractal analysis in the study of perfusion or ventilation heterogeneity is that the observed heterogeneity of either measure is complicated by dependence on the scale of resolution. With the use of fractal analysis, this problem is resolved because the scaledependent variability is described by a scaleindependent fractal dimension. Fractal analysis is useful for several other reasons. A fractal pattern in ventilation heterogeneity implies the presence of spatial clustering in which ventilation to a given region is correlated with that of neighboring regions (12); this, in turn, can give insight into the mechanisms of regional ventilation distribution. Glenny (8) suggested that spatial clustering of regional blood flow supports the hypothesis that regional perfusion is determined by resistive differences in the branching pulmonary vascular tree (3). Fractal analysis may also be useful for identifying the anatomic level at which gas exchange is determined. Ventilation heterogeneity is unlikely to continue increasing as resolution improves because of forces that promote mixing, such as gas diffusion, reinspiration of common dead space gas, and cardiogenic oscillations. At some regional volume, alveolar gas tensions will become uniform, despite any perfusion heterogeneity. The volume of this region, termed the unit of gas exchange, would be identified by a change in the slope of the fractal ventilation plot (Fig. 1). Finally, the scaleindependent fractal dimension (D) provides a way to compare measurements of regional ventilation by using different techniques with varying resolutions.
To determine whether regional ventilation has fractal properties similar to regional perfusion, we measured ventilation and perfusion with aerosolized and injected microspheres in six juvenile pigs. To determine if posture had similar effects on regional ventilation and perfusion, measurements were taken in supine and prone postures.
METHODS
Animal preparation.
The Animal Care Committee at the University of Washington approved these experiments. Six pigs, of either gender, weighing 18.5–25 kg, were studied. Anesthesia was introduced intramuscularly with ketamine and xylazine and followed by continuous intravenous thiopental sodium, at a rate sufficient to suppress spontaneous respiration. The animals were mechanically ventilated via tracheostomy with a tidal volume of ∼15 ml/kg and a respiratory rate sufficient to maintain an arterial carbon dioxide tension between 30 and 40 Torr. One carotid and one femoral artery were cannulated for continuous blood pressure monitoring and withdrawal of blood samples. Two femoral veins were cannulated for administration of anesthesia and injection of microspheres. A pulmonary artery catheter was inserted through an external jugular vein. A standard solution of six inert gases was infused, but resultant data were not used for this analysis.
Study protocol.
All animals were hyperinflated to twice the tidal volume every 10 min and before all measurements to minimize atelectasis. Data were collected twice in the prone posture and twice in the supine posture for each animal. Posture order was randomized for each experiment. Data collections were carried out at 30min intervals, with posture change occurring at the beginning of the interval. Each data collection included measurement of mean arterial pressure, pulmonary artery pressure, peak airway pressure, temperature, hematocrit, the average of three thermodilution cardiac output measurements, and blood gas determined from arterial and mixed venous samples. After each data collection, aerosolized 1μm fluorescent microspheres (FluoSpheres, Molecular Probes, Eugene, OR) were delivered over 10 min to measure regional ventilation as previously described (26). Simultaneously, 15μm fluorescent microspheres (FluoSpheres, Molecular Probes) were injected intravenously in multiple, small, evenly spaced increments to measure regional perfusion. One regional ventilation measurement and one regional perfusion measurement were randomly chosen for simultaneous duplicate measurement in each experiment. A total of 10 different fluorescent labels were used (1μm microspheres: yellowgreen, yellow, orange, orangered, red; 15μm microspheres: blue, bluegreen, green, crimson, scarlet). The fluorescentlabel orders for both ventilation and perfusion markers were independently randomized before each experiment. After the last data collection, 10,000 units of heparin and 1.5 ml of papaverine were intravenously injected, and the animal was killed by exsanguination under deep anesthesia.
Lung preparation and data collection.
A sternotomy was performed, the main pulmonary artery and left atrium were cannulated, and the aorta was ligated. The pulmonary vasculature was flushed with a dextran solution, and the lungs and trachea were dissected from the chest cavity and dried, inflated at 25cmH_{2}O pressure.
The dried lungs were fixed in a rapidsetting foam, sliced, mapped, and diced into cubes of 1.5 to 2.0cm^{3} volume. Each piece was weighed, visually scored for airway and blood content, and soaked for 4 days in 2ethoxyethyl acetate to extract the fluorescent dyes. Fluorescent signals for the 10 colors were measured in each piece with a fluorometer (LS50B, PerkinElmer, Beaconsfield, Buckinghamshire, UK). Spillover signals from colors adjacent in the spectrum were corrected by using matrix inversion of fixed wavelength intensities (28). Fluorescent signals were converted to number of microspheres by using the fluorescent intensity of reference samples containing a known number of microspheres. Pieces with airway content ≥25% of total piece volume were excluded from further analysis. Signals for each piece were normalized to that piece's weight, correcting for heterogeneity in ventilation and perfusion due to variation in piece size. The number of microspheres of each color in each piece was normalized to the mean number of microspheres of that color in all pieces, correcting for variations in the total number of microspheres of each given color.
Noise estimation.
Observed heterogeneity of injected or aerosolized microsphere deposition consists of both true heterogeneity of regional ventilation or perfusion and heterogeneity introduced by methodological error. Methodological error for measurement of regional blood flow by injected microspheres approximates a Poisson distribution; therefore, the standard deviation of multiple simultaneous measurements will equal the square root of the mean measurement
Fractal analysis and statistics.
The same method of fractal analysis is applied to characterize both regional ventilation and perfusion; therefore, for simplicity, all further discussion of the method will implicitly apply to perfusion as well as ventilation. Heterogeneity of regional ventilation can be characterized by the CV. Given a Poisson distribution of methodological noise, the true CV (CV_{true}) is estimated from the observed CV (CV_{obs}) by
We initially calculated the CV for a distribution of regional ventilation measured at a resolution of ∼2 cm^{3}. Estimates of CV for ventilation distributions measured at larger regional volumes were calculated by randomly choosing a starting piece and then combining its ventilation with the ventilation of adjacent 2cm^{3} pieces. The software performing these calculations was constrained as follows: 1) for each cluster size, the software formed as many clusters as possible without including any 2cm^{3} piece from more than one cluster, and 2) clusters cannot contain pieces from more than one lobe. These constraints resulted in fewer ventilation measurements at larger cluster sizes. Therefore, the software repeated the calculations, starting with a new 2cm^{3} piece for cluster sizes greater than three pieces. This allowed calculation of a mean CV and standard error. A weighted linear regression algorithm was used to calculate the slope of the fractal plot. The total number of measurements used to calculate a CV at a given cluster size was used as a weighting factor for the linear regression calculation. For example, at the highest resolution, the CV of ventilation may be calculated from 1,000 measurements and is therefore weighted by 1,000. At a cluster size of eight pieces, the CV may be calculated from three repetitions of the clustering algorithm, each using an average of 109 ventilation measurements; therefore, 3 × 109 or 327 would weight that CV. Because of the constraints of the clustering software, increasing numbers of 2cm^{3} pieces were excluded from the CV calculation of larger cluster sizes, resulting in decreased confidence in the accuracy of the CV and a corresponding decreased weighting in the linear regression calculation.
Measurements of regional ventilation and perfusion were taken twice in both the supine and prone postures. To look at the interanimal variability of ventilation and perfusion heterogeneity, a mean value for each posture was calculated for CV and D of both ventilation and perfusion in each animal. All values are presented as mean ± SD. Paired ttests were used for statistical comparisons, with P < 0.05 considered a significant difference.
RESULTS
Physiological response.
Tidal volumes and respiratory rates, once set, remained constant throughout the experiment, except for the second animal, in which respiratory rate was decreased after the second set of measurements to correct for respiratory alkalosis. There was no significant difference between prone and supine postures for airway, mean arterial, and mean pulmonary artery pressures. There was a trend toward a lower cardiac output in the supine posture, by a mean of 215 ml/min, but this did not reach statistical significance (P = 0.063). The average alveolararterial oxygen tension was significantly greater in the supine posture compared with the prone posture (mean difference = 7.38 Torr, P = 0.028).
Methodological noise estimate.
Two animals were studied to estimate the contribution of methodological noise to the observed heterogeneity of ventilation. In the second animal studied, the last 100 samples had significant degradation of yellowgreen intensity, likely attributable to exposure to a heat source. These 100 samples were excluded from the analysis, leaving 715 pieces. All 954 pieces of the first animal's lung were included in analysis. In both animals, the total number of microspheres administered varied from color to color due to imprecision in the administration of the quantities of aerosols. We compensated for this difference by averaging the total number of microspheres deposited between colors and normalizing the number of microspheres in each piece to this value. Correlations between a specific color and the mean of the other three colors are given in Table 1and averaged 0.995 ± 0.002. In both animals, the CV of regional deposition of four simultaneously administered aerosols decreased as a power function of the mean number of regional microspheres (Fig.2), with an average exponent of − 0.51 ± 0.11.
Fractal analysis.
The CV for both ventilation and perfusion at the smallest region size are shown in Table 2. There is no significant difference between CV for ventilation and perfusion in either posture, despite the wide range of CV observed between animals. CV is significantly greater in the supine posture compared with the prone posture for both ventilation (mean difference = 0.065; P= 0.04) and perfusion (mean difference = 0.105; P = 0.02). Despite the high degree of observed heterogeneity in regional ventilation and perfusion, a narrow distribution ofV˙a/Q˙ is preserved through close correlation between regional ventilation and perfusion (Table3).
Equation 4 provides a reasonable fit of observed heterogeneity for regional ventilation and perfusion as a function of region size (Fig. 3). The average coefficients of determination (r ^{2}) for ventilation fractal plots in the prone and supine postures are 0.80 ± 0.06 and 0.63 ± 0.18, respectively. The average r ^{2} for perfusion fractal plots in the prone and supine postures are 0.84 ± 0.06 and 0.80 ± 0.04, respectively. At clusters ≥32 × υ_{ref}, the standard deviation of the CV measurement increases significantly, and the data are not as well described byEq. 4 . Because of the constraint that all pieces in a cluster be within the same lobe, it is difficult to form more than a few clusters at sizes ≥32 × υ_{ref}. Given the finite ventilation or perfusion to a given lung, flow will be negatively correlated between large clusters, resulting in a poor fit to the linear model at the largest cluster sizes (clusters >32 ×n in animals of this size). This poor fit of the largest clusters to the fractal model has minimal effect on the calculation ofD because the linear regression is weighted by the number of measurements used to calculate CV at each resolution.
The D for individual animals is shown in Table4. The D of ventilation is less than that of perfusion in the prone posture, by a mean difference of 0.035 (P = 0.016). In the supine posture, the mean difference between the ventilation and perfusion D increases to 0.057 (P = 0.0001). The D in the supine posture is lower than in the prone posture for both ventilation (mean difference = 0.066,P = 0.003) and perfusion (mean difference = 0.044, P = 0.035).
DISCUSSION
This study uses aerosolized deposition of fluorescently labeled microspheres to measure regional ventilation with resolution similar to that of regional perfusion measurements. With the use of a clustering algorithm, the fractal characteristics of ventilation are examined in a fashion analogous to that applied to regional perfusion. The important findings from this study are that 1) regional ventilation and perfusion are similarly heterogeneous, but closely correlated, permitting efficient gas exchange; 2) regional ventilation and perfusion have similar fractal characteristics; 3) theD of both ventilation and perfusion is lower in the supine posture than in the prone posture; and 4) the error of measuring regional ventilation is small and partially explained by a Poisson distribution.
Implications for gas exchange.
Wilson and Beck (36) have mathematically shown thatV˙a/Q˙ heterogeneity and gas exchange are determined by the heterogeneity of regional perfusion, regional ventilation, and the correlation between ventilation and perfusion. This study demonstrates that, as the resolution of regional ventilation and perfusion measurement improves, the observed heterogeneity of both ventilation and perfusion increases. Despite this heterogeneity, a narrow distribution ofV˙a/Q˙ is preserved through close correlation of regional ventilation and perfusion (Fig.4). Because both observed perfusion heterogeneity (11) and variability of parenchymal expansion (27) increase at resolutions greater than those obtained in this study, gas exchange efficiency is likely determined by the degree of regional correlation between ventilation and perfusion at volumes <2 cm^{3}. At some level, mixing of alveolar gas by diffusion and reinspiration of common dead space will result in a homogeneous gas composition and uniform endcapillary oxygen contents within the volume of ventilation. Experiments measuring the change in physiological dead space after graded embolization suggest that homogeneous gas exchange occurs at the level of the acinus (37). However, modeling of nitrogen washouts within an acinus suggests that heterogeneity of regional gas composition is present at a subacinar level (7).
Implications for ventilation distribution.
A fractal pattern of ventilation heterogeneity implies that regional ventilation has spatial clustering characteristics similar to those of regional perfusion (Fig. 5) (12). This spatial pattern of regional ventilation promotes speculation regarding mechanisms that determine regional ventilation. Recent theoretical work by West et al. (32, 33) shows that fractal distribution of a substrate explains the 1/4 allometric scaling law observed throughout nature. Their model, based, in part, on the assumption that the energy required to distribute a substrate throughout a region must be minimized for maximal efficiency, predicts observed allometric exponents relating lung structure and function to body mass. Fractal regional ventilation must be determined by regional heterogeneity of lung structure. Given the fractal structure of the bronchial tree (17,31), it is tempting to attribute regional ventilation heterogeneity primarily to differences in regional airway impedance. Studies using both modeling and measurement of alveolar pressure demonstrate that tissue impedance comprises a significant component of total lung impedance at normal ventilation frequencies and is responsible for almost all of the frequency dependence of lung impedance (15, 16, 21). This would suggest that regional ventilation distribution is primarily determined by regional tissue impedance; however, these methods do not give adequate spatial information to draw this conclusion. Despite tissue impedance being greater than airway impedance, if tissue impedance is uniformly distributed, then airway structure may still be the principal determinant of regional ventilation heterogeneity.
Implications of posture effect on D.
The D of both perfusion and ventilation is lower in the supine posture compared with the prone posture. The significance of this is that spatial correlation must be greater for both ventilation and perfusion in the supine posture. This likely represents the superimposition of an organizing influence on the innate heterogeneity caused by pulmonary structure. Likely candidates for this effect are a hydrostatic gradient for pulmonary perfusion (35) and a topographically distributed change in compliance for ventilation (19). These mechanisms resulted in only a small increase in heterogeneity of both ventilation and perfusion (Table 1); however, the effects on ventilation and perfusion are apparently not spatially matched, resulting in increasedV˙a/Q˙ mismatch and, therefore, the increased alveolararterial oxygen difference observed in the supine posture.
Contribution of method error to regional heterogeneity measurement.
The aerosolized microsphere method has minimal noise and good reproducibility, as demonstrated by the high correlations between simultaneously aerosolized microspheres. The CV of si multaneously aerosolized microspheres decreases in proportion to the inverse square root of
In conclusion, efficient gas exchange is dependent on the close matching of regional ventilation and perfusion. Given the high degree of heterogeneity observed at small regional volumes in this study, ventilation and perfusion must be closely correlated to achieve this matching. In the normal lung, this must occur by one of the following three mechanisms: 1) active matching of regional perfusion to ventilation, 2) active matching of regional ventilation to perfusion, or 3) passive matching of ventilation and perfusion by innate pulmonary structure. In the normal lung, basal pulmonary vascular tone is minimal, suggesting that vasoregulation is of minor importance for maintaining closeV˙a/Q˙ matching in uninjured lungs (5, 6, 29, 30). In pigs, there is minimal ventilation redistribution after regional perfusion changes from microembolism, suggesting that active regulation of ventilation distribution is minimal (1). Passive matching of ventilation and perfusion by pulmonary structure is appealing because it requires the least amount of energy. An optimally engineered system requires no active feedback mechanisms during normal function. Pathological conditions, however, may require feedback mechanisms to correct instabilities. Regional ventilation and blood flow are both distributed through fractal structures that share similar geometry. It is tempting to assign responsibility for the fractal distribution patterns of ventilation and blood flow to the bronchial and pulmonary arterial trees, respectively. The high correlation between regional ventilation and perfusion that permits efficient gas exchange, despite high spatial heterogeneity, may be explained by the close correlation of the developing bronchial tree and pulmonary arterial tree during organogenesis, as first suggested by Weibel (31). Fractal distribution networks for ventilation and perfusion provide several inherent advantages. Fractal structures are the most efficient way to fill a threedimensional structure and provide the most energy efficient substrate transport (33). A fundamental characteristic of a fractal structure is that its basic form is replicated over a range of scales. This repetition would permit efficient genetic coding. The fractal nature of heterogeneity, as measured by our analysis, does not prove that regional ventilation and pulmonary perfusion distribution are determined by fractal pulmonary structure, but it is supportive of this idea. Further investigation into the determinants of regional ventilation and the matching between ventilation and perfusion is necessary to fully understand the gas exchange function of the lung.
Acknowledgments
We thank Dowon An and ShenSheng Wang for technical assistance and Dave Frazer for assistance with figure preparation.
Footnotes

Address for reprint requests and other correspondence: W. A. Altemeier, Div. of Pulmonary and Critical Care Medicine, BB1253 Health Sciences Building, Box 356522, Seattle, Washington 981956522 (Email:billa{at}u.washington.edu).

This work was supported by National Heart, Lung, and Blood Institute Grant HL10003 and by an American Lung Association Washington State Affiliate Research Grant.

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 Copyright © 2000 the American Physiological Society