Journal of Applied Physiology
Vol. 83, No. 4, pp. 1370-1382, October 1997
PULMONARY CIRCULATION AND LUNG FLUID BALANCE

Vascular tree structure affects lung blood flow heterogeneity simulated in three dimensions

James C. Parker¹, Chris B. Cave¹, Jeffrey L. Ardell¹, Charles R. Hamm², and Susan G. Williams³

Departments of ¹ Physiology, ² Pediatrics, and ³ Mathematics, University of South Alabama, Mobile, Alabama 36688

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

Parker, James C., Chris B. Cave, Jeffrey L. Ardell, Charles R. Hamm, and Susan G. Williams. Vascular tree structure affects lung blood flow heterogeneity simulated in three dimensions. J. Appl. Physiol. 83(4): 1370-1382, 1997.Pulmonary arterial tree structures related to blood flow heterogeneity were simulated by using a symmetrical, bifurcating model in three-dimensional space. The branch angle (), daughter-parent length ratio (r_L), branch rotation angle (), and branch fraction of parent flow () for a single bifurcation were defined and repeated sequentially through 11 generations. With  fixed at 90°, tree structures were generated with  between 60 and 90°, r_L between 0.65 and 0.85, and an initial segment length of 5.6 cm and sectioned into 1-cm³ samples for analysis. Blood flow relative dispersions (RD%) between 52 and 42% and fractal dimensions (D_s) between 1.20 and 1.15 in 1-cm³ samples were observed even with equal branch flows. When   0.5, RD% increased, but D_s either decreased with gravity bias of higher branch flows or increased with random assignment of higher flows. Blood flow gradients along gravity and centripetal vectors increased with biased flow assignment of higher flows, and blood flows correlated negatively with distance only when   0.5. Thus a recursive branching vascular tree structure simulated D_s and RD% values for blood flow heterogeneity similar to those observed experimentally in the pulmonary circulation due to differences in the number of terminal arterioles per 1-cm³ sample, but blood flow gradients and a negative correlation of flows with distance required unequal partitioning of blood flows at branch points.

regional pulmonary blood flow; pulmonary circulation; gravity gradients; fractal analysis; relative dispersion; computer simulation; distance correlation

INTRODUCTION

THERE IS MOUNTING EVIDENCE that pulmonary blood flow heterogeneity on a small scale is largely determined by anatomical features of the vascular tree (9, 11, 28). Because the pulmonary arteries parallel the airways, many of the structural properties that affect flow are common to both tree structures (39). Weibel (38, 39) proposed a model tree structure that branches by regular dichotomy, i.e., with equal branches, but noted that an irregular dichotomous tree, i.e., unequal branch model, would more accurately describe the observed anatomy. Some of the geometric properties that affect the spatial distribution of the branches of a vascular tree structure in space include the ratios of length and diameter of daughter branches to a parent segment, the number of branches at each branch point, the angle between daughter branches and their rotational orientation in space, and anatomic variations in the number of segments in a transit pathway (21, 22, 29, 33, 45). Additional variables that could affect blood flow distribution within the vascular tree are differences in conductance between vascular segments at a branch point and regional gas volumes and gravitational forces (5, 42). These variables can influence blood flow through individual vessel segments in addition to the number of vessel segments in a given volume (21, 27). Although several branching models of the airways and pulmonary circulation have been proposed, few investigators have extended these models to three-dimensional space or considered the effect of branching structure on flow within fixed sample volumes (12, 19, 20, 23, 25, 35, 38).

The morphometric structure of casts of sequentially branching structures such as airways and vessels has been described by using a number of classification methods, and segment length and diameter generally show a logarithmic relationship to branch generation (36, 38). Recently, the morphometric data from vascular and bronchial casts were reexamined by using a fractal analysis. Diameter and length measurements were found to correlate with generation over a larger range of generations when an inverse power function was used than when obtained by using a semilog relationship (22, 40). The coefficient of such a power function can be related to a fractal dimension (D_s), which is a measure of the complexity of the tree structure (D_s,t) and its ability to fill its topological space (3). A D_s can also be derived for regional blood flow heterogeneity, which describes how the relative dispersion (RD%; SD divided by the mean) changes as flows in adjacent pieces of tissue are aggregated (2, 3). Such a D_s will be independent of the scale of measurement and have a value between 1.0 (uniform) and 1.5 (random) when flows in adjacent pieces of tissue are positively correlated and >1.5 when flows in adjacent sample flows are negatively correlated (3). These regional correlations derive from a recursive branching pattern that distributes a given total flow unequally between branches such that an increase in flow to one branch of a bifurcation necessitates a decrease to the other branch (35). Whereas segment geometry can affect resistance, regional blood flow differences may also result from differences in the number of branches in each sample (21). The number of terminal branches in each sample, in turn, depends on the geometric branching features that determine how a vascular tree fills space (27). Analysis of acinar structure in humans, rabbits, and rats indicates an order of magnitude range of acinar volumes within the lung, and extremely variable branching patterns of airways (26, 30, 31). There is little data on the spatial distribution of arterioles, but the number in a given volume can also be expected to vary considerably.

The purpose of the present study was to model the distribution of regional pulmonary blood flow in three-dimensional space and determine how flow heterogeneity and D_s are affected by branch-point flow inequalities and differences in vascular tree geometry. Glenny and Robertson (12) have recently extended a branching model of the pulmonary circulation to three-dimensional space, but the effects of vascular branching geometry on regional flow distribution and D_s were not considered. We present here a model that incorporates a range of branch angles and daughter-parent length ratio (r_L) values to generate space-filling structures by using a dichotomously branching vascular tree. Inequality of branch flows and flow bias along gravitational and centripetal vectors on the D_s of blood flow dispersion as well as correlations of regional blood flows with distance are also considered. We used dimensions for the model that were approximately those of a dog lung so that regional flows could be sampled by using a three-dimensional grid divided into 1-cm³ cubes, similar to the method previously used for analysis of blood flow data in dog lung (28). The branching pattern and resultant tree-structure geometry as well as branch flow inequalities and directional gradients were observed to affect the D_s and regional blood flow heterogeneity. Whereas reasonable D_s and RD% values were produced by sampling a vascular tree with a fractal structure, negative correlations of regional blood flows with distance only occurred with unequal flow partitioning at branch points.

METHODS

Model Description

1

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Computations and Statistics

Fractal Analysis

(9)

(10)

RESULTS

Vascular Tree Morphology

The self-similarity D_s,t for the three-dimensional tree structure when using the cube-counting method (Eq. 9) was 2.798 (r² = 0.9999) for a tree generated by using  = 60° and r_L = 0.8, and it was 2.846 (r² = 0.9997) by using  = 70° and r_L = 0.8.

The effect of structural parameters on the D_s and RD% values of blood flow heterogeneity with homogeneous branch blood flows ( = 0.5) over ranges of  between 60 and 90° and r_L between 0.65 and 0.85 is shown in Fig. 6 and summarized in Table 1. Sample blood flow heterogeneity was present even when  = 0.5, because different numbers of vessel segments were included in each sample. Both D_s and RD% decreased markedly with increasing r_L, but the minimal values for both D_s and RD% were attained for  = 60° at a higher r_L (0.85) than for  = 90° (r_L = 0.80). At values of r_L >0.80 at  >70°, RD% and D_s apparently increased because of significant overlap of the tree structure at the midline.

Table  1.   Branch angle and length ratio effects on fractal dimensions of blood flow rL   60° 70° 80° 90° RD% Ds r2 RD% Ds r2 RD% Ds r2 RD% Ds r2 0.65 82.0 1.27 0.90 76.2 1.47 0.91 77.0 1.51 0.94 ND ND ND 0.70 58.4 1.24 0.88 60.3 1.33 0.96 56.0 1.34 0.96 64.4 1.20 0.83 0.75 52.8 1.22 0.91 52.2 1.22 0.89 45.1 1.22 0.85 47.9 1.18 0.90 0.80 52.2 1.20 0.93 42.4 1.19 0.89 43.9 1.17 0.95 41.8 1.15 0.90 0.85 42.1 1.14 0.77 49.2 1.19 0.95 55.0 1.16 0.91 75.3 1.21 0.99 , branch angle; rL, daughter-parent length ratio; RD%, blood flow relative dispersion; Ds, fractal dimension; r2, correlation coefficient; ND, not determined.

Values of D_s and RD% simulated by these tree structures are within the range reported for experimentally observed pulmonary blood flow heterogeneity. In five prone unanesthetized dogs, D_s for regional blood flow ranged from 1.111 to 1.148 (average 1.132 ± 0.006), and RD% ranged from 35.4 to 69.1% (average 47.3 ± 5.4%) at rest for total lung, but D_s values as high as 1.264 were obtained for single lungs (28). In 10 prone anesthetized dogs, D_s values between 1.08 and 1.16 and RD% values between 38.3 and 64.6% were reported (12). In five sheep lungs, D_s ranged from 1.07 to 1.17 and RD% from 48 to 86% (6).

Branch Blood Flow Inequality

The interactions of structural parameters, branch flow inequalities, and gravity gradients altered both RD% (Fig. 8) and D_s (Fig. 9). RD% increased as  decreased from 0.5 (unequal flow) and, to a lesser extent, as  decreased with (Fig. 8, bottom) or without (Fig. 8, top) a gravity bias of high-flow branches. Branch angle increases caused relatively large decreases in D_s between 60 and 90° with either random (Fig. 9, top) or gravity bias (Fig. 9, bottom) of high-flow branches. Unequal branch flows caused moderate increases in D_s with random flow assignment but a modest decrease with a gravity bias of high flows (Table 2).

Table  2.   Branch angle and flow fraction effects on fractal dimension of blood flow   rL   Random Gravity RD% Ds r2 RD% Ds r2 60 0.80 0.49 52.2 1.19 0.99 52.7 1.14 0.98 60 0.80 0.47 52.5 1.21 0.91 53.9 1.21 0.91 60 0.80 0.45 58.8 1.19 0.89 61.2 1.20 0.91 70 0.80 0.49 42.4 1.20 0.91 43.0 1.19 0.90 70 0.80 0.47 46.0 1.21 0.94 47.3 1.18 0.90 70 0.80 0.45 50.5 1.21 0.90 55.3 1.18 0.89 80 0.80 0.49 44.2 1.17 0.96 44.7 1.16 0.97 80 0.80 0.47 56.2 1.19 0.96 49.5 1.16 0.98 80 0.80 0.45 56.2 1.19 0.99 57.5 1.15 0.98 90 0.80 0.49 42.3 1.15 0.90 42.3 1.15 0.90 90 0.80 0.47 45.2 1.15 0.93 46.0 1.15 0.95 90 0.80 0.45 51.3 1.15 0.97 52.8 1.14 0.99 , parent branch length.

Correlation of Blood Flows With Distance

The flows generated by using  = 80°, r_L = 0.8, with  = 0.50 did not produce a correlation with distance. The use of the flows of individual branch points would obviously not show a correlation due to a homogeneous flow, but grouping flows into 1-cm³ cubes also failed to show a correlation, even though unequal flows were obtained in some sample cubes. Certain distances did show a modest correlation, possibly due to a repeating pattern of aggregated branch flows, but a graded negative correlation with distance was not present. Flows sampled from this model output had a D_s = 1.17 and a RD% = 43.9, indicating the presence of heterogeneity, but the heterogeneity was attributed to tree structure rather than generated by partitioning of flow between regions supplied by vessel branches.

Blood Flow Gradients

Table  3.   Branch flow inequality effects on gravity gradients   rL   Random Gravity Gravity vs. Random %/cm r2 %/cm r2 Slope r2 60 0.80 0.49 0.3 0.00 0.6 0.00 0.99 0.97 60 0.80 0.47 0.4 0.00 2.6 0.05 0.90 0.77 60 0.80 0.45 1.0 0.01 4.6 0.11 0.72 0.50 70 0.80 0.49  0.2 0.00 0.5 0.00 1.00 0.96 70 0.80 0.47  0.5 0.00 2.5 0.05 0.87 0.72 70 0.80 0.45 0.8 0.01 4.3 0.12 0.76 0.50 80 0.80 0.49  0.8 0.01 0.3 0.00 0.99 0.96 80 0.80 0.47 0.2 0.00 2.2 0.03 0.93 0.75 80 0.80 0.45  1.0 0.015 4.2 0.08 0.67 0.44 90 0.80 0.49 0.3 0.00 1.0 0.01 0.98 0.96 90 0.80 0.47 0.0 0.00 2.8 0.04 0.87 0.73 90 0.80 0.45 1.1 0.015 4.7 0.08 0.67 0.42 %/cm, blood flow %change per cm distance.

Table  4.   Effects of branch flow inequality on relative dispersion, fractal dimension, and centripetal blood flow gradients   rL   RD% Ds r2 %/cm r2 70 0.80 0.49 40.6 1.14 0.70 1.9 0.50 70 0.80 0.47 43.9 1.14 0.76 5.6 0.49 70 0.80 0.45 50.9 1.15 0.87 9.3 0.47 80 0.80 0.49 41.4 1.17 0.95 2.0 0.50 80 0.80 0.47 40.6 1.17 0.94 5.9 0.49 80 0.80 0.45 45.2 1.17 0.97 9.8 0.47

DISCUSSION

The pulmonary arterial circulation enters the lung at the hilus and largely parallels the bronchial tree, with the exception of small supernumery arteries that cross to different airways and gas-exchange units and comprise almost 25% of blood flow (39). Arterial branches tend to be asymmetrical at proximal bifurcations but more equal toward the periphery of the lung. The exact number of arterial branching generations depends on the ordering system used for classification, but there are ~17 Strahler orders in human lungs and 12 orders in the cat to reach the level of the arterioles (18, 19, 44). Whereas cast studies do not describe the spatial distribution of arterioles in three-dimensional space, the acinar volumes vary by an order of magnitude in lung: i.e., 0.5-5 mm³ in rat (26), 1-10 mm³ in rabbit (30), and 1.3-31 mm³ in humans (31). Distances between 30-µm diameter arterioles and venules varied threefold in rat acinus (26), and branching patterns of airways are extremely variable (31). Because recruitment for flow occurs downstream from the arterioles (16), the number of arteriolar segments per acinus should be a major factor in maintaining flow for each acinar volume within a limited target range under zone III conditions. Different acinar flows due to structural variation would impart a basic heterogeneity to regional blood flow. The vascularity within a sample volume could limit the range of possible flows and could account both for the high autocorrelation of individual lung pieces regardless of position and total flow and for the relatively small effect of gravity on overall heterogeneity, i.e., r² <11% (6, 8, 28).

Several models have been proposed to simulate the structural branching pattern of the pulmonary circulation (23, 27, 37, 39) or functional properties such as vascular impedance, vascular transit time distributions, or vascular volume-resistance and pressure-volume relationships (11, 12, 21-23). These models describe the circulation as a dichotomous branching structure with either equal (11, 35, 39) or unequal (35) branch lengths or as a more complex array of pathways using assorted segment lengths (21, 22). D_s values have been derived for both space-filling tree structures in two dimensions and probability density functions of blood flow heterogeneity (2, 11, 35), but only recently has a model of the pulmonary circulation been extended to three-dimensional space (12). The model proposed by Glenny and Robertson (12) represents the pulmonary circulation as an orthogonal, dichotomous branching structure that distributes flow to evenly dispersed terminal segments, so vascular tree structure was not a determinant of blood flow heterogeneity.

In the model presented here, we varied vessel branch angle and r_L to generate arterial tree structures with markedly different sizes and shapes. These three-dimensional trees were the approximate size of dog lungs, so pulmonary blood flow heterogeneity could be analyzed by dividing the lung into 1-cm cubes as previously done in experimental dogs (28). Differences in RD% and D_s of blood flow heterogeneity occurred even with equal flows at branch points because the number of terminal segments in each sample volume changed with shape (21). The number of segments per cubic centimeter ranged from 1 to 12 in some trees. Similarly, there was a fourfold difference in the number of 1-cm³ samples obtained from the vascular trees presented here. Values of RD% and D_s within an experimentally observed range could be obtained with trees generated by using  between 60 and 90° and r_L between 0.7 and 0.8, even without branch flow inequalities.

Whereas a regular dichotomous model differs significantly in structural detail from vascular casts of mammalian lungs, it retains many functional aspects of more complex vascular trees. Krenz et al. (21, 22) showed that a homogeneous dichotomous model could be obtained that was functionally equivalent to either irregular dichotomous branching models or models based on experimental vascular cast data with branching ratios >2.0. The exact number of branches in each generation and the number of generations of vessels in a vascular cast depend on the ordering system used for classification. However, the number of vessels of a certain diameter or the cumulative number of vessels at each diameter in all pulmonary vascular cast studies could be related to the same power function (₁), regardless of the ordering system (22). This relationship was maintained, even though vascular cast branch ratios were >2.0. Horsfield (18) observed an average daughter-to-parent branching number of 3.0 and length ratio of 0.63 in casts of human pulmonary arteries, whereas Yen et al. (44) found a branching ratio of 3.58 and a length ratio of 0.60 in casts of pulmonary arterial trees from cats. In both species, the log-to-log ratio of length to diameter (₂) approached 1.0, indicating that vessel diameters decrease as a power function of length at successive generations. Krenz et al. (22) demonstrated that a ₁ of ~2.5 was derived for pulmonary arterial tree casts of humans, dogs, and cats regardless of their classification system and that a homogeneous dichotomously branching model such as presented here could be found with the same value of ₁. The longitudinal distributions of vascular volume, resistance, and pressure would be equivalent in all models with the same exponent ₁ for the relationship of vessel diameter (D_j) and number (N_j) at each generation (j)

(11)

where D_a is the diameter of the first vessel (21), and ₁ is determined by the daughter-parent diameter ratio. Diameter and length are related with an exponent ₂ approximately equal to 1.0 (18, 21, 22, 44), so

(12)

where L_j is the length of generation j and L_a is the original segment length (21). Therefore, we can derive a comparable power coefficient, ₃ = ₁/₂, relating segment length to number, where

(13)

Thus r_L becomes the most critical parameter in the present model because it determines the space-filling properties of the vascular tree, blood flow dispersion, and D_s of flow and could be extrapolated to predict vascular hemodynamic properties.

Figure 15 demonstrates the effect on ₃ of changing r_L from 0.7 to 0.8 in the present model (solid symbols). Vessel lengths were normalized by dividing vessel lengths by the initial vessel length and plotting as a function of the cumulative vessel number. Also shown are normalized vessel length data from vascular casts of human () and cat () lungs by Horsfield (18) and Yen et al. (44), respectively. In Fig. 15, ₃ values of 1.95, 2.51, and 3.12, respectively, were produced by r_L values of 0.70, 0.758, and 0.80 (assuming ₂ = 1.0). Respective ₃ values from vascular cast data were 2.43 for cat and 2.96 for human lungs. Krenz et al. (21) obtained corresponding blood flow D_s values of 1.3, 1.2, and 1.15 from model trees with ₁ values of 2, 2.5, and 3. In our model, the r_L values of 0.70, 0.75, and 0.80 produced D_s values that varied with  but ranged between 1.20-1.34, 1.18-1.22, and 1.15-1.20 for the respective r_L values when  was 0.50. Thus a regular dichotomous model can simulate many of the structural effects on blood flow dispersion that occur in a pulmonary vascular tree structure having a higher average branching ratio and more irregular branch lengths and branch angles.

Vascular trees derived by using the present dichotomous model and pulmonary vascular casts, which are both characterized by a ₁ (or ₃) of 2.5, would possess similar longitudinal profiles of vascular resistance, vascular pressure, and vascular volume (22). In both such vascular trees, the smaller vessels would be the site of most of the vascular pressure drop and vascular resistance but contain little of the vascular volume. Larger pulmonary vessels in such a system would act as a pressure manifold with most of the vascular volume but with only a small drop in vascular pressure (22). Smaller values of ₁ (or ₃) would imply that relatively more of the total vascular resistance and less of the blood volume would reside in the smaller vessels, whereas values of ₁ (or ₃) closer to 3.0 would imply a more equal longitudinal distribution of resistance and volume. Therefore, using this dichotomous model, we could simulate basic hemodynamic properties of models that incorporated much more detailed morphometric cast data. Reasonably accurate pressure-flow relationships for the pulmonary circulation have been simulated for a variety of physiological conditions when using these more detailed models, and the longitudinal vascular pressure and volume profiles were predicted (17, 21, 22, 46). Even in the most detailed anatomic models, the accuracy for predicting vascular resistance effects is limited by the accuracy of the morphometric measurement of small-vessel diameters, because these vessels are critical determinants of overall pulmonary vascular resistance (21, 22). In the present model, we defined flow partition at bifurcations as  and 1  , which implies a structural difference between daughter branches sufficient to produce the defined flow differences. Whereas such partitioning is an oversimplification, the flow inequalities have a fractal pattern because flow from each segment is separately partitioned. In addition, a wide range of flow heterogeneities can be simulated by global changes in .

A novel feature of the present model is the use of different branch angles and length ratios to modulate three-dimensional space-filling properties and blood flow dispersion. Measurement of branch angles in vascular and bronchial casts has been difficult because of asymmetry and curved segments (29). Daughter branches are more asymmetric in branch angle and length at proximal branches, but both lengths and branch angles become more symmetric toward the periphery in human lung casts (29, 33). However, rotational angles of branches have not been systematically analyzed in casts. Previous model studies of the optimal branch angles for transport efficiency have been confined to two-dimensional space (1, 23, 27, 32, 34). In the present study, the rotational plane of the branches, , was fixed at 90° in all simulations because constant values other than 90° produced asymmetric, spiraled, or skewed tree structures. Varying the branch angle  from 50 to 90° produced marked differences in tree-structure shapes. A 90° angle from the midline produced a rectangular-shaped lung, whereas values between 60 and 80° produced rounded tree structures that more closely resembled casts of the pulmonary arterial tree. The structural fractal dimension D_s,t increased from 2.80 to 2.85 as  increased from 60 to 70°, indicating greater space-filling capacity of the structure as  increased. Zamir (45) examined the optimal branch angle and diameter ratio to obtain minimal values of surface area, blood volume, work, and drag at a branch point. The optimal branch angle for a single branch from a trunk was 90° and that for a symmetric bifurcation was 45-50° from the parent axis with a daughter-parent cross-sectional area ratio of 1.26. This cross-sectional area ratio would correspond to a r_L of 0.795 in our model, assuming that diameter and length changed proportionally (18, 22).

The spatial distribution of terminal branches was critically dependent on the r_L as the branches tended to clump around the initial branches at low length ratios. Branch points became more homogeneously dispersed as r_L increased from 0.60 to 0.80, but structures tended to overlap the midline at higher r_L values. Lefevre (23) optimized the two-dimensional geometric structure, vascular volume, and impedance properties in a model of the pulmonary circulation and obtained an optimal r_L (and a diameter ratio) of 0.78. When an r_L of 0.7937, or r_L, is used in a two-dimensional orthogonal (90°) branching model, the available space can be filled with an infinite number of branches without any branches crossing a previous branch (24). It should be noted that this same optimal r_L minimized impedance and volume and was required in the present study to simulate experimentally observed spatial flow distributions and D_s. It was also used by Krenz et al. (22) to simulate the observed longitudinal distribution of vascular resistance and pressure of isolated lungs.

Because both the branch angles  and length ratios r_L in the present model were determinants of the number of terminal vessels in each 1-cm³ of volume, these parameters affected flow heterogeneity, even without unequal flows at branch points (21). When  = 0.5 (Table 1), both D_s and RD% decreased as a function of increased branch angle and length ratio until the structure overlapped the midline at  >80° and r_L = 0.80. Van Beek et al. (35) also noted a decrease in D_s as the r_L increased. A homogeneous spatial flow distribution could not be obtained by  = 0.5 (equal flow) in the present model, but the minimal values of RD% = 42% and D_s = 1.15 obtained when  = 0.5 are within the range of values obtained experimentally in dog lungs (28). In the orthogonal model of Glenny and Robertson (12), spatial dimensions and structural geometry did not contribute to spatial flow variability. Blood flow heterogeneity was determined only by branch flow inequalities. Therefore, the lower limits of RD% and D_s would be 0.0 and 1.0, respectively, when  was 0.50. These minimal values for D_s and RD% could not be obtained in our model by using flows in 1-cm³ volumes due to a variable number of terminal vessel segments but could be attained by analyzing only segment flows (N) without considering structural patterns.

Whereas these simulations indicate that the RD% and D_s for measured regional flow do not specify a unique structure or branch flow inequality, structural parameters affected both RD% and D_s. In general, both RD% and D_s tended to decrease as  and r_L increased. Minimal values of RD% and D_s required higher values of r_L at lower , indicating the dependence of tree spreading on both  and r_L. Values of RD% between 40 and 50% and values of D_s between 1.14 and 1.22 were obtained for a range of  between 70 and 90° and r_L between 0.75 and 0.80 when  = 0.5. As branch flow inequality increased, RD% increased and D_s decreased when a gravity gradient in higher branch flows was present but not when flow inequality was randomly assigned or had a central bias. Krenz et al. (21) calculated a D_s of 1.2 and an RD% of 77.3% using a  of 0.42 in a homogeneous dichotomous model carried to 19 generations. Glenny and Robertson (12) obtained simulated RD% and D_s values of 46.7% and 1.13, respectively, using a random branch flow inequality with an SD of 0.05. D_s increased with increased inequality of randomly assigned branch flows. Our model values for RD% and D_s were within the upper range of respective average experimental values of total lung D_s and RD% of 1.132 and 47.3% (1.225 for single lung) reported by Parker et al. (28) for lungs of unanesthetized prone dogs, the 1.18 and 45.7% obtained by Glenny and Robertson (12) in anesthetized prone dogs, and the 1.14 and 64.0% obtained in isolated sheep lungs by Caruthers and Harris (6). D_s and RD% values comparable to those observed experimentally were obtained in the present model when values of  of >60°, r_L of >0.75, with  between 0.49 and 0.45 were used for simulations.

A correlation of blood flow with distance is a significant feature of pulmonary blood flow heterogeneity described by Glenny (7). A correlation that decreased with distance was observed, which became negative at a distance of 5-10 cm depending on the lobe in dog lung data (7). Glenny and Robertson (12) simulated this correlation in a three-dimensional flow model. We also show here that individual branch point flows correlate negatively with distance for a model where  = 80°, r_L = 0.80, and  = 0.45. The exponent for this relationship of 0.27 was similar to that reported for prone dogs (0.27) and a model with flow partitioning (7, 12). When individual flows were grouped into 1-cm³ samples, they retained this correlation with an exponent of 0.25. Both correlations became negative at a distance of 10 cm. However, the same model tree structure with  = 0.50 showed no correlation with distance when using either individual nodes or aggregating nodes into 1-cm³ samples. Some distances showed _xyz values of ~0.3 but no relationship to distance. The lack of correlation of individual nodes was to be expected because flows were equal. However, the aggregated model with  = 0.50 cut into cubes had a D_s = 1.17 and a RD% = 43.9, indicating the presence of flow heterogeneity. The heterogeneity described by D_s and RD% must represent heterogeneity due to tree structure and indicates a pattern complexity that varies with scale. Apparently only heterogeneity due to partitioning of flow can produce the correlation with distance and the negative correlation with distant regions, which implies a "steal" of flow from branches to distant regions to supply near regions. Thus D_s and RD% values appear to describe heterogeneity that varies with scale but are not as specific as the correlation with distance for the unique pattern of flow heterogeneity produced by a repetitive branching system that distributes a finite amount of flow to tissue segments. Tree structures other than  = 80°, r_L = 0.80 are expected to modify the shape of the curve relating correlation to distance when the structure is cut into 1-cm³ cubes, but the basic relationship is expected to persist for all reasonable tree structure with the same flow partitioning.

We also simulated the gravity-dependent and centripetal blood flow gradients previously observed in dog lungs using our model described here by assigning the higher flows to the daughter branches furthest along the gravity or centripetal axis. As expected, a directional blood flow gradient was produced, which increased as  decreased from 0.49 to 0.45. Gravity exerts a distending force on dependent vessels and reduces their resistance (41-43) but contributes a relatively small amount (<11%) to overall flow heterogeneity in small pieces of lung (8, 9, 28). A structural bias of high-flow regions in central-dorsal lung regions has also been observed, possibly due to shorter transit pathways for flow (4, 5, 13-15). This gradient also contributes <15% to overall flow heterogeneity of small tissue samples (28). Glenny and Robertson (12) simulated gravity gradients in evenly dispersed terminal nodes by adding a separate gravity term to flow partitioning, which biased a portion of flow down the gravity axis. Otherwise, flow inequalities were randomly assigned between daughter branches. Although both models produce gravity gradients, a systematic bias of all high flows along the gravity axis in their model would undoubtedly result in less overall heterogeneity than observed experimentally, whereas the present model includes an intrinsic heterogeneity based on vascular structure.

Gravity-dependent gradients ranged from 4.2 to 4.7%/cm vertical distance in simulations using  = 0.45 over a range of  from 60 to 90°. These values compared favorably to the average vertical blood flow gradient of 4.7%/cm (r² = 0.118) measured in the lungs of unanesthetized dogs at rest (28). The simulated gravity gradient of 2.5%/cm (r_L = 0.41) using  = 0.47 approached the 1.7%/cm (r² = 0.044) in lungs of experimental dogs during exercise as shown by Parker et al. (28). Centripetal gradients of 5.9 and 9.8%/cm were simulated by using respective  values of 0.47 and 0.45, which bracket the centripetal blood flow gradient of 7.2%/cm (r² = 0.108) observed in experimental animals (28). When high branch flows were randomly assigned, there were no consistent blood flow gradients. The stochastic model of Glenny and Robertson (12) randomly assigned unequal branch flows, so an additional flow factor was necessary at branch points to account for gravity. In the present model, structure provides basic heterogeneity, and   0.5 is required for gravity gradients. A decrease in from 0.49 to 0.45 also increased the RD% of blood flow and decreased D_s under the influence of a gravity bias in all vascular tree configurations. A similar decrease in D_s with gravity was also observed by Glenny and Robertson (12) in their three-dimensional orthogonal model. Apparently, the ordering of flows by a process other than a repetitive fractal branching reduces the correlation of flow in adjacent pieces and the relative influence of branching on heterogeneity (3).

In summary, a symmetrical, bifurcating model in three-dimensional space carried to 11 generations was sufficient to simulate the spatial heterogeneity, blood flow gradients, and D_s values of blood flow observed experimentally in 1-cm³ samples of dog lung. The use of  and r_L to produce tree structures with the same ₃ values as observed for vascular cast data simulated the spatial distribution, RD%, and D_s for blood flow heterogeneity within the range observed in experimental lungs. Values of RD% and D_s comparable to those observed experimentally were simulated with  >60°and r_L >0.75. Gravity and centripetal blood flow gradients comparable to those observed in dog lungs were simulated by nonrandom assignment of the higher of unequally partitioned branch flows along a gravity or centripetal vector. Although reasonable D_s and RD% values were obtained by aggregation of different numbers of vessels during sampling, D_s and RD% values closer to experimental values and a negative correlation of flows with distance was only obtained when unequal flow partitioning at branch points was included in the model.

ACKNOWLEDGEMENTS

This work was supported by Grant-in-Aid 94013094 from the American Hear