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 threedimensional space. The branch angle (Θ), daughterparent length ratio (r_{L} ), branch rotation angle (φ), and branch fraction of parent flow (γ) for a single bifurcation were defined and repeated sequentially through 11 generations. With φ fixed at 90°, tree structures were generated with Θ between 60 and 90°,r_{L} between 0.65 and 0.85, and an initial segment length of 5.6 cm and sectioned into 1cm^{3} samples for analysis. Blood flow relative dispersions (RD%) between 52 and 42% and fractal dimensions (D _{s}) between 1.20 and 1.15 in 1cm^{3}samples were observed even with equal branch flows. When γ ≠ 0.5, RD% increased, butD _{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 γ ≠ 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 1cm^{3} 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
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 threedimensional 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 threedimensional space and determine how flow heterogeneity andD _{s} are affected by branchpoint flow inequalities and differences in vascular tree geometry. Glenny and Robertson (12) have recently extended a branching model of the pulmonary circulation to threedimensional 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 daughterparent length ratio (r_{L} ) values to generate spacefilling structures by using a dichotomously branching vascular tree. Inequality of branch flows and flow bias along gravitational and centripetal vectors on theD _{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 threedimensional grid divided into 1cm^{3} cubes, similar to the method previously used for analysis of blood flow data in dog lung (28). The branching pattern and resultant treestructure geometry as well as branch flow inequalities and directional gradients were observed to affect the D _{s} and regional blood flow heterogeneity. Whereas reasonableD _{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
Geometry.
The pulmonary arterial tree was numerically simulated by using a threedimensional, symmetrical, dichotomously branching tree structure, as shown in Fig. 1. The algorithm computes a spacefilling vascular tree based on three model input parameters: the inplane rotation angle (branch angle) (Θ), the outofplane rotation angle (φ), and a length ratior_{L} , wherer_{L} = α/δ, i.e., the ratio of the daughter (α) to parent (δ) branch lengths, as shown in Fig. 2.
The computer algorithm computes ntotal generations, which results in 2^{n} ^{−1}terminal branches; n = 11 was used in all simulations. Herein a branch will be defined by the position vector of its terminal node. Daughter branch position vectors are computed by rotating and scaling the parent branch position vector about the origin and then adding the resultant vector to the parent vector to translate the daughter branch to its location in space.
The rotation/scaling operation is a vectormatrix multiplication of a position vector and a combined rotational matrix as follows. First, with the use of a righthanded, orthogonal coordinate system, let the origin of the branching structure reside in thexz plane. For an inplane rotation through a positive angle Θ, the rotational matrix is defined by
Blood flow. At each bifurcation, flow of the parent branch was divided between daughter branches with a fraction γ to one branch and 1 − γ to the other (35). Thus the flows for the branches at the first bifurcation were γF_{o}and (1 − γ)F_{o}, where F_{o} is the total flow. Each of these flows was then multiplied by γ and 1 − γ, and the flows and coordinates of the branch points were stored in matrices. In the absence of flow bias, the highest branch flow was randomly assigned by using a randomnumber generator. To simulate a gravity bias, the high flow was nonrandomly assigned to the branch with the greatest (most negative) Z value (gravity axis). To simulate the centripetal gradient, the highest flow was assigned to the daughter branch nearest the first branch point of the tree (X
_{m} = 0, Y
_{m} = 0,Z
_{m} = −5.6), which is near the center of volume for the tree. A vectorV from the midpoint to a branch node was calculated by using
Simulated regional blood flows were also correlated as a function of distance in three dimensions according to Glenny (7). Distance vectors were calculated between sampling regions usingx,y, andzaxis coordinates and a modification of Eq. 8 above. Linear correlation coefficient (ρ_{xyx}) values between regional flows were calculated for different distances independent of direction (7). The values of ρ_{xyx} obtained for groups of flows separated by increasing distances were plotted against distance, where correlation coefficients between 1.0 and −1.0 indicate respective positive and negative dependencies of regional flows on distance. Then a nonlinear curvefitting routine was used to obtain the parameters in ρ_{xyx}, resulting in the best curve fit, in the sense of least squares, to the data. Simulated flows for the 11th generation of a model where Θ = 80°, r_{L} = 0.8, and γ = 0.45 or 0.50 were analyzed. Flows were also analyzed as individual branch flows or after aggregation of flows within 1cm^{3} samples.
Computations and Statistics
The model was written in ASYST language and solved on a digital computer. The model output the branch point flows and threedimensional coordinates of each branch point to a LOTUS 1–2–3 spreadsheet for analysis. Statistical correlations, regressions, and descriptive statistical analyses were performed by using either Crunch or Statview statistical software. Values are expressed as means ± SE or as individual data points. A least squares regression was performed where indicated, and r ^{2}was used as an indicator of the influence of a variable on flow heterogeneity (7, 10).
Fractal Analysis
Blood flow heterogeneity. AD
_{s} of blood flow heterogeneity was performed by summing the flows within each 1cm^{3} sample volume (V_{o}) (35) and then by successively aggregating the adjacent V_{o} flows to obtain aggregate samples of 1, 2, 4, 8, 16, 32, and 64 cm^{3}. Sample volumes were paired along the Xaxis (n = 2),Yaxis (n = 4), andZaxis (n = 8), and the process was repeated to obtain n = 64. Mean flows and SD values were used to calculate the RD% [RD = (SD/mean) × 100] of each sample group, which was regressed on sample volume ratio (V/V_{o}) to obtain a D
_{s} by using (3)
Vascular tree structure.
A selfsimilarityD
_{s,t} was obtained by using a modified boxcounting method in three dimensions (3, 27). The number of cubes (n) that contained 11th generation branch points was calculated when the cubes were 1.0, 0.5, 0.33, and 0.25 cm on a side, i.e., had a scale factor (F) of 1, 2, 3, or 4, soD
_{s,t} was calculated by
RESULTS
Vascular Tree Morphology
Figures 3 and 4 show side and top views, respectively, of spacefilling patterns of the 11th generation branch points as a function of the branch angles Θ between 60 and 90° and length ratios r_{L} between 0.70 and 0.80. The branch rotation angle φ was maintained constant at 90°, as different values of φ resulted in tree structures that were skewed to one side. Smaller values of Θ andr_{L} resulted in 11th generation nodes that tended to clump around the initial branch points, whereas as Θ approached 90° andr_{L} approached 0.80 a rectangular structure was generated with uniform spacing of branch points. Such a rectangular tree structure (Θ = 90° andr_{L} = 0.7) was proposed for tracheal branching by Mandelbrot (25). Although 2,048 11th generation branches were generated in each simulation, the space filled by the vascular trees varied with branch geometry. The number of 1cm^{3} samples produced by the trees shown in Figs. 3 and 4 ranged from 234 (Θ = 60°;r_{L} = 0.7) to 1,175 (Θ = 90°;r_{L} = 0.8). The range of terminal segments (n) per 1cm^{3} sample wasn = 1 to betweenn = 4 (Θ = 90°;r_{L} = 0.8) andn = 12 (Θ = 90°;r_{L} = 0.7). A tree structure with Θ = 60–80° andr_{L} = 0.8 appeared to have a spacefilling tree structure with proportions similar to the dog lung. Figure 5 compares the spatial distributions of terminal branches in the model with 1cm^{3} tissue samples of a dog lung (28).
The selfsimilarityD _{s,t}for the threedimensional tree structure when using the cubecounting method (Eq. 9 ) was 2.798 (r ^{2} = 0.9999) for a tree generated by using Θ = 60° andr_{L} = 0.8, and it was 2.846 (r ^{2} = 0.9997) by using Θ = 70° andr_{L} = 0.8.
The effect of structural parameters on theD _{s} and RD% values of blood flow heterogeneity with homogeneous branch blood flows (γ = 0.5) over ranges of Θ between 60 and 90° andr_{L} between 0.65 and 0.85 is shown in Fig.6 and summarized in Table1. Sample blood flow heterogeneity was present even when γ = 0.5, because different numbers of vessel segments were included in each sample. BothD _{s} and RD% decreased markedly with increasingr_{L} , but the minimal values for bothD _{s} and RD% were attained for Θ = 60° at a higherr_{L} (0.85) than for Θ = 90° (r_{L} = 0.80). At values of r_{L} >0.80 at Θ >70°, RD% andD _{s} apparently increased because of significant overlap of the tree structure at the midline.
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, butD _{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
Structureinduced blood flow heterogeneity significantly limited the minimal heterogeneity that could be attained even with equal branch fractions of parent blood flow. However, when values of γ <0.5 were used, they introduced additional variability to blood flow. The contribution of tree structure to measurement of RD% andD _{s} can be seen in the fractal analysis shown in Fig. 7. A model simulation using Θ = 80°,r_{L} = 0.8, and γ = 0.45 with gravity bias is analyzed by using RD% as a function of either the sample volumes after sectioning the structure into 1cm^{3} cubes (V/V_{o}; Fig. 7,left) or the number of vessels in each generation, where individual branch flows were analyzed without aggregation into cubic samples (N/N _{o}; Fig. 7, right). Note the lowerD _{s} (1.079) and reference RD% (41%) when blood flow dispersion is determined only by the unequal flow fractions at branch points (γ = 0.45; Fig. 7,right) compared with the greater variability (D _{s} = 1.149; RD% = 57%) when using the same flow fractions (γ = 0.45) when the branching structure is included (Fig. 7,left). Minimal values ofD _{s} = 1.0 and RD% = 0.0 are produced by γ = 0.5 when they are analyzed by using RD% vs.N/N _{o}, whereas these minimal values were not attainable when using V/V_{o} with a defined tree structure.
The interactions of structural parameters, branch flow inequalities, and gravity gradients altered both RD% (Fig.8) andD _{s} (Fig.9). RD% increased as γ decreased from 0.5 (unequal flow) and, to a lesser extent, as Θ decreased with (Fig.8, bottom) or without (Fig. 8,top) a gravity bias of highflow branches. Branch angle increases caused relatively large decreases inD _{s} between 60 and 90° with either random (Fig. 9,top) or gravity bias (Fig. 9,bottom) of highflow branches. Unequal branch flows caused moderate increases inD _{s} with random flow assignment but a modest decrease with a gravity bias of high flows (Table 2).
Correlation of Blood Flows With Distance
Blood flows were correlated as a function of distance according to Glenny (7). The tree structure analyzed was produced by using Θ = 80° and r_{L} = 0.8 with either γ = 0.45 or 0.50. In Fig.10, the correlation coefficients, ρ_{xyz} as a function of distance between branch nodes are shown for Θ = 80°,r_{L} = 0.8, with γ = 0.45 (•). Individual 11th generation branch nodes of the simulation are distributed in space without sectioning the model into cubes, so each node represents a single branch node without cubic sampling. A leastsquares curve fit of the correlation coefficients is shown and becomes negative at a distance of ∼10 cm. The exponent of the fitted curve was −0.27 for this simulation withr = 0.98. Grouping the flows into 1cm^{3} cubes and again analyzing for distance correlation produced the relationship shown in Fig.11 (▪). The correlation again became negative at a distance of 10 cm but with an exponent of −0.25 andr = 0.97 for the line of best fit.
The flows generated by using Θ = 80°,r_{L} = 0.8, with γ = 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 1cm^{3} 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 aD _{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
The nonrandom bias of high branch flows along the gravityZaxis caused gradients in the blood flow distribution. Figure 12 compares the vertical distributions of segment blood flows in 11th generation vessels with biased assignment of higher branch flows down the gravity axis (Fig. 12,left) and random flow assignments (Fig. 12, right). Figure13 shows the distribution of flows as a function of distance from the first branch point when high branch flows were nonrandomly assigned along a centripetal vector. The residual scatter accounts for the lowr ^{2} values obtained with linear regression of gravity and centripetal gradients (Tables 3 and 4). As shown in Fig.14, gradients as percent total flow per centimeter increased as a linear function of branch flow inequality. Changes in branch angles had relatively minor effects on these gradients (Table3). There were no significant gravity or centripetal gradients when a random branch flow assignment was used. The gravitydependent blood flow gradients obtained in the model when using γ = 0.45 (slope = 4.2–4.6%/cm andr ^{2} = 0.075–0.113) were comparable to values previously measured in prone dogs at rest, where the gravitydependent slopes were ≤4.7%/cm with r ^{2} of ≤0.118 (28). The experimental centripetal gradient was 6.1%/cm in these dogs.
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 gasexchange 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 threedimensional space, the acinar volumes vary by an order of magnitude in lung: i.e., 0.5–5 mm^{3} in rat (26), 1–10 mm^{3} in rabbit (30), and 1.3–31 mm^{3} in humans (31). Distances between 30μm 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.,r ^{2}<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 volumeresistance and pressurevolume relationships (11, 12, 2123). 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 spacefilling 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 threedimensional 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 andr_{L} to generate arterial tree structures with markedly different sizes and shapes. These threedimensional trees were the approximate size of dog lungs, so pulmonary blood flow heterogeneity could be analyzed by dividing the lung into 1cm cubes as previously done in experimental dogs (28). Differences in RD% andD _{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 1cm^{3} samples obtained from the vascular trees presented here. Values of RD% andD _{s} within an experimentally observed range could be obtained with trees generated by using Θ 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 (β_{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 daughtertoparent 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 logtolog ratio of length to diameter (β_{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 β_{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 β_{1}. The longitudinal distributions of vascular volume, resistance, and pressure would be equivalent in all models with the same exponent β_{1} for the relationship of vessel diameter (D_{j}
) and number (N_{j}
) at each generation (j)
Figure 15 demonstrates the effect on β_{3} of changingr_{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, β_{3} values of 1.95, 2.51, and 3.12, respectively, were produced byr_{L} values of 0.70, 0.758, and 0.80 (assuming β_{2} = 1.0). Respective β_{3} values from vascular cast data were 2.43 for cat and 2.96 for human lungs. Krenz et al. (21) obtained corresponding blood flowD _{s} values of 1.3, 1.2, and 1.15 from model trees with β_{1} values of 2, 2.5, and 3. In our model, ther_{L} values of 0.70, 0.75, and 0.80 producedD _{s} values that varied with Θ but ranged between 1.20–1.34, 1.18–1.22, and 1.15–1.20 for the respectiver_{L} values when γ 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 β_{1} (or β_{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 β_{1} (or β_{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 β_{1} (or β_{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 pressureflow 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 smallvessel 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 γ and 1 − γ, 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 γ.
A novel feature of the present model is the use of different branch angles and length ratios to modulate threedimensional spacefilling 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 twodimensional space (1, 23, 27, 32, 34). In the present study, the rotational plane of the branches, φ, was fixed at 90° in all simulations because constant values other than 90° produced asymmetric, spiraled, or skewed tree structures. Varying the branch angle Θ from 50 to 90° produced marked differences in treestructure shapes. A 90° angle from the midline produced a rectangularshaped lung, whereas values between 60 and 80° produced rounded tree structures that more closely resembled casts of the pulmonary arterial tree. The structural fractal dimensionD _{s,t} increased from 2.80 to 2.85 as Θ increased from 60 to 70°, indicating greater spacefilling capacity of the structure as Θ 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 daughterparent crosssectional area ratio of 1.26. This crosssectional area ratio would correspond to ar_{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 asr_{L}
increased from 0.60 to 0.80, but structures tended to overlap the midline at higher r_{L}
values. Lefevre (23) optimized the twodimensional geometric structure, vascular volume, and impedance properties in a model of the pulmonary circulation and obtained an optimalr_{L}
(and a diameter ratio) of 0.78. When anr_{L}
of 0.7937, orr_{L}
Because both the branch angles Θ and length ratiosr_{L} in the present model were determinants of the number of terminal vessels in each 1cm^{3} of volume, these parameters affected flow heterogeneity, even without unequal flows at branch points (21). When γ = 0.5 (Table 1), bothD _{s} and RD% decreased as a function of increased branch angle and length ratio until the structure overlapped the midline at Θ >80° andr_{L} = 0.80. Van Beek et al. (35) also noted a decrease inD _{s} as ther_{L} increased. A homogeneous spatial flow distribution could not be obtained by γ = 0.5 (equal flow) in the present model, but the minimal values of RD% = 42% and D _{s} = 1.15 obtained when γ = 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% andD _{s} would be 0.0 and 1.0, respectively, when γ was 0.50. These minimal values forD _{s} and RD% could not be obtained in our model by using flows in 1cm^{3} 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% andD _{s} for measured regional flow do not specify a unique structure or branch flow inequality, structural parameters affected both RD% andD _{s}. In general, both RD% and D _{s}tended to decrease as Θ andr_{L} increased. Minimal values of RD% andD _{s} required higher values ofr_{L} at lower Θ, indicating the dependence of tree spreading on both Θ andr_{L} . Values of RD% between 40 and 50% and values ofD _{s} between 1.14 and 1.22 were obtained for a range of Θ between 70 and 90° andr_{L} between 0.75 and 0.80 when γ = 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 aD _{s} of 1.2 and an RD% of 77.3% using a γ 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% andD _{s} were within the upper range of respective average experimental values of total lungD _{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 Θ of >60°,r_{L} of >0.75, with γ 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 threedimensional flow model. We also show here that individual branch point flows correlate negatively with distance for a model where Θ = 80°,r_{L} = 0.80, and γ = 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 1cm^{3} 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 γ = 0.50 showed no correlation with distance when using either individual nodes or aggregating nodes into 1cm^{3} samples. Some distances showed ρ_{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 γ = 0.50 cut into cubes had aD _{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. ThusD _{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 Θ = 80°,r_{L} = 0.80 are expected to modify the shape of the curve relating correlation to distance when the structure is cut into 1cm^{3} cubes, but the basic relationship is expected to persist for all reasonable tree structure with the same flow partitioning.
We also simulated the gravitydependent 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 γ decreased from 0.49 to 0.45. Gravity exerts a distending force on dependent vessels and reduces their resistance (4143) but contributes a relatively small amount (<11%) to overall flow heterogeneity in small pieces of lung (8, 9, 28). A structural bias of highflow regions in centraldorsal lung regions has also been observed, possibly due to shorter transit pathways for flow (4, 5, 1315). 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.
Gravitydependent gradients ranged from 4.2 to 4.7%/cm vertical distance in simulations using γ = 0.45 over a range of Θ from 60 to 90°. These values compared favorably to the average vertical blood flow gradient of 4.7%/cm (r ^{2} = 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 γ = 0.47 approached the 1.7%/cm (r ^{2} = 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 γ values of 0.47 and 0.45, which bracket the centripetal blood flow gradient of 7.2%/cm (r ^{2} = 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 γ ≠ 0.5 is required for gravity gradients. A decrease in γ from 0.49 to 0.45 also increased the RD% of blood flow and decreasedD _{s} under the influence of a gravity bias in all vascular tree configurations. A similar decrease inD _{s} with gravity was also observed by Glenny and Robertson (12) in their threedimensional 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 threedimensional space carried to 11 generations was sufficient to simulate the spatial heterogeneity, blood flow gradients, andD _{s} values of blood flow observed experimentally in 1cm^{3} samples of dog lung. The use of Θ and r_{L} to produce tree structures with the same β_{3} values as observed for vascular cast data simulated the spatial distribution, RD%, andD _{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 Θ >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 reasonableD _{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.
Acknowledgments
This work was supported by GrantinAid 94013094 from the American Heart Association. J. L. Ardell is an Established Investigator of the American Heart Association.
Footnotes

Address for reprint request: J. C. Parker, Dept. of Physiology, MSB 3024, College of Medicine, University of South Alabama, Mobile, AL 36688.
 Copyright © 1997 the American Physiological Society