Abstract
Huang, W., R. T. Yen, M. McLaurine, and G. Bledsoe.Morphometry of the human pulmonary vasculature.J. Appl. Physiol. 81(5): 2123–2133, 1996.—The morphometric data on the branching pattern and vascular geometry of the human pulmonary arterial and venous trees are presented. Arterial and venous casts were prepared by the silicone elastomer casting method. Three recent innovations are used to describe the vascular geometry: the diameterdefined Strahler ordering model is used to assign branching orders, the connectivity matrix is used to describe the connection of blood vessels from one order to another, and a distinction between vessel segments and vessel elements is used to express the seriesparallel feature of the pulmonary vessels. A total of 15 orders of arteries were found between the main pulmonary artery and the capillaries in the left lung and a total of 15 orders of veins between the capillaries and the left atrium in the right lung. The elemental and segmental data are presented. The morphometric data are then used to compute the total crosssectional areas, blood volumes, and fractal dimensions in the pulmonary arterial and venous trees.
 pulmonary artery
 pulmonary vein
 diameterdefined Strahler system
 connectivity matrix
 vessel element
 vessel segment
 fractal dimension
the early history of the morphometry of pulmonary vasculature has been summarized by Miller (20). In 1963, Weibel (26) published his monumental work. Cumming et al. (1), Singhal et al. (22), Horsfield (79), and Horsfield and Gorden (11) then used Strahler’s (25) system to study the morphology of the human pulmonary arterial and venous trees. Using resin casts and vascular injections of human vascular trees, they measured the diameter, length, and order of all branches of blood vessels in the range of 13 μm–3 cm for arterial vessels and 13 μm–1.4 cm for venous vessels. Using the silicone elastomer casting method, Yen and Sobin (28) reported the diameter data in the range of 18.7–1,785 μm for human pulmonary arteries and 18.9–58.9 μm for human pulmonary veins; similar results were reported for the cat (29, 30). However, major gaps in knowledge remain, and a more complete set of morphometric data for the human lung is needed for medical applications.
Horsfield (79) encountered a number of difficulties with Strahler’s idea and introduced several improvements. In detail, there are already three versions of Strahler’s scheme (8). Yet, some major difficulties remain: 1) all vessels of the same order are treated as parallel, despite the fact that some are connected in series, and the seriesparallel feature is not given a quantitative expression;2) the range of diameters of the vessels in successive orders has extremely wide overlaps;3) the connectivity of asymmetric branching has not found a mathematical expression; and4) the long main pulmonary artery, which is tapered, has to be given a single order number. Obviously, these difficulties would create dilemmas when one applies morphometric data to hemodynamic circuits. Recently, three innovations (1214) were introduced to ameliorate these difficulties:1) a new criterion based on the vessel diameter change at points of bifurcation was adopted in Strahler’s ordering system; 2) the concept of segment and element was used to express the seriesparallel feature of blood vessels; and 3) the connectivity matrix was introduced to describe the connectivity of blood vessels among different orders. The details of these three innovations have been discussed by Jiang et al. (12) in their study of the rat pulmonary arterial tree. The new method is called the diameterdefined Strahler ordering system. Gan et al. (5) used the new system to describe the dog pulmonary venous tree.
The objective of this study is to describe the morphometry of human pulmonary vasculature by includinginnovations 1–3. The data are intended to provide a basis for formulation of hemodynamic circuits for the analysis of blood flow in the human lung. Although hemodynamic analysis is not presented, the data should help interpret clinical observations. Much literature is available on clinical investigations of the human lung. Additional theoretical analysis supported by morphometric data to the clinical researchers’ chest of tools will obviously be helpful.
METHODS
Specimen Preparation
This study was carried out on two postmortem human lungs (Table1). In both cases, the cause of death was accidental and did not involve the lung. The pulmonary arterial cast was obtained by antegrade perfusion in the left lung of a 44yrold man, and the pulmonary venous cast was obtained by retrograde perfusion in the right lung of a 24yrold man with the silicone elastomer casting technique, which was introduced by Sobin (23). This technique has been used in the study of the morphology of the pulmonary vascular trees of cats (24, 29, 30), dogs (5), and rats (12). The silicone elastomer used in the present study was the same as that used in the earlier studies; it is a clear and colorless fluid that can pass through the capillary bed (24) and has a low viscosity (4), a low surface tension (24), and a negligible volume change on catalysis (24). It is also nontoxic to the endothelium (23).
The casting procedure is briefly described below. The lung was placed in cold saline solution during cannulation and perfusion. The trachea was cannulated after gentle suction to remove retained secretions. The airway pressure was then held constant at 10 cmH_{2}O above the pleural pressure (Ppl), which was atmospheric. The pulmonary artery and pulmonary veins were cannulated. The lung was initially inflated to 20 cmH_{2}O and cyclically inflated from 5 to 15 cmH_{2}O with periodic inflation to remove areas of superficial atelectasis. After the lung was well inflated, a small amount of noncatalyzed fluid silicone elastomer with a low (20 cP) viscosity (Microfil CP101, Flow Tech, Boulder, CO) was perfused from the pulmonary artery to the pulmonary veins to establish vascular continuity across the lung. This step was followed by perfusion with silicone elastomer freshly catalyzed with 3% tin octoate (stannous 2ethyl hexoate) and 5% ethyl silicate. Before perfusion the catalyzed solution was well stirred, and no air bubbles remained. The perfusion was carried out under a pressure drop of 34 cmH_{2}O from inlet (34 cmH_{2}O) to outlet (0 cmH_{2}O) for 20 min, while the alveolar gas pressure was maintained at 10 cmH_{2}O and Ppl was zero (atmospheric). Then the perfusion pressure was lowered and maintained at 3 cmH_{2}O, and the left atrial cannula was closed. The time course of hardening and the flow behavior of the catalyzed silicone elastomer in the first 2 h are discussed in detail by Fung et al. (Appendix in Ref. 4). The hardening was so slow within 1 h after the flow stopped that it was possible for the fluid to redistribute itself with the requirement of equilibrium (4). Because the fluid pressure in the capillary blood vessels was lower than the alveolar gas pressure under this condition, all the capillary vessels were collapsed (4, 29, 30), separating the arterial and venous trees. After 3 h the cast lung was moved to a refrigerator and frozen for 2 wk to increase the strength of the silicone rubber. Then the lung was carefully removed and suspended in a 10% KOH solution for 2 wk to dissolve the lung tissue. Next, the cast was washed several times with water to remove any remaining tissue. The pulmonary arterial and venous trees were gently separated. Dimensional measurements were carried out on the pulmonary arterial cast of a left lung and the pulmonary venous cast of a right lung.
Our method of preparation relies on two facts:1) the pulmonary capillaries collapse when the pulmonary capillary blood pressure is lower than the alveolar gas pressure by ≥1 cmH_{2}O (27); and 2) the pulmonary arteries and veins do not collapse when the pulmonary blood pressure falls below the alveolar gas pressure (4). In fact, according to Yen and Foppiano (27), for the cat, the slope of the vessel diameterpressure difference ΔP = Pv − Pa (where Pv is the blood pressure and Pa is the alveolar gas pressure) does not change in the range of −10 to +10 cmH_{2}O. The slope of the normalized vessel diameterD/D
_{10}(vessel diameter divided by diameter at Pv − Ppl = 10 cmH_{2}O) depends somewhat on the Ppl and vessel diameter. The relationship can be expressed as
Morphometric Measurement of the Polymer Cast of the Vasculature
The pulmonary vascular casts were dissected and viewed with a zoom stereomicroscope (model SZH, Olympus). An imageanalysis system was set up to measure accurately the size of the vessels. The system consists of a Zenith computer with a DT2851 (Data Translation, Marlborough, MA), an inverted light microscope (model SZHILLB, Olympus), a video monitor (Sony Trinitron color video monitor), and a television camera (Cohu solidstate camera). The solid casts were viewed with the inverted light microscope and displayed on the video monitor through the television camera. The image was analyzed with the software package Optimas (BioScan). The Optimas computing program focuses on the image of a blood vessel chosen by the operator. By photo density contrast, the computer draws the boundary contours of the object. For diameter measurement, the program computes normal vectors to the contour, draws two neighboring normals to define an area, measures the area, and computes a width equal to the area divided by the length between normals. We use the word “diameter” to indicate the computed width of the vessel. Three diameter measurements were made along each vessel to obtain a mean diameter. A section normal to the vessel contour can be drawn on the screen. The centers of the normal sections are joined by the operator, and the line is considered to be the centerline of the vessel. This centerline is that of the twodimensional image. The intersection of the centerlines of two intersecting vessels is the bifurcation point. The vessel segmental length was obtained by measuring the length between two successive bifurcation points along the centerline of a vessel on the twodimensional image. Tacitly, we assumed that the blood vessels were round. Actually, the cross sections of the large pulmonary veins were found to be noncircular, but in the present study this matter was not pursued. An analysis of the “errors” caused by these projections is made by Yen et al. (30). They found that if the diameter of a blood vessel with an elliptical cross section was measured by projection from arbitrary directions, the mean diameter of a random sampling of the projected width is quite close to the diameter of a circular cylinder of the same circumference.
Because there are many branches in the human pulmonary arterial and venous trees, it was impossible to examine, measure, count, and list every branch. Therefore, pruning and statistical methods were used to obtain representative measurements (5, 1215, 29, 30). The backbone of the left pulmonary artery was sketched, and its segments were measured. The subtrees arising from the backbone were labeled, excised, and placed in separate dishes. To facilitate the measurement, daughter trees with a diameter of 600–800 μm were trimmed from each subtree. Daughter trees were randomly selected as statistical samples from each subtree and measured in detail. In the statistical samples of the daughter trees, branches with a diameter <100 μm were pruned. A small number of the branches with a diameter <100 μm from each lobe of the lung were randomly selected as small sample trees and measured in detail. This process was continued until the entire tree cast was sketched and the morphometric measurements were made. The same process was used to obtain measurements in the venous tree. With the morphometric data on the vascular geometry and branching pattern from the backbones, subtrees, daughter trees, and small sample trees, the left pulmonary arterial tree and the right venous tree were reconstructed.
Data Analysis
Assignment of order number to branches of the pulmonary vascular tree according to diameter.
We use the diameterdefined Strahler ordering system to describe the branching pattern of the pulmonary arterial and venous trees. This system modifies Horsfield’s Strahler system with the addition of a step to be described below.
Let us first explain the original Strahler ordering system. In the Strahler ordering system, the smallest noncapillary blood vessel is defined as of order 1. When two vessels of the same order meet, the order number of the confluent vessel is increased by 1. When a vessel of order n meets another vessel of order < n, the order number of the confluent vessel remains n. Strahler’s ordering system deals with the asymmetric bifurcations nicely; however, in the studies of the human and cat pulmonary vasculature, investigators (1, 7, 11, 22, 29, 30) found very large overlaps of the diameters in the successive orders of vessels by Strahler’s ordering system caused by the rather indiscriminating assignment of order numbers in very large trees. Additionally, other difficulties mentioned in the introduction were encountered.
To ameliorate these difficulties, Kassab et al. (14, 15) developed the diameterdefined Strahler ordering system. In this system, a new rule is added: when a vessel of order nmeets another vessel of order ≤ n, the confluent vessel is called of ordern + 1 only if its diameter is larger by a certain amount, which is determined by the statistical distribution of the diameters of each order, as discussed below. Figure1 shows a scheme of the diameterdefined Strahler ordering system.
Strahler’s system is applicable only to treelike structures. Pulmonary capillary blood vessels are not treelike in topology. Fung and Sobin (3) and Sobin et al. (24) showed that the pulmonary capillaries can be described by a sheetflow model. We designate an order number of 0 to the pulmonary capillaries. Strahler’s system begins with arterioles and venules. The smallest arterioles supplying the capillaries are assigned an order number of 1. The smallest venules draining the capillaries are assigned an order number of −1. Positive integers identify arteries; negative integers identify veins. This system has been used by Kassab et al. (14) for the pig coronary system. Use of positive and negative integers to differentiate arterial and venous trees is convenient in morphometry.
The mean and standard deviation of the diameters of the pulmonary arteries of an arbitrary order n are denoted by D_{n}
and SD_{n}, respectively. There are vessels of all sizes between
A process of iteration is used to determine the order number of the vessels. In this study the pulmonary vascular trees initially were assigned by the diameter ranges of orders 1–3 from Yen and Sobin (28) according to Strahler’s ordering system. Yen and Sobin measured the diameters and branching orders of the pulmonary microvasculatures with diameters <100 μm from the histological preparations of postmortem human lung prepared by perfusion with a silicone elastomer. Then, by the same Horsfield method, the order numbers of the vessels of orders 4–15 in the pulmonary arterial and venous trees were assigned. The mean and standard deviation of the diameters were calculated.
Once the initial mean and standard deviation of the diameters are obtained, we use Eqs. 2 and 3 to decide the order number of every vessel. Some changes will occur in the order number of some vessels in this process. After the change, new values ofD_{n} and SD_{n} are computed and used to reset the diameter ranges for order n. The process is repeated until the changes ofD_{n} and SD_{n} between successive iterations became <1%, at which convergence is considered achieved. By using the diameter criteria, the final diameter ranges of successive orders do not overlap and standard deviations are relatively small (Table2).
Vessel segment and vessel element.
In the first use of Strahler’s ordering system by Horsfield, no distinction is made between series and parallel vessels of the same order, nor is it possible to express the seriesparallel features in the circuit representing the pulmonary vasculature. To obtain a correct circuit model, Kassab et al. (14, 15) defined every vessel between two successive bifurcation points as a segment and combined those vessels of the same order connected in series as an element. Statistical data are obtained for elements and segments. Flow circuits are built of elements.
In this study we used the same terminology. The measurement of diameter and length were made in the twodimensional grabbing images. We found that the diameter of each segment is constant, and we measured it at the midpoint of each segment. The segment length is the distance between bifurcation points along the centerline of the segment. The diameter of an element was computed as the average of the diameters of the segments that make up the element. The length of an element was obtained by adding the lengths of the segments within the element. The relationship between the total number of segments of ordern, S(n), and the total number of elements of the same order, E(n), can be described by
Connectivity matrix.
We used the connectivity matrix to express how blood vessels of one order are connected to vessels of another order. Blood vessels of ordern not only arise from vessels of ordern + 1 but also originate from vessels of order n,n + 2,n + 3,.... There was no quantitative expression for the connectivity feature of blood vessels of one order to another until the connectivity matrix was first developed and used by Kassab et al. (14, 15). In the connectivity matrix, each component in the mth row and thenth column, designated as C(m,n), is expressed as mean ± SE. The mean value is the ratio of the total number of elements of order m sprung from parent elements of order n divided by the total number of elements of order n. The standard error of C(m,n),
In addition to expressing the branching pattern for the whole vascular tree, the connectivity matrix was used to calculate the total number of elements in each order in this study. The elements of ordern, n− 1,...,1 spring directly from the elements of ordern. Therefore, when the number of elements of order n,N_{n}
, is known, the total numbers of elements of ordern, n−1,..., are C(n,n)N_{n}
, C(n − 1,n)N_{n}
,..., respectively. Considering all the vessels in a tree, we see that the total number of elements of order m is given by
Many branches are missing, because the tree was pruned for practical counting or broken off, but the stubs were recognizable. The number of missing branches must be considered while the total number of elements is counted. The extrapolated number of the elements of each order in the missing subtrees is calculated from the number of cutoff and brokenoff subtrees and the connectivity matrix by the following equation
The number of cutoff elements in ordern,
RESULTS
This study shows that there are 15 orders of pulmonary arteries between the main pulmonary artery and the capillaries in the left lung of one man and 15 orders of veins between the capillaries and the left atrium in the right lung of another man.
The mean and standard deviation of the diameters and lengths of the elements in each order are listed in Table 2. Two significant digits after the decimal point for diameters and lengths are justifiable in Table 2. The diameters of the elements for arterial and venous trees are plotted in logarithmic scale against the order number in Fig.2. Straight regression lines were determined by the least squares method. Ify represents the logarithm of the diameter and x represents the order number, the regression lines for pulmonary arterial and venous trees are y = −1.84 + 0.19x andy = −1.74 + 0.20x, respectively. The antilog of the slope gives an average diameter ratio of 1.56 for all pulmonary arteries and 1.58 for all veins. The ratio of the diameter of the elements of order n to that of ordern + 1 is called the diameter ratio. The relationship between the logarithm of element length and the order number is depicted in Fig. 3. Similarly, the least squares fit regression line for pulmonary arteries isy = −0.95 + 0.17x and that for veins isy = −0.79 + 0.18x. The antilog of the slope yields an average length ratio of 1.49 for all pulmonary arteries and 1.50 for all veins. The length ratio is defined as a ratio of the length of the elements of order n to that ofn + 1.
The connectivity matrices of pulmonary arteries and veins are presented in Tables 3 and4, respectively. The corrected total number of elements of each order in the pulmonary arterial tree of the left lung of one man and in the pulmonary venous tree of the right lung of another man is computed from the connectivity matrices given in Tables3 and 4 starting from order 15 for arteries and veins and extrapolating downward. The results are shown in Table 5. The total number of intact elements and the total number of cutoff subtrees in each order are listed in columns 2 and 3, respectively, of Table 5. The values in column 4were computed by Eq. 7 . The last column shows the corrected total number of elements in each order, which is the sum of numbers in the previous three columns of the given order. If the logarithm of the corrected total number of elements (y) in each order is plotted against the order number (x) as displayed in Fig. 4, the relationship between the corrected total number of elements and the order number is given by a regression line y = 8.41 − 0.53x for pulmonary arteries andy = 7.66 − 0.52x for pulmonary veins. The antilog of the absolute value of slope yields an average branching ratio of 3.36 for all orders of pulmonary arteries and 3.33 for veins. The branching ratio is a ratio of the corrected total number of elements of order n to that of ordern + 1.
Data on the segmental diameter, length, and segmenttoelement ratio of each order in the pulmonary arterial and venous trees are listed in Table 6. Two digits after the decimal point in Table 6 are significant.
The average crosssectional area of vessel elements of ordern,a_{n}
, is calculated as
The average blood volume of vessel elements in ordern, v_{n}, is calculated as
DISCUSSION
The morphometric data of pulmonary vascular trees of the cat, dog, rat, and human are now available in various degrees of completeness. Table8 summarizes the total order number, the mean diameter of order 1 vessels, the diameter ratio, the length ratio, and the branching ratio of the pulmonary vascular trees of these animals. Fung (2) and Zhuang et al. (31) demonstrated the application of morphometric data in pulmonary hemodynamics. The data of the cat (29, 30) and human (7, 11, 22) by Horsfield, Singhal, and their colleagues were obtained by Strahler’s ordering system, whereas the other data were obtained by the diameterdefined Strahler system. The connectivity matrix is defined only in the latter system, for reasons to be explained.
Cumming and Horsfield (1) pioneered the use of Strahler’s ordering system in the morphometry of the lung. The first set of data on the human pulmonary arterial tree was published by Singhal et al. (22). Horsfield (7) and Horsfield and Gorden (11) amplified the data and used them to analyze pulmonary circulation.
The diameterdefined Strahler system used here modifies Horsfield’s Strahler system by adding a diameter judgment at the junction where two vessels meet to become one confluent vessel. Horsfield’s rule is to increase the order number of the confluent vessel by 1 indiscriminately. Our rule is to increase the order number of the confluent vessel by 1 if and only if the diameter of the confluence is greater than either of the two converging vessels by a certain amount specified by Eqs. 2 and 3 .
The first evident difference caused by this modification is that we now have a reasonable and systematic way to handle the most important pulmonary artery: the long tapered main artery from the pulmonary valve to the periphery. As we know, the most important formula for blood flow is that of Poiseuille. Poiseuille’s formula states that the flow rate of a Newtonian fluid, with coefficient of viscosity μ, in a circular cylindrical tube of lumen diameter Dand length L is related to a pressure drop ΔP. D is in the fourth power. A 10% error in D leads to a 46.4% error in flow (Q˙), whereas a 10% error in any other parameter, such as ΔP, μ, and L, leads to only a 10% error in Q˙. Obviously, among all parameters on the Poiseuille equation,D is the most demanding for accuracy of Q˙. In Horsfield’s way, the main pulmonary artery has to be designated with one order number. Because in Horsfield’s morphometry one vessel order is given one diameter and the main pulmonary artery has a fixed diameter, despite the tapered phenomenon of this vessel, it is difficult to account for the taper in hemodynamics regardless of how the diameter is chosen. In our system the order number increases with diameter in a systematic way, the long tapered vessel is divided into successive orders in a manner consistent with hemodynamics, and the application of Poiseuille’s formula (modified with proper Reynolds number and Womersley number effects) yields the correct hemodynamic formula automatically with regard to the changes of vessel diameter. Jiang et al. (12) provide a detailed discussion about handling of tapered vessels.
The second evident difference caused by this modification is the reduction of the standard deviation of the diameter of vessels of each order from a Horsfield value to ours; statistically this is achieved by eliminating the overlaps in the ranges of the diameters of vessels of successive orders. Jiang et al. (12) compared the statistical distributions of the diameters of each order in these two systems. The standard deviation of the diameter characterizes the dispersion in geometric size of the vessels, and it influences significantly the dispersion of blood flow in the lung (significant because diameter enters Poiseuille’s flow formula in the 4th power). The dispersion of flow is related to the heterogeneity of oxygen transport in the lung. Hence, the standard deviation of diameter is an important physiological parameter. We believe that the large standard deviation of diameters obtained by Horsfield’s method is an artifact caused by a definition of order without regard to the size of the vessels. This is obvious from the three pairs of inset boxes in which the differences of the two definitions are shown in Fig. 1. According to Horsfield’s Strahler ordering method, in two cases the vessels became larger but the order number did not increase. In the third case, the vessel remained the same size, but the order number increased. With Horsfield’s Strahler ordering method, it is difficult to account for the hemodynamics in these vessels when Poiseuille’s formula is applied.
The third evident difference of the two definitions is the way the lengths of vessels are computed. In our system the vessels of the same order connected in series are considered as one element, the length of which is the sum of the lengths of the vessels in series. In Horsfield’s method all vessels are considered parallel. The distinction of segment and element is an important step in the construction of analog hemodynamic circuits, because vessels connected in series and in parallel differ tremendously in resistance. The inverse of the length of a vessel equivalent ton vessels in parallel is equal to the sum of the inverse of the length of each vessel. This is evident in an analog electric circuit. Each segment is a resistor. If two resistors of resistance R each are connected in series, the total resistance is 2R. If they are connected in parallel, the total resistance isR/2. Hence, if several segments are connected in series into elements, they must not be mistaken as in parallel. In hemodynamics it is important to determine whether then vessels are connected in series or in parallel.
Hence, Horsfield’s method seems to emphasize the connectivity to the degree of ignoring diameter change at the points of connection, yet it ignores the connectivity by treating all vessels of the same order as if they are all parallel. The range overlap of the diameter and the large standard deviation are consequences of Horsfield’s basic definition of vessel order number. We believe that it is logical to introduce a modification of his definition. In 1991, Horsfield (9) recognized that all vessel segments of the same order are not all in parallel and suggested a “Strahler method stage 2,” which considers the segments of the same order in series if a tapering vessel is intersected only by the smaller branches; however, no data were reported.
How the connectivity feature among vessels of different order is described is very important for hemodynamic modeling. The importance has been extensively discussed by Jiang et al. (12). In 1971, Horsfield (810) developed a method of “delta” to describe the connectivity and asymmetry in a bronchial tree, where delta is a number designed to show the difference in the size of two vessels. A map and statistics of delta are needed. However, no data have been reported. In this study, we use the connectivity matrix to describe the connectivity of blood vessels from one order to another and to calculate the total element number for each order.
Our definition of order 1 vessels is different from Horsfield’s. We define order 1 vessels as the smallest noncapillary vessels (28). Yen and Sobin (28) showed the relationship between the capillary bed and the first several orders of arterioles and venules in detail. In Horsfield’s data (7), order 1 vessels are defined as those between 10 and 15 μm diameter that are first encountered going down any pathway. Horsfield’s order 2 vessels are equivalent to our order 1 vessels (28). According to our data, the total element number of order 1 arteries in the left lung is ∼5 × 10^{7}; if a similar number is assumed for the right lung, then the total number of elements of order 1 arteries for the whole lung is ∼1 × 10^{8}.
Because there is no way to obtain a complete set of human morphometric data on living persons, the use of postmortem material is the logical answer. In this study, the left lung was from a 44yrold man and the right lung from a 24yrold man. Unfortunately, we could not obtain a pair of lungs from either man. A study in a pair of lungs from one person would provide more information, but human lung specimens are extremely difficult to obtain. We hope for the opportunity to obtain more data to represent the human population.
A number of authors prefer to look at the lung structure as fractal. We do not think that the lung structure is fractal, because the total number of generations is only 15 and the order 1 vessels are of finite size, not infinitesimal. Nevertheless, we can use our morphometric data to compute the fractal dimension of human pulmonary vasculature (19) by the method of Nelson and Manchester (21), and the results are interesting in the interpretation of the overall structure of the lung. The fractal dimension (D
_{F}) is given by the formula (20)
The scope of this study is limited to morphometry. By itself, it is not sufficient for hemodynamics, for which data are also needed on the elasticity of pulmonary arteries and veins and the branching angles of blood vessels. Existing data on elasticity and branching angle are very sketchy and need systematic measurement. We expect the branching angle to be important where inertial force is significant, i.e., in large arteries and veins where the Reynolds and Womersley numbers are large. For smaller vessels, where the Reynolds and Womersley numbers are <1 and the viscous forces dominate, the branching angle is not expected to be a significant morphometric parameter. On the basis of the data of morphometry, including the connectivity matrix and the branching angle, and elasticity, the hemodynamic analyses will be done and compared with the theoretical models of Krenz et al. (16, 17) and explain the theory of Hakim et al. (6) on the regional distribution of pulmonary blood flow in humans.
Acknowledgments
Present address of W. Huang: Dept. of Bioengineering, University of California, San Diego, La Jolla, CA 920930412.
Footnotes

Address for reprint requests: M. R. T. Yen, Dept. of Biomedical Engineering, The University of Memphis, Memphis, TN 38152.
 Copyright © 1996 the American Physiological Society