Fractal
modeling
ROBB
of pulmonary
W. GLENNY
Department
AND
H. THOMAS
ROBERTSON
of Medicine, University of Washington, Seattle, Washington 98195
GLENNY, ROBB W., AND H. THOMAS ROBERTSONJNZC~CZZ modeling of pulmonary blood flow heterogeneity. J. Appl. Physiol. 70(3): 1024-1030, 1991.-The heterogeneity of pulmonary blood flow is not adequately described by gravitational forces alone. We investigated the flow distributions predicted by two fractally branching vascular models to determine how well such networks could explain the observed heterogeneity. The distribution of flow was modeled with a dichotomously branching tree in which the fraction of blood flow from the parent to the daughter branches was y and 1 - y repeatedly at each generation. In one model 7 was held constant throughout the network, and in the other model y varied about a mean of 0.5 with a standard deviation of 0. Both y and g were optimized in each model for the best fit to pulmonary blood flow data from experimental animals. The predicted relative dispersion of flow from the two model fractal networks produced an excellent fit to the observed data. These fractally branching models relate structure and function of the pulmonary vascular tree and provide a mechanism to describe the spatially correlated distribution of flow and the gravity-independent heterogeneity of blood flow. isogravitational;
blood flow distribution;
pulmonary
circulation
THE HETEROGENEITY of pulmonary blood flow distribution cannot be satisfactorily described by gravitational
influences alone (2, 4-6, 14, 15). While there remains a statistically significant contribution of the gravitational gradient to flow distribution, sections within isogravitational planes show heterogeneity of flow comparable to that observed in the whole lung (4). While this variability has traditionally been termed “random,” it is apparent from inspection of the flow distributions that regional flows are spatially correlated with neighboring regions tending to have similar magnitudes of flow. We have recently shown that the distribution of pulmonary blood flow, as measured by the relative dispersion (RD = standard deviation of blood flow/mean blood flow), is dependent on the scale of measurement. This observation suggests a fractal relationship between pulmonary blood flow and the scale of measurement (4). This relationship can be expressed analytically by the recursive definition RD(v) = RD(v,,)
u 1-Ds (1) ( V ref 1 where vrefis an arbitrary reference volume, v is a volume comprised of a number of contiguous vref units, and D, is the spatial fractal dimension. There is evidence to suggest that this observed relationship of flow distribution to piece size may be due to the fractal branching pattern of the pulmonary vascular tree. The anatomic branching pattern of the bronchial tree has been reanalyzed and determined to be fractal(9, 1024
blood flow heterogeneity
l
0161-7567/91
$1.50
19). A model of the pulmonary arterial tree based on a fractal structure was developed by Lefevre (11). He showed that this fractal model optimized the cost-function (energy-materials) relationship while closely approximating physiological and morphometric data. Van Beek et al. (18) have also developed fractal branching vascular models for the myocardium in which the radius and length of daughter branches are recursively defined by the parent branches. They concluded that the fractal nature of myocardial blood flow distribution can be explained by fractal vascular networks. The success of the models of van Beek et al. (18) as descriptors of myocardial blood flow distribution led us to apply two of their models to the distribution of blood flow within the lung and to compare the model predictions with observations from our previous measurements of the regional distribution of lung blood flow (4). MODELS
Constant asymmetry. This model is identical to model I of van Beek et al (18). The basic element is a dichotomously branching structure in which the distribution of blood flow between daughter branches is characterized by an asymmetry of flow parameter, 7. If flow in a parent branch is F,, the flow to one daughter branch is y F, and the flow to the other daughter branch is (1 - y) F, (Fig. 1, left). A y of 0.5 will produce equal flows in both daughter branches. As y diverges from 0.5, flow becomes more asymmetric. A second generation of branches is produced by appending this basic element to the end of each terminal branch (Fig. 1, right). The fraction of flow, y and (1 - y), distributed to the daughter branches remains constant at each bifurcation. The asymmetry of flow in a parent branch affects the flows in all subsequent generations down the vascular tree and the flow in each terminal branch can be determined (Fig. 1, right). A model of the pulmonary vascular tree can therefore be constructed with n bifurcations or generations and thus 2n terminal branches or perfused regions of lung. These terminal branches have discrete values of flow given by the equation l
l
F = +(l
- y)“-‘mFO
(2) after n number of generations. F, is the total flow into the vascular tree, k assumes integer values from 0 to n, and the frequency of each value is n!/[ k!( n - k)!] (18). Figure 2 shows two examples of the frequency distribution of flows for a y of 0.47 and 0.45 with n equal to 15. Fifteen generations produce a model with 32,768 terminal branches, which approximates the number of pieces in
Copyright 0 1991 the American Physiological
Society
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FRACTAL
‘y
l
MODELING
OF
PULMONARY
u
(1-y) ‘F.
FIG. 1. Dichotomously branching fractal model. Left: basic element in which fractions y and 1 - y of total flow F, are distributed to daughter branches. Right:flow at terminal branches in network of 2 generations.
each of the experimental data sets (4). The flows are presented as mean normalized flows where the mean flow after n generations is F,/2”. The flows generated by the network are similar to those obtained from experimental data in that they have a distribution which is skewed to the right. The flow distributions generated by this model will have some heterogeneity that can be characterized by the RD. An analytic solution for RD as a function of y and n has been obtained by van Beek et al. (18) and is given by the equation RD = \Jzn. [y2 + (1 - Y)~]” - 1
(3
Equation 3 can now be related to the fundamental fractal Eq. 1. The relative dispersion can be determined for any level of an n generation vascular tree. The RD after one branching is equivalent to partitioning the tree into two pieces. If the RD is determined after two branchings, the vascular tree is partitioned into four pieces. The volume of these pieces can be determined by arbitrarily selecting V ref equal to a single terminal branch. Because there are 2” terminal branches in an n generation tree, the size of the pieces (v) after one branching will be 2”/2 and the size of the pieces after two branchings will be 2”/4. Generalizing this for all of the branchings in the vascular model with n bifurcations, v = 2”/2’ for the ith generation where i assumes integer values from 1 to n and RD(v) = v2ia [y2 + (1 - r)“3’ - 1
(4)
Although Eq. 3 cannot be reduced to a fractal form, the relationship between the heterogeneity of flow from the
BLOOD
FLOW
1025
model (RD model) and the perfused piece size (v) appears to be fractal. Figure 3A shows the RDmodelas a function of piece size for a S-generation vascular tree with a y of 0.45. The individual points represent the relative dispersion calculated from Eq. 4 as a function of v. When plotted on a log-log scale after deleting the four largest pieces, the relationship appears linear, suggesting that it is fractal over the range of observations (4). The largest pieces are not used in the analysis because, as noted by van Beek and associates (18), they do not fit the theoretical fractal relationship of Eq. 1 as well as the smaller pieces. Equation 4 produces a slightly curved relationship between v and RD(v) while Eq. 1 predicts a linear relationship (18). The fractal dimension D,, defined by the slope of RD(v) vs. v, is theoretically constant over the range of observations. However, D, will gradually decrease as v decreases for the relationship of Eq. 4. To determine a single fractal dimension that best characterizes the distribution of flows, a least-squares linear fit to RD(v) is used to obtain a slope. The statistical certainty of RD(v) at large v values is low because the standard deviation is determined from a small sample size (only 2” observations after n bifurcations). The fractal dimension is therefore best defined by RD(v) for the smallest volumes. The decision to exclude the four largest pieces remains somewhat arbitrary but reflects the procedure used for estimating the fractal dimension of blood flow in experimental measurements on heart (1) and lung tissue (4). Random variation asymmetry. This model is identical to model III of van Beek et al. (18). The asymmetry of flow in the first model is the same at all bifurcations in the vascular network. This is not an accurate representation of the pulmonary vascular tree in that there is variability in the anatomical predicted asymmetry of flow (10). A more realistic model can be constructed by assigning a different y to each branch point. In this model the value of y is selected from a normal distribution with a mean 7 = 0.5 and a standard deviation 0. A vascular network with 0 = 0.0 produces a uniform distribution of flow to all branches with RD = 0.0. As CTdiverges from zero, so will RD. Unfortunately this model does not have an analytic solution (18). A computer simulation can be run that constructs a vascular network by randomly assigning y values with a standard deviation of 0 to each bifurcation and
B
0.20- A
15 generations y= 0.45 RD = 40.1%
15 generations
0.15 oh g 2 O.lO2 F4
y= 0.47
RD = 23.5%
I
FIG. 2. Frequency distributions of flow in network with 15 generations. Flows are normalized to mean flow.
0.050.00
0.0
-J--
1.0
2.0
3.0
Flow/Mean Flow
4.0
0.0
1.0
2.0
3.0
4.0
Flow/Mean Flow
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1026
FRACTAL
MODELING
OF
B
15 generations 0.45
-
1
2.0
.
1
.
4.0
1
.
6.0
I
1
‘*’
Ooo
In(u), Volume of Region
-
2*o
values are randomly chosen, the distribution of flow will be different each time the simulation is run. An average distribution of flow can be estimated by running the model several times. Figure 3B shows the average RD (after 10 runs) as a function of piece size for a E-generation vascular tree with a 7 = 0.5 and 0 = 0.05. When plotted on a log-log scale, the relationship again appears linear, suggesting that it is fractal. The distributions of flow are no longer discrete in this model. A pseudocontinuous distribution can be constructed by plotting frequency histograms of flow with narrow intervals. Frequency distributions of flow for two networks of 15 generations with 0 values of 0.03 and 0.05 are shown in Fig. 4. The flow distributions are skewed to the right and have RD values of 23.0 and 39.3%, respectively. To see how well these fractal vascular networks model the regional blood flow distribution in the lung, a fit of the relative dispersion predicted from the models can be compared to experimental observations. In the constant asymmetry model, the y values that produced the best fit of the modeled RD to the observed experimental RD were determined using Eq. 3 and a computer optimization algorithm (3, 18). The random variation asymmetry model was evaluated by running a computer simulation 10 times for a given value 0. The average RD at varying
0.08
1 i
w
0 = 0.03
RD = 23.0%
0.06 CT= 0.05
0.04
RD / = 39.7%
0.02 0.00
Flow/Mean
Flow
4. Frequency distributions of flow in random variation Simulated flows are generated only 1 time for each B. FIG.
, 4’o
-
I
-
‘*’
In(u), Volume of Region
determining the RD at each generation. Because the y
0.10
BLOOD
15 generations 0 = 0.05
y=
2.8! 0.0
PULMONARY
model.
I ‘*’
FLOW
FIG. 3. Relationship of relative dispersion (RD) to size of theoretical perfused region or generation level of the tree; 4 largest-pieces were excluded from fractal fit. A: log-log relationship of RD vs. v for network constructed with constant asymmetry. Line is least-squares linear fit to data and represents theoretical fractal relationship of Eq. 1. B: log-log relationship of RD vs. v for network constructed with random variation asymmetry with 7 = 0.5 and c = 0.025. Line again is least-squares linear fit to data and represents theoretical fractal relationship. SD bars represent variability in RD for each of 10 simulations .
piece sizes was determined for each G and compared with experimental data. The best 0 was found by minimizing the residual sum of squares between the experimental and modeled RD values. EXPERIMENTAL
METHODS
The experimental techniques have been described previously (4). Briefly, six dogs were anesthetized with intravenous Surital (thiamylal sodium) followed by LYchloralose-urethan, intubated, and mechanically ventilated on room air with a tidal volume of 15 ml/kg and rate sufficient to maintain a PCO, between 32 and 38 Torr. Then 25 mCi of ggmTc-labeled macroaggregated albumin (MAA) were rapidly injected via a central venous catheter while the dog was apneic at end expiration. A total of 2.24-4.5 X lo6 particles of MAA were injected. Ventilation of the dog was resumed 15-20 s after the injection. The animals were then deeply anesthetized, heparinized, and killed by exsangination. The endotracheal tube was clamped, the abdomen was entered, and the diaphragm was punctured. The recoil pressure of the lungs at functional residual capacity (FRC) was estimated by the increase in airway pressure when the diaphragm was punctured. A sternotomy was performed, the pulmonary artery and left atrium were plumbed with large bore catheters, and the aorta was tied off. After hyperinflation of the lungs, the airway pressure was set to the previously measured recoil pressure and the chest was closed with the vascular catheters exiting the thorax caudally above the diaphragm. The lungs were fixed by intravascular perfusion of glutaraldehyde and dehydrated with progressively increasing concentrations of ethyl alcohol. The lungs and heart were removed from the thorax, and the lungs were dissected free of the heart and dried for 12-18 h on continuous positive pressure equal to the previously measured recoil pressure at FRC. They were then embedded in rapidly setting urethane foam and cut in transverse H-mm slices. The individual lung slices were imaged on a planar gamma camera. A matrix size of 128 X 128 pixels was used, resulting in a voxel size of 1.48 X 1.48 X 11 mm. The square edges of the foam cast allowed accurate registration of the lung slices so that contiguous slices could be appropriately aligned. The pixel counts were proportional to the local ggmTc-labeled MAA determined blood
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FRACTAL
MODELING
OF
flow. The pixel values were transferred to an Apple Macintosh II computer where they were corrected for decay time and stored as a three-dimensional array. The most peripheral voxels and those adjacent to airways were excluded before numerical analysis. The heterogeneity or RD of the three-dimensional flow arrays was analyzed by initially subdividing the array along a sagittal plane that separated the right and left lungs and an RD was calculated for the mean flows of the two pieces. Each lung was then subdivided by transverse planes that produced equally sized “upper” and “lower” pieces and a relative dispersion was again calculated for the mean flows of the four pieces. The pieces were sequentially divided in half until each piece was a single voxel in size. As the pieces were subdivided, an RD was calculated at each step. An estimate of the RD due to measurement error was subtracted from the observed RD and a log-log plot of the corrected RD (in percent) as a function of the volume of pieces was constructed (4). A logarithmic fit to the data (excluding the first 4 points) was obtained using a least-squares approach. The spatial fractal dimension (D,) was determined from the slope of the regression line as defined by Eq. 1. STATISTICS
Data are given as means t SD. RD (which equals SD/ mean) characterized the flow distribution at various sample sizes. The goodness of fit of the model to the observed data was quantitated by the coefficient of variation (CV) cv
44
(5)
= CY
measured
n
is the experimental observations of RD, where Ymeaaured ymodelis the RD values from the model, n is the number of points, and df are the degrees of freedom, which is 2 for both models.
PULMONARY
BLOOD
1027
FLOW
metry and random variation asymmetry models is shown in Fig. 6. The solid lines of y = 0.452 and G = 0.0042 are the best fit to the experimental measurements of the second animal and represent the theoretical fractal relationship. The parameters y and c were varied about these values to demonstrate the sensitivity of the models to changes in these parameters. DISCUSSION
A useful model should have a plausible functional basis to explain the observations and should accurately predict those observations. An even better model fulfills these prerequisites while simultaneously generalizing and simplifying our understanding of the modeled system. Dichotomously branching fractal networks are simple yet powerful models that relate function and structure. They explain a number of observations concerning the heterogeneous and spatially correlated distribution of pulmonary blood flow. An assumption underlying the fitting of the experimental observations to the fractal model is that each piece of tissue (regardless of size) is supplied by a single terminal vessel. The validity of this assumption is not critical to the fractal analysis of flow distribution but its violation does have an effect on the estimated fractal dimension (4). If the pieces of tissue are not dissected along vascular boundaries, the boundaries imposed by the analysis will be “misregistered” from the true vascular partitions. This misregistration error will produce a smoothing effect, resulting in an underestimation of the true fractal dimension. The effect is small but will be more pronounced for more heterogenous flow distributions (4). The asymmetry of flow parameter y can be related to the radii and lengths of branches within the vascular tree and can be recursively defined from the parent branch. If the parent vessel has radius R and length L, the daughter vessels have radii scaled by ri and r2 and lengths that are scaled by I, and ZZof the parent dimensions. One daughter branch will have a radius of r1 R and a length of 1,. L, and the other daughter branch will have a radius of r2 R and a length of L, L. If the flow in the parent branch is F,, the flows in each daughter branch can be calculated from Poiseuille’s law. The flow to the first daughter branch, fi, is F, (/2 r:)l(Z, rt + 1, ri) and the flow to the other daughter branch, f2, is F, (1, ~-2)/( 1, rt + 1, ri). Van Beek et al. (18) stated that the fraction of flow to one daughter branch, previously defined as y, is equal to fi/(fi + f& and can be related to the radius and length scalars by the equation l
l
RESULTS
The fractal vascular models provide excellent fits to the observed distribution of pulmonary blood flow. Figure 5 is a representative fit of the models to the experimental \data. The four points representing the largest pieces were not included in the analysis of each animal because they did not fit the fractal relationship. The y values best characterizing RD(u) in the constant asymmetry model ranged from 0.447 to 0.468 with a mean for all animals of 0.459 t 0.009. The mean CV between this model and the experimental data was 0.069 t 0.038. The 0 values best characterizing RD(u) in the random variation model ranged from 0.033 to 0.057 with a mean for all animals of 0.041 t 0.01. The mean CV between this model and the experimental data was 0.070 t 0.042. The composite data for ail animals are presented in Table 1, showing the number of generations needed to fit the observed data, the y and 0 that best fit the experimental data, and the CV for each model and the experimental data. Sensitivity analysis for y and 0 in the constant asym-
l
l
l
l
l
l
l
l
l
(6) Successive branches have radii and lengths that are scaled by a constant fraction of the parent branches. The fractal nature of this model is apparent from the ratios of radii and lengths between parent and daughter branches that remain the same independent of scale. The asymmetry of flow value does not change, as it is a ratio of flows between parent and daughter branches. Although regional blood flow is determined by local resistance, this resistance is influenced by more factors
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1028
FRACTAL MODELING
OF PULMONARY
BLOOD FLOW
3.6 1 A o
l
0
.g 3.2 2 z 3.1pf z 30.
Experimental Modeled RD
8
RD
l
0 0
0
8
0
Q
15 generations o=o.O42
0 0
FIG. 5. Fit of modeled relative dispersion to experimental observations of 1 animal; 4 largest pieces were excluded from fit. A: constant asymmetry model. 23: random variation asymmetry model.
l 0
0 0
0 0
CV = 0.036 0
8
2.9 ‘1 0
I
I
12 In(u),
I
I
I
I
3
4
5
6
Volume
RD
0 0
0
15 generations y=o.451 cv = 0.035
Experimental Modeled RD
o
0
I
1
7
0
of Region
I
12 In(u),
I
I
3
4
Volume
than the radius and length of a vessel. Vasoactive factors, local alveolar volumes, and hydrostatic pressures affect the resistance in a given vessel and hence influence regional blood flow. The asymmetry parameter 7, should therefore be thought of as a general descriptor of flow distribution to daughter branches, which is determined by a number of factors. Blood flow within isogravitational planes is heterogeneous and spatially correlated (4-6, 14, 15). Nicolaysen and associates (14) suggested that this heterogeneity is a random process. The flow distributions in our fractal models are not influenced by gravitational factors. The asymmetry of flow at branches within the pulmonary arterial tree may therefore account for the heterogeneity of flow within isogravitational planes. These simple fractal networks also nicely explain the observation that pulmonary blood flow distribution is fractal and spatially correlated (4,6). A model of pulmonary blood flow distribution based solely on gravitational effects would not produce the fractal relationship we have observed. Blood flow distribution in the lung is spatially correlated in that high-flow regions tend to be near areas of high-flow and low-flow regions tend to be adjacent to other areas of low flow (4). This observation is accounted for by the dichotomously branching network in that flows to neighboring regions are more closely related (they share a parent or grandparent branch point) than are regions separated by a greater distance. The more closely related two regions
I
1
1
5
6
7
of Region
are in the vascular tree, the more likely their flows will be similar. These fractal networks obviously represent simplified models of the complex and dynamic nature of pulmonary blood flow, and as such they have a number of deficiencies. Morphometric analysis of the pulmonary arterial tree has shown that although the vast majority of branchings are dichotomous, there are occasional branch points with more than two daughter branches (7, 17). The asymmetry of branchings of the pulmonary arterial tree is not absolutely fixed as modeled by the first network. Measurements by Singhal et al. (17) have described the ratio of radii in daughter branches to range from 0.1 to 1.0 with a standard deviation of 0.20 in a given pulmonary arterial tree. The fractal models also have equal path lengths to all of the terminal branches which is unlike the pulmonary vascular tree (8). Horsfield (10) has analyzed morphometric data on the human lung to estimate the asymmetry of flow between daughter branches. Although he uses different nomenclature, it is possible to determined an estimate of 7 from his calculations. Using Poiseuille’s law and average measurements of daughter branch radii and lengths, Horsfield estimated the relative proportion of flows to be 0.635 and 0.365 to daughters of a parent vessel. If this estimate of 7 = 0.365 is used in the constant asymmetry model, the relative dispersion of flow after 15 generations would be 136.9%. This is considerably larger than
TABLE 1. Model parameters Constant Animal No.
Mean
Generation
1
14
2 3 4 5 6
14 15 14 15 14
-t SD
Asymmetry
0.468 0.457 0.447 0.464 0.451 0.468 0.459_+0.009
Asymmetry
Random Coefficient of variation
0.093 0.035 0.135 0.043 0.047 0.062 0.069+0.038
Standard deviation
0.033 0.042 0.057 0.036 0.048 0.033 0.041~0.010
Variation
Asymmetry Coefficient of variation
0.086 0.036 0.148 0.042 0.050 0.055 0.070+0.042
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FRACTAL
$4
.s
MODELING
OF
PULMONARY
BLOOD
1029
FLOW
3.6
-- ---, -&qo32
---- --
--Law ---,----I\ ---
2.8
! 0
1
1 2
3
1 4
‘i, 5
I 6
In(u), Volume of Region
4 7
I 0
1
2
--S ---
1 3
4
---- ----
Y
-1
=. -- ---.0 ---, -L ---, 5
I
1
6
7
FIG. 6. Sensitivity analysis of the parameters y and 0 in constant asymmetry (A) and random variation asymmetry (B) models. Solid lines, y = 0.452 and (T = 0.042, represent best fit to experimental data of 1 animal using 14 generations. y and 0 are varied about these values to demonstrate the sensitivity of variation in these parameters. Lines are least-squares fits to simulated data and represent theoretical fractal relationship of Eq. I.
In(u), Volume of Region
the relative dispersion measured in our experimental animals (mean RD = 35.4%) and would be even larger if the variability in y were considered. This sizable discrepancy may be due to the differences in experimental technique. Horsfield’s data were collected from casts of fully inflated saline-filled cadaver lungs in which the pulmonary arteries were injected with resin at a pressure of 40 cmH,O, and estimates of flow were calculated from these measurements. Our data are obtained from actual flow measurements from intact animals. A number of these differences may account for differences in our estimates of y, but it is intriguing to conjecture that within a fractal vascular structure there may be physiological mechanisms that regulate pulmonary blood flow toward homogeneity. Another interesting observation from the data of Horsfield et al. and from analysis of bronchial tree morphometric data by Nelson et al. (13) is that the first 3 generations of the trees do not conform to the fractal pattern of the rest of the lung (10,13). This suggests that the initial branchings of the pulmonary vasculature and bronchial tree are either not fractal or may have a different fractal dimension. Recent embryologic observations by Massoud and associates (12) have demonstrated that in fact there are two different branching patterns in the rat fetal lung, peripheral and central. The central branches of the pulmonary bronchial tree exhibit monopodial branching while the rest of the tree divides dichotomously. This does not mean that the initial branches of the bronchial tree are not fractal but rather they may follow a different fractal pattern of growth with a different fractal dimension. The constant asymmetry and random variation asymmetry models appear to describe the heterogeneity of pulmonary blood flow equally well (Fig. 5 and Table 1). The two models differ primarily in their spatial distribution of flow. The constant asymmetry model will have flows that are consistently biased to one side while the expected flows in the random variation asymmetry model will be equal distributed between both sides of the vascular tree. In this way, the random variation asymmetry model is a more realistic model. The constant asymmetry model has the advantages of being simple and having an analytic solution for the relative dispersion. Van Beek and associates (18) have presented two other related branching models. Their second model used a y
that decreased (asymmetry increased) with each generation by a fixed constant. This introduces another degree of freedom into the model, and optimization of this parameter will necessarily produce an equal or better fit to the observed data. This model uniformly fit their observed cardiac blood flow data better than the constant asymmetry or random variation asymmetry models. It should be noted that this model is no longer ideally fractal because the branching pattern is not constant across different scales of measurement. Of interest is that the mean optimized fixed constant was 0.993 and 0.996 for the two data sets they fit to the model, indicating that a constant y is a reasonable model. Because there are no morphometric studies to support a gradually diminishing y, we did not use this model. The other model they introduced differs from the random variation asymmetry model in that 0 decreases by one-half at each generation (18). This model did not fit their experimental data as well as the random variation asymmetry model. The fractal branching models are presented as concepts and are not meant to be a precise representation of anatomic structures. Rather we wish to emphasize the concept of how small degrees of asymmetry in flow can produce heterogeneous blood flow distributions similar to those seen in experimental studies. The precise value of 7 or 0 are of little importance relative to this idea. These models use the concepts of fractals to relate the function and structure of the pulmonary vascular tree and offer an explanation for the spatial distribution and gravity independent heterogeneity of blood flow. We acknowledge the valuable communications with Dr. Johannes van Beek, Dr. James Bassingthwaighte, and the personnel of the National Simulation Resource Facility for Mass Transport and Exchange (RR-01243) at the Center for Bioengineering, University of Washington. This work was supported by National Heart, Lung, and Blood Institute Grants HL-08155 and HL-38736 and Grant 89-WA-515 from the American Heart Association, Washington Afhliate. Address for reprint requests: R. W. Glenny, Div. of Pulmonary and Critical Care Medicine, Rm-12, Dept. of Medicine, University of Washington, Seattle, WA 98195. Received
11 June
1990; accepted
in final
form
11 September
1990.
REFERENCES 1. BASSINGTHWAIGHTE, nature of regional 65: 578-590, 1989.
J. B., R. B. KING, AND S. A. ROGER. Fractal myocardial blood flow heterogeneity. Circ. Res.
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FRACTAL
MODELING
OF
2. BECK, K. C., AND K. REHDER. Differences in regional vascular conductances in isolated dog lungs. J. Appl. Physiol. 61: 530~538,1986. 3. BEVINGTON, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969. 4. GLENNY, R. W., AND H. T. ROBERTSON. Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity. J. Appl. Physiol. 69: 532-545, 1990. 5. GREENLEAF, J. F., E. L. RITMAN, D. J. SASS, AND E. H. WOOD. Spatial distribution of pulmonary blood flow in dogs in left decubitus position. Am. J. Physiol. 227: 230-244, 1974. 6. HAKIM, T. S., R. LISBONA, AND G. W. DEAN. Gravity-independent inequality in pulmonary blood flow in humans. J. Appl. Physiol. 63: 1114-1121, 1987. 7. HORSFIELD, K. Morphometry of the small pulmonary arteries in man. Circ. Res. 42: 593-597, 1978. 8. HORSFIELD, K. Functional morphology of the pulmonary vasculature. In: Respiratory Physiology: An Analytic Approach, edited by H. K. Chang and M. Paiva. New York: Dekker, 1989, p. 499-531. (Lung Biol. Health Dis. Ser.) 9. HORSFIELD, K. Diameters, generations, and orders of branches in the bronchial tree. J. Appl. Physiol. 68: 457-461, 1990. 10. HORSFIELD, K., AND M. J. WOLDENBERG. Diameters and crosssectional areas of branches in the human pulmonary arterial tree. Anat. Rec. 223: 245-251, 1989. 11. LEFEVRE, J. Teleonomical optimization of a fractal model of the pulmonary arterial bed. Theor. Biol. 102: 225-248, 1983.
PULMONARY 12.
13.
14.
15.
16.
1%
18.
19.
BLOOD
FLOW
MASSOUD, E. A. S., A. ROTSCHILD, R. MATSUI, H. S. SEKHON, AND W. M. THURLBECK. In vitro airway branching morphogenesis of the fetal rat lung (Abstract). Am. Rev. Respir. Dis. 141: A340, 1990. NELSON, T. R., B. J. WEST, AND A. L. GOLDBERGER. The fractal lung: universal and species-related scaling patterns. Experientia BaseZ46: 251-254, 1990. NICOLAYSEN, G., J. SHEPARD, M. ONIZUKA, T. TANITA, R. S. HATTNER, AND N. C. STAUB. No gravity-independent gradient of blood flow distribution in dog lung. J. Appt. Physiol. 63: 540-545, 1987. REED, J. H., AND E. H. WOOD. Effect of body position on vertical distribution of pulmonary blood flow. J. Appl. Physiol. 28: 303-311, 1970. ROY, A. G., AND M. J. WOLDENGERG. A generalization of the optimal models of arterial branching. Bull. Math. Biol. 44: 349-360, 1982. SINGHAL, S., R. HENDERSON, K. HORSFIELD, K. HARDING, AND G. CUMMING. Morphometry of the human pulmonary arterial tree. Circ. Res. 33: 190-197, 1973. VAN BEEK, J. H. G. M., S. A. ROGER, AND J. B. BASSINGTHWAIGHTE. Regional myocardial flow heterogeneity explained with fractal networks. Am. J. Physiol. 257 (Heart Circ. Physiol. 26): H1670-H1680, 1989. WEST, B. J., V. BHARAGAVA, AND A. L. GOLDBERGER. Beyond the principle of simultude: renormalization of the bronchial tree. J. Appl. Physiol. 60: 1089-1097, 1986.
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modeling
ROBB
of pulmonary
W. GLENNY
Department
AND
H. THOMAS
ROBERTSON
of Medicine, University of Washington, Seattle, Washington 98195
GLENNY, ROBB W., AND H. THOMAS ROBERTSONJNZC~CZZ modeling of pulmonary blood flow heterogeneity. J. Appl. Physiol. 70(3): 1024-1030, 1991.-The heterogeneity of pulmonary blood flow is not adequately described by gravitational forces alone. We investigated the flow distributions predicted by two fractally branching vascular models to determine how well such networks could explain the observed heterogeneity. The distribution of flow was modeled with a dichotomously branching tree in which the fraction of blood flow from the parent to the daughter branches was y and 1 - y repeatedly at each generation. In one model 7 was held constant throughout the network, and in the other model y varied about a mean of 0.5 with a standard deviation of 0. Both y and g were optimized in each model for the best fit to pulmonary blood flow data from experimental animals. The predicted relative dispersion of flow from the two model fractal networks produced an excellent fit to the observed data. These fractally branching models relate structure and function of the pulmonary vascular tree and provide a mechanism to describe the spatially correlated distribution of flow and the gravity-independent heterogeneity of blood flow. isogravitational;
blood flow distribution;
pulmonary
circulation
THE HETEROGENEITY of pulmonary blood flow distribution cannot be satisfactorily described by gravitational
influences alone (2, 4-6, 14, 15). While there remains a statistically significant contribution of the gravitational gradient to flow distribution, sections within isogravitational planes show heterogeneity of flow comparable to that observed in the whole lung (4). While this variability has traditionally been termed “random,” it is apparent from inspection of the flow distributions that regional flows are spatially correlated with neighboring regions tending to have similar magnitudes of flow. We have recently shown that the distribution of pulmonary blood flow, as measured by the relative dispersion (RD = standard deviation of blood flow/mean blood flow), is dependent on the scale of measurement. This observation suggests a fractal relationship between pulmonary blood flow and the scale of measurement (4). This relationship can be expressed analytically by the recursive definition RD(v) = RD(v,,)
u 1-Ds (1) ( V ref 1 where vrefis an arbitrary reference volume, v is a volume comprised of a number of contiguous vref units, and D, is the spatial fractal dimension. There is evidence to suggest that this observed relationship of flow distribution to piece size may be due to the fractal branching pattern of the pulmonary vascular tree. The anatomic branching pattern of the bronchial tree has been reanalyzed and determined to be fractal(9, 1024
blood flow heterogeneity
l
0161-7567/91
$1.50
19). A model of the pulmonary arterial tree based on a fractal structure was developed by Lefevre (11). He showed that this fractal model optimized the cost-function (energy-materials) relationship while closely approximating physiological and morphometric data. Van Beek et al. (18) have also developed fractal branching vascular models for the myocardium in which the radius and length of daughter branches are recursively defined by the parent branches. They concluded that the fractal nature of myocardial blood flow distribution can be explained by fractal vascular networks. The success of the models of van Beek et al. (18) as descriptors of myocardial blood flow distribution led us to apply two of their models to the distribution of blood flow within the lung and to compare the model predictions with observations from our previous measurements of the regional distribution of lung blood flow (4). MODELS
Constant asymmetry. This model is identical to model I of van Beek et al (18). The basic element is a dichotomously branching structure in which the distribution of blood flow between daughter branches is characterized by an asymmetry of flow parameter, 7. If flow in a parent branch is F,, the flow to one daughter branch is y F, and the flow to the other daughter branch is (1 - y) F, (Fig. 1, left). A y of 0.5 will produce equal flows in both daughter branches. As y diverges from 0.5, flow becomes more asymmetric. A second generation of branches is produced by appending this basic element to the end of each terminal branch (Fig. 1, right). The fraction of flow, y and (1 - y), distributed to the daughter branches remains constant at each bifurcation. The asymmetry of flow in a parent branch affects the flows in all subsequent generations down the vascular tree and the flow in each terminal branch can be determined (Fig. 1, right). A model of the pulmonary vascular tree can therefore be constructed with n bifurcations or generations and thus 2n terminal branches or perfused regions of lung. These terminal branches have discrete values of flow given by the equation l
l
F = +(l
- y)“-‘mFO
(2) after n number of generations. F, is the total flow into the vascular tree, k assumes integer values from 0 to n, and the frequency of each value is n!/[ k!( n - k)!] (18). Figure 2 shows two examples of the frequency distribution of flows for a y of 0.47 and 0.45 with n equal to 15. Fifteen generations produce a model with 32,768 terminal branches, which approximates the number of pieces in
Copyright 0 1991 the American Physiological
Society
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FRACTAL
‘y
l
MODELING
OF
PULMONARY
u
(1-y) ‘F.
FIG. 1. Dichotomously branching fractal model. Left: basic element in which fractions y and 1 - y of total flow F, are distributed to daughter branches. Right:flow at terminal branches in network of 2 generations.
each of the experimental data sets (4). The flows are presented as mean normalized flows where the mean flow after n generations is F,/2”. The flows generated by the network are similar to those obtained from experimental data in that they have a distribution which is skewed to the right. The flow distributions generated by this model will have some heterogeneity that can be characterized by the RD. An analytic solution for RD as a function of y and n has been obtained by van Beek et al. (18) and is given by the equation RD = \Jzn. [y2 + (1 - Y)~]” - 1
(3
Equation 3 can now be related to the fundamental fractal Eq. 1. The relative dispersion can be determined for any level of an n generation vascular tree. The RD after one branching is equivalent to partitioning the tree into two pieces. If the RD is determined after two branchings, the vascular tree is partitioned into four pieces. The volume of these pieces can be determined by arbitrarily selecting V ref equal to a single terminal branch. Because there are 2” terminal branches in an n generation tree, the size of the pieces (v) after one branching will be 2”/2 and the size of the pieces after two branchings will be 2”/4. Generalizing this for all of the branchings in the vascular model with n bifurcations, v = 2”/2’ for the ith generation where i assumes integer values from 1 to n and RD(v) = v2ia [y2 + (1 - r)“3’ - 1
(4)
Although Eq. 3 cannot be reduced to a fractal form, the relationship between the heterogeneity of flow from the
BLOOD
FLOW
1025
model (RD model) and the perfused piece size (v) appears to be fractal. Figure 3A shows the RDmodelas a function of piece size for a S-generation vascular tree with a y of 0.45. The individual points represent the relative dispersion calculated from Eq. 4 as a function of v. When plotted on a log-log scale after deleting the four largest pieces, the relationship appears linear, suggesting that it is fractal over the range of observations (4). The largest pieces are not used in the analysis because, as noted by van Beek and associates (18), they do not fit the theoretical fractal relationship of Eq. 1 as well as the smaller pieces. Equation 4 produces a slightly curved relationship between v and RD(v) while Eq. 1 predicts a linear relationship (18). The fractal dimension D,, defined by the slope of RD(v) vs. v, is theoretically constant over the range of observations. However, D, will gradually decrease as v decreases for the relationship of Eq. 4. To determine a single fractal dimension that best characterizes the distribution of flows, a least-squares linear fit to RD(v) is used to obtain a slope. The statistical certainty of RD(v) at large v values is low because the standard deviation is determined from a small sample size (only 2” observations after n bifurcations). The fractal dimension is therefore best defined by RD(v) for the smallest volumes. The decision to exclude the four largest pieces remains somewhat arbitrary but reflects the procedure used for estimating the fractal dimension of blood flow in experimental measurements on heart (1) and lung tissue (4). Random variation asymmetry. This model is identical to model III of van Beek et al. (18). The asymmetry of flow in the first model is the same at all bifurcations in the vascular network. This is not an accurate representation of the pulmonary vascular tree in that there is variability in the anatomical predicted asymmetry of flow (10). A more realistic model can be constructed by assigning a different y to each branch point. In this model the value of y is selected from a normal distribution with a mean 7 = 0.5 and a standard deviation 0. A vascular network with 0 = 0.0 produces a uniform distribution of flow to all branches with RD = 0.0. As CTdiverges from zero, so will RD. Unfortunately this model does not have an analytic solution (18). A computer simulation can be run that constructs a vascular network by randomly assigning y values with a standard deviation of 0 to each bifurcation and
B
0.20- A
15 generations y= 0.45 RD = 40.1%
15 generations
0.15 oh g 2 O.lO2 F4
y= 0.47
RD = 23.5%
I
FIG. 2. Frequency distributions of flow in network with 15 generations. Flows are normalized to mean flow.
0.050.00
0.0
-J--
1.0
2.0
3.0
Flow/Mean Flow
4.0
0.0
1.0
2.0
3.0
4.0
Flow/Mean Flow
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1026
FRACTAL
MODELING
OF
B
15 generations 0.45
-
1
2.0
.
1
.
4.0
1
.
6.0
I
1
‘*’
Ooo
In(u), Volume of Region
-
2*o
values are randomly chosen, the distribution of flow will be different each time the simulation is run. An average distribution of flow can be estimated by running the model several times. Figure 3B shows the average RD (after 10 runs) as a function of piece size for a E-generation vascular tree with a 7 = 0.5 and 0 = 0.05. When plotted on a log-log scale, the relationship again appears linear, suggesting that it is fractal. The distributions of flow are no longer discrete in this model. A pseudocontinuous distribution can be constructed by plotting frequency histograms of flow with narrow intervals. Frequency distributions of flow for two networks of 15 generations with 0 values of 0.03 and 0.05 are shown in Fig. 4. The flow distributions are skewed to the right and have RD values of 23.0 and 39.3%, respectively. To see how well these fractal vascular networks model the regional blood flow distribution in the lung, a fit of the relative dispersion predicted from the models can be compared to experimental observations. In the constant asymmetry model, the y values that produced the best fit of the modeled RD to the observed experimental RD were determined using Eq. 3 and a computer optimization algorithm (3, 18). The random variation asymmetry model was evaluated by running a computer simulation 10 times for a given value 0. The average RD at varying
0.08
1 i
w
0 = 0.03
RD = 23.0%
0.06 CT= 0.05
0.04
RD / = 39.7%
0.02 0.00
Flow/Mean
Flow
4. Frequency distributions of flow in random variation Simulated flows are generated only 1 time for each B. FIG.
, 4’o
-
I
-
‘*’
In(u), Volume of Region
determining the RD at each generation. Because the y
0.10
BLOOD
15 generations 0 = 0.05
y=
2.8! 0.0
PULMONARY
model.
I ‘*’
FLOW
FIG. 3. Relationship of relative dispersion (RD) to size of theoretical perfused region or generation level of the tree; 4 largest-pieces were excluded from fractal fit. A: log-log relationship of RD vs. v for network constructed with constant asymmetry. Line is least-squares linear fit to data and represents theoretical fractal relationship of Eq. 1. B: log-log relationship of RD vs. v for network constructed with random variation asymmetry with 7 = 0.5 and c = 0.025. Line again is least-squares linear fit to data and represents theoretical fractal relationship. SD bars represent variability in RD for each of 10 simulations .
piece sizes was determined for each G and compared with experimental data. The best 0 was found by minimizing the residual sum of squares between the experimental and modeled RD values. EXPERIMENTAL
METHODS
The experimental techniques have been described previously (4). Briefly, six dogs were anesthetized with intravenous Surital (thiamylal sodium) followed by LYchloralose-urethan, intubated, and mechanically ventilated on room air with a tidal volume of 15 ml/kg and rate sufficient to maintain a PCO, between 32 and 38 Torr. Then 25 mCi of ggmTc-labeled macroaggregated albumin (MAA) were rapidly injected via a central venous catheter while the dog was apneic at end expiration. A total of 2.24-4.5 X lo6 particles of MAA were injected. Ventilation of the dog was resumed 15-20 s after the injection. The animals were then deeply anesthetized, heparinized, and killed by exsangination. The endotracheal tube was clamped, the abdomen was entered, and the diaphragm was punctured. The recoil pressure of the lungs at functional residual capacity (FRC) was estimated by the increase in airway pressure when the diaphragm was punctured. A sternotomy was performed, the pulmonary artery and left atrium were plumbed with large bore catheters, and the aorta was tied off. After hyperinflation of the lungs, the airway pressure was set to the previously measured recoil pressure and the chest was closed with the vascular catheters exiting the thorax caudally above the diaphragm. The lungs were fixed by intravascular perfusion of glutaraldehyde and dehydrated with progressively increasing concentrations of ethyl alcohol. The lungs and heart were removed from the thorax, and the lungs were dissected free of the heart and dried for 12-18 h on continuous positive pressure equal to the previously measured recoil pressure at FRC. They were then embedded in rapidly setting urethane foam and cut in transverse H-mm slices. The individual lung slices were imaged on a planar gamma camera. A matrix size of 128 X 128 pixels was used, resulting in a voxel size of 1.48 X 1.48 X 11 mm. The square edges of the foam cast allowed accurate registration of the lung slices so that contiguous slices could be appropriately aligned. The pixel counts were proportional to the local ggmTc-labeled MAA determined blood
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FRACTAL
MODELING
OF
flow. The pixel values were transferred to an Apple Macintosh II computer where they were corrected for decay time and stored as a three-dimensional array. The most peripheral voxels and those adjacent to airways were excluded before numerical analysis. The heterogeneity or RD of the three-dimensional flow arrays was analyzed by initially subdividing the array along a sagittal plane that separated the right and left lungs and an RD was calculated for the mean flows of the two pieces. Each lung was then subdivided by transverse planes that produced equally sized “upper” and “lower” pieces and a relative dispersion was again calculated for the mean flows of the four pieces. The pieces were sequentially divided in half until each piece was a single voxel in size. As the pieces were subdivided, an RD was calculated at each step. An estimate of the RD due to measurement error was subtracted from the observed RD and a log-log plot of the corrected RD (in percent) as a function of the volume of pieces was constructed (4). A logarithmic fit to the data (excluding the first 4 points) was obtained using a least-squares approach. The spatial fractal dimension (D,) was determined from the slope of the regression line as defined by Eq. 1. STATISTICS
Data are given as means t SD. RD (which equals SD/ mean) characterized the flow distribution at various sample sizes. The goodness of fit of the model to the observed data was quantitated by the coefficient of variation (CV) cv
44
(5)
= CY
measured
n
is the experimental observations of RD, where Ymeaaured ymodelis the RD values from the model, n is the number of points, and df are the degrees of freedom, which is 2 for both models.
PULMONARY
BLOOD
1027
FLOW
metry and random variation asymmetry models is shown in Fig. 6. The solid lines of y = 0.452 and G = 0.0042 are the best fit to the experimental measurements of the second animal and represent the theoretical fractal relationship. The parameters y and c were varied about these values to demonstrate the sensitivity of the models to changes in these parameters. DISCUSSION
A useful model should have a plausible functional basis to explain the observations and should accurately predict those observations. An even better model fulfills these prerequisites while simultaneously generalizing and simplifying our understanding of the modeled system. Dichotomously branching fractal networks are simple yet powerful models that relate function and structure. They explain a number of observations concerning the heterogeneous and spatially correlated distribution of pulmonary blood flow. An assumption underlying the fitting of the experimental observations to the fractal model is that each piece of tissue (regardless of size) is supplied by a single terminal vessel. The validity of this assumption is not critical to the fractal analysis of flow distribution but its violation does have an effect on the estimated fractal dimension (4). If the pieces of tissue are not dissected along vascular boundaries, the boundaries imposed by the analysis will be “misregistered” from the true vascular partitions. This misregistration error will produce a smoothing effect, resulting in an underestimation of the true fractal dimension. The effect is small but will be more pronounced for more heterogenous flow distributions (4). The asymmetry of flow parameter y can be related to the radii and lengths of branches within the vascular tree and can be recursively defined from the parent branch. If the parent vessel has radius R and length L, the daughter vessels have radii scaled by ri and r2 and lengths that are scaled by I, and ZZof the parent dimensions. One daughter branch will have a radius of r1 R and a length of 1,. L, and the other daughter branch will have a radius of r2 R and a length of L, L. If the flow in the parent branch is F,, the flows in each daughter branch can be calculated from Poiseuille’s law. The flow to the first daughter branch, fi, is F, (/2 r:)l(Z, rt + 1, ri) and the flow to the other daughter branch, f2, is F, (1, ~-2)/( 1, rt + 1, ri). Van Beek et al. (18) stated that the fraction of flow to one daughter branch, previously defined as y, is equal to fi/(fi + f& and can be related to the radius and length scalars by the equation l
l
RESULTS
The fractal vascular models provide excellent fits to the observed distribution of pulmonary blood flow. Figure 5 is a representative fit of the models to the experimental \data. The four points representing the largest pieces were not included in the analysis of each animal because they did not fit the fractal relationship. The y values best characterizing RD(u) in the constant asymmetry model ranged from 0.447 to 0.468 with a mean for all animals of 0.459 t 0.009. The mean CV between this model and the experimental data was 0.069 t 0.038. The 0 values best characterizing RD(u) in the random variation model ranged from 0.033 to 0.057 with a mean for all animals of 0.041 t 0.01. The mean CV between this model and the experimental data was 0.070 t 0.042. The composite data for ail animals are presented in Table 1, showing the number of generations needed to fit the observed data, the y and 0 that best fit the experimental data, and the CV for each model and the experimental data. Sensitivity analysis for y and 0 in the constant asym-
l
l
l
l
l
l
l
l
l
(6) Successive branches have radii and lengths that are scaled by a constant fraction of the parent branches. The fractal nature of this model is apparent from the ratios of radii and lengths between parent and daughter branches that remain the same independent of scale. The asymmetry of flow value does not change, as it is a ratio of flows between parent and daughter branches. Although regional blood flow is determined by local resistance, this resistance is influenced by more factors
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1028
FRACTAL MODELING
OF PULMONARY
BLOOD FLOW
3.6 1 A o
l
0
.g 3.2 2 z 3.1pf z 30.
Experimental Modeled RD
8
RD
l
0 0
0
8
0
Q
15 generations o=o.O42
0 0
FIG. 5. Fit of modeled relative dispersion to experimental observations of 1 animal; 4 largest pieces were excluded from fit. A: constant asymmetry model. 23: random variation asymmetry model.
l 0
0 0
0 0
CV = 0.036 0
8
2.9 ‘1 0
I
I
12 In(u),
I
I
I
I
3
4
5
6
Volume
RD
0 0
0
15 generations y=o.451 cv = 0.035
Experimental Modeled RD
o
0
I
1
7
0
of Region
I
12 In(u),
I
I
3
4
Volume
than the radius and length of a vessel. Vasoactive factors, local alveolar volumes, and hydrostatic pressures affect the resistance in a given vessel and hence influence regional blood flow. The asymmetry parameter 7, should therefore be thought of as a general descriptor of flow distribution to daughter branches, which is determined by a number of factors. Blood flow within isogravitational planes is heterogeneous and spatially correlated (4-6, 14, 15). Nicolaysen and associates (14) suggested that this heterogeneity is a random process. The flow distributions in our fractal models are not influenced by gravitational factors. The asymmetry of flow at branches within the pulmonary arterial tree may therefore account for the heterogeneity of flow within isogravitational planes. These simple fractal networks also nicely explain the observation that pulmonary blood flow distribution is fractal and spatially correlated (4,6). A model of pulmonary blood flow distribution based solely on gravitational effects would not produce the fractal relationship we have observed. Blood flow distribution in the lung is spatially correlated in that high-flow regions tend to be near areas of high-flow and low-flow regions tend to be adjacent to other areas of low flow (4). This observation is accounted for by the dichotomously branching network in that flows to neighboring regions are more closely related (they share a parent or grandparent branch point) than are regions separated by a greater distance. The more closely related two regions
I
1
1
5
6
7
of Region
are in the vascular tree, the more likely their flows will be similar. These fractal networks obviously represent simplified models of the complex and dynamic nature of pulmonary blood flow, and as such they have a number of deficiencies. Morphometric analysis of the pulmonary arterial tree has shown that although the vast majority of branchings are dichotomous, there are occasional branch points with more than two daughter branches (7, 17). The asymmetry of branchings of the pulmonary arterial tree is not absolutely fixed as modeled by the first network. Measurements by Singhal et al. (17) have described the ratio of radii in daughter branches to range from 0.1 to 1.0 with a standard deviation of 0.20 in a given pulmonary arterial tree. The fractal models also have equal path lengths to all of the terminal branches which is unlike the pulmonary vascular tree (8). Horsfield (10) has analyzed morphometric data on the human lung to estimate the asymmetry of flow between daughter branches. Although he uses different nomenclature, it is possible to determined an estimate of 7 from his calculations. Using Poiseuille’s law and average measurements of daughter branch radii and lengths, Horsfield estimated the relative proportion of flows to be 0.635 and 0.365 to daughters of a parent vessel. If this estimate of 7 = 0.365 is used in the constant asymmetry model, the relative dispersion of flow after 15 generations would be 136.9%. This is considerably larger than
TABLE 1. Model parameters Constant Animal No.
Mean
Generation
1
14
2 3 4 5 6
14 15 14 15 14
-t SD
Asymmetry
0.468 0.457 0.447 0.464 0.451 0.468 0.459_+0.009
Asymmetry
Random Coefficient of variation
0.093 0.035 0.135 0.043 0.047 0.062 0.069+0.038
Standard deviation
0.033 0.042 0.057 0.036 0.048 0.033 0.041~0.010
Variation
Asymmetry Coefficient of variation
0.086 0.036 0.148 0.042 0.050 0.055 0.070+0.042
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FRACTAL
$4
.s
MODELING
OF
PULMONARY
BLOOD
1029
FLOW
3.6
-- ---, -&qo32
---- --
--Law ---,----I\ ---
2.8
! 0
1
1 2
3
1 4
‘i, 5
I 6
In(u), Volume of Region
4 7
I 0
1
2
--S ---
1 3
4
---- ----
Y
-1
=. -- ---.0 ---, -L ---, 5
I
1
6
7
FIG. 6. Sensitivity analysis of the parameters y and 0 in constant asymmetry (A) and random variation asymmetry (B) models. Solid lines, y = 0.452 and (T = 0.042, represent best fit to experimental data of 1 animal using 14 generations. y and 0 are varied about these values to demonstrate the sensitivity of variation in these parameters. Lines are least-squares fits to simulated data and represent theoretical fractal relationship of Eq. I.
In(u), Volume of Region
the relative dispersion measured in our experimental animals (mean RD = 35.4%) and would be even larger if the variability in y were considered. This sizable discrepancy may be due to the differences in experimental technique. Horsfield’s data were collected from casts of fully inflated saline-filled cadaver lungs in which the pulmonary arteries were injected with resin at a pressure of 40 cmH,O, and estimates of flow were calculated from these measurements. Our data are obtained from actual flow measurements from intact animals. A number of these differences may account for differences in our estimates of y, but it is intriguing to conjecture that within a fractal vascular structure there may be physiological mechanisms that regulate pulmonary blood flow toward homogeneity. Another interesting observation from the data of Horsfield et al. and from analysis of bronchial tree morphometric data by Nelson et al. (13) is that the first 3 generations of the trees do not conform to the fractal pattern of the rest of the lung (10,13). This suggests that the initial branchings of the pulmonary vasculature and bronchial tree are either not fractal or may have a different fractal dimension. Recent embryologic observations by Massoud and associates (12) have demonstrated that in fact there are two different branching patterns in the rat fetal lung, peripheral and central. The central branches of the pulmonary bronchial tree exhibit monopodial branching while the rest of the tree divides dichotomously. This does not mean that the initial branches of the bronchial tree are not fractal but rather they may follow a different fractal pattern of growth with a different fractal dimension. The constant asymmetry and random variation asymmetry models appear to describe the heterogeneity of pulmonary blood flow equally well (Fig. 5 and Table 1). The two models differ primarily in their spatial distribution of flow. The constant asymmetry model will have flows that are consistently biased to one side while the expected flows in the random variation asymmetry model will be equal distributed between both sides of the vascular tree. In this way, the random variation asymmetry model is a more realistic model. The constant asymmetry model has the advantages of being simple and having an analytic solution for the relative dispersion. Van Beek and associates (18) have presented two other related branching models. Their second model used a y
that decreased (asymmetry increased) with each generation by a fixed constant. This introduces another degree of freedom into the model, and optimization of this parameter will necessarily produce an equal or better fit to the observed data. This model uniformly fit their observed cardiac blood flow data better than the constant asymmetry or random variation asymmetry models. It should be noted that this model is no longer ideally fractal because the branching pattern is not constant across different scales of measurement. Of interest is that the mean optimized fixed constant was 0.993 and 0.996 for the two data sets they fit to the model, indicating that a constant y is a reasonable model. Because there are no morphometric studies to support a gradually diminishing y, we did not use this model. The other model they introduced differs from the random variation asymmetry model in that 0 decreases by one-half at each generation (18). This model did not fit their experimental data as well as the random variation asymmetry model. The fractal branching models are presented as concepts and are not meant to be a precise representation of anatomic structures. Rather we wish to emphasize the concept of how small degrees of asymmetry in flow can produce heterogeneous blood flow distributions similar to those seen in experimental studies. The precise value of 7 or 0 are of little importance relative to this idea. These models use the concepts of fractals to relate the function and structure of the pulmonary vascular tree and offer an explanation for the spatial distribution and gravity independent heterogeneity of blood flow. We acknowledge the valuable communications with Dr. Johannes van Beek, Dr. James Bassingthwaighte, and the personnel of the National Simulation Resource Facility for Mass Transport and Exchange (RR-01243) at the Center for Bioengineering, University of Washington. This work was supported by National Heart, Lung, and Blood Institute Grants HL-08155 and HL-38736 and Grant 89-WA-515 from the American Heart Association, Washington Afhliate. Address for reprint requests: R. W. Glenny, Div. of Pulmonary and Critical Care Medicine, Rm-12, Dept. of Medicine, University of Washington, Seattle, WA 98195. Received
11 June
1990; accepted
in final
form
11 September
1990.
REFERENCES 1. BASSINGTHWAIGHTE, nature of regional 65: 578-590, 1989.
J. B., R. B. KING, AND S. A. ROGER. Fractal myocardial blood flow heterogeneity. Circ. Res.
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1030
FRACTAL
MODELING
OF
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