Quantifying the genetic influence on mammalian vascular tree structure
- *Division of Pulmonary and Critical Care Medicine, School of Medicine, Box 356522, and
- †Department of Physiology and Biophysics, University of Washington, Seattle, WA 98195; and
- §Mountain-Whisper-Light Statistical Consulting, 1827 23rd Avenue East, Seattle, WA 98112
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Edited by James H. Brown, University of New Mexico, Albuquerque, NM, and approved March 1, 2007 (received for review December 11, 2006)
Abstract
The ubiquity of fractal vascular trees throughout the plant and animal kingdoms is postulated to be due to evolutionary advantages conferred through efficient distribution of nutrients to multicellular organisms. The implicit, and untested, assertion in this theory is that the geometry of vascular trees is heritable. Because vascular trees are constructed through the iterative use of signaling pathways modified by local factors at each step of the branching process, we sought to investigate how genetic and nongenetic influences are balanced to create vascular trees and the regional distribution of nutrients through them. We studied the spatial distribution of organ blood flow in armadillos because they have genetically identical littermates, allowing us to quantify the genetic influence. We determined that the regional distribution of blood flow is strongly correlated between littermates (r 2 = 0.56) and less correlated between unrelated animals (r 2 = 0.36). Using an ANOVA model, we estimate that 67% of the regional variability in organ blood flow is genetically controlled. We also used fractal analysis to characterize the distribution of organ blood flow and found shared patterns within the lungs and hearts of related animals, suggesting common control over the vascular development of these two organs. We conclude that the geometries of fractal vascular trees are heritable and could be selected through evolutionary pressures. Furthermore, considerable postgenetic modifications may allow vascular trees to adapt to local factors and provide a flexibility that would not be possible in a rigid system.
Fractal vascular trees are found throughout the plant and animal kingdoms. The conservation of these structure across so may different organisms is thought to be due to evolutionary advantages conferred through efficient distribution of nutrients (1, 2). A central tenant of this theory is that the geometry of vascular trees is heritable. We hypothesized that a balance between genetic and nongenetic influences determines the geometries of these vascular trees and sought to quantify the relative contributions of each.
As demonstrated by classic monozygotic twin studies, phenotypes are determined by a combination of genetic and environmental (nongenetic) conditions (3). Although a number of signaling molecules, and presumably their associated genes, have been shown to control branching vasculogenesis (4, 5), the local cellular environment at the point of branching as well as the geometric space into which the vessels grow must also influence the vascular structure (6). We therefore hypothesized that the final geometry of vascular trees is determined by a balance between genetic and environmental influences and designed this study to quantify the relative contributions of each component. In addition, we also tested a prior observation that individual animals have a characteristic fractal dimension of blood flow across organs (7).
Because of the balance between genetic and environmental influences, determinants of the final vascular tree geometry are optimally studied in a whole-animal model to quantify the factors influencing vascular tree structures. Although the geometry of the vascular tree is of interest, it is the functional consequences of the tree structure that are of biologic importance. We therefore chose to quantify the degree of genetic influence on blood flow distribution within mammalian organs. We studied the nine-banded armadillo, Dasypus novemcinctus because it is the largest animal to routinely produce four monozygotic offspring (8). Although genetically identical, minor phenotypic variations occur within litters because of local influences during development (9) and maturation. By comparing the spatial distribution of organ blood flow within and between liters of armadillos, we are able to quantify the relative contributions of genetic influences within individuals and across the species. Blood flow distributions are compared across animals by determining the spatial distribution of regional blood flow in the lungs and across organs by using the fractal dimension of perfusion heterogeneity (10).
Results
Detailed spatial measurements were obtained from the lungs because they are the largest organ and provide the greatest spatial information. Because of one anesthetic death, a total of 15 armadillos were studied. The numbers of microspheres were determined within regions of interests (ROIs) from the same relative spatial locations across all animals (Fig. 1). The ROIs were spherical lung volumes 4 mm in radius. The coefficient of determination (r 2) between the counted microspheres in each lung region across animals (Fig. 2) were determined for all possible pairings of animals.
Spatial distribution of pulmonary blood flow in three armadillos. Fluorescent microspheres were injected intravenously to mark the regional distribution of blood flow in the lungs. The large spheres have radii of 4 mm and represent volumes of lung in which the numbers of microspheres were counted. The lung volumes are color-coded to indicate the numbers of microspheres counted in each region. The scales to the left of each lung provide the color key for the numbers of microspheres in each lung ROI. There are 69 ROIs in each lung set, and they are positioned in the same relative locations across the lungs. The spatial distribution of blood flow appears to be more similar between A1 and A4 (same litter) compared with A1 and C4 (different litters). See Fig. 2 for correlation plots.
Relationship between numbers of microspheres from the same lung regions in two different animals. (Left) Two monozygotic animals. (Center) Two genetically different animals. Note the much stronger correlation between the monozygotic animals. The data are from the same animals presented in Fig. 1. (Right) Distribution of correlations. The boxes represents interquartile distance, the horizontal line is the median, and the whiskers represent the range. Correlations were significantly stronger within litters than between litters. The range of correlations among littermates demonstrates the variable influences of nongenetic factors.
On average, the coefficient of determination was 0.56 between animals within litters and ranged from 0.37 to 0.72 (Fig. 2). The range of correlations demonstrates that the relative contributions of genetic and nongenetic determinants vary across individual animals. The mean coefficient of determination between animals from different liters was 0.38. Although statistically less than what is observed in genetically identical animals, the similarities in regional blood flow among unrelated animals was unexpectedly high and demonstrates that there is a pattern of blood flow shared among the armadillo population. The range of coefficients of determination was quite large across unrelated animals, ranging from 0.10 to 0.64 (Fig. 2).
We also used an ANOVA model to derive estimates of variance components (see Methods). This model partitions the overall variation in blood flow into between-litter, within-litter, and the combination of spatial and residual variation. The degree of genetic determination of regional blood flow is a sum of the fraction of the variation due to spatial location and the fraction due to litter effects, compared with the total variation across the entire data set. The total variation also includes environmental effects and measurement error. The variation due to spatial location is genetic because it captures the regional distribution of flow that is common among the armadillo species and which distinguishes this species from other species. The between-litter component of variation is genetic because it captures the litter-to-litter variation in the amount of blood flow to each lung region. Each litter has a genetically determined pattern of flow distribution that is common among members of the litter but differs from litter to litter, with each litter varying its distribution somewhat from the average regional distribution common to the species. This litter-to-litter component of variation does not include the variation in distribution from animal to animal within a litter, which arises from environmental effects unique to each animal. Using this ANOVA model, we estimate that 67% of the regional distribution of blood flow is determined by the genetic code in each animal. The residual variation, 33%, is ascribed to all nongenetic factors, including experimental method noise.
Because spatial locations cannot be compared across different types of organs, we used fractal analyses to characterize the spatial patterns and explore the genetic control of blood flow within the heart, lungs, and skeletal muscles (medical and lateral gastrocnemius of right leg). Fractal dimensions provide a scale-independent measure of spatial processes, ranging from 1.0 to 1.5 for uniform and random distributions, respectively (10, 11). Because of an anesthetic death and a missed left ventricular injection, n = 14 for these analyses (see Table 1). A fractal distribution of perfusion is expected when blood is distributed via a fractal vascular tree. We found that blood flow to the lungs, hearts, and skeletal muscles were all fractal in nature (Fig. 3) and ranged considerably across animals from 1.08 to 1.41 (Fig. 4).
Numbers of microspheres counted in each organ
Fractal plot of blood flow heterogeneity (coefficient of variation) as a function of the scale of resolution (sampled volume). Note that the relationship is linear on a log–log scale over a wide range of volumes. The slope of the relationship is a scale-independent measure of the fractal dimension and can be compared across organs and animals.
Fractal dimensions of blood flow to various organs: heart (Upper Left), right lung (Upper Center), left lung (Upper Right), muscle 1 (Lower Left), and muscle 2 (Lower Right). Fractal dimensions are statistically more similar within litters than across litters for hearts and lungs. Fractal dimensions are also positively correlated between hearts and lungs within litters (see Results).
We first sought to determine whether littermates tended to have similar fractal dimensions of blood flow. We used a permutation test to test the null hypothesis that fractal dimensions of organ blood flow do not cluster by litter and found that the fractal dimensions of blood flow were more similar between littermates than between animals from different litters, with P = 0.002 for lungs and P = 0.008 for hearts. The fractal dimensions of blood flow to muscle did not appear to be more similar within litters compared with that across litters (P = 0.6). These observations suggest that the spatial distribution of blood flow to the hearts and lungs are determined by a shared genetic instruction set.
We also explored the hypothesis that blood flow distribution across organs is influenced by a common genetic code. We therefore determined the mean fractal dimensions within litters for each organ and used the correlation coefficient of this value between organs as a measure of the genetic component shared by littermates. We observed a strong correlation among the heart, right lung, and left lung (median r = 0.8). In contrast, the correlation coefficients between the mean fractal dimensions of these organs and skeletal muscles were quite weak (r = 0.0). However, the correlation of the fractal dimensions between the lateral and medial gastrocnemius muscles were much stronger with r = 0.72.
Discussion
Classical twin studies have compared monozygotic and dizygotic twins to estimate the degree of genetic control for a large number of physical, cognitive, and behavioral characteristics (3). The present study deviates from the traditional approach by comparing groups of four monozygotic clones rather than large numbers of monozygotic and dizygotic twins. We adapted an ANOVA approach that estimates the relative contributions of genetic and nongenetic influence on regional blood flow (Methods). Using this ANOVA model, we estimate that 67% of the regional distribution of blood flow is determined by the genetic code and shared environmental factors in each animal. The residual variation, 33%, is ascribed to all unique environmental factors including experimental method noise. Although these estimates are derived from pulmonary vascular trees, we do not know of any reasons why the pulmonary vascular tree would be more or less heritable than other organs.
The degree of concordance between littermates is a measure of the genetic and shared environmental control of regional lung blood flow and by inference, the geometry of the pulmonary vascular tree. Metzger and Krasnow (5) used the tracheal system of Drosophila melanogaster to explore the genetic control of branching morphogenesis. Global embryonic patterning pathways establish an initial stereotypical branching pattern corresponding to the large airways. Subsequent branching is regulated by a general program in which fibroblast growth factor and fibroblast growth factor receptor are used repeatedly to control branching. This mechanism is similar to the recursive rules used to build fractal forms in that a simple rule is repeatedly applied at each branch point to build a complex structure. Although the genetic control of the mammalian respiratory system requires coordination of airway and circulatory systems and is therefore more complex, it continues to use a restricted number of proteins to guide its construction (4).
The degree of discordance within litters is a measure of the unique environmental influences on regional blood flow distribution. These factors include, but are not limited to, the local cellular environment during vasculogenesis, intrauterine factors, epigenetic modifications of DNA transcription, postnatal developments, and method noise. Growth discordance due to placental vascular supply can produce phenotypic differences among monozygotic twins (12). Methylation is an epigenetic mechanism by which sequences of DNA can be switched off, altering the expression of specific genes and creating phenotypic differences between monozygotic twins (13). Postnatal factors, such as accessibility to nutrition, infections, or trauma can also introduce discordance between monozygotic animals. The large range of correlation coefficients between the armadillo siblings suggests that these nongenetic factors have a variable and significant contribution to organ blood flow distributions across animals.
A number of factors may have both genetic and nongenetic influences on vascular growth. Although our statistical model dichotomizes effects into these two components, it is possible for any given influence to have both a genetic and nongenetic component. These effects are partitioned into each category in our model. For example, the local cellular environment at vascular branch points likely has both genetic and nongenetic components. Our statistical model proportions the relative contribution of each component into these two categories.
The average coefficient of determination of 0.56 among monozygotic animals is remarkably high given the confounding factors that can artificially decrease the estimate of the true concordance. The methods used to determine the numbers of microspheres in the same spatial locations across animals are imperfect. Microspheres are particulate and therefore have an associated Poisson noise in the estimates of regional blood flow (14). The microsphere method provides a snapshot in time of the regional distribution of blood flow. Because of vascular vasomotion, local blood flow varies over time (15, 16), and coefficients of determination for local lung perfusion within the same animal decreases to ≈0.8 when separated in time. Hence, the correlation of regional blood flow would be expected to be less than perfect, even in animals with identical vascular trees. All of these potential sources of noise work to degrade the true concordance; none can artificially increase the association. Hence the observed coefficients of determination of flow between animals are underestimates of the true concordance. Similarly, the degree of genetic control estimated from the ANOVA approach also underestimates the true degree of variance explained by genetic and shared environmental factors.
Our experimental methods and ANOVA model lump the genetic and the shared environmental factors into a single component. In classical twin studies, monozygotic twins that are separated at birth are used to estimate the influence of shared environmental factors. Unfortunately, we were not able to separate individual littermates or isolate them from their mothers. Our animals were weaned at the earliest possible age to minimize the shared environmental influence of their mothers. Vascular trees continue to grow and remodel after birth, and it is possible that a shared environmental exposure, such as a respiratory infection within a litter, may affect the geometry of the vascular tree. These shared environmental factors are more likely to increase the discordance in regional blood flow between littermates, and we believe that the shared environmental factors that increase the concordance between littermates must do so to a relatively small degree.
A surprising finding is that 38% of the spatial variability in pulmonary blood flow is shared across unrelated armadillos (Fig. 3). This common pattern is likely due to the stereotypical asymmetries of pulmonary arteries that are “hardwired” across animals. Visual inspection of the mean blood flow to lung regions across all animals reveals a tendency for blood flow to be greater in the right apical and bilateral anterior lung regions. Although five of the six microsatellites we measured were unique across the four litters, the animals were all captured within a limited area and likely shared some genetic material.
We are aware of only one previous study that attempted to quantify the genetic control of vascular trees by comparing the retinal vascular patterns in monozygotic and dizygotic twins (17). By using metrics of vessel length, numbers of crossings, regional branch points, and tortuosity, only the last measure was found to be genetically determined. In a qualitative study, Tower (18) found that retinal vascular patterns were more similar near the arterial origins than in the periphery of monozygotic twins, supporting the concept that the initial vascular branchings are under more strict genetic control than subsequent branchings. It is possible that our methods provide more quantifiable spatial information and allowed us to find a correlation between related individuals where prior studies have failed to find a strong correlation. It is also possible that the vascular patterns in the retina, much like fingerprints, are not heritable.
The correlation between fractal dimensions of hearts and lungs within animals and litters suggest a shared genetic instruction set or a similar mechanism for constructing the vascular trees in the heart and lungs, whereas skeletal muscles may use a different set of construction plans. Iversen and Nicolaysen (7) suggested a shared genetic component for the fractal distribution of blood flow across different organs within the same animal. Our study further strengthens this interpretation because it shows a strong correlation across different monozygotic animals, thereby removing the potential confounding influence of shared experimental conditions when studied within the same animal. Our study does differ from Iversen and Nicolaysen's in that we see a weak correlation between myocardial and skeletal muscles in our animals. A possible explanation for the tighter association between the heart and lungs is that the blood flow between these organs may be more tightly integrated than between the lungs and skeletal muscles.
Although the fractal nature of vascular and airway trees has been recognized (10, 19), the functional advantages of a fractal design over others was not initially appreciated. West and colleagues (1, 2) hypothesized that the fractal geometry of vascular trees provide an evolutionary advantage for organisms and thus have been retained with characteristic properties across the plant and animal kingdoms. Their implicit assertion was that the geometry of these fractal vascular trees is heritable and can be optimized through natural selection. This premise, however attractive, has not been directly tested. Our current work clearly demonstrates that the geometry of some mammalian vascular trees is strongly determined by genetic factors and is therefore heritable. West and colleagues also conjectured that the fractal geometries provided advantages through their efficiencies in delivering nutrients to a multicellular organism under specified constraints (e.g., space filling, invariant terminal units) by maximizing surface area for nutrient exchange, minimizing energy expenditure to distribute the nutrients, and minimizing materials needed to construct the trees. An additional advantage of a fractal vascular system is the efficient use of genetic code. There would be a high cost of genetic material if every branch in an organisms vascular system required a genetic code. By using a recursive construction mechanism that requires only a handful of proteins, the instructions for building fractal vascular trees can be efficiently coded in an organism's DNA. Although this efficient coding limits the potential sites for error in the DNA code, it also magnifies the importance of these few instructions. Furthermore, we have determined that there is considerable modification of these construction rules by nongenetic factors. These postgenetic modifications may be advantageous in that they allow the vascular trees to adapt to local factors and provide a flexibility that would not be possible in a rigid system.
Although fractal vascular trees may provide an efficient general solution, it is not possible to code for a vascular geometry that is truly optimized for all of the functions required of a vascular tree over time and space in a given organ. Therefore, mechanisms must exist that allow a vascular tree to adapt to local influences during construction (6). Furthermore, once established, the postnatal vascular tree is able to remodel over time in response to changes in local stimuli such as vascular pressures or flows. In addition, the geometry of vascular beds is not rigidly fixed, and vasomotion in response to local signals such as oxygen tension or adenosine can temporarily alter regional tree geometry and local blood flow. Hence, the genetic code is likely used to guide construction of a fractal vascular tree that provides a very good general solution to a number of vascular functions. Local factors influence these rules at the time of construction and then local signals alter the geometries of the tree through active feedback mechanisms.
A considerable strength of our methods is that we studied blood flow distribution in whole animals, where the entire scope of genetic and environmental influences is exhibited. Despite all of the potential factors that influence the structure of vascular trees, including local cellular environment during vasculogenesis, differences in intrauterine conditions, epigenetic modifications of DNA transcription, and postnatal events, regional blood flow is remarkably similar across monozygotic animals. The observation that 2/3 of regional blood flow distribution is determined by the genetic code strongly suggests that the structure of vascular trees is indeed an important heritable trait that has been used by evolutionary selection to optimize the delivery of nutrients.
Methods
Four litters of armadillos (n = 16 animals) were obtained from the University of the Ozarks (Clarksville, AR) after being birthed and raised in captivity for 4 months. The animals were housed as separate litters with their natural mothers until separation at 7–10 weeks of age and then raised in kennels with littermates. Weights of the animals ranged from 1.6 to 2.6 kg. An adult nine-banded armadillo weighs between 4 and 5 kg (20). The Animal Care Committee at the University of Washington approved these animal studies. All animals were cared for and handled in accordance with the guidelines established by the National Institutes of Health (21). Animals within a litter appeared identical, and their sibship was confirmed by genotyping six microsatellite markers. PCR primers were designed from Prodohl and colleagues (8), and DNA was isolated from blood by using PureGene DNA purification kits (Gentra, Minneapolis, MN). The DNA was amplified by using PCR, sequenced, and sized by Genescan analysis (Genescan, Metairie, LA). Allele calls were made by using Genotyper software (Applied Biosystems, Foster City, CA). Five of the six microsatellites were informative and demonstrated identical genotypes within each litter and differences across litters.
Anesthesia was induced with an i.p. injection of ketamine (20 mg/kg) and xylazine (2 mg/kg), and animals were mechanically ventilated through a tracheostomy. Anesthesia was maintained in a surgical plane with repeated injections of ketamine and xylazine in a 10:1 mixture. Catheters were placed in an internal jugular vein and a carotid artery. The animals were placed prone to minimize pleural pressure gradients and ventilated with 30% oxygen to minimize regional hypoxic pulmonary vasoconstriction. Arterial blood gases were obtained and demonstrated normal gas exchange in all animals. Regional blood flow was marked by 15-μm diameter fluorescent microspheres (22). Before injection, the microspheres were sonicated, vortexed, and suspended in 2 ml of saline. A total of 105 red microspheres, suspended in 6 ml of saline, were injected over five breaths in each animal via the central venous catheter. A sternotomy was performed, and 106 yellow microspheres, suspended in 1 ml of saline, were injected through the heart into the left ventricle over 30 seconds. The animals were then killed, their hearts, medial- and lateral-right gastrocnemius muscles, and lungs removed. The lungs were filled to total lung capacity with OCT media (Sakura Finetek, Torrance, CA) via the trachea and frozen. Fiducial markers, 45 on the right and 42 on the left lung, were placed at well defined anatomical locations on the surface of each lung. The heart cavities were filled with OCT media. All organs were frozen separately within blocks of OCT. One animal died of a surgical complication, resulting in 15 animals being analyzed.
The spatial locations of every fluorescent microsphere in all excised organs and the fiducial markers in each lung were located by using an Imaging Cryomicrotome (23). The Imaging CryoMicrotome (Barlow Scientific, Olympia, WA) is a device that determines the spatial distribution of fluorescent microspheres throughout an organ at the microscopic level. Details of the instrument configuration have been reported (23). The numbers of microspheres counted in each organ are shown in Table 1. Because of a dilution error, one armadillo (B1) received 10 times the numbers of microspheres via the central vein injection.
Random sampling without replacement is performed by choosing x, y, and z coordinates from a pseudorandom number generator (Unix, Berkeley, CA). If >95% of the ROI (defined by its center point and selected radius) lies within parenchyma, the ROI is considered adequate to sample. ROIs cannot overlap any portion of a previously sampled ROI. This sampling process continues until no other ROIs can be found within the organ. The sampling process is repeated 10 times at each given sampling volume. By using this approach, only a fraction of the organ is sampled at each iteration, but it is done so in an unbiased manner without overlap of sample volumes.
The spatial locations of microspheres are determined relative to each lung as it is imaged. To allow comparisons of regional blood flow between animals, coordinate systems relative to the fiducial markers of each lung were determined. The lungs from one animal (reference animal) were randomly sampled with ROIs that were spherical volumes with a radius of 4,000 μm (0.27 ml volumes). The spatial location of each ROI and the numbers of microspheres within each ROI were determined for the reference animal. Vectors from the center of each lung to the peripheral fiducial markers were mathematically determined in the reference animal. The spatial locations (x, y, and z coordinates) of the center of each ROI were used to identify the nearest three vectors for each ROI (Fig. 5). Sets of linear equations were solved to determine the spatial location of each ROI with respect to the nearest three vectors in the reference animal (24). Sets of three vectors and their scalars could therefore be used to describe the location of each ROI in the reference animal, providing a new coordinate system defined internally by the fiducial markers for that animal. Corresponding spatial locations within the second and all other animals were determined from the associated fiducial markers, vectors, and scalars for each animal. The number of microspheres in the corresponding ROIs were determined for each animal. Multiple data sets were created by using every animal once as the reference animal so as to negate any influence from the choice of any one reference animal.
Method for mathematically sampling two different lungs in the same relative locations. White dots are the fiducial markers on the surface of each lung. The yellow dot at the center of the right lung represents the centroid of the fiducial markers. The red sphere in the reference animal (Left) represents a single sampling volume (ROI). Its location is described by the three vectors emanating from the center of the lung to the three nearest fiducial markers on the lung surface. A system of three linear equations is solved to find the scalar components of each vector describing the location of the center of the spherical sampling volume. The location of the ROI in the second animal is determined from the corresponding fiducial markers and the same scalars from the reference animal.
The coefficient of determination (r 2) between the counted microspheres in each realized ROI was determined for all possible pairings of animals. Because microspheres are discrete particles, Poisson counting noise is superimposed on the expected distribution of microspheres to create the observed distribution (14, 25). An estimate of the correlation coefficient that would be observed in the absence of Poisson counting noise can be obtained (25).
We also used an ANOVA model to derive estimates of variance components. This model partitions the overall variation in blood
flow into between-litter, within-litter, and the combination of spatial and residual variation. The ANOVA model can be stated
as:
where Yijk is the blood flow to spatial location k, in sibling number j, within litter i normalized to the mean flow for that animal. The model includes a common fixed mean, μ0, random effects for spatial locations, αk
S, and litters within locations, αik
LS, and a residual component,εijk. The degree of genetic determination of regional blood flow is a sum of the fraction of the variation due to spatial location
and litter compared with the variation in the entire data set.
The general approach to determine the fractal dimension of regional blood flow in each organ is to randomly sample the organ space with ROIs of a set radius, determine the number of microspheres present in each ROI, and calculate a coefficient of variation (CV = SD/mean) for the sampled distribution. The starting radius in the lungs and heart is 1,000 μm, resulting in a smallest sampling volume of 4.2 mm3. The starting radius in the skeletal muscle is 400 μm, or a smallest sampling volume of 0.27 mm3. The random sampling is performed multiple times to obtain a precise estimate of the true CV and the standard error of the estimated mean CV. The radius of the ROI is incremented by 100 μm, and a new CV is determined at this new sampling volume. The noise due to Poisson error can be mathematically removed from the observed CV (27) at each step in the fractal algorithm.
The fractal dimension of perfusion is determined by
where CV(v) is the measured coefficient of variation when the organ is partitioned into regions of volume v, CV(vref) is the CV found at the smallest sampling size, and Ds is the derived spatial fractal dimension. For normal spatial dimensions, CV(vref)/v
ref = CV(v)/v, whereas for fractal dimensions CV(v)/v
(1−Ds) = CV(vref)/v
ref
(1−Ds). This relationship shows that, in a fractal distribution, the observed CV depends on the scale of resolution (volume of lung
observed) with a power function relationship. Fractal dimensions can also be determined from the number of ROIs obtained from
an organ at each sampling volume. We have therefore chosen to use the sample volume as the independent variable. A least-squares
fit to Eq. 2 is weighted by the total number of ROIs for all 10 sampling estimates obtained at each volume. This weighting favors the
small sampling volumes, so that the effect of negative spatial correlation at larger piece sizes is minimized (26).
We use a permutation test to test the null hypothesis that fractal dimensions of organ blood flow do not cluster by litter. The permutation test is performed 9,999 times, randomly permuting animals among litters. For each realization of the random permutation, the test statistics the absolute value of A − B, where both A and B are the mean of the absolute difference in the fractal dimension between members of a pair, for all possible pairs, with the restriction that members of the pair are drawn from (A) different litters and (B) the same litter.
Acknowledgments
We thank Dowon An for assistance with the animal experiments and H. Thomas Robertson for critical review of the manuscript. This work was supported by National Institutes of Health/National Heart, Lung, and Blood Institute Grant R01 HL56239.
Footnotes
- ‡To whom correspondence should be addressed. E-mail: glennny{at}u.washington.edu
-
Author contributions: R.G. designed research; R.G., S.B., B.N., and N.P. performed research; R.G., B.N., and N.P. analyzed data; and R.G. wrote the paper.
-
The authors declare no conflict of interest.
-
This article is a PNAS direct submission.
- Abbreviation:
- ROI,
- region of interest.
- © 2007 by The National Academy of Sciences of the USA
