Structural morphology of renal vasculature

 doi:10.1152/ajpheart.00814.2005 291:296-309, 2006. First published Jan 6, 2006;Am J Physiol Heart Circ Physiol

Nicolas P. Smith David A. Nordsletten, Shane Blackett, Michael D. Bentley, Erik L. Ritman andStructural morphology of renal vasculature

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Structural morphology of renal vasculature

David A. Nordsletten,1 Shane Blackett,1 Michael D. Bentley,2 Erik L. Ritman,3 and Nicolas P. Smith1

1Bioengineering Institute, University of Auckland, Auckland 1001, New Zealand; 2Departmentof Biological Sciences, Minnesota State University, Mankato, Minnesota; 3Department ofPhysiology and Biomedical Engineering, Mayo Clinic College of Medicine, Rochester, Minnesota

Submitted 31 July 2005; accepted in final form 20 December 2005

Nordsletten, David A., Shane Blackett, Michael D. Bentley, ErikL. Ritman, and Nicolas P. Smith. Structural morphology of renalvasculature. Am J Physiol Heart Circ Physiol 291: H296–H309, 2006.First published January 6, 2006; doi:10.1152/ajpheart.00814.2005.—Anautomatic segmentation technique has been developed and applied totwo renal micro-computer tomography (CT) images. With the use ofa 20-�m voxel resolution image, the arterial and venous trees weresegmented for the rat renal vasculature, distinguishing resolvingvessels down to 30 �m in radius. A higher resolution 4-�m voxelimage of a renal vascular subtree, with vessel radial values down to 10�m, was segmented. Strahler ordering was applied to each subtreeusing an iterative scheme developed to integrate information from thetwo segmented models to reconstruct the complete topology of theentire vascular tree. An error analysis of the assigned orders quantifiedthe robustness of the ordering process for the full model. Radial,length, and connectivity data of the complete arterial and venous treesare reported by order. Substantial parallelism is observed betweenindividual arteries and veins, and the ratio of parallel vessel radii isquantified via a power law. A strong correlation with Murray’s Lawwas established, providing convincing evidence of the “minimumwork” hypothesis. Results were compared with theoretical branchangle formulations, based on the principles of “minimum shear force,”were inconclusive. Three-dimensional reconstructions of renal vascu-lar trees collected are made freely available1 for further investigationinto renal physiology and modeling studies.

kidney; Strahler ordering; vascular statistics; renal modeling; vascularreconstruction

RENAL VASCULAR STRUCTURE has been the focus of a number ofstudies analyzing the affects of conditions such as hypertensionand diabetes (12, 17, 23, 24, 28, 29, 38) as well as studiesattempting to characterize renal function mathematically (10,26, 27). The kidney has been shown to play a pivotal role inblood volume, vascular tone, electrolyte control, as well asblood filtration. Strong structural heterogeneity and correlatesto functional physiology (3, 18) make the study of renalvasculature essential to understanding renal function and con-sequently, blood toxicity, volume, and pressure.

The current anatomical ontology characterizes renal vascu-lature based on its functional role in the circulation. Blood flowin the rat kidney enters through the renal artery splitting intotwo arterial trees that enter via the pelvic region dividing into6–10 (8 in the data set we segment below) interlobar arteries(IA). These vessels lead along the corticomedullary boundary(3) eventually feeding into the arcuate arteries (AA). Deriva-tives of these arteries are the cortical radial arteries (CRA),

which penetrate through the cortical labyrinth (34). Renalarterial anatomy is described in detail in Reference 3 and themicrovasculature in Reference 18. For standard nomenclature,refer to Kriz and Bankir (25).

Microcomputed tomography (micro-CT) provides the abilityto produce high resolution detailed images from which infor-mation on the vascular structures can be collected (4, 5, 6). Thedevelopment of automatic segmentation methods to extractrelevant information from these data sets and the use oforganization techniques provide a novel set of tools for ana-lyzing and quantifying the structure of vascular trees.

Significant work has gone into measuring and enumeratingthe topology of vascular trees (7, 37, 43, 45, 46), particularlyrecent work in the heart (20–22). Because of the amount andthe form of data in these trees, organizational techniques areessential to quantifying structural anatomy and gaining insightinto physiological processes. A number of ordering methodshave been proposed to tabulate and organize vascular data (19,44); however, the most commonly applied is the so-calledStrahler Ordering method proposed by Strahler (40) and Hors-field (15, 16). Strahler ordering begins by labeling all terminalarterioles as order 0. Following vessel segments upstream, iftwo order 0 vessels adjoin at a bifurcation, the order of theparent vessel is labeled 1, whereas if an order 0 vessel mergeswith a vessel of dissimilar order, the parent assumes the valueof the highest ordered vessel (40). This scheme, based entirelyon the vascular structure and connectivity, has been shown tocorrelate well with vessel radius (7, 19, 37, 45, 46). As bloodflow through the systemic circulation is, structurally, influ-enced most significantly by vessel diameters, Strahler orderingprovides insight into blood distribution and transport.

The focus in this study was the vasculature in the unilobarcortex of the rat kidney. Renal rat models are commonly usedfor developing a scientific understanding of renal function(2–4, 12, 17, 23, 24, 28, 29, 38). Raw data from micro-CTimages of vascular casts (11) were segmented by using amethod outlined below. The extracted three-dimensional arte-rial and venous trees were then analyzed for an entire renalcortex at a voxel resolution of 20 �m and for a cortical subtreeat a voxel resolution of 4 �m. Despite working with two partialimages, we developed an iterative technique to correlate or-dering and statistical data from both to assess the completevascular structure. Finally, a quantitative error analysis tech-nique is proposed to assess the validity of the applied methods.The three-dimensional data sets of these segmented vasculartrees are made available to both the biological modeling andphysiological communities1 for further investigation.

METHODS

Images and segmentation. Garcia-Sanz et al. (11), using micro-CTimaging, acquired scanned voxel images of rat renal vasculature.

1 http://www.physiome.org.nz/publications/nordsletten_blackett_ritman_bentley_smith_2005.

Address for reprint requests and other correspondence: D. A. Nordsletten,Bioengineering Institute, Univ. of Auckland, Uniservices, Level 6, 70 Sy-monds St., Auckland, 1001, NZ (e-mail: [email protected]).

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Renal circulation was infused at physiological pressures using theprotocol described by Bentley et al. (4). Briefly, the sample wasperfused using a radiopaque silicone polymer, effectively delineatingthe vasculature from the organ tissue via a uniform coloration specificto the vascular cast. The resulting voxel grayscale images were brokeninto two-dimensional images using the National Institutes of Healthsponsored program ImageJ.2 Renal images of 20-�m (showing anentire kidney) and 4-�m (showing a vascular subtree) voxel resolutionwere used for this analysis (4).

Further image processing was conducted in CMGUI, a three-dimensional graphical environment developed at the BioengineeringInstitute.3 The extracted images were uploaded, and isosurface ren-dering was used to locate vessel walls. Isosurfaces were stored as apoint group, consisting of up to a million points, dispersed on arterialand venous walls. With the use of this point group, the vascular treeswere skeletonized while approximate vessel radii were calculated, aprocess referred to in this paper as vessel emaciation, resulting invessel diameters, lengths, and connectivity calculations.

With the use of the point group generated from the kidney images,emaciation of vessels was conducted based on the assumption thatindividual vessel segments form elliptical cross sections, and onaverage the length between two nearby vessels is greater than that oftheir respective diameters. With these assumptions, peripheral pointson vessel walls were projected inward by surface projections, emaci-ating all vessels. This process is demonstrated in Fig. 1 on the 20-�mkidney image. Figure 2 provides a general schematic of the segmen-tation process where the detailed vector algebra for this process isoutlined in detail in APPENDIX A. The final result is a stream of pointsat the centerline of all the vessels as seen in Fig. 1F. Tracking, basedon a snake algorithm adapted to handle the three-dimensional pointgroup, was utilized to connect points and successfully recreate theoriginal vasculature. With the use of the original point group todefine the walls of the vasculature, the fully emaciated set of pointswere progressively linked and centered within the vessel crosssection. This process was repeated until all points had been trackedand associated to a vessel segment, resulting in a list of spatialpositions, radii, and local connectivity indicative of the originalvascular image.

Organizational technique. The scheme adopted for this work wasStrahler ordering (outlined in the introduction). Because of thebranching morphology of renal vasculature, the Strahler orderingtechnique required additional rules to handle higher-order junc-tions such as trifurcations (which occurred between 3% and 5.5%for all segmented branch points in this study across the differentnetwork subtrees) that were observed more often than reported inother ordering studies (20). These rules are illustrated in Fig. 3(dagger denotes location of trifurcations). The convention adoptedwas to apply Strahler ordering to those branching vessels withmultiple daughters by increasing the parent vessel segment anorder if two or more vessels of the same and highest order merge(†2), otherwise labeling the parent vessel as that of the highestorder daughter vessel (†1).

The Strahler ordering scheme works logically from capillaries orvenules upstream and is, consequently, sensitive to pruned subtrees. Eachimage in this study was incomplete due to resolution constraints andpartial casting of vessels. To accommodate this, an iterative techniquewas adopted to define the proper Strahler order of each vessel terminal.Under this scheme a conservative estimate of true order 0 terminalvessels was made while all other presumably pruned terminal vessels

were left undefined.4 Ordering proceeded through the subtrees untilundeclared terminal orders inhibited further progress. Based on thestatistics generated from this initial ordering, undeclared terminal vesselswere progressively defined according to their terminal bounds until theordering was complete.5 This process was iterated by reapplying statis-tical ranges of the initial ordering to all terminals and reordering untilconvergence was obtained (i.e., no statistical deviation from one iterationto the next). Statistical ranges from the high-resolution image (4 �m)were then used as an estimate for the bounds of terminal vesselssegmented in the low-resolution image (20 �m).

Ordering error analysis. Central to effective ordering of anyvascular tree is the proper assignment of vessel terminals. However,the assumption of a terminal vessel ordering profile, as the previous

2 ImageJ is an image processing tool sponsored by National Institutes ofHealth and downloadable at http://rsb.info.nih.gov/ij/

3 CMGUI or Continuum Mechanics Graphical User Interface is a opensource and available online at http://www.bioeng.auckland.ac.nz.

4 Estimates were determined based on fitting a histogram of terminal vesselradii to a normal distribution and ordering 0 for all radii within a standarddeviation (SD).

5 The high and low radial bounds for an order are defined as shown in Eq.1, where i indicates the order of interest.

Fig. 1. A: global surface point set �. B: results of surface projection. C–E:iteration of surface centering.

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section suggests, imparts error associated with commuting subtreeswith the entire vascular tree. A method was designed to assess howthese errors affect results of the Strahler classification. For example,Fig. 3 illustrates common artifacts seen in medical images of vascu-lature (denoted with ‡). The gray marker infers a bifurcation whereincorrect ordering has no effect on the upstream ordering, as changingthe first-order vessel to a terminal zero-order vessel results in the sameupstream ordering. Consequently, the probability that vessels wereordered incorrectly due to this terminal is zero. The dark markers,however, infer bifurcations where not applying the iterative techniqueentirely changes the ordering upstream as well as the calculatedstatistics (i.e., changing these terminal vessels to zero order causes arestructuring of the upstream ordering).

To assess these errors one might change terminal orders of thevascular network and compute the effect. This process, however,would be computationally intensive, especially for large-scale vascu-

lar trees. The framework developed provides a means for quantifyingthe sensitivity of a vascular tree to likely perturbations of terminalvessels, assuming the given ordering scheme is a close match to thecorrect ordering.

We define �i � [�Ii �S

i �Di ]T to be a vector of probable outcomes for

a vessel segment Si, where the components are the probabilities of thatsegment increasing (�I

i), remaining (�Si ), or decreasing (�D

i ) an order.The transference of probability through vessel segments (and conse-quently the overall effect) is mediated by the series of bifurcations, oroperators, and Strahler ordering rules at each. These operators sum-marize all possible outcomes given � for the daughter segments andmodify the upstream segment probabilities accordingly. Operators,denoted as �, have subscripts that label a specific branch point and thesuperscript T referring to the matrix transpose (for complete details onoperators, refer to APPENDIX B).

When calculating the probability profile of the jth terminal vesselsegment Sj, with radius rj that is of order i, we define the upper (rmax)bound by the same diameter defined criterion proposed by Jiang et al.6

(19) shown in Eq. 1, where ri and �i are the average radius andstandard deviation of order i vessels.

rmax � �ri � ri�1,i�1 � �i�1,i � �i,i�1�/2 (1)

Note that the lower bound for order i�1 is set as the upper bound fororder i. Using a normal distribution to describe error in orderingterminal vessels (with mean rj and standard deviation �rrj, where �r

is a constant), we have assumed that terminal probabilities can beassigned by the area under the curve delineated by the bounds in Eq.1 as seen in Fig. 4.

To calculate �i or the probability distribution of the ith networksegment (Si), the method requires the definition of a pathway ti.Considering the subtree beneath Si, ti is an index of segments thatdefines a single path down the ith subtree to a terminal vessel. Forexample, if one such path from Si to a terminal was given to gothrough segments S1, S2, and S3, ti � [S1 S2 S3]. Choice for definingti is arbitrary but requires only movement downstream. A differentpath for the same segment (Si) results in a modification of operators(which are defined by a chosen branch or branch pair) but no

6 Jiang et al. (19) applied Eq. 1 to reduce the radii overlap betweensuccessive orders and modify the process of assigning Strahler orders. It isimportant to note that Eq. 1 is applied, in this study, in a different context.

Fig. 2. A: a section of renal vasculature point set. B: examininga local section around pi from the point set. C: using � to definean outward and inward normal u. D: local cross sectiondivided into angle increments to define vascular boundaries.

Fig. 3. An example vascular tree using Strahler Ordering, where (†) indicateexamples of trifurcation ordering rules, and (‡) indicate junctions upstream ofvessel pruning.

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modification in the probability �i. If � is the probability of the terminalvessel by ti, the overall probability profile of Si can be written as seenin Eq. 2.

i � ��j�1

ti

�ti�j�T � � �ti�1�

T �ti�2�T . . .�ti�n�1�

T �ti�n�T (2)

The result is a directed product of operators acting on the terminalvalues of �. A sample calculation using the vascular tree from Fig. 3is presented in APPENDIX A. This calculation is a concise and efficientmeans for predicting how incorrectly ordering pruned terminals willalter ordering and, consequently, the statistics of the tree classified byorder.

Calculating lost networks. Kassab et al. (20, 21) presented the ideaof the connectivity matrix (CM). This matrix, C, is order dependentwith component Cmn representing the number of elements of order m,which arise from an order n vessel divided by the total number oforder n vessels. The CM gives insight into the connectivity of avascular tree, but perhaps more importantly it provides a means toapproximate the number of vessels in a subtree by order when theappropriate feeding vessel is known. This has implications on bloodflow and transport to downstream tissues. Unfortunately, in our studythe CM is based on incomplete images and is consequently notaccurate. However, by statistically defining terminal vessels, manypruned vascular trees can be identified. Let nv,O(n) refer to the numberof vessels of order n and nv,O(n)4O(m) refer to the number of vesselsof order m arising from a vessel of order n. If the prime superscriptuses the entire network as a sample, and the P superscript is used toidentify the subnetworks that consist of a pruned vessel (i.e., aterminal vessel with an order greater than the lowest vascular treeorder) and all subtrees that branch from the pruned vessel at nonter-minal positions. The CM is defined in Eq. 3.

Cmn �n v,O�n�4O�m� � nv,O�n�4O�m�

P

n v,O�n� � nv,O�n�P (3)

It is important to note that the lowest order vessels of a network arenot tabulated as part of the pruned vessels (as for within the image ofinterest, these vessels will not bifurcate). This formulation allowscomplete subtrees that arise from a pruned vessel to contribute to theglobal statistics of the network while eliminating the potential ofunderestimating the subtrees arising from that pruned vessel.

A complication of ordering terminals based on radial profile is thatspecific sets may lack low or high order terminal vessels. In this case,the 20-�m image lacked order 0 and 1 vessels, whereas the 4-�mimage lacked orders over 5 (arterial) and 7 (venous) under the Strahlerscheme. To properly describe the CM, a weighted average (shown inEq. 4) was used to amalgamate the results of both image sets based onthe estimated number of total vessels in each sample. In Eq. 4, theweight wm

i � nv,O(m)i, � nv,O(m)

i,P � Cmni nv,O(n)

i,P , and the superscripts refer toeither of the two image resolutions analyzed.

Cmn �wm

4�mCmn4�m � wm

20�mCmn20�m

wm4�m � wm

20�m (4)

It should be noted that this formulation was used when the valuesfrom both data sets for Cmn were not zero.

The number of vessels in each given order of a subtree defined asthe network downstream from an inlet (root vessel) can be determinedby using the connectivity matrix. This is done by calculating thenumber of daughter vessels branching form the inlet and then itera-tively progressing down the subtree. This process can be reduced tosolving the set of simultaneous equations given below in Eq. 5.

�C � I�N � � A (5)

Where C is the connectivity matrix calculated in Eq. 4, I is the identitymatrix, and A is the rth row of C, where r is the order of the rootvessel. Solving Eq. 5 for N provides as its components Ni, the numberof vessels of a given order i.

RESULTS

From the segmentation methods, both a full-scale rat kidneyat 20 �m voxel resolution (Fig. 5) and a portion of kidney at 4�m voxel resolution (Fig. 6) were reconstructed. The combi-nation of these two image reconstructions were used in theensuing statistical studies.

Each set was ordered using the Strahler method outlined.The average luminal radius of each order as well as theapproximate number of vessels is given for the venous tree(Table 1) and arterial tree (Table 2). Because of vessel pruningand significant vessel tapering in the kidney, giving equal

Fig. 4. Assignment of terminal probability profiles based on order interval.

Fig. 5. Reconstructed model of renal arterial and venous trees from 20 �mmicro-CT data.

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weight to complete vessels and proximal ends of prunedvessels can result in incorrect calculations. To account for this,the calculation of means and standard deviations was done byusing a weighted average based on vessel length. These dataare also given graphically for the venous tree in Fig. 7.

General vessel connectivity for the arterial tree is listed inTable 3, where the low-level vessel branching was includedfrom the high resolution set as prescribed in the methods. Thesame information is given for the venous tree (Table 4).

Vessel length data are given in Table 5 for all trees orga-nized according to Strahler ordering. Though vessel lengthoften has a poor correlation with order, this information can beuseful in calculating global vascular properties such as subtreevolume.

Because of the parallel architecture of the arterial andvenous trees, an empirical relation was derived to correlate thetwo. Figure 8 shows a relation between the parallel venous andarterial vascular trees. A power law relation is fitted to thesedata showing the ratio of arterial and venous radii for varyingvenous radii.

Error analysis of the ordering scheme is given in Fig. 9 foreach tree. Assignment of terminal probabilities are given indetail in METHODS and varied according to radial factor (�r).The results are shown graphically in Fig. 10.

Shown in Fig. 11 is the total downstream subtree volume ata specific bifurcation by the radius of the feeding vessel for thevenous tree. With the use of the CM (Table 4) to estimatenumber of subtree vessels and the vessels statistics (Table 5),an expected subtree volume was plotted by average radiusper order.

Fig. 7. Side view of rat renal venous tree with vessel thickness proportional toa tenth the true radial value. Color is an indicator of the vessel order.

Table 1. Renal venous tree

Order

Image Radius, �mEstimated No. of

Elements4 �m 20 �m

0 10.792.41 10.792.41 68,56416,6471 14.724.05 14.724.05 30,6592,0172 26.029.51 26.161.57 9,2586473 46.4614.32 40.138.51 2,926944 71.4216.34 50.3012.12 1,21055 123.9921.35 69.2221.18 4186 175.0723.35 114.0729.59 1397 205.414.38 177.0439.04 388 285.6352.86 99 428.0580.95 4

10 603.7794.52 2

Table 2. Renal arterial tree

Order

Image Radius, �mEstimated No.of Elements4 �m 20 �m

0 10.080.14 10.080.14 29,5665,9651 13.903.80 13.903.80 13,0702,2932 20.066.90 20.066.90 4,3736643 35.8511.66 29.870.35 1,2451984 64.348.82 39.291.08 578715 93.605.10 44.239.81 247236 53.8712.51 9067 86.1524.06 2418 139.8320.11 69 191.4217.79 3

10 216.104.74 1

Fig. 6. Reconstructed model of renal arterial and venous subtrees from 4 �mmicro-CT data.

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Cross-sectional area by order is plotted in Fig. 12 for thearterial and venous trees. The cross-sectional area was calcu-lated as using information about the average radius and numberof vessels from Tables 1 and 2.

A benefit over traditional cast studies is the quantification ofspatial aspects of vasculature, such as branching angles that wehave calculated from the two three-dimensional vectors, whichdefine the branch angle geometry at each bifurcation. Thisallows for spatial heterogeneity, which has a physiologicalbasis, but also allows for the experimental testing of hypothe-ses on vascular morphology. One such theory by Zamir (47)and Zamir and Bigelow (48) proposed that arterial branchingangles are designed to minimize total shear force based on theratio of daughter vessel radii (either linear or quadratic). Figure13 shows experimental data compared with these the proposedoptimums.

Another optimization theory is that proposed by Murray (31)where the fundamental structure of vascular trees is such thatit minimizes work. Under ideal conditions where flow rate isconserved, Murray’s Law states that the sum of each daughtervessel (rd) cubed is equivalent to the parent vessel (rp) cubed,or rp

3 � ¥rd3. Figure 14 shows the results for the arterial tree

compared with those predicted by Murray’s Law.

DISCUSSION

The applied mathematical reduction of micro-CT images isa useful and relatively straightforward means for converting animage into a quantified reconstruction of vascular structure.Though data were framed and handled discretely by usingpoint set representations, the same technique can be applied to

the voxel image itself, providing a major boost in computa-tional performance. The limitation of this process is that itrequires sufficiently clear imaging of the vascular walls andboundaries. This proposed method is not intended as an image-processing technique (because it does nothing to account oraccommodate for unclear imaging) but as a tool for automatedsegmentation.

A number of studies have looked at the effects of structureon function, primarily in the analysis of disease states. Recentstudies have assessed vascular changes in kidneys followingchronic bile duct ligation (33), hypercholesterolemia (5, 6),chronic nitric oxide inhibition (9), renal artery stenosis (50),and chronic potassium depletion (8). A significant setback toresearch in the morphological affects of physiological condi-tions is the lack of a concrete basis for comparison. Studieshave reported vascular diameters that vary significantly (24,38, 44, 11) for the same vessel structure. To integrate infor-mation from these sorts of studies, it is necessary to look atspecific regions of measurement to provide a context forevaluation of total renal physiological function.

Tables 1 and 2 give a quantitative correlation between radiusand order. The class of afferent arterioles is represented asorder 0 to 1 vessels. Published radial values for afferents havespanned from 7 �m (24) to 14 �m (38) with a number ofstudies reporting in that range (17, 28, 36). This is consistentwith vessels measured in the 4-�m data set. Cortical radialarteries span orders 2 to 6. These vessels bifurcate resulting ina significant span of orders as well as change in vessel radii.This is reflected in the literature, where deviations in radiusrange from 46.4 �m (42) to 68.2 �m (41). Because this

Table 3. Estimated connectivity matrix of renal arterial tree

Order

Estimated Connectivity Matrix

0 1 2 3 4 5 6 7 8 9 10

0 2.140.11 0.140.02 0.75 0.001 2.710.14 0.50 1.000.00 0.002 2.75 1.500.00 0.330.05 0.003 2.000.12 0.330.06 0.040.08 0.090.084 2.330.09 0.000.09 0.090.10 0.005 2.300.13 1.050.16 2.330.15 0.336 3.050.26 2.830.20 0.00 0.007 4.000.25 0.00 0.008 2.00 0.009 2.00

10

Table 4. Estimated connectivity matrix of renal venous tree

Order

Estimated Connectivity Matrix

0 1 2 3 4 5 6 7 8 9 10

0 2.140.11 0.150.01 0.310.00 0.310.00 0.400.00 0.001 2.690.02 1.200.00 1.380.00 1.200.00 0.00 1.002 2.740.00 0.620.01 1.000.00 0.50 0.00 0.003 2.350.04 0.190.00 0.04 0.00 0.00 0.004 2.730.01 0.41 0.18 0.22 0.49 1.465 2.61 0.76 1.86 1.97 0.986 3.07 2.30 0.25 0.497 4.00 0.25 0.508 2.25 0.009 2.00

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particular class of vessels is so broad, the terminology onlyprovides a loose way to correlate quantitative information.Orders 6 to 7 comprise the arcuate arteries running along thecorticomedullary boarder. As is the case with the cortical radialarteries, the arcuate arteries experience repeated bifurcationsand radial reduction. Because the kidney experiences no anas-tomoses between arterial branches, arcuate arteries eventuallyreduce to cortical radial arteries. This could be a possibleexplanation of deviation seen in the literature where radialaverages have been given as low as 96.5 �m (23) and as highas 153.5 �m (11). The interlobar arteries comprise orders 8 to9, whereas the feeding renal arteries are of orders 9 to 10. Fewother works have focused on the venous tree within thekidney (interlobular, orders 2–6; arcuate, orders 6–7; interlo-bar, order 8–9).

The estimated numbers of vessels by order are given inTables 1 and 2. Though the actual number of vessels in thekidney have yet been tabulated, the number of glomeruli in anormotensive rat kidney has been estimated at �30,000 (2, 29,32). With the assumption of an approximate 1:1 ratio betweenthe afferents and glomeruli, the predicted number of glomeruliwould be 29,566 5,965 according to our study. However,because of resolution, incomplete perfusion, and the visualiza-tion methods used, the majority of tracked afferents (4-�m

voxel image) and tracked cortical radial arteries (20-�m voxelimage) were lost and not linked to the arterial tree. Thoughaccounting for pruning in the connectivity matrix significantlyimproved results, to obtain a more accurate representation thearterial tree was correlated tightly to the venous tree.

A striking feature of renal vasculature is its parallel nature.In all images, arteries and veins could easily be distinguishedby their path duplicity and disparity in vessel radii. Parallelismwas maintained through the corticomedullary boundary be-tween arcuate arteries and veins. Orders are often seen runningin parallel (though the arterial orders tend to pair with veinstypically two orders higher). At higher resolution, deviationscould be seen for single segments where there appear to be twoarterial vessels following a vein. However, on bifurcation ofthe vein, proper ratios are restored. More often a direct parallelarchitecture was seen. Figure 8 shows the ratio of ra-to-rv forvarious values of rv. The fit, with constants � and , is of theform ra/rv � exp(� rv

�), chosen based on the observation that�ra/�rv 1 in all parallel vessels. That is, local change in arterialradii is small relative to the change in parallel venous radii.

At a fundamental level, parallelism in vascular trees hasbeen suggested as a means for prevention of vessel intersectionduring angiogenesis. However, functional hypotheses to vas-cular parallelism have been suggested. Thermoregulation hasbeen seen as an explanation for parallelism, occurring primar-ily in arteries and veins in the same sheath (35). “Cross talk” orthe so-called “counter-current exchange” hypothesis has beendemonstrated in renal vasa recta and tubules (13) as well as invilli of the small intestine (14). Bassingthwaighte et al. (1)suggest a mechanical advantage for distributing blood withinthe microvasculature of the myocardium as a possible expla-nation for parallelism. The reason for vascular parallelism inthe kidney has not been explained but could provide insightinto overall renal function.

From a technical standpoint, the parallel vessel architecturehad negative implications for both the imaging modality andsegmentation method. As a consequence of the renal arterialand venous interactions, often the spatial scale between these

Table 5. Vessel lengths

StrahlerOrder RV Length, mm RA Length, mm

0 0.1550.202 0.3120.2851 0.2480.230 0.4230.2832 0.3150.277 0.4040.3903 0.6250.434 0.6560.2864 0.8200.487 1.0010.2165 1.0540.626 0.5110.006 1.1470.761 1.0310.6747 1.6951.289 2.5162.0538 6.1312.251 8.9751.3319 3.0910.965 1.4400.647

10 3.123 0.185

Fig. 8. Empirical relation between arterial and venousradii (refer to DISCUSSION).

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parallel vessels was small as Fig. 2C demonstrates. Despite thefine voxel resolution of 4 �m, the distinction between arteriesand veins was lost. Furthermore, one may notice from Fig. 2Cthat the influence of an ambiguous boundary between thevessels has a greater impact on the total diameter of the arterythan the vein due to the shear size and shape of the two vessels.Thus extrapolating the arterial tree further became virtuallyimpossible. However, because of this parallelism, it was as-sumed that the arterial tree should continue to parallel thevenous tree. This assumption was implemented by way ofcross correlating the terminal ordering profile of the venous

tree with that of the arterial tree. This provides a more com-plete description of the renal arterial tree than could otherwisebe given by using the parallelism inherent to the vasculature.

Another complication of renal vascular parallelism is that amajor assumption of the segmentation technique was the spa-tial scale between vessels was greater than that of the vessellumen. This contradiction resulted in a number of seed pointsthat failed under tracking, requiring more computation time toobtain the entire network.

Figure 8 illustrates the average probability a vessel willremain its given order (�s) for varying allowed error modulated

Fig. 9. Network stability analysis plots based onarterial (A) and venous trees (B) for varying radialfactors (for more information, refer to APPENDIX A).

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by the radial factor (see METHODS). This process assumes thereis some error due to pruning or in the calculated radius fromsegmentation and consequently the iterative terminal assign-ment. The probability of �s for a specific order is not of generalconcern; however, it is the interplay of all orders that elude toerrors and/or significant changes in ordering given errors image

acquisition or methods applied. Because of the incomplete datasets, low �s suggests areas in the network requiring furtherinvestigation.

Figure 9, A and B, shows the profile for the arterial andvenous trees, respectively. This analysis predicts reasonablestability at high order even for large error due to a bufferingaffect of downstream operators, (i.e., the gray ‡ in Fig. 3). Thuswe have confidence that the method is robust for the ordersassigned via Strahler ordering. Specifically, these plots showthat the relative change in order of each tree is not prone todrastic changes in the ordering due to a different terminalordering profile.

Figure 11 illustrates the usefulness of statistically represent-ing structural data and the efficacy of the techniques applied.The plot shows subtree volume for both the image reconstruc-tions as well as a prediction based on the CM and vesselstatistics. The prediction shows agreement and integrationbetween both reconstructions.

The optimality comparison of Fig. 13 shows branch angledata from the three-dimensional anatomical data comparedwith the theories proposed by Zamir (47) and Zamir andBigelow (48). The significance of such data is in understandingthe means for which vessels branch within vascular systems.These results show a poor correlation with the “minimum shearforce” hypothesis set forth. In fact, the use of a power of zerofor the cost function resulted in a curve with a significantlylower mean standard error. These results suggest the need formore applicable cost functions, studies that are achievable withfull three-dimensional reconstructions.

Murray’s Law, which suggests that the relation betweendaughter and parent vessels is governed by minimizing work,was compared with data from both arterial trees. The deviationbetween the line of best fit and Murray’s Law was approxi-mately 1%. This provides convincing evidence that Murray’sLaw effectively characterizes the radii change across bifurca-tions over a range of spatial scales. Furthermore, this addsadditional support of the minimum work hypothesis (31).

Blood flow to glomeruli and the resulting glomerular filtra-tion rates remain difficult to calculate in vivo. However,

Fig. 10. Side view of rat renal vasculature with vessel thickness proportionalto a quarter the true radial value. Color of the arterial tree indicates vessel orderstability, white giving �s � 0 and red giving �s � 1. The venous tree exhibitsthe same information on a fading blue scale.

Fig. 11. Plot illustrating the successful merger of 4 �m and 20�m data sets by comparing downstream subtree volume both fromthe data itself and using average length and theoretical number ofvessels.

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estimates of renal perfusion are attainable through blood flowsimulations. Models such as Poiseuille flow (49), lumpedparameter models (30), and more complex models such asone-dimensional fluid flow through passive distensible vesselnetworks (39) have been proposed. These models requireinformation on structure due to the strong dependence ofvascular resistance on radius. Determining the blood flowthrough a vascular tree is dependent on connectivity, vessellength, and most significantl, radius. Correlations to totalcross-sectional area (determined entirely by vascular radius)determine the affect of flow through the vascular network,where increasing total area is proportional to decreasing flowand visa versa (Fig. 12). The data from this study provide botha specific three-dimensional data set and the statistical infor-mation required for reconstruction and morphologically basedblood flow analysis.

The renal vascular reconstructions developed through thecourse of this study will be freely available online for down-load and use, providing the academic community with astructurally accurate anatomical model. Not only useful forgeneral structural analysis and basis for comparisons, it pro-vides a modeling framework for further investigations intorenal hemodynamics.

Applied methods. Cast studies such as those by Kassab etal. (20, 21) have provided invaluable knowledge to thescientific community on vascular structure. Casting studiesare manually intensive; however, they provide a completedescription of vascular trees. The benefit in automatic seg-mentation techniques is the promise of a quick and efficientmeans for extracting relevant spatial and structural datafrom these images. At present, automatic segmentation ofvascular images can only hope to extract portions of mor-

Fig. 12. Plot showing the total cross sectional area by order forthe arterial and venous trees.

Fig. 13. Plot comparing ratio of daugther vessel radii at abifurcation with the ratio of cost function. H1 and H2 (the givencost functions) were determined experimentally for all structuralbifurcations and theoretically given proposed cost functions oflinear (Zamir,1) and quadratic (Zamir,2) relations.

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phological data due a lack of established techniques andproblems in processing medical images robustly. This paperattempts to address these issues that present significantchallenges to scientific research in vascular morphology andcardiovascular disease states.

Strahler ordering has been used in a number of studies toorganize structural information about vascular networks (7,37, 45, 46). Kassab et al. (20, 21) and Jiang et al. (19)proposed a modification to the Strahler scheme by modify-ing orders according to the bounds as those written in Eq. 1.This analysis was applied to the renal arterial tree, withsuccessful convergence; however, the 20-�m voxel venoustree did not achieve convergence after 20 iterations. Thiswas due to instances where the absolute difference inneighboring mean radii was less than or equal to the abso-lute difference in their standard deviations. Thus the mod-ification resulted in poor solutions or no solution at all. Forthese reasons, though the scheme worked well for thearterial tree, it was not used as it was not well suited for botharterial and venous trees.

The ordering error analysis applied to the renal vasculatureprovides a means for assessing the accuracy of the assignedorders by their probability of change. From a qualitativestandpoint, the ordering profile seemed to agree well withknown terminology and vessel morphology. However, becausemuch of the process relies on developed statistics for terminalordering, it is important to know the confidence in orderingresults. The drawback of this approach is the assumption thatthe network does not contain operators intermittently that werenot accounted for (i.e., vessel pruning occurred at terminals,not midway along network segments). Such added operatorscould alter the behavior of the upstream network but remaindifficult to account for.

The unique approach given to commute networks viadeclaring terminal orders using statistical profiles has itslimitations. Taking data on a specific subtree and applying it

to an entire vascular network assumes a limited sample sizeat higher orders within the subtree as well as insignificantregional variation. Because of the structure of renal vascu-lature and the agreement of collocated data with both sets(Fig. 11), sufficient overlap seems to exist to corroboratethis approach.

In conclusion, from micro-CT images of rat renal vascula-ture, high- and low-resolution data sets have been automati-cally segmented, and an anatomically accurate model of thearterial and venous trees was constructed. The segmentedmodels have been combined and classified using Strahlerordering and the statistics of radius, length, and connectivityreported by order. Further development in techniques forassessing vascular images and data were described. Thesemorphological data provide a statistical basis from whichrenal vascular topology can be generated. Furthermore, thethree-dimensional geometry (available online) enables fur-ther analysis of global spatial properties and local branchangle relations, as well as a basis for simulations of renalperfusion.

APPENDIX A

The notation used in the method below defines a3b as a length anddirection going from a to b, “ ˆ ” refers to a unit direction, and thesubscripts i, j, and k refer to specific points within a defined group.From the group of points (Figs. 1A and 2A), each point pi was selectedand a local neighborhood � of points near pi defined. Figure 2Billustrates the point pi and its neighborhood � seen in dark grey. If thetransmural direction u could be determined using the set �, the inwarddirection could be obtained for a particular point. The approximatetransmural direction u was determined as the average sum of crossproducts (formally stated in Eq. A1) between all combinations ofpoints in �. In Eq. A1, pi3j and pi3k refer to the direction from pi tothe points in �, pj, and pk, n� is the number of points in �, and APr

is an additive projection function ensuring all directions point inwardor outward from the vessel.

Fig. 14. Comparative plot of branching data. The normalizedparent vessel diameter cubed (2rp)3 is plotted against the sumof the cubed normalized daughter vessels ¥(2rd)3. Both axiswere normalized by the maximum cubed parent vessel diam-eter. Using linear regression, the deviation between the lineof best fit and that predicted by Murray’s Law was �1.0percent.

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u � �j�1

n��1

�k�j�1

n�

APr�j,k�� pi3j � pi3k

�pi3j � pi3k�� (A1)

APr�j,k� � 1, if �pi3j � pi3k� � �pi31 � pi32� � 0 APr�j,k� � � 1, if �pi3j � pi3k� � �pi31 � pi32� 0

Based on the assumptions outlined above, the closest point along uwas assumed to reside on the opposing vessel wall as seen in Fig. 2C.There are a number of cases where this assumption is invalid (i.e., ata bifurcation or near parallel vessels) that are dealt with below. Withthe knowledge of the direction and distance to the opposing wall, theith point is then projected to the center of the vessel as shown in Figs.1B and 2D.

The new partially emaciated group (best illustrated in Fig. 1B)consists of points that lie along the centerlines of all vessels and aresurrounded radially by the original point group. The further emacia-tion of the new point group is required to filter out errors in the initialprojection. It is important to note that large errors in the initialprojection lead to isolated or grouped points with little to no longi-tudinal neighbors, providing a quantitative means for error identifi-cation.

In the new partially emaciated point group, a neighborhood ofpoints is established based on the information stored about the initialprojection of pi to its new position pi. This group � is the collectionof points within the assumed “radius” of the vessel, which is simplycalculated as the distance of pi3pi. Applying Eq. A2, the resultingsum points transluminally. Although the group � contains scatteredradial noise due to error in the initial projection of all points, therandom occurrence of this error is relatively homogeneous and itseffects on the calculation of Eq. A2 approximately cancel.

uz � �j�1

n�i

APz�j, 1�Pi3j

�Pi3j�(A2)

APr�j, 1� � 1, if �pi3j� � �pi31� � 0 APr�j,1� � � 1,

if �pi3j� � �pi31� 0

The estimate of the axial direction at pi is useful because it not onlyorients the vessel direction, but defines the vessel cross section (i.e.,Fig. 2D is oriented so the reader is viewing along the transluminaldirection). Figure 2D illustrates the expected result when comparing

the new point group to the original, where the point pi is surroundedby the original set of vessel wall points. However, if the initialprojection was incorrect, the point pi would not have this relation tothe original set. To utilize this relationship, a section of the vessel(chosen as twice the assumed “radius”) was used. The axial normalfrom Eq. A2 was used to project points into a plane (as seen in Fig.2D) calculated using Eq. A3, where j subscript refers to the jth pointof the cross section from the original point group.

pj,2D � pj � �uz � pi3j�uz (A3)

Common to poorly projected points are holes in the wall cross section.Whereas the vessel bifurcations (seen in Fig. 2A) also produce holesin the wall, projecting points into a plane through a section of thebifurcation eliminates this problem. To identify holes numerically, theplanar cross section was divided into angles as indicated by Fig. 2D.The angle �j is defined by the direction pi3j and pi3� based on theirdot product as detailed in Eq. A4 and seen in Fig. 2D. However, thedot product only uniquely defines angles from 0 to �, thus it is alsonecessary to compare the vector from pi to an in-plane point with theresultant vector of pi3� and the axial vector uz (Eq. A4).

�j � cos�1�pi3j,2D � �� if �pi3j,2D � �pi3�,2D � uz�� � 0 (A4)

�j � 2� � cos�1�pi3j,2D � �� if �pi3j,2D � �pi3�,2D � uz�� 0

In each section, the closest set of points to pi were used to repositionpi to the centerline of the vessel, and the radius was recalculated as theaverage distance of pi3 pj,2D. Points that had holes or empty sectionswere eliminated from the new point group. By repeating this process,points converged toward the centerline of the vessels (shown in Fig.1, C–E), and erroneous points were removed.

APPENDIX B

Bifurcation operators are listed below for all possible combinationsof input orders. It is assumed that the vector of probable outcomesfrom i is being modified by the operator formed from j.

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To determine the probability �i that segment Si from Fig. 3 willincrease, remain, or decrease its order, we can express it in terms ofdownstream probabilities. Here the superscript indicates the vesselsegment from Fig. 3 seen as Eq. B1.

Labeling the operator � according to the upstream segment, theprobability can be rewritten (Eq. B2).

i � � iI

iS

iD� � �i

T�9T�6

T4 (B2)

If we identify the pathway for this calculation ti, which is identical tothe sequence Si,S9,S6, the total calculation is seen in Eq. B3.

i � ��j�1

3

�i�j�T �4 (B3)

This calculation also relies on the calculation of �3. Equation B4allows us to calculate the proper parameters for �i.

�iTf3 � �3

T1 (B4)

The combination of all these equations allows us to calculate how Si

will change in order due to any combination of terminal probabilities.

ACKNOWLEDGMENTS

The authors thank the reviewers for insightful and constructive commentsin regard to this work. Authors offer special thanks to P. E. Beighley and S. MJorgensen for contributions to data collection, J. Lam for background study,and A. S. Beatty for helpful comments.

GRANTS

D. A. Nordsletten is supported by scholarships from the New ZealandMathematics Institute and its Applications and the University of AucklandInternational Doctoral Scholarship. N. P. Smith acknowledges support by theMarsden Fund of the Royal Society of New Zealand through Grant 04-UOA-177 and the National Institutes of Health (NIH) through the NIH/NIBBMulti-Scale Grant RO1-EB-00582501. J. Carlos Romero’s lab, under thesponsorship of National Heart, Lung, and Blood Institute Grant HL-16496,performed the kidney preparation. High-resolution micro-CT data were ob-tained at the Brookhaven National Synchroton Light Source, BrookhavenNational Laboratory and was supported by the United States Department of

i � � Ii

Si

Di� � �I

3 S3 � D

3 00 I

3 � S3 D

3

0 I3 S

3 � D3� T� 1 0 0

I7 � I

8 � I7I

8 �S7 � D

7 ��S8 � S

8� 0I

7I8 1 � I

7I8 � D

7 D8 D

7 D8� T�I

5 S5 � D

5 00 I

5 � S5 D

5

0 I5 S

5 � D5� T� I

4

S4

D4� (B1)

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Energy, Division of Materials Science and Division of Chemical Sciences bycontract DE-AC02-76CH00016. NIH Grant EB000305 sponsored lower-res-olution micro-CT data collection.

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