Three-dimensional morphometric analysis of the renal vasculature · 3D morphometric analysis of the renal vasculature 9/14/2017 3 48 Introduction 49 The three-dimensional (3D) topology

3D morphometric analysis of the renal vasculature 9/14/2017 1

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Three-dimensional morphometric analysis of the renal vasculature 2

3

Zohaib Khan1, Jennifer P. Ngo2, Bianca Le2, Roger G. Evans2, James T. Pearson2,3, 4

Bruce S. Gardiner4 and David W. Smith5 5

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1School of Information Technology and Mathematical Sciences, University of South Australia, 7

Adelaide, Australia. 8

2Cardiovascular Disease Program, Biosciences Discovery Institute and Department of Physiology, 9

Monash University, Melbourne, Australia. 10

3Department of Cardiac Physiology, National Cerebral and Cardiovascular Center Research 11

Institute, Osaka, Japan. 12

4School of Engineering and Information Technology, Murdoch University, Perth, Australia. 13

5Faculty of Engineering and Mathematical Sciences, The University of Western Australia, Perth, 14

Australia. 15

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Running title: 3D morphometric analysis of the renal vasculature 17

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Author for correspondence: 19

Dr David Smith 20

Faculty of Engineering and Mathematical Sciences 21

35 Stirling Highway, The University of Western Australia, Western Australia 6009, Australia 22

Tel: 61 8 6488 2281 23

Fax: 61 8 64881089 24

Email: [email protected] 25

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Abstract 26

Vascular topology and morphology is critical in the regulation of blood flow and the transport of 27

small solutes including oxygen, carbon dioxide, nitric oxide and hydrogen sulfide. Renal vascular 28

morphology is particularly challenging as many arterial walls are partially wrapped by the walls of 29

veins. In the absence of a precise characterization of three-dimensional branching vascular geometry, 30

accurate computational modeling of the intrarenal transport of small diffusible molecules is 31

impossible. An enormous manual effort was required to achieve a relatively precise characterization 32

of rat renal vascular geometry, highlighting the need for an automated method for analysis of 33

branched vasculature morphology to allow characterization of the renal vascular geometry of other 34

species, including humans. We present a semi-supervised method for three-dimensional 35

morphometric analysis of renal vasculature images generated by computed tomography. We derive 36

quantitative vascular attributes important to mass transport between arteries, veins and the renal 37

tissue, and present methods for their computation for a three-dimensional vascular geometry. To 38

validate the algorithm, we compare automated vascular estimates with subjective manual 39

measurements for a portion of rabbit kidney. While increased image resolution can improve 40

outcomes, our results demonstrate that the method can quantify the morphological characteristics of 41

artery-vein pairs, comparing favorably with manual measurements. Similar to the rat, we show that 42

rabbit artery-vein pairs become less intimate along the course of the renal vasculature, but the total 43

wrapped mass transfer coefficient increases then decreases. This new method will facilitate new 44

quantitative physiological models describing the transport of small molecules within the kidney. 45

Abstract Word Count: 250 46

Keywords: X-ray imaging, computed tomography, kidney, vessels, segmentation, validation 47



Introduction 48

The three-dimensional (3D) topology and morphology of blood vessels is of interest in many fields 49

of renal anatomy and physiology. The renal artery originates from the abdominal aorta, while blood 50

returns to the inferior vena cava via the renal vein. The renal vessels branch multiple times in a 51

hierarchical manner forming arterial and venous trees. Larger intrarenal arteries and veins are 52

generally paired, and are often intimately associated, with veins often observed to partially wrap a 53

nearby artery (27). Further, a number of renal functional abnormalities can be associated with the 54

vessel-structure. An example is renal artery stenosis, due to the formation of atherosclerotic plaque in 55

the main renal artery, which can lead to increased renal vascular resistance, secondary hypertension 56

and chronic kidney disease (11, 15). It is also thought that abnormalities in the structure of the walls 57

of small arteries in the kidney might contribute to the pathogenesis of essential hypertension (3). 58

In order to characterize the morphology of vessels, volumetric imaging techniques such as computed 59

tomography (CT) and magnetic resonance imaging (MRI) can be used to generate 3D reconstructions 60

(33), with a resolution approaching micrometers under optimal conditions (37). Intravascular 61

administration of a contrast agent renders blood vessel lumens radio opaque, so the vascular structure 62

can be defined using X-ray CT (20). Initial studies of intrarenal vasculature were limited to 63

volumetric statistics over antomical regions of interest. For example Garcia-Sanz et al. quantified 64

vascular volume for cortical and medullary regions of rat kidney from synchrotron CT images (12), 65

while for a model of chronic nephropathy, Xie et al. used MRI to observe differences in vascular 66

volume in anatomical sub-regions of the rat kidney (34). 67

When attempting to computationally model transport in a large number of vessels in a highly 68

branched vascular structure, it is commonplace for vessels to be idealized as a series of cylindrical 69

tubes of varying radii and length (35). However the actual vascular geometry can be crucially 70



important, since even apparently minor differences in anatomical details of the vascular geometry 71

can have important physiological consequences, for example, modifying blood flow and the 72

advective transport of solutes along the vessels, and modifying diffusive transport through vessels 73

walls. 74

In the mammalian kidney a critical limitation in our current understanding of small molecule 75

transport arises from the imprecise knowledge of geometric relationships between paired arteries and 76

veins throughout the branched network (13, 14, 19, 20). Specifically, veins are often observed to 77

partially wrap arteries (Fig. 1). It has been previously proposed that this wrapping may facilitate 78

small molecule transport between these vessel pairs (27). But the resulting concave cross-section of 79

veins in the renal vasculature is not captured when models are based on cylindrical tubes, and this 80

can result in incorrect estimation of vessel separation distances and so mass transfer between vessels 81

(19, 20). 82

In a previous work to more precisely define the renal vascular geometry, Yoldas and Dayan 83

measured the topology and lengths of intrarenal arteries from endocasts of rat kidneys, obtained 84

using an injection-corrosion technique, defining between-animal variations in large renal vessel 85

branching and volumes (36). Sled et al. observed a high degree of similarity in the distribution of 86

vessel diameters within mouse kidneys based on micro-CT images (31). Marsh et al. scanned 87

polymer cast of a rat renal vasculature using low and high resolution micro-CT imaging (23). They 88

measured the segmental lengths of arteries and afferent arterioles, and distances between arteriolar 89

pairs to study vascular organizational patterns in the arterial-nephron network. However, three-90

dimensional structure of vessels was reconstructed by manual outlining. Most importantly, detailed 91

insights into structural aspects of intrarenal vasculature for a single rat kidney were presented by 92

Nordsletten et al. who combined high and low resolution micro-CT images to construct an 93

(estimated) tubular geometrical model (28). Nordsletten et al. analyzed length, radius, cross-section 94



area and connectivity based on cylindrical tubes for each discrete order of arterial and venous 95

vessels. Although cylindrical tubes are often useful to approximate hemodynamic vascular models 96

(e.g. defining flow rates and pressure drops along the renal vascular circuit) (18, 30), such an 97

approach averages out potentially important in vivo vessel morphology that influences local renal 98

flow characteristics (e.g. local wall shear stresses). Cylindrical tubes are also inadequate for renal 99

transport modeling. For example the actual structural morphology, peculiar to intrarenal vessels, 100

involves characteristic wrapping of veins around arteries. Precisely defining wrapping is essential to 101

our understanding of the physiological and pathophysiological importance of diffusive shunting 102

between arteries and veins (27), and exchange with nearby tissues. An enormous manual effort was 103

required to achieve a relatively precise characterization of the renal vascular geometry of the rat (19), 104

highlighting the need for an automated method for analysis of branched vasculature morphology to 105

facilitate characterization of the renal vascular geometry of other samples, and other species 106

including humans. 107

In this paper, we propose a method for measuring the detailed geometric relationships of intrarenal 108

blood vessels using micro-CT images, with a view to using these data in future computational models 109

of renal physiology. The method includes scanning and processing images, manually labeling arterial 110

and venous vascular networks, automatically extracting centerlines for the vascular topology and 111

then automatically defining the morphological characteristics of the vessels. In the current study we 112

used this approach to generate continuous quantitative information for the renal vasculature for a 113

sample volume of a synchrotron CT image of a rabbit kidney. We compute 3D morphometric 114

parameters including crucially important transport parameters such as arteriovenous (A-V) diffusion 115

distance and wrapping fraction identified by Gardiner et al. (14), and compare them with subjective 116

manual measurements. Unlike manual measurements based on 2D images, the automatic 3D 117

morphometric parameter measurements are observed continuously along the vessel network. Based 118



on vessel radius, vessel separation distance and wrapping fraction, for the first time we estimate the 119

mass transfer coefficient as a continuous function of distances from the origin of the renal artery. 120

Finally we discuss the implications of the proposed method and provide suggestions for its 121

improvement. 122

Methods 123

CT scanning: The kidneys of New Zealand White rabbits were perfusion fixed and the vasculature 124

filled with the radio-opaque silicone polymer Microfil®, as previously described (26). These studies 125

were approved in advance by the Animal Ethics Committee of the Monash University Animal 126

Research Platform. They conformed to the Australian Code of Practice for the Care and Use of 127

Animals for Scientific Purposes. 128

Images of the renal vasculature were generated at the Imaging and Medical Therapy Beam Line at 129

the Australian Synchrotron. Moderate resolution scans were captured by an X-ray source of 32 keV. 130

Scanning was performed by placing the specimen in the path of parallel beam and recording 131

projections at uniform angular increments of 0.1 degrees as shown in Fig. 2A. Cross-sectional images 132

were reconstructed by a (Linear-Ramp) Filtered Back-Projection algorithm using the XLI-XTRACT 133

software (CSIRO, Australia) at a voxel resolution of 15.6 µm in both the object plane and the sample 134

plane. The voxel values were converted from arbitrary units by linear scaling and normalization to 135

the range (0, 255). The reconstructed slices were stored as 8-bit TIFF volumetric images. 136

Vascular segmentation: Segmentation is the process of assigning image voxels to anatomically or 137

functionally meaningful volume categories. Much of the research in image segmentation has focused 138

on using 2D image segmentation (8). Segmentation methods can be categorized as supervised 139

(manual), semi-supervised (semi-automated) or unsupervised (completely automated). These 140

methods have the potential to be extended to segmentation of 3D images, such as CT and MRI. Here 141



we implement a new semi-supervised method for 3D image segmentation to reveal physiologically 142

important anatomic details of the renal vasculature. 143

A multi-seed ‘region growing’ algorithm (2) was implemented for segmentation of arterial and 144

venous voxels. Region-growing works by grouping together voxels with properties similar to the 145

seed voxels, so forming distinct regions (2). Multi-seed region growing takes into account the 146

presence of multiple structures with uniquely labeled seeds. The following summarizes the multi-147

seed region-growing algorithm we implemented in MATLAB (Mathworks Inc., USA). 148

Initially, seed voxels are manually distributed in the vascular regions corresponding to distinct types 149

of vessel (in our case, arteries and veins). The image voxels are then subjected to a grayscale 150

thresholding to distinguish ‘vascular’ voxels from ‘non-vascular’ voxels (the threshold chosen 151

depends on image contrast). Seeding and thresholding are the only essential manual steps required of 152

this method. This is followed by the computation of geodesic distances of the vascular voxels from 153

all seed voxels. Each vascular voxel is then designated a vessel-type based on the nearest seed voxel. 154

A volumetric label image is generated which maps the vascular voxels to their identification as either 155

an artery or vein. After smoothing the labeled image with an averaging filter, a 3D vascular geometry 156

is obtained using iso-surface generation by connecting neighboring voxels of the same label (Fig. 157

2C). The geometry is a tessellated triangular mesh, which defines the vascular surface. Each triangle 158

has a unique size (of the order of voxel resolution) and position which together vary to conform to 159

the local shape of a vessel. The triangular faces share edges with their neighboring triangles forming 160

a mesh in three-dimensional space. The generated mesh is simplified by reducing the number of 161

triangles while preserving the overall shape of the vessels. The key benefit of mesh simplification is 162

that it reduces the computational requirement for processing, analysis and visualization. 163



Vascular skeletonization: The vascular topology can be defined by a skeleton of vessel centerlines. 164

Vascular centerlines are extracted using a mesh contraction algorithm (6). The algorithm achieves 165

this by iteratively contracting the mesh until all vertices approximately converge to a curved 166

centerline. The centerline is pruned, smoothed and converted to a skeleton as shown in Fig. 2D. A 167

graphical representation of the skeleton allows identification of the branch intersections (nodes), 168

vascular branches (edges) and their connectivity (adjacency). The information contained within an 169

undirected graph (𝐺) of the centerline of the vessels is given in Table 1. 170

An undirected graph of vessel centerline can be converted to a directed graph 𝐺 = (𝑉, 𝐸) by defining 171

a reference node which denotes the origin of ‘(fluid) flow’, and finding paths to all neighboring 172

nodes using a graph search (16). The root node can be automatically detected as the vessel with 173

maximum thickness or it can be specified manually. A directed graph of the vessel centerline enables 174

determination of variation in vessel attributes in the direction of blood flow. 175

Morphometric analysis: We consider a vascular branch 𝑒𝑖, whose luminal surface is characterized 176

by tessellated triangular faces 𝐬𝑖 and a vessel centerline characterized by a series of points 𝐩𝑗 , 𝑗 =177

1,2, … , 𝑚, along the centerline of branch 𝑒𝑖, which can be analyzed to estimate explanatory variables 178

describing the vascular morphology. The procedure for computation of morphometric parameters is 179

as follows: 180

Distance from root: The distance of a branch from the root vessel is the length of its shortest path 181

along the vascular tree. It is a hierarchical indicator of a branch in terms of a continuous variable (i.e. 182

distance). The distance from the root is measured with the help of graphical representation of the 183

skeleton. The undirected graph edges are given weights equal to the vessel lengths (𝐿𝑖). The length 184

of a vessel segment is calculated as the node to node geodesic distance traversed over the vessel 185

centerline, viz, 186



𝐿𝑖 = ∑ ‖𝐩𝑗+1 − 𝐩𝑗 ‖2𝑗 = ∑ √(𝑥𝑗+1 − 𝑥𝑗)

2+ (𝑦𝑗+1 − 𝑦𝑗)

2+ (𝑧𝑗+1 − 𝑧𝑗)

2𝑛−1𝑗=1 (1) 187

Then, a graph search algorithm (9) determines the shortest route of each node (𝑣𝑘) from the root 188

vessel node (𝑣𝑟). The distance from the root node is defined as 𝐿𝑖∗ = ∑ 𝐿𝑗 ,𝑗∈𝐶 where 𝐶 is the set of 189

edge indices along the shortest path from root to edge 𝑒𝑖. 190

Radius: Radius is a well-defined attribute of cylindrical structures. However intrarenal vessels may 191

not always have a circular cross-section. This is particularly true for intrarenal veins, which can be 192

‘bean shaped’ (14, 25). Nevertheless an approximate feature, indicative of the diameter of an 193

arbitrary vessel can be estimated. The distance of each point along the vessel centerline 194

𝑝𝑗 , 𝑝𝑗+1, … , 𝑝𝑚 to the nearest vascular surface is computed as: 195

𝑟𝑗 = min𝑗 ‖𝐬𝑖 − 𝐩𝑗‖2

, 𝑗 = 1,2, . . . , 𝑚 (2) 196

so the ‘median radius’ of branch 𝑒𝑖 is given as 197

�̃�𝑖 = median(𝑟1, 𝑟2, … , 𝑟𝑚) (3) 198

Diffusion distance: The A-V ‘diffusion distance’ is estimated as the shortest distances from the 199

arterial vessel surface to the nearest venous vessel surface, viz, 200

𝑑𝑗𝐴 = min𝑖 ‖𝐬𝑖

𝑉 − 𝐬𝑗𝐴‖

2, 𝑗 = arg min𝑗 ‖𝐬𝑖

𝐴 − 𝐩𝑗𝐴‖

2 (4) 201

where 𝐝𝐴 ∈ ℝ𝑚 are the shortest diffusion distances transferred to the centerline. The superscripts 𝐴 202

and 𝑉 are symbols to denote intrarenal arteries and intrarenal veins, respectively. Then, the median 203

diffusion distance along a vessel is given by, 204

�̃�𝑖𝐴 = median(𝑑1

𝐴, 𝑑2𝐴, … , 𝑑𝑚

𝐴 ) (5) 205



Wrapping: The intrarenal arteries generally have a circular cross-section. However some inrarenal 206

veins exhibit a ‘bean shaped’ (convex-concave) cross-section (25). The side of the vein facing a 207

nearby artery is concave while the other side of the vein is convex. The extent of wrapping of an 208

artery by a vein can be described by the following three independent parameters; (i) proximity of an 209

artery to a nearby vein, (ii) curvature of the vein surface around the artery, and (iii) mutual 210

orientation of the arterial and venous surfaces. 211

Within the region of wrapping, the proximity of an artery to its wrapped vein is directly determined 212

by the diffusion distance. Surface curvature quantifies the convexity or concavity of a vessel surface. 213

Gaussian curvature, defined as a product of two principal curvatures, 𝐾 = 𝜅1𝜅2 determines the extent 214

of local curvature of a surface (1). It is positive for a convex surface (bulging out), negative for a 215

concave surface (bulging in) and zero for a planar surface. 216

𝐾𝑖𝐴 = 𝐾𝑗

𝑉 , 𝑗 = arg min𝑗 ‖𝒔𝑗𝑉 − 𝒔𝑖

𝐴‖2 (6) 217

where 𝐾𝑖𝐴 is the curvature at a point 𝑗 on the venous surface 𝒔𝑗

𝑉 transferred to the nearest point 𝑖 on 218

the arterial surface 𝒔𝑖𝐴. 219

The mutual surface orientation is defined as the scalar product of the arterial surface unit 220

normal (�̂�𝐴 ) at point 𝑖, to the venous surface unit normal (�̂�𝑉) at its nearest point 𝑗. 221

𝜃𝑖𝐴 = �̂�𝑖

𝐴. �̂�𝑗𝑉 , 𝑗 = arg min

𝑗 (𝒔𝑗

𝑉 − 𝒔𝑖𝐴) (7) 222

where 𝜃𝑖 ∈ [0,1]. 223

Wrapping is defined as the fraction of the arterial surface (𝑆𝑖𝐴) area within a specified angle and 224

proximity of a concave venous surface. 225



𝑊𝑖 =1

∑𝑖𝑆𝑖𝐴

{𝑆𝑖

𝐴, �̃�𝑖𝐴 < 𝜏𝑑 ∧ 𝐾𝑖

𝐴 > 𝜏𝑘 ∧ 𝜃𝑖𝐴 > 𝜏𝜃

0, otherwise (8) 226

where τ𝑑 ∈ [0, max𝑖(�̃�𝑖𝐴)], τ𝑘 ∈ [min𝑖(𝐾𝑖

𝐴) , max𝑖(𝐾𝑖𝐴)] and τ𝜃 ∈ [0,1]. The thresholds τ𝑑 , τ𝑘 and 227

τ𝜃 control the influence of diffusion distance, curvature and mutual surface orientation respectively, 228

on the calculation of 𝑊. 229

The local mass transfer coefficient (MTC), 𝑘𝑗𝐴𝑉, is a physiologically important parameter that can be 230

derived from the previously mentioned basic parameters The local mass transfer coefficient (local 231

MTC) from an arterial branch 𝑒𝑖 to its nearest vein is defined as, 232

𝑘𝑗𝐴𝑉/𝐷 = ∑

𝑊𝑗

𝐷𝑗𝑗 , 𝑗 = 1, ⋯ , 𝑚 (9) 233

where D is the diffusion coefficient (taken to be one for convenience), 𝑊𝑗 is the local wrapping 234

percentage (expressed in percentage of the local arterial perimeter) and 𝐷𝑗 is the local diffusion 235

distance. The total local mass transfer of all branches 𝑒𝑖 at some nominated distance from the root 236

artery is: 237

𝑘𝑖𝐴𝑉 = 𝛴𝑘𝑚

𝐴𝑉 (10) 238

where the sum is taken across all branches at the same distance from the root artery. 239

We validated the proposed estimation method by comparison with subjective expert estimation of 240

three manually measurable morphological parameters, namely the radius, diffusion distance and 241

wrapping fraction. We have recently demonstrated that 2D manual measurements of these parameters 242

from micro-CT images are in good agreement with manual measurements from the same vessel pairs 243

generated from 2D histological images and light microscopy (26). Thus, the aim of the current 244

comparative analysis was to observe the extent to which the techniques for 3D semi-automatic and 245

2D manual measurement of geometric parameters, from micro-CT, agree with each other. 246



Statistical methods 247

All statistical analysis were performed in MATLAB (Mathworks Inc., USA) using the Statistics 248

Toolbox and the function GMREGRESS by Trujilo-Ortiz and Hernandex-Walls (32). It is assumed 249

random errors are normally distributed, and all statistical tests employed a two-tailed 𝑝 = 0.05 level 250

of significance. To assess the validity of measurement methods, we employed two statistical analysis 251

methods: (i) Bland-Altman plots (5) and, (ii) a combination of a Pearson correlation coefficient (r) 252

and regression analysis. As both the manual and the semi-automatic measurements potentially 253

contain random errors of similar magnitude, regression lines were fitted to measurements based on a 254

geometric mean regression (otherwise known as ordinary least products regression) (22). The 255

Pearson correlation coefficient provides an assessment of measurement precision, while the 256

geometric mean regression assesses measurement bias (and so relative measurement accuracy). The 257

degree of fixed bias was assessed from the regression y-intercept and its 95% confidence interval 258

(CI), while the degree of proportional bias was assessed from the regression slope and its 95% CI. If 259

the CI on the y-intercept contains zero, there is no statistically significant fixed bias (null hypothesis 260

1), while if the CI on the slope contains one, there is no statistically significant proportional bias (null 261

hypothesis 2). Before beginning the regression analysis, we confirmed that the Pearson correlation 262

coefficient is significantly different from zero at the two-tailed 𝑝 = 0.05 level of significance (22). 263

Results 264

A 3D tissue region of 512×512×512 voxels (i.e. approximately 512 mm3 of tissue, or 5% of a whole 265

kidney (10), involving around 130 million voxels) was sampled from a single whole rabbit kidney 266

that had been perfusion fixed and filled with Microfil®. A grayscale threshold of 228 was applied to 267

the volume. For segmentation of vessels, a total of 131 seeds (Arteries: 59, Veins: 72) were 268

distributed in the volume. The arterial and venous geometries were composed of 129,875 and 269

165,583 triangles respectively, with an average triangular face of 325.3 μm2, none exceeding an 270



aspect ratio of approximately 30. The total length (i.e. sum of all vessel lengths) of arterial and 271

venous vascular networks computed along the centerline was 56.0 mm and 49.5 mm, and the total 272

vascular surface area was 42.4 mm2 and 53.7 mm2, respectively. For manual estimation by a single 273

observer, many 2D cross-section images were generated from the sampled 3D volume. Each cross-274

sectional image was positioned at the center of an arterial branch and oriented orthogonal to the 275

vascular centerline to facilitate manual measurement by the expert. A subset of 30 branches out of 47 276

were selected for manual measurements based on the image quality of the cross-section and the 277

proximity of the artery to a nearby vein. The arterial diameter (microns), diffusion distance (microns) 278

and wrapping fraction (%) were manually estimated for the 30 vessel branches of varying size and 279

structure. All automated measurements were independently computed at each cross-section in a 3D 280

region. The thresholds τ𝑑 = 200, τ𝑘 = 0.5 and τ𝜃 = 55 were empirically chosen for computation of 281

wrapping. 282

There was excellent agreement between automatically and manually measured arterial diameters, 283

with a correlation coefficient (r) of 0.99 (Fig. 3A). Agreement was generally better for larger vessels 284

than for smaller vessels, suggesting some proportional bias (Fig. 3D). Considering the image voxel 285

resolution of 15.6 μm, it is evident that the definition of the smaller vessels in the sample volume 286

becomes a challenging task. Regression analysis showed that the 95% CI for the intercept does not 287

include zero, suggesting some fixed bias (see Table 2). A closer investigation suggests that the 288

automatic segmentation of vessels underestimates the distance between vessel boundaries, leading to 289

smaller vessel diameter (we note that just two voxels explains nearly all the fixed bias). This fixed 290

bias is confirmed by examination of the Bland-Altman plot (Fig. 3D and Table 3). This had relatively 291

less impact on estimated diameter as the vessel diameter increased, which may explain some of the 292

proportional bias (Table 2). While it was possible to relax the segmentation algorithm’s criteria for 293



boundary definition, doing so lead to false segmentation results due to ambiguity in distinguishing 294

close vessel boundaries. Crucially, this tradeoff could be reduced with higher image resolution. 295

For estimation of the diffusion distance (Fig. 3B) the correlation coefficient (r) is again pleasingly 296

high at 0.98. Given the results for arterial diameter, unsurprisingly the regression analysis again 297

suggests some fixed bias (though now just one voxel explains nearly all the fixed bias). Observation 298

of the Bland-Altman plot for diffusion distances in Fig. 3E suggests that estimates by the two 299

methods are generally better at larger diffusion distances. In agreement with the regression analysis 300

(see Table 2), this is consistent with proportional bias. 301

The local wrapping of the artery by a nearby vein (expressed in percentage of the arterial perimeter) 302

is presented in Fig. 3C. The correlation coefficient (r) was significantly less (at 0.89) than for the two 303

other measures indicating less measurement precision. The automatic method resulted in under-304

estimation of the extent of wrapping compared to manual measurements. Nevertheless, there 305

appeared to be no statistically significant fixed or proportional bias in either the regression intercept 306

or slope (Table 2), suggesting that estimates generated by the two methods are in substantial 307

agreement. 308

To better understand the causes of differences in estimates by the two methods, it is helpful to 309

examine more closely individual cross-sectional micro-CT images. Two groups of micro-CT images 310

were chosen: (i) those where estimated morphological parameters were similar, and (ii), those where 311

estimated morphological parameters were dissimilar. For the majority of artery-vein pairs, there was 312

remarkable similarity in the measures of arterial diameter and diffusion distance generated by the two 313

methods (Fig. 4A-4C). However at some cross-sections the two methods provided disparate 314

estimates. This was most evident for smaller vessels (see Fig. 4D-4F). Fig. 4D shows an example 315

where the diameter is underestimated by the automatic method due to an incomplete labeling of the 316



artery because of low image contrast. For Fig. 4E, the automatic method estimates a much shorter 317

diffusion distance. In this case, this automatic estimate may be valid because there could be a nearby 318

vein, not visible in the 2D cross-section image. Fig. 4F is an example of disagreement in the 319

wrapping fraction simply because of differences in method, suggesting an issue with reproducibility. 320

We note that most of the discrepancies occur with the smaller vessels, which suggests agreement 321

could be improved with increased image resolution. 322

Finally we determined how vessel attributes change with respect to the hierarchy of branches and 323

distance from the root artery in the vascular network within the sample volume (Fig. 5). 324

Unsurprisingly, proximal vascular branches tended to be both longer and larger than the distal 325

branches (Fig. 5A). But counter-intuitively, the diffusion distance tended to be shorter for proximal 326

branches than for distal branches (Fig. 5B). Geometric wrapping was high for proximal to 327

intermediate branches, becoming low for distal branches (Fig. 5C). The local mass transfer 328

coefficient was highly variable in larger vessels (e.g. vessel 15 in Fig. 6A), and moderately variable 329

between large vessels (e.g. compare vessel 24 with vessel 29 in Fig. 6A). However the local MTC 330

became more uniform for smaller vessels (Fig. 6A). Fig. 6B shows the integral of the local MTC 331

along the segment, while Fig. 6C shows the integral of the local MTC along the vasculature tree. Fig. 332

6D shows the (smoothed) total local MTC as a function of distance from the origin of the root artery. 333

It clearly demonstrates that, within the portion of the vasculature we analyzed, the total local MTC 334

tended to increase and then decrease with distance along the vascular network. 335

Discussion 336

While cylindrical tubes may be useful for some blood flow applications, the actual local geometry is 337

needed for estimating velocity profiles within vessels, for estimating local wall shear stresses (24) or 338

estimating red blood cell distribution (7) (e.g. Fahraeus effect). Similarly, the actual local vascular 339



geometry is required to estimate the diffusive transport of small molecules to and from arteries, veins 340

and tissues (27). In a previous study (28) the branched vessel structures found in the kidney were 341

represented as a series of connected cylindrical tubes, but this approximation leads to incorrect 342

estimates for arterio-venous (A-V) shunting of dissolved gases (e.g. oxygen, carbon dioxide, nitric 343

oxide, hydrogen sulfide etc.). To illustrate one such problem, in some cases a cylindrical tube 344

representation results in negative diffusion distances, particularly when vessels were geometrically 345

wrapped (14). As geometric wrapping is a characteristic feature of A-V renal morphology, such 346

unrealistic representations of the vascular geometry becomes a substantial problem. In contrast, the 347

criteria proposed in this paper for measurement of diffusion distance is robust and always positive. 348

To our knowledge, the current study is the first attempt to semi-automatically quantify vascular 349

wrapping continuously in 3D, a unique structural attribute of intrarenal vessels. Such data are 350

particularly valuable for estimating diffusion of gases and other small molecules between arteries, 351

veins, and tissue. 352

Renal vascular segmentation offers several challenges: 353

Contrast: If the same contrast agent is used for arteries and veins, arteries cannot be 354

discriminated from veins based on voxel intensity; 355

Proximity: As veins are often seen to wrap arteries in the kidney, boundaries between arteries 356

and veins may be indistinguishable when their separation distance approaches voxel 357

resolution; 358

Size and number: The size and number of vessels varies substantially from the proximal to 359

the distal points in the pre-glomerular circulation. This poses significant challenges for 360

sampling, characterization and visualization of these structures. 361



However, provided there is sufficient image resolution, the method of analysis proposed here can 362

overcome all these challenges and provide quantitative morphological data regarding artery-vein 363

pairs in the renal vasculature. The region growing segmentation algorithm is able to delineate arterial 364

and venous vessels semi-automatically, without the specification of any parameters. The method only 365

requires the input image and initial seed locations. We also found the mesh-based vessel center-line 366

extraction algorithm (see Fig. 2D) to be reliable and robust, in contrast to reports of other image-367

based (e.g. homotopic thinning) algorithms (21). This image-based skeletonization method was 368

sensitive to noise and resulted in a large number of spurious branches. Furthermore it was 369

computationally expensive, with a processing time of the order of 𝑁3, where 𝑁 is a characteristic 370

dimension reflecting image volume. In contrast, the mesh-based algorithm operated on a smoothed, 371

simplified vascular mesh. This facilitated efficient extraction of the vessel topology, which is of 372

practical importance. 373

The methods described above for feature extraction from the vascular geometry were validated by 374

comparison with state-of-the-art, but nevertheless subjective, expert estimates of three manually 375

measureable morphological parameters (i.e. diameter, diffusion distance and wrapping fraction). A 376

comparatively simple but robust median distance approach (Eq. 5) was employed to estimate vessel 377

diameter. Though still a single statistic, this is a more realistic criterion compared to estimates based 378

on cylindrical or spherical approximations (29). We found the vessel diameter was well estimated 379

using the median distance measurement, as verified by the excellent correlation coefficient, though 380

there was fixed bias (about 2.0 voxels in size) and some proportional bias. We expect both would 381

reduce with improved image resolution. 382

Diffusion distances between renal A-V vessels are often small (of the order of 10 to 40 microns). 383

Given the resolution of the reconstructed scans was 15.6 microns, it is unsurprising that the diffusion 384

distances were less accurate than diameter estimations, and the correlation coefficient decreased 385



somewhat. While fixed bias was detected (about 1 voxel in size), proportional bias probably arose 386

because of increasing diffusion distance with diminishing vessel diameter. The criterion for 387

estimation of wrapping fraction reported here demonstrated no bias, but the correlation coefficient 388

was considerably less than those for arterial diameter and diffusion distance. In other words, the 389

limited precision reduced the ability of the regression analysis to statistically uncover any difference 390

in relative accuracy, though the relative accuracy is probably high given the regression slope is close 391

to one. 392

The small underestimation of wrapping by the automatic method relative to the manual method can 393

be attributed to several factors including segmentation irregularities. In addition, estimation of 394

wrapping fraction requires the simultaneous application of a relatively complex set of rules. These 395

rules are unambiguously stated mathematically in the automatic method of estimation, and so have 396

the advantage of being completely reproducible, which represents a significant advantage over 397

manual methods (4). These defined rules operating on a 3D vascular model are clearly very different 398

to a subjective expert assessment of wrapping based on measurement of 2D cross-sections extracted 399

from a volumetric image. Therefore, some differences between automated and manual methods are to 400

be expected on this basis alone. Clearly, a comparison of quantities from different dimensional 401

perspectives (2D-3D) is a source of divergence in estimates. 402

Based on our 3D analysis of the renal vascular morphology of the rabbit, unsurprisingly, we found 403

the proximal vascular branches tend to be longer than the distal branches. But counter-intuitively, we 404

found that the diffusion distance between arteries and veins tends to be shorter for proximal branches 405

compared to distal branches. Both these findings are consistent with those previously reported for the 406

rat kidney (20, 25). These previous studies were based only on an extremely time consuming, manual 407

2D analysis where the proximal-distal positions along the vascular network were inferred based on 408

vessel diameter alone. Unlike previous automated studies (28), here we were able to report on vessel 409



wrapping based on 3D morphology. The artery wrapping fraction is high for proximal to intermediate 410

branches, but reduces for more distal branches. Most importantly, we see that the physiologically 411

important mass transfer coefficient is highly variable between, and even along, individual vascular 412

branch segments. We also see that the total local MTC tends to increase with distance along the 413

vascular tree, and then decrease. It is possible that this total local MTC increases again at even 414

smaller vessel sizes (20), but this would require higher resolution images. 415

Importantly, we note that these observed continuous changes in 3D morphometric characteristics 416

along the vasculature are much less uncertain than those based on 2D morphometric analysis of 417

discrete cross-sections and interpolation between the sections. Although not reported here, there is 418

potential to gain other physiologically useful insights by observing other morphometric attributes in a 419

continuous hierarchical manner using a similar approach, including vascular volumes, branching 420

angles and vascular-tissue surface area. 421

We note the application of the analysis approach presented here is limited by two major factors, 422

moderate voxel resolution and relatively low vascular contrast, which together affect the performance 423

of inter-vessel and intra-vessel discrimination and subsequent segmentation. We also note that the 424

mass transfer coefficients are only calculated for a sub-region of the kidney centered on the renal 425

artery. This is a limitation of the current analysis, but not of the method proposed for calculating 3D 426

morphology of the arteries and veins in the pre-glomerular vasculature. It is expected that with 427

improved scanning resolution and segmentation of vessels, more accurate estimates will be made 428

possible on whole kidneys based on the method proposed. Finally we mention that if reference 429

images with labeled voxels of arterial and venous vessels are available, then training of learning-430

based segmentation algorithms could be employed to fully automate the semi-automatic method 431

described above, and potentially further improve accuracy (17). 432



Conclusions 433

Detailed vascular anatomy is required for physiological models of blood flow and small molecule 434

transport within the kidney. Obtaining these data is very difficult, requiring enormous manual effort, 435

which creates a serious practical bottleneck in development of computational models of renal 436

function. Further, it is highly desirable to mathematically define the parameter estimation methods, 437

as then precisely how estimates are made is unambiguous, thereby improving reproducibility of the 438

data analysis. To overcome these issues, in the current study we developed a semi-automatic method 439

for obtaining detailed topological and geometric data on the renal vasculature, and for the first time 440

obtained continuous 3D data on vessel diameter, A-V separation and the degree of venous wrapping 441

of arteries, a prominent feature in mammalian kidneys. These 3D data were validated using manual 442

estimates of variables at 2D cross-sections. This revealed high precision, but two of the parameters 443

automatically estimated showed some fixed and proportional bias. This bias is largely explained by 444

moderate image resolution. Obtaining continuous data along the vascular tree using the 445

semiautomatic method enabled the continuous estimation of the mass transfer coefficient for 446

shunting of small molecules (such as oxygen, carbon dioxide, nitric oxide and hydrogen sulfide) 447

between arteries and veins, along the vascular tree, albeit for only a small sub-volume of the kidney 448

of a single rabbit. These data are consistent with the proposition that the local mass transfer 449

coefficient varies significantly between branch segments, and even within branch segments, while 450

the total local mass transfer coefficient increases and then decreases with distance along the vascular 451

tree. 452

Acknowledgements 453

This work was supported by a grant from the Australian Research Council (DP140103045) and 454

merit-based access to the Australian Synchrotron (M8932). 455



Disclosures 456

None 457

References 458

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543


http://www.mathworks.com/matlabcentral/fileexchange/27918-gmregress


Table 1: Information from a skeletal graph. A graph is characterized by nodes (branching 544

points or end-points) and edges (centerlines). 545

Nodes Edges

𝑣𝑘 node of intersection 𝑒𝑖 edge connecting two nodes

𝑒1, 𝑒2, … edges common to node 𝑛𝑘 𝑣𝑠, 𝑣𝑡, source and terminal nodes of 𝑒𝑖.

𝑣1, 𝑣2, … neighbors of node 𝑣𝑘 ⟨𝐩1, 𝐩2, … , 𝐩𝑚⟩ physical coordinates of edge 𝑒𝑖

546

Table 2: Mean geometric regression analysis of fixed and proportional bias between manual 547

and automatic measurements for the diameter, diffusion distance and wrapping of arterial 548

vessels. 549

Regression Intercept (95%

CI), Fixed Bias

Regression Slope (95% CI),

Proportional Bias

Diameter (µm) -34.2 (-48.6,-19.9), Y 1.07 (1.02,1.13), Y

Diffusion Distance (µm) 13.0 (5.6,20.4), Y 0.74 (0.68,0.79), Y

Wrapping (%) 1.27 (-2.7,5.2), N 0.97 (0.8,1.1), N

CI = confidence interval 550

Table 3: Bland-Altman plots with means and 95% confidence intervals: analysis of differences 551

between manual and automatic measurements for the diameter, diffusion distance and 552

wrapping of arterial vessels. 553

Mean (95% CI)

Diameter (µm) 14.51 (-52.7,-23.7)

Diffusion Distance (µm) 20.90 (-173.3,-131.5)

Wrapping (%) -0.55 (-11.5,-10.4)

CI = confidence interval 554



Figure Legends 555

Fig. 1. The phenomenon of venous ‘wrapping’ of renal arteries. Panels A and B show two 556

examples of cross-sectional micro computed tomographic images of the characteristic partial 557

wrapping of the wall of a vein (V) around the wall of an artery (A) in the renal vasculature of a 558

rabbit. Pie-sectors visually indicate the approximate fraction of the arterial surface wrapped by a 559

nearby vein. 560

Fig. 2. Schematic of the renal vascular image acquisition and processing. (A) Renal sample is 561

scanned by a micro-CT X-ray scanner. (B) Scans are reconstructed to generate a three dimensional 562

volumetric image. (C) Image segmentation is carried out to delineate arterial and venous vessels from 563

the background. (D) Vascular geometry is analyzed to generate a 3D vascular skeleton which is 564

labelled with unique indices. 565

Fig. 3. Comparison of manual and automatic measurements. Panels A-C show pairwise scatter plots 566

of manually and automatically measured parameters (3A arterial diameter, 3B diffusion distance and 567

3C wrapping fraction). Regression lines of best fit were determined by ordinary least products (i.e. 568

geometric mean regression). 𝑟: correlation coefficient. All manual and automatic measurements have 569

𝑝 < 0.05 indicating a significant correlation. Panels D-F show Bland-Altman plots of differences 570

between pairwise manual and automatic measurements against their mean. Points grouped vertically 571

closer to the mean difference line are in better agreement than those more distant. 572

Fig. 4. Comparison of manual and automatic measurements of arterial diameter, arteriovenous 573

diffusion distance and wrapping for sample images. Panels A-C show examples of good agreement 574

between manual and automatic measurements (Panel A reused from Fig. 1B). Panels D-F show 575

examples of poor agreement between manual and automatic measurements. In each panel, an original 576

cross-sectional micro-CT image is presented alongside one with an overlaid arterial and venous 577



vessel. The morphometric measurements of both automatic and manual methods is also presented for 578

each sample. A: Artery, V: Vein, Dia.: diameter, Dif.: diffusion distance, Wrap.: wrapping. 579

Fig. 5. Variation in morphological parameters with the distance into the arterial vascular network 580

(distance from the origin of the root artery). (A) Vessel size (diameter) (B) Diffusion distance. (C) 581

Wrapping fraction. Each vessel is labelled with a branch identification number. Each branch 582

thickness and color is proportionally scaled to the parameter of interest (i.e. with vessel size, 583

diffusion distance and wrapping fraction). 584

Fig. 6. Mass Transfer Coefficients (MTCs). Branch ID labels same as Fig. 5. (A) Local MTC (Eq. 585

9) for each vessel branch showing local MTC variation along each branch. (B) Total MTC for each 586

individual branch, by integrating the local MTC along each branch segment. (C) Cumulative total 587

MTC, calculated by summing all the total MTCs for all branch segments shown in Fig. 6A up to a 588

nominated distance into the vascular tree. (D) Smoothed local total MTC, calculated by first 589

approximating the S-shaped curve shown in 6C with a cubic polynomial, and then differentiating to 590

obtain the local total MTC (Eq. 10). 591

592








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Three-dimensional morphometric analysis of the renal vasculature · 3D morphometric analysis of the renal vasculature 9/14/2017 3 48 Introduction 49 The three-dimensional (3D) topology