Three-dimensional analysis of complex branching vessels in confocal microscopy images

Three-dimensional analysis of complex branching vessels in confocal

microscopy images

Mahnaz Maddaha,b, Hamid Soltanian-Zadeha,b,c,*, Ali Afzali-Kushaa,

Ali Shahroknia,b,d, Zheng G. Zhange

aControl and Intelligent Processing Center of Excellence, Department of Electrical and Computer Engineering, University of Tehran, Tehran, IranbSignal and Image Processing Group, School of Cognitive Sciences, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran

cImage Analysis Laboratory, Department of Radiology, Henry Ford Health System, Detroit, MI, USAdComputer Vision Laboratory, Swiss Federal Institute of Technology, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland

eDepartment of Neurology, Henry Ford Health System, Detroit, MI 48202, USA

Received 1 November 2004; revised 12 March 2005; accepted 12 March 2005

Abstract

The characteristic of confocal microscopy (CM) vascular data is that it contains many tiny vessels with branching and complex structure.

In this work, an automated method for quantitative analysis and reconstruction of cerebral vessels from CM images is presented in which the

extracted centerline of the vessels plays the key role. To assess the efficiency and accuracy of different centerline extraction methods, a

comparison among three fully automated approaches is given. The centerline extraction methods studied in this work are a snake model, a

path planning approach, and a distance transform-based method. To evaluate the accuracy of the quantitative parameters of vessels such as

length and diameter, we apply the method to synthetic data. These results indicate that the snake model and the path planning method are

more accurate in extracting the quantitative parameters. The efficiency of the approach in clinical applications is then confirmed by applying

the method to real CM images. All three methods investigated in this work are accurate enough to correctly distinguish between normal and

stroke brain data, while the snake model is the fastest for clinical applications. In addition, three-dimensional visualization, reconstruction,

and characterization of CM vascular images of rat brains are presented.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Vascular analysis; Centerline extraction; Snake model; Path planning; Distance transform; Confocal microscopy

1. Introduction

The ability to describe the vascular network is crucial in

the diagnosis of vascular abnormalities, surgical planning,

and monitoring disease progress or remission [1]. Recent

advances provided by three-dimensional (3D) laser scan-

ning Confocal Microscopy (CM) leads to new perspectives

for quantitative analysis of vascular networks, provided that

efficient algorithms for processing of the data are developed.

The main advantage of CM is that it allows researchers to

evaluate tiny vessels and other biological structures in three

0895-6111/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compmedimag.2005.03.001

* Corresponding author. Address: Radiology Image Analysis Laboratory,

One Ford Place, 2F, Detroit, MI 48202, USA. Tel.: C1 313 874 4482;

fax: C1 313 874 4494.

E-mail address: [email protected] (H. Soltanian-Zadeh).

dimensions. Computer commands move the microscope

stage up and down under the control of a step motor in

increments smaller than 1 mm. A clear and fully focused

image is achieved every step of the way. This allows one to

collect large number of optical sections from the sample.

These images (sections) should be 3D processed for perfect

visualization and quantification.

For cerebral vessels, measurable geometrical changes in

the diameter, length, branch point density, or tortuosity

caused by disease have been defined [2,3]. Therefore,

reconstruction and measurement of them can be used to

quantify the severity of disease, as part of the process of

automated diagnosis of disease or in the assessment of the

progression of therapy [4]. The vessel centerline can serve

as a basis for the description of the vessels enabling the

clinician to obtain quantitative measures of the vessels of

interest [5]. A variety of approaches has been suggested in

order to obtain vessel centerlines from 3D images of

Computerized Medical Imaging and Graphics 29 (2005) 487–498

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M. Maddah et al. / Computerized Medical Imaging and Graphics 29 (2005) 487–498488

the vasculature. They include intensity-based methods [5,6],

generalized cylinder approximations [7], multi-scale [8,9]

and skeletonization schemes [10–12], tracking [13], region

growing [14], and deformable model approaches [15],

applied on successive two-dimensional (2D) slices or on

volumetric data. It should be noted that the imaging

modalities lead to very large data sets from which the

vessels and the quantitative parameters must be extracted

under very strict time constraints to meet the clinical and

research requirements.

In addition to speed, accuracy of detected centerlines is

of prime concern. In simple cases, where accurate

centerlines can be readily identified, quantification of

individual vessel segments is easily done from an

approximate position of the centerline. In complex images

with poor contrast that have touching or overlapping

structures, however, accurate centerline extraction is a

challenging task [16]. For 3D quantitative analysis, the

centerline accuracy becomes an even more critical issue.

The unavailability of manual intervention can also

introduce computational error. Generally, automated vessel

size measurements [17] are preferred over visual interpret-

ation because of the inter- and intra-observer variability [18,

19]. In most of the 3D imaging applications, the users are

interested in 3D visualization of the data, e.g. colonoscopy

or angiography, and sometimes in quantification of a tissue

or organ, e.g. a tumor in the brain or left ventricle of the

heart. However, in our application, all of the vessels should

be identified and quantified. Due to the large number of

vessels in the confocal microscopy images, supervised and

semi-supervised methods used in other applications are not

directly applicable to them. Therefore, we have developed,

evaluated, and compared specific methods for them.

Another issue is validation of the proposed methods

which for most algorithms is based on at least one of the

following items:

(1)
Comparing the results with specialists’ assessments; the
extracted centerlines are evaluated either visually or by

calculating distances between tracked and user-indi-

cated centerlines, which are considered as the ground

truth [16,19–21]. This can also be accompanied with a

subjective evaluation or a pilot study to validate the

results [4,16].

(2)
Generating phantoms of known parameters, taking their
image and comparing the results of processed images

with the known features [22–25]; or

(3)
Using synthetic images [13,26–28]; synthetic data used
in [26] is in 2D form and in [13] synthetic structures are

applied without providing any quantitative results.

The first evaluation method suffers from the lack of

objective criteria as well as inter-observer variability,

whereas the second requires a full imaging system to

evaluate the method. Also, the inherent complexity of CM

images of tiny vascular networks makes the use of the above

two validation strategies infeasible. The only existing option

is generating 3D synthetic images of tubular structures that

resemble the features of the vessels.

The objective of this work is to present a method for the

analysis of CM vascular images. It consists of two steps:

centerline extraction, where we compare three fully

automated centerline extraction methods, and quantitative

measurements, which estimate the length, diameter, and

other quantitative parameters of the vascular dataset. We

assess the accuracy of the estimated centerline and the

measured topological parameters of the corresponding

vessel using synthetic data. We also use 12 sets of real

CM images of vascular structures of normal and stroke rat

brains to evaluate and validate the methods.

The organization of the paper is as follows. The

segmentation and centerline extraction methods are intro-

duced in Section 2, while the estimation of the quantitative

parameters is described in Section 3. Validation methods

and comparison of three centerline extraction methods are

presented in Section 4. Quantitative analysis results

are presented in Section 5, followed by the summary and

conclusions in Section 6.

2. Segmentation and centerline extraction methods

2.1. Vessel segmentation

The focus of this paper is not on segmentation, so we

used the straightforward approach of applying a global

threshold, automatically determined from the image histo-

gram (one-half of the gray level corresponding to the second

major peak of the histogram). More sophisticated segmen-

tation algorithms can be employed to further improve the

performance of the methods when dealing with clinical data.

Median filtering is then used to remove the isolated islands

and fill small holes due to imaging distortion and noise. The

binary structure of interest is segmented using a region-

growing algorithm [29] and the last voxel marked, is chosen

as the Start Point (SP) of the vessel’s branches. The Start

Point is usually at the end point of a branch or very close to

an end point. Furthermore, as will be shown later, the snake

model allows the end points of the branches to move around

slightly toward the actual end point during the centralization

process.

2.2. Vessel centerline extraction

Once vessels are segmented, we need to extract 3D

centerline of the branching vessel. Three methods are briefly

described and compared here, namely the snake model, the

path planning approach, and the distance transform based

method. Fig. 1 schematically shows how these centerline

extraction algorithms work. The general concept of these

algorithms is well known and different groups have applied

them to different problems. We have previously applied

Fig. 1. A schematic view illustrating the essence of the three centerline extraction algorithms studied in this work: in the snake model (a), each initial path is

pushed towards the center of the vessel branch by the gradient field computed based on the boundary-seeded distance transform (d). In the path planning

approach (b), at each iteration, the surface of the vessel is removed and a new path is constructed, starting from the end point (EP) and following the shallowest

descent on the new surface based on the single-seeded distance transform from the start point (SP), until either SP or a point belonging to another path is

reached. In the DT-based approach, for a set of points with equal single-seeded DT from SP (c), the one with maximum boundary-seeded DT from the surface

(d) is selected as the next point of the path.

M. Maddah et al. / Computerized Medical Imaging and Graphics 29 (2005) 487–498 489

these algorithms to CM images with necessary modifi-

cations needed to handle such images [30–32]. The

modifications made the methods fully automated and

extended them to work for branching structures as well.

However, in the previous works, comprehensive evaluation

and quantification have not been performed. To the best of

our knowledge, no other method has been reported for

centerline extraction of CM vascular images.

2.3. Snake method

This approach [30] makes use of a variant of the snake

models for centerline extraction. The basic idea was first

proposed by Kass et al. [33] and since then it has been used

in many medical image analyses, mostly in segmentation

tasks. For centerline extraction, the snake method is a

generalized and extended version of the method proposed

by Cuisenaire et al. [15], which was proposed for virtual

endoscopy navigation. The method consists of two steps:

Initialization. Given the segmented vessel, a distance

map is computed for the voxels on the surface of the vessel,

using the Start Point, SP, as the reference point. The

endpoints of the branches are then defined as the voxels

having the local maximum distance. With the end points and

the start point determined, the initial paths can be generated

on the surface, starting from each end point, following the

steepest descent and finishing at SP. The first path or the

main branch is constructed by connecting the end point with

the maximum distance value to the selected SP by

monotonically descending through the distance map. For

the rest of the end points, if any, paths are constructed

starting from each end point and terminating when either SP

or a voxel on the previous paths is reached. At the end of this

stage, we have paths for the number of object branches

along with their corresponding end points and start points.

These initial paths are considered as snakes to be moved to

the center of the object.

Centralization. As shown in Fig. 1(a), the snakes are

evolved in order to minimize the image potential energy,

which is defined here as the computed distance map from

the boundary of the object (Fig. 1d). Forces are applied to

the snakes to shift them in the direction of decreasing the

potential energy. After a few iterations, the snakes will be

located at the center of the object, where the energy has its

minimum value. In order to avoid the coarse paths extracted

in [30], a set of equi-spaced voxels on the initial paths are

chosen as the landmarks. These landmarks are evolved in

order to minimize the image potential energy. When

displacing the voxels of the snakes, they might overlap

with or get far from each other. To obtain connected paths as

the centerlines of the object an up sampling is performed. A

simple algorithm is developed to march on the final

centerlines and update the start point of each branch.

Starting from an end point, each voxel of the path is checked

to assess if it belongs to the previous paths and if so, this

voxel is set as the start point of the path.

2.4. Path-planning method

This method [31] is based on thinning and path-planning

approaches. Path-planning method [34] has application in

virtual endoscopy and robotics and along with


morphological operations has been applied for centerline

extraction of vascular images in the following approach:

Initialization. Similar to the snake method, first the

endpoints of the branches are identified and successive set

of neighboring voxels on the surface of the vessel, called

initial paths, connect each endpoint to the SP, following the

steepest descent on the distance map.

Centralization. To move the initial paths toward the

centerline of the object, a thinning algorithm [31] is applied

to the segmented object. The surface of the object is

iteratively removed by morphological operations and after

each removal the distance map from SP is computed on the

new surface. All new paths are then constructed following

the shallowest descent on this map, starting from each end-

point and ending when SP or a voxel on previously

constructed branch (within the same iteration) is reached

(Fig. 1b). In case there is no voxel on the new surface to

preserve the connectivity of the path, a voxel of the previous

path is selected. This iterative process ends when there is no

surface to be removed. The last constructed path is

considered as the centerline of the vessel.

2.5. DT-based method

Distance transform, first introduced by Blum [35], has a

vast application in skeletonization. The DT-based thinning

method proposed by Shahrokni et al. [32] is a fast and

efficient implementation of the skeletonization method

proposed by Zhou and Toga [29], applied to vascular

centerline extraction. The algorithm is summarized below.

Initialization. Two distance maps, boundary-seeded (BS)

and single-seeded (SS), are generated to approximate the

distance of the object voxels from the boundary voxels and a

single reference point (SP, here), respectively (Fig. 1c and

d). A set of connected voxels having the same SS-field is

called a cluster and interconnection of these clusters form a

directed graph [29].

Centralization. To have a centerline from the graph, from

the voxels of the cluster, a voxel with the maximum value of

BS-field, called a medial voxel, is selected. The basic idea of

skeleton generation is to obtain and connect the medial

points of all object clusters. A path is defined as the smallest

set of object voxels connecting two points. A medial path is

derived from the original path by replacing its voxels by the

medial point of the corresponding clusters. The search for

the medial points starts from the reference point and

continues along cluster graph in ascending order with

respect to their SS-field value. At each algorithm iteration, a

code is associated with each of the medial points and

determines to which branch of the object they belong. A list

called ‘parent list’ is created which initially holds SP as the

single parent of the subsequent medial points. Any newly

found medial point corresponding to the next value of the

SS-field is saved in a list called ‘child list’. Then, the parent

of each voxel in this list is selected from voxels in the parent

list. Once all of the voxels in the child list have been

assigned parents, one of the following conditions occurs for

a parent voxel and its children. (1) A parent voxel has no

child voxel which means the parent voxel is a branch-ending

node. (2) A parent voxel has exactly one child voxel. This

happens for the middle voxels along an object branch. (3) A

parent voxel has more than one children voxels which

happens when a parent voxel is a branching node and

multiple branches emanate from a single stem. Then, new

codes are generated for the new branches. Finally the parent

list is replaced by the child list and the child list is reset. The

algorithm is repeated until the clusters associated with the

maximum value of SS-field are processed.

Fig. 2 illustrates the centerlines extracted for a vessel by

the three methods, namely, DT-based, path planning and

snake algorithm.

3. Quantitative measurements

Not only the extracted centerlines provide useful

information on the shape of the vessels but also some of

the parameters, which have significance in medical

applications, can be derived from them as we discuss here.

(A) Number of bifurcation points. In each of the three

centerline extraction methods, the number of bifurcation

points can be determined. In the snake and path-planning

methods, the numbers of refined (final) paths, along with

their corresponding start and end points are known. So, the

number of bifurcation points can be obtained by counting

the non-identical points in the set of start points. In the DT-

based method, the number of junctions can be determined

through the code assigned to each branch [32].

(B) Length. To estimate the length of vessel’s branches,

their corresponding digital centerlines are utilized. Many

algorithms have been proposed to estimate the length of

digital lines/curves, such as local metrics [36,37] and

polygonalizations [38,39]. Although it might seem a trivial

task, it is still an issue under study in order to extend the

methods to 3D, to suppress the error or decrease the

processing time. Coeurjolly and Klette [40] made a

comprehensive comparison between 2D digital length

estimators. Available 3D methods need a similar work

that is beyond the scope of this paper.

The approach we follow here is categorized as a 3D local

metric estimator, called sample-distance method [36], in

which the Euclidean Distances (ED) between some selected

voxels of the centerline are summed to estimate the length

of the path. The selected voxels include SP, endpoint of the

branch, and successive centerline’s voxels at a pre-defined

distance, for example, at every three or five voxels of the

centerline. This is more accurate than other methods to

estimate the length of a tubular structure from the extracted

centerline such as counting the number of voxels [41],

computing the ED between the extreme points of the path or

adding up the ED between all voxels of the path (counting-

distance) [4]. The errors of counting and sample-distance

Rotation Angle0 10 20 30 40 50 60 70 80 90

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Qua

ntiz

atio

n E

rror

(a)

(b)

(c)

-0.02

Fig. 3. Quantization error percentage in length estimation vs. the rotation

angle for a 100-pixel straight line. The Euclidean distance was computed at

every (a) 1, (b) 3 and (c) 5 pixels and the sum was considered as the

estimated length.

0 10 20 30 40 50 60 70 80 900

5

10

15

20L = 100L = 10L = 1000

Rotation Angle (degree)

Err

or (

%)

Fig. 4. Error in length estimation vs. the rotation angle for different lengths.

As can be seen, decreasing the length increases the standard deviation of

error not its average value.

Fig. 2. A cerebral vessel from CM images (a) and its extracted centerline

by: (b) DT-based [32], (c) path-planning [31], and (d) snake model

algorithm [30].


estimation are shown in Fig. 3 for a straight line with the

length of 100 units. In sample-distance method the farthest

the samples are selected, the less error we have for a straight

line but apparently not for curves. Therefore, a trade-off

should be made between bypassing some curvatures by

increasing the sampling distance and reducing the error by

decreasing it. Note that the length variation does not affect

the average quantization error but changes its standard

deviation as shown in Fig. 4.

When these lines are extracted by an algorithm as a

centerline of a vessel, the error will be increased with

respect to the algorithmic error. So, a family of straight

synthetic vessel with the same length and different angles in

range of 0–90 degrees to the x-axis were generated (Section

4). Each vessel is given to the centerline extraction methods

and the length of obtained centerline is computed. The

average of error in length estimation over the rotation angle

is given in Table 1 for each method. From Table 1 and also

the fact that the average quantization error is independent of

the length, we can conclude that the length of centerline

extracted for a straight vessel is over-estimated in average.

For example, for the path planning approach, there is 15%

over-estimate in average. Therefore, we subtract these

average error percents from the computed lengths, to

Table 1

The average error in length estimation of a rotated vessel for the three

centerline extraction methods with the sample distance of 3 and 5. Since the

length is overestimated by some percent for every method, the

corresponding average percent of overestimation is subtracted from the

computed length. The obtained mean and standard deviation values are

shown in the table below for each method

Methods Length estimation error (%)

Sample-distance 3 Sample-distance 5

DT-based C15 C4.9

Subtracting the average K1.7G7.9 K0.16G3.4

Path-planning C11 C5


Snake C4.2 C2.1



remove the estimation bias and minimize the average error

as the real vessels are in all directions (Table 1).

(C) Diameter. Using the segmented vessel data, diameter

is computed at each voxel of the centerline. First, the

perpendicular plane is obtained and then the number of

voxels belonging to both this plane and the vessel is

considered as the area of the cross-section and the

equivalent diameter is computed. This is done for a number

of the centerline’s voxels and the average value gives the

estimated diameter of the branch.

Since diameter is a quantity by which the cerebral

vessels are categorized, e.g. to capillary and pre-capillary

vessels, it requires an accurate estimation. We use the

above procedure to get a preliminary estimate of the

diameter. Then, we use vessel’s volume and length, as

described in part D of this Section, to improve accuracy of

diameter estimation.

Diameter estimation has been addressed in other work,

mostly on 2D processing (see e.g. [26]). Tata and Anderson

[3] conduct manual measurements by projecting sampling

grids into the microscopic images, so that the data can be

collected directly from the optical image. In the case of 3D

analysis, Wink et al. [20] proposed a method in which the

minimum distance from the border of the vessel is used. In

[41], the diameter is approximated from the distance map

computed from the boundary. This method runs fast but is

inaccurate due to the approximation of Euclidean distance

in the DT-mapping.

(D) Volume of vessel branches. The volume of each

branch is estimated by the multiplication of the obtained

values for the length and the area. Subtracting the sum of

branch volumes from the total volume of the vessel

represents the amount of error in the quantitative estimation.

We distribute this volume error among branch volumes

proportional to their lengths, so that the total of the new

branch volumes equals the vessel volume. Relying on the

accuracy of length approximation (which will be justified in

Section 4), a new estimation is made from the branch

volume and its length.

4. Validation method

Since the quantitative parameters for real datasets are not

known a priori, we employ 3D synthetic data to evaluate the

accuracy of the algorithms in estimating the quantitative

parameters. Note that there are two main steps involved in

the analysis of the CM images: ‘segmentation’ and

‘quantification’. The focus of this paper is on the

‘quantification’ part. Thus, we use the same real data and

segmentation method for the three methods so that our study

of the ‘quantification’ methods is not biased by the ‘image

acquisition’ and ‘segmentation’ methods. There may be

inaccuracies in the segmentation results due to the physical

characteristics and limitations of the imaging system or the

segmentation method used but their study and evaluation are

beyond the scope of the current paper. For the same reason

we use binary images as the synthetic data though it is

possible to add noise and intensity inhomogeneities to them.

After the vessels are segmented from the images (both

normal and abnormal cases) they can be considered as a

combination of simple structures like those considered in

our synthetic data. Thus, we would expect the ‘quantifi-

cation’ algorithms to provide accuracies similar to those

obtained for synthetic data when dealing with real data.

Vessel generation algorithm. An algorithm was devel-

oped to generate 3D vessels. A parametric and continuous

curve is given to the algorithm. This curve is quantized and

considered as the centerline of the synthetic vessel. For each

voxel on the centerline, a sphere volume with a predefined

radius is grown from the voxel as its center. This enables us

to have a synthetic vessel with known centerline, diameter

and volume. The length of the continuous centerline can be

calculated by integration, while the mean diameter of the

vessel can also be calculated if the radius is not constant. As

seen in Fig. 5, this algorithm can generate straight, curved,

and branching structures.

Centerline extraction. Centerlines of the simulated

vessels were extracted by the three methods: DT-based,

path planning, and snake. Four vessels along with their

representative centerlines extracted by the methods are

illustrated in Fig. 5. Object 1 is a nearly straight vessel with

a constant diameter of 7 and a real length of 109. Object 2 is

a curved structure with the same diameter of 7 and a real

length of 75. The DT-based method recognizes two

branches due to failure to track the centerline continuously.

However, the path-planning and snake methods generate

connected centerline as shown in Fig. 5(e). Object 3 is

similar to object 1 but with varied diameter, with an average

of 8.5 and a standard deviation of 1.5. Object 4 is a

branching structure with constant diameters of 5 and 3 for

the main and winging branches, respectively. In the path

planning and the snake methods, the number of extracted

endpoints defines the number of branches, and, hence, two

branches are recognized by these methods. In the DT-based

method, the bifurcation point is the important factor for

Fig. 5. Examples of synthetic vessels: (a) Object 1: a straight vessel with the length of 109 and constant diameter of 7; (b) Object 2: a curved vessel with the

length of 75 and constant diameter of 7; (c) Object 3: a straight vessel with the length of 109 and varying diameter of 8.5G1.5; (d) Object 4: a branching vessel

with the lengths of 109 and 25 for each branch and diameters of 5 and 3. The overlaid centerlines are extracted by DT-based method except for (e) where the

snake method is used.


specifying the number of branches, which for this object

leads to three branches.

Quantitative analysis. Table 2 represents the results of

the quantitative analysis for the four synthetic objects

(simulation studies). In this table, the real values for the

length and diameter are listed along with the results of

Table 2

Estimated length and diameter for four synthetic objects. The values obtained for

diameter of objects. As can be observed, in overall, sample-distance of 5 leads to

Real value DT-based

Sample-

distance

3

Sample-

distance

5

Prelimi-

nary est

mation

Object 1 Length 109 101.3 112.1 117.5

Diameter 7 7.3 6.97 6.5

Object 2 Length 75 – –

Diameter 7 – – –

Object 3 Length 109 110 115 114.3

Diameter 8.5G1.5 8.8 8.6 7.9

Object 4 Branch 1 Length 109 124 116 121.6

Diameter 5 4.85 4.7 4.7

Branch 2 Length 25 20 22 23.7

Diameter 3 3.1 3 3.3

the three centerline extraction methods. For object 2, the

quantitative parameters are not brought for the DT-based

method due to the lack of a fine centerline. The

disconnectivity of this extracted centerline and the fact

that two branches were recognized instead of one, falsely

increase the number of branches in a real data set.

sample-distances of 3 and 5 can be compared to the real value of length and

a better approximation

Path-planning Snake

i-

Sample-

distance

3

Sample-

distance

5

Prelimi-

nary esti-

mation

Sample-

distance

3

Sample-

distance

5

Prelimi-

nary esti-

mation

104 110 115.5 113 113 115.37

7.2 7 6.5 6.9 6.9 6.5

70.7 73.6 77.3 73.4 72 73.5

7.3 7.1 7.1 7.1 7.2 6.5

105 111 116.5 115 113 115.4

9 8.7 7.9 8.6 8.6 8.4

102.6 108.6 113.4 109.6 110 112.3

5.3 5.2 5.14 5.2 5.1 4.8

21.5 22.6 23.7 23 23 23.5

3.9 3.8 3.6 3.8 3.8 3.3

Table 3

Average errors in both length and diameter estimation of objects 1-4

Average error

Sample-dis-

tance 3

Sample-dis-

tance 5

DT-based Length 3.3% 2.3%

Diameter 5.3% 2.6%

Path-planning Length 4.2% 1.7%

Diameter 4.75% 1.0%

Snake model Length 1.5% 1.5%

Diameter 2.2% 2.5%


It, however, will not adversely affect the quantitative

analysis and the categorization of the vessels in terms of

their diameter, and therefore, the total length of the capillary

or pre-capillary vessels obtained through DT-based method

will be close to the results of the other two methods. For

object 4, since three branches are recognized by the DT-

based method, the average of the estimated diameter and the

sum of the lengths of the two branches were computed and

given in Table 2. These two branches correspond to the

main branch of the other two methods making the

comparison reasonable.

To summarize, the average of percent errors in

estimation of the length and diameter of the objects, for

each of the centerline extraction methods, are given in

Table 3. As can be inferred from Table 3, increasing the

samples distance not only reduces the quantization error but

also smoothes the extracted centerlines. This is especially

the case for the DT-based method whose extracted center-

line is accurately centered but is not smooth. In the case of

the snake model, the choice of samples distance does not

affect the results greatly because the centerlines are

smoother. The sample-distance value of 5 therefore, was

used for the quantitative analysis of real data. With this

distance, one overlooks the curvatures that may exist in

intervals with the length of less than five voxels. Path

planning seems to be most accurate of all for sample-

distance of 5. However, the aim of Table 3 is not just to

compare the error values for different algorithms. It also

compares the effect of sample-distance value and shows

that, with a value of 5, all algorithms generate reasonable

results.

5. Analysis of vascular images of confocal miroscopy

The real medical images collected for the study included

12 volumetric confocal microscopy image sets of vascular

structures from nine normal and three stroke rat brains. The

sections were imaged with a Bio-Rad MRC 1024 (argon and

krypton) laser scanning confocal imaging system mounted

onto a Zeiss microscope (Bio-Rad; Cambridge, MA). The

image size was 260.6!260.6 mm in the X and Y directions

and 1 mm increment in the Z direction.

Visualization and reconstruction. A 3D visualization of

vessels from a normal rat brain is illustrated in Fig. 6(a)

where the complex structure of the vessels is obvious. The

overlaid centerlines have been extracted by the modified

snake method. Using the estimated diameters, the vascular

network has been reconstructed as shown in Fig. 6(b). The

vessels with the length of less than 15 mm have been ignored

in the reconstruction process. Fig. 7 visualizes vessels of a

stroke rat brain.

Quantitative parameters for research applications. The

mean and standard deviation of quantitative values obtained

by the three methods of centerline extraction are shown in

Table 4. The corresponding correction factors for the length

estimation, extracted through synthetic data analysis, were

applied in each method. The total length and volume of

capillary and pre-capillary branches were extracted for each

dataset. The capillary type vessels are defined as ones that

have diameters in the range of less than 7.5 micron while

pre-capillary diameters are defined to be between 7.5 and

30 micron. The ratios of the capillary and pre-capillary

volumes to the tissue volume as well as the total volume of

vessels to tissue volume were also calculated. In the table,

the number of branch points in the nominal volume is the

number of capillary/pre-capillary branches over the total

volume multiplied by 1.8!106 mm3, which is the nominal

volume of the tissue volume imaged by the confocal

microscope. Since not all of the image sets cover the same

exact volume, to be able to compare the number of branch

points in the same volumes, we calculate the numbers for a

nominal volume.

Vascular characterization. Quantitative analysis is

beneficial for diagnosis and evaluation of therapy response.

In Table 4, the differences between the extracted parameters

for the normal and the stroke rat brains can be observed. An

expert physician selected the normal and stroke datasets for

processing and examined the segmentation and quantifi-

cation results in conjunction with the original CM images

and approved their overall quality.

With the existing CM technology, it is not feasible for an

expert to extract the centerlines from 3D CM images

correctly and get quantitative parameters. Thus, we used the

effectiveness of the algorithms in clustering of the

quantitative results as the best practical measure.

Fig. 8 illustrates the capillary length normalized by the

total volume vs. the pre-capillary length normalized by

the total volume. Based on these figures, the stroke and the

normal datasets can be easily differentiated. To show which

of the methods better clusters the datasets, we define a

representative for each cluster, which is the average of the

points belonging to each category. The standard deviation of

each cluster is also calculated. We use the ratio of the

Euclidean distance between the representatives to the

standard deviation of the normal cluster (since it is much

greater than that of the stroke cluster) for evaluation. The

obtained values are 3.26, 3.19 and 2.83 for the snake,

path-planning and DT-based methods, respectively.

Fig. 6. (a) 3D visualization of vessels from a 256!256!39 images of a normal rat brain with the overlaid centerlines; (b) Reconstructed dataset.


Thus, the snake method is superior to the other two for

automatic vascular characterization.

Processing time. The CPU time for processing the

images is obtained for the three methods. The codes were

implemented in MATLAB and were run on a PC with the

Fig. 7. 3D visualization of vessels from 256!256!29 ima

clock frequency of 1000 MHz CPU and 512 MB of RAM.

Fig. 9 illustrates the computation time vs. the number of

branches for each method. As can be seen, the relationship is

linear for the snake method and quadratic for the path

planning and DT-based approaches. Note that in the snake

ges of a stroke rat brain with the overlaid centerlines.

Table 4

Extracted quantitative parameters for CM images of 9 normal and 3 stroke rat brains by three methods. The number of images in normal datasets was 32G5 for

stroke datasets was 25G5. Vessels are categorized to capillaries and pre-capillaries based on their estimated value of diameter. Mean and standard deviation of

total length and volume of each category as well as specification of each dataset and processing time are given. Note that the given processing time for the snake

and path-planning methods are from un-optimized MATLAB codes

Number of branches Total length (mm) Total volume (mm3) Tissue volu-

me(mm3)Capillary Pre capillary Capillary Pre-capillary Capillary Pre-capillary

Snake Normal

Average 68.44 13 5186.51 655.00 108708.88 52415.38 2508016.7

Std. dev. 12.56 8.99 893.86 495.35 31746.07 42181.34 371978.88

Stroke

Average 42.33 3.66 1872.88 323.47 25676.77 26831.91 1945875

Std. dev. 6.60 1.70 369.81 71.97 4158.32 8474.32 386571.79

Path-planning Normal

Average 66.90 9.22 5602.54 551.13 119239.57 42078.97 2508016.7

Std. dev. 14.41 7.40 1048.90 495.73 37648.51 40720.95 394543.18

Stroke

Average 42 2 2092.46 258.82 24575.85 28336.66 1945875

Std. dev. 4.36 1 415.36 12.18 4411.30 11580.07 473451.82

DT-based Normal

Average 106.33 21.89 4726.72 711.18 105405.67 54284.51 2508016.7

Std. dev. 20 16.75 916.42 581.30 32479.53 41676.78 394543.18

Stroke

Average 59 4 1774.47 171.98 25193.92 27231.07 1945875

Std. dev. 8.72 0 300.07 64.63 2163.91 14782.68 473451.82

Capillary vol./

tissue vol. 3.

Pre-capillary

vol./tissue vol.

Vessel vol./

tissue vol.

Number of branch points/

volume

Total number

of vessels

Processing

time (s)

Snake Normal

Average 0.04 0.02 0.06 49.46 9.06 23 864.88

Std. dev. 0.01 0.014 0.02 7.37 5.66 5.25 260.94

Stroke

Average 0.01 0.013 0.03 39.56 3.47 56.67 793.33

Std. dev. 0.000 0.002 0.001 2.076 1.40 3.86 172.80

Path-planning Normal

Average 0.047 0.02 0.06 48.21 6.46 23 2553.55

Std. dev. 0.01 0.01 0.02 8.39 4.60 5.57 1798.30

Stroke

Average 0.01 0.01 0.03 39.93 1.78 56.66 874.33

Std. dev. 0.002 0.003 0.001 6.67 0.62 4.72 257.89

DT-based Normal

Average 0.04 0.021 0.06 76.72 15.21 22.33 421.78

Std. dev. 0.01 0.014 0.02 10.90 10.20 5.61 60.91

Stroke

Average 0.01 0.013 0.026 55.60 3.88 50 346.67

Std. dev. 0.003 0.005 0.002 6.36 1.10 4.36 95.89


and path-planning methods, the paths should be initialized.

The initialization time increases with the number of

branches but it is the same for both algorithms. In the

Snake method, each path is centered almost independently

of other branches. The time depends on the number of the

landmarks, which in turn depends on the number of

branches and the total length. In the path-planning

algorithm, in each iteration of thinning, all paths are

updated traversing the DT. In each iteration, when a path is

constructed, it is checked with all previously constructed

paths to see if they make a branching point. This step

depends on the number of branches squared. For DT-based

method, similar to the path-planning method, the curve is

quadratic. Note that the given time in Table 4 is from

running C codes of DT-based method, which is about 15–20

times faster than the corresponding codes in MATLAB.

6. Summary

Confocal Microscopic images of rat brain are three-

dimensional complex branching vasculatures, which are to

be fully automatically processed. In this paper, we

compared three centerline extraction methods applied to

these images: DT-based, path planning, and snake

approaches. In terms of accuracy, all three methods are

accurate enough to correctly cluster the images to the

normal and stroke cases (Fig. 8) but the results from the four

0 5 10 15 20 25 30 35 40 45 500

5

10

15

Capillary Length / Total volume1/3

Pre

-Cap

illar

y Le

ngth

/ T

otal

vol

ume1/

3

StrokeDatasets

NormalDatasets

Fig. 8. Pre-capillary length to the total tissue volume vs. capillary length to

the total volume for snake (,), path-planning (B) and DT-based (6)

methods. Filled marks represent the results of stroke datasets, which are

well separated from the results of normal ones.


synthetic objects studied in this work showed that in overall

snake and path-planning methods give more accurate results

for the length and diameter than the DT-based method

(Table 3).

In terms of speed, the snake approach is computationally

less expensive (Fig. 9). It does not have the iterative

thinning procedure as in the path-planning method and

requires less distance transform mapping compared to the

DT-based approach. In the DT-based method, two different

distance transforms are computed for all voxels of the object

while the snake algorithm uses one distance transform for

all voxels of the object and another one only for the voxels

on the surface of the object. In addition, the snake approach

showed a linear behavior in the processing time when

Path-Planing

102

103

104

Number of Branches

Pro

cess

ing

Tim

e (s

)

~ x 2.1

~ x 1.1

DT-Based

Snake

Fig. 9. The processing time vs. the number of branches for the DT-based

(6), path-planning (B) and snake (,) methods. The curve is nearly linear

in snake approach and quadratic in path-planning and DT-based methods.

the number of branches increases, while the other two

showed a quadratic trend (Fig. 9).

It is worth mentioning here that the DT-based method is

able to extract the centerline of structures with loops (like

the letter ‘p’), while the two other methods lack this

property. This is due to the end-point selection procedure

[30]. However, as far as we deal with tree-shape structures

like vessels, this is not a concern and the snake approach

would be the best possible choice in terms of accuracy and

processing time.

In this work, we also presented an approach for

quantitative analysis and reconstruction of cerebral images,

which is efficient for clinical and research applications.

Synthetic objects and real medical images of the confocal

microscopy were used to validate the results and confirm the

efficiency of methods for clinical applications.

The quantitative results from the analysis of the confocal

microscopy images of the normal and stroke rat brains

illustrated that the investigated methods were sensitive to

the differences between the two groups based on the

number, total length, and relative volumes of both

capillaries and pre-capillaries. The snake method was

superior to the other two methods considered in this study,

although the results of the path-planning and DT-based

methods also grouped into two distinct clusters. Important

applications of the methods presented in this paper include

evaluation of treatment responses and development of new

drugs.

Acknowledgements

This work was supported in part by grants from the

Research council of the University of Tehran and the

Institute for studies in Theoretical Physics and Mathematics

(ipm), Tehran, Iran. The authors would like to thank Ali

Khakifirooz for his helpful discussion and kind assistance.

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Three-dimensional analysis of complex branching vessels in confocal microscopy images

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