Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
.pdf
54 2 Morphometry of Coronary Vasculature
https://t.me/med1917
Fig. 2.18 Schematics of hierarchy of reconstructed tree. Epicard and endocard correspond to
epicardium and endocardium, respectively. Reproduced from Kaimovitz (2001) with permission
2. Transmural networks span orders 8 to order 5 which penetrate the myocardial
wall at a right angle to the epicardium towards the endocardium.
3. Perfusion networks start with vessels of order 4 and extend to the capillary level.
Each of these perfusing networks was positioned within a specific wall layer. The
following additional assumptions were made:
1. Epicardial and perfusion networks have a planar geometry in parallel to the wall
layers.
2. The networks have a diverging (arterial) and converging (venous) tree topology.
3. Collaterals (arterial) or anastomoses (venous) were not considered.
4. Each arterial tree was drained by two venal counterparts (Bassingthwaighte et al.,
1974).
5. Length of sequential segments were not correlated, nor were the lengths of sister
segments, nor were expansion ratio (VanBavel & Spaan, 1992 ).
6. Bifurcation and trifurcation branching were considered for arteries (Kassab et al.,
1993) and bifurcations up to quantifications were considered for the venous
system consistent with measurements (Kassab et al., 1994b).
2.4 Integration of 3D Coronary Vasculature 55
https://t.me/med1917
2.4.5 Reconstruction Approach
The reconstruction procedure was composed of the following seven steps:
1. Partition into functional sub-networks: epicardial, transmural, and perfusion.
2. Reconstruction of a population of “primitive” trees, i.e., topological trees without
geometric features by repeated stochastic generation of networks based on
Kassab’s topological data (Kassab et al., 1993, 1994a, 1994b; Kassab,
Pallencaoe, et al., 1997).
3. Selection of physiologically compatible tree based on the criterion of total branch
length/depth, initial vessel homogeneity, and total number of vessels.
4. Assignment of 3D structure to the tree and geometrical optimization was based on
iterative simulated annealing scheme. During this stage, the purely topological
primitive trees were transformed into a network with spatial 2D or 3D geometric
characteristics. The optimization was performed based on criterion for transmural
span and homogeneity including boundary avoidance, local bifurcation geome-
try, and avoidance of segment intersections.
5. Diameter assignment was based on iterative algorithm by imposing monotonic
longitudinal diameter change (intra-element constraint) and Murray’s law of
bifurcations (inter-element constraint).
6. Transformation from a rectangular slab geometry into prolate spheroid surface
through optimization implemented using Genetic Algorithm method.
7. Venous trees were generated based on the following princ iples:
(a) Venous tree reconstruction was similar to the arterial trees.
(b) The epicardial functional sub-network complied with criterion that each
arterial capillary was drained by two venal counterparts (Bassingthwaighte
et al., 1974). The venous transmural and perfusion sub-networks were
required to encompass a similar number of equivalent capillaries and to
extend over similar span to their arterial counterpart.
(c) The venous tree reconstruction was subject to an additional constraint of
compatibility with the number of arterial capillaries in each of the wall layers.
2.4.6 Geometric Optimization
The geometrical optimization was done iter atively based on simulated annealing
algorithm. The scheme for the annealing procedure includes random selection of a
segment from an entire population of network segments, perturbation of the
branching geometry, and objective function evaluation. This scheme was repeated
subject to the “cooling” down of the annealing ensemble. The perturbation of a
bifurcation angle was subject to the following geometrical constraints:
56 2 Morphometry of Coronary Vasculature
https://t.me/med1917
1. Mother and daughter vessels were positioned on the same plane (Zamir et al.,
1983).
2. Bounds on branching angles were imposed.
3. Constraints on cumulative element angle were also imposed where the upper
bound decreased with increase in element order number.
4. Avoidance of angle overlap.
2.4.7 Verification of Coronary Network
The validity of the reconstructed stochastic coronary trees was confirmed based on
two criteria:
1. Overall visual similarity to native trees as observed, for example, from corrosion
casting and other methodologies (Fig. 2.19),
2. Agreement of the reconstructed tree with the statistical database of Kassab et al.
(1993, 1994b) Kassab, Pallencaoe, et al. (1997), and VanBavel and Spaan (1992)
in terms of:
(a) Statistics of segmental and elemental diameters and lengths
(b) Statistics of S/E ratio
(c) Connecti vity and longitudinal position (CM and LPM data)
(d) Adherence to Murray’s law (see Chap. 7)
Once the annealing phase had been completed, the primitive trees were turned
into a network having a geometrical visual appearance which resembled those of
native coronary networks (Fig. 2.19a). The arterial epicardial tree shown in
Fig. 2.19a is two dimensional and includes orders 11–8. The 3D transmural trees
span over the entire wall and consists of vessel orders 8–4 (e.g., representative
examples for arterial network is shown in Fig. 2.19b). After transformation into
prolate spheroid surfaces, the results for LAD, LCx, and RCA representing the
isolated crown and the major branches imposed on the left and the right ventricles
are shown in Fig. 2.19c and d, respectively.
Figure 2.20 shows a fully reconstructed coronary vasculature ranging from the
largest coronary arteries to the largest veins. The total number of reconstructed
venous segments was about 17 million which spans orders 12 (coronary sinus)
to 0v capillaries (first segment of venous capillary). Combined with the
reconstructed arterial network, the number of vessel segments for the entire coronary
network adds up to about 27 million vessels. The reconstructed full coronary
vascular network agrees with the gross anatomy of coronary networks in terms of
structure, location of major vessels, and measured morphometric statistics of native
coronary networks.
2.4 Integration of 3D Coronary Vasculature 57
https://t.me/med1917
Fig. 2.19 (a) Reconstructed arterial epicardial subtree for the LAD compared to native branch. (b)
Reconstructed arterial transmural branches for the LAD. (c) Isolated crown of the three major
epicardial branches: (a) Top view demonstrating the septum coverage, (b) Lateral posterior view
demonstrating the RCA (c) Lateral anterior view demonstrating the LAD and LCx branches. (d)
Major arterial epicardial branches imposed on the left and right ventricles: (a) Lateral posterior view
demonstrating the RCA branch, (b) Lateral anterior view demonstrating the LAD and LCx
branches. Reproduced from Kaimovitz et al. (2005) by permission
58 2 Morphometry of Coronary Vasculature
https://t.me/med1917
Fig. 2.20 Rendering of the reconstructed arterial and venous trees (orders 1 to 11 and 1 to 12)
as viewed from two different aspects (anterior and posterior). Red corresponds to veins and blue
corresponds to veins. Reproduced from Kaimovitz et al. (2010) by permission
2.5 Non-tree Structures
Although tree structures are prevalent in nature for distribution and collection of
nutrients, non-tree-like structures such as arcades, anastomoses, and interconnections do exist in biological organs. Vascular trees appear to be the rules of
architecture for three-dimensional (3D) organs (Kassab, 2000), while the existence
of arcades seem to be restricted to 2D organs or surfaces of 3D organs, e.g.,
mesentery and omentum, surface of the heart, inner ear, retina, surface of the small
intestine and colon, iris, diaphragm, thin skeletal muscle, surface of uterus, and
surface of the elbow (see review in Kassab (2000)). A set of scaling laws that can
explain the design of non-tree structures remain to be elucidated.
2.6 Labor Savings in Morphological Reconstruction
Besides the extensive morphometric data on the porcine coronary vasculature
presented in this chapter, there is a paucity of data on the branching pattern and
dimensions of the blood vessels in various organs including the human coronary
vasculature. The reason for this lack of data is undoubtedly the tremendous effort
2.6 Labor Savings in Morphological Reconstruction 59
https://t.me/med1917
needed to obtain the morphometric data. The author of this book can verify the
countless hours needed over 4 years that was required to obtain a complete set of
morphometric data on the porcine coronary vasculature.
For labor saving, pruning of the pulmonary vasculature has been introduced in
morphometric data collection: cutting off branches at successive generations, measuring what remained, using the statistical data to estimate what were cut off, and
adding the estimated data to the measured data to obtain the final results (Horsfield,
1978; Horsfield & Gordon, 1981; Singhal, Henderson, Horsfield, Harding, &
Cumming, 1973; Yen et al., 1983, 1984). In these studies, evaluation of the effects
of pruning was not possible because a full set of precise data did not exist. With a
complete set of unpruned coronary morphometric data is available (Kassab et al.,
1993), the effect of a pruning procedure on accuracy can be evaluated (Kassab et al.,
1994a). A particular pruning procedure was found to reduce the labor by 79% when it
was applied to the LAD artery of the pig; however, it introduced the following errors
based on comparison with the unpruned morphometric data (Kassab et al., 1994a):
1. The largest error incurred in the mean diameters of all orders of tree was 7.6%.
2. The corresponding maximum errors in the length and number of elements in all
orders were 9.8% and 30.0%, respectively.
3. The estimated error of the total equivalent Poiseuille’ s resistance for the LAD
artery computed from pruned data was 25.2% when compared with that com-
puted from unpruned data.
Since pruning of trees introduces significant error that may not be acceptable for a
particular analysis, the use of automation or semi-automation to reduce the workload
may be necessary. In this regard, Spaan and colleagues (Spaan et al., 2005; van
Horssen, Siebes, Hoefer, Spaan, & van den Wijngaard, 2010) have developed a
cryomicrotome approach to automate the reconstruction of coronary vasculature.
Their approach consists of 3D reconstruction of the coronary arterial tree from
frozen sections of the myocardium whose vasculature is filled with fluorescent
polymer. This approach allows the reconstruction of coronary vasculature down to
20 μm. In parallel with this development, computerized tomographic (CT) imaging
technology has provided important data on the three-dimensional branching patterns
of vascular trees (Beighley, Thomas, Jorgensen, & Ritman, 1997; Garcia-Sanz,
Rodriguez-Barbero, Bentley, Ritman, & Romero, 1998; Jorgensen, Demirkaya, &
Ritman, 1998). Ritman and colleagues (1997) have successfully used μCT imaging
to provide the 3D geometry of the entire coronary arterial tree of the rat heart
(capillary level resolution). In conjunction with volumetric imaging, it is necessary
to have accurate and reproducible algorithms to extract detailed morphometric data
on the diameters, lengths, number of vessels, connectivity, and branching angles
from the CT or μCT reconstructed tree.
The sections that follow discuss image processing, centerline detection, and grid
generation. Image processing, particularly image segmentation, is a necessary first
step for both centerline detection and grid generation. The centerlines allow for the
computation of various quantitative measurements, such as vessel length, vessel
radius, and bifurcation angles.
60 2 Morphometry of Coronary Vasculature
https://t.me/med1917
2.7 Automation: Segmentation and Centerline Detection
2.7.1 Image Processing
In quantify the geometry of the vasculature, CT or MRI is typically used, resulting in
a volumetric image. A volumetric image consists of voxels aligned along a regular
3D grid. It is generally not likely that the boundaries of the vessels are exactly
located at these voxels. Better precision can be achieved by finding the exact location
in between a set of voxels. Since an accurate representation of the object boundary is
crucial to any further processing of the data, improvement of the precision is an
essential step. Different approaches are available depending on the need of the
algorithm used to further process the result. Some algorithms for computing the
centerline only require an accurate representation of individual points. On the other
hand, grid generating algorithms typically require a surface representation of the
boundary, i.e., the points need to be connected by some geometric primitive. The
following subsections provide examples of both types of algorithms to extract
morphometric data from medical images.
2.7.2 Segmentation of Vessel Boundary
The method described here uses similar techniques as described by Canny’s
non-maxima suppression (Canny, 1986) but extended to 3D. First, the image
gradient is computed for every voxel. Using an experimentally determined threshold, all voxels with a gradient length below this threshold are neglected. The
advantage of this gradient-based thresholding is that it is less sensitive to the selected
threshold compared to intensity-based segmentation algorithms. This is particularly
relevant for smaller vessels (one voxel in diameter or less) that can be missed due to
partial volume effects when using intensity segmentation.
2.7.3 Segmentation Under Topological Control
In order to create a volume grid that corresponds to the medical image, it is necessary
to produce a triangulated isosurface from a segmentation of the data. It is important
to produce such an isosurface while preserving correct vessel topology. From a
topological point of view, an arterial tree (excluding the capillary bed) is homeomorphic with a sphere. Due to finite resolution, isosurfacing algorithms such as
Marching Cubes (Lorensen & Cline, 1987) are unable to determine whether voxels
that connect only by a corner or by an edge should truly be connected. This
ambiguity can give rise to multiple handles that disrupt segmentations. Therefore,
segmentations must be performed under topological control (Carson et al., 2010).
2.7 Automation: Segmentation and Centerline Detection 61
https://t.me/med1917
To segment the data, a fuzzy connected-threshold algorithm is applied to the
image in order to convert the series of grayscale images into a binary volume.
Connectedness is restricted to face connectivity to prevent ambiguous representations of the surface between vessels and background. Face connectivity is accomplished both by restricting the region growing algorithm to faces and by a postsegmentation connectivity check that reassigns voxels found to possess vertex or
edge connectivity. Subsequently, loops are removed to bring the face-connected
segmentation into proper topology by using an automated approach based on
skeletonization, loop detection, loop cutting, and cleanup. A breadth-first search of
branches in the skeleton is applied, starting at the top of the coronary ostia. To find
the optimal cutting location within the loop, a test cut is performed separately for
each skeleton voxel belonging to the loop. Cuts are then affected at the region of
minimum cross-sectional area and maximum path length from the ostia.
To extract the isosurface from the segmented image, the Marching Tetrahedra
variant of the popular marching cubes algorithm can be applied. This produces a
closed triangulated surface, without boundary patches at the inlets and outlets, and
whose surface density is a function of resolution of the underlying data.
2.7.4 Centerline Detection
Numerous algorithms for extra cting centerlines from volumetric data sets are available. An overview of the various techniques can be found in the paper by Cornea,
Silver, and Min (2005). Some methods begin with all voxels of a volumetric image
and use a thinning technique to shrink down the object to a single line (Bertrand &
Aktouf, 1994; Brunner & Brunnett, 2004; Dyedov et al., 2009; Lee, Kashyap, &
Chu, 1994; Lohou & Bertrand, 2004; Palágyi & Kuba, 1996; Saha, Chaudhuri, &
Majumder, 1997; Tsao & Fu, 1981). Ideally, the topology of the object should be
preserved (Lobregt, Verbeek, & Groen, 1980) which is the basic technique used in
commercial software systems. Luboz et al. (2005) used a thinning-based technique
to determine vessel radii and lengths from a CT scan. A standard deviation of
0.4 mm between the computed and the actual measurements was reported for a
scan with a resolution of 0.6 mm. The disadvantage of thinning algorithms is that
they can only be applied to volumetric data sets and the centerlines are described at
voxel precision resulting in somewhat jagged lines, which do not allow accurate
measurements of branch angles.
For extracting centerlines from volumetric images, geometry-based approaches
are preferable over voxel-based approaches. Due to the discrete nature of a voxel of
the volumetric image, the location of the centerline can have an error of half a voxel.
Geometry-based methods do not have this shortcoming. Nordsletten, Blackett,
Bentley, Ritman, and Smith (2006) determined normal vectors based on an
isosurface computed using the volumetric image. These normal vectors are projected
inward. The resulting point cloud is then collected and connected by a snake
algorithm.
62 2 Morphometry of Coronary Vasculature
https://t.me/med1917
2.7.5 Vector Field
The method for centerline detection described in the following subsections follows
an algorithm developed by Wischgoll, Choy, Ritman, and Kassab (2008). A major
advantage of this approach lies in the demonstrated accuracy based on actual
validations between computed vessel diameters and optical measurements for porcine hearts. This algorithm consists of several steps. Since the object is given as a
volumetric CT-scanned image, the object boundary is extracted as previously
described. A vector field is then computed that is orthogonal to the object boundar y
surface. Once the vector field is computed, the centerlines can be determined by
applying a topological analysis to this vector field. As a last step, gaps between
segments of the centerlin es can be closed automatically and vessel diameters can be
computed. The following subsections explain these steps in detail.
The method presented above computes the centerlines by applying a topological
analysis to a vector field that is determined based on the geometric configuration of
the object of which the centerlines are to be determined. The vector field is computed
at the identified points on the vessel boundary in such a way that the vectors are
orthogonal to the vessel boundary surface. Based on these vectors, the vector field
inside the vessels is computed by using linear interpolation. Since the vasculature is
given as a volumetric data set, the image gradients can be used to define these vectors
on the boundary surface. These image gradients are previously determined as they
are needed for extracting the boundary. Since the points are only moved along the
direction of the image gradient when determining the sub-voxel precision, this image
gradient is still orthogonal to the boundary surface and therefore represents a good
approximation for the desired vector field.
2.7.6 Determination of Centerlines
In order to determine the centerlines of the object, a tetrahedrization of all points on
the object boundary is computed first. For this, Si’s(2004) fast implementation of a
Delaunay tetrahedrization algorithm is used. Tetrahedra outside of the vessels are
removed based on the gradient vectors. This step also closes small gaps that may
exist since tetrahedra covering these gaps will still have vectors attached to the
vertices which point inward. Since vectors are known for each vertex of every
tetrahedron, the complete vector field can be computed by using this tetrahedrization
by linear interpolation within each tetrahedron. This vector field is then used to
identify points of the centerlines which are then connected with each other.
Points on the centerlines can be identified by computing the singularities within
the vector field interpolated within every face of the tetrahedrization. For example,
for a perfectly cylindrical object, the vector boundary points directly at the center of
the cylinder. When examining the resulting vector field at a cross section of the
cylinder, a focus singularity is located at the center of the cylinder within this cross
2.8 Grid Generation 63
https://t.me/med1917
section. The location of this focus singularity resembles a point on the centerline of
the cylinder. Hence, a singularity of type of node, focus, or spiral within a face of a
tetrahedron indicates a point of the centerline. Since not all objects are cylindrical in
shape and given the numerical errors and tolerances, points on the centerlines can be
identified from singularities that resemble focus and spiral singularities.
After computing the center points, the vessel diameters are computed for each
center point and all points within the vicinity are ident ified. From this set of points,
only the ones that are within the slice of the vessel used to determine the center point
are selected to describe the boundary. The radius is then computed as the average of the
distances between the center points and the points on the boundary of the vessel slice.
Once individual p oints of the centerlines (including the corresponding vessel
diameters) are computed by identifying the focus and spiral singularities within the
faces of the tetrahedra, this set of points must be connected in order to retrieve all
centerlines. Since the tetrahedrization describes the topology of the object, the
connectivity information of the tetrahedra can be used. Thus, identified points of
the centerlines of neighboring tetrahedra are connected with each other, forming the
centerlines. In some cases, gaps will remain due to the choice of thresholds which
can be closed using the method described in the next section.
2.7.7 Geometric Reconstruction
Based on the centerlines extracted from the volumetric image (Fig. 2.21), various
measurements can be extracted, such as vessel radius or bifurcation angles
(Wischgoll, Choy, & Kassab, 2009). A comparison of the computed radii, which
were measured as the distance between centerline and vessel wall, and optical
measurements of the radii for the main trunk of five porcine hearts show an excellent
accuracy with an average error of 0.7% and root mean square (rms) error of 1.1% of
the radii . Using the centerline and radii information, conic cylinders can be formed to
represent the individual vessel segment. By representing every segment in this way,
the vascular tree can be reconstructed. Since the vasculature is represented as
geometry, the visualization software not only facilitates the gathering of statistical
information about the morphometry but it also allows a user to perform various
measurements, such as distances or bifurcation angles.
2.8 Grid Generation
A grid or mesh generation is a mathematical process of generating a polygonal or
polyhedral mesh that approximates the geometric object of interest (e.g., coronary
vessel). The process of generating a grid or mesh of the segmented object is
necessary to solve the conservation or field equations on the geometric boundary
of interest. Accurate grid generation is key to obtaining reliable numerical solutions
Соседние файлы в папке Библиотека им академика М.И. Перельмана