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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
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64 2 Morphometry of Coronary Vasculature
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Fig. 2.21 (a) Volume
rendering (with shading
enabled) of a Microfilperfused porcine heart,
which was scanned using a
computed tomography
(CT) scanner. (b)
Centerlines (solid lines)
extracted from the CT scan
shown in (a). The
boundaries of the vessel
wall identified by the
algorithm are shown in light
gray. Reproduced from
Wischgoll et al. (2009) with
permission
to the boundary value problems (Chap. 8). The numerical solution depends on the
quality of grid or mesh. Hence, the grid must be of sufficient resolution and quality to
capture the physical phenomena of interest. The assessment of grid density or size on
the solution of an equation is a standard sensitivity analysis in numer ical simulations.
With surfaces derived from imaging data, the organization and densi ty of the
original surface triangles depend on the resolution of the digital data. The characteristic dimension of the surface triangles is on the order of one voxel. Simply
generating a volume grid from the original surface could result in grossly underresolving the computed field, where the surface density is close to that of the local
feature size or conversely over-resolving the computed field where the surface
density is much finer than that of the local feature size. These issues lead to a
consideration of the local feature size as an important criterion for sizing and
gradation control of the surface that is complementary to criteria that attempt to
2.8 Grid Generation 65
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preserve surface features, topology and curvature. Moreover, the local feature size in
vessel geometry is related to the local diameter. Thus, a measure of the local feature
size can also provide a guide for organizing elements radially in layers. This
approach has the advantage of creating elements that are mostly parallel to the
wall, which reduces discretization error in flows that are predominately axial. At
the same time, it essentially decouples strategies for controlling grid density in the
normal and tangential directions. It also directly embeds a local understanding of
scale into the grid, since the local diameter is related to the local scale.
A robust and computationally efficient metric for local scale is the so-called
gradient-limited feature size (GLFS) (Kuprat & Einstein, 2009). Unlike other measures of the local feature size, the GLFS (Fig. 2.22) can be defined directly on a
triangulated surface mesh without a background grid and without referencing the
medial axis. Thus, determination of the GLFS is not only computationally efficient,
but also robust in the sense that it is Lipchitz continuous and does not change
unreasonably under perturbation of the surface mesh. Grids that are organized
according to GLFS, such that roughly the same number of layers of elements can
be found at all resolved scales, are said to be scale-invariant. Scale-invariance is
critical in grids of vascular trees because it assures that discretization error at the
smallest scale does not unduly affect solution error at the highest scale, i.e., the
discretization error is equilibrated at all resol ved scales. Combined with the GLFS,
the idea of scale-invariance enables the automatic generation of quality anisotropic
unstructured grids, while keeping the overall computational cost of the problem
tractable. This approach has been adopted in two complementary scale-invariant
gridding algorithms for quality-layered tetrahedra (Kuprat & Einstein, 2009) and
quality hybrid prismatic/tetrahedral grids (Dyedov et al., 2009). These algorithms
have been implemented in two software frameworks, Lagrit-PNNL and MeshMagic.
The defined GLFS in these two algorithms serves three functions as follows:
1. A field for tangential adaptation of the surface grid
2. A metric for creating layered tetrahedra
3. A speed function for construction of a prismatic boundary layer by application of
the Generalized Huygens’ Principle (Jiao & Zha, 2008).
Further discussion of GLFS can be found in Wischgoll, Einstein, Kuprat, Jiao,
and Kassab (2010). Examples of various tetrahedra (e.g., layered anisotropic tetrahedra, hybrid prismatic/tetrahedral grids) used for grid generation are given in
Wischgoll et al. (2010).
2.8.1 Element Quality
Discretization error can have two sources: (1) Insufficient grid density to resolve
computed gradients, and (2) “Badly” shaped elements. What exactly constitutes a
“badly” shaped element is somewhat applicati on dependent. It is generally accepted
that an isotropic element (i.e., an element with nearly equal internal angles and
66 2 Morphometry of Coronary Vasculature
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Fig. 2.22 Gradient-limited feature size (GLFS) and first principal curvature (top panel) defined on
a mouse coronary arterial tree from computed tomography. Efficient computation of these sizing
fields was performed in less than 5 s for this geometry on a laptop. Based on the GLFS modulated
by the curvature, the original surface mesh from Marching Cubes is selectively refined and
de-refined. The bottom panel shows the tangential adaption of the triangulated surface mesh for
(a user definable parameter) values of 0.6 (152,282 triangles) and 1.2 (129,366 triangles). The
c
t
curvature field for linear values of c
applications, it may make sense to convolve the GLFS with a nonlinear function that weights higher
or lower scales. These operations are supported in Lagrit-PNNL and MeshMagic. Reproduced from
Wischgoll et al. (2010) by permission
prevents further de-refinement of the surface grid. For certain
t
2.9 Visualization of Reconstructed Network 67
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Fig. 2.23 Complete representation of the vasculature of a heart and close-up view depicting the
large amount of detail in the model. Reproduced from Wischgoll et al. (2007) with permission
approximately equal edge lengths) is “good” and a highly skewed element is “bad.”
For certain classes of problems such as computational fluid dynamics, however,
isotropic elements may neither be necessary nor particularly appropriate. Nevertheless, the accuracy or speed of some applications can be compromised by just a few
“bad” elements. Hence, it is important to have a validated criterion for element
quality by some standard measure.
2.9 Visualization of Reconstructed Network
The 3D models of coronary vasculature can not only serve as powerful models for
hypothesis generation, but they can also be used for virtual interventions and
educational training. For such applications, fast and efficient visualization algorithms are essential. The complexity of these models (Fig. 2.23), which include
vessels from the large proximal coronary arteries and veins down to the capillary
level (3 orders of magnitude difference in diameter), is a challenging visualization
problem since the resulting geometrical representation consists of millions of vessel
segments. An interactive model has been proposed as an interactive method for
rendering the entire porcine coronary arterial tree down to the first segments of
capillaries which employs geometry reduction and occlusion culling techniques
(Wischgoll, Meyer, Kaimovitz, Lanir, & Kassab, 2007). Due to the tree-shaped
nature of the vasculature, these techniques exploit the geometrical topology of the
object to achieve a faster rendering speed while still handling the full complexity of
the data. A significant increase in performance combined with a more accurate,
gapless representation of the vessel segments was found. This resulted in a more
68 2 Morphometry of Coronary Vasculature
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Fig. 2.24 (a) Example of CT image segmentation and the 3D geometric reconstruction of LAD in
one patient. (b ) IVUS (intravascular ultrasound) example from one frame of the LAD artery. Lumen
region is marked in polygon. (c) A frame of IVUS. (d) A Third frame. (e) The corresponding CSA
location to IVUS frame in (b). (f) A second cross-sectional area (CSA). (g) A Third CSA.
Reproduced from Luo et al. (2014) by permission
interactive visualization and analysis tool for the entire coronary tree. The proposed
techniques can also be applied to similar data structures, such as neuronal trees,
airway structures, bile ducts, and other tree-like structures.
2.10 Patient-Specific Coronary Morphometry
Accurate computed tomography (CT)-based reconstruction of coronary morphometry (diameters, length, bifurcation angles) is important for construction of patientspecific models to aid diagnosis and therapy. Luo, Wischgoll, Koo, Huo, and Kassab
(2014) validated the accuracy of patient coronary artery lumen area obtained from
CT images based on intravascular ultrasound (IVUS) as shown in Fig. 2.24. In this
study, morphometric data of 5 patient CT scans with 11 arteries from IVUS were
reconstructed by including the lumen cross-sectional area (CSA), diameter, and
length. The volumetric data from CT images were analyzed at sub-pixel accuracy
to obtain accurate vessel centerlines and CSA. A centerline extraction approach was
used where an initial estimated skeleton in discrete value was obtained using a
traditional thinning algorithm. The CSA was determined directly without any circular shape assumptions to provide accurate reconstruction of stenosis. The root mean
square error (RMSE) for CSA was 16.2% and for diameter was 9.5%. The image
segmentation and CSA extraction algorithm for reconstruction of coronary arteries
proved to be accurate for determination of vessel lumen area even in the presence of
coronary artery disease. This approach provides fundamental morphometric data for
patient-specific models to diagnose and treat coronary artery disease.
Appendix 1: Diameters, Lengths, and S/E for Segments... 69
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Appendix 1: Diameters, Lengths, and S/E for Segments
and Elements of Arteries and Veins
Table 2.1 Diameters and lengths of vessel segments and elements in each order of vessels in
pig RCA
Segments Elements
Diameter,
μm n Length, mm n
Order
1 9.6 0.97 1033 0.069 0.046 510 9.3 0.84 146 0.125 0.084 146
2 13.2 1.6 741 0.083 0.070 441 12.8 1.4 136 0.141 0.103 136
3 19.1 2.7 490 0.085 0.061 328 17.7 2.1 79 0.178 0.105 79
4 34.1 6.0 1189 0.118 0.113 202 28.6 5.4 36 0.253 0.174 36
5 64.4 15.1 2594 0.449 0.350 526 63.1 11.3 91 0.545 0.415 91
6 137 31.5 2142 0.748 0.654 1506 132 22.2 431 1.64 1.13 428
7 265 45.2 1173 0.986 0.810 1066 256 30.1 303 3.13 2.11 299
8 438 64.7 536 1.26 1.10 524 428 47.5 108 5.99 3.53 108
9 730 129 177 1.62 1.31 174 706 75.2 33 9.06 5.56 33
10 1430 379 85 1.89
11 3218388 26 3.24 2.09 26 3218 1 84.1 1
Values are means SD; n no. of vessels measured. RCA right coronary artery
Reprinted with permission from Kassab et al. (1993)
1.38 85 1302 239 10 16.1 13.3 10
Diameter,
μm n Length, mm n
Table 2.2 Diameters and lengths of vessel segments and elements of each order in pig LAD
Segments Elements
Diameter,
μm n Length, mm n
Order
1 9.2 0.94 835 0.056 0.038 506 9.0 0.73 139 0.115 0.066 139
2 13.0 1.7 539 0.072 0.045 326 12.3 1.3 115 0.136 0.088 114
3 18.7 2.6 266 0.072 0.049 177 17.7 2.2 54 0.149 0.104 54
4 34.6 7.4 841 0.112 0.10 108 30.5 6.0 22 0.353 0.154 22
5 71.6 17.2 2171 0.454 0.33 435 66.2 13.6 80 0.502 0.349 78
6 150 35.8 1627 0.609 0.48 1017 139 24.1 253 1.31 0.914 252
7 303 54.5 1000 0.920 0.79 901 308 56.6 223 3.54 2.11 222
8 467
9 715 130 193 1.54 1.25 191 714 81.8 33 9.03 6.13 33
10 1492 365 54 2.26 1.56 54 1573 361 7 20.3 17.9 6
11 3176 654 17 2.82 1.96 17 3176 1 47.9 1
Values are means SD; n no. of vessels measured. LAD left anterior descending coronary artery
Reprinted with permission from Kassab et al. (1993)
56.1 459 1.09 0.83 437 462 40.9 96 4.99 3.02 95
Diameter,
μm n Length, mm n
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Table 2.3 Diameters and lengths of vessel segments and elements of each order in pig LCx
Segments Elements
Diameter,
μm n Length, mm n
Order
1 9.2 0.94 835 0.056 0.038 506 9.0 0.73 139 0.115 0.066 139
2 13.0 1.7 539 0.072 0.045 326 12.3 1.3 115 0.136 0.088 114
3 18.7 2.6 266 0.072 0.049 177 17.7 2.2 54 0.149 0.104 54
4 33.2 9.1 294 0.190 0.097 93 27.5 6.1 14 0.405 0.170 14
5 76.3 14.5 513 0.615 0.508 75 73.2 14.2 22 0.908 0.763 22
6 143 30.5 575 1.11 0.983 276 139 26.2 76 1.83 1.34 76
7 285 53.3 323 1.60 1.33 283 279 38.4 89 4.22 2.26 89
8 468 78.8 199 1.78 1.46 198 462 56.1 49 6.98 3.92 49
9 1025 273 66 3.18 2.41 66 961 193 10 21.0 15.6 10
10 2603 337 14 3.54
Values are means SD; n no. of vessels measured. LCx left circumflex coronary artery
Reprinted with permission from Kassab et al. (1993)
Table 2.4 S/E in pig RCA, LAD, and LCx
RCA LAD LCx
Order
1 1.88 0.99 155 2.30 1.4 134 2.30 1.4 134
2 1.88 1.0 138 1.79 0.95 117 1.79 0.95 117
3 2.20 1.2 89 2.00 1.1 54 2.00 1.1 54
4 2.30 1.8 50 2.28 1.3 18 2.06 1.2 16
5 2.00 0.91 125 2.02 1.2 63 2.20 1.3 20
6 2.30 1.3 503 2.23 1.3 266 2.11 1.1 122
7 3.23 2.1 324 3.89 2.1 216 2.75 1.6 95
8 4.68 2.7 111 4.69 3.0 98 4.22 2.4 46
9 5.38 3.6 34 6.06 4.2 32 6.60 4.0 10
10 8.5 7.2 10 9.0 7.0 6 14 1
11 26 1 17 1
Values are means SD; n no. of observations.
ratio of total no. of segments in a given order to total no. of elements in that order. This ratio is also
average no. of vessel segments in series for each order of vessels
Reprinted with permission from Kassab et al. (1993)
S/EnS/EnS/En
2.00 14 2603 1 49.6 1
Diameter,
μm n Length, mm n
S/E segment-to-element numbers ratio, which is
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Table 2.5 Major axis, major-to-minor axis ratio, and lengths of vessel segments and elements in each order of vessels in pig coronary sinusal veins
Segments Elements
D, μm nL,mm nD, μm nL,mm nD/minor axis
Order
1 10.8 1.7 872 0.051 0.041 251 10.6 1.6 121 0.079 .054 121 1.32 0.047
2 17.6 3.0 624 0.056 0.041 313 16.5 2.7 115 0.092 .065 115 1.38 0.082
3 30.0 4.3 425 0.063 0.043 263 29.6 3.2 88 0.117 .071 88 1.48 0.100
4 55.5 13.5 13,719 0.223 0.179 1948 57.5 11.8 435 0.350 .277 435 1.70 0.060
5 117 25.1 10,238 0.302 0.239 6408 117 18.7 2069 0.698 .649 2069 1.83 0.05
6 206 42.4 6611 0.367 0.297 6175 205 25.6 1713 1.26 1.13 1711 1.91 0.069
0.376 3435 317 32.7 768 2.08 1.87 768 1.96 0.058
7 321 63.0 3572 0.447
8 487 97.5 1785 0.591 0.474 1705 488 46.5 282 3.62 3.19 282 1.90 0.101
9 770 154 675 0.857 0.743 662 773 62.4 98 6.07 5.11 98 1.82 0.110
10 1192 185 213 1.26 0.967 200 1165 88.2 39 10.2 8.32 39 1.68 0.148
11 1999 731 117 1.72 1.37 106 1804 464 16 25.8 24.2 14 1.45 0.053
12 59192353 24 2.84 1.48 23 5919 1 71.9 1 1.25 0.145
D major axis (means SD), L length of segment (means SD), n no. of vessels measured, D/minor axis major-to-minor axis ratio (means SD)
Reprinted with permission from Kassab et al. (1994b)
72 2 Morphometry of Coronary Vasculature
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Table 2.6 Major axis, major-to-minor axis ratio, and lengths of vessel segments and elements in each order of vessels in pig Thebesian veins
Segments Elements
D, μm nL,mm nD, μm nL,mm nD/minor axis
Order
1 10.8 1.7 872 0.051 0.041 251 10.6 1.6 121 0.079 .054 121 1.32 0.047
2 17.6 3.0 624 0.056 0.041 313 16.5 2.7 115 0.092 .065 115 1.38 0.082
3 30.0 4.3 425 0.063 0.043 263 29.6 3.2 88 0.117 .071 88 1.48 0.100
4 54.4 15.2 4635 0.235 0.204 1681 54.8 11.5 492 0.384 0.327 490 1.70 0.060
5 110 24.0 2668 0.314 0.228 1967 111 17.2 718 0.684 0.580 716 1.83 0.052
6 190 38.8 1552 0.372 0.323 1427 189 22.5 399 1.25 1.11 397 1.91 0.069
0.402 760 292 30.6 140 2.44 1.99 138 1.96 0.058
7 299 69.7 789 0.443
8 549 116 331 0.682 0.616 318 549 53.5 54 4.21 3.36 62 1.90 0.101
9 820 135 156 0.812 0.681 154 813 55.6 24 6.23 5.51 24 1.82 0.110
10 1171272 67 0.970 0.587 64 1184230 11 9.35 6.67 11 1.68 0.148
Values for D and L are means SD and for D/minor axis are means SE
Reprinted with permission from Kassab et al. (1994b)
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Table 2.7 Segment-to-element numbers ratio for each order of vessels in sinusal and coronary
Thebesian veins of pig
Sinusal veins Thebesian veins
Order
1 1.77 0.90 121 1.77 0.90 121
2 1.83 1.0 115 1.83 1.0 115
3 1.91 1.0 88 1.91 1.0 88
4 1.87 1.2 605 1.88 1.3 518
5 2.33 1.7 2382 2.36 1.6 686
6 3.47 2.7 1912 3.67 2.8 327
7 4.71 3.8 786 5.41 3.9 123
8 6.16 5.0 286 6.05 3.8 42
9 7.63 5.3 98 7.14 6.1 22
10 7.66 5.3 38 8.5 4.5 8
11 10.5 9.0 13
12 24 1
Values are means SD. S/E series-to-element numbers ratio, n no. of observations
Reprinted with permission from Kassab et al. (1994b)
S/EnS/En
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