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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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34 2 Morphometry of Coronary Vasculature
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Fig. 2.3 (a)
Photomicrographs of
coronary vasculature pig left
ventricle (LV). A 70 μmthick section taken at
3.9 mm from the epicardial
surface shows several
neighboring arterioles
(a) feeding a capillary bed
and a nearby venule
(v) draining it. (b) Cast of
pig left anterior descending
artery (LAD). Reproduced
from Kassab et al. (1993)
with permission
2.3.5 Morphometric Measurements
The distinction between arterioles and venules is possible because of the differences
in the branching pattern. The topology of the arterioles is tree-like whereas that of
venules is ginger-root-like resembling fingers collecting into a hand (Fig. 2.4a, b)
(Kassab et al., 1993, 1994b). Three criteria were used to distinguish arterial capillaries from arterioles:
1. Muscle fiber orientation: Most capillaries are oriented in the direction of the
muscle fiber.
2. Tortuosity: Arterioles are tortuous (i.e., make many twists and turns), whereas
capillaries are nearly straight due to their connections to the myocytes through
collagen struts (Arts, 1978; Caulfield & Borg, 1979).
3. Capillary topology: Branching pattern of the capillaries is very different from the
branching pattern of arterioles, as will be discussed below.
It should be noted that Criteria 1 and 3 can also be used to distinguish venous
capillaries from venules. Criterion 2, however, cannot be used for venous capillaries
because venules are not tortuous.
2.3 Reduction of Coronary Vasculature 35
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A
B
Fig. 2.4 (a) Photomicrographs of coronary vasculature in pig LV. A 80-μm-thick section taken at
4.2 mm from the epicardial surface. It shows several adjacent venules draining into a small vein. (b)
Cast of coronary sinus veins taken through a stereo-dissecting microscope. Reproduced from
Kassab et al. (1994b) with permission
The morphology of the coronary microvessels was recons tructed through optical
sectioning, as described by Kassab et al. (1993). Briefly, the microvessels were
viewed with an inverted microscope and displayed on a video monitor through a
television camera. The image was grabbed by computer software and analyzed with
a digitizing system. Several lumen diameter measurements (in μm) were made along
each vessel segment to obtain a mean diameter, and the vessel segmental length
(in μm) was obtained by measuring between bifurcation points along the centerline
of the vessel as shown in Fig. 2.5.Avessel segment is defined as the distance
between two consecutive bifurcation points (Fig. 2.5). Several hundred arteriolar and
venular trees, smaller than 40 μm in diameter, were thus reconstructed down to the
capillary level.
The RCA, LAD, and LCx arterial trees and Thebesian and sinusal venous casts
were dissected and viewed with a stereo-dissection microscope and displayed on a
36 2 Morphometry of Coronary Vasculature
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Fig. 2.5 (a)
Photomicrograph of
microvessel illustrating
morphometric
measurements of diameter
(i.e., mean of several lumen
diameter measurements) and
length of a vessel segment
(i.e., distance between two
consecutive bifurcations).
Reproduced from Kassab
et al. (1993) with permission
video monitor as described above. Morphometric measurements of segment diameter and length were made along with a sketch of each vascular system. Initially, the
trunks of the coronary arteries and veins were sketched and their segments measured.
The subtrees arising from the trunk were then labeled, excised, placed in separate
jars, further sketched and measured. This process was continued until the entire
RCA, LAD, and LCx arterial trees and Thebesian and sinusal venous casts were
sketched and their morphometric measurements made, down to 40 μm in diameter.
The branching pattern for the whole arterial tree and venous trees was quantified.
The branching of the whole arterial tree was found to be 98% bifurcations and 2%
trifurcations; that of venous trees was found to be 86% bifurcations, 12.8% trifurcations, 1% quadrifications, and 0.2% quintifications. Therefore, trifurcations,
quadrifications, and quintifications were found to be more frequent in venules than in
arterioles.
2.3.6 Mathematical Description of Branching Pattern
The largest coronary arteries and veins are relatively few, and hence their geometric
characteristics can be considered individually. Flow in these large vessels can be
analyzed in a subject-specific manner in conventional ways, e.g., by methods of
computational fluid mechanics in a subject-specific manner (Chap. 8). Small coronary arteries and veins are very large in number and are organized topologically like
trees except at the (a) Epicardial surface, where arcades (i.e., arching connecting
vessels) are found connecting the sinusal veins, and (b) Endocardial surface, where
arcades are found connecting Thebesian veins. Analysis of blood flow in the smaller
tree-like blood vessels is best done statistically as described below.
The best known mathematical models of statistical tree-like structures in physiology are Weibel’s(1963) bifurcation model, and Strahler’s(1952) model of rivulets
2.3 Reduction of Coronary Vasculature 37
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Fig. 2.6 (Left) Strahler’s vessel order scheme. (Right) Diameter-defined Strahler order scheme.
The lower (R
Δn)]/2 and RU¼ [(Dn+ Δn)+(D
+(D
n
modifications of order number according to diameter-defined rule. Reproduced from Kassab et al.
(1993) with permission
) and upper (RU) range of the diameters are defined as RL¼ [(D
L
n+1
Δ
)]/2, respectively. The orders with “*” denote
n+1
n1
+ Δ
n1
collecting into a river. In Weibel’s model, each branch is assigned a “generation”
number. If the largest trunk is designated as generation 0, then there are two branches
of generation 1 and so on. The Strahler system was originally introduced by Horton
in 1945 for geographical study of rivers and Strahler later introduced some variations
(Horton, 1945; Strahler, 1952). Strahler’s model has several versions. The first
version of Strahler’s model is the reverse of Weibel’s. Each branch is assigned an
“order” number and the smallest branches are assigned the smallest order number. In
the second version of Strahler’s model , the rule is imposed that when a vessel of
order 2 meets a vessel of order 1, the order number remains unchanged. Similarly, if
a vessel of order 3 meets a vessel of order 2 or 1, the resulting vessel order number
remains 3 (Fig. 2.6). In a final version, the successive branches of vessels of the same
order number connected in series are considered as one vessel of a greater length.
The bifurcations of coronary blood vessels are usually asymmetric, i.e., the two
daughter vessels are of unequal diameters and lengths. Weibel’s model handles
asymmetry with difficulty, but the Strahler model is designed to handle the asymmetry. Shortcomings with Strahler’s system arise when (a) Daughter vessels are no
smaller than the mother vessel but are assigned two different order numbers; (b) A
large trunk has many small branches, so that the trunk appears tapered but is
assigned only one order number; and (c) Two trunks of similar diameters from
two subtrees meet, but the order number of these two trunks is different.
A serious shortcoming of both Weibel and Strahler systems is apparent when the
ranges of diameters of the vessels of successive orders or generations are very broad,
as is the case in the coronary vasculature. In a quantitative examination of the
)
38 2 Morphometry of Coronary Vasculature
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histograms of the diameters of vessels of successive orders, it was found that the
ranges of successive orders of vessels overlap extensively. The overlap in the
diameters of consecutive orders renders a non-uni que relation between diameter
and order number. Thus, the models fail to simulate a well-known feature of blood
flow, i.e., blood flows from larger arteries into smaller arteries. Alternatively, one
may look at hemodynamics from the point of view of equivalent circuits and
conclude that these overlaps make the calculation of the resistance to flow in
successive orders of vessels excessively inaccurate. To remedy these difficulties,
five innova tions were proposed by Kassab and colleagues (see Review in Kassab
(2000)) to construct a more accurate mathematical model of the coronary vasculature, as described below.
2.3.7 Diameter-Defined Strahler System
The first innovation is a rule for assigning the order numbers of the vessels based on
diameter ranges (Kassab et al., 1993), known as the diameter-defined Strahler
system, which eliminates the overlap in diameters between successive orders of
vessels (Fig. 2.6). In this scheme, analogous to the Strahler system, (Fig. 2.6, left
panel), the capillary vessels are defined as vessels of order 0. The smallest arterioles
supplying blood to the capillaries are assigned an order of 1. When two arterioles of
order 1 meet, the confluent vessel is given an order 2 and so on in line with Strahler’s
system. Once the tree is assigned order numbers and the corresponding diameter is
determined, the ordering is revis ed according to a diameter-defined rule as follows:
Let D
denote the mean diameter of the vessels of order j, Δjdenotes the standard
j
deviation of the diameters of order j, whereas the subscript j denotes the order
number of the vessel. Then, the diameter-defined rule specifies that the dividing
point of the diameters of vessels of the order j and those of vessels of order j+1 lies
midway between D
of order 1 meet, the confluent vessel maintains an order 2 if its diameter exceeds the
mean diameters of the order 1 vessels by an amount specified by the diameter rule
(Kassab et al., 1993), or remains as order 1 if the diameter of the confluent is not
larger than the amount specified by the rule (Fig. 2.6, Right). When an order 2 artery
meets another order 2 or order 1 artery, the order number of the confluent is 3 if its
diameter is larger than the mean value of order 2 by an amount specified by the
formula or remains at 2 if its diameter does not increase sufficiently. This process is
continued until all arterial segments are arranged in increasing diameter and assigned
the order numbers 1,2,3, ... n, ... Although an iterative process is required to
determine the order number of all vessels of a given tree, the process has been
found to converge rapidly (within 2 or 3 iterations). Similarly, the veins are assigned
the order numbers 1, 2, 3 ..., n, ... according to the diameter-defined
Strahler system.
+ Δjand D
j
j+1
Δ
(see Fig. 2.6 legend). When two arterioles
j+1
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2.3.8 Meshing of Histological and Cast Data
Arterial and venous vessels were assigned order numbers first according to the
Strahler rules and then iterated according to the diameter-defined rule. The histological sections provided the data for the first four orders of the arterial and venous
vessels (<40 μm in diameter). Based on the means and standard deviations of the
vessels of orders 3 and 4 in the histological slides, order numbers were assigned to
the vessels in the tree casts. The assum ption is that the data from the slides and casts
are equally valid. This is plausible because the casts and slides were made of the
same polymer under the same protocol.
2.3.9 Segments and Elements
The second innovation of Kassab et al. (1993) was to introduce a clarifying terminology. Each blood vessel between two successive points of bifurcation is called a
segment. If several segments of a given order j are connected in series, then they are
lumped together into a unit called an element of order j (Fig. 2.7). Thus, the total
number of elements of order j is smaller than the total number of segments of order j.
The length of an element of order j is equal to l (if there is only one segment in an
element) or larger than the segment of order j. The lengths of segments and elements
are related by the segment-to-elem ent (S/E) number ratio (1). This second innovation was simultaneously proposed by VanBavel and Spaan (1992). This distinction between the series from the parallel arrangement of vessel segments is essential
in constructing an equivalent electrical circuit of the vascular system.
Figure 2.8 shows the diameters, lengths, and S/E number ratio for the entire
coronary arterial and venous vessels. A linear relation on a semi-log plot suggests
that the diameter is a geometric sequence with the order number in accordance with
Horton’s law (1945). Horton’s law implies that the ratio of the diameters of vessels
of successive orders is a constant independent of order number (Horton, 1945;
Strahler, 1952). The slope of Fig. 2.7 is referred to as the diameter ratio which is
the ratio of diameters of consecutive order numbers. The diameter ratio has a range
of 1.8–1.9 for the coronary arterial trees (RCA, LAD, and LCx) and 1.6–1.7 for
venous trees (Kassab et al., 1994b). Hence, the diameter decreases (on average) at
the rate of this ratio at each consecutive order number during descent from larger to
smaller vessels. The mean element length also obeys Horton’s law but with a
discontinuity at order 3. The element length ratio was about 2 (orders 1–3) and
1.1–1.3 (orders 4) for the various coronary arterial trees (Kassab et al., 1993) and
about 1.18 (orders 1to3) and 1.6–1.8 (orders 4) for the venous trees (Kassab
et al., 1994b). Appendix 1 summarizes the statistical data of segment and element
diameters and lengths as well as the segment-to-element ratio for the arterial and
venous trees.
40 2 Morphometry of Coronary Vasculature
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Fig. 2.7 Definition of
elements and segments to
identify analogous parallelseries network
characteristics. Rows
represent daughter elements
(m), and columns represent
mother elements (n). In this
illustration, 1.0 order
2 daughter elements arise
from the order 3 mother
element, 0.2 0.2 order
1 daughter elements arise
from another order
1 element, and 2.0 order
1 elements arise from each
element of order 2 or 3. For
capillaries of order
0, 2.6 0.89 arises from
each order 1 element and 1.0
arises from each order 2 or
3 element. The matrix below
summarizes the connectivity
matrix for the example tree
above. Reproduced from
Kassab et al. (1993) with
permission
3
0
3
1
3
1
1
0
1
0
0
0
0
0
2
2
1
11
-2
Element Connectivity Matrix, C (n,m)
12 3
0 2.6±0.4 1.0 1.0
1 0.2±0.2 2.0 2.0
0
1
0
1
0
0
0000
-1-1
2 0 1.0
30
2.3.10 Connectivity Matrix
The third innovation of Kassab et al. (1993) was to describe the connectivity
between the mother and daughter vessels of the various elements by a connectivity
matrix C
elements of order m that spring directly from mother elements of order n . The
connectivity matrix quantifies the connection of mother vessel element to diameter
vessels. Figure 2.7 shows an illustration of connectivity matrix for a sample tree. If
the tree was fully symmetric, only the diagonal entries would exist. The existence of
off diagonal entries indicates the asymmetry of the connectivity of the coronary
branches (e.g., smaller twigs arising from the tree trunk).
The connectivity relationship is necessary for understanding the hemodynamics
of the coronary blood flow. For example, the flow in a daughter vessel element of
order m that springs from a mother vessel of order n dictates the pressure at the point
of connectivity. Because this particular element of order m is connected to a larger
vessel of order n, the inlet pressure to this element is the exit pressure of order n, with
whose element in row m and column n is the ratio of the total number of
mn
2.3 Reduction of Coronary Vasculature 41
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Fig. 2.8 Average diameters (μm), lengths (μm), and segment-to-element number ratio of vessel
elements in successive orders and order number of vessels in pig RCA, LAD, and LCx arteries
(positive orders); and Thebesian and sinusal veins (negative orders). Reproduced from Kassab et al.
(1993) with permission
n > m. The connectivity matrix is also essential for extrapolating the total number of
vessel in the entire tree as described below. Appendix 2 summarizes the statistical
connectivity data tables for the coronary arterial and venous trees.
42 2 Morphometry of Coronary Vasculature
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Fig. 2.9 Schematic of a
vessel element of order
n and its branches (orders m,
k, etc.). Direction of blood
flow is from inlet at left to
outlet on right. A local
dimensionless curvilinear
coordinate x is shown along
the vessel elements with
x ¼ 0 at inlet and x ¼ 1at
outlet. Reproduced from
Kassab, Pallencaoe, et al.
(1997) with permission
2.3.11 Longitudinal Position Matrix
The fourth innovation by Kassab, Pallencaoe, Schatz, and Fung (1997) was to
describe the longitudinal position of the bifurcation of the daughter vessels along
the length of the mother vessel by a longitudinal position matrix LPM
mn
whose
element in row m (daughter elements) and column n (mother elements) is the
fractional longitudinal position of the vessels of order m which spring directly
along the length of vessels of order n. The longitudinal position matrix quantifies
the distribution of daughter vessels axially along the length of the mother element, as
shown in Fig. 2.9. For the order two element shown in Fig. 2.7, the capillary vessel
(order 0) would have an LPM (0,2) corresponding to 0.5 if the two order 2 segments
in series have the same length, i.e., the longitudinal position of the capillaries arises
midpoint along the order 2 arteriole. Appendix 3 summarizes the statistical data for
the LPM of coronary arteries and veins.
2.3.12 Asymmetry Ratios
The fifth innovation from Kalsho and Kassab (2004) was to describe the local
asymmetry of the coronary vasculature since the branching pattern of the coronary
arteries and veins is asymmet ric, i.e., many small vessels branch off a large trunk,
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Fig. 2.10 (a) Diameter asymmetry ratio and (b) Length asymmetry ratios of daughter vessels in pig
RCA, LAD, and LCx arteries (positive orders); and Thebesian and sinusal veins (negative orders)
plotted as a function of order number of mother vessels. Reproduced from Kalsho and Kassab
(2004) by permission
such that the two daughter vessels at a bifurcation are of unequal diameters and
lengths. To document the asymmetric branching pattern of the coronary vessels, an
asymmetry ratio was computed for the diameters and lengths of all vessels, defined
as the ratio of the daughter diameters and lengths, respectively. It was found that the
largest orders of arterial and venous vessels are more asymmetric and the degree of
asymmetry decreases towards the smaller vessels (Fig. 2.10). Furthermore, the
diameter asymmetry at a bifurcation is significantly larger for the coronary veins
(1.7–6.8 for sinus veins) than the corresponding arteries (1.5–5.8 for LAD artery) for
orders 2–10, respectively, as shown in Fig. 2.10. The diameter asymmetry at a
bifurcation can lead to significant heterogeneity of blood flow at a bifurcation, as
will be described in Chap. 5. Appendix 4 summarizes the statistical data for diameter
and length ratios for the coronary arteries and veins.
An alternative approach to describe the asymmetry of bifurcations is to define the
diameter ratio of the daughters relative to mother vessel (e.g., D
the ratios of the diameters of larger (D
) and smaller (Ds) daught ers, respectively,
l
and Ds/Dm, i.e.,
l/Dm
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