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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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7.8 Scaling Laws of Blood Flow Rate, Vessel Blood Volume, Vascular... 475
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Fig. 7.11 Relationship between normalized stem flow (Qs/Q
capillaries (N
scatter plot. RCA right coronary artery, LAD left anterior descending artery, LCx left circumflex
artery, PA pulmonary artery, PV pulmonary vein, SMA sartorius muscle arteries, MA mesentery
arteries, OV omentum veins, BCA bulbar conjunctiva arteries, BCV bulbar conjunctiva veins, RMA
retractor muscle artery. Reproduced from Razavi et al. (2018)
) for the symmetric trees of various species and organs shown in a log-log
c/Nc,max
) and normalized number of
s,max
malnutrition, atrophy, or apoptosis of cells. Conversely, the number density of
capillaries may be increased in exercise or tumors in line with increase in blood
flow to enhance the growth of the tissue. The number density can be determined
from histological sections of biopsy specimens of animals and patients.
The flow perfusion-number density scaling relation can also be used for drug dose
determination. The dose can be titrated b etween species as the number density
reflects perfusion (flow per mass) of tissue. Adequate perfusion (volumetric flow
per mass of tissue) is essential for any organ because it affects its health and function.
The linearity between stem flow and the number of capillaries the functional
capillary density can be obtained from the length of vascular network noninvasively
from standard medical imaging.
7.8.2 Crown Volume Scales with Capillary Number
Since crown volume (Vc) is cumulative blood volume within a network, a derivation
based on the average branching ratio and scaling of vessel diameters and lengths in
476 7 Scaling Laws of Coronary Vasculature
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each branching level results in a relationship between crown volume and number of
capillaries as shown in Appendix 13. If we normalize Eq. (7.94) in Appendix 13,
with respect to maximum crown volume and number of capillaries in the entire
vascular network, a general form of scaling relationship is obtained as:
where V
c,max
and N
V
V
c, max
are the total crown volume and the total number of
c,max
c
N
N
c
c, max
λ
capillaries in a vascular system, respectively.
The crown volume obeys a power-law (Eq. 7.15) as shown by analysis of
morphometric data of full asymmetric arterial trees of porcine RCA, LAD, and
LCx (Fig. 7.12). The scaling exponents (λ) are 1.48 (R
(R
2
2
2
total numbe r of data points shown in Figs. 7.12a–c are the same as those in
Figs. 7.10a–c; respectively.
The exponents in the symmetric analys is for the RCA, LAD, and LCx are 1.45
(R
2
2
2
similar to the asymmetric tree analysis and close to the theoretical value of 3/2
(Fig. 7.13). The mean exponent across various species and organs are 1.37 0.166
2
(R
> 0.92), which is within the theoretical range predicted by “2/3” and “3/4”
scaling laws. The scaling exponents, confidence intervals, and R
2
associated with
scaling exponent for various species and organs are summarized in Appendix 12.
7.8.3 Crown Length Scales with Capillary Number
Since crown length is simply cumulative length of blood vessels of all branching
levels within the network, a derivation based on the average branching ratio and
scaling of average length of blood vessels in each level of branching (Huo & Kassab,
2012), results in a direct relationship between crown length and number of capil-
laries (Appendix 13). If Eq. (7.92) of Appendix 13 is normalized with respect to an
entire tree, the following relation holds:
where L
c, max
capillaries in the entire tree. The hypothesis that λ is equal to 1 is confirmed below.
The crown length linearly scales with the number of capillaries for all stem-crown
units of the full asymmetric coronar y arterial trees (Fig. 7.14). The values of scaling
exponent λ (Eq. 7.16) obtained from Figs. 7.14a–c were 1.03 (R
2
(R
respectively (as compared to a theoretical value of unit y, Eq. (7.16)).
and N
L
L
c, max
are the maximum crown length and the number of
c, max
2
c
N
N
c
c, max
λ
2
7.8 Scaling Laws of Blood Flow Rate, Vessel Blood Volume, Vascular... 477
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Fig. 7.12 Relationship
between normalized crown
volume (V
c/Vc,max
) and
normalized number of
capillaries (N
c/Nc,max
) for
the full asymmetric porcine
arterial tree shown in a
log-log density plot: (a)
RCA right coronary artery;
(b) LAD left anterior
descending artery; (c) LCx
left circumflex artery. The
total number of data points
are the same as those in
Fig. 7.11. Reproduced from
Razavi et al. (2018) with
permission
478 7 Scaling Laws of Coronary Vasculature
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Fig. 7.13 Relationship between normalized crown volume (Vc/V
capillaries (N
scatter plot. RCA right coronary artery, LAD left anterior descending artery, LCx left circumflex
artery, PA pulmonary artery, PV pulmonary vein, SMA sartorius muscle arteries, MA mesentery
arteries, OV omentum veins, BCA bulbar conjunctiva arteries, BCV bulbar conjunctiva veins, RMA
retractor muscle artery. Reproduced from Razavi et al. (2018) with permission
The exponents in the symmetric analysis for the RCA, LAD, and LCx is 1.02
2
(R
) for the symmetric trees of various species and organs shown in a log-log
c/Nc,max
2
2
) and normalized number of
c,max
similar to the asymmetric tree analysis and close to the theoretical unity (Fig. 7.15).
The average scaling exponent (Eq. 7.16) for all species and organs is 1.08 0.102
2
(R
> 0.9). Table 7.9 (Appendix 12) summarizes the least squares power-law
relation for each of the vascular trees, including exponent, confidence interval, and
2
R
. The exponents are nearly unity and the R2is highly significant.
The conformity between scaling of crown length and number of capillaries
among various species and organs reveals another salient proportionality law
between form and function of the vascular system. The blood vessels are known to
adapt to physiological demands and altered homeostatic conditions. The capability
of vascular trees to deliver oxygen tissue and nutrients to serve metabolism strongly
depends on the number of capillaries. The capillary density changes in response to
conditions like hypoxia. In the context of new blood vessel formation and vascular
sprouting, length of the perfused blood vessel is a key determinant of growth and
development. It has been shown that length of blood vessels adapts to changes in
homeostatic conditions (Humphrey, Eberth, Dye, & Gleason, 2009; Lehman,
Owens, Kassell, & Hongo, 1991; Sho et al., 2004). The scaling law links vascular
length to respective capillaries through a structure–function relation.
7.8 Scaling Laws of Blood Flow Rate, Vessel Blood Volume, Vascular... 479
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Fig. 7.14 Relationship
between the normalized
crown length (L
c/Lc,max
) and
normalized number of
capillaries (N
c/Nc,max
) for
the full asymmetric porcine
arterial tree shown in a
log-log scatter plot: (a) RCA
right coronary artery, LAD
left anterior descending
artery, LCx left circumflex
artery. The total number of
data points are the same as
those in Fig. 7.10.
Reproduced from Razavi
et al. (2018) with permission
480 7 Scaling Laws of Coronary Vasculature
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Fig. 7.15 Relationship between the normalized crown length (Lc/L
capillaries (N
RCA right coronary artery, LAD left anterior descending artery, LCx left circumflex artery, PA
pulmonary artery, PV pulmonary vein, SMA sartorius muscle arteries, MA mesentery arteries, OV
omentum veins, BCA bulbar conjunctiva arteries, BCV bulbar conjunctiva veins, RMA retractor
muscle artery. Reproduced from Razavi et al. (2018) with permission
) for the full asymmetric porcine arterial tree shown in a log-log scatter plot.
c/Nc,max
) and normalized number of
c,max
7.8.4 Transit Time Scales with Crown Volume and Length
Because of the structural heterogeneity of vascular networks and hence heterogeneous perfusions, the particles traverse various paths in the network. Hence, the
mean transit time (T
vascular network over period of time. Appendix 14 demonstrates a derivation for a
relation between mean transit time and crown volume and length (i.e., Eq. 7.97).
Equation (7.97) (Appendix 14) can be presented in a general normalized form as:
where T
c, max
, L
volume in the entire tree, respectively.
The crown volume and the product of crown length and mean transit time of
asymmetric coronary arterial trees follows a scaling relationship (Fig. 7.16). The
) is the average time required for blood to travel through the
c
c, max
T
T
c, max
, and V
c, max
c
L
c
L
c, max
V
V
c
c, max
λ
are the crow n transit time, crown length, and crown
7.9 Other Design Features of Vascular Trees 481
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scaling exponent λ (Eq. 7.17) are 0.981 (R
2
2
2
for the porcine RCA, LAD, and LCx, respectively, as compared to a theoretical
value of unity reflected by Eq. (7.17).
The transit time is a fundamental physiological parameter in biological transport
phenomena and has critical implications for vascular disease. Too low or too high of
transit times may not be physiological. For example, no-capillary flow and altered
blood volume conditions occur under pathophysiological conditions. The scaling
relation between mean transit time, blood volume, and the number capillaries can be
used as a theoretical basis to understand the distribution of oxygen and nutrients
under physiological conditions and microvascular failure under pathological conditions. It was shown by Razavi et al. (2018) that the estimation of transit time based
on the crown length and volume hold for proximal trees (down to 1 mm diameter
vessels which can be observed in angiograms). Hence, standard clinical imaging of
blood vessel anatomy may yield functional data on the transit times through the
organ of interest.
Similarly, the exponents in the symmetric analysis were 1.01 (R
2
(R
similar to the exponents related to the asymmetric data (Fig. 7.17). The mean
exponents for all species and organs is 1.03 0.082 (R
data). The exponents for various species and organs along with the associated
confidence interval and R
2
2
2
were summarized in Table 7.9 (Appendix 12).
2
> 0.98 for symmetric
7.9 Other Design Features of Vascular Trees
Horton’s law states that there exists a geometric relationship between the number of
vessels of a given order and the corresponding order where the parameter of this
geometric relationship is the branching ratio (Chap. 2). Although biological trees do
not strictly obey Horton’s law because the various branching ratios vary somewhat
with order number (Kassab, Lin, & Fung, 1994), a mean ratio may be defined over
the entire range of orders such that Horton’s law is a good approximation. Table 7.8
(Appendix 8) shows the mean diameter, length, and number ratios over a range of
order numbers for various organs and species. Horsfield (1980) suggested that for
laminar flow with minimal resistance and minimal entropy production, the number
ratio should be equal to the cube of the diameter ratio R
ratio should be equal the diameter ratio (R
L
that the length ratio is within 10% of the diameter ratio for most organs and species,
while the number ratio deviates significantly from the cube of the diameter ratio.
Barker, Cumming, and Horsfield (1973) observed that the quotient of the diameter
and number ratios is 0.5 for the human pulmonary arterial and bronchial trees and
birch trees. The mean SD of the R
ratio for the various organs and species
D/RN
listed in Table 7.8 is 0.48 0.057. This implies a coefficient of variance
). Table 7.8 (Appendix 8) shows
D
3
N
, and the length
D
482 7 Scaling Laws of Coronary Vasculature
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Fig. 7.16 Relationship
between normalized crown
volume (V
c/Vc,max
) and
multiplication of crown
mean transit time and crown
length (T
s/Ts,max*Ls/Ls,max
)
for the full asymmetric
porcine arterial tree shown
in a log-log density plot: (a)
RCA right coronary artery;
(b) LAD left anterior
descending artery; (c) LCx
left circumflex artery. The
total number of data points
are the same as those in
Fig. 7.10. Reproduced from
Razavi et al. (2018) with
permission
7.10 Fractal Description of Branching Pattern 483
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Fig. 7.17 Relationship between normalized crown volume (Vc/V
mean transit time and crown length (T
and organs shown in a log-log scatter plot. RCA right coronary artery, LAD left anterior descending
artery, LCx left circumflex artery, PA pulmonary artery, PV pulmonary vein, SMA sartorius muscle
arteries, MA mesentery arteries, OV omentum veins, BCA bulbar conjunctiva arteries, BCV bulbar
conjunctiva veins, RMA retractor muscle artery. Reproduced from Razavi et al. (2018) with
permission
s/Ts,max*Ls/Ls,max
) for the symmetric trees of various species
) and multiplication of crown
c,max
(SD/mean 100) of approxi mately 10% over a wide variety of organs and species.
The constancy of this ratio is suggestive of a design criterion for tree structures.
7.10 Fractal Description of Branching Pattern
A system obeying Horton’s law is said to be “fractal” according to the mathematical
terminology introduced by Mandelbrot (1977). Mandelbrot classified fractals
according to their dimensions. A fractal structure exhibits a power-law relationship
between total length of a morp hological feature (e.g., total length of elements or sum
of diameters of elements) and basic length of measuring unit (e.g., length or diameter
of a single element). Nelson (1988) has shown that for a tree structure, the fractal
dimension can be computed as one minus the exponent of the power-law relationship. Gan, Tian, Yen, and Kassab (1993) have determined that the fractal dimensions
of diameter and length are 2.49 and 2.99, respectively, for the dog pulmonary venous
tree. Huang et al. (1996) obtained diameter and length fractal dimensions of 2.71 and
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2.97, respectively, for the human pulmonary arterial trees; the corresponding fractal
dimensions for the human pulmonary venous trees are 2.64 and 2.86, respectively.
The coronary data show that the diameter fractal dimensions for the RCA, LAD, and
LCx are 2.09, 2.06, and 1.99, respectively, while the corresponding length fractional
dimensions are 1.89, 1.96, and 1.78, respectively. These and other results summarized in Table 7.8 (Appendix 8) suggest that the various vascular trees possess the
characteristics of a fractal structure.
7.11 Intraspecific Scaling Laws of Vascular Trees
The fractal dimension defined by Mandelbrot has been shown to characterize natural
phenomena with remarkable simplicity (Mandelbrot, 1977). Vascular
(Bassingthwaighte, Van Beek, & King, 1990; Family, Masters, & Platt, 1989;
Matsuo, Okeda, Takahashi, & Funata, 1990; Sernetz, Wübbeke, & Wlczek, 1992),
bronchial (Kitaoka & Itoh, 1991; Nelson & Manchester, 1988), and botanical trees
(Kamiya & Takahashi, 2007; Matsuo, Nakakubo, & Yamamoto, 1997; Morse,
Lowton, Dodson, & Williamson, 1985; Shibusawa, 1994) have been found to
have fractal-like features. West et al. (1997) proposed a mathematical model referred
to as the WBE (West, Brown, and Enquist) model to support the allometric 3/4
scaling law of metabolism for interspecific (from species to species) scaling, based
on three major assumptions in a fractal-like cardiovascular system:
1. Area-preservation for branching patterns of vessel diameter >1 mm and cubed-
law for smaller vessels as wel l as space-filling branching pattern.
2. Size-invariant capillaries.
3. Minimum energy loss.
In the process, they assumed that the fractal-like networks in life have a fourth
spatial dimension (West, Brown, & Enquist, 1999), which served as the basis of the
WBE model (West et al., 1997). There is much debate on the WBE model, however,
especially for the assumption of the space-filling branching pattern in the cardiovascular system (i.e., LR ¼ BR
1
3
, where BR and LR are the branching ratio and length
ratio, respectively. Kozlowski and Konarzewski (2004, 2005).
To derive the scaling laws within a species (intraspecific scaling, within an organ
of given species), the branching ratio, diameter ratio, and length ratio in a fractal-like
tree structure are defined as: BR ¼ n
where n
i ¼ 1, ... , N
, Di, and Liare the number, diameter, and length of vessels in level i,
i
(Huo & Kassab, 2012). Level 0 is the most proximal stem for an
total
entire tree or a stem for the stem-crown unit and level N
arterioles or venules. Based on the assumptions of LR ¼ BR
i/ni 1
,DR¼ Di/D
, and LR ¼ Li/L
i 1
refers to the smallest
total
1
3γ
and DR ¼ BR
i 1
2þE
(where γ ¼ 0 represents space-filling, γ ¼ 1 area-filling, and γ ¼ 2 lengthpreservation; ε ¼ 0 represents area-preservation and ε ¼ 1 Murray ’s law) for a
tree structure, fundamental derivations of the volume–diameter and flow–length
,
1
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