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7.4 Validation of Scaling Laws in Other Vascular Trees 465
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data to the scaling power-law relation expressed by Eq. (7.25) is excellent. This
implies that the equivalent resistance of a complex arterial tree can be described by a
relatively simple scaling expression. The values of ε
, along with the correlation
coefficients of the least squares fit for each organ are listed in Table 7.1 of Appendix
3. The linear flow–length relationships hypothesized by Eq. (7.26) of Appendix 2
2
(R
2006). The crown volume–length, diameter–length and flow–diameter relationships
are also summarized in Table 7.1. The major conclusion of this analysis was that the
design of various vascular trees of different organs and species can be deduced on
the basis of the minimum energy hypothesis and conservation of energy under
steady-state conditions. The study revealed the similarity of nature’s scaling laws
that dictate the design of various vascul ar trees and the underlying physical and
physiological principles.
It should be noted that the scaling laws are not sensitive to the asymmetry of the
tree, as symmetric or asymmetric flow analysis have led to similar scaling exponents
(Zhou et al., 1999). Kassab et al. (1997) have previously shown that the distribution
of mean flow rate in various orders of a tree is not strongly dependent on the
asymmetry of the tree; albeit the dispersion of flow cannot, by definition, be
predicted by the symmetric model. Since the scaling laws only represent the mean
parameters, it is not surprising that the asymmetry does not influence the relations.
The symmetric model simulates the mean statistical data of the trees and is equivalent to assuming that all the vessel elements in any order are of equal diameter and
length, and are arranged in parallel, and the blood pressures at all of the junctions
between specific orders of vessels are equal. Consequently, the symmetric model
only provides the mean data at each order number (total number of 11, 11, and
10 orders for the RCA, LAD, and LCx trees, respectively). This contrasts the
asymmetric full model which provides the huge data set shown in Figs. 7.5, 7.6,
and 7.7 which demonstrates the dispersions of the various parameters. In summary,
the scaling laws which provide relations between mean parameters can be adequately represented by symmetric tree models albeit the dispersion of parameters
require asymmetric tree representations.
7.4.1 Optimal Power Dissipation
The optimum (minimum) power consumption normalized with respect to the metabolic requirements of blood volume is expressed by Eq. (7.39) in Appendix 2 as
shown in Fig. 7.9 for various organs and species. Appropriately, the power dissipation is strongly dictated by ε
process. Borders and Granger (1986) have previously shown that power dissipation
is related to the flow rate through a power-law relation in the microcirculatory bed of
normal and hypertensive rat cremaster muscle. Since the flow rate is proportional to
crown length, their experiments vali date the form of our theoretical prediction. Our
results also show that for an entire crown, L ¼ L
; i.e., power dissipation is directly related to the resistive
, the total dissipation power is
max
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Fig. 7.9 Relationship between normalized minimum power consumption and crown length for
various organs and species. Reproduced from Kassab (2006) by permission. RCA right coronary
artery, LAD left anterior descending, LCx left circumflex, PA pulmonary artery, PV pulmonary vein,
SKMA skin muscle arteries, SMA sartorius muscle arteries, MA mesentery arteries, OV omentum
veins, BCA bulbar conjunctiva arteries, BCV bulbar conjunctiva veins
proportional to the metabolic power. The proportionality constant is given by
metabolicþviscous
metabolic
power dissipation
whose mean value for the various organs and
ε
ε
species was found to be 1.34 0.031. It should be noted that the exponent of the
volume–length relation is also the exponent of the power dissipation–length relation
(Eqs. 7.29 and 7.39, Appendix 2). This is expected since the volume–length exponent is a direct result of the minimum energy principle. It should also be noted that
when Murray’s law holds (δ ¼ 3, i.e., ε
highest value but decreases as ε
increases beyond 2 as is the case for the organ
systems shown in Table 7.1 (Appendi x 3).
7.4.2 Vascular Metabolic Dissipation of Blood Vessel Wall
The metabolic dissipation in Murray’s minimum energy hypothesis includes only
the blood metabolism. The metabolic dissipation of the vascular tree, howe ver,
should also include the metabolism of passive and active components of the vessel
wall. Liu and Kassab (2007b) extended the metabolic dissipation to include blood
metabolism, as well as passive and active components of the vessel wall. The
analysis was extended to the entire vascular arterial tree rather than a single vessel
as in Murray’s formulation. The calculations were based on experimentally measured morphological data of the coronary artery network and the longitudinal
7.5 Scaling Law of Flow Resistance 467
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distribution of blood pressure along the tree. While the model includes multiple
dissipation sources, the total metabolic consumption of a complex vascular tree was
found to remain approximately proportional to the cumulative arterial volume of the
stem-crown unit (Appendix 4). This implies that the scaling relations for the various
morphological features (volume, length, diameter, flow) remain unchanged under
the generalized condition of metabolic requirements of blood and blood vessel wall
(passive and active).
7.5 Scaling Law of Flow Resistance
Although the validated analysis in Appendix 2 provided an analytical equation for
the equivalent resistance of a tree or crown, it involved the use of an empirical
parameter ε
empirical parameter and subsequently expresses all scaling laws in terms of invariant
exponents that can be tested against experimental data.
Several concepts first need to be defined to formulate the resistance scaling law
(Huo & Kassab, 2009b ). A vessel segment is defined as a stem and the tree distal to
the stem is defined as a crown (see Fig. 7.1). Obviously, an entire tree consists of
many stem-crown units down to the smallest arterioles or venules. The capillary
network (vessel diameter <8 μm) is excluded from the present analysis because it is
not tree-like in structure. A stem is assumed to be a cylindrical tube with no
consideration of vessel tapering and other nonlinear effects because they play a
relatively minor role in determining the hemodynamics of the entire tree. Through
the well-known Hagen–Poiseuille law, the resistance of the steady laminar flow in a
stem of an entire tree, R
and Q
. In this section, a derivation will be provided that eliminates the
ΔP
s
s
is volumetric flow rate through the stem), can be written as:
s
(where ΔPsis the pressure gradient along the stem
Q
s
where D
and Lsare the diameter and length of the stem, respectively. The fluid
s
viscosity, μ, and K
resistance of a crown, R
the stem to the terminal vessels), is proposed as follows (see Appendix 5 for
derivations):
where L
is the crown length that is defined as the sum of the lengths of each vessel in
c
the crown and D
constant that depends on the branching ratio, diameter ratio, total number of tree
128μL
R
s
s
is the diameter of the stem vessel proximal to the crown. Kcis a
s
ΔP
c
(where ΔPcis the pressure gradient in the crown from
c
Q
s
4
πD
s
R
c
L
s
c
s
s
4
D
s
L
c
4
D
s
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generations, and viscosity in the crown. Since Eq. (7.2a) is applicable to any stem-
L
max
crown unit, we can obtain: R
and R
correspond to the most proximal stem diameter, the cumulative vascular
max
max
4
c
D
max
such that K
c
R
max
L
max
4
max
, where D
max
, L
max
length, and total resistance of the entire tree. In the nondimensional form, Eq. (7.5a)
can be written as:
R
Parameter A
in Eq. (7.5b) should be equal to one. Table 7.3 (Appendix 5) shows
1
validation of the form of Eq. (7.5b) where the values of A
R
max
c
4
D
s
D
max
L
c
1
L
max
1
are reasonable close to
1 for a variety of organs and species (mean SD of 1.01 0.063).
From Eqs. (7.4)to(7.5a), the desired resistance scaling relation between a single
vessel (a stem) and the distal crown tree can be obtained as:
R
s
R
c
K
s
K
c
L
s
L
c
Equations (7.4–7.6) relate the resistance of a single vessel to the corresponding distal
tree. The form of Eq. (7.6) was validated as the ratios K
constant (i.e.,
R
L
s
s
) for various organs and species, as given by the significant
R
L
c
c
were found to be
s/Kc
correlation coefficient of the linear least squares fit (Appendix 5).
,
7.6 Scaling of Myocardial Mass
The scaling laws of morphometry (diameter, length, volume, etc.) and flow must be
related to the size of the organ they serve. Allometric scaling laws describe how
biologic parameters vary with scale, regardless of the differences among the organisms or species. Scaling laws are independent of the specific nature of an organism
and originate from common underlying mechanisms. The most well-known allometric scaling law dates back to 1932, when Kleiber showed that the standard metabolic
rates among mammals varied with the 3/4 power of body mass, the so-called
elephant to mouse curve (Kleiber, 1932). Scaling laws arise from common underlying mechanisms that are independent of the specifi c nature of individual organisms.
In particular, hierarchical fractal-like branching networks, which distribute energy
and materials, are considered to play a central role (West & Brown, 2005; West,
Brown, & Enquist, 1997).
Allometric scaling phenomena have been investigated widely, and the dependence
of a biologic variable Y on body mass M is characterized by the allometric scaling law
Y ¼ Y
Mb, where b is the scaling exponent and Y0is a constant characteristic of the
0
kind of organism (West et al., 1997). For example, the radii of aortas scale as M
3/8
,
7.6 Scaling of Myocardial Mass 469
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cardiac frequency scales as M
–1/4
, blood circulation time scales as M
output and number of capillaries scale as M
3/4
(West et al., 1997).
1/4
, and cardiac
Several explanatory models have been proposed for the allometric scaling with
body mass (Agutter & Wheatley, 2004). The model proposed by West and colleagues, WBE model (West & Brown, 2005; West et al., 1997) addres ses the supply
of materials (e.g., oxygen) to cells through hierarchical networks of branching tubes
(e.g., the circulatory system). Although this model has invited critical remarks when
extended to all biologic levels, its validity for mammals is widely accepted. It should
be noted that the WBE model is an inter-species scaling law as opposed to Murray or
ZKM models which represent intraspecific scaling laws (formulations within the
heart or other organs of a given species).
West, Brown, and Enquist proposed that the 3/4 rule arises from optimized
dissipation of energy within vascular networks and independent terminal capillaries
that do not vary with body size wi thin the mammalian vasculature (West et al.,
1997). Based on a surface-to-volume ratio, some investigators have argued an
exponent of 2/3 rather 3/4 (White & Seymour, 2003). It has been also debated that
the scaling relation cannot be purely described by a power-law, and exponent
changes depending on other factors like mass or environment (Kolokotrones, Savage, Deeds, & Fontana, 2010). West, Brown, and Enquist derived the 3/4 power-law
based on the concepts of fractal and transport networks. They used the assumption
that flow rate scales with total capillary number while total capillary number scales
with metabolism. Further, the total volume of network and branching pattern were
used to derive the 3/4 scaling law.
There is evidence indicating scaling relationships between tissue mass and
respective vasculature within an organ. For example, the sum of arterial branch
lengths distal to the point of occlusion has been proposed for estimating the
corresponding regional myocardial mass at risk (Seiler et al., 1993). The total
volume of blood in mammals has been found to scale proportionately with body
mass (Stahl, 1967; West et al., 1997). More recently, Karalis and colleagues (Karalis,
Claret, Iliadis, & Macheras, 2001) have indicated that the fractal volume of blood
scales proportionally to mass.
Choy and Kassab (2008) hypothesized that myocardial mass scales with V, L, D,
and Q, where V and L correspond to cumulative arterial volume and length,
respectively; D is arterial diameter, Q the volumetric flow. To delineate various
myocardial regions perfused by different coronary arterial trees, multiple
cannulations with different cast colors were made. Two cannulations were made at
different points (proximal and distal) along the main trunk of the LAD, three in the
RCA, and two in the LCx artery. The seven vessel segments were individually
perfused at pressure of 100 mmHg with cardioplegic solution and the corresponding
flows measured by a flowmeter. The seven vessel segm ents were then perfused at
pressure of 100 mmHg with seven different colors of a liquid polymer mixed with
Cab-O-Sil to block the capillaries, resulting in the perfusion of the entire arterial tree
down to pre-capillary levels. After the cast material was completely hardened, the
myocardium was cut in various regions delineated by each color of the Micro fil
compound. Each region was weighted and macerated in 30% potassium hydroxide
470 7 Scaling Laws of Coronary Vasculature
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solution for 5–7 days to remove the cardiac tissue and obtain a cast of the coronary
arteries and their branches. Each cast corresponding to the arterial tree segment was
weighted, dissected into smaller pieces, and photographed using a stereomicroscope.
The coronary arterial segments were reconstructed completely with the measured
proximal lumen diameters and the cumulative lengths of the tree. The volume (V)
corresponding to each arterial tree was calculated from the weight of the casts and the
density of the Microfil compound. Two vascular circuits were considered. One, called
the truncated tree model, was an actual reconstruction of the coronary arterial tree
down to approximately 1 mm in diameter. This model corresponds to what would
typically be observed in an angiogram (spatial resolution of approximately 1 mm), and
hence has obvious clinical implications. The other model, called an extrapolated full
tree model, is an idealization of the entire tree down to the capillary level. This model
was generated by using a combination of data obtained from the arterial casts and the
extrapolation of data from the terminal diameters of the casts based on tree growth
algorithms of coronary arteries described above (Mittal et al., 2005).
A scaling model of the form Y ¼ Y
cumulative volume, length, branch diameter, or flow; Y
mb, was fitted to the data where Y is the
0
is a normalization constant,
0
and b is the power-law exponent. The findings are summarized along with a least
squares fit model in Table 7.4 (Appendix 6). The experimental data validate the
following myocardial mass scaling laws:
1
m
0
3=8
m
0
3=4
m
0
where V
V ¼ V
D ¼ D
Q ¼ Q
, D0, and Q0are proportionality constants. These relations constitute
0
intraspecific allometric scaling laws in the heart that have numerous utility as
shown below.
7.7 Scaling Law of Vascular Blood Volume
Vascular volume is of fundamental significance to the function of the cardiovascular
system. An accurate prediction of blood volume in patients is physiologically and
clinically significant. To formulate a theory for scaling of blood volumes, a vessel
segment is defined as a stem and the tree distal to the stem is defined as a crown
(Fig. 7.1) similar to above (Huo & Kassab, 2009a). Then, V
perfused by the stem-crown unit) as shown above (Eq. 7.7), where V
volume (i.e., the sum of all vessel volumes in the crown). Therefore, V
represented as V
mass–morphometry relations given above (D
L
m
c
l
c
3/4
, Eq. (7.9)), the following can be obtained:
m
v
1/4m3/4
where Cvis a volume–mass constant. Based on the
s
3/8
m
d
c
, Eq. (7.8); Q
is the crown
c
c
m
s
Q
can be
3/4
and
7.7 Scaling Law of Vascular Blood Volume 471
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2=3
D
L
s
d
2=3
max
c
C
l
L
max
2=3
D
v
s
such that K
L
c
V
max
v
, where
2=3
D
L
max
max
1=4m3=4
m
v
C
is a constant. Since Eq. (7.10a) is applicable to any stem-
l
where K
V
c
2=3
= C
v
v
d
crown unit, the following holds: V
, L
max
, and V
D
max
correspond to the most proximal stem diameter, the cumulative
max
max
v
C
D
v
vascular length of entire tree, and the cumulative vascular volum e of entire tree,
respectively. Equation (7.10a) can be made nondimensional as:
V
c
V
max
D
s
D
max
2
3
L
c
L
max
The validity of Eq. (7.10b) is examined in the entire asymmetric (down to the
pre-capillary vessel segments) and epicardial (vessel diameter 1 mm) LAD, LCx,
and RCA trees of pig, as shown in Table 7.5 (Appendi x 7). Equation (7.10b) is also
validated in symmetric trees for various organs and species, as listed in Table 7.6
(Appendix 7).
7.7.1 Comparison with ZKM Model
An additional validation for volume scaling law in relation to minimum energy
hypothesis is described in the Appendix 7. The use of the volume and resistance
scaling laws leads to the derivation of relations similar to Eqs. (7.1)–(7.3 ) but
without the resistance exponent ε
:
V
c
V
max
D
s
D
max
Q
s
Q
max
2
1
7
L
c
L
max
3
7
L
c
L
max
1
2
3
D
s
D
max
The ZKM model predicted the exponents χ ¼
diameter–length, volume–length, and flow–diameter relations, respectively. Based
on the newly proposed Eqs. (7.11)–(7.13), the corresponding exponents are χ ¼
2
, and δ ¼ 2
β ¼ 1
7
values over all organs and species are 0.43 0.02, 1.28 0.09, and 2.33 0.11 for
exponents χ, β, and δ, respectively, which agrees well with the present predictions,
1
. With the respective ε
3
4 ε
3ε
, β ¼
5
ε
, δ ¼
4 ε
for
3ε
(see Table 7.1 Appendix 3), the mean
3
,
7
472 7 Scaling Laws of Coronary Vasculature
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i.e.,
3
7
2
7
1
3
mean SD value of 2.98 0.34 for volume–diameter relation with the respective ε
which is consistent with the exponent value of 3 in Eq. (7.66) (Appendix 7). This
provides further validation for the proposed volume scaling law as demonstrated in
Appendix 7. Table 7.7 (Appendix 7) provides validation of Eqs. (7.11)–(7.13)of
vascular trees of various organs and species.
7.8 Scaling Laws of Blood Flow Rate, Vessel Blood Volume,
Vascular Lengths, and Transit Times with Number
of Capillaries
The mean transit time (MTT), defined as the time required to transport blood within
the vascular network, plays a vital role in the physiologic al function of the circulatory system (Crumrine & LaManna, 1991; Derdeyn, Grubb, & Powers, 1999). The
vascular network has structural heterogeneity, the complexity of spatial arrangement
of vessels, and adaptation of vascular anatomy in response to hemodynamic and
metabolic stimuli (Pries & Secomb, 2009). Hence, developments of structure–
function relations which relate the MTT to vascular morphology are fundamental
to understanding the interplay between vascular form and function, and thus provide
a better rationale for clinical diagnostics and therapies.
An adequate tissue perfusion (volumetric blood flow per unit mass of tissue) to
match metabolic requirements of an organ such as the heart is essential for normal
function of an organism across all species. Too low of tissue perfusion may cause
hypoxia, ischemia, cell death, and ultimate loss of organ function. Histological
assessment of biopsy tissues, including capillary density measurements, is common
but invasive measurements and the connection with the flow and hence function is
empirical and qualitative. Since there is no equivalent relation between flow (and
related parameters such as vascular volume, length, and transit times) and capillarity
(i.e., number or density of capillaries), derivation of such relations is of significance
as outlined below (Razavi, Shirani, & Kassab, 2018).
,
7.8.1 Flow Scales with Capillary Numbers
A direct relation between flow through a branch (i.e., stem flow) of an organ vascular
system and the respective number of capillaries through which the blood flow
distributes should follow from the conservation of mass as outlined in Appendix
11. Briefly, if we normalize the flow at the inlet of the tree or crown (Q
and capillary numbers perfused by a given stem (N
the following is obtained:
stem flow)
st
) with respect to an entire tree,
c
7.8 Scaling Laws of Blood Flow Rate, Vessel Blood Volume, Vascular... 473
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st
N
N
c
c, max
where Q
st,max
and N
Q
Q
st, max
are the inlet flow and the total number of capillaries in a
c,max
vascular system, respectively.
The normalized flow and number of capillaries for all stem-crown units of the full
asymmetric coronary arterial trees obeys a power-law (Fig. 7.10). The values of
scaling exponent obtained from nonlinear regression are 1.005 (R
(R
2
2
2
Hence, the relation is line ar as theoretically derived. The total number of data points
shown in Figs. 7.10a – c are 838,462, 950,014, and 575,868, respectively.
Analysis of normalized stem flow-crown capillaries for symmetric trees for
various vascular trees of various species including the coronary arterial trees also
shows a linear relation between perfusion flow and the respective number of crown
capillaries (Fig. 7.11). The exponents in the symmetric analysis for all species and
organs are equal to a theoretical value of unity. Appendix 12 summarizes the least
squares power-law relation for each of the vascular trees, including the coefficient,
exponent, and R
2
. The exponents are nearly unity and the R2is highly significant.
Since a linear relationship between flow rate and capillary number is expected
based on the conservation of mass, such relation was verified for both symmetric and
asymmetric morphometric data sets as shown in Figs. 7.10 and 7.11. Flow–length and
flow–diameter relations were demonstrated in earlier sections of this chapter based on
the minimum energy hypothesis. This analysis suggests that the number of capillaries
in the length–capillary relation (form–form relation) and flow–capillary relation
(form–function relation) relates flow to length (Huo & Kassab, 2012). The analysis
takes into account the effect of heterogeneity in vessel geometry and hemodynamic
parameters. The physical basis of the flow–number relation is the relative uniformity
of the diameter of arterial capillaries which has been previously shown by Kassab and
Fung (1994) for the coronary vasculature. The coefficient of variation (CV ¼ SD/
Mean) is 0.15 and 0.18 for the right and left ventricle walls, respectively. Furthermore, it is well recognized that the capillary dimensions are generally conserved
across species (e.g., capillary diameters are similar in rat and human (Karbowski,
2011)). The upstream blood vessels and variation of pressure at the capillary bed,
however, can lead to variation in terminal flow. Hence, scaling relationships for flow–
capillary, and subsequently flow–length and flow–diameter relations, provide a better
fit for larger vessels where many stem-crowns are included.
Perfusion is expressed as flow per mass and hence relates proportionally to the
number of capillaries per mass. Since mass is equal to the volume and density of
tissue, the perfusion increases with the increase in the number of capillaries per
volume of tissue or number density as can be determined histologically. Hence, the
linear scaling allows a direct connection between structure (number density) and
function (perfusion). This relation may be used to understand the borderline between
physiology and pathophysiology. When the number density of capillaries is
decreased due to infarction, hypertension, or obesity, etc., this may lead to
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Fig. 7.10 Relationship
between normalized stem
flow and normalized number
of capillaries 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 shown in panels
(a), ( b), and (c) are 838,462,
950,014, and 575,868,
respectively. The dash lines
correspond to the theoretical
exponent of unity.
Reproduced from Razavi
et al. (2018) with permission
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