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7.3 Zhou, Kassab, and Molloi (ZKM) Model 455
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Fig. 7.1 A schematic
illustration of the definition
of stem-crown unit.
Reproduced from Kassab
(2007) by permission
7.3 Zhou, Kassab, and Molloi (ZKM) Model
Both Murray ’ s formulation (Appendix 1) and Uylings’ modification are focused on
an individual vessel branch. The flow rate through a vessel branch, however,
depends not only on the resistance of that branch but also on the total resistance of
the distal tree. Hence, the formulation of an optimization principle requires the
treatment of a tree structure as an integrated system. Zhou, Kassab, and Molloi
introduced the ZKM model, which formalized the “minimum energy hypothesis” to
an entire vascular system (Zhou, Kassab, & Molloi, 1999). In the process, a vessel
segment was defined as a stem and the entire tree distal to the stem was defined as a
crown (Wahle et al., 1993). Obviously, the entire tree consists of many stem-crown
units down to the capillary vessels as show n in Fig. 7.1. At each bifurcation, there is
a unique stem-crown unit which continues down to the smallest unit, i.e., the
capillary. Functionally, each stem supplies or collects blood from the crown for an
arterial or venous tree, respectively.
The ZKM model is based on the minim um energy hypothesis along with conservation of energy for steady-state flow and leads to power-law relationships between
(1) vascular length and volume of arterial tree, (2) diameter and length of vessel
branches, and (3) lumen diameter and blood flow rate in each vessel branch (see
derivation in Appendix 2). If A and Q represent the mean cross-sectional area and
blood flow rate of a stem, respectively, and V and L represent the cumulative arterial
volume and length of a crown, respectively, the ZKM model predicts the following
relationships (Appendix 2):
V
V
max
5
ε
L
L
max
456 7 Scaling Laws of Coronary Vasculature
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3ε
4 ε
L
L
max
4 ε
3ε
D
D
max
where D
max
, Q
max
, V
max
, and L
D
D
max
Q
Q
max
correspond to the diameter and flow rate of the
max
most proximal stem, the volume of the entire crown, and the cumulative arterial
length of the crown. The parameter ε
relates to the crown flow resistance and is
equal to the ratio of metabolic-to-viscous power dissipation (Appendix 2). The form
of relation 3 (Eq. 7.3) is equivalent to Murray’s law with the important distinction
that the exponent is not equal to 3. Relations or Eqs. (7.2) and (7.3) are novel (Zhou
et al., 1999 ) and advance Murray’s original formulation (Murray, 1926).
7.3.1 Validation of ZKM Model
Equations (7.1)–(7.3) have been determined and validated for the coronary trees in
the following three ways:
1. In vivo data on coronary stem flow rate, crown length, and volume using digital
subtraction angiography has verified the relationships for vessels proximal to
0.5 mm in diameter as observed in an angiogram.
2. Hemodynamic analysis of coronary blood flow rate based on detailed anatomical
data yields these relationships over the entire arterial network.
3. Derivation of scaling laws is based on fractal geometry.
7.3.2 Experimental Validations
Equations (7.1)–(7.3) were validated based on angiographic measurements of the
epicardial arteries (Zhou, Kassab, & Molloi, 2002). The 3D lengths of epicardial
coronary arteries were measured from bi-plane angiography after power injection of
contrast material into the left main coronary artery. A video densitometry technique
that quantifies the absolute cross-sectional area including small vessel diameter and
any complex shape of the vessel cross-section was used for lumen cross-sectional
area measurement (Molloi, Ersahin, Hicks, & Wallis, 1995; Molloi, Ersahin, Roeck,
& Nalcioglu, 1991). Briefly, a cylindrical vessel phantom, consisting of plastic
tubing with different inside diameter s (0.95–4.75 mm in diameter) filled with
contrast material, was imaged over the heart region for calibration purposes. The
integrated gray levels in the vessel profiles were related to the known CSA of the
vessel phantoms. This information was used to directly convert the integrated gray
levels to CSA for the coronary angiograms.
7.3 Zhou, Kassab, and Molloi (ZKM) Model 457
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A video densitometry technique for quantification of lumen volume was also
used, where phase-matched subtracted images were used to quantify regional lumen
volume (Molloi, Kassab, & Zhou, 2001). Briefly, temporal angiographic subtraction
images were formed after the images were corrected for scatter and veiling glare
(Ersahin, Molloi, & Yao-Jin, 1995). A region of interest (ROI), approximately
outlining the visible epicardial arteries, was manually drawn. A narrow background
shell was drawn just outside the arterial ROI. The background ROI was corrected for
the iodine signal in the myocardium. The integrated videodensitometric signal was
converted to iodine mass by using the system iodine calibration curve. The calculated iodine mass was then converted to volume by using the known iodine concentration of contrast material.
Finally, a first pass analysis technique in conjunction with phase-matched temporal subtraction images were used to measure regional coronary blood flow in the
branches of the LAD (Molloi, Bednarz, Tang, Zhou, & Mathur, 1998; Molloi,
Ersahin, Tang, Hicks, & Leung, 1996). Briefly, the arterial ROI identified the
coronary artery of interest. A narrow background shell, next to the arterial ROI,
was used to subtract the contribution of iodine signal from the myocardium to the
arterial ROI. The arterial ROI limits the flow measurement to the time interval that it
takes to opacify the arterial branch of interest. Arterial blood volume was determined
as per phantom calibration described above. The measured volume difference and
the known time be tween subsequent images were used to calculate absolute volumetric flow. Flow measurements were made by using phase-matched temporal
subtraction images.
The normalized cumulative arterial volume of each crown, defined as the ratio of
the crown volume to the volume of the entire tree, was significantly correlated with
the corresponding normalized cumulative arterial length, as shown in Fig. 7.2a
(Eq. (7.1); r ¼ 0.98). The normalized arterial diameter of each stem was also related
to the corresponding normalized cumulative arterial length (Fig. 7.2b) through a
power-law (Eq. (7.2); r ¼ 0.89). Finally, the normalized coronary blood flow
entering each defined crown was correlated with the normalized arterial diameter
of the stem feeding the crown, as shown in Fig. 7.2c (Eq. (7.3); r ¼ 0.90).
7.3.3 Computational Validations
The formulation of coronary blood flow under steady-state conditions is outlined in
Chap. 5. Accordingly, a network flow analysis was performed in the entire coronary
arterial trees with the total numbe r of vascular branching in a given tree equal to
858,353 (RCA), 936,013 (LAD), and 572,631 (LCx) (Kassab, 2007). These numbers include only the arterial branches and not the capillaries. Given the large
number of data points, the figures below are expressed as iso-density plots showing
five layers of frequency. As expected, the majority of vessels are in the smaller
arterioles diameter range.
458 7 Scaling Laws of Coronary Vasculature
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Fig. 7.2 (a) Normalized
cumulative arterial volume
with the corresponding
crown length (r ¼ 0.98), (b)
Normalized stem diameter
and crown length (r ¼ 0.89),
and (c) Normalized stem
flow and stem diameter
(r ¼ 0.90) on a log-log scale.
Reproduced from Zhou et al.
(2002) with permission
7.3 Zhou, Kassab, and Molloi (ZKM) Model 459
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30
10
24
10
3
)
MAX
) / (L/L
MAX
Frequency
0.000001
0.000010
0.000100
0.001000
0.010000
10
10
18
12
(Rc/R
6
10
0
10
10
–10
10
–8
10
–6
10
–4
10
–2
10
0
Normalized Cumulative Arterial Volume of Crown
Fig. 7.3 The relationship between the ratio of the equivalent resistance and the cube of the length
of crown versus the corresponding arterial volume of the crown for RCA arterial tree as an
iso-density plot showing five layers of frequency. The plots for LAD and LCx arterial trees are
similar to those of RCA. Reproduced from Kassab (2007) by permission
The inlet or maximum flow in the most proximal stem was found to be 0.53
(RCA), 0.63 (LAD), and 0.32 (LCx) mL/s. The cumulative crown lengths were
volumes were 1.3 (RCA), 0.97 (LAD), and 0.56 mL (LCx).
The validity of Eq. (7.25) in Appendix 2 for the crown resistance was examined
for the RCA, LAD, and LCx arterial trees as illustrated in Fig. 7.3. It is apparent that
Rc=R
the relation between
L=L
max
max
and V/V
3
is a power-law whose exponent is equal to
max
. The data conforms to the scaling power-law relation (Appendix 2) which
implies that the equivalent resistance of a complex coronary arterial tree can be
described by a relatively simple scaling expression. The values of ε
LAD, and LCx trees are 2.71 (R
2
(R
2
2
for the RCA,
Figure 7.4 validates the linear flow–length relationship hypothesized by
Eq. (7.26) in Appendix 2 (also see Appendix 9). The data are shown on a log-log
plot to highlight the large span of flow rate and length (appro ximately eight to nine
orders of magnitude) from small microcirculatory units (as small as 8 μmin
diameter) to large epicardial arteries (4–5 mm ). The exponents of the power-law
plot are 0.984 (R
2
(R
2
2
different from one, the relationship between flow and length can be considered
linear.
460 7 Scaling Laws of Coronary Vasculature
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–8
10
Normalized Stem Flow
–6
10
Normalized Crown Length
10
–4
10
–2
Frequency
0.000001
0.000010
0.000100
0.001000
0.010000
10
0
0
10
–3
10
–6
10
–9
10
Fig. 7.4 The relationship between stem flow and cumulative length of crown for the RCA arterial
tree as an iso-density plot. The plots for LAD and LCx arterial trees are similar to those of RCA.
Reproduced from Kassab (2007) by permission
The crown volume–length relationship for the various coronary arterial trees
follows a power-law relation predicted by Eq. (7.1) as shown in Fig. 7.5. The
exponents are 1.43 (R
2
2
2
RCA, LAD, and LCx arterial trees, respectively.
Figure 7.6 shows the cross-sectional area–length relation (Eq. 7.37a in Appendix
2) with exponents of 0.937 (R
2
(R
2
2
the diameter–length relation are simply one half of those for the area–length relation
(i.e., 0.47 (RCA), 0.46 (LAD), and 0.46 (LCx)). In agreement, Seiler, Kirkeeide, and
Gould (1992, 1993) found a power-law relationship between the stem crosssectional area and the crown cumulative arterial length for both canine and human
studies. They reported a value of 0.82 for in vivo human studies of left coronary
arteries and similar values for the canine studies based on angiographic measurements. The agreement is reasonable despite (1) the experimental data are partial
(vessels larger than approximately 1 mm in diameter at the resolution of an angiogram), and (2) the studies correspond to different species.
Finally, the flow–diameter relationship of a stem is governed by Eq. (7.38)of
Appendix 2 as shown in Fig. 7.7 . The exponents for the RCA, LAD, and LCx are
2.09 (R
2
2
2
seen, the exponents deviate significantly from Murray’s value of 3. An alternative
formulation to evaluate the Murray’s exponent for flow–diameter relationship is
7.3 Zhou, Kassab, and Molloi (ZKM) Model 461
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10
Normalized Crown Volume
–8
10
–6
10
–4
10
–2
Frequency
0.000001
0.000010
0.000100
0.001000
0.010000
10
0
0
10
–2
10
–4
10
–6
10
–8
10
–10
10
Normalized Crown Length
Fig. 7.5 The relationship between crown volume and length of crown for the RCA arterial tree as
an iso-density plot. The plots for LAD and LCx arterial trees are similar to those of RCA.
Reproduced from Kassab (2007) by permission
10
–8
10
–6
10
–4
10
–2
10
0
0
10
Normalized Stem Cross–Sectional Area
Fig. 7.6 The relationship between stem cross-sectional area and cumulative length of crown for the
RCA arterial tree as an iso-density plot. The plots for LAD and LCx arterial trees are similar to those
of RCA. Reproduced from Kassab (2007) by permission
Normalized Crown Length
Frequency
0.000001
0.000010
0.000100
0.001000
0.010000
10
10
10
–2
–4
–6
462 7 Scaling Laws of Coronary Vasculature
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10
Normalized Stem Diameter
–3
10
–2
10
–1
Frequency
0
10
0
10
Normalized Stem Flow
–3
10
–6
10
–6
10
–5
10
–4
10
–3
10
–2
10
10
–9
Fig. 7.7 The relationship between stem flow and diameter for the RCA arterial trees as an
iso-density plot. The plots for LAD and LCx arterial trees are similar to those of RCA. Reproduced
from Kassab (2007) by permission
through the conservation of mass at a bifurcation. The flow through a bifurcation
must satisfy conservation of mass as Q
m
volumetric flow rates of mother and daughter vessels. If Murray’s law (Q ¼ kD
substituted for each flow in the various branches, the well-known diameter relation is
obtained D
δ
δ
δ
m
l
where D
s
, Dl, and Dsare the diameters of mother and
m
+ Qs, where Qm, Ql, and Qsare the
l
δ
)is
daughter vessels, respectively, and δ is Murray’s exponent equal to 3). In a simulation study, Karau, Krenz, and Dawson (2001) have shown that when the value of δ
was distributed, and the wall shear stress distribution was virtually independent of
the mean δ value. Hence, they concluded that significant heterogeneity in δ as
observed experimentally along bifurcations (see Fig. 7.8) renders knowledge of
the mean and local values of δ to have little effect on the uniform shear hypothesis.
Kaimovitz, Huo, Lanir, and Kassab (2008) determined the values of the exponent
according to the diameter relation as shown in Fig. 7.8 for various vessel orders of
the LAD, LCx, and RCA trees. There is a significant variation in the parameter along
with a drastic change at order 5 for the arterial trees, which may signify structural and
functional transition at the microvasculature as discussed above.
The uniformity of WSS throughout the vascular system is not supported experimentally in the coronary circulation (Stepp, Nishikawa, & Chilian, 1999) or other
organs (Lipowsky, Kovalacheck, & Zweifach, 1978; Lipowsky & Zweifach, 1974;
Pries, Secomb, & Gaehtgens, 1995). The WSS has been show n to vary by as much as
an order of magnitude with the amplification occurring in the microcirculation. The
WSS predicted by our computer models has varied nearly two orders of magnitude
between the largest and smallest vessels (larger in the microcirculation). It should be
7.3 Zhou, Kassab, and Molloi (ZKM) Model 463
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3.2
3
2.8
2.6
2.4
2.2
2
1.8
Bifurcation Exponent δ
1.6
1.4
1.2
0123456789101112
RCA
LCx
LAD
Mother vessel order
Fig. 7.8 Relationship between parameter δ D
δmδlδ
and mother vessel order number for
s
arterial trees. Reproduced from Kaimovitz et al. (2008) by permission
noted that the anatomical measurements used for the computer simulation were
obtained from porcine hearts with vasodilated vessels. Thus, the vessel wall lacks
the active smooth muscle component which regulates the shear stress in small
vessels. It is expected that vasoconstriction would decrease the value of ε
the viscous dissipation would increase ε
4 ε
δδ¼
. Although vasoconstriction would decrease the wall shear stress in
3ε
KmV
max
. This would in turn increase
2
Q
R
max
max
because
the microcirculation, it is unlikely that the shear will be completely uniform as
required by Murray’s exponent of 3.
An additional validation of these scaling laws can be confirmed by examination
of the exponents β, χ, and δ for the relationships
D
D
max
χ
L
L
max
(Eq. 7.37b) and
Q
Q
max
δ
D
D
max
(Eq. 7.38), respectively, in Appendix
V
V
max
β
L
L
max
(Eq. 7.29),
2. It is clear that the theoretical formulation predicts a dependence of the expo-
nents on the resistance parameter ε
(i.e., β ¼
ε
5
, χ ¼
4 ε
3ε
, and δ ¼
4 ε
3ε
Hence, the regression values can be well compared with the theoretical predictions. The β, χ, and δ exponents predicted from theory are 1.35, 0.41, and 2.41
(RCA); 1.33, 0.42, and 2.40 (LAD); and 1.34, 0.41, and 2.42 (LCx), respectively.
The values determined from a curve fit are within 15% of those predicted from
theory. Additionally, Eqs. (7.29), (7.37b), and (7.38) reveal two sets of constraints
for the three exponents β, χ, and δ, namely, χδ ¼ 1 and β +4χ ¼ 3. These two
quantities have values of 0.98, 3.31 (RCA); 0.97, 3.25 (LAD); and 0.97, 3.26
(LCx), respectively. There is good agreement between experimentation and theory,
as the differences are within 10%.
).
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7.4 Validation of Scaling Laws in Other Vascular Trees
If the ZKM scaling laws are universal, they should apply not only to coronary
arteries but also to other vascular trees. Since the extensive morphometric data for
the coronary arteries is not available for other vascular trees, a simplified symmetric
flow analysis has been adopted for those vascular trees for which there exists
statistical morphological data in the lung, various skeletal muscles (e.g., sartorius,
retractor, and skin muscles), omentum, mesentery, and bulbar conjunctiva of various
species. Briefly Singhal, Cumming, Horsfiled, and Harding (1973), Singhal, Henderson, Horsfield, Harding, and Cumming (1973), and Horsfield and Gordon (1981)
used Strahler’ s system to study the pulmonary arteries and veins of humans whereas
Yen et al. (1983, 1984) used Strahler’s system to study the cat pulmonary arterial
and venous trees. Strahler’s system has also been used to study the microcirculation
of cat sartorius muscle (Koller, Dawant, Liu, Popel, & Johnson, 1987), hamster
retractor muscle (Ellsworth, Liu, Dawant, Popel, & Pittman, 1987), hamster skin
muscle (Bertuglia, Colantuoni, Coppini, & Intaglietta, 1991), rat mesenteric
microvessels (Ley, Pries, & Gaehtgens, 1986), rabbit omentum (Fenton & Zwei fach,
1981), and human bulbar conjunctiva microvessels (Fenton & Zweifach, 1981).
Huang, Yen, McLaurine, and Bledsoe (1996) used the diameter-defined Strahler
system to model the human pulmonary arterial and venous trees, while Jiang,
Kassab, and Fung (1994) used it to describe the rat pulmonary arterial tree.
Based on this existing statistical morphometric database of various organs in
various species, a symmetric flow analysis was performed (Kassab, 2006). The
symmetric model simulates the mean statistical data of the trees described in the
previous section. Physically, the symmetric model 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 equ al (Kassab, Berkley, & Fung, 1997). In this simplified circuit, the flow
rate in each element of order n is Q
coronary arterial tree and N
is the total number of vessels at order n. The Q
n
determined as the ratio of pressure drop and equivalent resistance of the entire tree
(Kassab et al., 1997). The resistance, R, is computed by using Poiseuille’s equation
128μl
R ¼
, where μ represents the viscosity of blood; and l and D represent the length
4
πD
and diameter of a vessel segment. The equivalent resistance of a crown or the entire
tree is then determined by the summation for the vessel segment depending on the
series or parallel arrangement. The apparent viscosity of blood was determined as a
function of vessel diameter, hematocrit, and shear strain rate (Pries et al., 1994).
Boundary conditions were imposed on the inlet pressure of systemic and pulmonary
vessels (100 and 35 mmHg, respectively). The pressure at the inlet of the capillary
vessels was taken as 25 mmHg and the post-capillary pressure and venous outlet
pressures were considered as 15 and 5 mmHg, respectively, for both systemic and
pulmonary circulation.
The validity of Eq. (7.25) (Appendix 2) for the crown resistance was examined
based on the symmetric models of the various organs and species. The conformity of
max/Nn
, where Q
is the total flow rate into the
max
max
is
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