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104 2 Morphometry of Coronary Vasculature
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Yen, R. T., Zhuang, F. Z., Fung, Y. C., Ho, H. H., Tremer, H., & Sobin, S. S. (1983). Morphometry
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Yen, R. T., Zhuang, F. Y., Fung, Y. C., Ho, H. H., Tremer, H., & Sobin, S. S. (1984). Morphometry
of cat’s pulmonary arterial tree. Journal of Biomechanical Engineering, 106, 131–136. https://
doi.org/10.1115/1.3138469
Zamir, M. (1978). Nonsymmetrical bifurcations in arterial branching. Journal of General Physiol-
ogy, 72, 837–845. https://doi.org/10.1085/jgp.72.6.837
Zamir, M. (1990). Flow strategy and functional design of the coronary network. Tokyo: Springer.
Zamir, M. (1999). On fractal properties of arterial trees. Journal of Theoretical Biology, 197,
517–526. https://doi.org/10.1006/jtbi.1998.0892
Zamir, M. (2001). Arterial branching within the confines of fractal L-system formalism. Journal of
General Physiology, 118, 267–276. https://doi.org/10.1085/jgp.118.3.267
Zamir, M., & Brown, N. (1982). Arterial branching in various parts of the cardiovascular system.
The American Journal of Anatomy, 163, 295–307. https://doi.org/10.1002/aja.1001630403
Zamir, M., Phipps, B. L., & Wonnacott, T. H. (1984). Branching characteristics of coronary arteries
in rats. Canadian Journal of Physiology and Pharmacology, 62, 1453–1459. https://doi.org/10.
1139/y84-241
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of rat. Journal of General Physiology, 81, 325–335.
Chapter 3
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Mechanical Properties and Microstructure
of the Coronary Vasculature
3.1 Introduction
An understanding of the mechanical properties of blood vessels is fundamental to
understanding the hemodynamics of normal blood flow as well as the initiation and
progression atherosclerosis (Vito & Dixon, 2003). The vascular mechanical
properties largely stem from microstructural components, such as elastin and collagen fibers, smooth muscle cells, and ground substance (Azuma & Hasegawa, 1971;
Azuma & Oka, 1971; Kassab & Molloi, 2001; Oka, 1972; Oka & Azuma, 1970; Vito
& Dixon, 2003). Thus, the relation between the microstructure and macroscopic
mechanical properties of the vessel is essential in both biomedical research and
clinical practice. The accurate determination of microstructural deformation and
stress, and in turn function of the blood vessel, has resulted in a new level of
understanding of the blood vessel tissue.
In this chapter, we present the biomechanical properties of coronary arteries, the
role of radial constraint by surrounding tissue and the pressure–volume relationship;
the zero-stress state and vessel wall strain distribution; mechanical testing of coronary arteries; the acti ve mechanical properties of the vessel (i.e., vasoreactivity); and
the ultrastructure of the coronary arteries, including the morphometry of collagen
and elastin in the various layers of the vessel and the smooth muscle cells in the
media. This data will serve as the basis for the constitutive modeling that is the focus
of Chap. 4 .
3.2 Compliance, Distensibility, and Stiffness
The pressure–diameter relation is extremely important in vascular physiology
because it plays a crucial role in the pressure–flow relationship of blood flow through
the vessel, and hence, blood flow through the organ (see Chaps. 5 and 6). In fact, the
© Springer Science+Business Media, LLC, part of Springer Nature 2019
G. S. Kassab, Coronary Circulation, https://doi.org/10.1007/978-3-030-14819-5_3
105
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compliance of the vasculature (i.e., the slope of the pressure–diameter relation) is an
important determinant of the nonlinearity of the pressure–flow relationship (Kassab,
2001). Furthermore, the pressure–diameter–length relation can be transformed into a
biaxial (circumferential and longitudinal) stress–strain relation, wherein the mean
circumferential stress is computed from pressure, diameter, and wall thickness as per
Laplace’s equation and circumferential and axial strains are computed from circumference (or diameter) and axial measurements, respectively; in reference to the zerostress state. The major advantage of the pressure–diameter test protocol is that it
preserves the physiological coupled nature (i.e., circumferential and axial) of
mechanical loading of blood vessels unlike uniaxial strip experiments as described
below.
3.2.1 Epicardial Arteries
Kassab and Molloi (2001) determined the cross-sectional area (CSA) relation of the
first several generations of the in situ pig coronary arteries (vessels >0.7 mm in
diameter) using a videod ensitometric technique. The coronary arteries of
KCl-arrested, maximally vasodilated pig hearts are perfus ed with iodine and 3%
Cab-O-Sil. Since the Cab-O-Sil (large molecule) occludes small arteries, the flow
can be stopped and the pressure is maintained while the coronary arteries are imaged
using digital angiography. The pressure is varied while the absolute CSA of each
vessel and the total arterial volume are calculated using videodensitometry in
conjunction with digital subtraction angiography. A video densitometry technique
that quantifies the lumen cross-sectional area (CSA) is used, as described in Chap. 2.
Once the CSA is measured, the circular diameter (D ¼ (4CSA
various pressures. The Pressure–Cross-Sectional Area (P-CSA) relationship for an
epicardial artery that reflects the compliance of the vessel is shown in Fig. 3.1a. The
hysteresis loop that reflects viscoelasticity can be seen during the loading and
unloading ramps of pressure. The loading P-CSA relationship for the first several
generations of left coronary arteries is shown in Fig. 3.1b.
The Pressure–Diameter (P-D) relation can be calculated from the P-CSA relation
by assuming that the normal coronary arteries have a circular cross section. The
results show that the P-D relationship is nonlinear over the full range of pressure
(0–160 mmHg) but linear in the 60–140 mmHg pressure range. The P-D relationship
will be linear if D/2hE (where D, h, and E are diameter, wall thickness, and Young’s
modulus) remains constant as the pressure varies (Chap. 1, Appendix 2). The P-D
relationship will also remain linear if the changes in D and hE are proportional. The
linearity between pressure and diameter, for the coronary arteries, has been previously reported by other investigators in a similar pressure range (Gow & Hadfield,
1979; Gow, Schonfeld, & Patel, 1974; Patel & Janicki, 1970; Tomoike, Ootsubo,
Sakai, Kikuchi, & Nakamura, 1981).
The mean SD of slope of the P-D relation, α (compliance, which is the change
in diameter per change in pressure) is computed using linear least squares fit, in the
1/2
/π)) is computed for
3.2 Compliance, Distensibility, and Stiffness 107
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Fig. 3.1 (a) Relationship between pressure (P) and cross-sectional area (CSA) for an epicardial
artery. The least squares curves are of third-order polynomials (CSA ¼ 1.0e6P
4.3e2P + 2.7; R
2
¼ 0.987 for unloading). (b) Relationship between pressure (P) and CSA for the first several
R
generations of left coronary arteries. All curves correspond to a loading pressure ramp. Reproduced
from Kassab and Molloi (2001)
60–140 mmHg pressure range, and is summarized in Table 3.1 (Appendix 1) for the
three largest orders. It is apparent that the compliance of the coronary arteries is
small, i.e., the diameter of the coronary artery changes by less than 15% (5.4% for
order 11, 9.6% for order 19, and 13% for order 9 in the 80 mmHg pressure range.
3
2
¼ 0.991 for loading and CSA ¼ 2.1e6P3 5.3e4P2+ 5.7e2P + 2.7;
4.2e4P2+
108 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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A summary of diameter distensibility of the large coronary vessels in the literature
is listed in Table 3.2. The results agree with the in vivo and in vitro data from the dog
(Gow & Hadfield, 1979; Patel & Janicki, 1970; Tomoike et al., 1981). There are
differences, however, between the elasticity of human coronary arteries, which show
less distensibility and those o f the dog and pig (Gow et al., 1974). The decreased
distensibility is likely due to postmortem changes since the human coronary arteries
in these studies are stored overnight prior to measurements.
Generally, the compliance data of various species shows a statistically significant
decrease in distensibility as the vessel diameter increases. This may be either due to a
change of diameter-to-wall thickness ratio, or a change of Young’s modulus (see
Chap. 1 and Appendix 2) likely because of changes in the proportion of various
microstructural components (e.g., elastin, collagen, smooth muscle cells, ground
substance) that is observed when vessel diameter increases. The variation of distensibility of coronary arteries with diameter is similar to that of the pulmonary veins in
the cat (Yen & Foppiano, 1981).
The determination of coronary arterial volume using digital angiography as an
additional measure of coronary compliance has been validated (Molloi, Kassab, &
Zhou, 2001). Kassab and Molloi (2001) used this method to determine the pressure–
volume (P-V) relation of coronary arterial tree and found it to be linear in the same
pressure range as the P-D relation. The mean SD of the slope of the P-V relation as
the volume compliance (change of volume per change in pressure) is found to be
(1.1 0.45) 10
distensibility is reported as (1.1 0.36) 10
3
mL/mmHg (R2¼ 0.965–0.999). The corresponding volume
3
1/mmHg (R2¼ 0.972–0.999).
Figure 3.2a illustrates the hysteresis loop of the P-V relationship for the main
branches of the left anterior descending (LAD) arterial tree (vessels >0.7 mm in
diameter). Figure 3.2b shows the loading P-V relationships of swine, with the
volume normalized with respect to the volume at 100 mmHg. These data mirror
those of Salisbury, Cross, and Rieben (1961) who found that the coronary arterial
blood volume is a linear function of coronary arterial pressure between 30 and
125 mmHg. Morgenstern, Holjes, Arnold, and Lockner (1973) also found that the
total blood volume varies linearly with intravascular pressure in the 70–170 mmHg
pressure range. The first estimate of arterial volume compliance in the passive,
arrested heart is provided by Gregg, Green, and Wiggers ( 1935). They obtained a
static compliance value of approximately 1 10
80 mmHg. Subsequently, Patel and Janicki (1970), using a similar method, obtained
a value of 0.5 10
(1.1 0.36) 10
3
3
mL/mmHg. These data agree with the value of
, in the pressure range of 60–140 mmHg, found by Kassab
3
mL/mmHg at a mean pressure of
and Molloi (2001).
3.2.2 Capillaries
To examine the distensibility of the epicardial capillaries, the surface of the isolated
heart preparation is transilluminated and viewed with an intravital microscope
3.2 Compliance, Distensibility, and Stiffness 109
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Fig. 3.2 (a) Relation between pressure (P) and volume (V ) for the main branches (vessels
>0.5 mm in diameter) of the left coronary artery. The least squares curves are of third-order
polynomials (V ¼ 1.1e7P
V ¼ 2.1e7P
3
8.0e5P2+ 1.0e2P + 0.77; R2¼ 0.999 for unloading). (b) Relation between
arterial volume (normalized with respect to volume at 100 mmHg) and pressure. All curves
correspond to a loading pressure ramp. The least squares curves are of third-order polynomials
¼ 9.56e8P3 4.09e5P2+ 6.58e3P + 0.660; R2¼ 0.984). Reproduced from Kassab
(V/V
100
and Molloi (2001)
3
6.0e5P2+ 7.9e3P + 0.76; R2¼ 0.998 for loading and
110 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Fig. 3.3 (a) Pressure–diameter relationship for 12 capillary vessels measured at epicardial surface.
(b) Relationship between pressure (P) minus physiological pressure (P*, 30 mmHg) and diameter
(D) minus diameter at physiological pressure (D*) for the capillary vessels in (a). Reproduced from
Kassab et al. (1999) by permission
(Kassab, Le, & Fung, 1999). The coronary arteries are perfused with a colored
Microfil (inert, fluid silicone as described in Chap. 2) to visualize the epicardial
surface microvessels. The pressure is regulated in the entire vasculature by clamping
off the coronary sinus and establishing a static pressure throughout the vasculature.
Diameters of epicardial surface capillaries are recorded in the pressure range from
0 to 60 mmHg as shown in Fig. 3.3a (Kassab et al., 1999). Figure 3.3b shows that, in
the pressure range (10–50 mmHg), the elastic deformation can be described by the
equation D D* ¼ α (P P*), where D is the diameter at a given intravascular
pressure P, D* is the diameter corresponding to the physiological pressure P*
(30 mmHg), and α is the compliance constant of the vessel. The mean SD of α
is found to be 1.7 0.91 10
with the intercept set at zero, as shown in Fig. 3.3b, in the 10–50 mmHg pressure
range with a mean R
2
¼ 0.92.
6
cm/mmHg by a linear least squares fit of the data
The distensibility of capillary blood vessels is previously determined using
several methods including airtight pressure chamber (Davis, 1988), microannulation
and injection of oil drops (Swayne, Smaje, & Bergel, 1989), micro-occlusion wi thin
the limits of a pulse pressure range (Smaje, Fraser, & Clough, 1980), and elastomer
perfusion under known hydrostatic pressures (Sobin & Tremer, 1966) in various
species and organs. The epicardial coronary capillaries are among the least distensible vessels in various organs (Kassab et al., 1999) likely due to the mechanical
support from the surrounding tissue.
3.3 Effect of Surrounding Tissue: Radial Constraint and Tethering 111
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3.3 Effect of Surrounding Tissue: Radial Constraint
and Tethering
The mechanical properties of blood vessels depend not only on the microstructural
components of the vessel wall such as collagen and elastin fibers, smooth muscle cells,
and ground substances but also on the properties of neighboring tissue. All blood
vessels receive some perivascular support from the surrounding tissue. Some vessels
such as pulmonary arteries receive little support while myocardial, skeletal, or vertebral vessels are much more constrained (See review in Kassab and Navia (2006)).
Anatomically, the coronary arteries originate from the aortic ostia, just above the
aortic valve, and continue along the surface of the heart as they penetrate into the
myocardium, where they deliver blood throughout the thickness of the heart
(Kassab, 2000). The posterior component of the proximal coronary artery is partially
embedded into the myocardium, while the anterior portion is surrounded by the
serous visceral pericardium. As the coronary artery descends along the ventricle, it
becomes fully embedded into the myocardium. Several studies have examined the
mechanical properties of coronary arteries under in vitro conditions, i.e., after
dissection of the vessels from the myocardium (review in Hamza et al. (2003)).
Although those studies provide a wealth of data on the compliance and material
properties of blood vessels, they did not consider the mechanical contribution of the
surrounding medium.
It has generally been difficult to determine the compliance of the same blood
vessel both with (in situ) and without (in vitro) the surrounding tissues. Consequently, very little data can be found in this regard. Furthermore, the compliance of
the coronary arteries has been previously determined primarily under distension.
This is surprising since it is well recognized that the myocardium may exert
compressive stresses on the embedded blood vessels during the cardiac cycle (see
Chap. 6). Hamza et al. (2003) addressed this issue to investigate the impact of
surrounding tissues on the compliance of coronary vessels as described in the section
below.
3.3.1 Pressure–Cross-Sectional Area Relation
Hamza et al. (2003) determined the effect of passive surrounding tissue on the
mechanical properties of coronary blood vessels both under positive and negative
transmural pressures. The heart is placed into a saline filled Lucite box as shown in
Fig. 3.4. The Lucite box contained two side openings and a third opening on the top
cover. The coronary artery cannula is connected to one of the side openings and is
used to regulate the intravascular pressure. The second side opening is used to
regulate and measure the box pressure. The top cover of the box contained a
ventilation hole, connected to a stopcock, which is normally closed during pressurization of the box.
112 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Fig. 3.4 Schematic representation of the experimental setup used for testing the mechanical
properties of LAD artery under pressure differences in the 150 to 150 mmHg range. Reproduced
from Hamza et al. (2003) by permission
The pressure–diameter relation of the coronary arteries with in situ diameters
>0.5 mm (appro ximate resolution of the imaging system; (Molloi et al., 2001)is
determined by using quantitative coronary angiography. After perfusion with
cardioplegic solution, the LAD is filled with iodinated contrast material and 3%
Cab-O-Sil to ensure uniformity of the imposed pressure throughout the imaged
coronary arterial tree (Kassab & Molloi, 2001). To vary the ΔP (intravascular
pressure–box pressure), the box pressure is, in turn, set at four different pressures
(0, 50, 100, and 150 mmHg), while the LAD pressure is ramped between 0 and
150 mmHg in a triangular form with a slope of ~3 mmHg/s. Different box pressures
are used to allow the generation of a range of ΔP from 150 to +150 mmHg. To
ensure the reproducibility of the mechanical properties of the arteries, the vessels are
preconditioned with several cyclic changes in pressure, between 0 and 150 mmHg
(Fung, 1993). Following the completion of the in situ mechanical testing, a 2 cm
proximal segment of the LAD artery is dissected out from the heart, and every
bifurcation is ligated. The cannulated coronary artery is stretched to its in situ length
and anchored to the two cannulas in line with the two side holes of the saline filled
Lucite box, where the above mechanical testing procedure is repeated.
The loading ΔP-CSA (cross-sectional area) relationships for vessels representing
the largest several orders of the LAD artery of a single heart are shown in Fig. 3.5a.
A nonlinear equation is proposed (Eq. (3.1), Appendix 2) to curve fit the data over
the entire ΔP range (150 to +150 mmHg), and the empirical constants α, β, γ, and δ
are determined. These constants are expressed in terms of CSA
1/2
ΔP
, according to Eq. (3.3) (Appendix 2) and are summarized in Table 3.3 of
Appendix 2 for the three largest orders of vessels. Generally, as the ΔP increases the
CSA of the coronary vessel reaches an asymptotic value CSA
+
, CSA, CSA,and
+
. At pressure of
3.3 Effect of Surrounding Tissue: Radial Constraint and Tethering 113
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)
2
Cross-sectional Area (mm
-150 -100 50 100 150
-50 0
Pressure Difference (mmHg)
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
A
)
2
Cross-sectional Area (mm
-200 -100 0 100 200
Pressure Difference (mmHg)
8
7
6
5
4
3
2
1
0
In vitro
In situ
B
D=2.36mm
D=2.13mm
D=1.91mm
D=1.65mm
D=1.33mm
D=0.94mm
Fig. 3.5 (a) Relation between pressure difference (ΔP) and cross-sectional area (CSA) for the first
several generations of the LAD artery, over the full range of pressure difference ( 150 to
150 mmHg range). (b) Comparison between the in vitro and in situ ΔP-CSA relationships of a
proximal LAD artery. Reproduced from Hamza et al. (2003) by permission
100 mmHg, the CSA in situ is 34% smaller than that at the in vitro state. This
corresponds to a 19% decrease in diameter due to the surrounding tissue constraint.
Hence, the coronary arteries are radially constrained by the surrounding tissue and
myocardium. Moreover, as the ΔP increases in the negative direction the CSA
reaches an asymptotic value of CSA
vessel under compression is always non-zero. In the negative ΔP range, where box
pressure exceeds intravascular pressure, the ratio of in vitro to in situ CSA reduces to
zero. That is, in contrast to the in situ vessel, the isolated artery collapses under
compression. Hence, the large coronary arteries (diameter >0.5 mm) do not collapse
in situ when they are tethered by the surrounding myocardium. The mechanical
implications of these observations are addressed in Chap. 6.
. The results show that the CSA for an intact
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