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114 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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The CSA-compliance of the LAD artery is also calculated (as
ΔCSA
ΔP
a
) for values
corresponding to box pressure of zero. The CSA-compliance values (at 100 mmHg)
for the three largest generations of the LAD artery are summarized in Table 3.4 in
Appendix 2. Figure 3.5b presents a comparison between the ΔP-CSA relationship of
the proximal LAD artery in vitro (isolated segment) and in situ over the entire ΔP
range. In the positive ΔP range, the CSA attained in vitro is significantly larger than
that in situ. The in situ CSA
+
is 41% smaller than that at the in vitro state. Similarly,
at pressure of 100 mmHg, the CSA in situ is 43% smaller than that at the in vitro
state. Furthermore, the CSA-compliance of the most proximal (largest) LAD artery
in situ is 71% smaller than that of the in vitro state. Therefore, the myocardium limits
the CSA expansion and compliance of the coronar y arter ies.
3.3.2 Pressure–Volume Relation
Digital angiography is used to determine the coronary arterial volume of all vessels
with diameters >0.5 mm as shown in Fig. 3.6a. A manually drawn region of interest
approximately outlined the epicardial arteries as shown in Fig. 3.6a. The in situ P-V
relationship of the entire LAD arterial tree (vessels >0.5 mm in diameter) and the P-V
relationship of the main LAD trunk (vessels >1.0 mm in diameter) are determin ed.
The ΔP-V relationship is similar in shape to the ΔP-CSA relation as shown in
Fig. 3.6b for the arterial tree volume and the main trunk. Hence, a similar equation
is employed and the empirical constants are expressed in terms of V
1/2
ΔP
. It is found that V+, V, V0, and ΔP
1/2
have mean values of 1.41 0.3 mL,
0.70 0.3 mL, 0.85 0.3 mL, and 24 13 mmHg, respectively (R
The mean arterial volume at 100 mmHg is 1.36 0.3 mL, which is approximately
twice as large as the mean volume of the trunk (0.75 0.2 mL). The mean arterial
volume compliance at 100 mmHg (2.6 1.8 10
found to be very similar to that of the trunk (2.5 2.2 10
3
mL mmHg1), however, is
3
The LAD arterial tree retains a significant amount of volume (0.70 0.3 mL)
under external compression in an arrested, vasodilated heart. These results are
consistent with the CSA data, which confirm that vessel collapse does not occur
under compression. Furthermore, it is interesting to note that the two curves (arterial
tree and trunk) can become very similar when they are normalized by their respective
volumes at zero pressure, i.e., the difference in compliance or distensibility is not
significant.
+
, V, V0, and
2
¼0.980–0.990).
mL mmHg1).
3.3.3 Slackness Between Vessels and Myocardium
Since all mechanisms of coronary autoregulation (Chap. 6) depend on active
changes (vasoconstriction and dilatation) in vessel diameter, it is essential that
coronary arterioles can change their diameters without impediment from the
3.3 Effect of Surrounding Tissue: Radial Constraint and Tethering 115
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A
1.8
1.6
1.4
1.2
1
Volume (ml)
-200 -150 -100 -50 0 50 100 150 200
Pressure Difference (mmHg)
0.8
0.6
0.4
0.2
Arterial Tree
Trunk
0
B
Fig. 3.6 (a) A typical arteriogram used to obtain cross-sectional area (CSA) and lumen volume of
the left anterior descending (LAD) artery and its branches (left side of photograph). The arterial and
background region of interest used for lumen volume measurements are superimposed on the image
(right side of photograph). (b) Relation between pressure difference (ΔP) and arterial volume (V )
for the total LAD arterial tree in comparison to that of the main trunk. Reproduced from Hamza
et al. (2003) by permission
surrounding myocardium. Tone-regulation in coronary microvessels, however, has
largely been studied in isolated vessels in the absence of myocardial tethering (Liao
& Kuo, 1997). Although the histology of blood vessels has been well documented
(Clark & Glagov, 1985; Rhodin, 1980; Wasano & Yamamoto, 1983), there has been
less detail on the radial tethering and cross-talk between the myocardium and
microvessels (Borg & Caulfield, 1980; Westerhof, Boer, Lamberts, & Sipkema,
2006). Moreover, there is a paucity of data on the nature of the connective tissue
in the interstitial space between the vessel and myocardium (interstitial space
connective tissue, ISCT, see Fig. 3.7a). Although there have been a number of
studies which hypothesize the nature of the effect of axial and circumferential
116 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Fig. 3.7 (a) Representative
image of the lumen (white),
wall (black), and interstitial
space connective tissue
(ISCT) gap (gray) of an
arteriole (Order 4, about
35 μm). (b) Measured ISCT
gap size between the porcine
coronary vessels and
myocardium, from order
0 to 10 vessels of
endocardial
(Gap ¼ 0.181e
2
¼ 0.934; n is order
R
number), midmyocardial
(Gap ¼ 0.239e
2
¼ 0.959), and epicardial
R
(Gap ¼ 0.292e
2
¼ 0.989) regions. Lines
R
represent a log-linear fitto
the mean gap-size and the
error bars are 1SD. There
is no statistical difference in
gap-size between each layer.
Reproduced from Young,
Choy, Kassab, and Lanir
(2012) with permission
0.444n
0.459n
0.467n
,
A
,
,
Gap (μm)
100
10
Subepi
Mid
Endo
1
0
012345678910
Order Number
B
tethering on pulsatile flow in passive microvessels (Cinthio et al., 2006; Hodis &
Zamir, 2009; Humphrey & Na, 2002; Misra & Choudhury, 1984; Steelman, Wu,
Wager, Yeh, & Humphrey, 2010), the radial component of tethering and the size of
the ISCT gap between coronary vessels and myocardium, have been less studied. It
is important to note that these issues are essential for understanding flow-regulation
in the intramyocardial microvessels. Blood pressure differences may activate the
myogenic control mechanism (Chap. 6) and decrease the diameter of a microvessel
by up to 60% in order to keep blood flow constant within the vessel (Liao & Kuo,
1997); however, the ability of that vessel to “pull” the myocardium (if rigidly
tethered) is not well understood.
To understand the interaction between passive myocardium and vasoreactivity of
coronary vessels, Young et al. (2012) studied the potential effect of radia l tethering
and ISCT between coronary microvessels and the surrounding myocardium. A rigid
tethering between microvess els and myocardium would constrain the active contraction of arterioles and is not compatible with the observed tone-regulation. The
ISCT between coronary vessels and myocardium in swine is found to increase
exponentially from 0.22 0.02 μm in capillaries (diameter-defined Strahler order
0) of the endocardium to 34.9 7.1 μm in epicardial vessels (Order 10) as shown in
Fig. 3.7b. Microvessels with both soft-tethering and an ISCT gap are capable of
3.4 Zero-Stress State 117
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significant changes in vessel resistance, consistent with experimental measurements
of high coronary flow reserve. Additionally, the mechanical energy required for
myogenic contraction analysis indicates that rigid tethering requires up to four times
more mechanical energy than soft-tethering in the absence of a gap (Young et al.,
2012). Hence, the experimental measurements and model predictions suggest that
effectiveness and efficiency in tone-regulation can only be achieved if the vessel is
both softly teth ered to and separated from the myocardium in accordance with the
experimental findings of ISCT gap. This seminal axiom provides the basis for the
analysis of coronary autoregulation presented in Chap. 6.
The presence and arrangement of connective tissue in skeletal and cardiac
muscles is first described by Homgren (1907) who elaborated on an extracellular
system of fibrils that interconnected capillaries to myocytes and myocytes to each
other. This system has been divided into three major components:
1. Collagen network that surround groups of myocytes
2. Network of collagen struts (bundles of small collagen fibers, 120–150 nm in
diameter) that extends from the basal laminae of a myocyte to the basal laminae of
all contiguous myocytes
3. Network of similar size collagen struts that extends from the basal laminae of the
capillaries to the basal laminae of the myocytes
Myocyte-to-myocyte struts are thought to prevent slippage of adjacent cells
during the cardiac cycle to ensure an equal stretch of adjacent myocytes during
diastole (Borg & Caulfield, 1981; Borg, Sullivan, & Ivy, 1982; Caulfield & Borg,
1979). Myocyte-to-capillary stru ts may be important in maintaining capillary
patency during the early phases of systole. The measurements in Fig. 3.7b are the
first systematic quantification of the ISCT gap between myocytes and arterioles or
larger coronary arteries that generally consists of collagen fibers, glycoprotein s, and
glycosaminoglycans. These fibers are likely slack to allow free contraction of the
microvessels from the myocytes. Further studies of gap thickness in the
vasoconstricted state are needed to experimentally verify the slack hypothesis used
in Chap. 6.
3.4 Zero-Stress State
3.4.1 Circumferential Residual Strain
The stress–strain relation and hence material properties must be determined in
reference to the zero-stress state (Fung, 1990). The stress and stra in that remain in
an organ when the external load is removed are called residual stress and strain,
respectively. Prior to 1983, it is believed that there are no residual stress or strain in
the vessel wall, and the zero-stress state is equivalent to the no-load (zero transmural
pressure) state. The outcome of this assumption is the existence of a stress concentration at the intima which resulted in a much higher circumferential stress at the
118 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Intima
Intima
o
θ< 180
Fig. 3.8 Photographs of coronary rings cut radially to reveal sectors with opening angles <180
(left) and >180(right). Reproduced from Guo, Xiao, and Kassab (2005) by permission
θ> 180
o
inner than the outer vessel wall (Chuong & Fung, 1986); such a stress concentration
would imply higher energy consumption and oxygen demand at the inner wall which
has not been corroborated experimentally.
The assumption of lack of residual strain is challenged simultaneously and
independently by Fung (1983) and by Vaishnav and Vossoughi (1983). A radial
cut of a blood vessel ring demonstrated the existence of residual strain which
changed the no-load state (circular geometry) into the zero-stress state (an open
sector). The vessel is first reduced from the loaded (pressurized) state to a ring in the
no-load state by making two transverse cuts to the long axis of the vessel. A
subsequent radial cut causes the vessel ring to spring open into a vessel sector that
can be characterized by an opening angle defined as the angle subtended by two radii
drawn from the midpoint of the inner wall (endothelium) to the tips of the inner wall
of the open sector as shown in Fig. 3.8. The discovery of circumferential residual
stress removed the concept of stress concentration at the inner wall of the vessel in
the in vivo state (Chuong & Fung, 1983). The circumferential residual strain led to
the “transmural uniform stress ” hypothesis proposed by Fung (1983), i.e., the
circumferential stresses at the inner and outer wall are nearly equal.
The zero-stress state is also the best state to study tissue remodeling since any
change in structure can be documented without the effect of stress or strain (Fung,
1993). The changes in the zero-stress state and opening angle provide a simple index
of the non-uniformity of growth and remodeling. Numerous publications have been
written on the subject, as detailed in Fung (1990) and reviewed by Rachev and
Greenwald (2003).
There is significant literature on the opening angle of blood vessels for different
vessels and species. Previous studies have included opening angle measurements
along the aortic tree in rats (Liu & Fung, 1988), rats, pigs, and canines (Han & Fung,
1991), coronary arterial tree of pigs (Frobert, Gregersen, Bjerre, Bagger, & Kassab,
1998; Guo & Kassab, 2004; Rehal, Guo, Lu, & Kassab, 2006) and dogs (Jiang, Ji, &
Dong, 1995),systemicarteriesof cows (Vaishnav & Vossoughi, 1987), systemic veins
of rats (Xie, Liu, Yang, & Fung, 1991), left ventricles of rats (Omens & Fung, 1990)
and dogs (Nevo & Lanir, 1994), duodenum of guinea pigs (Gregersen et al., 1997),
3.4 Zero-Stress State 119
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trachea of pigs and dogs (Han & Fung, 1991), and pulmonary arteries and veins of
humans (Huang & Yen, 1998). In summary, the opening angle varies significantly
along the arterial tree with large scatter; typical values for opening angles along the
porcine aortic tree vary between 20
1988) and between 89
and 128along the human arterial tree (Huang & Yen, 1998).
and 160(Guo & Kassab, 2004; Liu & Fung,
The opening angle has been measured for the six largest orders of the coronary
arterial tree of the pig (Frobert et al., 1998). The data show that the mean opening
angle for the largest order (main coronary artery) is approximately 170
decreases linearly towards the smaller orders (a slope of 7.3
/order in the range of
and
the six largest orders). Orders 10 and 11 are epicardial vessels and do not have as
much tethering by the myocardial tissue as the lower orders. Hence, a larger opening
angle is needed given the higher stress in those unsupported vessels.
3.4.2 Longitudinal Distribution of Opening Angle
To extend the database on the zero-stress state, the variations of opening angles are
examined for vessel diam eters that span over three orders of magnitude (10 μm
arterioles to 3 mm coronary arteries). The left common coronary artery is cannulated
and perfused with 6% dextran solution (Guo & Kassab, 2004). The LAD artery is
then perfused at a physiologi cal pressure (100 mmHg) with catalyzed silicone
elastomer as described in Chap. 2. After the elastomer hardened in 45 min, the
LAD artery is carefully dissected down to small branches with diameters of about
10 μm. The vessel is cut perpendicular to the longitudinal axis into rings with
segmental lengt h of approximately one-fourth to one-half of the radius. Each ring
is transferred to a Ca
photographed in the loaded state.
The elastomer is then pushed out and the rings are cut radially to obtain the zerostress state. The elastomer is gently remo ved and the morphological data of the
coronary vessels in the zero-stress states are obtained. The zero-stress state is
characterized by the opening angle (OA) whose variation with order number is
shown in Fig. 3.9. A linear least squares fit is used to describe the data over the
entire range of arterial orders (n)asOA¼ 10.2n + 63.4 (R
variation of opening angle with order number implies a logarithmic variation with
diameter given that diameter and order number are related as a geometric sequence
(Chap. 2).
2+
-free Krebs solution, aerated with 95% O2and 5% CO2and
2
¼ 0.989). The linear
3.4.3 Transmural Wall Strain Distribution
Chuong and Fung (1986) showed that the existence of circumferential residual strain
(opening angle) reduces the transmural gradients of stress and strain, i.e., the inner
(intima) and outer (adventitia) circumferential stresses and strains are similar at the
120 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Fig. 3.9 Variation of
opening angle (θ) with order
number (n) in the loaded
state for LAD arterial tree.
Reproduced from Guo and
Kassab (2004) with
permission
250
200
150
100
50
0
1234567891011
Opening Angle(degrees)
Order Number
in vivo state. Their computational approach is based on a constitutive equati on
whose constants are determined experimentally. The stresses and strains used in
the constitutive equation are based on the zero-s tress state which is characterized by
an opening angle. Takamizawa and Hayashi (1987) solved the inverse problem, i.e.,
they showed that under the uniform strain hypothesis, the thin-wall theory can be
used to predict the material constants in the constitutive equation. The first direct
experimental evidence for the uniform transmural strain hypothesis at the in vivo
state is provided by Fung and Liu (1992) on small vessels, where they measured the
circumferences in the loaded and zero-stress state and computed the corresponding
strains at the inner and outer wall.
In both computational and experimental studies, the vessels studied had θ < 180
Although the majority of vessels fall into this category, there are regions of rat and
human aorta, rat pulmonary artery, porcine coronary artery, and rat ileal arterioles
that have θ > 180
(Frobert et al., 1998; Fung & Liu, 1992; Kassab et al., 2002; Liu
& Fung, 1992; Saini , Berry, & Greenwald, 1995). Furthermore, the opening angle is
known to increase beyond 180
in hypertension, cigarette-smoke, and diabetesinduced remodeling (Fung & Liu, 1991, 1993; Fung, Liu, & Zhou, 1993; Liu &
Fung, 1992). Finally, other tubular organs such as the dog trachea and guinea pig
small intestines are known to have opening angles well in excess of 180
et al., 1997; Han & Fung, 1991). For cases where θ > 180
, the uniform transmural
(Gregersen
strain hypothesis cannot apply when the vessels turn inside out (see theoretical
arguments in Appendix 3). In such cases, the loaded circumferential strain on the
inner wall will become smaller than the circumferential strain on the outer wall as
described below, which is the converse of the case where the residual circumferential
strain is ignored.
Using porcine hearts, Guo et al. (2005) experimentally considered the issue of
θ > 180
in 387 vessels with diameters greater than 50 μm from the coronary arterial
tree. The data from the same hearts are used to determine the longitudinal distribution of strain and stress. The inner and outer strains are listed in Table 3.5 (Appendix
3) for the coronary arterial tree. The data are classified according to order number
and range of opening angle (in increments of 45
) for the coronary arterial tree. The
outer strain is significantly larger than the inner strain for orders 6–11% by 7–45%,
respectively. When compared with respect to opening a ngle, the outer strain is
significantly large r than the inner strain for θ > 135
. The variation of the first
term on the right-hand side of Eq. (3.5) (Appendix 3) with opening angle (θ)is
.
3.4 Zero-Stress State 121
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)
1.8
zs2
i
-C
2
1.6
)/(Ci
1.4
zs2
1.2
-Co
2
1
(Co
0 45 90 135 180 225 270 315 360
Opening Angle (Degrees)
A
1.2
1.1
2
)
1
zs
0.9
/Co
zs
0.8
(Ci
0.7
0.6
0 45 90 135 180 225 270 315 360
Opening Angle (Degrees)
B
1.8
1.6
i
1.4
/ε
ο
ε
1.2
1
0.8
0 45 90 135 180 225 270 315 360
Opening Angle (Degrees)
C
Fig. 3.10 Relation between the ratio of outer to inner vessel circumferences in the loaded state (a),
inner to outer vessel circumferences in the zero-stress state (b), and outer to inner vessel stretch ratio
(c) and the opening angle. The data correspond to the LAD arterial tree; orders 5–11. Reproduced
from Guo et al. (2005) by permission
shown in Fig. 3.10a. This ratio is always greater than one for all values of θ.
Figure 3.10b shows data for the second term on the right -hand side of Eq. (3.5)
(Appendix 3). When θ is equal to 180
and hence the ratio is one. When θ < 180
the outer circumference and the ratio is less than one. The converse is true when
θ > 180
as shown in Fig. 3.10b. The ratio of outer to inner strain (i.e., product of the
two ratios shown in Fig. 3.10a, b) is demonstrated in Fig. 3.10c. An interesting
pattern is revealed where the non-uniformity of strain increases with an increase in
opening angle, especially when θ > 180
The coronary opening angles reported in the Guo et al. (2005) study are significantly larger than those of Frobert et al. (1998) described above. The reason for this
discrepancy is that in the Guo et al. (2005) study, the vessel rings are distended with
, the inner and outer circumferences are equal
, the inner circumference is smaller than
.
122 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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elastomer at physiological pressures in order to obtain strain at the loaded condition.
The ring is cut open 1 or 2 min after the elastome r is pushed out of the vessel. In the
earlier study of Frobert et al. (1998), the isolated heart vessels are not pressurized and
are maintained at the no-load state prior to the radial cut. This suggests that the initial
state of stress and history in the vessel wall affect the opening angle and reflects the
viscoelastic proper ties of the vessel wall which is described in the next section. The
main issue is that for vessels with θ > 180
, the strain distribution cannot possibly be
uniform theoretically or experimentally. Hence, although the unifo rmity of
transmural stress distribution is still possible (Chap. 8 ) because of the composite
nature of the blood vessel wall (i.e., the intima-medial layer is stiffer than the
adventitial layer), the strain distribution cannot be uniform when θ exceeds 180
.
3.4.4 Effect of No-Load Duration on Opening Angle
Prior studies (Fung, 1993) have determined the zero-stress state or opening angle
with no regard to the duration of the no-load state, i.e., the period between the first
two transverse cuts and the radial cut. At some unspecified time after the two
transverse cuts, the vessel ring is cut in radial direction and the resulting sector is
considered as the zero-stress state typically 30 min after the radial cut. Given the
viscoelastic properties of blood vessels, the zero-stress state may have “memory” of
prior circumferential and axial loading, i.e., the duration of the no-load state will
influence the opening angle. To test this hypothesis, Rehal et al. (2006) considered
ring pairs of porcine coronary arteries to examine the effect of duration in the no-load
state following circumferential distension. Two different conditions are considered:
(1) Circumferential loading, and (2) Axial loading; both conditions are shown in
Fig. 3.11. In both cases, two adjacent rings are cut and tested (one within 30 s of the
radial cut and another at different durations in the no-load state).
The data show coronary arteries that are reduced to the zero-stress state directly
from the loaded state attain much larger opening angles at 30 min after the radial cut
than those rings that are in the no-load state for various durations as shown in
Fig. 3.12a. The time course of the difference in opening angle, ΔOA (between
loaded and no-load states) is shown in Fig. 3.12a (for three orders of epicardial
and one order of intramyocardial arteries) and fi tted with a Kelvin Model (see
Appendix 4) along with the curve fit model parameters (Appendix 4, Table 3.6).
In addition, the axial data (Fig. 3.12b) show a similar trend as the circumferential
data albeit the effect is more modest. Collectively, the zero-stress state depends on
the time from initial circumferential and axial loading. The circumferential effect is
larger than the axial (Fig. 3.12a and b, respectively) since the open sector primarily
reveals circumferential residual strain. Furthermore, the effect is stronger in epicardial vs. intramyocardial vessels (EPCA vs. IMCA), given that the former deforms
more in the absence of surrounding tissue. These data may explain the significant
variability in the previous opening angle measurements reported in the literature
(Rachev & Greenwald, 2003) since the duration of the no-load state is not controlled.
3.4 Zero-Stress State 123
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Fig. 3.11 Schematic of experimental protocol for (a) Circumferential loading: vessel distended
(shaded lumen) with hardened elastomer, adjacent rings cut, and reduced to zero-stress state from
loaded and no-load states. (b) Axial loading: vessel axially stretched to in situ length, adjacent rings
cut and reduced to zero-stress state from in situ and no-load states. Reproduced from Rehal et al.
(2006) with permission
The opening angle decreases with time for a vessel ring cut from the loaded state
while the openin g angle increases with time for a vessel ring cut from the no-load
state (Fig. 3.13). The decrease in the opening angle over time for a vessel ring cut
from the no load state is a novel observation while the latter is well documented (e.g.,
Frobert et al., 1998). The data suggests that the opening angle will reach the same
value after 3 h despite the duration in the no-load state (1 h vs. 3 h). Hence, it can be
concluded that the opening angle value depends on two factors: the time in the
no-load state and the time after the radial cut. While the latter is well docum ented, it
is important to call atte ntion to the former.
Despite the duration in the no-load state, a unique opening angle can be achieved
after 3 h. Since it is common practice to consider the zero-stress state 30 min after the
radial cut, the no-load state duration will impact the resulting opening angle. Hence,
if the opening angle is measured 30 min after the radial cut, it is essential to specify
the duration in the no-load state. After 3 h in the zero-stress state, the duration of the
no-load state becomes insignificant. Henc e, a possible recommendation would entail
the measurement of the opening angle after 3 h rather than 30 min which would
minimize the effect of the duration of the no-load state and serve to standardize
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