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1Department of Biomedical Engineering, University of California, 92697-2715; and 2Department of Radiological Science, University of California, Irvine, California 92697-5000
Submitted 30 January 2003 ; accepted in final form 25 April 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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P =
intravascular pressure – box pressure). The P-cross-sectional
area relationship of the first several generations of porcine coronary
arteries and the P-volume relationship of the coronary arterial tree
(vessels >0.5 mm in diameter) were determined using a video densitometric
technique in the range of +150 to –150 mmHg. The vasodilated left
anterior descending (LAD) coronary artery of six KCl-arrested hearts were
perfused with iodine and 3% Cab-O-Sil. The intravascular pressure was varied
in a triangular pattern, whereas the absolute cross-sectional area of each
vessel and the total arterial volume were calculated using video densitometry
under different box pressures (0, 50, 100, and 150 mmHg). In the range of
positive P, we found that the compliance of the proximal LAD artery in
situ (4.85 ± 3.8 x 10–3
mm2/mmHg) is smaller than that of the same artery in vitro (16.5
± 6 x 10–3 mm2/mmHg;
P = 0.009). Hence, the myocardium restricts the compliance of the
epicardial artery under distension. In the negative P range, the LAD
artery does not collapse, whereas the same vessel readily collapses when
tested in vitro. Hence, we conclude that myocardial tethering prevents
collapse of large blood vessel under compression.
digital subtraction angiography; video densitometry; mechanical properties; mechanics; tethering
A number of previous studies (1, 2, 9, 10, 13, 15, 21) have examined the mechanical properties of coronary arteries under in vitro conditions; i.e., after dissection of the vessels from the myocardium. For example, Patel and Janicki (21) determined the in vitro pressure-diameter relationship for isolated segments of the dog's left circumflex. The pressure-diameter relationship of excised coronary arteries from dogs and humans was measured by Gow et al. (10) and Gow and Hadfield (9), respectively. More recently, several in vitro inflation and extension tests on isolated passive human, porcine, and bovine coronary arteries have been performed by Carmines et al. (1) and Kang et al. (15). Although those studies provide a wealth of data on the compliance and material properties of blood vessels, they do not take into account the mechanical contribution of the surrounding medium. Our hypothesis is that the mechanical properties of the coronary artery stems not only from the intrinsic properties of its wall (collagen, elastin, ground substance, etc.) but also stems from the surrounding medium (myocardium including serous pericardium, fat tissue, etc.). 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. We are unaware of any data on the
compressive mechanical properties of the coronary arteries. This is surprising
because it is well recognized that the myocardium may exert compressive
stresses on the embedded blood vessels during the cardiac cycle. Indeed, the
compression between the heart muscle and coronary blood vessels may be an
important determinant of the zero-flow pressure in the coronary circulation.
Hence, in the present study, we wanted to determine the effect of passive
myocardium on the mechanical properties of coronary arteries both under
distension and compression. Specifically, our objectives are 1) to
determine the cross-sectional area (CSA) of the first several generations of
coronary arteries in the range of –150 to +150 mmHg pressure difference
(P); 2) to determine the P-volume relationship of the
proximal coronary arterial tree (vessels >0.5 mm in diameter) under
distension and compression; and 3) to compare the compliance of the
proximal coronary artery with and without the myocardium.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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Isolated heart preparation. The studies were performed on six male Duroc swine weighing 34 kg ± 5 kg (range 30–39 kg). Surgical anesthesia was induced with ketamine (20 mg/kg im) and atropine (0.04 mg/kg im) and maintained with isofluorane (1–2%). Ventilation with 100% O2 was provided. Blood gas values were measured, and ventilation was adjusted to maintain normal values of PO2 and PCO2. A midline sternotomy was performed, and an incision was made in the pericardium to support the heart in a pericardial cradle. Anticoagulation was induced with heparin (100 U/kg) followed with the injection of pentobarbital (80 mg/kg iv) and then followed by a saturated KCl solution, through the jugular vein, to euthanize the animal and arrest the heart. The heart was then excised, and the aorta was clamped to keep air bubbles from entering the coronary arteries. The right coronary artery and the left circumflex were ligated, while the left anterior descending (LAD) was cannulated while immersed under saline to avoid air bubbles. Immediately afterward, an isosmotic cardioplegic rinsing solution (composed of 6% dextran in saline, including 80 mg/l of adenosine and 1.5 g/l of 2,3-butanedione monoxime) was perfused through the LAD artery to keep the myocardium relaxed and the vasculature vasodilated.
The heart was then placed into a saline-filled Lucite box with dimensions of 20 cm x 20 cm x 10 cm. A schematic of the experimental setup is shown in Fig. 1. Before the heart was placed into the box, paper towels were added to fix the position of the heart to ensure a nonoverlapping projection of the LAD arterial tree. The LAD artery was positioned to face the bottom of the box to prevent it from being pushed up against the top of the box as the box pressure increased. We maintained some air in the ventricles of the heart to make it buoyant so that the LAD artery was not compressed by the heart weight. The Lucite box contained two side openings and a third opening on the top cover. The coronary artery cannula was connected to one of the side openings and was used to regulate the intravascular pressure. The second side opening was used to regulate and measure the box pressure. The top cover of the box contained a ventilation hole, connected to a stopcock, which was normally closed during pressurization of the box. The stopcock was only opened to the atmosphere before the imaging process when the box pressure was set equal to zero or to remove air bubbles from the box. The post-mortem imaging experiments were completed within 1 h of euthanasia.
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Determination of P-CSA relationship. The
elasticity of the coronary arteries with in situ diameters >0.5 mm was
determined using quantitative coronary angiography. After perfusion with
cardioplegic solution, the LAD was filled with iodinated contrast material
(Omnipaque; Nycomed Amersham; Princeton, NJ) and 3% Cab-O-Sil (Eastman Kodak).
Because Cab-O-Sil is a colloidal silica that forms agglomerated particles with
effective diameters exceeding those of the capillaries, it was used to prevent
the flow of the iodinated contrast material into the capillaries, thus
ensuring the uniformity of the imposed pressure throughout the imaged coronary
arterial tree (18).
To vary the P (intravascular pressure – box pressure), the box
pressure was, in turn, set at four different pressures (0, 50, 100, and 150
mmHg), whereas the LAD pressure was ramped between 0 and 150 mmHg in a
triangular form with a slope of 
3 mmHg/s. Hence, there was some overlap
in the range of P achieved at different box pressures. Because the
results depend only on the P, different box pressures were used to
allow the generation of a range of P from –150 mmHg to +150 mmHg.
To ensure the reproducibility of the mechanical properties of the arteries,
the vessels were preconditioned with several cyclic changes in pressure
between 0 and 150 mmHg (6).
Coronary arteriograms were acquired at 1 frame per 5 s, as described in Kassab
and Molloi (17). The pressure
imposed and the X-ray tube voltages were recorded continuously (Biopac MP100
Systems; Santa Barbara, CA), thus the intravascular and box pressures for each
acquired image were determined.
For calibration purposes, a cylindrical vessel phantom, which consists of contrast-filled plastic tubing with different inside diameters (between 1.01 and 3 mm), was imaged on top of the Lucite box over the heart region. The integrated gray levels in the vessel profiles were related to the known CSA of the vessel phantoms, thus the integrated gray levels were directly converted to CSA values for the coronary angiograms. Corrections were made for magnification differences between the calibration phantom and the arterial segment of interest.
Determination of P-CSA relationship for the proximal
artery in vitro. After the completion of the in situ mechanical testing,
the heart was removed from the Lucite box and placed in a saline bath. A 1.5-
to 2-cm proximal segment of the LAD artery was dissected out from the heart,
and every bifurcation was identified and each branch was ligated with 6-0
suture. Before dissection, a suture was used to measure the length of the LAD
artery by tracing along the vessel's longitudinal axis. The cannulated vessel
was then 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 was repeated. The same iodine 3% Cab-O-Sil
solution was used as the contrast agent for imaging of isolated coronary
artery.
Determination of the P-V relationship. Digital
angiography was used to determine the coronary arterial volume of all vessels
with diameters >0.5 mm, which is the approximate resolution of the imaging
system (20), as shown in
Fig. 2B. After the
images were corrected for scatter and veiling glare, temporal subtraction
images were formed. A manually drawn region of interest (ROI) approximately
outlined the epicardial arteries as shown in
Fig. 2B. A second
narrow shell (a background ROI) was drawn outside the arterial ROI
(Fig. 2B). The
background ROI was used to correct the iodine signal in the myocardium. The
system iodine-calibration curve was used to convert the integrated video
densitometric signal to iodine mass, which was then converted to volume by
using the known iodine concentration of the contrast material
(20). The in situ
pressure-volume relationship of the entire LAD arterial tree (vessels >0.5
mm in diameter) and the pressure-volume relationship of the main LAD trunk
(vessels >1.0 mm in diameter) were determined.
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Image acquisition and processing. All images were acquired using a conventional X-ray tube (Dynamax 79-45/120, Machlett Laboratories; Stamford, CT), a constant potential X-ray generator (Optimus M200, Philips Medical Systems; Shelton, CT), a 23/15 cm CsI image intensifier, a focused grid (8:1 grid ratio, 36 lines cm–1), and a CCD camera (Multicam MC-1134GN, Texas Instruments; Dallas, TX). An adjustable aperture controlled the camera's light intensity. A Matrox Pulsar frame grabber (Matrox Electronics Systems; Dorval, Quebec, Canada) and a Pentium III computer were used to linearly digitize the video signal to 640 x 480 x 8-bit precision (17).
The images were obtained by using the large (1.2 mm nominal) focal spot and the 15-cm image-intensifier mode. A convolution filtering technique was used to estimate the scatter-glare distribution in images. Exposure parameters and the detected intensity distribution were used to estimate the scatter-glare intensity by predicting the total thickness at every pixel in the image. The thickness information was used to estimate scatter glare on a pixel-by-pixel basis (4).
Zero-stress state. After the completion of the in vitro mechanical testing, the proximal segment of the coronary artery was cut into several consecutive rings. Methods for the determination of coronary zero-stress state have been previously described in our laboratory (5). After a ring is cut, it was transferred to a Ca+-free Krebs solution (composition in mM: 117.9 NaCl, 4.7 KCl, 1.2 MgCl2, 25 NaHCO3, 1.2 NaH2PO4, 0.0027 EDTA, 0.1 ascorbic acid, and 11 glucose) bath aerated with a gas mixture of 95% oxygen-5% carbon dioxide at room temperature. This represents the no-load state of the vessel. One more cut was then made radially, which caused the ring to open into a sector that represents the zero-stress state of the vessel. The opening angle of each sector, defined as the angle subtended by two radii connecting the midpoint of the inner wall, was measured from photographs using an image analysis system (Optimas). The morphological measurements of inner and outer circumference, and area in the no-load and zero-stress state were also made and used for the analysis of arterial mechanics as outlined below.
Analysis of stress and strain. The internal diameter was
calculated from the CSA measurements (D =
where D is the
luminal dimension) for the in vitro segment assuming that the coronary artery
is cylindrical. The wall thickness (h) was also calculated based on
the following assumptions: 1) the wall is incompressible; i.e., the
volume of the wall does not change during distension, and 2) the
shape of the vessel is cylindrical. The total wall area
(Ao) was measured from the no-load pictures of the rings.
The incompressibility condition for a cylinder was used to compute h
as given by
 | (1) |
z =
l/lo where l and lo
are the vessel lengths in the loaded and no-load state, respectively.
The circumferential strain (
) is defined according to Green (see Fung
in Ref. 6)
 | (2) |
is the circumferential stretch ratio
calculated at the midwall as the ratio of the circumferences in the
pressurized state to that in the zero-stress state. The circumferential stress
(S) is expressed by Kirchhoff stress as
 | (3) |
Data analysis. The P-CSA relationship for various vessels
with diameter >0.5 mm was determined. The vessels were grouped in the
following diameter ranges: 0.5–1.0 mm, 1.01–2.0 mm, and
2.01–3.5 mm, which roughly correspond to orders 9, 10, and 11,
respectively, as previously reported by Kassab et al.
(17). For each experiment, the
P-CSA measurements were taken for seven segments along the main LAD
trunk and three segments along the side branches. The P-CSA
relationship in the range of –150 to +150 mmHg pressure difference were
curve fitted using nonlinear regression, according to the following
relationship
 | (4) |
P =
intravascular pressure – box pressure), and 
, 
, 
,
and 
are curve-fit constants. Equation 4 can be expressed in
terms of four physical constants as
 | (5) |
P
direction (below yield pressure where the vessel may undergo plastic
deformation and rupture), CSA– is the asymptotic value of the
CSA in the negative P direction, CSA0 is the CSA value at
P = 0, and P1/2 is the pressure difference
corresponding to the average of CSA+ and CSA–
[i.e., (CSA+ + CSA–)/2]. The empirical curve fit
constants (, , , and ) are related to the physical
constants (CSA+ and CSA–, CSA0, and
P1/2) as follows
 | (6) |
The volume data were defined similarly where CSA+,
CSA–, and CSA0 were replaced with V+,
V–, and V0, respectively, with similar
definitions. The P-V relationships of the entire LAD arterial tree, as
well as that of the main LAD trunk were determined.
The compliance of the coronary arteries was determined as the change in
luminal dimension (D, CSA, or V) per change in
arterial pressure (Pa) (where box pressure = 0 mmHg; i.e.,
P > 0 mmHg). In the negative pressure difference (P < 0
mmHg) where the vessels were under compression, the compliance was not
defined. The CSA compliance for the first several generations of the LAD
artery was calculated, as well as the volume compliance of the total arterial
tree (vessels >0.5 mm in diameter).
Statistical analysis. All data given in the text and tables were expressed as means ± SD, whereas the data in the figures were expressed as means ± SE. Analysis of variance was used to detect differences in compliance, as well as, in CSA and volume among the different-sized vessels within each heart and among various hearts. For all analyses, P < 0.05 level was used to indicate statistical significance.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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P. The loading P-CSA relationships for
vessels representing the first several generations of the LAD artery of a
single heart are shown in Fig.
3. A nonlinear equation was proposed (Eq. 4) to curve fit
the data over the entire P range (–150 to +150 mmHg), and the
empirical constants , , , and were determined.
These constants were expressed in terms of CSA+,
CSA–, CSA0, and P1/2 according
to Eq. 6 and are summarized in
Table 1 for the three largest
orders of vessels.
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The CSA compliance of the LAD artery was also calculated
(CSA/Pa) for values corresponding to box pressure of
zero. The CSA-compliance values (at 100 mmHg) for the three largest
generations of the LAD artery, from all six hearts, are summarized in
Table 2.
Figure 4 presents a comparison
between the P-CSA relationship of the proximal LAD artery in vitro
(isolated segment) and in situ over the entire P range. At 100 mmHg,
the in situ mean CSA and CSA compliance are 4.26 ± 1.8 mm2
and 4.85 ± 3.8 x 10–3
mm2/mmHg, respectively. These values are significantly smaller than
their corresponding in vitro values of 7.12 ± 1.6 mm2 and
16.5 ± 6 x 10–3 mm2/mmHg,
respectively (P = 0.014 and P = 0.009, respectively). The
CSA+, CSA–, CSA0, and
P1/2 for the proximal LAD artery in situ are 4.3 ±
2.1 mm2, 1.09 ± 0.6 mm2, 1.4 ± 0.6
mm2, and 43 ± 22 mmHg (R2 =
0.988–0.999). Their corresponding values in vitro are 7.24 ± 1.6
mm2, 0.15 ± 0.2 x 10–3
mm2, 1.3 ± 0.7 mm2, and 21 ± 3 mmHg,
respectively (R2 = 0.996–0.999). The differences
were statistically significant for CSA+ and CSA–
(P = 0.02 and 0.007, respectively), whereas the difference between
the in situ and in vitro CSA0 and P1/2 were not
statistically significant.
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The P-V relationship was similar in shape to the P-CSA
relationship as shown in Fig. 5
for the arterial tree volume and the main trunk. Hence, a similar equation was
employed, and the empirical constants were expressed in terms of
V+, V–, V0, and P1/2.
It was found that V+,V–,V0, and
P1/2 have mean values of 1.41 ± 0.3 ml, 0.70 ±
0.3 ml, 0.85 ± 0.3 ml, and 24 ± 13 mmHg, respectively
(R2 = 0.980–0.990). 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); and the differences are
statistically significant (P = 0.00007). However, the mean arterial
volume compliance at 100 mmHg (2.6 ± 1.8 x
10–3 ml/mmHg) is found to be very similar to that
of the trunk (2.5 ± 2.2 x 10–3
ml/mmHg).
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The circumferential stress-strain relationship for the in vitro vessel was computed according to Eqs. 2 and 3 and shown in Fig. 6. The tangent modulus of the vessel was calculated as the change of stress per change of strain from the data in Fig. 6. Our results show that the mean value of the tangent modulus at 100 mmHg is 220 ± 74 kPa (2.2 ± 0.74 x 106 dyn/cm2).
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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P-CSA relationship: in situ properties. It is evident that
the P-CSA relationship is nonlinear over the full range of
P(–150 to +150 mmHg) and can be fitted by a nonlinear function
(Eq. 4) as shown in Fig.
3. Our results, in the positive P range, agree with those
previously reported by Kassab and Molloi
(17). However, the
P-CSA relationship in the negative P range has not been
previously determined. Generally, as the P increases in the positive
direction, the CSA of the coronary vessel reaches an asymptotic value
CSA+. This pressure is well below the rupture pressure of the
vessel. Moreover, as the P increases in the negative direction, the CSA
reaches an asymptotic value of CSA–. Our results show that
the CSA for an intact vessel under compression is always nonzero; thus the
large coronary arteries (diameter >0.5 mm) do not collapse. In two of six
hearts, however, we did observe partial collapse of the most proximal artery.
In those cases, we confirmed that the proximal segment was more superficial;
i.e., less tethered than the corresponding vessels of other hearts. Similar to other vessels, the compliance of the coronary arteries decreases as the pressure is increased. We found that at 100 mmHg, the CSA compliance decreases as the vessel branching order decreases (Table 2).
Effect of myocardium on the CSA and compliance of coronary artery.
To determine the effect of the myocardium on the CSA and compliance of the
coronary artery, the P-CSA relationship of the proximal LAD artery was
compared in the in situ and in vitro states. Our data show that there are
significant differences in the P-CSA relationship between the two
states, as shown in Fig. 4. In
the positive P range, the CSA attained in vitro is significantly larger
than that in situ (P = 0.02). The in situ CSA+ is 41%
smaller than that at the in vitro state. Similarly, at a 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 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 coronary artery in the positive P regime.
One of the hypotheses of the present study was that the myocardium provides
mechanical support for the blood vessels during compression. To test this
hypothesis, the proximal artery was dissected away from the myocardium and its
mechanical properties were retested. Our results show that in the negative
P regime, 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. Therefore, the
myocardium tethers the vessel and prevents it from collapse.
P-volume relationship. The relationship between P
and arterial volume for the coronary arterial tree (vessels >0.5 mm in
diameter) is shown in Fig. 5.
It can be seen that 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
confirms that vessel collapse does not occur under compression. Furthermore,
it is interesting to note that the two curves (arterial tree and trunk) become
very similar when they are normalized by their respective volumes at zero
pressure; i.e., the difference in compliance or distensibility is not
statistically significant.
Comparison with other works. The diameter was computed from the
CSA to calculate diameter-compliance values (D/P) for
comparison with the literature. It is found that the in vitro diameter
compliance of the proximal LAD is 6.51 ± 7.0 x
10–3 mm/mmHg (at 100 mmHg), which is in agreement
with previously published data. For example, using microscopy, Patel and
Janicki (21) obtained diameter
compliance of a dog's left circumflex coronary artery of 4 x
10–3 mm/mmHg (at a pressure range of 107–135
mmHg). Furthermore, Gow and Hadfield
(9), using an electrical
caliper, calculated the diameter compliance of the left common coronary artery
(LCCA) in humans (D = 4.9 ± 0.3 mm) at pressure range of
70–110 mmHg and found it to be 10.8 x
10–3 mm/mmHg. However, because human arteries were
stored overnight before experiments, it is difficult to make any direct
comparison. Furthermore, the same investigators
(10) calculated the diameter
compliance in dogs (D = 3.1 mm) at a pressure range of 60–140
mmHg and obtained a value of 2.10 x 10–3
mm/mmHg. The variations may be due to species differences.
Tomoike et al. (26), using an ultrasonic dimension gauge, measured the diameter of a dog's coronary arteries in situ in the beating heart and calculated the diameter compliance of LCCA as 2.50 x 10–3 mm/mmHg (at pressure range of 60–140 mmHg), which is in agreement with our in situ diameter-compliance value of 2.42 ± 2.3 x 10–3 mm/mmHg (at 100 mmHg) in the arrested heart. Furthermore, our data are in agreement with earlier swine studies in situ (17).
The in situ CSA compliance data determined in this study are also similar
to those obtained in in vivo studies on calves and human subjects
(12,
22,
27). Gross et al.
(12) used ultrasonic crystals
to measure the coronary artery diameter changes during the cardiac cycle. They
also made simultaneous measurements of coronary blood pressure and computed a
pressure elastic modulus. Their modulus can be converted into a diameter
compliance to give an approximate value of 5 x
10–3 mm/mmHg. Williams et al.
(27) obtained CSA compliance
of 20 ± 12 x 10–3
mm2/mmHg (at 100 mmHg) using intravascular ultrasound, which is
similar to the values obtained in the present study (13.5 ± 9.7 x
10–3 mm2/mmHg, for order 11). Using a
similar method, Shaw and colleagues
(22) calculated the
CSA-compliance at the diastolic blood pressure to be 17.5 ± 5
x 10–3 mm2/mmHg.
Volume distensibility is defined as the volume compliance normalized with respect to its volume at zero pressure. We found the volume distensibility data to be in reasonable agreement with earlier studies. Gregg et al. (11) estimated the LAD arterial volume distensibility in a passively arrested dog's heart to have a value of 5.4 x 10–3 mmHg–1, and a mean pressure of 120 mmHg. Using a similar method, Patel and Janicki (21) obtained volume distensibility values of 4.5 x 10–3 mmHg–1 at a mean pressure of 120 mmHg. Furthermore, using radiogram studies of LCCA under static distension in dogs, Douglas and Greenfield (3) evaluated the volume distensibility at a mean pressure of 100 mmHg to be 5.8 x 10–3 mmHg–1. Those values are somewhat higher than those obtained in the present study (2.9 ± 1.3 x 10–3 mmHg–1 at a pressure of 100 mmHg).
Elasticity of coronary artery and passive myocardium. It is well known that blood vessels in vivo are mechanically supported by the surrounding tissue. Previously, Fung et al. (8) evaluated the effect of mesentery on the elasticity of its capillary blood vessel by measuring the elasticity of the tissue and computing the contribution under the hypothesis that the vessel is in direct contact with the tissue. This formulation led to the well-known tunnel-in-gel concept; i.e., the elasticity of the capillary blood vessel is derived almost entirely from the surrounding tissue. Fung et al. (8) showed that the extent of tissue contribution on the elasticity of the blood vessel depends on the size of the tissue relative to the vessel. For a capillary vessel, the relative size of tissue to vessel is very large, and hence the contribution to elasticity is dominated by the tissue. For a coronary artery, the contribution may be smaller as demonstrated below.
An idealized version of the tunnel-in-gel model can be proposed to
determine the relative elasticity of coronary artery and surrounding
myocardium (8). The coronary
artery is assumed to be uniform, isotropic, linear (i.e., obeys Hooke's law),
and infinitely long so that a plane state of strain exists. It follows from
the infinitesimal theory of elasticity that the change of radius, R,
with internal pressure, P, of a thin wall vessel is given by
 | (7) |
 | (8) |
and Emyo correspond to the Poisson's ratio and
Young's modulus of the myocardium, respectively. The ratio of these two rates
is given by
 | (9) |
is 0.5 (i.e., an incompressible material), the computed mean value
of Emyo (Eq. 9) at 100 mmHg is 140 kPa. These
values are in agreement with the biaxial measurements of passive myocardium
reported by Yin et al. (28) in
the low stress-strain regime. Effect of myocardium on stress distribution in the coronary artery wall. It is well accepted that fluid and solid mechanics of the blood vessel are important determinants of the health and disease of the cardiovascular system (7). One serious disease that seems to be uniquely associated with abnormal stress and strain is atherosclerosis. A compelling observation is that the epicardial arteries develop atherosclerosis, whereas the intramural arteries do not (19). Atherosclerotic changes involving the epicardial portion of the coronary artery abruptly stop at the point where the artery enters the myocardium. It has been previously proposed that the sites of prevalent atherosclerosis are sites of high wall stress (24). Hence, it would be expected that intramural vessels that have support from the myocardium would have a lower wall stress. In the context of the present study, we would hypothesize that the myocardial side of the epicardial artery would be less susceptible to atherogenesis. The hypothesis of circumferential variation of atherosclerosis in the epicardial coronary arteries remains to be tested.
Physiological implications. As the heart muscle contracts, the
cells and collagen fibers bear stresses, whereas the fluid in the tissue
compartments are exposed to hydrostatic pressure. The hydrostatic pressure in
the soft tissue surrounding the myocardial fibers is referred to as the
intramyocardial pressure (IMP) and may compress the coronary vessels. Numerous
methods have been used to estimate hydrostatic pressure throughout the
thickness of the heart muscle during the cardiac cycle. Although there is some
disagreement about the absolute values measured, all studies observed a
gradient in IMP across the wall during systole, with endocardial values being
larger (see review in Ref.
14). Hence, the intramural
arteries (orders 
10) experience compression during the cardiac cycle.
Although the mechanical properties of the coronary blood vessels are well
understood under distension, there are no comparable data under compression.
Such data are necessary to understand the physiology of coronary blood flow
during the cardiac cycle. Both the vascular waterfall mechanism and the
intramyocardial pump model presuppose the existence of an IMP acting on the
external surface of the deformable vessel wall to explain the impediment of
coronary flow during cardiac contraction [see review in Smith and Kassab
(23)]. Although we do not
suggest that the present preparation is a model for heart contraction, it is
very interesting that the larger arteries are tethered by the passive
myocardium and do not collapse under significant compression.
In summary, the passive myocardium limits the expansion of coronary arteries during distension and prevents collapse of vessels during compression. Hence, the myocardium is an important determinant of the mechanical properties of coronary arteries. This suggests that the results of in vitro measurements must be interpreted with caution. These conclusions apply to coronary arteries >0.5 mm in diameter. The vessel-myocardium interaction, however, may be even more significant for the smaller vessels, i.e., the microvasculature. Thus further studies are needed to understand the effects of the myocardium on the mechanical properties of smaller vessels. Furthermore, the active myocardium with its increased stiffness may affect the mechanical properties of coronary arteries to a greater extent. Finally, the effect of vessel tone may play an important role in vessel-myocardial interaction, which is not considered in the present study. These issues remain as topics for future studies.
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DISCLOSURES |
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FOOTNOTES |
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  Y. Liu, W. Zhang, and G. S. Kassab Effects of myocardial constraint on the passive mechanical behaviors of the coronary vessel wall Am J Physiol Heart Circ Physiol, January 1, 2008; 294(1): H514 - H523. [Abstract] [Full Text] [PDF]
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Y. Liu, C. Dang, M. Garcia, H. Gregersen, and G. S. Kassab Surrounding tissues affect the passive mechanics of the vessel wall: theory and experiment Am J Physiol Heart Circ Physiol, December 1, 2007; 293(6): H3290 - H3300. [Abstract] [Full Text] [PDF]
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 W. Zhang, Y. Liu, and G. S. Kassab Flow-induced shear strain in intima of porcine coronary arteries J Appl Physiol, August 1, 2007; 103(2): 587 - 593. [Abstract] [Full Text] [PDF]
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 N. Westerhof, C. Boer, R. R. Lamberts, and P. Sipkema Cross-talk between cardiac muscle and coronary vasculature. Physiol Rev, October 1, 2006; 86(4): 1263 - 1308. [Abstract] [Full Text] [PDF]
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J. S. Choy, Q. Dang, S. Molloi, and G. S. Kassab Nonuniformity of axial and circumferential remodeling of large coronary veins in response to ligation Am J Physiol Heart Circ Physiol, April 1, 2006; 290(4): H1558 - H1565. [Abstract] [Full Text] [PDF]
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G. S. Kassab Scaling laws of vascular trees: of form and function Am J Physiol Heart Circ Physiol, February 1, 2006; 290(2): H894 - H903. [Abstract] [Full Text] [PDF]
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 X Lu and G. S Kassab Nitric oxide is significantly reduced in ex vivo porcine arteries during reverse flow because of increased superoxide production J. Physiol., December 1, 2004; 561(2): 575 - 582. [Abstract] [Full Text] [PDF]
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