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1Department of Biomedical Engineering, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana; 2Center for Sensory-Motor Interaction, Aalborg University and Center of Excellence for Visceral Biomechanics and Pain, Aalborg Hospital, Aalborg, Denmark; and 3Department of Surgery, 4Department of Cellular and Integrative Physiology, and 5Indiana Center for Vascular Biology and Medicine, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana
Submitted 8 June 2007 ; accepted in final form 12 September 2007
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ABSTRACT |
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 TOP ABSTRACT GlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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20% lower than in the untethered state and the average circumferential stress was reduced by 70%. For femoral arteries, the reductions were 15% and 70%, respectively. These experimental data support the proposed hypothesis and suggest that in vitro and in vivo measurements of the mechanical properties of vessels must be interpreted with consideration of the constraint of the surrounding tissue.
stress analysis; strain; impedance planimetry; carotid artery; femoral artery
Clinically, changes in arterial stiffness represent an early risk factor for cardiovascular diseases; i.e., increased aortic stiffness is associated with aging, hypertension, diabetes, hyperlipidemia, atherosclerosis, heart failure, and smoking (19). Current methods to assess local vessel stiffness in vivo are based on measurement of luminal area and pressure (30), without consideration of the surrounding tissue. Furthermore, Laplace's equation for determination of wall tension or stress is based on the model of an untethered cylinder or vessel.
For biomechanical simulations, there are two natural ways of modeling the effects of perivascular constraints. The mechanically accurate approach requires appropriate description of the mechanical properties of the surrounding tissue and the interfacial and boundary conditions, which are usually unclear or unattainable in vivo. For example, Misra and Singh (27) assumed that the perivascular tissue exhibits linear elastic properties. The simpler approach introduces a lumped parameter on the outer vessel surface such as an effective perivascular pressure (EPP) (14, 42, 43) to simulate the radial constraint. For example, Zhang et al. (42) predicted that even a small external compression (10%) causes a large reduction in circumferential stress and tends toward a biaxially (circumferential and axial) uniform strain state for the coronary artery and aorta. The question arises: what is the relation between the EPP and the intravascular blood pressure? To the best of our knowledge, there has not been a previous experimental study in this respect.
The goal of the present study was to assess the ubiquity of the significance of surrounding tissue in other vessels, i.e., carotid and femoral arteries, and to quantify the EPP. We used an impedance planimetry method in which the impedance electrodes are encased in a saline-filled balloon containing saline at the tip of a catheter. The vessels were distended in vivo at the desired pressure, and the corresponding luminal CSA values were determined based on an electrical impedance principle (18). Hence, a pressure-CSA relation was determined for the blood vessel with the surrounding tissue intact and after dissection of the tissue away from the vessel wall. The EPP was calculated, and the circumferential wall stretch and strain were calculated in both states. The average circumferential stress in the vessel wall was analyzed for both states based on continuum mechanics analysis.
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Glossary |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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P
z, , r
s
rr,
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MATERIALS AND METHODS |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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The impedance planimetry (IP) probe used in these studies was described previously in detail by Frøbert et al. (7) and Tanko et al. (37). Figure 1 schematically shows the construction of the catheter. The impedance catheter was constructed from three-lumen polyethylene tubing (Microtube Extrusions) with an outer diameter of 1.5 mm. The tip of the IP catheter was fitted with a saline-filled cylindrical balloon. The balloon was made of 25-µm-thick polyurethane (ST-1183-85, Stevens Urethane). The balloon was designed to be 40% larger than the vessel so that it did not take up any tension within the testing pressure range. The balloon was connected through an infusion channel to a level container with electrically conducting saline (0.9% NaCl). Intravascular pressure was applied by inflating the balloon with saline.
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 | (1) |
The other two lumens of the IP catheter had exit holes between the electrodes. One lumen was considered the inlet port, through which saline was infused to inflate the balloon. The other lumen was considered the outlet port, which allowed the balloon to be flushed of air bubbles and was connected to a pressure transducer.
Calibration of the CSA measurement system was performed before experiments. The detailed calibration procedure was described previously by Kassab et al. (18). In this study, the IP probe was calibrated at body temperature (37°C). It was inserted into holes of known radii, ranging from 1.5 to 3 mm (CSA from 7.06 to 28.26 mm2), drilled in a solid polyphenolenoxide block. Impedance measurements from each hole were obtained at a balloon pressure of 100 mmHg. The pressure transducer used to monitor the balloon pressure was calibrated against a hydrostatic mercury column while applying pressures of 0 and 100 mmHg. A representative calibration curve is given in Fig. 2.
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Ketamine (20 mg/kg) and atropine (0.05 mg/kg) were used to preanesthetize the animals. Endotracheal intubation was performed, and surgical anesthesia was mechanically maintained through inhalation of isofluorane (1–2%) and oxygen. A 1-cm section of the carotid artery on either side of the larynx was chosen for testing. The skin directly above the carotid section was marked for reference. A small incision 10 cm proximal to the tested region was made sufficiently distant from the tested carotid segment not to disturb the surrounding tissue. An IP catheter was gently inserted into the carotid artery through a puncture hole at the incision site and advanced to the tested region. A purse suture placed around the puncture hole was tightened to secure the catheter and to seal the vessel. Carotid arteries of five animals were tested and labeled as C1–C5 in chronological order.
For the femoral artery, a 1-cm section of the left or right femoral artery was marked for testing. An incision 10 cm proximal to the tested region was made in the lower abdominal region. The incision exposed the iliac artery, which provided catheter access to the femoral artery downstream. A purse suture on the iliac artery was used to secure the IP catheter and to prevent leaks. A short period was allowed to attain equilibrium before testing the vessels. Femoral arteries of five animals were tested and labeled as F1–F5 in chronological order.
Pressure-Impedance Measurements
Sodium nitroprusside (SNP, 100 µmol/l) was used to relax the active stress generated by the smooth muscle cells, such that the response of the vessel was passive during the test. A controlled saline infusion pump was utilized to vary the pressure in the balloon, starting at 20 mmHg. An equal amount of saline was pumped during inflation and deflation to ensure zero net volume in the balloon at the end of each cycle. The pressure was monitored by a low-compliance external pressure transducer coupled to the infusion channel of the catheter. In the intact vessel, the balloon was inflated and deflated for a number of cycles sufficient to precondition the vessel. A vessel was determined to be preconditioned when the CSA reached the same value (within 5%) at 120 mmHg (femoral artery) or 150 mmHg (carotid artery). Thereafter, balloon pressure and the voltage between the two inner electrodes were continuously recorded, and data of 10 additional cycles were used for the pressure-CSA relation with the surrounding tissue intact.
The surrounding tissue was then dissected away, leaving only a fascia layer, denoted as the "untethered" state, and the above measurements were repeated. We were careful not to disturb the fascia layer, because this can cause vasospasm.
At the conclusion of in vivo studies the animal was killed with pentobarbital sodium (120 mg/kg) injected through a jugular vein, and the vessel segment was excised. Its length was measured before and after excision for determination of in vivo axial stretch ratio. Three rings were cut from the middle of the vessel to measure the wall area in the no-load state (axial stretch ratio = 1, pressure = 0). The rings were then cut radially, and the opening angle and inner/outer radii were measured in the zero-stress state (Fig. 3A).
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Pressure-radius relation.
We approximate the vessel to have a circular cross section at all pressures, and the inner radius rin at different pressures P is calculated by
 | (2) |
 | (3) |
 | (4) |
= 
/( – ), with the opening angle. The wall thickness is h = rout – rin.
Stretch and strain.
The in vivo axial length of the vessel shows little change with applied pressure. The in vivo axial stretch ratio, z, is obtained as the ratio of axial length before and after excision, and the axial Green strain is Ezz = (
– 1)/2.
The circumferential stretch ratio, , varies across the wall thickness. Consider a material point with radial coordinate r in the deformed vessel wall, the circumferential stretch ratio with respect to the zero-stress state is (33)
 | (5) |
on the inner vessel surface is
 | (6a) |
)in with respect to a reference physiological state, which is the intact vessel at 75 mmHg intravascular pressure for this study, such that
 | (6b) |
= (2 – 1)/2. Finally, the radial stretch ratio is r = (z)–1.
Radial constraint of surrounding tissue.
For a vessel in situ, the mechanical behaviors of vessel and surrounding tissue are well approximated as axisymmetric. As derived in the APPENDIX, the radial constraint can be quantified as an effective perivascular pressure (EPP), P, applied on the outer surface of the vessel (see 
Fig. 9, state B). At intravascular pressure P, the intact vessel deforms to inner radius r
. First a pressure PU* is determined from the pressure-radius curve of the same vessel in the untethered state such that the corresponding radius of the vessel, 
(PU*), equals to 
(see Fig. 10). Then,
 | (7) |
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is approximately uniform across the wall thickness under physiological conditions. Hence, we consider only the average circumferential Cauchy stress in the vessel wall.
For the vessel in the untethered state, the average circumferential Cauchy stress, denoted as 
, is estimated according to Laplace's law as
 | (8a) |
For the vessel in the intact state, effects of the surrounding tissue must be taken into account. Under assumptions of axisymmetric deformation and incompressibility, we derived analytical formulas (see APPENDIX) for the average circumferential Cauchy stress, 
, as
 | (8b) |
and at the same pressure can be estimated with Eq. A18 without measurement of the wall thickness. It is noted that Eq. 8b is accurate under the assumption of axisymmetric deformation, regardless of the actual strain and stress in the surrounding tissue. Statistical Analysis
Ten sets of pressure impedance data were obtained for each vessel in the intact and untethered states, respectively, corresponding to continuous inflation and deflation cycles. For each state, the data were first processed with Eq. 2 and the calibration curve in Fig. 2 for pressure-radius relation. The resulting 10 sets of pressure-radius relation were fit with polynomial functions. In Fig. 4, the polynomial fit is plotted as a solid line, together with the first three sets of experimental data points. It is noted that these three sets of data largely represent the variation of the measurements, which were very repeatable. The strain and stress were calculated, by Eqs. 5–8, with both the discrete experimental data points and polynomial fit of the pressure-radius relation, as shown in Figs. 5, 7, and 8. The mean values and SD of the radius and EPP at 100 mmHg were calculated from the discrete data points, as summarized in Table 1.
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RESULTS |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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The zero-stress state of vessel tissue is described by the opening angle of the radially cut ring and the inner and outer radii (Fig. 3A), Rin and Rout, respectively. The detailed method and error analysis are presented in Wang et al. (40). For carotid artery C4, we have = 29.5°, Rin = 1.32 mm, and Rout = 1.80 mm. The wall area is A0 = 3.94 mm2. The in vivo axial stretch ratio was measured as z = 1.59. For carotid artery C5, we have = 17.5°, Rin = 1.04 mm, Rout = 1.70 mm, A0 = 5.13 mm2, and z = 1.63, and so on. During the in vivo distension, no change of the axial length was visible.
Pressure-Radius Relation
Figure 4 shows typical impedance planimetry measurements of the pressure-radius relation of carotid and femoral arteries, in the intact and untethered states. The radius is calculated from CSA by Eq. 2. Radii at 100 mmHg of intact and untethered vessels are given in Table 1.
Carotid arteries. As shown in Fig. 4, A and B, for carotid arteries C4 and C5, the radius significantly increases at low distension pressure, and tends to an asymptotic value at higher pressure in the untethered state. Of the five carotid arteries tested, some pressure-radius curves become asymptotic at lower pressure (e.g., C4), while some like C1 continue to increase at higher pressure in the intact state. In addition to the differences between vessels and the surrounding tissue, this may also be partly contributed to by the active contraction stress from smooth muscle.
The pressure-radius relations of the intact and untethered states are significantly different, as shown in Fig. 4, A and B, for C4 and C5. Table 1 lists the radius at 100 mmHg. The dissection of the surrounding tissue significantly increases the radius by 20.1 ± 6.99% compared with the intact state. These increases are significant, since the intact vessels cannot be distended to the radius of the untethered state at 100 mmHg, even when subjected to pressure of 200 mmHg. This result suggests that the surrounding tissue plays an important role in preventing the vessel from overstretch.
Femoral arteries. In general, the responses of femoral arteries, as shown in Fig. 4, C and D, for F1 and F4, appear more linear than those of the carotid arteries in both states. The radius increases more gradually with increase in pressure, and does not tend to an asymptotic value at 120 mmHg. Table 1 lists the radius at 100 mmHg. The dissection of the surrounding tissue significantly increases the radius by 15.4 ± 9.15% compared with the intact state.
The intravascular pressure was controlled in the experiments, and thus the accuracy of this study can be quantified with the coefficient of variation (CV = SD/mean) of the measured voltage, or equivalently the radius. The calibration data (Fig. 2) show very little variation of the voltage, with a typical CV <1%, demonstrating the repeatability of the impedance measurement. The CV did increase for in vivo measurements, probably because of the variation of the mechanical environment during the experiments. As shown by the pressure-radius relation in Fig. 4 and Table 1, the CV was estimated as <3% at physiological pressure, which is consistent with previous measurements (14). The CV was significantly lower than the increase of vessel radius because of dissection of the surrounding tissue, i.e., 20.1 ± 6.99% for carotid arteries and 15.4 ± 9.15% for femoral arteries at 100 mmHg.
Stretch and Strain
Carotid arteries.
The stretch ratio and Green strain of vessels C4 and C5 are calculated with respect to the zero-stress state. The axial stretch ratio z is independent of pressure. We will focus on the circumferential stretch ratio (Eq. 6a) and the corresponding Green strain on the inner wall surface. Figure 5A gives representative results of carotid arteries C4 and C5. The mechanical constraint from the surrounding tissue significantly reduces the circumferential stretch ratio and strain. When the tissue is removed, the stretch ratio increases by 25% at 75 mmHg, 21% at 100 mmHg, and 15% at 120 mmHg pressures. Correspondingly, the strain increases by 75%, 56%, and 35%, respectively. The magnitude of radial strain, which is always negative during vessel distension, is also reduced.
Femoral arteries. We calculated the circumferential relative Green strain of femoral arteries with respect to the intact state at 75 mmHg (Eq. 6b). Figure 5B gives results of arteries F1 and F4. When the tissue is removed, the strain also shows a large increase, by 0.16–0.33 in magnitude for F1 and 0.16–0.20 for F4.
Radial Constraint of Surrounding Tissue
Effect of the surrounding tissue is quantified as an EPP P on the outer surface of the intact vessel. As listed in Table 1 for the vessels at pressure P = 100 mmHg, P is estimated as 64.6 ± 14.8 mmHg for carotid arteries and 52.4 ± 13.4 mmHg for femoral arteries; that is, EPP is at least half of the intravascular pressure at physiological loading. The relation between P and P varies among vessels, and the statistical results are shown in Fig. 6, A and B, for carotid and femoral arteries, respectively. When P increases from 40 to 180 mmHg, P increases linearly. The ratio between P and P is 0.5 at the physiological pressure range from 60 to 120 mmHg, and increases gradually with P. At low pressure, the ratio is lower than 0.5. In fact, the previous experiments (11) suggest that the constraint of the surrounding is minimal at 0 mmHg pressure. On the basis of these observations, we propose an empirical relationship
 | (9) |
0.55 for 40 
P < 60 mmHg, k 0.5 for 60 P 120 mmHg, and k 0.45 for 120 < P 180 mmHg. Circumferential Wall Stress
The average circumferential wall stress was calculated with Eqs. 8a and 8b for the untethered and intact states, respectively. Figures 7A and 8A plot the pressure-stress relation of arteries C4 and C5, respectively. The radial constraint of surrounding tissue significantly reduces the circumferential stress . At 75 mmHg pressure, is estimated as 55 and 150 kPa for intact and untethered states of C4, respectively, and 27 and 131 kPa for C5. At 100 mmHg, is 63 and 211 kPa for C4 and 49 and 187 kPa for C5. At 120 mmHg, is 70 and 258 kPa for C4 and 70 and 233 kPa for C5, and so on.
Figures 7B and 8B show the relation between and the circumferential Green strain on the inner surface. The large difference shown in the pressure-stress relation (Figs. 7A and 8A) becomes less apparent. Within the range of intact strain, the untethered stress is consistently and slightly higher than the intact stress.
We also computed the ratio between the stress of untethered and intact carotid and femoral arteries as / with Eq. A18. It is noted that Eq. A18 yields a lower estimate of the ratio. At 100 mmHg intravascular pressure, the ratio is 3.8 ± 0.8 for carotid arteries and 3.2 ± 1.7 for femoral arteries (Table 1).
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DISCUSSION |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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P applied on the outer vessel surface. This pressure is determined from the pressure-radius relation of the same vessel in the intact and untethered states. For carotid and femoral arteries, P can be as high as 50% of the intravascular pressure under physiological loadings. For intact arteries, the circumferential stretch ratio is reduced by an average of 15–25% under the radial constraint at physiological pressure, and the average circumferential wall stress is reduced to 30% or less of the untethered stress. The stress reduction is due to the following: 1) the radial constraint limits vessel distensibility so that the radius-to-thickness ratio rin/h for intact vessels is lower than that of untethered vessels at the same pressure, and 2) the EPP adds a uniform compressive stress in the wall and decreases the tensile circumferential stress. Laplace's Law
Stress in the vessel wall has important physiological implications for understanding vessel growth and remodeling and vessel metabolism (23, 36), as well as clinical assessments of arterial stiffness (19, 21) and vulnerability of plaque (22, 41). To access the local arterial stress and stiffness, Laplace's law in its original form, Eq. 8a, has been used in a large body of literature; i.e., the calculation is directly based on the luminal radius and blood pressure (12, 19, 20, 30), without considering the constraint of surrounding tissue. The findings of this study show clear limitations of this approach as outlined below.
To investigate how ignoring the radial constraint may affect the calculation of the intact stress, we conducted stress computation using Laplace's law without the correction term P, i.e., as
 | (10) |
is higher than the actual circumferential stress as predicted with the modified Laplace's law Eq. 8b, since P is found to be about twice PU* (= P – P) in Eq. 8b, as shown in Fig. 6. The experimental results are plotted as "Intact*" in Figs. 7 and 8 for carotid arteries C4 and C5. This clearly shows that the radial constraint of surrounding tissue cannot be neglected for the intact wall stress. In other words, direct use of Laplace's law for vessels in situ, as in most clinical applications, may significantly overestimate the circumferential stress and give an incorrect stress-strain relation or stiffness of the vessel wall. In the stress-strain relation (Figs. 7B and 8B), the "Intact" curve calculated with "modified" Laplace's law Eq. 8b is consistent with the "Untethered" curve, which reflects the continuum mechanics theory that constitutive relation is independent of the deformation and boundary conditions. The "Intact*" curve, however, is much higher.
Therefore, modified Laplace's law Eq. 8b must be used for tissue-surrounded vessels. Unfortunately, the relation between intravascular pressure P and the EPP P may vary among vessels and subjects, and cannot be obtained exactly without dissection of the surrounding tissue. For pig carotid and femoral arteries under physiological loading, the modified Laplace's law can be approximated with use of the empirical formula Eq. 9, as
 | (11a) |
0.5 for 60 P 120 mmHg. Considering that the radius-to-thickness ratio is much higher than 1 for most vessels and k (1 – k), further approximation can be made as
 | (11b) |
Vascular Mechanics
Factors that affect the deformation and stress/strain of a vessel include geometry, intravascular pressure, passive/active mechanical properties of the vessel wall, and the boundary with surrounding tissues. In recent years, refined models and parametric identification methods for the vessel tissues have been developed (13, 32, 40), and accurate geometry and intravascular pressure have been obtained. The majority of vascular mechanics simulations (31, 33, 35), however, considered the outer surface of the vessel free of external constraint. Some recent simulations (14, 43) formulate the radial constraint with an equivalent pressure on the outer vessel surface, as the EPP P here. However, determination of the equivalent pressure was ad hoc. The present study provides experimental data that call attention to the role of the surrounding tissue.
Elastography, or identification of the nonlinear material parameters of tissues from in vivo deformation measured with ultrasound, computerized tomography (CT), MRI, or other imaging techniques, is an exciting development. It is also a refinement of the present clinical methods that yield simple indices like linear stiffness and Young's modulus. Elastography has been applied for cancerous tissue, myocardium, etc. (24, 25, 28, 29). Because of rapid development of high-resolution imaging techniques such as intravascular ultrasound (IVUS) and optical coherence tomography (OCT), vascular elastography has also been attempted (1, 3, 6). Our results, however, suggest that development of vascular elastography requires special caution, at least for its quasi-static approach, since the constraint of surrounding tissue cannot be quantified without tissue removal, nor can it be ruled out. Figures 7B and 8B further suggest that the intact strain-stress range at physiological pressure, 70–120 mmHg for carotid artery, may be too narrow for characterization of the full nonlinear tissue properties.
Vascular Remodeling
In a previous study (4) of large coronary veins in response to pressure overload, significant intimal hyperplasia (IH) was observed in the vessel wall that was not tethered to the myocardium. For vessels that were completely surrounded by myocardium, no IH was observed. Our hypothesis was that the remodeling may be triggered by excessive circumferential wall strain or stress, which does not occur in tethered vessel wall. The present study quantifies the effects of surrounding tissue on the intramural stress and strain, and provides further evidence for the proposed hypothesis.
Kamm et al. (16) examined the carotid arteries of control and deoxycorticosterone acetate-hypertensive swine and reported that chronic elevation of the mean arterial pressure from 100 to 135 mmHg caused substantial media thickening. Some related studies (5, 35) suggested that the remodeling is initiated by the high strain induced by elevated pressure. Figures 4, A and B, and 5A show for carotid arteries that the strain of the untethered vessels at pressure of 40–60 mmHg easily exceeds that of the intact state at 140 mmHg. Therefore, the surrounding tissue prevents the vessels from overstretch that causes growth and remodeling.
Critique of Methods
For some carotid arteries such as C4 (Fig. 4A) there exists a low-pressure zone, where the radius shows no change with the balloon pressure. The reason may be that the balloon pressure was not high enough to overcome the blood pressure, so that the balloon had not been inflated to contact the vessel wall. On the other hand, the behaviors of carotid and femoral arteries at low pressure are considered less important.
The present analyses are based on an assumption of axisymmetric mechanical behaviors. The exact boundary condition between the vessel and the surrounding tissue, however, is not uniform circumferentially, and the material properties of the tissue may be anisotropic. Our recent finite-element simulation of effects of asymmetric tissue constraint suggests that the present assumption is a close approximation for vessels that are completely embedded into the tissue such as carotid and femoral arteries (unpublished observations). The present results of the EPP, strain, and stress are thus considered as average behaviors of the vessel wall. On the other hand, the agreement of the stress-strain relations in intact and untethered states (Figs. 7B and 8B) indicates that the assumption of axisymmetric deformation and the subsequent modified Laplace's law (Eq. 8b) are reasonable. For partially constrained vessels, analyses should be conducted on the detailed strain and stress fields.
In the untethered state, fascia was left intact to prevent vasospasm. While the fascia is very soft, it can still take up some loads at high strain. This contributes partially to the slightly higher untethered strain-stress curves as in Figs. 7B and 8B. Ideally, the intact stress should be compared with the fascia-removed state, where the active stress caused by fascia removal is completely relaxed.
Relaxant SNP was used so that response of the vessel was passive in the intact and untethered states. This suggests that the constitutive model of the vessel wall is identical for both states, as we have used in the stress analysis (see APPENDIX). Under physiological conditions, the active response may also contribute to the stress-strain relation. The most commonly employed mechanical models calculate the active stress as a function of the strain (see Refs. 15 and 32 for examples). With this concept in mind and the assumption that the calcium kinetics remains similar at intact and untethered states, the active stress can be absorbed in the term s(X, F) of Eq. A1, and all the present formulas hold for vessels involving active response, since the strain in a vessel is determined only by the deformed radius.
While active stress was not the focus of this study, we estimated its in vivo magnitude in some vessels. After the test on the untethered state of carotid artery C4, SNP was washed out with saline for 30 min and the fascia was then removed to cause vasospasm. The active stress resulted in a much stiffer vessel, whose distensibility was reduced to about the same level as the tissue-surrounded intact state. This indicates that effects of the active stress can be as strong as the radial constraint of the surrounding tissue. We assume that the active stress is generated perpendicular to the radial direction, i.e., in axial and circumferential directions (15, 26, 32). Considering the pressure-radius relation of fascia-removed vessel in passive (relaxed) and active states, the average circumferential active stress can be estimated as
 | (12) |
from untethered and intact pressure-radius relations. The active stress is indeed profound and increases with pressure (thus circumferential stretch), consistent with previous studies (15, 26, 32). Future Studies
The radial constraint of the surrounding tissue may be more significant for veins (thinner and softer) and smaller vessels like the coronary arteries that are deeply embedded into the myocardium. Thus further studies are needed to quantify the constraint on veins and smaller vessels. While we consider the passive effects of the radial constraint by the surrounding tissue, it should be noted that the in vivo distensibility of vessel is affected by both the active and passive stresses. The present experiment and analysis assume quasi-static deformation. The effects of tissue motion and contraction may play an important role in the vessel-tissue interaction. These issues remain as important topics for future studies.
In summary, the present findings have important implications for vascular mechanics, mechanobiology, and clinical assessments of arterial stiffness. It is clear that effects of the radial constraints must be considered in simulation of vessel deformation and interpretation of the in vivo and in vitro measurements. High stresses and strains can induce growth, remodeling, and atherosclerosis, and hence a reduction of stress and strain by the surrounding tissue may be atheroprotective. Although the test of this hypothesis remains a task of future investigation, the relevance to pedicled vessels in vascular surgery is obvious.
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APPENDIX: STRESS ANALYSIS |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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 | (A1a) |
 | (A1b) |
, X is the spatial coordinate in the reference configuration (RC) and F is the deformation gradient with respect to RC at point X. The Cauchy stress at point X has two contributions: s(X, F) due to deformation and hydrostatic compression stress –p(X)I to maintain nearly incompressible deformation. It is noted that hydrostatic pressure p changes with location X in the vessel wall and the external loading. The circumferential Cauchy stress component is calculated as Eq. A1b. Here, we use superscript I to denote quantities of a vessel when it is in the intact state tethered to the surrounding tissue, and superscript U to denote those of the untethered state. Typical experimental pressure-radius curves of a vessel at the two states are shown in Fig. 9.
Circumferential Stress and Wall Tension of Untethered Vessel
Assuming that the vessel deforms axisymmetrically, the circumferential wall tension force, T, is the integral of stress across the deformed wall thickness h, i.e.,
 | (A2) |
 | (A3) |
 | (A4) |
Circumferential Stress and Wall Tension of Intact Vessel
For an intact vessel, the circumferential wall tension force and the average circumferential Cauchy wall stress can be calculated with Eqs. A2 and A3. Equation A4 needs modification, however, to take into account the effects of the surrounding tissue. Assuming that the material properties of the surrounding tissue are symmetric with respect to the vessel axis, the effect on the vessel deformation (state A in Fig. 9) is equivalent to an EPP (effective perivascular pressure), denoted as P, applied on the outer surface of the vessel (state B), whose value is determined below.
At intravascular pressure P and EPP P, the intact vessel deforms to an inner radius r(state B). The same deformation (displacement and strain) can also be achieved by applying an internal pressure PU* = P – P and leaving the external surface free (state C). The latter is possible because for an incompressible vessel the axisymmetric deformation can be described with only one variable, i.e., the inner radius.
Therefore, the displacement, deformation gradient, and strain of an intact vessel at blood pressure P and deform to inner radius r are identical to those of its untethered state at blood pressure PU* and deforms to the same inner radius, that is,
 | (A5) |
 | (A6a) |
 | (A6b) |
= r
(PU*) (Fig. 10). The hydrostatic compression stress –pI(X) must be determined to satisfy the boundary conditions on the inner and outer surfaces as:
 | (A7a) |
 | (A7b) |
 | (A8a) |
 | (A8b) |
 | (A9a) |
 | (A9b) |
 | (A10a) |
 | (A10b) |
Since a uniform hydrostatic compression stress field is self-equilibrating without introducing any additional deformation when material is incompressible (state D), we find that
 | (A11) |
 | (A12a) |
 | (A12b) |
 | (A13) |
 | (A14) |
P = P – PU*. Equation A14 indicates that can be accurately calculated once the pressure-radius curves are experimentally measured of the vessel intact and untethered, without need of the mechanical properties and strain/stress states of the vessel wall and the surrounding tissue. It should be noted that PU* is always lower than P, meaning that when a vessel deforms to a given inner radius, the circumferential wall tension and stress are lower for the intact state than for the untethered state. Furthermore, at the same blood pressure P, the wall tension and circumferential stress are lower for the intact state than the untethered state. Two factors contribute to this reduction: 1) r(P) > r(P) and hU(P) < hI(P), so that
 |
) < P and P > 0. Therefore, according to Eq. A14, we have
 | (A15) |
 | (A16) |
P > 0. In physiological condition, the wall thickness h is typically <20% of the radius. It can be derived from Eq. 3 that
 | (A17) |
=A0/z. Finally,
 | (A18) |
2 and r/r is 1.2 or higher, which suggest that is about three times or more of (P).
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FOOTNOTES |
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TOPABSTRACTGlossaryMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: STRESS ANALYSISREFERENCES |
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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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