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Department of Biomedical Engineering, University of California, Irvine, California 92697-2715
Submitted 13 June 2003 ; accepted in final form 16 August 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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stress; strain; elastic modulus; zero-stress state; strain energy function
The data on the elasticity of the aorta are enormous (see reviews in Refs. 11, 12, and 27). One of the reasons for the great interest stems from the observation that increased stiffness of large elastic arteries represents an early risk factor for cardiovascular diseases (2, 19). Specifically, increased aortic stiffness is associated with aging (3, 14, 28), hypertension (10, 18), diabetes (1, 26, 29), hyperlipidemia (23), atherosclerosis (6), heart failure (21), and smoking (25, 31). Furthermore, arterial stiffness has also been shown to be an independent risk factor for cardiovascular events such as primary coronary events, stroke, and mortality (5, 22). Therefore, the assessment of aortic mechanical properties is particularly important in understanding the mechanisms of cardiovascular disease.
Although many species have been studied (see review in Ref. 27), there are no data on the variation of mechanical properties along the entire length of the in situ aorta of any species. We are interested in such data in the mouse model because the mouse is being used with increasing frequency in experimental studies of many aspects of cardiovascular health and disease. In the present study, we focus on the C57BL/6J strain because it is the most widely used and hence the best known of all inbred strains. Furthermore, it has substrains that are predisposed to hypertension, diabetes, and hypercholestermia, all of which are important risk factors for coronary artery disease (9, 28). The major goal of the present study is to characterize the mechanical properties of the mouse aorta to understand its physiological implications. Furthermore, the present study provides a phenotypic model for geneotypic manipulation.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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A group of seven homozygous inbred male mice (C57BL/6 strain) with body weight of 24.7 ± 1.49 (means ± SD) g and age 68.3 ± 3.2 (means ± SD) days was used in this study. The mice were preanesthetized with intraperitoneal injections of ketamine (56.2 mg/kg) and midazolam (3.75 mg/kg) to achieve requisite immobilization. The mice were then anesthetized with an intraperitoneal injection of pentobarbital sodium (40 mg/kg). Aortic blood pressure was measured by a catheter inserted into the common carotid artery. Heparin (200 U/ml) was used to prevent blood clots in the heart and the vessels via a jugular vein catheter. A tracheostomy was performed, and the mouse was ventilated with a rodent respirator. The chest was opened with a midline sternotomy. The mouse was euthanized with pentobarbital sodium (120 mg/kg iv), and the apex of the heart was excised. All experiments were performed in accordance with national and local ethical guidelines, including the Institute for Laboratory Animal Research Guide, National Institutes of Health Policy, Animal Welfare Act, and University of California Institutional policies regarding the use of animals in research.
Pressure-Diameter-Length Relation
The ascending aorta was cannulated by a 23-gauge needle through the apex of the left ventricle. The aorta from the aortic valve to the common iliac bifurcation was then carefully exposed on the anterior portion (dissected away the surrounding connective tissue) leaving the posterior portion attached. The aorta was then perfused with 6% dextran solution at the various perfusion pressures. Cab-O-Sil (0.35% by weight), a colloidal silica, was mixed into the Dextran solution to prevent flow through the microvessels and hence attain zero-flow distensions. Water-resistant carbon particles were used to mark the aorta, dividing it into a series of 10–11 short segments (about 3–4 mm per segment). The aorta was preconditioned with three cyclic changes in pressure from 0 to 150 mmHg in a triangular form. The perfusion pressure was increased in 30-mmHg step increments from 0 to 150 mmHg in a staircase pattern. The external geometry of each segment of the aorta, at the pressurized state, was photographed to obtain the loaded outer diameter along the trunk of the aorta. The outer diameter-length-pressure relation for each of the 11 segments was measured serially. A local dimensional coordinate x was introduced along the loaded vessel trunk, with x = 0 at the aortic valve and x = 1 at the point of common iliac bifurcation as shown in Fig. 1. The coordinate x represents the length measured from the aortic valve, along the centerline of the aorta, divided by the total length at the pressurized state.
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No-Load and Zero-Stress State
After the mechanical testing, the aorta was carefully dissected and placed into a Ca2+-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). The anterior position of the aorta was marked with carbon from the aortic valve down to the common iliac artery. The entire length of the aorta was then cut into 50–60 segments. Each segment was a ring with length of 0.3–0.5 mm (thoracic aorta: 0.4–0.5 mm, abdominal aorta: 0.3–0.4 mm). Each ring was transferred to a Ca2+-free Krebs solution, aerated with 95% O2-5% CO2, and photographed in no-load state. Each ring was then cut radially by a scissor at the anterior position labeled with carbon black particles. The ring opened into a sector and gradually approached a constant opening angle defined as the angle subtended by two radii connecting the midpoint of the inner wall. The cross section of each sector was photographed 30 min after the radial cut. The morphological measurements of inner and outer circumference, wall thickness, and area in the no-load and zero-stress state were made from the images using a morphometric analysis system (Sigma Scan). The length of each ring was measured at the no-load state, and a local dimensional coordinate a was introduced along the vessel trunk, with a = 0 at the aortic valve and a = 1 at the bifurcation of the common iliac artery as shown in Fig. 1. The coordinates x and a are referred to as the fractional longitudinal position (FLP) in the loaded and no-load states, respectively (20).
Biomechanical Analysis
Incompressibility condition. The incompressibility condition for a cylindrical vessel can be expressed as
 | (1a) |
z = l/lo is the stretch ratio in the axial direction where l and lo are the vessel length in the loaded and zero-stress state, respectively; and Ao is the wall area in the no-load state. The vessel length (l) may correspond to either a segment (each of the 11 segments marked along the length of the aorta) or to the entire length of the aorta (distance from aortic valve to common iliac bifurcation). In the former we define a local and in the latter a global longitudinal stretch ratio.
It should be noted that ri and Ao are functions of coordinate axes x and a, respectively. Because a = x/z, Eq. 1a can be rewritten as
 | (1b) |
Strain and stress. The circumferential deformation of the artery may be described by the Green strain (
), which is defined as follows
 | (2a) |
i,o is the inner or outer circumferential stretch ratio (i,o = 
); ci,o refers to the inner or outer circumference of the vessel in the loaded or no-load state; and 
refers to the inner or outer circumference in the zero-stress state as described in the previous section. In the no-load state, the strain given by Eq. 2 is referred to as the residual strain. Similarly, the longitudinal Green strain is given by
 | (2b) |
z is the local longitudinal stretch ratio as defined above.
At an equilibrium condition, the average circumferential Kirchhoff stress in the vessel wall was computed, with an assumption that the shape of the artery is cylindrical
 | (3a) |
and h are the circumferential stretch ratio and wall thickness of the vessel, respectively. The longitudinal Kirchhoff stress, under similar assumptions, is given by
 | (3b) |
Equations 2 and 3 allow us to compute the circumferential and longitudinal stresses and strains, respectively, for different pressure distensions.
Elastic moduli. A pressurization of a blood vessel induces biaxial (circumferential and longitudinal) stresses and strains. The increase in the circumference during pressurization will tend to shorten the vessel longitudinally because of incompressibility (Eq. 1a) but also tend to lengthen the vessel because of the longitudinal stress arising from pressurization (Eq. 3b). The Poisson's ratio (
= –
/z) and the elastic moduli (E and Ez) determine whether the vessel shortens or lengthens. In analogy to an isotropic tube, Dobrin and Doyle (8) expressed the relationship between the circumferential and longitudinal strains for an anisotropic vessel as
 | (4a) |
 | (4b) |
signifies a change in a quantity. Equations 4a and 4b can be solved for the respective moduli as
 | (5a) |
 | (5b) |
It should be noted that as Ez tends to infinity, E tends to 
/, which is the definition of the tangent modulus (or Young's modulus if the stress-strain relation is linear) for a uniaxial experiment.
Compliance. The lumen cross-sectional area (CSA) was computed from the lumen radius as given by Eq. 1b (CSA = 
). The lumen volume (V) of the entire aorta can be consequently computed as
 | (6) |
The volume compliance (CV) of the aorta is given as the slope of the pressure-volume relationship; i.e., CV = V/P. The CSA compliance (CCSA) is defined similarly as CCSA = CSA/P. The relationship between CV and CCSA can be expressed through the chain rule of differentiation as
 | (7) |
Constitutive equation. A stress-strain relation for an artery can be determined from a strain energy function, which represents stored energy per unit volume of arterial wall. For a linear stress-strain relation, the strain energy function, in a two-dimensional analysis, can be expressed as a simple second-order polynomial
 | (8) |
0 is the mass per unit volume of the artery, W is the strain energy per unit mass, and the constants b1, b2 and b3 are material coefficients. Under the assumptions that the arterial wall is homogeneous and pseudoelastic (i.e., the loading and unloading curves of the stress-strain relation are considered to represent properties of two materials with different elasticity), the strain energy function can be applied to loading and unloading processes separately, and stress components are expressed as follows
 | (9) |
ij and Eij are components of Kirchhoff stress and Green strain, respectively. The stress and strain components are measured experimentally, and the coefficients of the strain energy function, b1, b2, and b3 can be determined by using a linear least square fit as outlined in the APPENDIX. Data Analysis
The position along the aorta was normalized with respect to the total length. Hence, the results were expressed in terms of the FLP, ranging from 0 to 1. The data for both the independent (FLP) and dependent variables (diameter, wall thickness, area, stress, etc.) were then divided into 10 equal intervals: 0–0.1, 0.11–0.2, 0.21–0.3,..,0.91–1.0. The results are expressed as means ± SD over each interval. In some cases, we further combined the intervals into anatomic regions. The thoracic aorta was divided into proximal, middle, and posterior regions, whereas the abdominal aorta was divided into proximal and distal regions. Anatomically, the proximal, middle, and posterior thoracic aorta corresponds to the following: aortic valve to first pair of intercostal arteries, first pair to sixth pair of intercostal arteries, and sixth pair of intercostal arteries to subcostal artery (diaphragm), respectively. The proximal and distal abdominal aorta corresponds to the subcostal artery to the left renal artery and the left renal artery to the common iliac artery, respectively.
Either linear or nonlinear regressions were used to curve fit the data. The longitudinal variations were examined with a one-way ANOVA. Student's t-test was also used to detect possible differences between animals.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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The outer diameter increases linearly with an increase in pressure for all aortic regions. The data for each region were fitted by a linear least-square fit, and the empirical constants are summarized in Table 1. The loaded inner radius was calculated from the incompressibility condition (Eq. 1). The wall thickness at the loaded state was computed as the difference between the outer and inner radius of the vessel at different pressures. The wall thickness of the aorta decreases linearly with an increase in pressure, and the least-squares fit constants are also summarized in Table 1. The variations of diameter and wall thickness along the length of the aorta were found to be statistically significant as were the variations with pressure.
The open sector in the zero-stress state was characterized by the opening angle. The longitudinal variation of the opening angle is shown in Fig. 3A. The inner and outer strains were computed in reference to the zero-stress state. The variation of inner and outer residual strain, over the length of the aorta, at a given pressure was not found to be statistically significant. The mean values of the inner and outer residual strains over the length of the aorta were –0.071 ± 0.015 and 0.058 ± 0.012, respectively. At all distension pressures, the variation of inner and outer strain was found to be statistically significant for the loaded inner and outer strains, respectively. Hence, the mean values of inner and outer strains over the entire length of the aorta were computed at each pressure. Figure 3B shows the variation of the difference in mean strains 
with pressure. The difference between the inner and outer strain is statistically significant at 0, 30, and 150 mmHg.
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The variation of the average circumferential and longitudinal Green strains and Kirchhoff stresses with pressure were computed, according to Eqs. 2 and 3, respectively, for various segments of the aorta. The relationships were found to be linear and were fitted with a least-square fit as summarized in Table 2. Figure 4A shows the circumferential stress-strain relationship for various segments of the aorta in the pressure range of 30–120 mmHg. Figure 4B shows the corresponding data for the longitudinal stress-strain relation. The circumferential and longitudinal elastic moduli were computed according to Eq. 5. The variation of the circumferential elastic modulus along the length of the aorta is shown in Fig. 5. The changes in the longitudinal elastic modulus along the length of the aorta were not found to be statistically significant (P = 0.8). The longitudinal elastic modulus was averaged over the entire length of the aorta and was found to have a value of 20 ± 15 MPa.
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The variation of lumen CSA with pressure for the various segments of aorta was found to be linear. The relationship is curve fitted with a linear least-square fit, and the empirical constants are summarized in Table 1. Figure 6 shows the CCSA of the various aortic segments (i.e., slope of the CSA-P relation). The pressure-volume relationship of the entire aorta is shown in Fig. 7A. The slope of the curve corresponds to the compliance of the aorta, which has a value of 2.67 x 10–4 ml/mmHg. Finally, Fig. 7B shows the volume compliance of thoracic (sum of proximal thoracic, middle thoracic, and distal thoracic) and abdominal (sum of proximal abdominal and distal abdominal) aorta.
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It was observed that portions of the in situ aorta become curved at higher distension pressures. The degree of curvature was estimated by the angle relative to the long axes of the aorta. Our results show that in comparing pressures of 150 to 90 mmHg, the curvature angles for the proximal thoracic, middle thoracic, proximal abdominal, and middle abdominal were 14 ± 2.8, 6.2 ± 0.96, 9.3 ± 4.0, and 7.4 ± 2.4°, respectively. We found no change of curvature of the distal thoracic region during distension presumably because of the mechanical constraint from the diaphragm.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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Numerous estimates of compliance have been made in vivo based on the windkessel model by calculation of the time constant (product of total arterial resistance and compliance) of pressure decay or some modification thereof (see review in Ref. 30). Data on the compliance of pig, dog, and rat have been reported as 
0.5 ml/mmHg (30), 0.1 ml/mmHg (32), and 5 x 10–3 ml/mmHg (24), respectively. In the dog experiment, the reported compliance corresponds to the proximal descending aorta, which accounted for 60% of the total arterial compliance. It should be noted, however, that one of the limitations of the windkessel model is that it does not account for wave reflections. This limitation should be taken into account when interpreting the data.
In the present study, we found the pressure-volume relationship to be linear as shown in Fig. 7A with a slope of 2.67 x 10–4 ml/mmHg. The slope corresponds to the static volume compliance of the entire aorta, which is independent of pressures in the range of 30–150 mmHg. We also evaluated the volume compliance of thoracic and abdominal aorta as shown in Fig. 7B. Because the volume compliance is additive, the total volume compliance, obtained from Fig. 7A, is equal to the sum of volume compliance of thoracic and abdominal aorta. It is interesting to note that the abdominal compliance is only 15% of the thoracic compliance.
Alternatively, the compliance can be expressed in terms of CSA (CSA/P). It is much easier to measure the CSA compliance; particularily, in vivo. Hence, the majority of data in the literature is on the CSA compliance, and it is often used interchangably with volume compliance. The relation between volume compliance and CSA compliance is given by Eq. 7. The general practice is to assume that the axial deformation with distension is negligible; i.e., ignore the second term in Eq. 7. In the present study, we found that the second term contributes 10% to the volume compliance because the axial deformation with distension is small as confirmed in Figs. 1, A and B. Hence, as an approximation the volume compliance is the product of CSA compliance and length of the vessel. We should caution, however, that even when the axial deformation is small, the second term can be significant for a vessel that has a small length-to-CSA ratio. The results of the mouse CSA compliance are shown in Fig. 6. Both CSA and volume compliance show that the proximal aorta, near the heart, is most compliant with the compliance decreasing toward the distal aorta. Hence, the mouse aorta shows geometric and compliance taper similar to other species (27).
We can estimate the degree of accomodation that the aorta provides to each ejection of the heart. The stroke volume can be determined as the quotient of cardiac output and heart rate, whereas the change of volume of the aorta during each cardiac cycle can be estimated from the product of compliance and pulse pressure. For the mouse, at a cardiac output of 12.5 ml/min and heart rate of 622 beats/min, we obtain a stroke volume of 0.02 ml (4). For a pulse pressure of 25 mmHg, the computed compliance yields a change in aortic volume of 0.0067 ml. Hence, the aorta can accommodate about one-third of the stroke volume. It should be noted that this is an estimate because the dynamic compliance may be somewhat different, which would affect the value. Similar calculations for the pig yield a stroke volume of 20 ml assuming a cardiac output of 2 l/min and heart rate of 100 beats/min. The change in aortic volume during each cardiac cycle can be estimated as 12.5 ml assuming a pulse pressure of 25 mmHg and dynamic compliance of 0.5 ml/mmHg. Hence, the aorta accommodates about two-thirds of the blood pumped in one stroke. Incidentially, this is the same estimate made by Hales (17) nearly three centuries ago.
At physiological pressure, the mean aortic lumen volume (from aortic valve to common illiac bifurcation) of the mouse, with a mean body weight of 25 g, is 0.035 ml. Hence, the aortic blood volume is equal to 0.14% of the mouse body weight. Barbee et al. (4) have previously shown that the entire circulating blood volume is 2 ml. Hence, only 2% of the entire mouse blood volume resides in the aorta.
Other Biomechanical Characteristics
The circumferential and longitudinal deformations were measured as a function of pressure along the entire length of the aorta. Figure 1A shows that the longitudinal stretch ratio increases along the length of the aorta (larger near the common illiac bifurcation) and increases with pressure at any longitudinal position. The former observation is very pronounced where the stretch ratio is near 1 at the ascending aorta and 1.6 near the common illiac bifurcation. This deformation is due to longitudinal stretch from the zero-stress state to the in situ state. Consequently, when the vessel is excised, the various regions will shorten according to Fig. 1A where the abdominal aorta shortens the most. Han and Fung (16) measured the longitudinal stretch ratio of canine and porcine aortas in situ (zero pressure) relative to the zero-stress state. The similarity of the logitudinal stretch ratio of the mouse aorta to that of dog and pig is striking. They did not examine the effect of pressure on the longitudinal stretch ratio, however, which was found to be fairly small in the present study (Fig. 1, A and B).
The linearity between pressure and various morphometric parameters (diameter, wall thickness, lumen CSA, and volume) in a large range of pressure (30–150 mmHg) is remarkable. The wall thickness decreases, whereas the diameter, lumen CSA, and volume increase distally toward the common iliac bifurcation. The pattern holds for all segments of the aorta. Similarly, the computed stress and strain vary linearly with pressure. The resulting stress-strain relationship is also linear in the pressure range of 30–120 mmHg. The observed simplicity of mechanical properties along the length of the aorta allows the use of a simple strain energy function (Table 2), which will simplify future mechanical modeling of the cardiovascular system of the mouse.
The linearity of the stress-strain relation is limited to the range of pressure used. Figure 8 shows the stress-strain relation for the middle thoracic and abdominal aorta in the pressure range of 30–210 mmHg. It can be seen that the relationship is linear up to 120 mmHg (i.e., physiological range) beyond which it becomes nonlinear. It is also interesting to note that the relationship in the two regions is essentially the same below 120 mmHg. Above this pressure, the abdominal aorta attains larger stresses and strains. This observation may be due to the fact that the abdominal aorta has fewer bifurcation constraints and is less tethered by surrounding tissue. This observation underscores an interesting possible correlation between higher stresses and strains with the predisposition of disease in the abdominal aorta in hypertension (7).
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The circumferential and longitudinal moduli were computed along the length of the aorta. The circumferential modulus attains its largest value near the diaphragm; i.e., the aorta is most stiff there. Thirty years ago, Tanaka and Fung (33) cut the aorta into strips along the length of the aorta. They tested each strip uniaxially either in the circumferential or longitudinal direction. In the process, they obtained the Lagrangian stress-stretch ratio relation, which was found to have an exponential character. The tangent modulus was computed (slope of the stress-strain relation) for the various segments. They showed a similar peak in the tangent modulus near the diaphram, which is consistent with our present findings. The mean longitudinal modulus was found to be fairly constant along the length of the mouse aorta. The value of the longitudinal modulus was approximately two orders of magnitude larger than in the circumferential direction. It should be noted that the aorta is constrained along its entire length through numerous bifurcations. Hence, the measured modulus is not only of the aorta but also of the surrounding tissue.
Uniform Transmural Strain Hypothesis
The existence of the opening angle, as shown in Fig. 3A, makes the transmural strain distribution more uniform (13). This was previously demonstrated along the entire length of the mouse aorta at physiological pressure (15). In the present study, we can examine the effect of pressure on the uniformity of transmural strain. The difference between outer and inner strain was computed and shown in Fig. 3B. At zero pressure, the values correspond to the residual strain. Outer residual strain is tensile, whereas inner strain is compressive. Hence, the difference is fairly large as shown in Fig. 3B. As the vessel is pressurized, the outer strain remains larger than the inner strain at a pressure of 30 mmHg. At 60 mmHg, the difference in transmural strains is not statistically significant. This remains to be the case until the pressure is increased to a value of 150 mmHg where the inner strain exceeds the outer strain. These results have important implications for hypertension where the increase in pressure may be well above 150 mmHg. The resulting gradient in transmural strain may be a stimulus for growth and remodeling (20).
Future Studies
The structural components of the vessel wall are altered in the various conditions (e.g., aging, diabetes, hypercholestermia, hypertension, smoke exposure, etc.) that pose a risk factor for arteriosclerosis due to the mechanical or metabolic abnormality through various mechanisms. One of the implications of the structural changes is the change in mechanical properties of the blood vessel; i.e., the compliance or modulus. Numerous studies on animal and human tissue have demonstrated increased vascular stiffness during these risk factors. In the present study, the mechanical properties of the entire aorta are described quantitatively. This model will serve as a physiological reference state for future studies of risk factors in the mouse model.
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APPENDIX |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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 | (A1) |
 | (A2) |
A residual or error, e, can be defined for the stress as the square of the difference of the curve fit and experimental value as
 | (A3) |
 | (A4) |
The above set of equations can be represented in terms of matrix algebra as follows
 | (A5) |
The solution to this problem yields values of b1, b2, b3 for each set of experimental data on stress and strain. The set of constants for the strain energy function is summarized in Table 2 for each region of the aorta, respectively.
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DISCLOSURES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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) with the pressures (P). Solid line is a linear least-squares fit of the form 
, * Value that is statistically different from zero.
