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Department of Biomedical Engineering, University of California, Irvine, California 92697-2715
Submitted 1 December 2003 ; accepted in final form 29 January 2004
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ABSTRACT |
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) and midwall stretch ratio (
) showed a linear variation with order number (n) as follows: = 10.2n + 63.4 (R2 = 0.989) and = 4.47 x 10–2n + 1.1 (R2 = 0.995). Finally, the stretch ratio and stress varied between 1.2 and 1.6 and between 10 and 150 kPa, respectively, along the aorta and left anterior descending arterial tree. The relative uniformity of strain (50% variation) from the proximal aorta to a 10-µm arteriole implies that the vascular system closely regulates the degree of deformation. This suggests a homeostasis of strain in the cardiovascular system, which has important implications for mechanotransduction and for vascular growth and remodeling.
uniform strain hypothesis; opening angle; homeostasis; wall thickness
The goal of the present study is to evaluate the distribution of the circumferential stresses and strains along the length of aorta and throughout the coronary arterial tree in a porcine model to assess the homeostasis hypothesis. The variations of stress and strain were examined for vessel diameters that span three orders of magnitude (10 µm–18 mm), and it has been hypothesized that the variation of stress and strain in the cardiovascular system is relatively narrow. Furthermore, we hypothesize that the variation of strain is much more uniform than the variation of stress in the coronary arterial tree. The implication is that strain may play a key role as a mechanosensor during vascular growth and remodeling in response to changes in mechanical stimuli.
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METHODS |
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Ten normal farm pigs weighing 28.6 ± 3.2 kg were used in this study. Surgical anesthesia was induced with ketamine (33 mg/kg) and atropine (0.05 mg/kg) and maintained with isoflurane (1–2%). Ventilation with 100% O2 was provided with a respiratory pump, and a midline sternotomy was performed. After induction of anticoagulation with heparin (100 U/kg), pentobarbital sodium (80 mg/kg iv) was injected to ensure deep anesthesia. The heart was then arrested with a saturated KCl solution administered through the jugular vein. The heart was excised and used for another experiment, whereas the aorta was used for the present study. All animal experiments were performed in accordance with national and local ethical guidelines, including Institute of Laboratory Animal Research guidelines, United States Public Health Service policy, the Animal Welfare Act, and University of California-Irvine policies regarding the use of animals in research.
Preparation of the Aorta
The ascending aorta was cannulated and perfused with 6% dextran solution to flush out the blood. In 6 of the 10 animals, the aorta was perfused at 100 mmHg with catalyzed silicone elastomer containing 1% Cab-O-Sil, which is a colloidal silica that forms agglomerated particles to prevent flow through the small arteries and capillaries. Hence, the pressure was maintained at a constant level throughout the aorta. After hardening of the elastomer, the entire aorta, down to the common iliac and femoral artery and their branches, was dissected, and the anterior portion was marked with carbon black particles. The entire aorta and its major branches were placed in a Ca2+-free Krebs solution aerated with 95% O2-5% CO2. The vessels were then cut transversely into rings with lengths that were 
4–5 mm at the thoracic aorta, 2–4 mm at the abdominal aorta, and 1–3 mm at the iliac and femoral arteries and their respective branches. All rings were photographed transverse to the long axis of the vessel in the loaded state, with the hardened elastomer maintained in the lumen. The elastomer was then pushed out of each ring, and a radial cut was made at the anterior position as labeled with carbon particles. This process causes the ring to open up into a sector. The cross section of each sector was photographed 30 min after the radial cut. Morphological measurement of inner and outer circumference, wall thickness, and opening angle in the loaded and zero-stress states was made from the images using an image analysis system (Sigma-scan).
In the above-described preparation, the transmural pressure was set at 100 mmHg (external pressure was atmospheric). In vivo, however, the intrathoracic pressure may be negative and abdominal pressure positive. Hence, the transmural thoracic and abdominal pressures may be above and below 100 mmHg, respectively. In 4 of the 10 animals, we cannulated the ascending aorta and abdominal aorta (just below the diaphragm) separately and perfused them with elastomer and Cab-O-Sil at 120 and 95 mmHg, respectively. The rest of the protocol was identical to that described above.
Preparation of Coronary Arteries
Ten hearts weighing 241 ± 27.2 (SD) g from normal farm pigs of either gender were obtained from a local slaughterhouse on the morning of the experiment. Immediately after the pig was killed, the heart was placed in a cold saline bath and transported to our laboratory. The left common coronary artery was cannulated and perfused with 6% dextran solution. The blood was immediately flushed out of the heart. The left anterior descending (LAD) artery was then perfused at a physiological pressure (100 mmHg) with catalyzed silicone elastomer (18). After the elastomer was allowed to harden for 45 min, the LAD artery was carefully dissected down to 10-µm-diameter branches. The LAD artery was then placed in a Ca2+-free Krebs solution, and the anterior position of the artery was marked with carbon particles. The vessel was cut perpendicular to the longitudinal axis into rings with segmental length of approximately one-fourth to one-half of the radius. Each ring was transferred to a Ca2+-free Krebs solution, aerated with 95% O2-5% CO2, and photographed in the loaded state.
For the larger vessels, the inner and outer dimensions (and, hence, wall thickness) were measured from transverse sections. The elastomer was then pushed out, and the rings were cut radially to obtain the zero-stress state. For the smaller (<40-µm) vessels, the outer diameter and wall thickness were measured from a longitudinal view, because the vessel is transparent and the wall thickness can be visualized. Furthermore, we could not find a scissors small enough to cut those vessels. Hence, the loaded vessel with the elastomer in the lumen was held with a very fine forceps transverse to the long axis of the vessel. A very thin blade (double-edged Schick razor) was then passed through the space of the forceps to cut the anterior surface of the vessel. The elastomer was gently removed to obtain the zero-stress state. The morphological data of the coronary vessels in the loaded and zero-stress states were obtained with the same method described for the aorta.
To assess the effect of pressure distribution in the coronary arterial tree, we perfused three additional hearts with elastomer and Cab-O-Sil to ensure a uniform pressure (100 mmHg) throughout the coronary arterial tree. The rest of the protocol was identical to that described above. To assess the effect of silicone elastomer, we perfused two additional hearts with gelatin (6% by weight). The gelatin was dissolved in water by heating the solution to 50–60°C and allowed to cool to room temperature before perfusion into the coronary arterial tree. Once perfused, the heart was placed in ice-cold (0°C) saline to polymerize the gelatin. The blood vessels were dissected in cold saline to maintain a firm gelatin. Otherwise, the determination of loaded and zero-stress states was identical to that described above.
Biomechanical Analysis
The circumferential deformation of an artery may be described by the midwall circumferential stretch ratio (), which is defined as follows
 | (1) |
At an equilibrium condition, the average circumferential Cauchy stress (
) in a cylindrical vessel wall was computed as follows
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 | (3) |
Data Analysis
Classification of aortic data. The aorta was subdivided into thoracic aorta (descending aorta, just below the arch, to the diaphragm), abdominal aorta (diaphragm to the common iliac artery), and >1.5-mm-diameter secondary branches (e.g., femoral and renal). The data were grouped together for the thoracic and abdominal aorta and secondary branches, and means ± SD are reported.
Ordering of coronary arterial branches. We previously developed and implemented an ordering system to classify vessels of various sizes into orders on the basis of a diameter-defined Strahler system (18). This classification has resulted in a unique relation between diameter and order number for the entire coronary arterial tree. The relation between the diameter range and order number obtained from the previous study is as follows: order 1 (8.3–10.2), order 2 (10.3–15.4), order 3 (15.5–24.2), order 4 (24.3–48), order 5 (48.1–101 µm), order 6 (102–217 µm), order 7 (218–384 µm), order 8 (385–554 µm), order 9 (555–986 µm), order 10 (987–2,189 µm), and order 11 (>2,190–4,500 µm). In the present study, we determined the relation between stress, strain, wall thickness, opening angle, and diameter throughout the coronary arterial tree. Hence, by using the relation between diameter and order number from the previous study, we determined the relation between stress, strain, wall thickness, opening angle, and order number.
Statistical Analysis
Each arterial ring was considered an independent statistical sample. Values are means ± SD. The differences in aortic segments and LAD arterial order numbers were examined with two-way ANOVA. The results were considered significant when P < 0.05.
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RESULTS |
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Data for the aortas (n = 6) perfused at the same pressure (100 mmHg) and for abdominal and thoracic aortas (n = 4) perfused at 95 and 120 mmHg, respectively, were analyzed separately. We found no statistically significant differences in the strain or stress (computed at the appropriate pressure) between the two groups. Hence, the preceding data were pooled from all 10 animals, regardless of perfusion pressure.
In total, 572 aortic (186 thoracic and 136 abdominal rings) and secondary arterial branches (250 rings from iliac, femoral, and other branches) were measured. Measured wall thickness for the aorta and its branches is shown in Fig. 1A. The data were fitted with an exponential relation with least-square fit constants. The data from Fig. 1A are partitioned into thoracic and abdominal aorta and secondary branches, and means ± SD are shown in Fig. 1B. The variation in wall thickness from the thoracic aorta to the secondary branches was statistically significant (P < 0.001). Figure 2A shows opening angle data grouped into thoracic and abdominal aorta and branches of the aorta. The opening angle is smaller for thoracic segments than for the abdominal aorta and branches (P < 0.001), which are not statistically different from each other (P = 0.766). Figure 2B shows the computed relation between the midwall stretch ratio and diameter in the loaded state for the aorta and its branches. Partition of data into thoracic and abdominal aorta and secondary branches is shown in Fig. 2C. The midwall stretch ratio is significantly less in the thoracic region (P < 0.001). The stretch ratio in the abdominal aorta and branches is not statistically different (P = 0.437). Similarly, data for the mean hoop stress are shown in Fig. 3A. The variation of mean wall stress, from the thoracic aorta to the branches, is statistically significant (Fig. 3B; P < 0.01).
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We found that the reproducibility of the elastomer preparations was superior to that of gelatin preparations. The elastomer was more firm and did not require cooling of the heart. Hence, the data are entirely from the elastomer preparations and exclude the gelatin data, even though we found no statistically significant differences between the elastomer and gelatin heart preparations with respect to the parameters of interest.
Table 1 shows a summary of morphometric (order number, diameter, and wall thickness) and mechanical (stress and strain) data. The relation between vessel wall thickness and diameter is shown in Fig. 4A. Data were fitted with an exponential curve. Diameters of vessels of various sizes were grouped according to order number and are shown in a semilogarithmic plot in Fig. 4B. The variation of wall thickness with order number is a geometric sequence that can be fitted by an exponential relation. Figure 4C shows the relation between wall thickness-to-radius ratio and order number. The increase in the wall thickness-to-radius ratio toward the smaller arteries is statistically significant (P < 0.01). There is a fourfold increase in the wall thickness-to-radius ratio from the proximal LAD artery to the smallest arteriole. Hence, the vessel lumen radius decreases faster than the wall thickness when the vessel diameter is <400 µm (order 7). The zero-stress state may be characterized by the opening angle, which varies with order number (Fig. 5A). A linear least-square fit was used to describe the data over the entire range of arterial orders. The slope of the line is 10.1°/order.
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50% larger than the strain of the smallest arteriole (10 µm). Similarly, the relation between mean Cauchy stress and diameter is shown in Fig. 7A. The decrease in mean hoop stress with a decrease in order number was statistically significant (P < 0.001; Fig. 7B). There is an 15-fold decrease in hoop stress from the LAD artery to the smallest arteriole.
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DISCUSSION |
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The cardiovascular system is constantly loaded by blood pressure and blood flow, which induce circumferential and longitudinal stresses and strains in the vessel wall and shear stress on the glycocalyx of endothelial cells, respectively. It is well accepted that a "homeostatic" state of stress exists in the cardiovascular system. Such a homeostatic state has been well established in relation to blood flow and wall shear stress. Kamiya et al. (13) collected diameter and velocity data from the literature and estimated the shear stress in the arterial system. They found that, despite a difference of 8 orders of magnitude in flow rate in the arterial tree (aorta to precapillary arterioles), the shear stress varies by a factor of 2 (10–20 dyn/cm2). The uniformity of wall shear stress is restored, even when blood flow is perturbed, such as in flow overload (14) or flow reduction (19). In summary, it has been observed that the vessel will remodel its luminal diameter so as to return the wall shear stress to the homeostatic value (for review, see Ref. 20). The mechanotransduction responsible for this regulation has been a subject of great interest (for review, see Ref. 3).
The goal of the present study was to search for similar principles of homeostasis with respect to circumferential stress or strain, which are the major mechanical consequences of blood pressure. We found the strain to vary within a relatively narrow range (1.2–1.6; Figs. 2 and 6), whereas the stress varies more widely (10–150 kPa; Figs. 3 and 7) over the entire aorta and coronary arterial tree (difference of >3 orders of magnitude in vessel diameters). The Green strain (
), which is commonly used to describe large deformation [ = 0.5(2 – 1)], varies from 0.16 to 0.75, which is still significantly smaller than the variation in stress. The uniformity of strain in the arterial system is intriguing.
Uniform Circumferential Stress Hypothesis
The blood pressure is primarily opposed by the forces of elastin, collagen, and smooth muscle cells that are oriented to form well-defined layers (for review, see Ref. 24). Thick elastin bands form concentric lamellae, whereas finer elastin fibers form networks between lamellae. The collagen fibers are distributed circumferentially in the interstices. In a comparative study of aorta from various species ranging from mouse to pig, Wolinsky and Glagov (29) found that the total number of medial lamellar units is proportional to the aortic diameter. Hence, despite a large variation in aortic diameter, the average tension per lamellar unit of an aortic media was fairly constant.
The concept of wall stress, i.e., tension ÷ wall thickness, is theoretically more appropriate than tension for a thick-walled vessel. Hence, the "uniform tension" hypothesis was generalized into a "uniform stress" hypothesis as the principle that dictates the remodeling of arteries in hypertension (21, 26–28). It has been observed that the wall thickness-to-radius ratio increases in proportion to the increase in pressure, such that the circumferential wall stress is restored after some period of growth and remodeling. Hence, this hypothesis presupposes a homeostatic state of stress.
It is well known that the material properties and, hence, constitution of the vessel wall change longitudinally along the arterial tree (24). The larger arteries consist of intima, media, and adventitia; the smaller arteries and arterioles consist primarily of intima and media. Hence, the proportion of collagen and elastin is greater in the larger arteries, and the proportion of smooth muscle cells is greater in the smaller arteries. Consequently, the elastin and collagen fibers can bear the stress in the larger arteries, whereas the smooth muscle cells have to bear a greater fraction of the stress in the periphery. This might explain the longitudinal variation in mean stress along the arterial tree.
Uniform Circumferential Strain Hypothesis
Our group has recently examined the uniform stress hypothesis in a flow-overload model of an arteriovenous fistula (20) and a pressure-overload model of common bile duct after obstruction (4, 23). The time course of change of stress and strain was determined over a 12-wk period in the arteriovenous fistula and 32-day period in the obstructed common bile duct. Indeed, we found that the remodeling principle is consistent with normalization of circumferential stress in the vessel wall. However, we found that strain reached its peak sooner and normalized faster than stress. Hence, we concluded that the vessel seems more "sensitive" to changes in strain. This underscores the existence of a homeostatic state of strain, which has not been previously reported. Our current finding that strain varies by a maximum of 50% throughout the coronary arterial tree and aorta suggests a possible mechanism of strain regulation throughout the arterial system.
In the heart, Grossman (9) proposed that the increase in myocardial wall thickness normalizes the systolic wall stress in pressure overload. Nguyen et al. (22) proposed that end-diastolic, rather than end-systolic, stress is normalized in pressure-overload hypertrophy and, hence, is the more likely stimulus. Similarly, end-diastolic stress was proposed to stimulate growth in volume-overload hypertrophy (6, 9). In those studies, however, strain was not measured. Recently, Emery and Omens (5) measured strain in their arteriovenous fistula rat model and reported that midwall strains at end-diastolic pressure were normalized, but ventricular wall stresses remained substantially elevated in conjunction with an order-of-magnitude increase in stiffness 6 wk after volume hypertrophy. Hence, they suggested end-diastolic fiber strain as the likely stimulus for remodeling.
Finite-element models of the heart have shown substantial regional heterogeneity of ventricular mechanics, even under normal conditions (2), with the notable exception of the fiber strain distribution, which is remarkably uniform, despite significant variations in fiber stress and the strains in other directions. One mechanism of this fiber strain uniformity is the torsional deformation that results from the helical fiber orientations and the anisotropy of the myocardium (10). The homeostasis of strain in the heart and blood vessels is remarkable.
Is Circumferential Stress or Strain the Stimulus for Mechanotransduction?
It has not been possible to establish whether blood vessels respond to stress or strain, because it is difficult to experimentally separate the two mechanical stimuli. Stress and strain are intimately related through the material properties of the vessel wall (constitutive relation); hence, it is difficult to change one without affecting the other. We postulate that strain or, possibly, strain rate is the stimulus for mechanotransduction on the basis of the premise that forces transmitted via individual proteins cause conformational changes that alter their binding affinity to other intracellular molecules. The force transmission may occur at the site of cell adhesion or within the stress-bearing members of the cytoskeleton. The altered equilibrium state can subsequently initiate a biochemical signaling cascade or produce a local structural change. Because conformational change of a molecule or enzyme is inherently related to deformation, it is likely that chemical kinetics are affected by deformation. Stress, on the other hand, is a driving force for transport processes.
There are significant data in the literature that support the existence of stretch- or deformation-activated ion channels in the vascular system (for review, see Ref. 25). Experiments that use micropipettes to stretch an endothelial cell membrane show that the degree of stretch or deformation is related to the opening of transmembrane cation channels. The major effect of the activation of these mechanosensitive ion channels is the influx of Ca2+ and, hence, the depolarization of the cell.
Comparison with Other Studies
We previously measured the opening angles of the five largest orders (orders 6–11) of the coronary arterial tree and found a linear relation similar to that shown in Fig. 5. The slope from our previous publication (7) was 9.7°/order (orders 6–11) compared with 10.2°/order (orders 1–11) in the present study. The relation remains linear down to order 1 arterioles. Hence, even a 10-µm-diameter vessel has residual stress and strain. No previous studies have reported opening angles for vessels of these small dimensions. The smallest vessels previously investigated were 50 µm diameter in the mesentery with an opening angle of 100° (8), which is similar to our order 5 vessels. Hence, the residual stress-and-strain is a ubiquitous phenomenon that exists throughout the coronary arterial system.
Han and Fung (12) measured the distribution of opening angle in the pig aorta. They found that the opening angle was 60° in the thoracic aorta and varied from 60° to 80° in the abdominal region. In the present study, the mean opening angle was 52.5° and 84.5° in the thoracic and abdominal aorta, respectively.
The stress and strain distribution in the aorta of the mouse was previously investigated (11). We found the mean values (averaged over the length of the aorta) of Cauchy stress and Green strain to be 155 kPa and 0.48, respectively, compared with 97 kPa and 0.39 in the present study. Hence, the mouse appears to bear higher mean stresses and strains along its aorta.
Critique of Methods
Silicone elastomer was used to distend the vessels at physiological pressure to compute the stress and strain at a loaded condition. Elastomer is an inert material mixed with catalyst (7% ethyl silicate and 3.5% tin octate) to induce polymerization or hardening of the material. We evaluated the effect of the elastomer solution by comparing the opening angles and morphometry with hearts prepared with gelatin. Gelatin consists of a protein matrix that is dissolved in water by heating and solidifies on cooling. We found no significant differences in any of the parameters investigated.
Stress and strain are mathematical concepts based on the continuum model developed to describe the force intensity (force per area) and the deformation (change of length) in the vessel wall, respectively. As such, there are many ways to define stress and strain. In the present study, we chose Cauchy's stress or true stress, which refers to the force per deformed area, and stretch ratio or Green's strain, which refers to the zero-stress state. However, these are simply mathematical concepts that are intended to describe a physical quantity. Hence, biological cells may not "sense" strain or stress directly but, rather, may be stimulated by more fundamental quantities, such as conformational changes or molecular forces. However, because it is difficult to measure the latter quantities, the measurements of stress and strain as quantities averaged over a continuum are very useful.
In vivo, the stress and strain can be modulated by vasoactivity of smooth muscle cells. Because the present data were obtained in dilatation, they represent the largest values of stress and strain. In the small arteries and arterioles, smooth muscle contraction can significantly reduce wall strain and stress. The effect on stress is more profound, however, because the radius is decreased, whereas the wall thickness is increased. Hence, although the strain may have a larger variation in vivo, i.e., smaller vessels would have smaller strains than depicted by our data, its relative variation will still be significantly smaller than the variation of stress. Furthermore, the pressure is pulsatile in vivo; hence, the stress and strain vary throughout the cardiac cycle. In the present study, we present the stress and strain values at the mean pressure. Hence, our conclusions of strain homeostasis are based on the mean values. It may be, however, that the pulsatile features are important and that the amplitude and rate of deformation have physiological significance.
The longitudinal pressure distribution given by Eq. 3 was obtained from analysis of blood flow based on viscosity of blood (15). Because silicone elastomer is much more viscous than blood (18), the pressure distribution may be different. To assess the effect of pressure distribution on the conclusions made in the present study, we examined three hearts perfused with elastomer and Cab-O-Sil, such that the pressure was uniform (100 mmHg) throughout the coronary arterial tree. Although the values of mean stress were higher in the smaller vessels because of higher pressures, the values of mean strain were not statistically significantly different from those in hearts that were perfused with elastomer alone. This is a reasonable conclusion, because it has been shown that the compliance of the coronary vessels is relatively small; hence, the diameters or circumferences in Eq. 1 do not change significantly (for review see Ref. 17).
Significance of the Study
The data on the zero-stress state, wall thickness, stress, and strain presented here characterize the mechanical status of the aorta and coronary arterial tree. The data are important for analysis of wave reflection, mass transport, and elasticity of the blood vessel wall. Furthermore, the data will serve as a reference state for future studies of growth and remodeling in response to changes in mechanical stimuli (e.g., hypertension or flow overload).
The major finding of the present study is that the mean wall stress varies by a factor of 15 (10–150 kPa) over vessel dimensions of 10–18,000 µm (the range of the arterial tree), and the stretch ratio and Green strain variations are 1.2–1.6 and 0.16–0.75, respectively. Although these results cannot determine which mechanical factor is regulated in the arterial wall, we hypothesize that strain is an important mechanosensor involved in vascular regulation because of the relatively small variation in strain compared with stress. The molecular mechanism responsible for the transduction of physical stimulus into a biochemical event is one of the most exciting areas of biomechanics research.
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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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