INTRODUCTION

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THE ZERO-STRESS STATE of a blood vessel is the state at which the vessel is stress-free everywhere. Before 1983 it had been assumed that blood vessels were in their zero-stress state when free of external loads, such as at zero blood pressure and zero axial tethering forces. Calculations based on this assumption showed that, under physiological conditions, the blood creates a tensile circumferential stress, which varies from a maximum at the inside of the vessel wall to a minimum at its outer margin. Stress values have been calculated as being up to seven times greater at the inner wall compared with average values (1, 20). Since Vaishnav and Vossoughi (20) and Fung (3) found that arterial segments sprang open into sectors when cut radially, the existence of residual stress and strain in the blood vessel wall has been recognized. Fung and Liu (5, 16) have shown that the sectors are in a state of zero stress to the first order of infinitesimals. To verify that the stress throughout the opened sector was zero, Han and Fung (10) made further arbitrary cuts and showed that no subsequent changes in strain resulted. The difference between stress at the state in which the internal pressure, external pressure, and longitudinal stress are all zero and that at the zero-stress state is the residual stress. Calculations that take the observed residual stress into account show that its presence makes the transmural stress distribution more uniform at the in vivo state and that the stress gradient that exists from the inside to the outside of the vessel at the in vivo state is attenuated by the residual stress (1, 20).

It is clinically important to better understand how the structure and mechanical properties of pulmonary blood vessels change under physical stress such as hypertension and how these changes affect the pressure and flow in the lung. The zero-stress state is the best state in which to study tissue remodeling, because at that state any change in structure is exhibited without deformation (5). Changes in the zero-stress state of the tissue can be observed in association with material and structural remodeling. Therefore, observation of changes in zero-stress state provides the simplest method to demonstrate the remodeling of the vessel wall.

The zero-stress state of a vessel can be characterized by its opening angle (Fig. 1). Current literature contains data on the opening angles of pulmonary arteries in normal, hypertensive, and diabetic rats (6, 17) and of pulmonary arteries in canines (2). To our knowledge, however, no investigation of opening angles within the human pulmonary vasculature has been reported. The objective of this study is to investigate the opening angles of human pulmonary arteries and veins.

View larger version (19K): [in this window] [in a new window]   Fig. 1.   Definition of opening angle (). Sector represents circumferential cross section of a blood vessel at zero-stress state. Opening angle is an angle subtended between 2 lines originating from midpoint to tips of inner wall.

	MATERIALS AND METHODS

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Human lungs from five men and one woman were collected from subjects within 15 h postmortem (Table 1). The subjects' age ranged between 25 and 37 yr. Six lungs were obtained from individuals who died from accidental causes. In these individuals, neither lung injury nor gross evidence of lung pathology was observed at the time of autopsy, nor was there any reason to suspect lung pathology as a cause of death. The blood vessels along the pulmonary arterial tree and venous tree were dissected while the lung floated in cold normal saline solution, then were transferred into dishes containing normal saline solution at room temperature (20°C). No significant difference was found between the opening angles of pulmonary arterial segments in aerated Krebs and normal saline solutions (6). The small vessels were dissected down to ~50-µm lumen diameter with the aid of a stereomicroscope. The blood vessels were cut transversely into short ring-shaped segments. The segments were then cut radially, causing them to spring open into sectors. All radial cuts were made on the anterior line. Segments containing branches were excluded.

Changes of the opening angle with time were observed within 100 min of the radial cut of the ring-shaped segment. Photographs of cross-sectional views of the segments were taken from 1 to 120 min after the segments were cut radially. The opening angles were measured from the photographs. The photographs were displayed on the video monitor through a TV camera (Cohu solid-state camera). The image was analyzed with the Optimas software package (BioScan). A digital image of each ring segment at no-load state was captured. The cross-sectional area of the ring along the inner wall was measured and used to calculate the lumen diameter. The cross-sectional area (A) of wall and middle arc length (l_m) were determined. The radius of the intact ring was taken as l_m/2, and the mean wall thickness was calculated as A/l_m. The opening angle was defined as the angle subtended by two radii drawn from the midpoint of the inner wall to the tips of the inner wall of the open sector, as shown in Fig. 1. The mean and SD of the opening angles of the pulmonary arterial and venous segments from each order were computed and used to describe the zero-stress state in the corresponding order. The circumferential lengths along the inner and outer walls at the state in which the internal pressure, external pressure, and longitudinal stress are all zero and at the zero-stress state were measured for the computation of residual strain.

In this paper, each pulmonary vessel was assigned an order number in accordance with its diameter under the rules of the diameter-defined Strahler method. A brief review of the present understanding of the morphometry of pulmonary vasculature is given in Huang et al. (12). Weibel (22) initiated precise morphometry and used a bifurcation model. Horsfield and Gorden (11) used Strahler's method. Kassab et al. (15) introduced the diameter-defined Strahler's method. The differences between results in Refs. 11 and 15 have been discussed by Jiang et al. (14) and Huang et al. (12). The rule of assigning the order number in Refs. 12, 14, and 15 differs from that of Ref. 11 by an additional criterion that when two vessels of order n meet, the confluent vessel's order number is increased by one if the diameter of the confluent is larger than those of order n by a certain amount. That amount is based on the rule that the interval between the diameters of vessels of order n and n + 1 is split between n and n + 1 in proportion to the relative sizes of the SDs of the order n and n + 1 vessels. Thus, if the mean and SD of the diameters of the pulmonary blood vessels of order n are denoted by D_n and SD_n, respectively, the diameter range of the pulmonary arteries of order n is between

(1a)

and

(1b)

For the human lung, the data on the diameter range of successive order have been reviewed by Huang et al. (12), and the result is given in Table 2. For example, the diameter range of the pulmonary arteries of order 12 is between D_12,left = [(D₁₁ + SD₁₁) + (D₁₂  SD₁₂)]/2 = 2.151 mm, and D_12,right = [(D₁₂ + SD₁₂) + (D₁₂₊₁  SD₁₂₊₁)]/2 = 3.310 mm. Similarly, a vein of order n is bounded by

(2a)

and

(2b)

Therefore, the diameter range of the pulmonary veins of order 12 is between D_12,left = [(D₁₁ + SD₁₁) + (D₁₂  SD₁₂)]/2 = 3.381 mm, and D_12,right = [(D₁₂ + SD₁₂) + (D₁₃  SD₁₃)]/2 = 4.905 mm.

	RESULTS

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Table 2 shows the diameter ranges for each order of human pulmonary arteries and veins. The diameter ranges were calculated based on the mean and SD of the diameters from Ref. 12. In our previous study (12), the morphometric data on the branching pattern and vascular geometry of the pulmonary arterial and venous trees were collected from two silicone elastomer casts of human lungs (a left lung from a 44-yr-old man and a right lung from a 24-yr-old man). A total of 15 orders of arteries were found between the main pulmonary artery and the capillaries, and a total of 15 orders of veins were found between the capillaries and the left atrium in the human lungs. The pulmonary vessels were assigned order numbers according to the diameter-defined Strahler's method. Taking into account the number of vessels measured, we computed the SE of the mean diameter for each order as shown in Table 2.

The material of the arterial and venous wall is viscoelastic, as all biological materials are, to some extent (3, 4). Hence, on a sudden removal of the residual stress in the vessel wall to zero by cutting, the following deformation takes place (5, 6, 23): a quick elastic springing apart, followed by a creep deformation, which is revealed as a continuous change of the opening angle. Figure 2 shows the change of opening angle of human pulmonary arteries and veins (35-yr-old man, left lung; 26-yr-old man, right lung) at 1, 5, 10, 20, 60, and 120 min. Figure 3 shows the change of opening angle of human pulmonary arteries and veins (37-yr-old man, right lung; 25-yr-old man, right lung) at 1, 5, 10, and 20 min. Records of other specimens are similar. The curves in Figs. 2 and 3 are fitted with the formula

(3)

in which _o is the opening angle at steady state, t is time after cutting (in min), (t) is the opening angle at time t, c is a constant, and k is the rate (in min¹). For each experimental curve, the constants _o, c, and k can be determined by minimizing the least squares errors. The good fitting between Eq. 3 and the experimental data is illustrated in Figs. 2 and 3. The values of the material constants _o, c, and k, determined by the least squares method from curves shown in Figs. 2 and 3 and from other specimens, are listed in Table 3. One can see that all the curves are well fitted by Eq. 3. The goodness of fit is characterized by the correlation coefficient (r), and the SE of estimate of the opening angle at various instants of time (s). The values of r and s are listed in Table 3.

View larger version (35K): [in this window] [in a new window]   Fig. 2.   Course of change of opening angle of human pulmonary arteries and veins (35-yr-old man, left lung; 26-yr-old man, right lung) at 1, 5, 10, 20, 60, and 120 min. Curves are fitted with Eq. 3. Mathematical values are listed in Table 3.

View larger version (49K): [in this window] [in a new window]   Fig. 3.   Course of change of opening angle of human pulmonary arteries and veins (37-yr-old man, right lung; 25-yr-old man, right lung) at 1, 5, 10, and 20 min. Curves are fitted with Eq. 3. Mathematical values are listed in Table 3.

In all cases, the deformation is essentially completed at 20 min after cutting. The corresponding equilibrium (steady state) opening angles are 92.7 ± 3.2 (SD)°, 119.0 ± 33.2°, 134.6 ± 32.3°, 129.3 ± 21.3°, 109.9 ± 16.2°, 149.8 ± 23.9°, and 163.3 ± 16.2° for orders 3, 4, 5, 6, 7, 8, and 9 of the pulmonary arteries, respectively; for orders 3, 4, 5, 6, 7, 8, and 9 of the pulmonary veins, the opening angle values are 93.8 ± 33.2°, 89.9 ± 43.6°, 101.7 ± 37.3°, 93.6 ± 27.8°, 121.8 ± 30.4°, 123.0 ± 13.1°, and 127.8 ± 7.7°, respectively. The opening angle varies with the location on the pulmonary vascular tree. There is a tendency for opening angles to increase as the sizes of the arteries and veins increase. A statistical method, one-way analysis of variance, used for multiple comparisons, was employed to determine the significance of differences in the opening angles for different orders (25). A difference was considered statistically significant at P < 0.05. For the pulmonary arteries (orders 3-9), the differences in the opening angles for different orders were statistically significant (P < 0.05); but the differences in the opening angles of veins (orders 3-9) for different orders were not significant (P > 0.05). The main pulmonary artery at the arch region has the largest opening angle in the pulmonary arterial tree (6). We were unable to collect the human main pulmonary arteries and other large pulmonary arteries and veins in this study, and hence the data for the zero-stress states for both arteries and veins are presented from order 9 to order 3.

A comparison of the equilibrium opening angles (i.e., 20 min after radially cutting) between humans (our data) and rats (6) is presented in Fig. 4. Fung and Liu (6) divided the main pulmonary arterial tree, from the pulmonary valves to vessels with lumen diameter ~60 µm, into eight "regions." Each region was defined as a segment of the artery between two consecutive lateral side branches. Fung and Liu (6) presented the opening angles of pulmonary arteries at 20 min in the eight regions. The correlated order number of each region can be found in Ref. 14. Comparing Refs. 6 and 14, we found that regions 3, 4, 5, 6, 7, and 8 are orders 10, 9, 8, 7, 6, and 5, respectively. In Fig. 4, each point is presented by the mean value of opening angles. A flag at each point signifies the SD, drawn upward for the arteries (* next to the error bar for the rats) and downward for the veins. The differences between the opening angles of small arteries measured in the humans and the rats are small; however, big differences are found in the large arteries. We do not know the causes of these differences. The differences in vessel geometry and morphology may contribute to the differences of opening angles among different species.

View larger version (20K): [in this window] [in a new window]   Fig. 4.   Comparison of opening angles at 20 min after being cut on human pulmonary arteries and veins (our study) and rat pulmonary arteries (6). Each point is presented by mean value of opening angles. A flag at each point signifies SD, drawn upward for the arteries (* next to error bar for rats) and downward for the veins.

	DISCUSSION

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Our study shows that the change in opening angle with time after the cut is a viscoelastic property, which has been suggested by many authors (5, 6, 13, 21). The time constants (1/k) data in our study as shown in Table 3 are consistent with previous reports of vascular creep responses (5, 6, 13, 21). The opening angles of human pulmonary arteries and veins vary with the location, the vessel size, and curvature of the vessel centerline. The large SDs of the opening angles at given localities and the large regional variations are linked to the nonuniformity of structure (6) and nonuniform remodeling under stress. However, the variation with circumferential location of cutting was statistically insignificant for the pulmonary artery of the rat (6).

Postmortem changes in elasticity (8) are thought to be due to a gradual loss of vasomotor tone. Vaishnav and Vossoughi (21) have found a small but insignificant decrease in the residual strain of bovine and porcine aortas after cold storage for between 1 and 12 days. Sugihara and co-workers (19) found no difference between human autopsy material up to 36 h postmortem and surgical specimens in tension-length studies of parenchymal strips of human lungs. Saini et al. (18) used the aortas of rats and humans to study the effect of time after death on opening angle. They found there was a significant fall during the first 24 h (~10%) and no significant change within the next 48 h. Also, they saw a small but insignificant decrease in the opening angle of human aorta between 24 and 144 h. However, the mechanical and biological properties of the aorta and pulmonary vessels may be different; we cannot make a similar assumption on pulmonary vessels based on the data from the aorta. In our study, all human lungs were collected around 15 h postmortem. We were unable to collect the fresh lung specimens in a short time after death. Therefore, we cannot examine the effect of time after death on opening angle of pulmonary blood vessels.

The strains at the state in which the internal pressure, external pressure, and longitudinal stress are free are called residual strains. From the difference of the vessel geometry between the zero-stress state and the no-loaded state, we can compute the residual strains (the simplest cases being illustrated in Ref. 3). There are many strain measures. For constitutive equations of soft tissue involving finite strain (see chapter 10 in Ref. 4), Green's strain is used (1, 2, 4, 7, 23). In circumferential direction, Green's strain is related to the circumferential stretch ratio by the equation: strain = (stretch ratio²  1)/2. The circumferential stretch ratio for the inner wall is the ratio of circumferential length of inner wall at no-load state and circumferential length of inner wall at zero-stress state; similarly, the circumferential stretch ratio for the outer wall is the ratio of circumferential length of the outer wall at no-load state and circumferential length of the outer wall at zero-stress state. Hence, the deformation of pulmonary arteries and veins can be analyzed either in terms of strains or in terms of stretch ratios. We have listed both stretch ratio and strain in Table 4. However, it was not possible to compute the homeostatic strains, since we do not have the circumferential lengths at the homeostatic condition.

The residual stress distribution in the pulmonary arterial and venous vessels can be estimated on the basis of the constitutive equation of the wall material. Debes and Fung (2) have found that the stress-strain relationship of the pulmonary arteries in dog is linear in the physiological range and beyond. Yen et al. (24) showed that the pressure-diameter relationship of pulmonary arteries and veins in humans is linear up to 50 cmH₂O. Hence, Hooke's law  = Ee (where  denotes the stress, e the strain, and E Young's modulus) seems justified. The residual stress is obtained by the multiplication of the residual strains by Young's modulus E. Greenfield and Griggs (9) gave E of human pulmonary artery as 2.6 × 10⁶ dyn/cm². For pulmonary veins, E is not available. If we assume that the same E applied to both pulmonary arteries and veins, then the estimated residual stresses for pulmonary arteries and veins are presented in Table 4. For all orders of pulmonary arteries and veins, the inner wall of the vessel is under compression, whereas the outer wall is in tension. In general, the magnitude of compressive stress was greater than the magnitude of tensile stress. For the arteries, the compression lies between 0.341 × 10⁶ and 0.876 × 10⁶ dyn/cm², and the tension lies between 0.216 × 10⁶ and 0.401 × 10⁶ dyn/cm². For the veins, the compression is between 0.402 × 10⁶ and 0.824 × 10⁶ dyn/cm², and the tension is between 0.087 × 10⁶ and 0.404 × 10⁶ dyn/cm². However, these values were calculated under the assumption that the vessel was homogeneous. A recent study by Xie et al. (23) found that Young's modulus of the inner layer (intima and media) in rat's ascending aorta is four times bigger than that of the outer layer (adventitia). To determine the stress-strain relationship of the inner (intima and media) and outer (adventitia) layers of blood vessels in the neighborhood of the zero-stress state, they performed the bending experiments on aortic strips of rats. The bending experiment provides a method to investigate the mechanical properties of different layers of blood vessels without physically separating the layers. Human pulmonary artery has not been analyzed as bilayered so far. If a similar ratio of four were assumed, then Young's moduli of the inner layer and outer layer in human pulmonary artery might be ~4.16 × 10⁶ dyn/cm² and 1.04 × 10⁶ dyn/cm², respectively. Then the compressive residual stress at the inner wall will be 60% larger than the values listed in Table 4, whereas the tensile residual stress in the outer wall will be smaller by a factor of 2.5.

To compute the stress and strain in a tissue in vivo, one must know the zero-stress state. Data on opening angles at the zero-stress state have clinical relevance because they change by diseases, e.g., hypoxic hypertension (6) and diabetes (17). When the physiological set point changes, the residual stresses will change. Information on these changes will be helpful in evaluating the relationship between stress and growth in living organs. Fung (4) has proposed a "stress growth law," which suggests that nonuniform distribution of stress across a vessel wall in vivo is responsible for a remodeling process, which tends to reduce this nonuniformity.

In conclusion, this paper presents the measurements of the opening angle of the zero-stress state in normal human pulmonary arteries and veins. The tissues were collected and studied at ~15 h postmortem. The zero-stress configuration of arteries and veins varied with the location on the pulmonary vascular tree. For the pulmonary arteries (orders 9-3) the differences in the opening angles for different orders were statistically significant; but the differences in the opening angles of veins for different orders were not significant. Based on the experimental measurements, we obtained the residual strains for pulmonary arteries and veins and estimated the residual stresses according to Hooke's law. We found that the inner wall of vessel at no-load state was under compression and the outer wall was in tension and that the magnitude of compressive stress was greater than the magnitude of tensile stress.