## Abstract

The transmural distributions of stress and strain at the in vivo state have important implications for the physiology and pathology of the vessel wall. The uniform transmural strain hypothesis was proposed by Takamyzawa and Hayashi (Takamizawa K and Hayashi K. *J Biomech* 20: 7–17, 1987; *Biorheology* 25: 555–565, 1988) as describing the state of arteries in vivo. From this hypothesis, they derived the residual stress and strain at the no-load condition and the opening angle at the zero-stress state. However, the experimental evidence cited by Takamyzawa and Hayashi (*J Biomech* 20: 7–17, 1987; and *Biorheology* 25: 555–565, 1988) to support this hypothesis was limited to arteries whose opening angles (θ) are <180°. It is well known, however, that θ > 180° do exist in the cardiovascular system. Our hypothesis is that the transmural strain distribution cannot be uniform when θ is >180°. We present both theoretical and experimental evidence for this hypothesis. Theoretically, we show that the circumferential stretch ratio cannot physically be uniform across the vessel wall when θ exceeds 180° and the deviation from uniformity will increase with an increase in θ beyond 180°. Experimentally, we present data on the transmural strain distribution in segments of the porcine aorta and coronary arterial tree. Our data validate the theoretical prediction that the outer strain will exceed the inner strain when θ > 180°. This is the converse of the gradient observed when the residual strain is not taken into account. Although the strain distribution may not be uniform when θ exceeds 180°, the uniformity of stress distribution is still possible because of the composite nature of the blood vessel wall, i.e., the intima-medial layer is stiffer than the adventitial layer. Hence, the larger strain at the adventitia can result in a smaller stress because the adventitia is softer at physiological loading.

- opening angle
- aorta
- coronary arteries
- stress

the recognition of circumferential residual strain by Fung (5) and Vaishnav and Vossoughi (30) has placed vascular biomechanics on firmer grounds. Previous to what has now become axiomatic, it was thought that the circumferential strain (and consequently stress) are greatest at the inner wall and decrease towards the outer wall at the in vivo state. In a computational study, Chuong and Fung (3) showed that the existence of circumferential residual strain reduces the transmural gradients of stress and strain, i.e., the inner (intima) and outer (adventitia) circumferential stresses and strains are similar at the in vivo state. Their computational approach was based on the existence of a strain energy density function whose constants were determined experimentally. The stresses and strains used in the strain energy function were based on the zero-stress state, which was characterized by an opening angle (θ; defined as the angle subtended by two radii connecting the midpoint of the inner wall). Takamizawa and Hayashi (28, 29) solved the inverse problem, i.e., they showed that under the uniform strain hypothesis, the thin-wall theory can be used to predict the material constants in the strain energy density function.

The first direct experimental evidence for the uniform transmural strain hypothesis at the in vivo state was provided by Fung and Liu (10) on small vessels where they measured the circumferences in the loaded and zero-stress state and computed the corresponding strains at the inner and outer wall. In both the computational and experimental studies, the vessels studied had θ < 180°. Although the majority of vessels fall into this category, there are regions of the rat and human aorta, rat pulmonary artery, porcine coronary artery, and rat ileal arterioles that have θ > 180° (4, 6, 9, 10, 18, 27). Furthermore, θ is known to increase beyond 180° in hypertension-, cigarette smoke-, and diabetes-induced remodeling (8, 9, 11, 13). Finally, other tubular organs such as the dog trachea and guinea pig small intestines are known to have θ well in excess of 180° (14, 16).

The objective of the present study is to examine the validity of the uniform transmural strain hypothesis at the in vivo state along the aorta and coronary arterial tree. Our hypothesis is that the uniform transmural strain hypothesis cannot apply to those vessels that turn inside out (θ > 180°). In those cases, we will show that the loaded circumferential strain on the inner wall will become smaller than that at the outer wall, which is the converse of the case where the residual strain is ignored. The mechanical and physiological implication of these observations will be discussed.

## METHODS

### Existing Data

The method of animal preparation and the measurement of strain in the aorta and coronary arterial tree have been recently described in detail (15). In that study, the distribution of mean stress and strain throughout the porcine coronary arterial tree and in the aorta and its branches were presented. The same data were used in the present study to determine the inner and outer strain, and the discussion herein is focused on the transmural variation of loaded strain. In total, 572 aortic (186 thoracic and 136 abdominal rings) and secondary arterial branches (250 rings from iliac, femoral, and other branches) were analyzed. Additionally, 387 vessel rings from the coronary arterial tree were considered with diameters >50 μm.

### Aorta and its Primary Branches

The surgical preparations of the 10 pigs used for the aorta have been described in Guo and Kassab (15). Briefly, the ascending aorta was cannulated and perfused with catalyzed silicone elastomer at 100 mmHg after the animal was euthanized (18). The aorta, with the solidified elastomer in its lumen, was dissected and cut transversely into rings. 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. This process causes the ring to open up into a sector. The cross section of each sector was photographed 60 min after the radial cut. The morphological measurement of inner and outer circumference, wall thickness, and θ in the loaded and zero-stress states were made from the images using an image-analysis system (SigmaScan). θ was defined as the angle subtended by two radii connecting the midpoint of the inner wall.

### Coronary Arterial Vessels

Ten hearts were obtained from a local slaughterhouse on the morning of the experiment (15). A cast of the left anterior descending (LAD) arterial tree at 100 mmHg was made with silicone elastomer (18), and the vessels were carefully dissected down to small branches with diameters of ∼50 μm. Each vessel segment along the LAD arterial tree was cut perpendicular to the longitudinal axis into rings. A photograph of every transverse section was taken, and the inner and outer dimensions (and hence wall thickness) were measured. The elastomer was then pushed out, and each ring was cut radially to obtain the zero-stress state, as shown in Fig. 1. The morphological data of the coronary vessels in the loaded and zero-stress states were obtained with the same method described for the aorta.

### Biomechanical Analysis

Let the circumference of a deformed vessel in the loaded state be designated by *C* and that of the undeformed vessel in the zero-stress state be designated by *C*^{zs}. Hence, the circumferential deformation of a cylindrical can be described by Green's strain (ε), which is defined as follows: (1) where λ_{i,o} = *C*_{i,o}/*C*and is the stretch ratio; *C*_{i,o} refers to the inner or outer circumference of the vessel in the loaded state, and *C*refers to the corresponding inner or outer circumference in the zero-stress state. To assess the degree of nonuniformity of transmural strain, we can evaluate the ratio of outer to inner strain as (2) Hence, the product of the first and second terms of *Eq. 2* gives the ratio of outer to inner Green strain. *Equation 2* can be simplified if we consider the quotient in terms of λ, as follows: (3) The first and second terms become linearized and are easier to interrupt physically, as discussed later.

Because the aorta and coronary arteries have different wall thicknesses, it is useful to define a normalized parameter for the transmural strain. As such, the strain distribution in the arterial wall can be characterized in analogy to a coefficient for stress distribution (*v*_{ε}) as proposed by Rachev et al. (25), namely, (4) This parameter will be used an index of nonuniformity of strain across the wall thickness.

#### Classification of aortic data.

The aorta was subdivided into thoracic (descending aorta, just below the arch, to the diaphragm), abdominal (from diaphragm to the common iliac artery), and secondary branches >1.5 mm in diameter (femoral, renal, etc.). The data were grouped together for the thoracic, abdominal, and secondary branches, respectively.

#### Ordering of coronary arterial branches.

We have previously developed and implemented an ordering system to classify various size vessels into orders based on a diameter-defined Strahler system (18). This has resulted in a unique relationship between diameter and order number for the entire coronary arterial tree. The relationship between the diameter range and order number obtained from the previous study is as follows: *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 relationship between inner and outer strains and diameter for coronary vessels >50 μm in diameter. Hence, by using the relation between diameter and order number from the previous study, we determined the relationship inner and outer strains and the order number.

### Statistical Analysis

A Student's *t*-test was used to compare inner with outer strain for coronary and aortic vessels. A *P* value of <0.05 was indicative of statistical significance.

## RESULTS

The variation of the first term on the righthand side of *Eq. 2* with θ is shown in Fig. 2*A*. It can be seen that this ratio is always greater than one for all values of θ. Figure 2*B* shows data for the second term on the righthand side of *Eq. 2*. When θ is equal to 180°, the inner and outer circumferences are equal and hence the ratio is one. When θ < 180°, the inner circumference is smaller than the outer circumference, and the ratio is <1. The converse is true when θ > 180°, as shown in Fig. 2*B*. The ratio of outer to inner strain, i.e., product of the two ratios shown in Fig. 2, *A* and *B*, is demonstrated in Fig. 2*C*. An interesting pattern is revealed where the nonuniformity of strain increases with an increase in θ, especially when θ > 180°. Figure 3 shows the corresponding data for the aorta and its primary branches. Because the wall thickness for the aorta is significantly larger than that for the coronary vessels, the ratio of inner to outer circumference was significantly smaller for the smaller θ. This gave rise to a greater nonuniformity of strain for the aorta, i.e., the ratio of outer to inner strain was smaller for the smaller angles (e.g., θ < 45°). It should be noted that the largest angle for the aortic vessels did not exceed 180°, whereas significantly smaller angles were found for the aorta compared with the coronary vessels.

To assess the nonuniformity of strain distribution for the coronary artery and aorta that have very different wall thicknesses, we defined a strain parameter as per *Eq. 4*. The data are shown for the coronary and aortic vessels in Fig. 4, *A* and *B*, respectively. It is apparent that the transmural strain is nonuniform for the coronary vessels when θ > 180° and for the aorta when θ < 45°.

The inner and outer strains are listed in Tables 1 and 2 for the coronary arterial tree and aorta, respectively. In Table 1, the data are classified according to order number and range of θ (in increments of 45°) for the coronary arterial tree. The outer strain is significantly larger than the inner strain for *orders 6–11* by 7–45%, respectively. When the outer strain was compared with respect to θ, it was significantly larger than the inner strain for θ > 135°. In Table 2, the data are classified for different segments of the aorta (thoracic, abdominal, and primary branches) and range of θ. The inner and outer strains are not statistically different for the abdominal aorta and branches. In the thoracic aorta, the inner strain is 27% larger than the outer strain. In relation to θ, the inner strain is larger than the outer strain for θ < 90°, whereas the converse is true for θ > 135°.

## DISCUSSION

### Circumferential Residual Strain

Before 1983, every study pointed to the existence of a stress concentration at the intima of the blood vessel and the subendocardium of ventricle, to the extent that the circumferential tension at the inner wall was much higher than that at the outer wall (2). The stress concentration implied high local energy consumption by the vessel or ventricle and consequently a high oxygen demand at the inner wall. The stress concentration at the inner wall was a direct consequence of the starting assumption that the unloaded (zero transmural pressure) blood vessel or ventricle is at the zero-stress state. Knowledge of the zero-stress state is of vital significance in mechanics because all calculations of stress and strain are made in reference to such state. Simultaneously and independently, Fung (5) and Vaishnav and Vossoughi (30) challenged the starting assumption that the unloaded blood vessel is at the zero-stress state. A radial cut of a vessel ring relieved the residual stress and strain and changed the no-load circular geometry into an open sector (5, 30). The open sector was quantified by θ. The recognition of residual stress and strain reduced the stress concentration problem and simplified the stress-strain relation because it referred to a well-defined state. Rachev and Greenwald (24) provide a thorough review of the literature on residual strain of blood vessels.

### Uniform Transmural Strain Hypothesis

The physiological implication of θ was investigated theoretically by Chuong and Fung (3) based on an experimentally determined constitutive equation, the geometry of the zero-stress sector, and a θ of 108.7°. They showed that the undesired stress concentration at the physiological loading conditions was significantly reduced (3). Fung and Liu (10) later reached a similar conclusion experimentally in regard to strain distribution in small arteries. The intima and adventitia strains were computed in reference to the zero-stress state in small arteries, and it was found that the transmural strain was uniform. Although θ was not specified, the calculations of inner and outer circumferences were based on vessels whose inner circumferences were smaller than the outer and suggested θ < 180°.

Fung and Liu (10) examined the transmural variation of stress and strain at the homeostatic condition of 120 mmHg in the saphenous artery in response to diabetes. The control vessel had θ < 180° but increased well above 180° during diabetogenesis. They showed that the transmural strain and stress were fairly uniform for the control vessel but were higher at the outer wall for the remodeled vessel with θ > 180°. These are similar observations to our normal vessels with θ > 180°.

In the present study, our contention is that the intimal strain cannot equal to the adventitial strain when θ > 180°. This point can be simply illustrated if we consider the deformation in terms of λ as given by *Eq. 3*. The first term, the ratio of outer to inner circumference in the loaded state, is physically always >1. The second term, the ratio of inner to outer circumference in the zero-stress state, is <1 if θ < 180° and >1 if θ > 180°. Hence, when θ < 180°, the product of the two terms (the first term is >1 and the second is <1) can be approximately equal to 1 and hence implies uniformity of strain. On the other hand, when θ > 180°, both terms are >1 and hence their product must further deviate from unity. The same argument applies to ε, as shown in Fig. 2, *A-C*, respectively. The degree of deviation from unity increases with an increase in θ. Hence, we find that theoretically and experimentally, the strain cannot be transmurally uniform when θ > 180°. We shall explore the experimental evidence in more detail below.

### Experimental Evidence for the Nonuniformity of Transmural Strain

Experimentally, the nonuniformity in strain occurs well below the 180° angle for the coronary arteries. Table 1 shows that the outer strain is greater than the inner strain for θ > 135°. This occurs because the second term in *Eq. 2* (Fig. 2*B*) is disproportionately smaller than 1 compared with the first term in *Eq. 2* being >1 (Fig. 2*A*). Hence, the product is smaller than 1 (Fig. 2*C*). Although the results for the aorta were similar to those of coronary arteries, there were some interesting differences. Similar to the coronary vessels, the first term was >1 (Fig. 3*A*), and the second term was <1 for θ < 180° (Fig. 3*B*). The ratio of outer to inner strain (product of Fig. 3, *A* and *B*) was smaller than 1 for θ < 90° and >1 for θ > 90°. Below 90°, the ratio was <1, implying that inner strain is greater than outer strain, whereas above 90°, the converse was true, as shown in Table 2. The differences between the aorta and coronary data are in part due to the thickness difference. Because the aorta is significantly thicker than the coronary arteries, the ratio of inner to outer circumference in the zero-stress state is reduced more than the increase in the ratio of outer to inner circumference in the loaded state and hence the product is <1. Furthermore, the θ for the aorta and branches reach smaller values than those of the coronary vessels, which amplifies the ratios described.

In addition to examining the difference between inner and outer strain, it is important to consider the gradient of strain, *v*_{ε}, which vanishes when the strain distribution is uniform across the arterial wall and approaches 1 as the circumferential strain at the outer wall becomes much larger than the strain at the inner wall. Conversely, the parameter will approach −1 as the inner strain becomes much larger than the outer strain. Despite the differences in wall thicknesses, we find that the nondimensionalized gradient parameter has similar values for the coronary arteries and aorta (Fig. 4, *A* and *B*). For the coronary vessels with θ > 180°, it is clear that the adventitial strain is significantly larger than the intimal strain.

### Differences Between Elastic and Muscular Arteries

Arteries are generally subdivided into two types: elastic (e.g., aorta, carotids, and pulmonary arterial vessels) and muscular (e.g., coronary, femoral, and cerebral arteries) (1). The wall structure of both types of arteries consists of intima, media, and adventitia. The aortic media contains many elastic laminae and relatively few smooth muscle cells. There is a gradual transition in structure and function between the elastic and muscular arteries. The amount of elastic tissue decreases as the vessels become smaller and the smooth muscle component becomes more prominent. Hence, the proportion of smooth muscle cells is much greater in the coronary arteries than in the aorta. Consequently, the strain in the coronary artery can be regulated by muscle contraction, which can affect the transmural deformation reported in the present study. Indeed, Matsumoto et al. (22) have shown that θ increases with contraction and decreases with relaxation in the rat thoracic aorta. Contrary to these findings, Zeller and Skalak (35) reported an increase in θ due to vasodilatation in small rat arteries. The discrepancy may be due to the difference in location of the neutral axis relative to the smooth muscle cells in large and small vessels. If the neutral axis is closer to the intima, smooth muscle contraction will increase θ. Conversely, if the neutral axis is closer to the adventitia, muscle contraction should decrease θ. Finally, if the neutral axis coincides with the smooth muscle layer, contraction will not affect θ. The location of the neutral axis is unknown for the coronary arteries and should be determined in future studies.

### Implications of Nonuniformity of Transmural Strain on Stress Distribution

Physiologically, it may not be disconcerting that the outer strain (at the adventitia) is larger than the inner strain (at the intima). Our group has recently determined the incremental moduli of intima-media and adventitial layers for the proximal LAD artery (20). We found that the modulus of the intimia-media is greater than that of the adventitial layer, which is true for other vessels as well (31, 33, 34). Hence, it may be advantageous that the outer strain is larger than the inner strain because of the composite nature of the vessel wall. For example, if the strain distribution was uniform, the stress would be significantly higher at the inner wall because the intima-medial layer is significantly stiffer than the adventitial layer under in vivo loading conditions. A larger strain at the adventitia can translate into a lower stress because of the smaller modulus.

We can demonstrate this quantitatively for the incremental, linear elasticity case where the circumferential stress (*S*_{11}) can be expressed as follows: (5a) and (5b) where *E*_{11} is the incremental Young's modulus in the circumferential direction and *E*_{12} denotes the cross-modulus (12). The superscripts im and a represent intima-media and adventitial layers, respectively. Lu et al. (20) obtained data from LAD arteries of 10 porcine hearts. In five hearts, the biaxial incremental moduli of intact wall and intima-media layer were measured, and those of the adventitia layers were computed. In five additional hearts, the biaxial incremental moduli of intact wall and adventitia were measured, and those of intima-media were computed. The mean circumferential modulus of 10 intima-media and adventitia layers (5 measured and 5 computed) were 259 (*E*) and 107 kPa (*E*), respectively (20). The mean cross-moduli for the intima-media and adventitia were 47.6 (*E*) and 104 kPa (E), respectively. The mean ε at in vivo loading for the LAD artery is 0.62 (inner) and 0.90 (outer), as listed in Table 1 for *order 11* vessels. The axial stretch ratio for the LAD artery is ∼1.4, which gives a ε of 0.48 for both the inner and outer layers. Given these parameters, we can compute the ratio of outer (*Eq. 5b*) to inner (*Eq. 5a*) mean circumferential stress, which yields a value of 0.80. Hence, although the ratio of outer to inner strain is 1.45, the stress gradient is significantly smaller. If the outer and inner strains were uniform at 0.62, similar calculations would yield a stress ratio of 0.63 and hence a less uniform transmural stress distribution.

### Critique of Method

We have previously observed θ in the proximal coronary arteries that exceed 180° (4, 17). In the present study, we obtained θ that were significantly larger than those of the previous studies. The reason for this discrepancy was that in the present study, the vessel rings were distended with elastomer to obtain strain at the loaded condition. The ring was cut open 1 or 2 min after the elastomer was pushed out of the vessel. In the previous study, the vessel was maintained at no-load state for longer duration before the radial cut. Hence, we found that the initial state of stress in the vessel wall and its time history affects θ. This reflects the viscoelastic properties of the vessel wall. These interesting features will be described in a future publication and are beyond the scope of the present study. The main issue here is that θ > 180° have been documented not only for the coronary arteries but also for the aorta, pulmonary artery, and ileal arterioles (6, 9, 10, 27). For those vessels, the strain distribution cannot possibly be uniform theoretically or experimentally.

The definition of θ as described in Fig. 1 assumes a circular geometry for the vessel sector. This is, of course, only an approximation because the majority of vessel sectors are not exactly circular. The utility of this definition lies in its simplicity. More complex approaches can be found in the literature. Matsumoto et al. (21) proposed a method by which the vessel sector is cut into numerous segments to eliminate the noncircularity of the whole sector, which is particularly important for diseased vessels. The residual strain of the whole sector was then estimated from the curvature and dimensions of each segment. Alternative measures of θ based on measurement of edge angle and tangent rotation angle have also been introduced in the literature (14, 19).

A question arises as to whether a single radial cut of vessel ring is sufficient to ensure a zero-stress and zero-moment state of the vessel sector. In theory, this can only occur if the tissue is cut into infinite surfaces, each of which has zero traction. In reality, Fung and Liu (7) have demonstrated that subsequent additional radial cuts do not affect the vessel sector configuration. In the myocardium, Omens and Fung (23) found that a second radial cut produced deformations significantly smaller than those produced by the first cut. In contrast, Vossoughi et al. (32) found that the sector geometry changes when an additional circumferential cut is made to separate the vessel wall into two layers. Similarly, Lu et al. (20) found that when the vessel is separated into intima-media and adventitial layers, θ of the intima-media sector increases, whereas that of the adventitial layer decreases relative to the intact wall.

The present study does not take into account the effect of muscle tone. The in vivo and the residual stress in arteries have been found to be strongly dependent on muscle tone (22, 26). Although the present results were obtained under passive state, they provide the framework for exploring the effect of active contraction in future studies.

### Significance of Study

The uniform transmural strain hypothesis has been generally accepted for all blood vessels. The significance of the present study is to provide a clarification that the strain cannot be transmurally uniform for vessels whose zero-stress state reveals θ > 180°. For the coronary vessels, the transmural strain was uniform only for θ < 135°, whereas for the aorta and its primary branches, the transmural strain was uniform in the range of 135 > θ > 90°. The nonuniformity of transmural strain, however, may lead to uniformity of transmural stress because the vessel is a composite structure with different material properties in each layer as discussed above. Indeed, if the vessel were homogenous with uniform material properties, the nonuniform strain would lead to a very nonuniform stress because of the highly nonlinear stress-strain relation. This underscores the importance of considering the composite nature of blood vessels for a realistic mechanical analysis. A full mechanical analysis of the coronary artery that accounts for the composite nature of the vessel remains a worthwhile task for the future.

## GRANTS

This research was supported in part by National Heart, Lung, and Blood Institute Grant 2 R01 HL-055554-06. G. S. Kassab was the recipient of an American Heart Association Established Investigator Award.

## Acknowledgments

We thank Professor Y. C. Fung for the kind review of the manuscript.

## Footnotes

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- Copyright © 2005 by the American Physiological Society