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Department of Biomedical Engineering, University of California, Irvine, California
Submitted 29 June 2004 ; accepted in final form 8 January 2005
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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stress; strain; constitutive equation; material constants; two-layer model
A common feature of blood vessels is the layered construction of their walls, which consist of intimal, medial, and adventitial layers. Each layer has a zero-stress-resultant and zero-stress-moment state and a set of material properties. If we consider these layers as elastic shells, then these states of zero-stress-resultants and zero-stress-moments can be referred to as the "states of zero stress" (11). The entire wall is a composite material made up by these layers with different elastic constants and zero-stress states. The passive stress and strain distribution in the whole wall can be computed from the elasticity and zero-stress states of the individual layers.
There are currently three views on the stress distribution in the vessel wall: 1) the classic view where the vessel wall is considered as made of homogeneous material and that the no-load state is equal to the zero-stress state (see review in Ref. 3); 2) the 1980s view where the vessel wall is considered as homogeneous, the state of zero-stress-resultant and zero-stress-moments has been measured and has been found to be quite different from the no-load state (10, 28); 3) the 1990s and present view where the vessel wall is made of several layers each of which has its own elasticity constants and its own state of zero-stress-resultants and zero-stress-moments (8, 16, 24, 25, 29, 30).
In the present study, we wanted to pursue the latest view for the coronary arteries. We treat the vessel wall as a two-layered structure consisting of the intima-media (endothelial cells and vascular smooth muscles, including elastin and some collagen) and adventitia layer (collagen, fibroblasts, and elastin). The vessel wall was initially mechanically tested as a whole and subsequently as the intima-media or adventitial layer. Two experiments were done for each layer, which included inflation and axial stretch. An exponential strain energy function was used to curve fit the experimental data on the intact wall and individual layers for the porcine right coronary artery (RCA) and left anterior descending (LAD) artery. Two methods were explored for the determination of material constants: the classic Marquardt-Levenberg (12, 21) and a more recent method using genetic algorithm (6, 26). The major conclusion of the present study is that the coronary artery is a composite material of which the intima-medial and adventitial layers have different material properties. These features are important to understand coronary vasa vasorum, baroreceptors, atherogenesis, restenosis, stenting, and many other clinically relevant issues.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Biaxial test: intact vessel.
The triaxial (inflation, extension, and torsion) testing of coronary arteries has been recently described by Kassab's group (20). Although the experimental protocol included torsion, we limited the present analysis to inflation and axial extension at zero torsion. Briefly, 12 hearts from Danish Landrace-Yorkshire pigs were obtained from a local slaughterhouse. The hearts used for testing of whole wall and media had a mean weight of 463 ± 57 g (n = 7), whereas the hearts used for testing of adventitia weighed 245 ± 32 g (n = 5). The RCA and LAD artery were dissected and tested biaxially within 2–3 h of heart harvest. The arterial specimen was mounted on the cannula horizontally on both ends in an organ bath at room temperature (22°C). The organ bath chamber was designed to maintain the specimen immersed in a physiological Krebs solution. The cannulas were made to pass through the holes on the side walls of the chamber, and then a circulative pump was used to maintain the volume of the solution in the chamber. One side of the cannula was connected with a pressure transducer (Summit Disposable Pressure Transducer, Baxter Healthcare; error of ± 2%) through a Y tube. A pressure regulator was used to control the luminal pressure in the arterial specimen. The cannula was mounted on a linear stage (Daedal Positioning Table, type 402006 LNMS D L2 C2 M1, Parker Hannifin) where the motor for linear motion (type SDL 603, Parker Hannifin) drives the stage and records the axial force by a load cell (Sensotec model; Sensotec error at 0.15–0.25% at full scale). The load cell was calibrated with a series of weights, and a linear relation (correlation coefficient of 1.000) in the range of 0–2.45 N was confirmed. The axial stretch ratio (
z) was varied from 1.1 to 1.4 in increments of 0.1. The transmural pressure was varied from 20 to 140 mmHg in increments of 20 mmHg at every z using a Ca+2-free Krebs solution containing EGTA (2.6 µM) to prevent vessel tone. Microbeads (stainless steel, 60–100 µm in diameter) were dispersed on the outer surface of the segment, which were videotaped during testing to measure displacement and outer diameter.
Biaxial test: dissected vessel. Each LAD artery segment (2 cm in length) was removed from the biaxial machine and transferred to a dish in cold Krebs solution. Two rings were cut from the two ends of the segment for the measurements of no-load and zero-stress state of the intact vessel as described in No-load and zero-stress state. In seven vessels, the adventitia of the arterial segments was carefully dissected away from the media at the external elastic laminae with the aid of a stereomicroscope. The intima-medial layer of the arterial segments remained intact and was then tested in the biaxial machine according to the same protocol as the intact LAD artery. In the remaining five vessels, the vessels were inverted inside out, and the media was dissected away, leaving the adventitia intact. The adventitia was then reinverted and tested in the biaxial machine using the same protocol as described previously. The same procedure was also repeated for five intact RCA segments, and their medial layers in the same hearts were used for the LAD artery. The adventitial layer was only tested for the LAD artery.
No-load and zero-stress state. After the mechanical tests, the coronary segment was 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, and 11 glucose). Each ring of the intact and dissected coronary artery 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, which was previously 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 Pro.5.0).
Theoretical Formulation
Biomechanical principles.
The artery is considered to be a two-layer cylinder. Because the intima is relatively thin in the normal artery, the intima-media is considered as the inner layer and the adventitia as the outer layer. The circumferential deformation of the entire artery or its individual layers may be described by the midwall circumferential Lagrangian Green's strain, which is defined as follows:
 | (1a) |
is the midwall stretch ratio ( = c/C), c refers to the midwall circumference of the vessel in the loaded state, and C refers to the corresponding midwall circumference in the zero-stress state. Similarly, the axial Green strain is given by:
 | (1b) |
z is the local axial stretch ratio defined as the change in axial length between loaded and no-load state.
The midwall circumference in the loaded state was computed from the average of inner and outer radius. The inner radius (ri) of the vessel can be computed from the incompressibility condition for a cylindrical vessel as:
 | (2) |
The mean second-Piola Kirchhoff stresses in the circumferential (S) and axial (Sz) directions are given by:
 | (3) |
 | (4) |
Equations 1–4 were also applied individually to each separate layer. If the radii of the interface of the media and adventitia and the outer boundary of the adventitia are denoted by rm and ra, respectively, the two components of the mean stresses were computed according to Eqs. 3 and 4 using the respective radii and wall thicknesses. Similarly, the strain was computed with the respective circumference as given by Eq. 1. Finally, the wall thickness of each individual layer was similarly determined by Eq. 2 with the appropriate loaded radius and no-load wall area.
Strain Energy Function
A well-known approach to elasticity of bodies capable of finite deformation is to postulate the form of an elastic potential or strain energy function (14). Following the arguments made by Fung (11), we use the following form of strain energy function:
 | (5a) |
 | (5b) |
z = 1.4 axial stretch). Here the superscript "2" over 
0W(2) signifies that this is a two-dimensional approximation, treating the arterial wall as a membrane and ignoring the radial stress. The symbol W represents the strain energy per unit mass of the material, and 0 is the mass density at zero stress. C has the units of stress (force/area); al, a2 and a4 are dimensionless constants. Although Eq. 5 applies either to the loading or the unloading curve with different set of constants, we shall focus on the former. The differentiation of the strain energy equation leads to the stress-strain relationship as:
 | (6) |
,z) is a set of local right-handed cylindrical coordinates with an origin lying on the neutral surface of the blood vessel wall, the axis pointing in the circumferential direction, and z in the axial direction. The strains are finite and referred to the zero-stress state; E and Ezz are normal strains, and ez = ez are shear strains taken as zero because of the axisymmetric loading conditions. The subscripts i and j range over 1, 2; with 1 referring to ; 2 referring to z. When Eqs. 5 and 6 are applied to intima-media, every symbol should have a superscript (im). Similarly, adventitial layer should have a superscript (ad). Note that the quadratic form of Q is written for two dimensions in the spirit of the theory of thin shells in classic mechanics. The blood vessel material is incompressible (2, 4). In two dimensions, however, it is not incompressible (5). Determination of Elastic Constants
If we combine Eqs. 5 and 6, we obtain a constitutive relation that relates the circumferential and axial stresses to strains as:
 | (7a) |
 | (7b) |
) and axial, S
stresses as:
 | (8) |
Marquardt-Levenberg Method
In this approach, the determination of the material constants of the strain energy function was carried out using Mathematica, which utilizes the Marquardt-Levenberg (M-L) method for nonlinear optimization. The optimized cost function is expressed by Eq. 8. Initially all four parameters (C, a1, a2, and a4) were evaluated. The value of a4 was then fixed at the predetermined mean for the intact wall, and each respective layer and the three parameters (C, a1, and a2) were reevaluated.
Genetic Algorithm Method
The genetic algorithm (GA) is a stochastic global search that attempts to mimic biological evolution. GA operates on a population of potential solutions applying the principle of survival of the fittest to produce the best approximation to a solution. At each generation, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators in analogy to biological genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than their parents, similar to natural selection. The objective is to minimize the cost function in Eq. 8 based on a probabilistic rather than numerical approach (6, 26).
Briefly, we developed a simple code to implement the GA using the MATLAB Genetic Algorithm Toolbox (27). A number of parameters were selected, including the size of the population, probability of crossover, and mutation; scale for mutation and Tournament probability; initial guess values for lower and upper limit of C, a1, a2, and a4; and number of generations. The error function (Eq. 8) was evaluated based on the initial parameters. These values were stored in an array, and the search was initiated for the best "individuals" or material constants. We set an elitism and recombinant individual as per Tournament algorithm or Roulette Algorithm based on the probability of subsequent tournament algorithm. We selected two parents based on Tournament Algorithm and roulette principle; i.e., the lower the value, the better chance. The selection of children from the two parents was made using crossover, mutations, and elitism. This process was repeated a number of times, and the fitness value was computed at each cycle. The converged values represented a minima for Eq. 8. We then fixed a4 and redetermined the values of C, a1, and a2 by using the GA.
Statistical Analysis
The "goodness" of fit was determined by a correlation coefficient R2 for the relation between calculated and experimental values. A P value <0.05 indicated that the differences between RCA and LAD artery and their respective layers were statistically significant.
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RESULTS |
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z = 1.4). These seven LAD arteries had their adventitia dissected away and their media tested. The material constants for those hearts are listed in Table 2. Five other LAD vessels were tested for their adventitia once their media was dissected away, and their material constants are shown in Table 3.
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DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The constitutive equation of the blood vessel wall was a much-debated subject in earlier years (see review in Ref. 11). Later, the following observation by Fung (11) was generally accepted: Soft tissues have a special kind of viscoelasticity where the percentage loss of energy by hysteresis per cycle in a cyclic loading and unloading process is only a few percent, and this percentage does not vary more than a factor of two over a frequency range of five decades. However, at any particular frequency, it is not easy to predict the exact value of the loss per cycle. Fung (11) and others showed that this is consistent with a model of viscoelasticity with a continuous relaxation spectrum. In cyclic loading and unloading, the stress-strain relationship is unique in the loading stroke and is also unique in the unloading stroke (although the two are somewhat different). Fung called such a material "pseudoelastic," a term that is now widely used.
Alternative forms for the strain energy function abound (1, 11), especially the polynomial form of Patel et al. (23) and the logarithmic form by Hayashi (15). But the exponential form Eq. 5 is accepted widely, mainly because it is the only form that we know how to invert; i.e., to derive analytically the tensorial strain-stress relationship from the stress-strain relationship by a method proposed by Fung (9). Many validation experiments have been done (11). Shear has been studied in the coronary arteries by Lu et al. (19), but it is zero if the deformation is axisymmetric as in the present study. More recently, Holzapfel et al. (16) introduced a strain energy function for three-dimensional, two-layer artery previously used in mechanics of fiber-reinforced composites. The three-dimensional model has the advantages of a few material parameters (five in each layer) and is partly based on the histological fiber structure. As a microstructural model, the constitutive relation remains to be validated histologically. As a phenomenological model, it appears promising for the characterization of three-dimensional, anisotropic, and nonlinear mechanical properties of arteries.
Parameter Estimation
Determination of constitutive equations is an inverse problem. We assume a form of the constitutive relation, and based on computed values of stress and strain from loading and deformation experiments, we determine the material properties that give satisfactory agreement between the theorized form and the experimental data. It should be noted, however, that there are inherent difficulties with nonlinear least squares procedures in that large differences in the values of the material constants may result in very small differences in the data. This is a well-known fact about class of nonlinear "inverse" problems (18). The difficulty is that the error function given by Eq. 8 has many relative minima. A simple solution to this difficulty was presented by Zeng et al. (31). Their approach involved the initial determination of the four constants followed by fixing one of the constants, namely a4, which reduces the degree of freedom by one and makes the determination of the remaining three constants rather stable. The choice of fixing a4 stems from the observation that variation of this parameter has the smallest effect on the curve fit.
Comparison of M-L and GA Methods
There are several significant differences between the traditional nonlinear least squares method (e.g., M-L) and the GA that deserve mention. These differences include 1) GA searches a population of points in parallel, not a single point; 2) GA does not require derivative information or other auxiliary knowledge; only the objective function and the corresponding fitness levels influence the directions of search; 3) GA uses probabilistic transition rules, not deterministic ones, 4) GA works on an encoding of the parameter set rather than the parameter set itself (except where real-valued individuals are used); and 5) traditional methods may miss a local minimum depending on the optimization method employed. It is also important to note that the GA provides a number of potential solutions to a given problem, and the choice of final solution is left to the user. In cases where a particular problem does not have one individual solution (for example, a family of Pareto-optimal solutions), the GA is then potentially useful for identifying these alternative solutions simultaneously. In the present study, we did not find any statistically significant differences between the two methods of parameter estimation; hence, we gained confidence that the parameters estimated minimize the error between the proposed theoretical function and experimental data.
Convexity of Strain Energy Function
Although we did not put any restrictions on the choice of material constants, we verified that the material constants do not violate the physical requirements of hyperelasticity; i.e., the condition that the material must be stable under loading (22). Accordingly, we confirmed that the second derivative of strain energy function with respect to strain is positive definite. Hence, the proposed strain energy functions are convex, which ensures physically meaningful mechanical behavior. This is easy to verify because of the quadratic form of Q in Eq. 5. Indeed, it can be shown that Eq. 5 is locally convex if a1 > 0, a2 > 0 and a1a2 > a
for C > 0 (16).
A Two-Layer Model of Coronary Artery
The coefficient C dictates the stress scale, the product C·a1 dictates the nonlinear rate of change in the circumferential stress with respect to strain, the product C·a2 has the same meaning in the axial direction, and a4 indicates the interaction between terms in both circumferential and axial directions. For the RCA, for example, C·a1 is significantly smaller for the intact wall than the media for both M-L and GA methods (P = 0.01 for both methods) as shown in Table 6. For the LAD artery, the parameter C·a1 was found to be significantly smaller for the intact wall compared with the intima-media (P = 0.022 for both methods).
An inspection of Fig. 2, A–C, shows that the vessel stiffens circumferentially (leftward shift) with increase in axial stretch ratio. This behavior has been previously reported (e.g., 7, 17). Furthermore, at in vivo axial stretch ratio of 1.4, the slope of the circumferential stress-strain relation is larger for the intima-media layer than the intact wall, which is larger than the adventitial layer. This is consistent with our previous data on the incremental homeostatic elastic moduli: i.e., intima-media > intact > adventitia (19).
Limitations of Proposed Model
There are two approaches for the description of the biomechanics of vessel wall. In one approach, the vessel wall is regarded as a three-dimensional incompressible material. A second approach relies on the axisymmetric, thin-walled assumptions and regards the wall as a two-dimensional compressible membrane structure. The latter approach yields the mean stresses over the wall thickness in the circumferential and axial direction. Such a two-dimensional approach cannot describe the behavior of a thick-walled cylindrical tube under, for instance, inflation and torsion. The proposed form is limited to a membrane description that involves the mean quantities of stresses.
Significance of Study and Future Direction
Despite the recognition of the significance of stress and strain in gene expression, cell biology, cell adhesion, and cell differentiation and proliferation, we are still unable to determine the internal stresses and strains because we do not know the constitutive equations for the different layers of the blood vessel wall. Most of the past determinations of the constitutive equation have treated the vessel wall as a homogeneous material. In the past two decades, advances in vascular mechanics have resulted from theory and experiment that demand greater detail. The separation of a vessel wall into several layers is a step toward the next level in the hierarchy of structure. Each level has a continuum model, and different levels have different models of continua. Clearly, the layered concept is not the end, and successively lower levels can be explored. Idealization of the coronary artery as composed of two layers each of homogenous materials may serve as a beginning for further analysis. More detailed modeling should be done if greater details are required; i.e., the modeling of the intima-media layer as composed of elastin and collagen fibers and muscle cells. The next logical step is to determine the three-dimensional constitutive equations; i.e., the stress-strain-history relations, of the individual layers that take into account the circumferential, axial, and radial stresses and strains and their respective shear components. Furthermore, the active smooth muscle stress must also be considered in future models to understand important clinical issues such as coronary vasospasm.
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GRANTS |
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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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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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