The normal coronary artery consists of two mechanically distinct layers: intima-media and adventitia. The objective of this study is to establish a two-layer three-dimensional (3-D) stress-strain relation of porcine coronary arteries. Experimental measurements were made by a series of biaxial tests (inflation and axial extension) of intact coronary arteries and, subsequently, their corresponding intima-media or adventitia layer. The Fung-type exponential strain energy function was used to describe the 3-D strain-stress relation for each layer and the intact wall. A genetic algorithm was used to determine the material constants in the Fung-type constitutive equation by curve fitting the experimental data. Because one layer must be sacrificed before the other layer can be tested, the material property of the missing layer was computed from the material constants of the intact vessel and the tested layer. A total of 20 porcine hearts were used: one group of 10 hearts for the left anterior descending artery and another group of 10 hearts for the right coronary artery. Each group was further divided into two subgroups of five specimens tested for the intact wall and the intima-media layer and for the intact wall and the adventitia layer. Our results show statistically significant differences in the material properties of the two layers. The mathematical model was validated by experimental stress-strain data for individual layers. The validated 3-D constitutive model will serve as a foundation for formulation of layer-specific boundary value problems in coronary physiology and cardiology.
- stress-strain relation
- constitutive relation
- strain energy function
cardiovascular disease is responsible for about one-third of all deaths globally, and coronary artery disease accounts for 42% of cardiovascular disease-related deaths (29a). Atherosclerosis, a major disease of coronary arteries, may lead to myocardial ischemia and infarct. The mechanical properties of the artery are important determinants of stress and strain distribution in the vessel wall, which in turn affects vascular health and disease. Atherosclerosis is associated with abnormal strain and stress in the vessel wall (19), and changes in material properties are typically observed in diseased arteries (29).
A three-dimensional (3-D) mechanical model of the coronary artery is necessary for analysis of the strain and stress distributions in the vessel wall. Numerous studies of blood vessel mechanical properties have been summarized by Vito and Dixon (25). In most of the early studies, the vessel wall was considered to consist of homogeneous material and the mechanical properties were considered to be uniform throughout the vessel wall (6, 21, 24). Although this is a reasonable first approximation, it is not consistent with the microstructure. A better model must consider the normal wall to consist of two distinct layers: intima-media and adventitia. Experimentally, this is feasible, inasmuch as we previously found that the vessel can be dissected at the cleavage plane that separates intima-media from adventitia at the external elastic lamina (13).
A number of studies have considered the artery to be a two-layered structure theoretically and experimentally (5, 8, 9, 12, 13, 15, 17, 18, 26, 27, 30). Despite the significant contributions of the various studies, there is still no two-layer 3-D model of the coronary artery based on cylindrical biaxial experiments of intact vessel and individual layers that takes into account the zero-stress state of each layer. The objective of this study was to fill the gap. Specifically, we present an experimentally validated two-layer 3-D model of the passive mechanical properties of coronary arteries. The present model will be central to the formulation of various boundary value problems in coronary physiology and cardiology.
MATERIALS AND METHODS
Hearts of 20 pigs (243 ± 54 g body wt) were obtained from a local slaughterhouse. Ten hearts were used for study of the left anterior descending (LAD) artery and another 10 for study of the right coronary artery (RCA). In each group, five hearts were used for study of the intact vessel and the adventitia layer and the other five for study of the intact vessel and the media layer. The hearts were transported to the laboratory in saline solution within several hours after the animal was killed. A 3- to 4-cm segment of the LAD artery or the RCA was dissected carefully from its emergence at the aortic ostia. Every bifurcation was identified, and each branch was ligated with a 6-0 suture to prevent leakage.
Preconditioning and biaxial test: intact vessel.
The triaxial (inflation, extension, and torsion) testing of coronary arteries has been previously described by our group (13
The vessel segment was first preconditioned to obtain reproducible mechanical data. The procedures of preconditioning were as follows: The pressure was set to zero, the axial stretch ratio was ramped from 1 to 1.5 over 10 cycles and then fixed at 1.5, and the luminal pressure of the vessels was ramped 10 times from 0 to 130 mmHg. The stretch ratio of 1.5 and pressure of 130 mmHg were used to avoid any reference length change in the biaxial tests (see discussion). After the vessels were preconditioned, two 1- to 2-mm-long rings were cut each from proximal and distal LAD artery and RCA for measurement of the no-load and zero-stress states of the intact vessel (see below).
The protocol for biaxial measurements was as follows: The axial stretch ratio (λz) was varied from 1.2 to 1.4 in increments of 0.1. Because λz < 1.2 can lead to a curved vessel during inflation, only λz ≥ 1.2 were considered. The transmural pressure (P) was varied from 0 to 120 mmHg at every λz with use of saline solution. Outer diameter of the vessel, transmural pressure, and axial force were recorded. Only data from the loading process of straight segments were used for analysis.
Biaxial test: individual layer.
After the initial mechanical test on the intact vessel was completed, each 3- to 4-cm-long RCA/LAD artery segment was removed from the biaxial machine and transferred to a dish in saline solution. From five vessels, the adventitia of the arterial segments was carefully dissected with the aid of a stereomicroscope, with care taken to ensure that the intima-media layer remained intact. In another five hearts, the vessels were inverted inside out and the intima-media was dissected away, leaving the adventitia intact. Two 1- to 2-mm-long rings were then cut each from proximal and distal ends of the adventitia or intima-media layer for measurements of no-load and zero-stress state for the individual layer (see below). The intima-media or adventitia layer segment was tested biaxially according to the protocol for the intact artery described above.
No-load and zero-stress state.
The rings from the vessel segment and individual layers were placed in a saline solution and photographed in the no-load (P = 0 mmHg) state. To obtain the zero-stress state, we cut ring radially with scissors, and 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 in the zero-stress state. A morphometric analysis system (Sigma Scan Pro version 5.0) was used for morphological measurements of inner, middle (i.e., interface between media and adventitia, for intact vessel segments only), and outer circumference and area of the intact wall and corresponding layers in the no-load and zero-stress state from the images.
Determination of Elastic Constants
The theoretical formulation for the mechanics of the blood vessel and the solutions of the equilibrium equations is described in the appendix. Here we focus on determination of the elastic constants. Equations A11 and A13 are integral equations from which we can determine the material constants given the values of transmural pressure (pi), axial stretch force (F), external radius (re), internal radius (ri), and distribution of Green’s strain components (Eθ, Ez, and Er). F, pi, and re are direct experimental measurements, whereas ri was calculated from the outer radius and the no-load wall area on the basis of the incompressibility assumption. Green’s strain at inner and outer surfaces was computed using Eqs. A1–A5. Because the circumferential strain [Eθ(r)] between the two surfaces is a function of the radius, we can simplify Eqs. A11 and A13 if we assume a linear distribution of Eθ(r) between two surfaces along the radial direction as follows (1) Ez(r) and Er(r) can be computed using Eqs. A2–A5. We verified that the computed strain distribution is very close to linear, as given by Eq. 1.
The goal of an algorithm to determine the material constants C, b1, b2, b3, b4, b5, and b6 is to minimize the square of the difference between theoretical (see Eqs. A11 and A13) and experimental values of internal pressure (pie) and axial force (Fe) as follows (2) where N represents the total number of experimental data points used to determine the material constants for each curve. The strain energy function must be positive definite, which means any possible strain configuration, except zero strain, should have positive strain energy. Although this condition was not applied as a constraint on the material constants, all the material constants were verified to satisfy this condition.
Determination of Elastic Constants of the Dissected Layer
One layer must be dissected or sacrificed to allow testing of the other layer. Here we propose a novel method to determine the material constants of the dissected layer from those of the intact vessel and the measured layer. We assume that the total strain energy of the intact vessel segment (VI) is the sum of the strain energy of the intima-media (VIM) and adventitia (VA) layers. Furthermore, total axial force of the intact vessel (FItotal) must be the sum of the axial force of the intima-media (FIM) and adventitia (FA) layers. The two assumptions can be mathematically stated as follows (3a) (3b) The energy Vj (j = I, IM, and A) can be computed by taking an integral of strain energy density given by Eq. A6 over the respective wall volume as follows (4) where ri, rm, and re are radii for inner, middle (the interface between the 2 layers), and outer vessel wall, respectively, h is the deformed length of the arterial specimen, and Er can be determined by Eθ(r) and Ez on the basis of the incompressibility condition. The total force Fjtotal (j = I, IM, and A) is the integral of axial Cauchy stress (σz) over the wall area expressed as follows (5) Equations 4 and 5 are also simplified by the assumption of a linear distribution of Eθ(r) along the radial direction expressed in Eq. 1. Material constants bi,j (i = 1–7 and j = I, IM, and A) of the intact vessel and the measured layer were obtained by minimization of the objective function (Eq. 2). Hence, the material constants of the dissected layer can be determined by minimization of the following error function (6) Equation 6 is the square of the difference between theoretical values of strain energy and axial force of the sacrificed layer and the values determined by the material constants of the intact wall and the tested layer. Equation 6 can be used to determine the material constants of the intima-media layer. To determine the material constants of the adventitia layer, the terms of the two layers are interchanged. A genetic algorithm (GA) was adopted for determination of material constants as described below.
Convergence of the material constants for a global minimum is difficult, and the final results strongly depend on the initial guess of the parameters when a traditional optimization method, such as the Marquardt-Levenberg (M-L) algorithm (14), is used. Therefore, a GA (3) was used to minimize the error expressed by Eq. 2 as outlined below.
The GA is a global search that attempts to mimic biological evolution. It begins with a population of potential solutions and evaluates their level of fitness in the problem domain and creates a new set of approximation based on breeding the better evaluated solutions. The cycles of selection lead to the evolution of populations of individuals that are better suited, on average, to their environment than their parents, similar to natural selection. The objective is to minimize the cost function in Eqs. 2 and 6 on the basis of a probabilistic, rather than numerical, approach (14). This method was chosen, because it is more likely to reach global minimum than the traditional M-L method. More importantly, it was more stable than the M-L method for our data analysis; i.e., the material constants appeared to be in a more similar range than the those computed by the M-L method. Using the MATLAB Genetic Algorithm Toolbox (14a) as described previously (17), we developed a simple code to implement the GA.
The “goodness” of fit was determined by a correlation coefficient for the relation between calculated and experimental total force (sum of force applied by internal pressure and external axial force) and inner pressure values. The percentage of the root-mean-square error with respect to the mean value was also calculated to evaluate the goodness of the fit. Stress values were grouped at various strain intervals from 0.05 to 0.6 at increments of 0.05 (i.e., 0.05–0.1, 0.101–0.15,…, 0.501–0.55, and 0.551–0.6) for comparison between two sets of material constants. The stress values at each of these strain intervals were then compared by t-test. P < 0.05 indicates statistically significant difference.
The material constants for the 10 RCAs are listed in Tables 1 and 2. Table 1 shows five intact vessels and their intima-media layers. There were statistically significant differences between the intact wall and the intima-media layer (P = 0.001). Table 2 summarizes the other five intact RCA vessels and their adventitia layers. There were no statistically significant differences in the intact RCA between the two groups of five hearts (Tables 1 and 2; P = 0.06). There were statistically significant differences, however, between the intact wall and the adventitia layer (P = 0.018). Table 3 lists the predicted material constants of the dissected intima-media layer on the basis of the material constants of the intact vessel and their adventitia layers. We found no statistically significant differences between predicted and measured values (P = 0.11).
Similarly, the material constants for the 10 LAD arteries are listed in Tables 4 and 5. The differences between the intact LAD artery and the intima-media (Table 4) and adventitia (Table 5) layers were statistically significant (P = 0.02 and P = 0.048, respectively). The differences between the two groups of intact LAD arteries in Tables 4 and 5, however, were not statistically significant (P = 0.36). Table 6 summarizes the predicted material constants of the dissected intima-media layer on the basis of the material constants of the intact vessel and their adventitia layers. A comparison of the predicted values in Table 6 with experimental measurements in Table 4 shows agreement (P = 0.3).
A statistical comparison between the LAD artery and RCA was made. Ten sets of data were used to compare the intact vessel and five sets to compare the individual layers. There were no statistically significant differences between the measured intact vessels (P = 0.4) and the intima-media (P = 0.104) and adventitia (P = 0.602) layers of RCA and LAD arteries.
Figure 1A shows the circumferential stress-strain curves of the intact RCA and the media layer of the first five hearts (Table 1) obtained from the mean material constants. Similarly, Fig. 1B shows the circumferential strain-stress curves of the intact RCA and the adventitia layer of the second five hearts (Table 2) obtained from the mean material constants. It is apparent that the media is stiffer and will sustain most of the circumferential force, whereas the adventitia is softer, in the normal physiological loading state (100 mmHg, λz = 1.4).
The strain-stress behavior of the predicted material constants of the intima-media layer of the RCA is compared with experimental data in Fig. 2. The differences between theoretical predictions and experimental data were not statistically significant (P > 0.48).
The circumferential strain-stress curves of the intact LAD artery and the media layer of the first five hearts (Table 4) obtained from the mean material constants is shown in Fig. 3A. Similarly, the circumferential strain-stress curves of the intact LAD artery and the adventitia layer of the second five hearts (Table 5) obtained from the mean material constants are shown in Fig. 3B. The strain-stress relations of the LAD artery are similar to those of the RCA.
The strain-stress behavior of the predicted material constants of the intima-media layer of the RCA is compared with experimental data in Fig. 4. The differences between the theoretical predictions and experimental data were not statistically significant (P > 0.39).
The local convexities of the strain energy function (Eq. A6) for the intact LAD artery and the intima-media and adventitia layers are shown in Fig. 5. Similar patterns were observed for the RCA (data not shown).
In this study, the material constants for a 3-D model of intact vessel and intima-media and adventitia layers of the RCA and LAD artery were determined on the basis of the cylindrical biaxial mechanical test (Tables 1, 2, 4, and 5, Figs. 1 and 3). It was demonstrated that the material constants of the dissected layer can be determined on the basis of the experimental data of the intact vessel and the intact layer. The predicted material constants were similar to those directly determined from the experimental data (Tables 3 and 6, Figs. 2 and 4). This finding validates the feasibility of the proposed two-layer model and the reliability of the material constants.
Comparison With Previous Studies
Previous mechanical studies of coronary arteries were not based on experimental data of individual layers (8, 9, 18, 26). Although previous publications from our group were based on experimental data of individual intima-media and adventitia layers (12, 13, 17), some focused on the incremental moduli of coronary arteries (12, 13) and others on a two-dimensional (2-D) thin-shell model (17). This is the first study to provide validated material constants of individual layers of porcine coronary artery in a 3-D model based on experimental data of individual layers.
The two-layer 2-D model reported that the slope of the circumferential stress-strain relation is larger for the intima-media layer than for the intact wall, which is larger than the adventitial layer at an in vivo axial stretch ratio of 1.4 (17). This is consistent with previous data on the incremental homeostatic elastic moduli: intima-media > intact > adventitia (12). It is also consistent with the present findings (Figs. 1 and 3).
The interaction between the axial and circumferential directions is shown in Fig. 1 for the RCA. The strain-stress curve shifts significantly leftward when the axial stretch increases. The same behavior was observed for the LAD vessels. Thecoefficient b4 (cross term for the circumferential-axial direction) can be considered as an index of this interaction. In most results, b4 is significantly larger than b5 and b6, which implies a stronger interaction than those of other directions. These findings are consistent with the previously reported results of the 2-D model study (17).
It is generally accepted that preconditioning of biological tissue is necessary to obtain a reproducible stress-strain relation. It has been reported that the reference length of the sheep tendon will increase during preconditioning (20). We observed the same phenomenon in the axial and circumferential reference length of the coronary arteries. In the axial direction, the increase was ∼3–7% of the original reference length. In the circumferential direction, the reference length change is smaller. The reference length increase may be due to some irreversible vessel microstructure change caused by the preconditioning. Without consideration of this effect on the reference length, the strains (especially axial) will be overestimated. Because the stress-strain relations of the coronary arteries are sensitive to axial strain (17), it is important to acquire accurate reference lengths. Hence, the reference lengths should be determined after preconditioning and mechanical testing of the vessels. An additional observation from the tendon study (20) is that no further change in reference length was observed if the stretch ratio during preconditioning is set higher than the actual test. For this reason, the upper limits of the axial stretch ratio and the pressure during preconditioning were set higher than those of the biaxial tests (1.5 and 130 mmHg, respectively).
Error functions were minimized to determine the material constant (17). In this study, the external axial force and internal pressure were used as the minimized objective functions. Because both ends of the vessel specimen were cannulated, the total axial force (FT) applied on the specimen is the sum of the force applied by the internal pressure and the external axial force. Under most of the experimental conditions, the external force was much smaller than the force imposed by the internal pressure. Hence, the goodness of fit was generally better for the internal pressure than for the external force. In Tables 1, 2, 4, and 5, the correlation coefficient of FT was calculated.
We have used M-L and GA methods to compute the material constants. We found that the M-L method results in a larger variation of the material parameters and is more sensitive to the initial prediction. The GA generates parameters with less variance and is more robust to local minima “traps.”
Convexity of the Strain Energy Function
The values of the material constants in the strain energy function expressed in Eqs. A1 and A2 must be such that the strain energy is convex, i.e., the material must be stable under loading (16). Although the convexity condition was not applied as a constraint in the GA, all the material constants were verified to satisfy this condition. Because of the quadratic form of Q in Eq. A2, we verified that the strain energy is locally strictly convex when the eigenvalues of the matrix are positive. The eigenvalues were confirmed to be positive for all the material constants. Examples of contour plots of the convex strain energy for intact LAD artery and intima-media and adventitia layers are shown in Fig. 5.
Experimental Limitations of the Study
It is assumed that axial stretch is unchanged during the inflation process. The gauge length increases in the axial direction. Some beads were used as surface markers, and images were recorded to track the axial change during inflation. It was observed that axial stretch during the inflation is small. The axial stretch can be closely followed locally by a video-camera system and by changes in the outer diameter. The current version of the video diameter-tracking software (Diamtrak, T. Neild, Flinders University) is capable of tracking movements in only one direction. Future improvements will allow tracking of movements in two directions (diameter and axial stretch).
Inner diameter was calculated on the basis of the outer diameter data, the vessel wall area of the no-load state, and the incompressibility assumption. The wall area data in the no-load state are the least accurate, because the rings were taken from the two ends of the segment and the average of the values was used. Because the shape of the vessel is not a perfect cylinder or cone and the outer diameter was often not measured exactly in the middle of the arterial segment, there may be some error. Future developments should allow us to determine the inner diameter directly at the center of a specimen. An impedance-catheter system incorporated into the triaxial machine allows direct measurement of the inner diameter change (11).
Significance of the Study and Future Directions
The constitutive equations for the intima-media and adventitia layers of the coronary artery are fundamental to the study of strain and stress distribution in the blood vessel wall. The present two-layer 3-D model of the coronary artery will serve as a basis for formulation of various boundary value problems. For example, we can determine the stress and strain distributions in the layered vessel. There should be continuity of strain, but not stress, at the external elastic lamina (junction between the intima-media and the adventitia). This will have important implications for baroreceptors and the vasa vasorum. We can also consider the effect of different degrees of myocardial support on the intramural distribution of stress and strain on the media and adventitia. For the adventitia, it would be interesting to relate the microstructure of the predominantly collagenous fibers to the observed macromechanics. For the media, the active properties will be incorporated by the addition of active stress. The solutions of these and many other boundary value problems in cardiology and coronary physiology become possible when we have knowledge of the constitutive equation.
The general formulation presented here can be found in standard biomechanics textbooks (10). Briefly, the artery is considered a two-layer cylindrical model. Because the intima is relatively thin in the normal artery, the intima-media is considered the inner layer and the adventitia the outer layer. The arterial wall material is assumed to be incompressible for the intact artery and its individual layers (1). The circumferential deformation of an arbitrary point on the artery may be described by the circumferential Lagrangian Green’s strain as defined by (A1) where λθ is the stretch ratio (λθ = c/C, where c and C represent circumference in the loaded and zero-stress states, respectively). Similarly, the axial and radial Green’s strains are given by (A2) where λz and λr are the local axial stretch ratio (change in axial length between the loaded and the no-load state) and the radial stretch ratio (change in wall thickness between the loaded and the no-load state) respectively.
The assumption of incompressibility requires the following relation between the principal stretch ratios (2) and (A3) where R denotes the radius of an arbitrary point in the zero-stress state (an open sector) as a reference, r is the radial coordinate in the deformed configuration, and χ = π/(π − φ) is a factor that depends on the opening angle φ, defined as the angle subtended by two radii connecting the midpoint of the inner wall of the open sector. Although the axial and circumferential stretch ratios were measured independently, the radial stretch ratio was computed from the incompressibility assumption (Eq. A3). The inner and outer radii in the no-load and zero-stress states were obtained from measurements of the vessel rings to determine the residual circumferential strains in the no-load state. The outer radius of the vessel segment in the loaded configuration was measured directly (see above), whereas the inner radius was computed from the incompressibility condition for a cylindrical vessel as follows (A4) where A0 is the cross-sectional area of vessel wall in the no-load state. The reference radius (R) in the zero-stress state can also be determined by the incompressibility condition as follows (A5) The stretch ratios in Eq. A3 can be written as a function of radius r. Equations A1–A5 can be combined to give the components of Green’s strain tensor at any given deformed state. Equations A1–A5 were also applied individually to each separate layer. The wall thickness of each individual layer was determined as the difference between the inner radius and the outer radius.
Strain Energy Function
A well-known approach to elasticity of finite deformation is to postulate the form of an elastic potential or strain energy function (7). Following the arguments of Chuong and Fung (1), we use the following form of the strain energy function (A6) where where W represents the pseudostrain energy per unit volume, C has the units of stress (force/area), and b1, b2, b3, b4, b5, and b6 are dimensionless constants.
The vessel wall is assumed to be incompressible. This constraint is added to the strain energy function by the method of a Lagrangian multiplier (H) to yield (A7) It is known that −H has the significance of a hydrostatic pressure. The Cauchy stresses (σ) can be related to the strain energy function as follows (A8) where xi and Xa denote coordinates and ρ and ρ0 represent densities in the deformed and reference states, respectively. The summation convention is used in these expressions. The principal stress components can be determined from Eqs. A1–A3 and Eqs. A7–A8 to yield (A9) Equation A9 will be incorporated into the equilibrium equation to obtain the desired results.
Equation of Equilibrium and Boundary Conditions
As shown by Chuong and Fung (1), the problem of a prestrained thick wall vessel under transmural pressure and axial force can be solved by substitution of Eq. A9 into the equation of equilibrium as follows (A10) We consider the following boundary conditions that simulate the experimental protocol. 1) On the external surface (r = re), the pressure is zero. On the internal surface (r = ri), an internal pressure (pi) is imposed. Integration of Eq. A10 along with this boundary condition yields (A11) 2) On the two ends of a blood vessel segment, there exists an external axial force (F). For static equilibrium, the sum of the axial and pressure force (F + piπri2) equals the integral of axial stress over the vessel wall cross section; i.e., (A12) Use of Eqs. A9 and A11 in A12 yields (A13) which represents a force-displacement relation.
This research was supported in part by National Heart, Lung, and Blood Institute Grant 2 R01 HL-055554-06.
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