Passive and Active Triaxial Wall Mechanics in a Two-Layer Model of Porcine Coronary Artery

Triaxial active and passive mechanical properties of coronary arteries are needed for understanding arterial mechanics in health and disease. The aim of the study was to quantify both active and passive strain energy functions in circumferential, axial and radial directions based on the experimental measurement. Moreover, a two-layer computational model was used to determine the transmural distribution of stresses and strains across the vessel wall. The first Piola-Kirchhoff stresses in the three normal directions had the approximate relationship as:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{T}_{\theta \theta }\cong 2{T}_{zz}\cong 5|{T}_{rr}|$$\end{document}Tθθ≅2Tzz≅5|Trr|. The two-layer model showed that circumferential Cauchy stresses increased significantly from the intima layer to the interface between media and adventitia layers (from ~80 to 160 kPa), dropped abruptly at the interface (from ~160 to <5 kPa), and increased slightly towards the outer boundary of the adventitia layer. In contrast, absolute values of radial Cauchy stress decreased continuously from the inner to outer boundaries of the vessel wall (from ~11 kPa to zero). Smooth muscle cell contraction significantly increased the ratio of radial to circumferential Cauchy stresses at the intima-media layer, which had the highest values at the intima layer.

Coronary artery disease (CAD) is the major cause of mortality and morbidity in the World 1,2 . Although substantial studies have emphasized the role of hemodynamic stimuli on CAD 3,4 , the mechanical properties of coronary artery wall are fundamental for understanding the basic mechanisms of these diseases [5][6][7] . Since the intima layer does not contribute to mechanical properties of the vessel wall 8 , the arterial mechanics is generally performed in a two-layer model; i.e., intima-media (IM) and adventitia layers. Collagen fibers and elastin networks in the entire vessel wall determine passive mechanical properties and smooth muscle cells (SMCs) in the IM layer dictate active properties.
Uniaxial studies of circumferential passive and active mechanical properties of the vessel wall are very prevalent 9-16 while a few biaxial models have been proposed in the circumferential and axial directions [17][18][19] . The transmural changes of radial stresses and strains can also contribute to vascular function (e.g., SMC migration) in response to abnormal hemodynamic stimuli (e.g., high blood pressure). To our knowledge, there is still no three-dimensional (3D) mechanical analysis of both active and passive stress-strain relations in a two-layer model of coronary arteries.
The objective of this study was to determine triaxial (circumferential, axial and radial) active and passive mechanical properties in the entire vessel wall and IM layer of porcine coronary artery. Based on the constitutive relation, a two-layer model was developed to compute the transmural distribution of circumferential and radial stresses and strains across the vessel wall. A large-scale global search algorithm (i.e., genetic algorithm) was developed to quantify material constants of 3D active and passive strain energy functions based on the experimental measurements 17,18 . The significance, implications and limitations of the study are discussed to improve the understanding of coronary arterial mechanics.
Mathematical models. Fung's 3D model without shear deformation [20][21][22] was selected to characterize passive mechanical properties and a previous 2D active model 17,18 is extended to describe active mechanical properties. Hence, 3D passive and active strain energy functions (W passive and W active , respectively) are written as: , where l and l 0 are circumferential lengths in loaded and zero-stress states; and L and L 0 are axial lengths in loaded and no-load states (the axial length at the no-load state is assumed equal to that at the zero-stress state) 9 . Although an actively contracting system is not necessarily described by a strain-energy function, this approach is used here to obtain an empirical fit to the observed active mechanical properties. Moreover, the first Piola-Kirchhoff (1 st PK) stresses ( θθ T , T zz and T rr ) are computed from the 3D strain energy functions theoretically (see Appendix A).
Determination of material constants. Passive material constants are determined by minimizing the square of difference between theoretical and experimental values of passive 1 st PK stresses as:  where N is the total number of experimental points when a vessel is at the maximal vasoconstriction, based on which θθ T total experiment , T rr total experiment , and T zz total experiment are determined. A large-scale global search algorithm, genetic algorithm (GA) 23 , was used to minimize the error function as expressed by Eqs [2] and [3] given that − > to satisfy the hyperelasticity condition 24,25 . Briefly, a MATLAB code was developed to determine material constants based on the MATLAB Genetic Algorithm Toolbox, similar to previous studies 26,27 . A number of parameters were selected, including the size of the population, probability of crossover, and mutation; scale for mutation and Tournament probability; and number of generations. The search was initiated for the best material A two-layer computational model. Similar to a previous study 17 , a two-layer model (see Appendix B) was demonstrated to compute the transmural distribution of stresses and stretches across the vessel wall at passive and active states. Material constants of the 3D passive strain energy function for IM and adventitia layers as well as stress-free geometry were consistent with those in Tables A1 and A2 of ref. 17 . Material constants (C 2 = 28.92, b 1 = 0.43, b 2 = 1.62, b 3 = 4.38, and b′ = 4.87) of the 3D active strain energy function in the IM layer were determined by the optimal fit of all six RCA IM layers. The transmural distribution of stresses was computed at healthy and pressure-overload states. Data Analysis. ANOVA (SigmaStat 3.5) was used to compare 3D experimental and theoretical results of passive and active mechanical properties in circumferential, axial and radial directions, where p value <0.05 represented the statistically significant difference. Moreover, we carried out the sensitivity analysis of material constants b b 1 3 in the 3D active strain energy function to determine the effects on the active 1 st PK stresses, which was compared with the 2D active model 17,18 .

Results
Tables 1 and 2 list material constants of Fung's passive and K + -induced active 3D strain energy functions in the entire vessel wall and IM layer of each RCA, respectively. Good agreement is demonstrated between experimental  IM in active and passive states similar to the entire vessel wall. At a circumferential stretch ratio (e.g., at λ θ of 1.4), the passive 1 st PK stresses have higher values in the IM layer than the entire vessel wall despite similar active values. Furthermore, Fig. 4A and B show the sensitivity analysis of material constants b b 1 3 in the 3D active strain energy function for active 1 st PK stresses θθ T active , and T rr active , , respectively, in the vessel wall. Figure 4C presents the corresponding sensitivity analysis in the 2D active model. The plots in Figs 1-4 show the averaged values over the entire wall or IM layer thickness. Figure 5A and B show the transmural distribution of circumferential and radial Cauchy stresses, respectively, across the normalized vessel wall at λ z of 1.3, transmural pressure of 80 mmHg, and wall thickness of 0.22 mm. Circumferential Cauchy stresses between IM and adventitia layers are discontinuous with significantly higher values in the IM layer. Figure 5C and D show the transmural distribution of circumferential and radial stretch ratios, both of which present discontinuity at the interface between IM and adventitia layers. In contrast, radial Cauchy stresses are continuous across the entire vessel wall. In particular, K + -induced vasoconstriction significantly decreases circumferential Cauchy stresses (~15%, p value < 0.05) in the IM layer, but has relatively negligible effects on radial Cauchy stresses, which results in an increased ratio of radial to circumferential Cauchy stresses in the IM layer. We investigated the effects of pressure overload on the transmural distribution of Cauchy stresses. Figure 6A and B show the transmural distribution of circumferential and radial Cauchy stresses, respectively, across the normalized vessel wall under the active state. The transmural pressure and wall thickness increase to 160 mmHg and 0.44 mm (twofold increase from those in Fig. 5), respectively, to mimic pressure overload. This significantly increases the radial Cauchy stresses despite the relatively unchanged circumferential values. Moreover, an increase in the thickness ratio of IM layer to the entire vessel wall leads to an increase and decrease of radial and circumferential Cauchy stresses, respectively.
We demonstrated a comparison of computational Cauchy stresses from the two-layer model with the 2D or 3D active strain energy function in the IM layer as well as the same 3D passive strain energy function in the vessel wall. Figure 7A shows the relative difference of circumferential and radial Cauchy stresses (i.e.,

Discussion
The study demonstrated a 3D analysis of active and passive mechanical properties of coronary arteries in normal directions, based on the experimental measurements 17,18 . A two-layer model was used to compute the transmural distribution of stresses and stretches across the vessel wall, based on 3D active and passive strain energy functions. The major findings are reported as follows: 1) Good agreement is found between 3D theoretical predictions and experimental measurements; 2) There is a gradual decrease of the 1 st PK stresses in a sequence of circumferential, axial and radial directions under active and passive states; and 3) The ratio of radial to circumferential Cauchy stresses in the IM layer has the highest value at the intima layer, which is significantly increased by contraction.
3D active strain energy function. We previously carried out biaxial mechanical tests in the entire vessel wall and IM layer of porcine RCAs, based on which 2D active and passive models were determined in circumferential and axial directions 17,18 . The present study extends the previous model to 3D active and passive strain energy functions including the radial stress-strain relationship. The GA method is more likely to reach a global minimum than the traditional L-BFGS method (i.e., limited-memory quasi-Newton method for large-scale optimization) 28 . We further enhanced the GA method by estimation of the initial material constants from the M-L method, which significantly reduced the spread of material constants in Tables 1 and 2. The improved GA method has the advantage of accuracy over the L-BFGS method in previous studies 17,18 albeit it is much more time-consuming. Based on the GA method, the error function for total 1 st PK stresses in Eq. [3] was minimized to determine material constants of the 3D active strain energy function (R 2 > 0.92 in the three normal directions) instead of the error function for active 1 st PK stresses in the 2D studies (R 2 <0.85 in both circumferential and radial directions) 17,18 . Minimizing the error function in Eq. [3] by the proposed GA method significantly improved the accuracy of the optimal fit of experimental measurements to the 3D active strain energy function. Kassab and his colleagues investigated passive mechanical properties of porcine coronary arteries and reported a larger slope of the circumferential stress-strain relation in the physiological pressure range for the IM layer than the entire vessel wall 26,27,29 , which agrees with Fig. 1A and B. This study further shows larger passive elastic moduli in the physiological pressure range in all three directions of the IM layer than the entire vessel wall, as shown in Figs 1-3. Since SMCs mainly reside in the IM layer, the peak magnitudes of active stress in the IM layer were similar to those in the entire vessel wall (Figs 1A, 2A and 3A vs. Figures 1B, 2B and 3B). Material constants b 1 to b 3 represent the relative curve width of active stresses as a function of circumferential, axial, and radial stretches, respectively. The sensitivity analysis in Fig. 4 showed that the change of b 3 affects the peak magnitude of the radial active stress and the corresponding stretch while b 1 and b 2 only led to the stretch change in correspondence to the peak value. Similarly, parameters b 1 and b 2 in the 3D active model altered circumferential and axial active stresses, respectively, consistent with the 2D active model despite the absence of radial parameters. Since a previous study only carried out the optimal fit of experimental data in the physiological range 18 IM  IM  IM  1  2 3 , respectively. Circumferential 1 st PK stresses were approximately 5-fold or more of radial values.

Transmural distribution of stresses and stretches.
A two-layer model was previously developed to compute the transmural distribution of circumferential Cauchy stresses based on the 2D active strain energy function and 3D passive strain energy function 17 . Here, the two-layer model was used to determine the transmural distribution of normal Cauchy stresses in all three directions from 3D active and passive strain energy functions. We selected the axial stretch of 1.3, transmural pressure of 80 mmHg, and wall thickness of 0.22 mm to mimic the physiological state of the RCA, similar to a previous study 17 . Since the opening angle and zero-stress circumferential length of the IM layer were larger than those of the adventitia layer 17 , this led to discontinuous circumferential and radial stretches (despite continuous displacements) at the interface between IM and adventitia layers as well as higher λ θ and lower λ r in the IM layer given that λ = λ λ θ r 1 z at λ z of 1.3, as shown in Fig. 5C and D. On the other hand, circumferential Cauchy stresses increased significantly from the intima layer to the interface between media and adventitia layers, dropped abruptly at the interface, and increased slightly towards the outer boundary of the adventitia layer. In contrast, absolute values of radial Cauchy stress decreased continuously from the inner to outer boundaries of the entire vessel wall albeit the slope in the IM layer was much higher than that in the adventitia layer. There were the highest radial Cauchy stress and stretch, but the lowest circumferential Cauchy stress and stretch at the inner wall of the IM layer. Hence, it is necessary to consider the mechanical stimuli of both circumferential and radial stresses and stretches to the atherosclerosis-prone intima layer.
K + -induced SMC contraction significantly reduced circumferential Cauchy stresses in both IM and adventitia layers, but had relatively negligible effects on radial Cauchy stresses. Microstructure observations have shown that radial tilt angle of SMCs is about 8° and much smaller than tilt angles in other directions 5,[30][31][32] , which supports the present theoretical predictions. The increased ratio of radial to circumferential Cauchy stresses in active state further indicated the significance of radial stresses in the study of vascular remodeling.
Potential implications for coronary artery mechanics in pressure overload. Wang and Kassab have shown that an increase in opening angle due to acute pressure increase shifts excessive circumferential Cauchy stresses from the IM layer to the adventitia layer 33,34 . Here, we used the two-layer model with 3D active The results were computed when the 3D active strain energy function was applied to the IM layer and the 3D passive strain energy function was applied to the entire vessel wall (h = 0.22 mm with the thickness of IM layer equal to h and passive strain energy functions to compute the transmural distribution of circumferential and radial active stresses. Given the uniform circumferential stress hypothesis 35,36 , the transmural pressure and wall thickness in pressure overload were set to 160 mmHg and 0.44 mm (twofold increase from those at the normal state), respectively. This significantly increased the ratio of radial to circumferential stresses (an increase from 1 7 to 1 2 at the intima layer) owing to the constant circumferential stress and the increased radial Cauchy stress that is proportional to the increase in blood pressure, as shown in Fig. 6. Moreover, we simulated the transmural distribution of active stresses with the increased ratio of IM to Adventitia layer thicknesses, which further increased the ratio of radial to circumferential stresses. This illustrates the importance of radial stresses to coronary artery mechanics in pressure overload.
A comparison with the 2D active strain energy function. Wang et al. showed the 3D passive mechanical properties in the IM layer and entire vessel wall of porcine coronary arteries and indicated that the 3D passive model served as a foundation for formulation of layer-specific boundary value problems 26 . Here, we showed the 3D active mechanical properties in the IM layer. In comparison with the 2D active model 17,18 , the 3D active model revealed the following features: 1) The fit to experimental measurements was improved; 2) The radial term significantly affected the peak magnitude of the radial active stress and the corresponding stretch, but only led to a slight stretch change in correspondence to the peak circumferential active stress ( Fig. 4; 3) σ σ <  ).
advantage of the 3D active strain energy function over the 2D active model in a previous study 17 in the study of vascular mechanics under pressure overload (i.e., mimic hypertension).
Critique of the study. Although this study investigated 3D active and passive mechanical properties in normal directions, the stresses in shear directions were neglected. Multiple CADs (e.g., atherosclerosis, aneurysm, and vessel dissection) change both morphometry and components of vessel wall and affect the distribution of stresses in all three directions. The altered ratio of radial to circumferential stresses may result in the relief or deterioration of these CADs. Hence, the following studies should include stresses in all normal and shear directions of healthy and diseased vessels.

Conclusions
Material constants of 3D active and passive strain energy functions were determined in the entire vessel wall and IM layers of porcine RCAs. A 3D mechanical analysis demonstrated a gradual decrease of active and passive 1 st PK stresses in the order of circumferential, axial and radial directions. Moreover, a two-layer model showed that Cauchy stresses were discontinuous in the circumferential direction, but continuous in the radial direction. The ratio of radial to circumferential Cauchy stress was the highest at the inner boundary of the IM layer, which significantly increased in contraction. This study enhances our understanding the distribution of intramural stresses in vessel wall.