Two-layer model of coronary artery vasoactivity

Abstract

Since vascular tone is regulated by smooth muscle cells in the media layer, a multilayer mechanical model is required for blood vessels. Here, we performed biaxial mechanical tests in the intima-media layer of right coronary artery to determine the passive and active properties in conjunction with the passive properties of adventitia for a full vessel wall model. A two-layer (intima-media and adventitia) model was developed to determine the transmural stress and stretch across the vessel wall. The mean ± SE values of the outer diameters of intima-media layers at transmural pressure of 60 mmHg in active state were 3.17 ± 0.16 and 3.07 ± 0.18 mm at axial stretch ratio of 1.2 and 1.3, respectively, which were significantly smaller than those in passive state (i.e., 3.62 ± 0.19 and 3.49 ± 0.22 mm, respectively, P < 0.05). The inner and outer diameters in no-load state of intima-media layers were 1.17 ± 0.09 and 2.08 ± 0.09 mm, respectively. The opening angles in zero-stress state had values of 159 ± 21° for intima-media layers and 98 ± 15° for adventitia layers, which suggests a residual strain between the two layers. There were slightly decreased active circumferential stresses (<10%), but significantly decreased active axial stresses (>25%) in the intima-media layer compared with those in the intact vessel. This suggests that the adventitia layer affects vascular contraction. The two-layer analysis showed that the intima-media layer bears the majority of circumferential tensions, in contrast to the adventitia layer, while contraction results in decreased stress and stretch in both layers.

Keywords: constitutive stress-strain relation; contraction; vessel mechanics.

Fig. 1.

Circumferential first Piola-Kirchhoff stress (T_θθ) as a function of circumferential stretch ratio (λ_θ) at axial stretch ratio (λ_z) of 1.2 (A) and 1.3 (B) measured at K⁺-induced contraction and Ca²⁺-free-induced vasodilation. Shown are total stress (□): (Pr_i/hλ_θ)_total at contraction; passive stress (△): (Pr_i/hλ_θ)_passive at vasodilation; and active stress (thin solid line) =(Pr_i/hλ_θ)_total − (Pr_i/hλ_θ)_passive. The thick dotted, dashed, and solid lines refer to the theoretical total, passive, and active stresses, respectively.

Fig. 2.

Axial first Piola-Kirchhoff stress (T_zz) as a function of λ_θ at λ_z of 1.2 (A) and 1.3 (B) measured at K⁺-induced contraction and Ca²⁺-free-induced vasodilation. Shown are total stress (□): ${\left[\frac{F}{{\lambda }_z\pi \left(r_o^2-r_{\text{i}}^2\right)}+\frac{{\mathrm{Pr}}_i^2}{{\lambda }_{z}h\left(r_{\text{o}}+r_{\text{i}}\right)}\right]}_{\text{total}}$ at contraction; passive stress (△): ${\left[\frac{F}{{\lambda }_z\pi \left(r_o^2-r_{\text{i}}^2\right)}+\frac{{\mathrm{Pr}}_i^2}{{\lambda }_{z}h\left(r_{\text{o}}+r_{\text{i}}\right)}\right]}_{\text{passive}}$ at vasodilation; and active stress (thin solid line): ${\left[\frac{F}{{\lambda }_z\pi \left(r_o^2-r_{\text{i}}^2\right)}+\frac{{\mathrm{Pr}}_i^2}{{\lambda }_{z}h\left(r_{\text{o}}+r_{\text{i}}\right)}\right]}_{\text{total}}{\left[\frac{F}{{\lambda }_z\pi \left(r_o^2-r_{\text{i}}^2\right)}+\frac{{\mathrm{Pr}}_i^2}{{\lambda }_{z}h\left(r_{\text{o}}+r_{\text{i}}\right)}\right]}_{\text{passive}}$. The thick dotted, dashed, and solid lines refer to the theoretical total, passive, and active stresses, respectively.

Fig. 3.

Circumferential active first Piola-Kirchhoff stress, T_θθactive^theory = $\frac{2C_2}{b_1\sqrt{\pi }}\exp \left[-{\left(\frac{{\lambda }_{\theta }}{b_1}+\frac{{\lambda }_z}{b_2}-b{\prime }^{\prime }\right)}^2\right]$ (A), and axial active first Piola-Kirchhoff stress, T_zzactive^theory = $\frac{2C_2}{b_2\sqrt{\pi }}\exp \left[-{\left(\frac{{\lambda }_{\theta }}{b_1}+\frac{{\lambda }_z}{b_2}-b\prime \right)}^2\right]$ (B), as a function of λ_θ at λ_z of 1.3. Parameters, C₂, b₁, b₂, and b′, have values of 7.88, 0.24, 0.43, and 8.60 for intima-medial layer (Table 2) and 4.18, 0.12, 0.18, and 20.3 for intact vessel (Table 2 in Ref. 10).

Fig. 4.

Computed transmural distribution of λ_θ and Cauchy stress (σ_θ) across the vessel wall at λ_z of 1.3 and transmural pressure of 80 mmHg using the two-layer model.

Fig. A1.

Schematic representation is shown of an intact vessel wall at no-load or loaded states (A) and zero-stress state (B). Schematic representation is shown of a single intima-media layer at no-load or loaded states (C) and zero-stress state (D). Schematic representation is shown of a single adventitia layer at no-load or loaded states (E) and zero-stress state (F). The solid lines refer to the measurements in an intact vessel wall, while the dash lines refer to those in a single layer, intima-media layer (forward slashes), or adventitia layer (backward slashes). See text for definition of terms.