A Structural Constitutive Model For Collagenous Cardiovascular Tissues Incorporating the Angular Fiber Distribution

Contribution of the hydrostatic pressure (p), matrix (τm) and fibers (τf) to the total Cauchy stress (σ) in the circumferential (a) and radial (b) direction. These curves are obtained by simulating an equibiaxial test (σc=σr=400kPa) using the model parameters for the 0 mm Hg fixed leaflet (Table 2). The matrix and fiber contribution are here defined as τm=τ̂−∑i=1Nϕfi(e⃗fi∙τ̂∙e⃗fi)e⃗fie⃗fi and τf=∑i=1Nϕfiψfie⃗fie⃗fi. From these curves, it can be seen that the presence of the hydrostatic pressure and matrix material only slightly affects the mechanical response. Note that in the model of Billiar and Sacks (24), from which our parameters are obtained, only the fiber contribution is present.

Stress-free reference configuration of the arterial wall. Indicated are the opening angle (α), the inner (Ri) and outer radius (Ro), and the thickness of the medial (hM) and adventitial layer (hA).

Finite element mesh of the stented valve geometry. Because of symmetry only one half of a leaflet is used in the FE computations. This part of the geometry is subdivided into 224 hexahedral elements with 2847 nodes. The stent, a frame on which the valve is mounted to provide support for the leaflets, is assumed to be rigid.

Sketch of the aortic valve showing the loading and boundary conditions. The gray area represents the aortic side of the leaflet to which the diastolic pressure is applied. A contact surface is defined to model coaptation of adjacent leaflets.

Schematic representation of the fiber direction (e⃗f0) in a local coordinate system, spanned by the vectors v⃗1,v⃗2 and v⃗3. β denotes the angle between e⃗f0 and v⃗3, whereas the angle between e⃗f0 and v⃗1 in the plane spanned by v⃗1 and v⃗2 is given by γ.

Schematic representation of the modeled fiber architecture in the arterial wall. Two normal probability distributions are used for both the media (top) and the adventitia (bottom). The fibers represent symmetrically arranged helices. ϕf is the fiber content and γ denotes the angle with the first local direction v⃗1, which coincides with the circumferential direction.

Schematic representation of the modeled fiber architecture in the aortic valve. The first and second local directions are indicated in the figure [v⃗1 (right) and v⃗2 (left), respectively]. The first local direction is calculated from the principal stretch directions in an isotropic leaflet. The third local direction (v⃗3) is assumed to coincide with the normal direction of the leaflet. The second local direction (v⃗2) is subsequently constructed perpendicular to v⃗1 and v⃗3. γ denotes the angle with v⃗1 and one single probability distribution is defined for the fiber content (ϕf) at each location of the leaflet.

Internal pressure pi (a) and axial force Fz (b) as a function of the inner radius ri during inflation of the artery. The axial force is calculated using Eq. 17.

Principal Cauchy stresses σrr,σθθ and σzz as a function of the deformed radius r at pi=13.33kPa. (a) is obtained with the standard values of the parameters (μ1M=−μ2M=30°, μ1A=−μ2A=60°, σ1M=σ2M=σ1A=σ2A=10° and α=0°), (b) with increased standard deviations of the fiber distributions (σ1M=σ2M=σ1A=σ2A=40°), (c) with decreased mean values (μ1M=−μ2M=15° and μ1A=−μ2A=30°) and (d) with the presence of residual stresses (α=160°).

Evolution of the leaflet’s symmetry line (x=0cm in Fig. 2) in the (y,z) plane during the pressure application. The maximum and minimum value of y correspond to the fixed edge and free edge (nodulus of Arantius), respectively. (a) is obtained with the parameters for the 0 mm Hg fixed leaflet (k1=5.35kPa, k2=5.85[−] and σ=16.1°), (b) with the parameters for the 4 mm Hg fixed leaflet (k1=55.3kPa, k2=5.75[−] and σ=14.9°) and (c) with the parameters for the fresh native leaflet (k1=0.7kPa, k2=9.9[−] and σ=10.7°).

Evolution of the stretches and stresses in the circumferential and radial directions for a node in the belly region of the valve leaflet. (a) is obtained with the parameters for the 0 mm Hg fixed leaflet, (b) with the parameters for the 4 mm Hg fixed leaflet and (c) with the parameters for the fresh native leaflet.

Distributions of the maximum principal stretch [−] (left) and Cauchy stress [kPa] (right) at the aortic side of the leaflet for p=12kPa. (a) is obtained with the parameters for the 0 mm Hg fixed leaflet, (b) with the parameters for the 4 mm Hg fixed leaflet and (c) with the parameters for the fresh native leaflet.