Comparison and critical analysis of invariant-based models with respect to their ability in fitting human aortic valve data

Abstract

With the increase of life expectancy and population average age, heart valve diseases have become more frequent, representing an always increasing percentage among cardiovascular diseases, which are the predominant cause of death in the western country. For this reason, research activities within such a context and, in particular, computer-based predictions of valve behavior are strongly motivated. Consequently, the study of the tissue mechanical response and the constitutive relationships for modeling material behavior represent crucial a aspect to be investigated in order to perform realistic simulations. The mechanical response of the aortic valve tissue depends on the contribution, composition, and interaction of different constituents, such as collagen fibers and elastin network. Accordingly, constitutive laws including non-linearity and anisotropy are necessary. Clearly, the complexity of a constitutive model increases more as it takes into account the histological structure of the tissue. Numerous constitutive models have been developed to describe arterial tissue, but relatively few models have been calibrated specifically for the aortic valve. This study focuses on the investigation of constitutive models so far proposed in the literature which could be suitable to capture the mechanical behavior of the aortic valvular tissue. To make the right choice, the comparison between these constitutive models is done in terms of the fitting quality achieved with respect to human aortic valve data proposed in the literature. For this purpose, an optimization technique based on the nonlinear least square method is used. The obtained material parameters could be later used in finite element analysis adopted, in this last decade, as an innovative approach to support the operation planning procedure and the design of artificial grafts.

A stress components

1.1 A.1 Stresses in aortic leaflets

Aortic leaflet tissue is a material reinforced with a single class of fibers circularly oriented. We select a base vectors (e ₁,  e ₂,  e ₃) with the plane e ₁–e ₂ coincident with the plane of the specimen. The unit vector e ₃ is normal to the sheet plane. In the plane e ₁–e ₂, the unit vector e ₁ lies coaxial with the mean fiber direction a ₀ (i.e., the circumferential direction) while the unit vector e ₂ = e ₃ × e ₁. We assume the one-fiber family embedded in the plane e ₁–e ₂, so that the fiber direction is a ₀≡e ₁, in the reference configuration, and a = F e ₁, in the current configuration.

In terms of the principal stretches, λ₁, λ₂, λ₃, which satisfy the incompressibility constraint λ₃ = (λ₁λ₂)⁻¹, the invariants I ₁ and I ₄ are given by:

$$ I_1=\lambda_1^2+\lambda_2^2+(\lambda_1\lambda_2)^{-2}, \quad\quad I_4=\lambda_1^2. $$

(13)

The non-zero components of the Cauchy stress tensor, σ₁₁ and σ₂₂, are given by:

$$ \sigma_{11}= 2{\Uppsi}_1\left[\lambda_1^2-(\lambda_1\lambda_2)^{-2}\right] + 2{\Uppsi}_4\lambda_1^2, \quad \quad \sigma_{22}= 2{\Uppsi}_1\left[\lambda_2^2-(\lambda_1\lambda_2)^{-2}\right]. $$

(14)

Noting that the Lagrange multiplier $p=2{\Uppsi_1}(\lambda_1\lambda_2) ^{-2}$ has been determined from the plain stress condition σ₃ = 0.

In the particular case of uniaxial tensile test, the stress σ₂₂ = 0, then, it follows from Eq. (14)₂ that λ₂ = λ ^−1/2₁ and from Eq. (14)₁ that the only one non-zero stress component is $\sigma_{11}=2{\Uppsi}_1\left[ \lambda_1^2-\lambda_1^{-1}\right]+2{\Uppsi}_4\lambda_1^2$.

1.2 A.2 Stresses in aortic sinuses

Aortic sinuses tissue is a material reinforced with two class of fibers symmetrically oriented with respect to the circumferential direction at angles ±ϑ in the reference configuration.

We select a base vectors (e ₁, e ₂, e ₃) so that the e ₁–e ₂ plane coincides with the sheet plane where the fibers are embedded and the unit vector e ₃ is normal to this plane. Hence, in the reference configuration, the fiber directions are: ${\bf a}_0= \cos\vartheta{\bf e}_1+\sin\vartheta{\bf e}_2$ and ${\bf b}_0=\cos\vartheta{\bf e}_1-\sin \vartheta{\bf e}_2,$ whereas their spatial counterparts become: ${\bf a}=\lambda_1 \cos\vartheta{\bf e}_1+\lambda_2\sin\vartheta{\bf e}_2$ and ${\bf b}=\lambda_1\cos \vartheta{\bf e}_1-\lambda_2\sin\vartheta{\bf e}_2$, respectively.

In terms of the principal stretches, λ₁,  λ₂,  λ₃ = (λ₁λ₂)⁻¹, the invariants I ₁ I ₄ and I ₆ are given by:

$$ I_1=\lambda_1^2+\lambda_2^2+(\lambda_1\lambda_2)^{-2}, \quad\quad I_4=I_6=\lambda_1^2\cos^2\vartheta+\lambda_2^2\sin^2\vartheta. $$

(15)

Assuming that the to fiber classes are mechanically equivalent such that ${\Uppsi}_1={\Uppsi}_2$, the non-zero components of the Cauchy stress tensor, σ₁₁ and σ₂₂, are given by:

$$ \sigma_{11}= 2{\Uppsi}_1\left[\lambda_1^2-(\lambda_1\lambda_2)^{-2}\right] + 2{\Uppsi}_4\lambda_1^2\cos^2\vartheta, \quad\quad \sigma_{22}= 2{\Uppsi}_1\left[\lambda_2^2-(\lambda_1\lambda_2)^{-2}\right] + 2{\Uppsi}_4\lambda_1^2\sin^2\vartheta. $$

(16)