Effect of Nonlinear Hyperelastic Property of Arterial Tissues on the Pulse Wave Velocity Based on the Unified-Fiber-Distribution (UFD) Model

Abstract

Pulse wave velocity (PWV) is a key, independent risk factor for future cardiovascular events. The Moens–Korteweg equation describes the relation between PWV and the stiffness of arterial tissue with an assumption of isotopic linear elastic property of the arterial wall. However, the arterial tissue exhibits highly nonlinear and anisotropic mechanical behaviors. There is a limited study regarding the effect of arterial nonlinear and anisotropic properties on the PWV. In this study, we investigated the impact of the arterial nonlinear hyperelastic properties on the PWV, based on our recently developed unified-fiber-distribution (UFD) model. The UFD model considers the fibers (embedded in the matrix of the tissue) as a unified distribution, which expects to be more physically consistent with the real fiber distribution than existing models that separate the fiber distribution into two/several fiber families. With the UFD model, we fitted the measured relation between the PWV and blood pressure which obtained a good accuracy. We also modeled the aging effect on the PWV based on observations that the stiffening of arterial tissue increases with aging, and the results agree well with experimental data. In addition, we did parameter studies on the dependence of the PWV on the arterial properties of fiber initial stiffness, fiber distribution, and matrix stiffness. The results indicate the PWV increases with increasing overall fiber component in the circumferential direction. The dependences of the PWV on the fiber initial stiffness, and matrix stiffness are not monotonic and change with different blood pressure. The results of this study could provide new insights into arterial property changes and disease information from the clinical measured PWV data.

Appendix: Stress Calculation

The Cauchy stress of the arterial tissue can be expressed as [34]

$${\varvec{\sigma}}=2\varvec{F}\frac{\partial \overline{\Psi } }{\partial \overline{\varvec{C}}}{\varvec{F} }^{T}-p\varvec{I},$$

(22)

where p is a Lagrange contribution to the hydrostatic pressure, and

$$\frac{\partial \overline{\Psi } }{\partial \overline{\varvec{C}} }= {\overline{\Psi } }_{1}\varvec{I}+ {\overline{\Psi } }_{4}{\varvec{a}}_{\theta }\otimes {\varvec{a}}_{\theta }+ {\overline{\Psi } }_{4\perp }{\varvec{a}}_{z}\otimes {\varvec{a}}_{z},$$

(23)

where

$${\overline{\Psi } }_{1}=\frac{\partial \overline{\Psi } }{\partial {\overline{I} }_{1}}, {\overline{\Psi } }_{4\theta }=\frac{\partial \overline{\Psi } }{\partial {\overline{I} }_{4\theta }}, {\overline{\Psi } }_{4z}=\frac{\partial \overline{\Psi } }{\partial {\overline{I} }_{4z}}.$$

(24)

Substituting Eqs. (1)–(3) into Eq. (24), we have

$$\begin{gathered} \overline{\Psi }_{1} = \frac{c}{2} , \hfill \\ \overline{\Psi }_{4\theta } = \delta_{\theta } k_{1} \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right) \cdot \exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}, \hfill \\ \overline{\Psi }_{4z} = \delta_{z} k_{1} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right) \cdot \exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}. \hfill \\ \end{gathered}$$

(25)

For the cylindrical artery tube under inner pressure with fixed axial pre-stretch, the deformation gradient can be expressed as $\varvec{F}=\mathrm{diag}[1/({\lambda }_{\theta }{\lambda }_{z}),{\lambda }_{\theta },{\lambda }_{z}]$. Then we have ${\overline{I} }_{4\theta }={\lambda }_{\theta }^{2}$ and ${\overline{I} }_{4z}={\lambda }_{z}^{2}$. Substituting ${\overline{I} }_{4\theta }$ and ${\overline{I} }_{4z}$ into Eq. (25) and then into Eqs. (22)–(24), together with the condition of ${\sigma }_{rr}=0$, the non-zero components of the Cauchy stress can be expressed as

$$\begin{gathered} \sigma_{\theta \theta } = c\left( {\lambda_{\theta }^{2} - \frac{1}{{\lambda_{\theta }^{2} \lambda_{z}^{2} }}} \right) + 2\delta_{\theta } k_{1} \lambda_{\theta }^{2} \zeta^{2} \left( {\lambda_{\theta }^{2} - 1} \right)\exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}, \hfill \\ \sigma_{zz} = c\left( {\lambda_{z}^{2} - \frac{1}{{\lambda_{\theta }^{2} \lambda_{z}^{2} }}} \right) + 2\delta_{z} k_{1} \lambda_{z}^{2} \left( {1 - \zeta } \right)^{2} \left( {\lambda_{z}^{2} - 1} \right)\exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}. \hfill \\ \end{gathered}$$

(26)

Further, the non-zero components of the 2nd P–K stress can be expressed as

$$\begin{gathered} S_{\theta \theta } = c\left( {1 - \frac{1}{{\lambda_{\theta }^{4} \lambda_{z}^{2} }}} \right) + 2\delta_{\theta } k_{1} \zeta^{2} \left( {\lambda_{\theta }^{2} - 1} \right)\exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}, \hfill \\ S_{zz} = c\left( {1 - \frac{1}{{\lambda_{\theta }^{2} \lambda_{z}^{4} }}} \right) + 2\delta_{z} k_{1} \left( {1 - \zeta } \right)^{2} \left( {\lambda_{z}^{2} - 1} \right)\exp \left\{ {k_{2} \left[ {\delta_{\theta } \zeta^{2} \left( {\overline{I}_{4\theta } - 1} \right)^{2} + \delta_{z} \left( {1 - \zeta } \right)^{2} \left( {\overline{I}_{4z} - 1} \right)^{2} } \right]} \right\}. \hfill \\ \end{gathered}$$

(27)