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Three-Dimensional Contractile Mechanics of Artery Accounting for Curl of Axial Strip Sectioned from Vessel Wall

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Abstract

Purpose

It is well known that a sliced ring of arterial wall opens by a radial cut. An axial strip sectioned from arterial wall also curls into an arc. These phenomena imply that there exist residual strains in the circumferential and axial directions. How much do the axial residual strains affect the stress distributions of arterial wall? The aim of the present study is to know stress distributions of arterial wall with the residual strains under the passive and constricted conditions.

Methods

We analyzed the stress distributions under passive and constricted conditions with considering a Riemannian stress-free configuration. In the analysis, we used strain energy functions to describe the passive and active mechanical properties of artery.

Results

The present study provided distributions of stretch ratio with reference to the stress-free state (Riemannian stress-free configuration) and stress with and without the curl of axial strip of a homogenous cylindrical arterial model under the passive and constricted smooth muscle conditions. The circumferential and axial stresses with activated smooth muscle (noradrenaline 10−5 M) at the intraluminal pressure 16 kPa and the axial stretch ratio 1.5 with reference to the unloaded vessel decreased by 3.5 and 13.8% at the inner surface with considering the axial residual strain, respectively.

Conclusions

We have shown that the Riemannian stress-free configuration is appropriate tool to analyze stress distributions of arterial wall under passive and activated conditions with the residual stresses.

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Correspondence to Keiichi Takamizawa.

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Conflict of interest

This work was conducted by a now retired researcher of National Cerebral and Cardiovascular Center Research Institute in Osaka without support from any funding organization. The author declares that he has no conflict of interest.

Ethical Approval

All care and use of laboratory animals followed JALAS (Japanese Association for Laboratory Animal Science) Guidelines on Animal Experimentation (in Japanese).15 The experiments were performed by the author in 1992 at National Cardiovascular Center Research Institute (National Cerebral and Cardiovascular Center Research Institute since 2010).

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Associate Editor Hwa Liang Leo oversaw the review of this article.

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Appendix

Appendix

In the present analysis, the Riemannian stress-free configuration23,27,28 was used for the reference state. This is not the necessary choice to analyze the strains and stresses in the artery because the local stress-free configurations are also available. It may be decided by using the Riemann–Christoffel tensor whether the Euclidean global stress-free configuration exists or not. If the axial strip sectioned from the vessel wall does not curl, there exists the global Euclidean stress-free configuration, i.e., a straight tube with an axial slit is that.23

The covariant components of the Riemann-Christoffel tensor21 for the stress-free configuration are expressed as follows:

$$ {{R}_{\alpha \beta \mu \nu}}=\frac{1}{2}\left(\frac{{{\partial}^{2}}{{\eta}_{\alpha \nu}}}{\partial {{\xi}^{\beta}}\partial {{\xi}^{\mu}}}+\frac{{{\partial}^{2}}{{\eta}_{\beta \mu}}}{\partial {{\xi}^{\alpha}}\partial {{\xi}^{\nu}}}-\frac{{{\partial}^{2}}{{\eta}_{\alpha \mu}}}{\partial {{\xi}^{\beta}}\partial {{\xi}^{\nu}}}-\frac{{{\partial}^{2}}{{\eta}_{\beta \nu}}}{\partial {{\xi}^{\alpha}}\partial {{\xi}^{\mu}}} \right)+{{\eta}^{\delta \varepsilon}}({{\varGamma}_{\delta \alpha \nu}}{{\varGamma}_{\varepsilon \beta \mu}}-{{\varGamma}_{\delta \alpha \mu}}{{\varGamma}_{\varepsilon \beta \nu}}) $$
(20)

where \( {{\eta}^{\delta \varepsilon}} \) represent the reciprocal components of the metric tensor. In the three-dimensional space, there exist distinct covariant components of the Riemann–Christoffel tensor \( {{R}_{1212}}={{R}_{\vartheta \zeta \vartheta \zeta}},\ {{R}_{1313}}={{R}_{\vartheta \rho \vartheta \rho}},{{R}_{2323}}={{R}_{\zeta \rho \zeta \rho}},{{R}_{1213}}={{R}_{\vartheta \zeta \vartheta \rho}},{{R}_{2123}}={{R}_{\zeta \vartheta \zeta \rho}} \).21 Therefore, we must check these components to know whether the Riemannian stress-free configuration is Euclidean or not. If there is a global Euclidean stress-free configuration, all components vanish everywhere.

The length s of a curve \( {{\xi}^{\alpha}}={{\xi}^{\alpha}}(\tau)\ (\alpha =1,\ 2,\ 3) \) in the Riemannian stress-free configuration is defined as follows:

$$ s=\int_{{{\tau}_{1}}}^{{{\tau}_{2}}}{\sqrt{{{\eta}_{\alpha \beta}}\frac{d{{\xi}^{\alpha}}}{d\tau}\frac{d{{\xi}^{\beta}}}{d\tau}}}d\tau $$
(21)

Here, \( {{\eta}_{\alpha \beta}} \) are functions of only \( \rho \in [{{\rho}_{i}},\ {{\rho}_{o}}] \) in the present case. The length of \( {{\Theta}_{0}}R \) with the circular arc is provided as follows (see Fig. 1 and Eq. (1)):

$$ {{\Theta}_{0}}R=\int_{0}^{2\pi}{\sqrt{{{\eta}_{\vartheta \vartheta}}}}d\theta =2\pi \sqrt{{{\eta}_{\vartheta \vartheta}}} $$
(22)

In the same way, we obtain as follows:

$$ \begin{aligned} {{\Psi}_{0}}S&=\int_{0}^{{{l}_{u}}}{\sqrt{{{\eta}_{\zeta \zeta}}}}d\zeta =\sqrt{{{\eta}_{\zeta \zeta}}}{{l}_{u}},\\ R-{{R}_{i}}={{S}_{i}}-S&=\int_{{{\rho}_{i}}}^{\rho}{\sqrt{{{\eta}_{\rho \rho}}(\rho^{\prime})}}\ d\rho^{\prime} \end{aligned} $$
(23)

The components of the metric tensor for the Riemannian stress-free configuration are provided from Eqs. (23) and (24):

$$ \begin{aligned} {{\eta}_{\vartheta \vartheta}}&={{\left(\frac{{{\Theta}_{0}}R}{2\pi} \right)}^{2}},\ {{\eta}_{_{\zeta \zeta}}}={{\left(\frac{{{\Psi}_{0}}S}{{{l}_{u}}} \right)}^{2}},\\ {{\eta}_{\rho \rho}}&={{\left(\frac{dR}{d\rho} \right)}^{2}}={{\left(\frac{dS}{d\rho} \right)}^{2}},\\ {{\eta}_{\alpha \beta}}&=0\quad (\alpha \ne \beta) \end{aligned} $$
(24)

Is the coordinate system for the Riemannian stress-free configuration \( \left\langle {{\xi}^{\alpha}} \right.;\ \left. {{\eta}_{\alpha \beta}} \right\rangle \) Euclidean? We calculated six components of the Riemann–Christoffel tensor. The result was as follows:

$$ {{R}_{\vartheta \zeta \vartheta \zeta}}={{\left(\frac{{{\Theta}_{0}}{{\Psi}_{0}}}{2\pi {{l}_{u}}} \right)}^{2}}RS>0 $$
(25)

The other five components were equal to zero. Therefore, the Riemannian stress-free configuration \( \kappa (B) \) with the metric tensor \( {{\eta}_{\alpha \beta}} \) is non-Euclidean although the other five components vanish. This means that there is no global Euclidean stress-free configuration. If there is no curling of axial strip, Eq. (25) becomes zero. It is evident that \( {{R}_{\vartheta \zeta \vartheta \zeta}} \) approaches to zero because \( {{\Psi}_{0}} \) approaches to zero, S infinitely increases, and the product \( {{\Psi}_{0}}S \) is almost maintaining \( {{l}_{u}} \). In this case there is a global Euclidean stress-free configuration, i.e., a tubular segment with a slit may be the stress-free state.

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Takamizawa, K. Three-Dimensional Contractile Mechanics of Artery Accounting for Curl of Axial Strip Sectioned from Vessel Wall. Cardiovasc Eng Tech 10, 604–617 (2019). https://doi.org/10.1007/s13239-019-00434-1

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