Determination of the Critical Buckling Pressure of Blood Vessels Using the Energy Approach

Abstract

The stability of blood vessels under lumen blood pressure is essential to the maintenance of normal vascular function. Differential buckling equations have been established recently for linear and nonlinear elastic artery models. However, the strain energy in bent buckling and the corresponding energy method have not been investigated for blood vessels under lumen pressure. The purpose of this study was to establish the energy equation for blood vessel buckling under internal pressure. A buckling equation was established to determine the critical pressure based on the potential energy. The critical pressures of blood vessels with small tapering along their axis were estimated using the energy approach. It was demonstrated that the energy approach yields both the same differential equation and critical pressure for cylindrical blood vessel buckling as obtained previously using the adjacent equilibrium approach. Tapering reduced the critical pressure of blood vessels compared to the cylindrical ones. This energy approach provides a useful tool for studying blood vessel buckling and will be useful in dealing with various imperfections of the vessel wall.

Appendix

For cylindrical arteries with an orthotropic nonlinear elastic wall described by the Fung-type strain energy function given in Eq. (34a), integrating the equation of equilibrium and applying the boundary conditions yield the lumen pressure as:

$$ p = \int\limits_{{r_{\text{i}} }}^{{r_{\text{e}} }} {\left[ {\left( {1 + 2E_{\theta } } \right)\left( {b_{1} E_{\theta } + b_{4} E_{z} + b_{6} E_{\text{r}} } \right) - \left( {1 + 2E_{\text{r}} } \right)\left( {b_{6} E_{\theta } + b_{5} E_{z} + b_{3} E_{\text{r}} } \right)} \right]b_{0} e^{Q} {\frac{d\xi }{\xi }}} $$

(A1)

and the axial tension N as

$$ N = \pi r_{\text{i}}^{2} p + \pi \int\limits_{{r_{\text{i}} }}^{{r_{\text{e}} }} {\left[ {2\left( {1 + 2E_{z} } \right)\left( {b_{4} E_{\theta } + b_{2} E_{z} + b_{5} E_{\text{r}} } \right) - \left( {1 + 2E_{\text{r}} } \right)\left( {b_{6} E_{\theta } + b_{5} E_{z} + b_{3} E_{\text{r}} } \right) - \left( {1 + 2E_{\theta } } \right)\left( {b_{1} E_{\theta } + b_{4} E_{z} + b_{6} E_{\text{r}} } \right)} \right]b_{0} e^{Q} rdr} $$

(A2)

When arteries buckle (bend), the flexural rigidity EI is given by13,14

$$ EI = H = \pi \int\limits_{{r_{\text{i}} }}^{{r_{\text{e}} }} {\left[ {J_{z} - \frac{1}{3}J_{\theta } } \right]b_{0} e^{Q} r^{3} dr + {\frac{{\pi \, r_{\text{i}}^{3} }}{3}}\int\limits_{{r_{\text{i}} }}^{{r_{\text{e}} }} {\left[ {J_{\theta } } \right]b_{0} e^{Q} dr} } $$

(A3)

wherein

$$ \begin{gathered} J_{z} = 2g_{z} \left[ {\left( {1 + 2E_{z} } \right)g_{z} - \left( {1 + 2E_{\text{r}} } \right)g_{r} } \right] + 2g_{z} + \left( {1 + 2E_{z} } \right)b_{2} - \left( {1 + 2E_{\text{r}} } \right)b_{5} \hfill \\ J_{\theta } = 2g_{z} \left[ {\left( {1 + 2E_{\theta } } \right)g_{\theta } - \left( {1 + 2E_{\text{r}} } \right)g_{\text{r}} } \right] + \left( {1 + 2E_{\theta } } \right)b_{4} - \left( {1 + 2E_{\text{r}} } \right)b_{5} \hfill \\ \end{gathered} $$

with

$$ \begin{array}{*{20}c} {g_{\theta } = b_{1} E_{\theta } + b_{4} E_{z} + b_{6} E_{\text{r}} } \\ {g_{z} = b_{4} E_{\theta } + b_{2} E_{z} + b_{5} E_{\text{r}} } \\ {g_{\text{r}} = b_{6} E_{\theta } + b_{5} E_{z} + b_{3} E_{\text{r}} } \\ \end{array} . $$

(A4)

Note that there was an error in one of the previous publications13 and the error has been corrected in the current equations (A3) and (A4).