Analytical and numerical investigations of the collapse of blood vessels with nonlinear wall material embedded in nonlinear soft tissues

In this paper, shapes of nonlinear blood vessels, surrounded by nonlinear soft tissues, and buckled due to radial pressure are solved for analytically and numerically. The blood flow rates through the bucked shapes are then computed numerically. A Fung-type isotropic hyperelastic stress-strain constitutive equation is used to establish a nonlinear mathematical model for radial buckling of blood vessels. The surrounding tissues are modeled as non-linear springs. Novel formulas for critical buckling pressures are derived analytically from the bifurcation analysis. This analysis shows that the nonlinearity of vessel's wall increases the critical buckling pressure. A numerical differential correction scheme is introduced to solve for post-buckling shapes. And the corresponding blood flow rates are provided before touching of the collapsed walls. The blood flow rate through a one-point wall-touching case is also provided. Numerical results show that both vessel's wall and soft tissues nonlinearities increase, locally, the flow rate through the buckled blood vessels. More importantly, a nonlinear relation between blood flow rate and the soft tissue spring constants is found.