A new mathematical model for the interaction of blood flow with the arterial wall surrounded by cerebral spinal fluid is developed with applications to intracranial saccular aneurysms. The blood pressure acting on the inner arterial wall is modeled via a Fourier series, the arterial wall is modeled as a spring-mass system incorporating growth and remodeling, and the surrounding cerebral spinal fluid is modeled via a simplified one-dimensional compressible Euler equation with inviscid flow and negligible nonlinear effects. The resulting nonlinear coupled fluid-structure interaction problem is analyzed and a perturbation technique is employed to derive the first-order approximation solution to the system. An analytical solution is also derived for the linearized version of the problem using Laplace transforms. The solutions are validated against related work from the literature and the results suggest the biological significance of the inclusion of the growth and remodeling effects on the rupture of intracranial aneurysms.
In this work, we will present a mathematical model that describes a coupled fluid-structure interaction between the arterial wall, the blood flow inside the wall and the cerebral spinal fluid outside the wall with applications to intracranial saccular aneurysms. The governing system of differential equations includes a nonlinear power-law fluid equation coupled with a nonlinear elasticity equation describing the wall in conjunction with blood pressure that is modeled via a Fourier series. The thrust of this work involves the analysis and simulation of the associated mathematical model using classical differential equation techniques. Besides proving existence and uniqueness of the solution to the related system, analytical expressions for the associated traveling wave solutions for the governing nonlinear differential equation is also derived for a special class of problems that is biologically tractable.
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