Applied Math Colloquium
Nonlinear Dynamics of Aortic Prostheses Conveying Pulsatile Flow
The nonlinear dynamics of thin-walled shells conveying pulsatile flow is of particular interest in cardiovascular biomedical applications.
In this talk, I will discuss the dynamical behavior of woven Dacron thoracic prostheses subjected to physiological pulsatile blood flow and pressure. The artificial vessel is modeled with the nonlinear Novozhilov shell theory applied to an orthotropic circular cylindrical slightly corrugated shell. In Dacron implants, surface waves of the corrugation are in longitudinal direction and they can be modelled introducing a sinusoidal geometric imperfection on the circular cylindrical shell geometry for both axial and radial displacement. A pulsatile time-dependent blood flow model is considered by applying physiological waveforms of velocity and pressure during the heart beating period approximated through Fourier series with eight harmonics. The fluid is assumed to be Newtonian and the pulsatile flow is formulated using a hybrid model that contains the unsteady effects obtained from the linear potential flow theory and the pulsatile viscous effects obtained from the unsteady time-averaged Navier-Stokes equations. Residual stresses because of pulsatile pressurization are evaluated and included in the model. Coupled fluid-structure Lagrange equations of motion for a non-material volume with wave propagation in case of pulsatile flow will be presented for shells with both fixed boundary conditions and boundary conditions allowing radial displacement at both ends. The last ones boundary conditions are meant to reproduce the simple interrupted suture technique that can be performed by surgeons.
Several superharmonic resonance peaks appear in the physiological frequency range by including higher harmonics in the Fourier expansion of the physiological waveforms of pressure and velocity. Large amplitude oscillations are observed in the physiological frequency range during exercise conditions. Since vibrations of the artificial vessel walls are activated for certain heart rates, the related high stress concentration combined with the fatigue cycles of the heart beats, could reduce the long term patency of the prosthesis.
For very low damping values, flow-induced asymmetric vibration of the aortic prosthesis is possible in case of fixed boundary conditions. A period-doubling bifurcation appears at HR = 191.4 bpm giving a dynamic instability characterized by a periodic response with two times the excitation period (2T). This vibration can cause high stress concentration which, combined with the fatigue cycles of the heart beats, could contribute to material deterioration. In addition, by simply approximating the pulsatile blood flow velocity and pressure with only the first term of the Fourier series, geometrically nonlinear vibrations show interesting and intricate nonlinear dynamics (chaos, amplitude modulation and period-doubling bifurcations) for frequencies out of the physiological range. Results will be presented via frequency-response curves, time histories, bifurcation diagrams and Poincaré maps.
The growing understanding of the dynamic behavior of vascular prostheses currently used in clinical practice could help controlling their common long-term adverse effects and could inspire the design of a new generation prosthesis with physiological properties more similar to the host arteries.