Faculdade de Motricidade Humana, Department of Mathematics, TU Lisbon, Lisbon, Portugal.
Biomech Model Mechanobiol. 2009 Dec;8(6):457-71. doi: 10.1007/s10237-009-0149-2. Epub 2009 Feb 15.
We consider the model originally proposed by Shah and Humphrey (J Biomech 32:593-599, 1999) for a class of intracranial saccular aneurysms and show that for constant pressure the addition of the viscoelastic term corresponding to the presence of cerebral spinal fluid outside the membrane, no matter how small, does ensure convergence to an equilibrium. Our arguments apply to a general equation of this type, and thus also hold for variations of this model such as that proposed by David and Humphrey (J Biomech 36:1143-1150, 2003). On the other hand, it is known that the presence of damping may destabilize periodic orbits of periodically forced systems or even prevent them from existing altogether. We present numerical simulations showing that for some forcing terms the high-frequency oscillations do not die out with time, and a much more complex behaviour may emerge as a discontinuous forcing term is approached. The key point for this situation to occur is related to the derivative of the forcing term, supporting the hypothesis that sharper rises (or falls) in blood pressure may increase the risk of aneurysm rupture.
我们考虑了 Shah 和 Humphrey 最初提出的用于一类颅内囊状动脉瘤的模型，并表明对于恒定压力，即使添加了对应于膜外存在脑脊髓液的粘弹性项，无论多么小，都能确保收敛到平衡。我们的论点适用于这种类型的一般方程，因此也适用于这个模型的变体，例如 David 和 Humphrey 提出的模型（J Biomech 36:1143-1150, 2003）。另一方面，已知阻尼的存在可能会使周期性强迫系统的周期轨道失稳，甚至完全阻止它们的存在。我们提出了数值模拟结果，表明对于某些强迫项，高频振荡不会随时间衰减，并且随着接近间断强迫项，可能会出现更复杂的行为。这种情况发生的关键点与强迫项的导数有关，支持了这样一种假设，即血压的急剧上升（或下降）可能会增加动脉瘤破裂的风险。
