Stability of membrane elastodynamics with applications to cylindrical aneurysms. (English) Zbl 1330.74101

Samuelson, A.; Seshaiyer, P.

J. Appl. Math. 2011, Article ID 906475, 24 p. (2011).

Summary: The enlargement and rupture of intracranial and abdominal aortic aneurysms constitutes a major medical problem. It has been suggested that enlargement and rupture are due to mechanical instabilities of the associated complex fluid-solid interaction in the lesions. In this paper, we examine a coupled fluid-structure mathematical model for a cylindrical geometry representing an idealized aneurysm using both analytical and numerical techniques. A stability analysis for this subclass of aneurysms is presented. It is shown that this subclass of aneurysms is dynamically stable both with and without a viscoelastic contribution to the arterial wall.

MSC:

74K15	Membranes
34D20	Stability of solutions to ordinary differential equations
76Z05	Physiological flows
92C35	Physiological flow

References:

[1]	J. D. Humphrey, Cardiovascular Solid Mechanics: Cells,Tissues, and Organs, Springer, 2002.
[2]	Cerebral Aneurysm Fact Sheet: National Institute of Neurological Disorders and Stroke (NINDS), 2011, http://www.ninds.nih.gov/disorders/cerebral_aneurysm/detail_cerebral_aneurysms.htm.
[3]	J. C. Daniel, A. Tongen, D. P. Warne, and P. G. Warne, “A 3D nonlinear anisotropic spherical inflation model for intracranial saccular aneurysm elastodynamics,” Mathematics and Mechanics of Solids, vol. 15, no. 3, pp. 279-307, 2009. · Zbl 1197.74073
[4]	J. D. Humphrey and C. A. Taylor, “Intracranial and abdominal aortic aneurysms: similarities, differences, and need for a new class of computational models,” Annual Review of Biomedical Engineering, vol. 10, pp. 221-246, 2008.
[5]	M. J. H. Wermer, I. C. van der Schaaf, A. Algra, and G. J. E. Rinkel, “Risk of rupture of unruptured intracranial aneurysms in relation to patient and aneurysm characteristics: an updated meta-analysis,” Stroke, vol. 38, no. 4, pp. 1404-1410, 2007.
[6]	P. N. Watton and N. A. Hill, “Evolving mechanical properties of a model of abdominal aortic aneurysm,” Biomechanics and Modeling in Mechanobiology, vol. 8, no. 1, pp. 25-42, 2009.
[7]	G. M. Austin, W. Schievink, and R. Williams, “Controlled pressure-volume factors in the enlargement of intracranial aneurysms,” Neurosurgery, vol. 24, no. 5, pp. 722-730, 1989.
[8]	N. Akkas, “Aneurysms as a biomechanical instability problem,” in Biomechanical Transport Processes, F. Mosora, Ed., pp. 303-311, Plenum Press, 1990.
[9]	J. D. Humphrey and P. B. Canham, “Structure, mechanical properties, and mechanics of intracranial saccular aneurysms,” Journal of Elasticity, vol. 61, no. 1-3, pp. 49-81, 2000. · Zbl 0973.92016
[10]	A. D. Shah and J. D. Humphrey, “Finite strain elastodynamics of intracranial saccular aneurysms,” Journal of Biomechanics, vol. 32, no. 6, pp. 593-599, 1999.
[11]	G. David and J. D. Humphrey, “Further evidence for the dynamic stability of intracranial saccular aneurysms,” Journal of Biomechanics, vol. 36, no. 8, pp. 1143-1150, 2003.
[12]	A. E. Green and W. Zerna, Theoretical Elasticity, Clarendon Press, Oxford, UK, 1954. · Zbl 0056.18205
[13]	H. Kraus, Thin Elastic Shells, John Wiley and Sons, New York, NY, USA, 1967. · Zbl 0266.73052
[14]	J. G. Simmonds, A Brief on Tensor Analysis, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1982. · Zbl 0482.53011
[15]	G. G. Ferguson, “Direct measurement of mean and pulsatile blood pressure at operation in human intracranial saccular aneurysms,” Journal of Neurosurgery, vol. 36, no. 5, pp. 560-563, 1972.
[16]	W. R. Milnor, Hemodynamics, Williams and Wilkins, Baltimore, Md, USA, 2nd edition, 1989. · Zbl 0354.76079
[17]	Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York, NY, USA, 1993.
[18]	P. Freitas, “On the effect of sharp rises in blood pressure in the Shah-Humphrey model for intracranial saccular aneurysms,” Biomechanics and Modeling in Mechanobiology, vol. 8, no. 6, pp. 457-471, 2009.
[19]	P. Seshaiyer and J. D. Humphrey, “On the potentially protective role of contact constraints on saccular aneurysms,” Journal of Biomechanics, vol. 34, no. 5, pp. 607-612, 2001.