Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture


 Elastodynamics, Filtration, Navier-Stokes, Numerical Solutions, Permeability, Poisson Ratio, Aneurysm.

How to Cite

Tor A. Kwembe, & Ashley M. Sanders. (2014). Modeling Nontraumatic Aneurysm Evolution, Growth and Rupture. Journal of Basic & Applied Sciences, 10, 393–413.


We have presented a mathematical model to study the evolution, growth and risk rupture of nontraumatic aneurysms contained within a cylindrical region of blood vessels. Analytical and numerical solutions are studied. Results affirmed that the intra-aneurysmal pressure and bloodstream flow account for the evolution and growth of aneurysms, and we find that an aneurysm may rupture when the ratio of the lateral membrane contraction to longitudinal membrane extension approaches one. Numerical properties of intra-aneurysmal pressure, impact fluid velocity, membrane displacement and the deformed radius with respect to the Poisson ratio, membrane thickness and extensional rigidity are studied. The importance of the findings is rested on the fact that they can be used to improve noninvasive means for predicting aneurysm rupture, and treatment and management decisions after rupture.


Camenschi G. A Mathematical Model of the Permeable Transporting Systems. Rev. Roum. Math. PUres et Appli, Tome XXVIII, Bucarest 1983. 1981; 4: 275-82.

Fung YC. Biomechanics mechanical properties of of living tissues. Springer Verlag, New York, Inc. 1993.

Ferguson GG. Physical factors in the initiation, growth, and rupture of human intracranial saccular aneurysms. J Neurosurg 1972; 37: 666-77.

Kwembe TA, Jones SN. A mathematical model of cylindrical shaped aneurysms. Biomat 2006, Proceedings of the International Symposium on Mathematical and Computational Biology. Rio de Janeiro, Brazil. world Scientific Publishers 2006; pp. 35-48.

Sekhar LN, Heros RC. Origgin, growth, and rupture of saccular aneurysms: a review. Neurosurgery 1981; 8: 248-60.

Shah AD, Humphrey JD. Finite strain elastodynamics of intracranial saccular aneurysms. J Biomech 1999; 32: 593-500.

Stehbans WE. Etiology of intracranial berry aneurysms. J Neurysurg 1989; 70: 823-31.

Steiger HJ. Pathophysiology of development and rupture of cerebral aneurysms. Acta Neurochir Suppl 1990; Wein: 48: 1-57.

Isaken JG, Bazileus Y, Kuamsdal T, Zhang Y, Kaspersen JH, Waterloo K, Romner B, Ingebrigtsen T. Determination of wall tension in cerebral artery aneurysm by numerical simulation. Stroke 2008; 39: 3172-78.

Shojima M, Oshima M, Takagi K, Torii R, Nagata K, Ichiro S, Morita A, Kirino T. Role of the bloodstream impacting force and the local pressure elevation in the rupture of cerebral aneurysms. Stroke 2005; 36: 1933-38.

Taylor AJ, Bobik A, Berndt MC, Ramsay D, Jennings G. Experimental rupture of artherosclerotic lesions increases distal vascular resistance: A limiting factor to the success of infarct angioplasty. Arterioscler Thromb Vasc Biol 2002; 22: 153-60.

Michel CC, Curry FE. Microvascular Permeability. Physiol Rev 1999; 79: 703-61.

Moftaker R, Aagaard-Kienitz B, Johnson K, Turki PA, Turk AS, Niemann DB, Consigny D, Grinde J, Wiebeno, Mistretto CA. Noninvasive Measurement of Intra-aneurysmal Pressure and Flow Pattern using Phase Contrast With Vastly Undersampled Isotropic Projection Imaging. AJNR Am J Neuroradiol 2007; 28: 1710-14.

Darcy H. Les Fontaines Publiques de la ville de Dijon, Dalmont, Paris 1856.

Lee T, Lakes RS. Anisotropic polyurethane foam with Poisson’s ratio greater than 1. J Mater Sci 1997; 32: 2397-401.

Mazumdar J. An introduction to mathematical physiology and biology, 2nd ed. New York 1999; pp. 136-161.

Benderson J, Connolly E, Batjer H, et al, Guidelines for the management of aneurysmal subarachnoid hemorrhage: a statement for healthcare professionals from a special writing group of the Stroke Council, American Heart Association. Stroke 2009; 40(3): 994-1025.

Ting TCT, Chen T. Poisson’s ratio for anisotropic elastic materials can have no bounds. Q J Mechanics Appl Math 2005; 58(1): 73-82.

Brisman JL, Song JK, Newell DW. Cerebral aneurysms. N Engl J Med 2006; 355(9): 928-39.

Greaves GN, Greer AL, Lakes RS, Rouxel T. Poisson’s Ratio and Modern Materials. Nat Mater 2011; 10: 823-36.

Lakes RS, Wineman A. On Poisson’s Ration in Linearly Viscoelastic Solids. J Elasticity 2006; 85: 45-63.

Norris AN. Extreme values of Poisson’s ratio and other engineering moduli in anisotropic materials. J Mechan Mater Struct 2006; 1(4): 793-12.

Olafsson E, Hauser WA, Gudmundsson G. A population-based study of prognosis of ruptured cerebral aneurysm: mortality and recurrence of subarachnoid hemorrhage. Neurology 1997; 48(5): 1191-95.

West J. Respiratory Physiology: the essentials – 9th edition. Baltimore: Lippincott Williams & Wilkins 2012; p. 177.

Michel CC. Fluid exchange in the microcirculation. Physiol 2004; 557(3): 701-702.

Michel CC. Microvsular permeability, ultrafiltration, and restricted diffusion. Am J Physiol Heart Circ Physiol 2004; 287: H1887-H1888.

Fick A. Annalen der Physik 1855; 170(1): 59-86.

Nagy JA, Benjamin L, Zeng H, Dvorak AM, Dvorak HF. Vascular permeability, vascular hyperpermeability, and angiogenesis. Angiogenesis 2008; 11: 109-19.

Sample C, Golovin AA. Morphological and chemical oscillations in a couple-membrane system. SIAM J Appl Math 2011; 71: 622-34.

Sample C, Golovin AA. Nonlinear dynamics of a double bilipid membrane. Phys Rev E 2007; 76: article 031925.

Kyriacou SK, Humphrey JD. Influence of size, shape and properties on the mechanics of axisymmetric saccular aneurysms. J Biomech 29(8): 1996; 1015-22.

Milnor WR. Hemodynamics. Williams & Wilkens, Baltimore 1982.

Bender CM, Orszag SA. Advanced mathematical methods for scientists and engineers. McGraw Hill, Inc. 1978; pp. 319-361.

Segers P, Carlier S, Pasquet A, Rabben SI, Hellevik lR, Remme E, De Backer T, De Sutter J, Thomas JD, Verdonck P. Individualizing the aorta-radial pressure transfer function: feasibility of a model based approach. Am J Physiol Heart Circ Physiol 2000; 279: H542-H549.

Greenberg MS. Subarachnoid hemorrhage and aneurysms. In: Greenberg, M.S, ed. Handbook of Neurosurgery, 5th ed. Thieme Medical Publishers, Newy York 2001; pp. 754-803.

Morey SS. AHA recommendations for the management of intracranial aneurysms. Agency for Health Care Policy and Research. Am Fam Physician 2001; 63(12): 2465-66.

Fisher CM, Kristler JP, Davis JM. Relation of cerebral vasospasm to subarachnoid hemorrhage visualized by computerized tomographic scanning. Neurosurgery 1980; 6(1): 1-9.

Barker FG, Ogilvy CS. Efficacy of Prophylactic nimodipine for delayed ischemic deficit after subarachnoid hemorrhage: a metaanalysis. J Neurosurgery 1996; 84(3): 405-14.

Evans E, Mohandas N, Leung A. Static and Dynamic Rigidities of Normal and Sickle Erythrocytes: Major Influence of Cell Hemoglobin Concentration. J Clin Invest 1984; 73: 477-88.

Mohandas N, Evans E. Mechanical Properties of the red cell membrane in relation to Molecular Structure and Genetic defects. Annu Rev Biophys Biomol Struct 1994; 23: 787-18.

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