A mathematical model for the growth of the abdominal aortic aneurysm
We present the first mathematical model to account for the evolution of the abdominal aortic aneurysm. The artery is modelled as a two-layered, cylindrical membrane using nonlinear elasticity and a physiologically realistic constitutive model. It is subject to a constant systolic pressure and a physiological axial prestretch. The development of the aneurysm is assumed to be a consequence of the remodelling of its material constituents. Microstructural ‘recruitment’ and fibre density variables for the collagen are introduced into the strain energy density functions. This enables the remodelling of collagen to be addressed as the aneurysm enlarges. An axisymmetric aneurysm, with axisymmetric degradation of elastin and linear differential equations for the remodelling of the fibre variables, is simulated numerically. Using physiologically determined parameters to model the abdominal aorta and realistic remodelling rates for its constituents, the predicted dilations of the aneurysm are consistent with those observed in vivo. An asymmetric aneurysm with spinal contact is also modelled, and the stress distributions are consistent with previous studies.
KeywordsCollagen Fibre Arterial Wall Abdominal Aorta Abdominal Aortic Aneurysm Fibre Angle
P. N. Watton gratefully acknowledges the award of a Research Studentship funded by the UK Medical Research Council. The authors are indebted to the Consultant Vascular Surgeons, Mr S. Dodds (Good Hope Hospital, Sutton Coldfield, UK) and Mr D.A.J. Scott (St. James’s University Hospital, Leeds, UK) for many helpful discussions about the clinical aspects and physiology of abdominal aortic aneurysms. We also acknowledge the Harwell Software Library (http://www.hsl.ac.uk ) for granting UK academics the free use of its Fortran subroutines in non-commercial applications. MA38 was employed to solve the linear system that arises in the Newton iteration, which is required to update the deformation at successive timesteps.
- Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson JD (1994) Molecular biology of the cell, 3rd edn. Garland Publishing, New YorkGoogle Scholar
- Armeniades CD, Lake LW, Missirlis YF (1973) Histological origin of aortic tissue mechanics: the role of collagenous and elastic structures. Appl Polym Symp 22:319–339Google Scholar
- Armentano R, Levenson J, Barra J, Fischer E, Breitbart G, Pichel R, Simon A (1991) Assessment of elastin and collagen contribution to aortic elasticity in conscious dogs. Amer J Physiol 60:H1870–H1877Google Scholar
- Fukui T, Matsumoto T, Tanaka T, Ohashi T, Kumagai K, Akimoto H, Tabayashi K, Sato M (2002) Biaxial tensile properties of aortic aneurysm tissues under equibiaxial stress. In: Proceedings of the world congress of biomechanics, Calgary, AlbertaGoogle Scholar
- He CM, Roach M (1993) The composition and mechanical properties of abdominal aortic aneurysms. J Vasc Surg 20(1):6–13Google Scholar
- Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, ChicesterGoogle Scholar
- Humphrey JD (2002) Cardivascular solid mechanics. Springer, Berlin Heidelberg New YorkGoogle Scholar
- Lever MJ (1995) Mass transport through the walls of arteries and veins. Biological flows. In: Jaffrin MY, Caro CG (eds) Plenum Press, New York, pp 177–197Google Scholar
- Raghavan ML, Webster M, Vorp DA (1999) Ex-vivo bio-mechanical behavior of AAA: assessment using a new mathematical model. Ann Biomed Eng 24:573–582Google Scholar
- Watton PN (2002) Mathematical modelling of the abdominal aortic aneurysm. PhD thesis, Department of Applied Mathematics, University of LeedsGoogle Scholar
- Wempner G (1973) Mechanics of solids. McGraw-Hill, New YorkGoogle Scholar