A mathematical model for the growth of the abdominal aortic aneurysm
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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.
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