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Mathematical Medicine and Biology Advance Access originally published online on February 20, 2009
Mathematical Medicine and Biology 2009 26(2):133-164; doi:10.1093/imammb/dqp001
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Modelling the growth and stabilization of cerebral aneurysms

Paul N. Watton{dagger} and Yiannis Ventikos

Institute of Biomedical Engineering, Department of Engineering Science, University of Oxford, Parks Road, OX1 3PJ Oxford, UK

Gerhard A. Holzapfel

Institute of Biomechanics, Center of Biomedical Engineering, Graz University of Technology, Kronesgasse 5-I, 8010 Graz, Austria and Royal Institute of Technology (KTH), Department of Solid Mechanics, School of Engineering Sciences, Osquars Backe 1, 100 44 Stockholm, Sweden

{dagger} Email: paul.watton{at}eng.ox.ac.uk

Received on May 1, 2008. Revised on August 14, 2008. Accepted on January 12, 2009.

Experimental and theoretical guidance is needed to understand how the collagen fabric evolves during the development of aneurysms. In this paper, we model the development of an aneurysm as a cylindrical/spherical membrane subject to 1D enlargement; these conceptual models reflect the development of fusiform and saccular cerebral aneurysms. The mechanical response is attributed to the elastin and collagen. We introduce variables which define the elastin and collagen fibre concentration; these evolve to simulate growth/atrophy of the constituents. A hypothetical aneurysm model is analysed: collagen stretch is constant, elastin degrades and collagen fibre concentration can adapt to maintain mechanical equilibrium. An analytic expression for the rate of evolution of the fibre concentration is derived. The functional form is dependent on (i) the current collagen fibre concentration, (ii) the deviations in the collagen fibre stretch from the attachment stretch, (iii) the rate of change of fibre stretch, (iv) the rate of loss of elastin and (v) the ratio of load borne by elastinous and collagenous constituents. Finally, numerical examples of aneurysm development are considered. Suitable candidates for the fibre concentration evolution equations are identified that yield stabilization of the aneurysm even when there is complete loss of elastin. This theoretical analysis provides the basis for the development of physiologically realistic models of aneurysm development.

Keywords: aneurysm; artery; cerebral; collagen; elastin; growth; remodelling


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