Biomedical Engineering Reference
In-Depth Information
Fig. 1 Fluid-Solid-Growth computational framework for modelling aneurysm evolution con-
sisting of (i) structural analysis, (ii) computational fluid dynamics (CFD) analysis and (iii) G&R
algorithms. Further details are provided in Sects. 2.1 , 2.2 and 2.3 , respectively
2.1 Structural Model of Aneurysm Evolution
A geometric nonlinear membrane theory (see, for example, [ 42 ]) is adopted to model
the steady deformation of the arterial wall. The unloaded internal abdominal aorta is
treated as a thin cylinder of undeformed radius R ; length L 1 ; and thickness H : The
thickness of the media H M is assumed to be equal to 2/3 the thickness of the arterial
wall, i.e. H M ¼ H = 3 ; and thus the thickness of the adventitia H A ¼ H = 3 : The artery is
subject to a physiological axial pre-stretch k z and pressure p which causes a cir-
cumferential stretch k : A body fitted coordinate system is used to describe the
cylindrical membrane with axial and azimuthal Lagrangian coordinates h 1 0 ; L 1
and h 2 0 ; 2pR Þ; respectively. Formation and development of the aneurysm is
assumed to be a consequence of G&R of the material constituents of the artery. The
principle of stationary potential energy is the governing equation for the steady
deformation of the arterial wall. It requires that the first variation of the total potential
energy vanishes,
dP int dP ext ¼ 0 ;
ð 1 Þ
where dP int represents the variation of the internal potential energy P int stored in the
arterial wall, whilst dP ext is the variation of the external potential energy P ext caused
by the normal pressure that acts on the artery. Appropriate functional forms for the
spatially and temporally heterogeneous strain-energy functions (SEFs) for the media
P M ; and the adventitia P A must be specified so that dP int can be computed. Details of
the theoretical formulation to describe the deformation of the arterial wall and the
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