A model of growth and rupture of abdominal aortic aneurysm
Introduction
The condition of a focal dilation of the infrarenal aorta—abdominal aortic aneurysm (AAA)—is found in ≈2% of the elderly population, with ≈150,000 new cases diagnosed each year, and the occurrence is increasing (Bengtsson et al., 1996; Ouriel et al., 1992). In many cases AAA gradually expands until rupture causing a mortality rate of 90%. The AAA rupture is considered the 13th most common case of death in the US (Patel et al., 1995). Since the AAA treatment is expensive and bears considerable morbidity and mortality risks it is vital to predict when the risk of rupture justifies repair. Such a prediction should be based on a biomechanical model of growth and rupture of the aneurysm.
Very few models of aneurysmal growth have been developed: Humphrey and Canham (2000); Watton et al. (2004); Baek et al. (2006); and some empirical criteria of the aneurysm rupture were proposed: Elger et al. (1996); Li and Kleinstreuer (2005). Though the models incorporate growth descriptions, none of them couples growth with rupture in the theoretical setting. The latter is important in order to provide an objective criterion of the material failure, which is a part of the model formulation and not an external condition imposed on the stress/strain field. Besides, the insertion of the failure description makes the model more physical because no real material can sustain large enough strains—it should fail. Existing models of soft tissues including AAA do not describe failure. According to the traditional models, the material is always intact independently of the amount of the accumulated energy or strain. The latter is unphysical, of course. The coupling of growth and rupture is the primary goal of the present work.
Our description of rupture is based on the idea that a small (infinitesimal) material volume possesses a limited capacity of accumulating energy under increasing strain. Volokh (2007) shows that the average energy of the atomic/molecular bonds sets the failure energy limit, which is a material constant. This new failure constant controls softening in the constitutive law. An application of the average bond energy to analysis of the overall strength of an intact arterial wall can be found in Volokh (2008). In the present work, we introduce the AAA model in the form of isotropic and incompressible Neo-Hookean type material with softening. All material constants are calibrated in the uniaxial tension (UT) test for an AAA sample. The calibrated rupture model is further enhanced with a description of growth (Volokh, 2004, Volokh, 2006b), which incorporates the law of mass balance/evolution with a source term related with the tissue remodeling and an additional term in the stored energy expression related with the material expansion under the mass supply. The proposed coupled model is used to simulate the evolution and rupture of an idealized spherical AAA. The results of the simulation encourage further development and calibration of the model, concerning growth, when more data is collected.
Section snippets
AAA model with softening
We consider the classical formulation of nonlinear elasticity according to which a generic material particle of body Ω occupying position X in the reference configuration moves to position x(X) in the current configuration. The deformation of the particle is defined by the tensor of deformation gradient, F=∂x/∂X. The equilibrium equation, div σ=0 in Ω, and boundary conditions on ∂Ωx or on ∂Ωt, should be obeyed, where ‘div’ operator is with respect to the current position, x; σ is the
Growth and remodeling
In order to include growth and remodeling in a mathematical description of the AAA evolution it is necessary to couple equations of the mass and momentum balance. The full-scale coupling should generally require the account of mass diffusion. However, we simplify the general model assuming that AAA is very thin and the mass supply is volumetric and homogeneous. Thus, we ignore the diffusion of mass. Under the mentioned assumption the mass balance equation reduces to the following evolution
Results
We use the numerical procedure ‘NDSolve’ of Mathematica 5.2 (Wolfram, 2003) for solving (20), (21). This procedure allows solving a system of differential-algebraic equations based on the IDA package, which is a part of the software developed at the Center for Applied Scientific Computing of Lawrence Livermore National Laboratory. It solves the system of differential-algebraic equations by combining the Backward Differential Formula methods and Newton-type methods. The procedure provides the
Discussion
We proposed a mathematical model describing growth and rupture of the AAA. This model includes equations of the momentum and mass balance and the constitutive law is enhanced with the influence of the mass change. The evolution of the material constants during growth is also taken into account. Failure is described by inducing the average bond energy limit, φ, in the stored energy function. Such a limit controls material softening indicating failure. The initial values of the AAA material
Conflict of interest
None declared.
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