Estimation of the distributions of anisotropic, elastic properties and wall stresses of saccular cerebral aneurysms by inverse analysis

A new method is proposed for estimating the elastic properties of the inhomogeneous and anisotropic structure of saccular cerebral aneurysms by inverse analysis. The aneurysm is modelled as a membrane and the constitutive response of each individual layer of the passive tissue is characterized by a transversely isotropic strain energy function of exponential type. The collagen fibres in the aneurysm wall are assumed to govern the mechanical response. Four parameters characterize the constitutive behaviour of the tissue: two initial stiffnesses of the collagen fabric in the two in-plane principal directions, one parameter describing the degree of nonlinearity that the collagen fibres exhibit and the other structural parameter, i.e. the angle which defines the orientation of the collagen fibres. The parameter describing the fibre nonlinearity is assumed to be constant, while all others are assumed to vary continuously over the aneurysm surface. Two model aneurysms, with the same initial geometry, boundary and loading conditions, constitutive behaviour and finite-element discretization, are defined: a 'reference model' with known distributions of material and structural properties and an 'estimation model' whose properties are to be estimated. An error function is defined quantifying the deviations between the deformations from the reference and the estimation models. The error function is minimized with respect to the unknown parameters in the estimation model, and in this way the reference parameter distributions are re-established. In order to achieve a robust parameter estimation, a novel element partition method is employed. The accordance between the estimated and the reference distributions is satisfactory. The deviations of the maximum stress distributions between the two models are below 1%. Consequently, the wall stresses in the cerebral aneurysm estimated by inverse analysis are accurate enough to facilitate the assessment of the risk of aneurysm rupture.