A volumetric model for growth of arterial walls with arbitrary geometry and loads

Abstract

Stress and deformation in arterial wall tissue are factors which may influence significantly its response and evolution. In this work we develop models based on nonlinear elasticity and finite element numerical solutions for the mechanical behaviour and the remodelling of the soft tissue of arteries, including anisotropy induced by the presence of collagen fibres. Remodelling and growth in particular constitute important features in order to interpret stenosis and atherosclerosis. The main object of this work is to model accurately volumetric growth, induced by fluid shear stress in the intima and local wall stress in arteries with patient-specific geometry and loads. The model is implemented in a nonlinear finite element setting which may be applied to realistic 3D geometries obtained from in vivo measurements. The capabilities of this method are demonstrated in several examples. Firstly a stenotic process on an idealised geometry induced by a non-uniform shear stress distribution is considered. Following the growth of a right coronary artery from an in vivo reconstructed geometry is presented. Finally, experimental measurements for growth under hypertension for rat carotid arteries are modelled.

Introduction

Models for simulating the behaviour of soft tissue such as blood vessel walls must incorporate an adequate representation of the highly nonlinear material behaviour with large deformations, strains and rotations (Fung, 1993, Humphrey, 2001). Additionally, for patient-specific analysis the models should be applicable to arbitrary 3D geometries, one of the main motivations for this work. An adequate geometrical reconstruction of cardiac arteries for our purpose needs not only the data for the lumen but also for the inhomogeneous wall thickness. This may be obtained based on bi-plane X-ray angiography and intravascular ultrasound (IVUS) methods (Sanmartín et al., 2006, García et al., 2003), using similar procedures to those proposed by Wentzel et al. (2003).

The incremental mechanical (stress–strain) response of soft tissue depends significantly on the initial stress level, due to the nonlinear behaviour (Humphrey, 2001). In turn, the initial stresses may be explained as originating from growth and remodelling processes (Takamizawa and Hayashi, 1985, Fung, 1993). As a consequence, determining biomechanical growth laws constitutes an important feature to understand how loading conditions affect blood vessels. Moreover, this knowledge is useful for replacement of blood vessels through the techniques of tissue engineering.

In the present paper we deal with biological growth processes in arterial walls in response to mechanical stimuli. Previous work along this line includes that of Taber (1998), Rodríguez et al. (1994) and Fung (1990). Recently Kuhl and Steinmann (2004) have proposed a model for growth using the material force concept. Other features such as structural remodelling, involving alteration in the material structures are not considered here. Further studies of biological remodelling related mainly to bone tissue may be found in the work of Carter and Beaupré (2001), Cowin (2001) and Huiskes et al. (1987).

Blood flow originates two main mechanical actions on vessel walls. Firstly, the internal pressure, which produces a state of wall (hoop) stress, which in general will be non-uniform across the thickness. Secondly, the viscous nature of the flow transmits a shear stress to the endothelium (Milnor, 1989). Regarding wall stress, a common hypothesis for remodelling is to consider that the arterial wall will modify its geometry in order to recover a given uniform stress level (Takamizawa and Hayashi, 1985, Fung, 1990, Rachev and Greenwald, 2003). With regard to the shear stress, a number of authors have found correlations between low shear stress and thickening of the arterial wall (Zhao et al., 2002).

Here a model for incorporating volumetric remodelling of tissues is proposed for general 3D geometries, extending the model of Taber (1998), which considers the shear sensed by the endothelial cells in the intima and local stresses in the arterial wall. We have programmed this model in a finite element code (FEAP, Taylor, 2002) through user material subroutines. Three representative numerical examples are presented: a stenotic process in a simplified (human) geometry, the growth of a patient-specific right coronary artery reconstructed from medical images, and an application to model experimental observations for hypertensive growth in rat carotid arteries.