A Nonlinear Elastic Soft Tissue Model for Craniofacial Surgery Simulations

Biomechanical modeling of soft tissue is essential to many computer assisted medical applications. Because of complexity of soft tissue behavior and a lack of exact and comprehensive data on material properties of living tissues, simplified models such as widespread linear elastic models are usually applied. However, large deformations often occurring in craniofacial surgery can not be accurately modeled via linear elastic approach, since it is generally limited by the assumption of small deformations. The investigations carried out in the present work shows that the nonlinear elastic model yields more realistic results in simulations of large soft tissue deformation. Introduction The finite element method (FEM) becomes in the last three decades a state of the art approach for the realistic simulation of the soft tissue deformation in the computer assisted surgical applications. However, the consistent physically based modeling of soft tissues is substantially limited by the complexity of their biomechanical properties. In general, soft tissue exhibits anisotropic, nonhomogeneous, nonlinear plastic-viscoelastic mechanical behaviour. The complex models, which consider all these properties, are difficult to implement and to control. Thus, the simplified constitutive models usually have to be applied. The goal of the soft tissue modeling within the scope of craniofacial surgery planning is the ’long term’ prediction of the patient’s postoperative appearance, i.e., the quasi-static deformation of relatively thin facial tissue under the impact of forces indirectly given by the prescribed displacements of bone structures has to be calculated. Thereby, the deformations are usually performed by surgeons (i) ’at ones’, in the case they are small, or (ii) over a longer period of time, if they are large. From this problem definition, a linear elastic model of soft tissue appears a natural approximation. In our previous works [6], we have investigated the scope of validity of a linear elastic model and have found that it produces a substantial error in the case of large deformations. In this work, we present the results of the application of a nonlinear elastic FE-based model, where we primarily focus on the investigation of the effects of the so-called geometrical nonlinearity. Being independent of material properties, this kind of nonlinearity is relevant for any large deformed physical body, including soft tissue. 1. Nonlinear Elasticity In this section, we briefly explain the basics of elasticity theory. 1 Zuse Institute Berlin (ZIB) Takustr.7, 14195 Berlin, Germany, email: {gladilin,zachow,deuflhard,hege}@zib.de c © EDP Sciences, SMAI 2003 62 P. DEUFLHARD AND H.-C. HEGE 1.1. Geometrical Nonlinearity Under external forces, physical bodies are deformed, which means that they change both their shape and volume. The deformation of a body is completely described if the displacements ui of all its points are known: ui = xi − xi , (1) where xi and xi denotes coordinates of a point before and after the deformation. The strain tensor εij that describes the deformation of an infinitesimal volume is generally a nonlinear function of displacements εij = 0.5 (∂jui + ∂iuj + ∂iul∂jul) , (2) which is known as geometrical nonlinearity. In the case of small deformations, the derivatives of the displacement vector, which represent the relative displacements, are significantly smaller than 1: