Techniques for the Generation of 3D Finite Element Meshes of Human Organs

Introduction

Continuum mechanics (CM) is a branch of mechanics that deals with the analysis of the kinematic and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids (i.e., liquids and gases). In a nutshell, CM assumes that matter is continuous (ignoring the fact that matter is actually made of atoms). This assumption allows the approximation of physical quantities over the materials, such as energy and momentum, at the infinitesimal limit. Differential equations can thus be employed in solving problems in CM.

Let Ω be a volumetric domain defined in 3D and P a point inside Ω. A deformation occurs in Ω as external (surface or volumetric) forces ƒ are applied to some P in Ω. If Ω is considered as an elastic body, the deformation is characterized by:

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  A displacement vector field u caused by ƒ applied over one or more P.

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  An internal state of deformation, mathematically defined by a “strain tensor”for each P in Ω.

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  The “force reaction” of the body to the external forces; this is mathematically defined by a “stress tensor” for each P in Ω.

The Local Equilibrium of the Medium (LEM) is an expression defined for each P in Ω that links the displacement, the strain and the stress tensor through Partial Differential Equations (PDE). In order to do so, an analytical relationship is assumed between the strain and the stress tensors, known as the constitutive law of the material.

The difference between displacement and deformation is made in function of the Reference System (RS) used to measure each one of them. The first one is made regarding the overall RS. The second one, uses a relative RS associated to the domain. Figure 1 illustrates the initial state of the system at the top; at the bottom left panel a displacement is produced as the coordinates of point P change to P' regarding the x and y axis, however no deformation is presented in the r and s axis. At the bottom right panel the inverse situation is produced as no displacement of P is presented in the x and y axis. In the other hand, point P has a severe deformation regarding the r and s axis.

Figure 1.

Displacement vs deformation. Top, initial system. Bottom left, displacement (a difference regarding the x and y axis). Bottom right. deformation (a difference regarding the r and s axis).

The strain level measures the domain deformation regarding the relative position of P in Ω. Let vector a be the initial position of P and vector b be the position of P after the deformation in the relative axis: r, s. The strain level is then defined as:

The importance of measuring the strain level is that when it is inferior to 10% the “small strain” hypothesis can be made, which assumes a linear geometrical resolution of the PDEs. Otherwise, a “large strain” framework is necessary, which means a much more complex resolution of the system.

If the constitutive law can be assumed to be linear, another simplification ca be made: an elasticity tensor is computed to link the stress and strain tensors. On the contrary, if a linear law does not account properly for the mechanical behavior of the tissues, a non-linear law can be chosen to account for the hyperelastic behavior of soft tissues (see Bonet and Wood (1997) for more details).

Most of the modeling works proposed in the literature assumed both, the small strain and the linearity of the constitutive law hypotheses. The term “linear elasticity” is then commonly used, i.e. assuming linearity of the geometry and the mechanics. In that case, it can be shown that the constitutive behavior of the material can be characterized by only two parameters: the Young's modulus that depends on the stiffness of the material, and the Poisson's ratio that is related to the compressibility of the material.