Page 31 - Advances in Biomechanics and Tissue Regeneration

2.4 BIOMECHANICAL MODEL OF THE SEMICIRCULAR DUCTS 25

It uses a Lagrangian formulation in which the particles are constant mass particles and particle mesh is attached to the
material and particles deform as the material starts to deform. The SPH method works by dividing a continuous field
into a set of discrete particles. These particles have a spatial distance over which their properties are “smoothed” by a
kernel function. The particles are identified with some characteristics, such as mass, position, and velocity, but particles
can also carry estimated physical properties depending on the problem, like mass density, temperature, and pressure.
This method is best suited for problems in which the mesh deformations are large [29].
The SPH was first developed to simulate astrophysical fluid dynamics. Since then, it has been successfully applied
to a vast range of problems, as fluid flows. It was developed by Gingold and Monaghan (1977) and Lucy (1977).
A computational domain of this method is represented by computational points—particles.
This method has some advantages over grid-based techniques because its concept is simple and it is relatively easy
to incorporate complicated physical effects into the SPH formalism [30, 31].
During 1995 Liu and his team [32], after some tests with SPH conditions, proposed a reproducing kernel particle
method that improves the accuracy of SPH approximation.
SPH method was initially developed as a probabilistic meshless particle method and was later modified to a deter-
ministic meshless method. Over the years the method has been optimized to increase the accuracy solution and to be
applied in most of the cases.
In the next subchapter, there will be detailed numerical models and simulations performed with the meshless
methods on the vestibular system of the inner ear.

2.4 BIOMECHANICAL MODEL OF THE SEMICIRCULAR DUCTS

In this section the computational model developed within the scope of this work is presented.
Regarding the computational analysis software, it was found that ABAQUS possesses enough flexibility to build
and analyze such complex systems. ABAQUS is a commercial software, widely used in several computational
mechanic fields, to build and simulate numerical models. Furthermore, it allows combining the FEM with the SPH
meshless method. The first circular model built to study the vestibular system using the SPH method to simulate
the endolymph is represented in Fig. 2.2. This model comprises the membrane of the duct discretized with shell ele-
ments (S3R) and the inside endolymph discretized with particle (PC3D) for the SPH method [33].
In the present model, the shell was defined as a rigid body because it does not present significant deformations. The
fluid particle distribution is regular, and the sum of the volume of the particles is equal to the volume of the SCC with
the cupula. The dimensions of the duct represented in Fig. 2.2 were obtained from a 3-D model of the complete ves-
tibular system that can be found in the work of O.W. Henson et al. [34], which was constructed based on magnetic
resonance imaging. The considered endolymph material properties were obtained from the literature, being
3
1.0 10 3 kg/m and 4.8 10 3 Pas, the density and viscosity of the endolymph, respectively [35]. Additionally, to

FIG. 2.2 Three-dimensional model of the semicircular duct of the vestibular system.

I. BIOMECHANICS