Biomedical Engineering Reference
In-Depth Information
medical application. Physically based methods lead to more realistic simulation and
also provide the capability to study a case as a mechanical simulation and to find
the real behavior of tissues during deformation. In this method,
partial differential
equations
(PDEs) which govern the evolving shape of the deformable objects and
their motion through space should be solved. The major difficulty in these methods
lies in the complexity of the physical phenomena that should be simulated and com-
putationally solved from the PDEs. To overcome this difficulty, one should simplify
the model and apply numerical techniques to solve the PDEs. We describe hereafter
two relevant methods based on mass-spring systems and finite elements.
Mass-spring systems
Amass-spring system is a physically based technique that has been widely and effec-
tively used for modeling deformable objects. An object is modeled as a collection of
point masses connected by massless springs in a lattice structure. Springs connecting
point masses exert forces on neighboring points when a mass is displaced from its
rest position. The elastic force acting on mass
i
connected by a spring to mass
j
is
given by:
k
x
ij
−
l
ij
x
ij
x
ij
f
ij
=
where
x
ij
=
x
i
and
x
i
,
x
j
are the locations of point masses
i
and
j
, respectively,
l
ij
is the rest length between them and
k
is the spring's stiffness. Applying this
equation on all the points leads to a differential system of ordinary equations that
can be solved explicitly using various algorithms [
18
]. Mass-spring systems are easy
to construct, and both interactive and real-time simulations of mass-spring systems
are possible. Also it has the ability to handle large deformations. As a disadvantage,
usually spring constants are approximated from measured material property and
allocating suitable constants that express all tissue properties in a natural way is
difficult.
Mass-spring systems have been widely used in facial animation. As an ex-
ample in biomechanical modeling, mass-spring systems were used by Nedel and
Thalmann [
70
] to simulate muscle deformation. Muscles were represented at two
levels, action lines and muscle shapes. The muscle shapes were deformed using a
mass-spring mesh. They used angular spring to control the volume of muscles during
deformation and smooth out mesh discontinuities.
x
j
−
Finite element method
Finite element method (FEM) is another physically-based technique that has been
widely used in soft tissue modeling. Contrary to mass-spring systems that treat the
mechanics as a discrete process, FEM views it as a continuum (see sect.
6.2.1.1
).
For that reason, FEM usually leads to more accurate physical models compared to
