Biomedical Engineering Reference
In-Depth Information
1.4 Evaluation of Some Kinematic Parameters
The position of an object
is established by the position of its center of
mass
X
1
a
volume
x
CM
(t) =
x;
(1.11)
(t)
x
2
where a
volume
(t) is its actual volume.
The instantaneous velocity of
is calculated as the velocity of its center
of mass:
x
CM
(t) x
CM
(t t)
t
v
(t) =
;
(1.12)
over t = 1 MCS, as done in [261, 309, 350].
Indeed, if t > 1 MCS, we measure a mean velocity of the individual. In
particular, the average velocity of
over an entire simulation is given by
x
CM
(t
final
) x
CM
(0)
v
(t) =
;
(1.13)
t
final
where x
CM
(0) is the initial position of its center of mass and t
final
corresponds
to the final time of the observation period of interest.
The mean square displacement (MSD) at time t of an individual
, <
d
2
(t) >, is calculated as
< d
2
(t) >=< (x
CM
(t) x
CM
(0))
2
>;
(1.14)
where x
CM
(0) is defined in Equation (1.13). Following [106, 421], the square
displacements are averaged over all previous time steps in order to take into
account the back and forth motions exhibited by the moving individual.
It is useful to emphasize that the above definition of kinematic parameters
can be easily extended to the compartmentalization approach that will be
introduced in Chapter 4, by noting that the center of mass of the individual
will have to be calculated for the entire element and not for each single objects.
1.5 Some Illustrative Simulations
In order to clarify the applicative significance of the specific terms in the
Hamiltonian introduced in the previous section, we now illustrate some bio-
logically relevant test simulations.
The elastic moduli
i
and
j
;
0
in Equations (1.6) and (1.7) essentially
ensure that the relative individuals maintain their biophysical attributes close
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