Biomedical Engineering Reference
In-Depth Information
the perimeter) of the cytosol and the surface (respectively, the perimeter) of
the nucleus. These dimensions, given in Table C.10, reflect the mean measures
of typical eukaryotic cells except white blood cells [7].
In our model, we set the length of a collagen-like fiber equal to 15 lattice
sites ( 20 m). Its thickness would generally range between 100 nm and 0.5
m [44, 325, 331], and therefore it would be substantially smaller than the
grid resolution. Modeling them with the real size would considerably increase
the computational cost of the simulations. Indeed, we here follow most CPM
applications (see, for example [162, 334]), where a fiber is modeled as a single
site thick. Each simulated fiber therefore is assumed to contain nearly 10
6
collagen-like molecules, given that a single matrix protein is approximately
300{600 nm long and 1.5{5 nm wide [7].
For the sake of simplicity, we will use the term fiber for both the basic short
ECM structure ( 20 m long threads) simulated for the 2D condition, and
the long structure crossing the entire spatial domain of the 3D cubic network.
The m
i
gratory properties of cells are quantified by evaluating their average
velocity (v
defined as in Equation (1.13) where t
final
= 12 h, 21600 MCS),
and their mean square displacement (d
2
defined as in (1.14)).
In particular, as demonstrated in a number of experimental [106, 177] and
computational [106, 421] studies, at suciently long times the mean square
displacements vary approximately linearly with the number of time steps.
It can therefore be related to the cell instantaneous velocity (v
, defined in
(1.12)) and its persistence time (p
, which quantifies the directional productive
motion) with the so-called persistence-random-walk (PRW) law:
< d
2
(t) >= 2v
2
(t)p
(t)[tp
(t)(1 e
t=p
(t)
)]:
(9.2)
In particular, at still longer observation periods, (9.2) reduces to
< d
2
(t) > 2v
2
(t)p
(t)t;
(9.3)
and the persistence time of a moving individual can be directly calculated as
p
(t)
< d
2
(t) >
2v
2
(t)t
:
(9.4)
The PRW relation has been demonstrated to characterize the cells' migratory
behavior more properly than other common methods, which calculate the av-
erage distance migrated by biological individuals in an arbitrary time interval,
as commented in [121].
The quantitative analysis of cell morphological changes is carried out by
evaluating the evolution of the cell aspect ratio, given by the ratio between
the actual cell surface (respectively, perimeter in 2D) and the surface of the
sphere having the same volume (respectively, the perimeter of the circle having
the same area in 2D). It is useful to underline that in our model cell volume
(respectively, area in 2D) is kept nearly fixed by high values of
volume
;N
=
(respectively
surface
;N
=
surface
volume
;C
;C
in 2D) Therefore, the aspect ratio
gives a quantitative measure of cell membrane ruing.
Search WWH ::
Custom Search