Biomedical Engineering Reference
In-Depth Information
We consider
k
(•) of all unordered pairs involving the following sets:
P
i
(
t
),
P
i
(
t +
1),
N
i
(
t
),
N
i
(
t +
1),
P
j
(
t
),
P
j
(
t +
1),
N
j
(
t
),
N
j
(
t +
1). Since
()
= , there are 28
such pairs that need to be mutually exclusive to prevent interference between
the droplets. Pairs of cell sets that must be mutually exclusive for noninterfer-
ence must be contained in this pool of unordered pairs because these eight
sets include all cells currently occupied by droplets and all of their neigh-
boring cells. A total of 22 pairs can be quickly removed from consideration,
leaving only 6 pairs of sets that need to be closely examined to determine if
their mut
u
al exclusion is required for noninterference. For simplicity, the pin
operator
k
(•) is left implicit in the following discussion. Mutual exclusion
always refers to the
pins
controlling the cells, and not the cells themselves.
We can analytically confirm that the control pins for these six pairs must be
mutually exclusive to prevent interference.
8
28
2
Pair 1
{
P
i
(
t
),
N
j
(
t
)}: if
N
j
(
t
) contains a cell that shares the same pin as
P
i
(
t
),
then
D
j
may be between
P
j
(
t
) and
P
j
(
t
+ 1) at time
t
. If this is not the
case,
D
j
will not be able to move to
P
j
(
t +
1),
because it will no longer
overlap with
P
j
(
t +
1).
Pair 2
{
P
i
(
t +
1),
N
j
(
t
)}: if
N
j
(
t
) contains a cell that shares the same pin as
P
i
(
t +
1),
D
j
will not move properly at time
t +
1 unless
P
j
(
t +
1)
is the
cell in
N
j
(
t
) that shares the same pin as
P
i
(
t +
1).
Pair 3
{
P
i
(
t +
1),
N
j
(
t +
1)}: if
N
j
(t +
1
)
contains a cell that shares the same
pin as
P
i
(
t +
1),
then
D
j
will drift after moving to
P
j
(
t +
1)
so that it
ends up between
P
j
(
t +
1)
and a cell that is an element of
N
j
(
t +
1).
Pair 4
{
N
i
(
t
),
P
j
(
t
)}: if
N
i
(
t
) contains a cell that shares the same pin as
P
j
(
t
),
then at time
t
,
D
i
will drift between
P
i
(
t
) and the cell of interest in
N
i
(
t
).
D
i
may no longer overlap with
P
j
(
t +
1); if so, it will not be able
to move to
P
j
(t +
1
)
.
Pair 5
{
N
i
(
t
),
P
j
(
t +
1)}: if
N
i
(
t
) contains a cell that shares the same pin as
P
j
(
t +
1),
D
i
will not move properly at time
t +
1 unless
P
i
(
t +
1)
is the
cell in
N
i
(t)
that shares the same pin as
P
j
(
t +
1).
Pair 6
{
N
i
(
t +
1),
P
j
(
t +
1)}: if
N
i
(
t +
1)
contains a cell that shares the same
pin as
P
j
(
t +
1),
then at time
t +
1,
D
i
will drift between
P
i
(
t +
1)
and
the cell of interest in
N
i
(
t +
1).
It can be easily shown that the control pins for these six pairs must be mutu-
ally exclusive to prevent interference between droplets
D
i
and
D
j
. These lead
to the following necessary and sufficient interference constraints:
