Biomedical Engineering Reference
In-Depth Information
( )
∩
( )
=
( )
∩
(1)
kPt
{()}
k
Nt
()
φ
i
j
( )
=
( )
∩
( )
=
(2)
kPt
{(
1
)}
k
N
(
t
1 φ
)
i
j
(3)
kNt
()
kPt
{()}
φ
i
j
( )
=
( )
∩
(4)
kkNt
(
1
)
k
{(
Pt
1
)}
φ
i
j
(
)
=
( )
∩
()
5
kPt
{(
1
)}
k
NNt
() {(
− +
Pt
1
)}
φ
i
j
j
( )
∩
(
)
=
(6)
kPt
{(
1
)}
k
Nt
()
−−+
{(
Pt
i
1 φ
)}
j
i
Moving one droplet and stalling the other is a special case that can be used
when concurrent movement of two droplets leads to violation of the interfer-
ence constraints. In this situation,
P
j
= P
j
(
t
)
= P
j
(
t +
1),
and the interference
constraints reduce to
( )
∩
()
=
( )
∩
()
=
(1)
kPt
{()}
k
N
φ
i
j
(2)
kPt
{(
1
)}
k
N
φφ
i
j
( )
∩
( )
=
( )
∩
(3)
kNt
()
kP
{}
φ
i
j
( )
=
( )
∩−
( )
=
(4)
kNt
(
1
)
kkP
{}
φ
i
j
()
5
kPt
{(
1
)}
k
NP
{}
φ
i
j
j
( )
∩−
( )
=
{{(
Pt
1
)}
kN
{}
P
φ
(6)
k
j
i
i
The preceding constraints must be satisfied during droplet routing [27].
Therefore, they are enumerated here for the sake of completeness.
3.1.2 Array Partitioning and Pin-Assignment Methods
In this subsection, we present a pin-constrained design method for digital
microfluidic biochips based on array partitioning. The key idea is to “virtu-
ally” partition the array into regions. Similar partitioning techniques have
been used for very large scale integration (VLSI) circuits and for microarray
biochips [58-60].
Mutually exclusive sets of pins are utilized for different partitions.
Therefore, if we can partition the array so that droplets are in different parti-
tions, interference between them can be avoided. Partitions can be viewed as
subarrays that can contain at most one droplet at any given time. Hence, the
partitioning criterion here is to ensure that at most one droplet is included
in each partition. However, partitions with no droplets (at any point in time)
should be avoided because no droplet manipulation is done in this region
with the additional set of pins assigned to it. Hence, it is best to ensure that
each partition has exactly one droplet in it.
