Information Technology Reference
In-Depth Information
1
.
0
.
0
1
.
0
.
0
1
.
0
b
N
D
Inputs:
CF
Level
CF
A
C
h
= 100%
CF
D
C
=0%
27
32
5
4
b
1
c
A
Related Inputs:
(
A
4
)
1
16
A
1
=
2
=
16
A
32
A
CF
(
A
3
)
8
3
b
2
C
t
=
2
32
=0
.
11011
2
A
2
=
3
4
=
24
A
32
A
3
=
3
8
=
12
A
32
A
A
(
A
2
)
4
2
D
Output Parameters:
(
A
1
)
2
1
0
A
4
=
11
A
16
=
22
A
32
0
.
0
0
.
0
0
.
0
0
.
0
0
.
0
d
m
W
I
(
A, D
)=
A
D
=
0
32
A
1
A
2
A
N
D
Target
Solution
D
A
N
related input
solutions of
(b)
A
(a)
Fig. 2.
(a) Related input and target
CF
s of same fluid in the scale 0 to 1. (b) An example of
producing related input fluids as by-products of dilution process for
C
t
=
2
32
by
twoWayMix
[6].
3
Dilution from Multiple Arbitrary
CF
s of the Same Fluid
In this Section, we present a scheme for generalized dilution of a sample fluid from
the supply of multiple arbitrary concentrations (referred as
related
inputs) of the same
fluid. The related inputs are the same fluid diluted with the same buffer solution to have
different
CF
s. We formulate the generalized dilution problem as follows.
Inputs:
(a)
N
related inputs
A
1
,A
2
,...,A
N
(
N
2) of a biofluid
A
with
CF
s
b
1
,b
2
,...,b
N
are
supplied, each diluted with the buffer solution
D
,where
b
i
s are positive real numbers
and 0
<b
i
≤
≥
1. (b) buffer solution
D
with
CF
=0is available in supply. Thus, the
total number of related inputs including
D
is (
N
+1),where(
N
+1)
≥
3.(c)desired
CF C
A
of the biofluid
A
,where
C
A
is a positive real number and 0
≤
C
A
≤
1.(d)
the integer
d
determines the accuracy level of
C
A
. Thus, the maximum error in
C
A
is
bounded by
1
2
d
+1
.
Output:
a dilution/mixing tree of depth
d
needed to produce droplets
with target
CF
of fluid
A
as
C
A
.
If
C
A
>
max
i
{
2
d
(
i
=1to
N
+1), the desired
CF
is not reachable from the
+
0
.
5
b
i
}
inputs. Again, if
C
A
is in the range of
b
i
to (
b
i
+
0
.
5
2
d
)
for any
i
(
i
=1to
N
+1), then
the desired
CF
is reachable and the input droplet with
CF
=
b
i
can be treated as the
target droplet. Otherwise, we need to solve this problem and the target droplet contains
some part of fluid
A
coming from some of the
N
related inputs
A
1
,A
2
,...,A
N
along
with a part of
D
and shown in Fig. 2(a). Hence, the
CF
of
D
in target droplets becomes
C
D
=1
1) and it may come
from the related inputs only (as each of them are diluted with
D
)oritmayrequire
mixing of some additional amount of
D
.
We can have different optimization criteria to decide upon an optimal dilution/mixing
tree as follows: (i) the total number of non-leaf nodes (
m
) in the dilution/mixing tree
is minimized, or (ii) the height (
d
) of the dilution/mixing tree is minimized, or (iii)
the lower-bound of number of mixers required for earliest completion, i.e.,
M
lb
of the
dilution/mixing tree is minimized.
−
C
A
(where
C
D
is a positive real number and 0
≤
C
D
≤
