Optimization of On-Chip Polymerase Chain Reaction - Hardware/Software Co-Design and Optimization for Cyberphysical Integration in Digital Microfluidic Biochips - page 123

where A j

k is the event that “no signal has been observed from a good droplet with j
DNA strands (j M min )afterk thermal cycles”. It is easy to note that the events
A k , A k , , A k are mutually exclusive.
From the definitions of G i and A j

k , it follows that P.G i \ A j k / D 0,ifi ¤ j or

i D j<M min .Wheni D j M min ,thevalueofP.G j \ A k / can be empirically
derived from calibrating the amplification of DNA. The probabilities P.G \ A k /
and P.G c \ A k / in ( 5.3 ) can be calculated as follows:

P.G \ A k / D P.G \ .A k [ A k [ A k [ A k

[ A k

//

D P.G M min \ A M min

/ C P.G M min C1 \ A M min C1 /

k k

C P.G M min C2 \ A M min C2

/ CC P.G 1 \ A k

/

k

P.G c \ A k / D P.G 0 / CC P.G M min 1 /:

We assume that the probability of detecting the fluorescence signal (i.e., the DNA
has been amplified) is p i at the i th thermal cycle when we run DNA amplification
on a good droplet that contains m (m M min ) DNA strands. Then the value
of P.G \ A k / and P.G c \ A k / can be derived in terms of p i . Therefore, if
the fluorescence signal is not detected at the kth thermal cycle, the conditional
probability that “the droplet is an empty droplet” as well as “the droplet is a good
droplet” can be analyzed in a similar way as the method presented in Sect. 5.2.2 .
During DNA amplification on a cyberphysical PCR biochip, the control software
calculates the values of P.G \ A k / and P.G c \ A k / according to the feedback from
the sensor at each thermal cycle. When the value of P.G c \ A k / reaches a threshold
(e.g., 95 %), the control software will conclude that the droplet is empty. Then it
will initiate a new set of electrode-actuation sequences to discard the empty droplet
and dispense a new droplet into the biochip. The PCR procedure will therefore be
continued in an adaptive manner on the biochip.

5.3 Optimized Resource Placement Under Proximity

Constraints

In this section, we present our algorithm for optimizing the layout under design
constraints. The device-proximity constraints for the placement of the devices
on the PCR biochip are introduced in Sect. 5.3.1 , and the objective function for
this placement is presented in Sect. 5.3.2 . The geometry-based device placement
algorithm is discussed in Sect. 5.3.3 . Finally, Sect. 5.3.4 introduces two approaches
that can be used to select the optimal result of device-placement.