Hardware Reference
In-Depth Information
5.2.1
Statistical Model for the Number of DNA Strands
in a Droplet
When droplets are dispensed from the reservoir into the DMFB, the number of DNA
strands contained in a droplet can be considered to be a random variable [
15
,
20
].
Based on the statistical data derived from several biochemistry experiments, we
note that the dispensing process follows Poisson statistics when the DNA strands in
the biological sample has low density. Therefore, the number of DNA strands in a
droplet is derived as [
15
,
20
]:
P.X
c
D
k/
e
k
kŠ
(5.1)
where k is the number of DNA strands in the droplet and is the average number
of DNA strands per droplet [
15
,
20
].
When the number of DNA strands in a droplet reaches a low threshold, the PCR
procedure cannot be carried out successfully [
21
]. The droplets that contain low
amounts of DNA strands are referred to as “empty droplets”. If empty droplets
are detected at the end of the PCR, the time spent on running the thermal
cycles is wasted. In order to investigate this problem, the feedback signal from
the sensors, which are incorporated on the biochip, can be used for making an
online probabilistic decision whether a droplet is empty. As noted in [
6
,
7
], during
the execution of the PCR, the intensity of fluorescence in the droplets can be
monitored by on-chip detectors. The droplet will be discarded if the probability
that “the droplet is empty” is high. A new droplet will then be dispensed into the
biochip for PCR processing.
In the next part of this section, we introduce statistical models of the PCR
bioassay based on which an on-line decision making system will be designed.
5.2.2
A Simplified Statistical Model for Amplification of DNA
During the calibration of the dispensing operation, the probability of generating a
“good droplet” can be derived [
15
]. A good droplet is one that contains a sufficient
number of DNA strands for running the PCR. Let G denote the event “a good droplet
is dispensed into the biochip”. The complement of G is the event G
c
, i.e., “an empty
droplet is dispensed into the biochip”.
We assume that when running the PCR on a good droplet, the probability of
detecting a fluorescence signal (which indicates that the DNA has been amplified)
at the i th thermal cycle is p
i
.LetA
k
denote the event “no signal is obtained from
the droplet at the kth thermal cycle”. Therefore, the joint probability of G and A
k
can be written as:
