Biomedical Engineering Reference
In-Depth Information
leads to the form:
(numerical methods). This section shows some examples
of how biomedical engineers use numerical analysis and
numerical methods.
D ¼
KT
m
q
2.1a.4.1 Modeling rtPCR efficiency
in which
K
is Boltzmann's constant (1.38
10
23
J/K),
T
is absolute temperature in Kelvin, and
q
is the mag-
nitude of the electric charge (1.61
10
19
C).
Both Fick's Law and Ohm's Law express
conservation of mass and can be combined by the
engineering principle of superposition, yielding an
expression for total flow:
PCR is an acronym that stands for polymerase chain re-
action, a technique that will allow a short stretch of DNA
(usually fewer than 3,000 base pairs) to be amplified
about a million fold so that one can determine its size and
nucleotide sequence.
The particular stretch ofDNAto be amplified, called the
target sequence,
is identified by a specific pair of DNA
primers, which are usually about 20
nucleotides
in length.
For convenience, these four nucleotides are called dNTPs.
There are three major steps in a PCR reaction, which
are repeated for 30 or 40 cycles. This process is done on
an automated cycler, which can heat and cool the tubes
with the reaction mixture in a very short time. The three
steps of a PCR reaction (illustrated in
Fig. 2.1a-1
) are:
1.
Denaturation at 94
C: During the denaturation, the
double strand melts open to single-stranded DNA
and all enzymatic reactions stop.
2.
Annealing at 54
C: Bonds are constantly formed and
broken between the single-stranded primer and the
single-stranded template. When primers fit exactly,
the bonds are more stable and last a bit longer. On
that piece of double-stranded DNA (template
and primer), the polymerase can attach and starts
copying the template. Once there are a few bases
built in, the ionic bond is so strong between the
template and the primer that it does not break.
3.
Extension at 72
C: The primers, where there are a
few bases built in, already have a strong attraction to
the template. Primers that are on positions with no
exact match get loose again and don't give an exten-
sion of the fragment. The bases complementary to
the template are coupled to the primer on the 3
0
side.
Because both strands are copied during PCR, there is an
exponential increase of the number of copies of the gene.
Real-time PCR (rtPCR) is used to determine gene ex-
pression over and above size and sequence information. In
rtPCR, it is assumed that the efficiency is constant; how-
ever, analysis of data shows that the efficiency is
not
con-
stant, but is instead a function of cycle number (
Gevertz
et al., 2005
). Based on this observation, it seems reason-
able that rtPCR quantification techniques can be im-
proved upon by understanding the behavior of rtPCR
efficiency. A mathematical model of rtPCR will provide
this understanding andwill lead tomore accuratemethods
to quantify gene expression levels from rtPCR data.
The mathematical model of annealing and extension is
as follows: after the double-stranded DNA is denatured,
J ¼ J
diffusion
þ J
drift
¼D
d½K
þ
dx
m
Z½K
þ
dv
dx
which, in the steady state, is 0 (that is, no net trans-
port). The conservation of momentum principle,
Einstein's relationship, relates the two rates on the
right-hand side:
q
m
d½K
þ
0
¼ J
diffusion
þ J
drift
¼
KT
m
Z½K
þ
dv
dx
dx
for
K
+
,Z¼
1 and after integrating the potential across
the cell boundary:
ð
v
i
ð
K
i
d½K
þ
½K
þ
dv ¼
KT
q
v
o
K
o
which yields:
ln
½K
þ
i
½K
þ
o
v
i
v
o
¼
KT
q
known as the Nernst equation, expressing the poten-
tial difference across a cell membrane as a function of
chemical and electrical forces driving ion transport.
2.1a.4 Examples of solving
BME models by computer
The materials presented in the previous section are the
basic tools that a biomedical engineer will use to solve
a problem or design a new diagnostic or therapeutic
device. However, an analytic solution is not sufficient;
the device will be designed and/or controlled using a
computer. This means that one must know how to
transform the model from continuous mathematics to
discrete mathematics (numerical analysis) and then write
a computer program to implement the new equation
