Biology Reference
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intensity profile F ( t ), which is dependent on both the fluorescence sample
and the instrumentation, is defined by Eq. (5.26) .
This fitting procedure is usually performed with the least squares method
(LSM), which consists in minimizing an error function w
2
defined as the total
squared differences between experimental data points x i and theoretical
values s i deduced from Eq. (5.26) :
p X
N d
2
1
N d
ð
s i
x i
Þ
2
LSM
w
¼
½
5
:
35
x i
i
¼
1
where N d is the number of data points and p is the number of fitting param-
eters. The minimization of this error function is generally performed with
the Levenberg-Marquardt algorithm, which has been implemented in most
of the commercially available FLIM analysis software. This commonly used
FLIM image analysis strategy is robust and gives reliable results when the
number of photons is large 8,11,23,82 (cf. Fig. 5.18 ).
However, if the total number of photons is low, the LS method becomes
inaccurate because its error function assumes that the noise is described with
a Gaussian instead of a Poisson distribution, which is incorrect. In the case of
a small number of photons, a solution named “adaptive Monte Carlo data
inflation” (AMDI) algorithm is used, which consists in inflating statistically
the number of photons for being compatible with the LS method. 11 Another
possibility, which is called the “maximum likelihood estimation” (MLE)
method, 82,83 is to modify the error function to take into account the
Poisson noise distribution. This error function is now defined as 83
p X
p X
N d
N d
2
N d
2
N d
s i
x i
2
MLE
w
¼
ð
s i
x i
Þ
x i ln
½
5
:
36
1
1
It was demonstrated that both solutions (AMDI and MLE methods) give
an accurate lifetime component estimation of multiexponential intensity de-
cays with a reduced number of photons, in comparison with the LS
method. 11,82 Finally, it has been recently demonstrated that an alternative
method based on Bayesian analysis enables the correct estimation of the
lifetime of monoexponential decays with a few photons. 84 However, this
method was never applied with multiexponential decays, which limits its
application in the context of biosensor activity measurements.
5.2.2.2 Exploiting data
As already mentioned, the fluorescence intensity decay of a single-chain bio-
sensor is described with multiexponential terms (cf. Eq. 5.10 ).
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