Principles of Fluorescence Spectroscopy

130 TIME-DOMAIN LIFETIME MEASUREMENTS
Figure 4.45. Intensity decay of [Ru(bpy) 3 ] 2+ measured with a photon
counting multiscalar. Excitation was accomplished using a 440 nm
laser diode with a repetition rate of 200 kHz. The lifetime was 373 ns.
Revised from [169].
method, 178–179 Prony's method, 180 sine transforms, 181 and
phase-plane methods. 182–183 These various techniques have
been compared. 184 The method-of-moments (MEM) and the
Laplace methods are not widely used at the current time.
The maximum entropy method is newer, and is being used
in a number of laboratories. The MEM is typically used to
recover lifetime distributions since these can be recovered
without assumptions about the shape of the distributions.
During the 1990s most studies using TCSPC made
extensive use of NLLS and to a somewhat lesser extent the
MEM. This emphasis was due to the biochemical and biophysical
applications of TCSPC and the need to resolve
complex decays. At present TCSPC is being used in analytical
chemistry, cellular imaging, single molecule detection,
and fluorescence correlation spectroscopy. In these applications
the lifetimes are used to distinguish different fluorophores
on different environments, and the number of
observed photons is small. A rapid estimation of a mean
lifetime is needed with the least possible observed photons.
In these cases maximum likelihood methods are used (eq.
4.20). In the following sections we describe NLLS and the
resolution of multi-exponential decay.
4.9.1. Assumptions of Nonlinear Least Squares
Prior to describing NLLS analysis, it is important to understand
its principles and underlying assumptions. It is often
stated that the goal of NLLS is to fit the data, which is only
partially true. With a large enough number of variable
parameters, any set of data can be fit using many different
mathematical models. The goal of least squares is to test
whether a given mathematical model is consistent with the
data, and to determine the parameter values for that model
which have the highest probability of being correct. Least
squares provides the best estimate for parameter values if
the data satisfy a reasonable set of assumptions, which are
as follows: 185-186
1. All the experimental uncertainty is in the dependent
variable (y-axis).
2. The uncertainties in the dependent variable (measured
values) have a Gaussian distribution, centered
on the correct value.
3. There are no systematic errors in either the dependent
(y-axis) or independent (x-axis) variables.
4. The assumed fitting function is the correct mathematical
description of the system. Incorrect models
yield incorrect parameters.
5. The datapoints are each independent observations.
6. There is a sufficient number of datapoints so that
the parameters are overdetermined.
These assumptions are generally true for TCSPC, and
least squares is an appropriate method of analysis. In many
other instances the data are in a form that does not satisfy
these assumptions, in which case least squares may not be
the preferred method of analysis. This can occur when the
variables are transformed to yield linear plots, the errors are
no longer a Gaussian and/or there are also errors in the xaxis.
NLLS may not be the best method of analysis when
there is a small number of photon counts, such as TCSPC
measurements on single molecules (Section 4.7.1). The
data for TCSPC usually satisfy the assumptions of leastsquares
analysis.
4.9.2. Overview of Least-Squares Analysis
A least-squares analysis starts with a model that is assumed
to describe the data. The goal is to test whether the model is
consistent with the data and to obtain the parameter values
for the model that provide the best match between the
measured data, N(t k ), and the calculated decay, N c (t k ), using
assumed parameter values. This is accomplished by minimizing
the goodness-of-fit parameter, which is given by
χ 2 ∑ n
k1 σ2 k
∑ n
k1
1
N(t k) N c(t k) 2
N(t k) N c(t k) 2
N(t k)
(4.21)