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Anal. Chem. 1989, 61, 2310-2315
Quantitative Estimation of Component Amplitudes in Multiexponential Data: Application to Time-Resolved Fluorescence Spectroscopy A. L. Wong and J. M. Harris* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Ouantltatlve Information about lndlvldual component contrlbutions to mutllexponentlal data Is obtalned by a reiterative, regresskn algorlttun that employs a llnear least-squares determlnatlon of component amplltudes within a nonlinear least-squares search for the exponential decay tlmes. Uncertainty In the parameter estlmates, arlshg from uncertalnty In the data and overlap In the response, is pedlcted from first prlndples. The analysts method Includes weighthy to account for the Pdsson error distrlbutlon arislng from shot-nolse-lbnlted signals, which Increases the accuracy of the amplitude estlmates. While the algortthm is applicable to a variety of klnetlc methods, ll Is applied In the present work to the analysis of tlme-resolved fluorescence decay curves. A fluorescence decay curve, wrltten as a row vector, D, Is decomposed into two factors: A, a column vector Containing the amplitude contribution of each component, and [ C ] , a matrix which contalns temporal behavlor of each component In Its rows. The analysls uses llnear least squares to obtain estimates of A, which Increases the efflclency by reducing the number of parameters that are searched. The theory of error In linear least squares allows the uncertalnty of the component amplitudes to be determlned from the [ C ] matrix, derived from best estlmates of the temporal behavlor of the sample.
Quantitative determination of individual component contributions in multicomponent spectroscopic data is a common and difficult analytical problem. For data that derive from first-order kinetic reactions, such as time-resolved fluorescence, metal ion complexation reactions, and the decay of radioisotopes, component contributions to the observed amplitude are particularly difficult to retrieve due to the similar temporal behavior of exponential decay curves. Computational approaches to reaction-rate methods of chemical analysis have been compared by Wentzell and Crouch (1) for reactions following first-order and pseudo-first-order kinetics, and the computational difficulties commonly encountered were described. Mieling and Pardue (2) have developed a multiplelinear regression procedure to obtain the amplitudes of sample components simultaneously reacting at different rates. The method was similar to that examined by Willis et al. ( 3 ) to follow complexation kinetics of alkaline-earth complexes. The approach modeled the product concentration as the sum of the integrated first-order rate law for each component plus any other sources of product that are time-independent. The expression that is fit to the data is a sum of an offset and two or three exponentials of known decay constant. where the amplitudes are extracted by linear regression. Time-resolved fluorescence spectroscopy, where the decay of intensity following pulsed excitation is generally obtained by time-correlated single photon counting techniques, produces data that also follow exponential relationships ( 4 , 5 ) . Most data analysis schemes for time-resolved fluorescence (6-9) follow an approach similar to the method described
above for chemical kinetics, except that the sensitivity of fluorescence lifetimes to the sample matrix does not generally allow the decay constants to be known in advance. As a result, fitting the data requires a nonlinear least-squares method since the observed intensity depends exponentially on the unknown decay constants; these nonlinear parameters appear in the argument of the exponential time dependence. In nonlinear least squares, optimal values for each parameter are found by directly searching a parameter space while minimizing the squared deviations between theory and experimental results. A sequence of error-reducing steps is chosen where the best direction is reappraised after each step. A complication arises with this approach for fitting multicomponent systems, which results in large increases in computation time and uncertainty in evaluating parameters. Since each resolvable component in the fluorescence decay signal is described by an amplitude and a lifetime or decay constant, the number of parameters that must be optimized grows twice as fast as the number of components in the system. As a result, a fluorescence decay signal containing three resolvable components requires that a six-dimensional parameter space be searched. In this work, we present a more efficient approach to the resolution of multiexponential data, where the decay constants are unknown. The method combines a linear regression step to obtain the amplitudes, similar to that described for mul, a nonlinear ticomponent chemical kinetic data ( 2 , 3 ) within least-squares algorithm to determine the best fit decay constants. This approach was first used to resolve multicomponent fluorescence spectra from emission wavelength-decay time data matrices (10). We have modified the method to obtain single preexponential factors for each component and to account for the nonhomogeneous variance of shot-noiselimited data. Proper weighting of the observations is found to significantly influence the accuracy of the results. The theory of error in linear regression has allowed us to develop an analytical expression that predicts quantitatively the precision of determining individual component contributions. Synthetic data, generated with multiexponential decay curves, and experimental fluorescence decay transients, obtained by time-correlated single photon counting, were used to evaluate the efficiency and accuracy of weighted regression algorithm and its predictions of the uncertainty for the component amplitudes.
THEORY Modeling Fluorescence Decay Data. In absence of excited state-excited state interactions, the decay of fluorescence after pulsed excitation follows first-order kinetics, and the response of an n-component sample is d, = c ( a 1 / 7 )exp(-jAt/Ti)*Il
(1)
r=l
where the intensity, d,, at time interval j is the sum of contributions of the n fluorophores emitting in the sample. The parameters T~ and a, are the lifetime and total intensity of the
0003-2700/89/0361-2310$01.50/0 (C 1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 20, OCTOBER 15, 1989
ith component, respectively, and At is the time interval between data points. The asterisk, “*”, represents convolution (11) of the exponential decay of fluorescence with the instrument response function, I? This fluorescence decay data can also be represented as a row vector, D, where t is the number of time intervals over which the fluorescence is recorded. The data can be expressed as a product of a vector and a matrix D = A[C] R (2)
+
where A is a row vector of length n containing the amplitudes of the fluorophores in the sample and [C] is an (n X t ) matrix containing the time decay behavior of the excited-state components in its rows. Since the observed data, D, contains error associated with the observation, there is a difference between the observation and the theory, given by a vector of residuals,
R. Combining Linear and Nonlinear Least Squares. An efficient method of utilizing linear least squares in the analysis of time-resolved fluorescence decays has been described (IO), based on assuming uniform variance in the residuals, R. Given a time-resolved data vector D, the method factors this matrix into A and [C] which provide quantitative and lifetimc information, respectively. To analyze D, a trial matrix [C] of decay curves is constructed row by row, by convolution of the exponential decay of the ith component having an trial lifetime, ii,with the measured instrument response, I,
cIij = Ij*(1/?J expGAt/.ri)
(3)
This approach requires the assumption that the excited states of each component decay by first-order kinetics, an excellent approximation for dilute solutions and high repetition rate excitation. The unweighted, linear least-squares solution for the best estimate of the component amplitudes A is given by (12,13)
[e]’
A = D[6]’([6][C]’)-l
(4)
where signifies the transpose of [e] and the superscript “-1” repJesents the matrix inverse operation. A model data vector D is calculated from the product of the estimated amplitudes, A, and the trial time decay matrix, [C]
D = A[C] Optimal parameter values are those which minimize
are reoptimized upon each move of the simplex in the nonlinear parameter space. Weighting Shot-Noise-LimitedData. Application of the linear least-squares method of eq 4 and minimization of the unweighted squared error, eq 6, require that the error in the data be described by an error distribution of constant variance (12,13). In the case of shot-noise-limited,fluorescence decay data, typical of single photon counting experiments ( 4 ) )the dependence of the signal variance on amplitude requires that the solution be modified to account for differences in the size of the residuals. Regression methods for exponential decay data have been shown to be sensitive to inhomogeneous variance (16),where the efficiency of extracting parameters is degraded if weighting is neglected. The expected variance of the individual observations, u’j, required to weight the observations is readily available for intensity measurements made by single photon counting (17). Under such conditions, the error is Poisson distributed, approaching a Gaussian distribution for a large number of counts (18). The variance, a2j, of the data in the jth time interval has an expected value equal to the mean number of counts for that interval (4). Although the mean number of counts is not known from single observation, if the number of counts is sufficiently large ( X O O ) , the variance may be estimated (with
