2.1. Fluorescence decays¶

2.1.1. Analysis¶

2.1.1.1. Phasor¶

Changing the data representation from the classical time delay histogram to the phasor representation provides a global view of the fluorescence decay at each pixel of an image. In the phasor representation reveal the presence of different in a a fluorescence decay. The analysis of time-resolve fluorescence data in the phasor space is done by observing clustering of populations in the phasor plot rather than by fitting the fluorescence decay using exponentials. The analysis is instantaneous since is not based on calculations or nonlinear fitting. Thus, the phasor approach can simplify the way data are analyzed and makes FLIM technique accessible to the nonexpert in spectroscopy and data analysis :cite:DIGMAN2008L14.

The phasor transformation can use data collected in either the time or the frequency domain. In the time-correlated single-photon counting (TCSPC) technique, the histogram of the photon arrival time at each pixel of the image is transformed to Fourier space, and the phasor coordinates are calculated using the following relations:

\[g_{i,j}(\omega) = \int_{0}^{T} I(t) \cdot \cos(n\omega t) dt / \int_{0}^{T} I(t) dt\]

\[s_{i,j}(\omega) = \int_{0}^{T} I(t) \cdot \sin(n\omega t) dt / \int_{0}^{T} I(t) dt\]

in which gi,j(ω) and si,j(ω) are the x and y coordinates of the phasor plot, n and ω are the harmonic frequency and the angular frequency of excitation, respectively, and T is the repeat frequency of the acquisition :cite:ranjit2018fit.

2.1.1.2. Fluorescence lifetime analysis¶

2.2. Maximum likelihood fits¶

tttrlib provides a set of precompile models with fits (e.g. fit23, fit24, and fit25) that have their own model function, target (objective) function, fit function, and parameter correction function. Briefly, the purpose of these functions is as follows.

The model function computes for a set of input parameters the corresponding realization of the model (the model fluorescence decay curve(s)). The target function (more often called objective function) computes the disagreement of the data and the model function for a given set of model parameters. The fitting function optimizes a set of selected input parameters to the data by minimizing the disagreement of the data and the model. Finally, the estimates of the model parameters are corrected by a correction function. These functions have names that relate to the name of the model, e.g., DecayFit23.target corresponds to fit23. For computationally efficiency, the functions are hardcoded in C.

The models of the different fits differ in their parameters. The parameters of the fits are described in detail below. The target functions explicitly consider the counting noise and provide an estimate of the likelihood of the parameters. The initial values of the parameters are specified by the user. Next, the likelihood of the target is maximized by the Limited-Memory Broyden-Fletcher-Goldfarb-Shanno Algorithm (L-BFGS). L-BFGS is an iterative method for solving unconstrained nonlinear optimization problems. Gradients for the L-BFGS algorithm are approximated by linear interpolation to handle the differentiable components, i.e., the model parameters that are free. The parameters of a model ca be either specified as free or as fixed. Only free parameters are optimized.

Python ^^^^^^ For a use in python the tttrlib module exposes a set of C functions that can be used to (1) compute model and (2) target/objective function and (3) optimize model parameters to experimental data. Besides the exposed functions the fit models are accessible via a simplified object-based interface that reduces the number of lines of code that need to be written for analyzing fluorescence decay histograms. The code blocks that are used below to illustrate the tttrlib functionality are extracts from the tests located in the test folder of the tttrlib repository. The test can be used as a more detailed refernce on how to use tttrlib.

Model functions can be computed using for instance the DecayFit23.modelf function of the tttrlib module. Here, the 23 represents a particular model function. The use of a tttrlib model function is for the model function 23 (DecayFit23.model) below.

To compute a model, first a set of model parameters and a set of corrections need to be specified. All input parameters for DecayFit23.model are numpy arrays. In addition to the model parameters and the corrections DecayFit.23.model requires an instrument response function (irf) and a background pattern. The model functions operate on numpy arrays and modify the numpy array for a model in-place. This means, that the output of the function is written to the input numpy-array of the model. In the example above the output is written to the array model.

To compute the value of a target for a realization of model parameters DecayFit23 provides the functions DecayFit23.target. The inputs of a target function (here DecayFit23.target) are an numpy array containing the model parameter values and a structure that contains the corrections and all other necessary data to compute the value of the objective function (for fit23 i.e. data, irf, background, time resolution).

The data needed in addition to the model parameters are passed to the target function using fit2x.MParam objects that can be created by the fit2x.CreateMParam function from numpy arrays. The return value of a target function is the score of the model parameters

Model parameters can be optimized to the data by fit functions for fit 23 the fit function fit2x.DecayFit23.fit is used.

The fit functions takes like the target function an object of the type fit2x.MParam in addition to the initial values, and a list of fixed model parameters as an input. The array containing the initial values of the model parameters are modified in-place buy the fit function.

Alternatively, a simplified python interface can be used to optimize a set of model as illustrated by the source code embedded in the plot below. The simplified interface handles the creation of auxiliary data structures such as tttrlib.MParam.

Analysis result of fit23 by the simplified python interface provided
by ``tttrlib``

In the example shown above, first a fit object of the type fit2x.DecayFit23 is created. All necessary data except for the experimental data for a fit is passed to the fit object when it is created. To perform a fit on experimental data for a set for a set of initial values, the fit object is called using the inital values and the data as parameters.

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