Fluorescence Decay Fitting

AlliGator offers basic capabilities to analyze data using standard techniques of fluorescence decay fitting:

• • Individual decays can be fitted with a single or double-exponential decay convolved with an experimental instrument response function (IRF).

  • Time series analysis of a ROI in the image source can use "Decay Fittting" instead of "Phasor Analysis" (each time point decay is fitted with the same model, and selected parameters are output as time-series).

  • Selected decays can be globally fit to a single or double-exponential decay convolved with an experimental IRF, some parameters of the model being common to all decays.

These different capabilities are described next.

1. Single decay fit

1.1. Overview

A single plot can be fitted by a model function convolved with the selected IRF by right-clicking on its legend (or close to it in the graph) and selecting Fit Decay. The parameters specifying the type of fit and constraints used are defined in the Settings>>Fluorescence Decay>>Fitting panel reproduced below:

Several fitting options, discussed below, are available:

• • Fit Model

  • Weighted Fit

  • Show Fit Residuals

  • Type of Residuals

  • Max & Min Decay Percentile

  • Show Full Decay

Fit with constraints applied on individual parameters is handled by an array of Fit Parameter Constraints specifying the:

• • Parameter

  • Min & Max Value

  • whether or not it is a Global parameter (applicable in case of multiple fits only) 

The Displayed Fit Parameters array only applies to time-series analysis and will be discussed in that context in a later section.

1.2. Fit Options

• • Fit Model: Two models are currently available.

A single exponential (1-Exponential) model defined by

,where b is the baseline and the IRF is offset by an amount (i.e. centered at) .

A double exponential (2-Exponentials) model defined by:

,

• • Fit Weight: Two types of fits can be performed: an unweighted fit where all data points are equally weighted in the minimization function (sum of difference squared), or a weighted fit, where each data point i is weighted by 1/

  (or 1 if  = 0), where  is the function value.

• • Residuals: The fit residuals (difference between the original decay and its fit) can be optionally plotted in addition to the fit itself. Several weighting schemes can be chosen:

The standard residual is the mere difference between the original decay and its fit, while the normalized residual is the difference divided by the function value. The reduced residual is the difference divided by the square root of the absolute value of the function value.

• • Min & Max Decay Percentile: The fit can be performed over the whole decay or limited to the "tail" part of the decay. The latter is defined as the part of the decay located between XX% of the decay maximum (max percentile) and YY% (0 

 YY  XX  1) of the decay maximum (min percentile).

• • Show Full Decay: When only part of the decay is fitted, it is possible to show the fitted curve (and residuals, optionally) calculated over the full decay range by checking this checkbox. The default (unchecked) is to only show the decay over the selected range.

  • Parameter Uncertainties: Because parameter uncertainty calculation involves computing the covariance matrix of all parameters, this can be very memory consuming in the case of global fit of large data sets. In that case, it might be desirable to skip calculation of parameter uncertainties by leaving this checkbox unchecked.

  • Periodic Boundaries: This option is available in beta mode and allows enforcing periodic boundary conditions (and provide a value for the repetition period).

This is mostly useful for large gates (e.g. SwissSPAD data) for which the resulting decay does not look anymore as a sharp rise followed by a tail decaying to background level, but instead as a continuous "wave". In these conditions, it is advantageous to treat the decay as periodic. Note that the recorded decay needs to be no longer than the provided period for the fit to be any good (it can be shorter, i.e. truncated). This option has no effect on global fits at this time.

1.3. Fit Parameter Constraints

Fit parameters can be constrained within a specified range defined by the min (-Inf if unconstrained) and max value (Inf if unconstrained).

The list of actual parameters that can be constrained depends on the chosen model:

For instance, choosing tau_2 as a constrained parameter in a 1-Exponential model will have no effect.

If a parameter is unconstrained, it is possible to remove it from the array of constrained parameters by right-clicking on it and choosing Delete Element. If no parameter is constrained, it is possible to delete all elements of the array by right-clicking on the scrollbar and choosing Empty Array.

1.4. Fit Results

In addition to the plot output(s) in case of a successful fit, the fit results are output to the Notebook. A typical output will read:

2-Exponentials unweighted fit of XXXXX

Fit range: 0%-100%

Fitted Parameters:

Offset: 0.003166 ± 0.001487 [-0.1, 0.1]

Baseline: -0.003543 ± 0.004714

A1: 0.26873 ± 0.140506 [0, Inf]

tau 1: 0.446588 ± 0.151041 [0.005, 1]

A2: 1.122693 ± 0.144699 [0, Inf]

tau 2: 1.13671 ± 0.078737 [0, 2]

R^2: 0.999408

Reduced Chi^2: 0.007177

Standard residuals

where XXXXX is the decay name.  and the reduced  as well as the 68% confidence intervals (errors) are defined according to the definitions provided here.If the fit fails, an error message will be displayed instead (and not plot added to the Decay Graph).

2. Time-series decay fit

In the case of a time-series analysis, decay fits can be performed instead of phasor analysis, by choosing Time-Series Analysis (Fitting) in the Analysis menu:

Each time point decay is fitted separately, following the protocol described previously for single decays. In addition, it is possible to generate one or more plots of the evolution of selected fit parameters across the series, using the Displayed Fit Parameters array. These plots will be output in the Lifetime Graph of the Lifetime Analysis panel (see corresponding manual page). Parameters that can be displayed can be chosen from the following list:

This list includes the fit parameters and derived quantities, such as the mean lifetime <tau> or fraction f1 and f2 (for the 2-Exponentials model, defined below), or the  and  outputs.

3. Global fit of selected decays

To constrain multiple decays to use the same fitted parameter(s), a global fit is needed. To achieve this, the different decays to be fitted need to be present in the Decay Graph and selected. Additionally, the common parameters need to be specified in the Fit Parameter Constraints array of the Settings>>Fluorescence Decay>>Fitting panel.

Note that the parameters themselves do not need to be constrained at all (i.e. the Min and Max values can be set to -Inf and Inf respectively). However, they need to be added to the array and the corresponding Global checkbox checked.

To start the global fit, right-click the Decay Graph and choose Global Fit of Selected Decays or use the Analysis>>Global Fit of Selected Decays menu item.

A "busy" icon appears at the bottom of the AlliGator window, as well as an Abort Fit button, as this type of analysis can take a long time for large numbers of decays. The fit status can be checked by opening the Fit Status Window via the Analysis>>Fit Status Window menu item (see below).

While the results of individual fits are exported to the Notebook, it can be convenient to have all parameters exported in a tabulated form at the end of the analysis. Check the Export Tabulated Results checkbox in the Settings>>Fluorescence Decay>>Fitting panel to select this option.

Additionally, as for a time series decay fit analysis (Section 2 above), it is possible to export the series of selected fit parameters as plots in the Lifetime Graph. For this to happen, define which parameters to represent in the Displayed Fit Parameters control.

4. Fit Status Window

As a global fit of a large number of decay can take a long time, it can be useful to check its progress with the AlliGator Fit Status Window, accessible via the Analysis menu.

Note that this window also provides information during standard fit operations, but is generally less useful in these cases, as results are obtained relatively rapidly and immediately exported to the Notebook.

The Fit Status Window is shown below:

The update frequency is user-adjustable, but is limited by the actual computational speed (at most one update per iteration can be provided, and such an iteration can be quite long when a global fit with 100's of parameters is performed).

3 types of fits are performed successively (least squares, least absolute residuals, bisquare), the best of 3 (the fit with the largest R2)  being retained as the final result. The ongoing type of fit is indicated by the Method indicator, while # Iterations, # Function Calls, etc., report on current parameters of the ongoing fit. The last computed parameters (Global Parameters and Decay Specific Parameters) are also indicated, the Decay control allowing to select which plot parameters are displayed. Parameter names are indicated to the left and their value to the right.

Once a method is completed, its statistics (Residue, R2 and RMSE) are visible on the right, different results being accessible via the Method pull-down list at the top. If a method hasn't been tried yet, the corresponding statistics are NaN (not a number).