{'fraction_non_alphanumeric': 0.01722736876320494, 'fraction_numerical': 0.003087924589631074, 'mean_word_length': 4.8833652007648185, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fluorescence decay data analysis correcting for detector pulse pile-up at very high count rates

Using Time-Correlated Single Photon Counting (TCSPC) for the purpose of fluorescence lifetime measurements is usually limited in speed due to pile-up. With modern instrumentation this limitation can be lifted significantly but some artefacts due to frequent merging of closely spaced detector pulses (detector pulse pile-up) remains an issue to be addressed. We propose here a data analysis method correcting for this type of artefact and the resulting systematic errors. It physically models the photon losses due to detector pulse pile-up and incorporates the loss in the decay fit model employed to obtain fluorescence lifetimes and relative amplitudes of the decay components. Comparison of results with and without this correction show a significant reduction of systematic errors at count rates approaching the excitation rate. This allows quantitatively accurate fluorescense lifetime imaging (FLIM) at very high frame rates.

In order to acquire a FLIM image with a confocal microscope, one typically scans the excitation and detection foci across the sample [Kobe 03, Ortm 04]. In case of a fast scanning system, this very often leads to only a few detected photons per pixel, which will not allow accurate lifetime analysis. It is therefore necessary to increase the number of detected photons per pixel either by reducing the scan speed or by accumulating several frames to sum up the photons. This is why it typically takes several seconds up to half a minute to obtain a good FLIM image. A possible path to speedier FLIM is to perform TCSPC with the highest possible photon rate.
Even though it is feasible to correct for dead-time [Patt 07, Isba 17] the ultimate key to high throughput TCSPC is eliminating dead-time. However, the highest time resolution and lowest differential nonlinearity (DNL) in timing can typically only be obtained with dedicated time-todigital converters that incur a dead-time on the order of some tens of nanoseconds after each photon detection. It is possible to build faster timing circuits that allow effectively zero dead- One issue that remains with the approach sketched so far is that of detector pulse pile-up. This effect is caused by the fact that individual detector pulses have a certain width that cannot be made infinitely small. With existing hardware as introduced above this is about 500 ps. At the desired high count rates the actual photon statistics imply that there will quite frequently be photon emissions closer than this, which means that successive detector pulses may overlap and merge into one. Since the TCSPC electronics cannot distinguish the original pulses they will be counted as only one. This causes another form of histogram distortion where early photons are under-represented. In the following we introduce a correction scheme for the systematic errors resulting from this type of distortion. More precisely, we show how the underlying physical mechanism of photon losses due to pulse pile-up can be incorporated in the fit model so that the fluorescence lifetime estimate is made robust against this type of histogram artefact, even at very high count rates.

Proposed Correction Metod
A key parameter for the photon loss model is the closest pulse spacing in time that the particular combination of TCSPC electronics and detector can still resolve. Let this pulse pair resolution be called Dt here. Assuming Poisson statistics and a count rate n, the time intervals between successive photons will be exponentially distributed and the fraction of intervals shorter than Dt will be 1 -e -nDt . A photon falling into Dt after the detection of a previous photon will be lost. The fraction of remaining photons will therefore be e -nDt [Reed 72].
TCSPC with pulsed excitation samples the photon distribution as a function of elapsed time t after excitation. Let Dec(t) be the decay curve. With known excitation rate f exc and known pulse count N each discrete time t i can be associated with a "differential" or instantaneous count rate dn = f exc Dec(t i ) / N . ( This model curve is fitted to the data as usual by means of a nonlinear least squares algorithm or maximum likelihood estimation. The pulse pair resolution Dt can either be determined in advance as an instrument specific constant or alternatively be obtained as an additional fit parameter. The latter allows a calibration measurement of Dt to be performed in reverse without additional expenditure of apparatus or method. Nevertheless it must be noted that despite the correction, the loss of photons due to detector pulse pile-up will cause some loss in accuracy of the curve fitting and parameter estimation. These losses in accuracy, however, emerge only as statistical errors, which is a substantial improvement over the uncorrected situation, where systematic errors dominate over statistical errors.
A visual explanation of the proposed correction method is shown schematically in Fig. 1. For simplicity it uses a single-exponential decay and for clarity the photon losses due to detector pulse pile-up are exaggerated.  Above 60 Mcps there appears to be a slight trend towards systematically increased lifetimes even for the corrected lifetimes, indicating that the correction starts to become less precise. The reason for this lies in count rate dependent systematic errors in the decay curves that probably originate from the front end analog electronics or the detector and cannot be described in terms of detector pulse pile-up. Nevertheless, the relative error associated with the trend is still smaller than 5%.
In order to demonstrate the effectiveness of the correction also in

Conclusion
We have shown a correction method for detector pulse pile-up in fluorescence lifetime measurements with TCSPC that allows for data collection at very high count rates. Experimental results indicate that this allows measuring at count rates approaching the excitation rate with lifetime errors below the 5% level. This facilitates quantitatively accurate confocal FLIM measurements at very high frame rates.