3.2 Multi parameter fluorescence detection
3.2.3 Detection pathway
3.2.4.6 Lifetime fit and implementation
The microtime (i.e. the relative arrival time of the photons after a syncpulse, see chapter 3.2.3) contains information about the fluores- cence lifetime. This microtime is recorded for every photon with a
2For the experiments presented here the microtime histograms consisted of 4096
32 Experimental setup and methodology
resolution of 4096 bins spanning a timerange from sync pulse to sync pulse of 37.5 ns (∼10 ps time resolution). In order to extract the lifetime information one has to fit a model function, usually a single exponential decay. If many photons are recorded (e.g. in an ensemble measurement) the bins are sufficiently populated and can be easily fit using a least squares algorithm. In contrast for burst analysis experi- ments with only 10-200 total photons per channel most of the bins will be empty and hence a more complicated approach has to be used. To this end a maximum likelihood estimator (MLE) was applied which is explained in full detail in [111] [60].
In brief, only a single exponential decay is assumed for the fluo- rescence intensity since information contained in a single burst is too low (i.e. there are not enough photons) to allow for a robust multi ex- ponential description. The time-dependent model of the signalM(τ) is defined as the sum of the fluorescent componentF(τ)and the back- ground signal from scattered lightB(τ)(Eq. 3.16)
M(τ) = (1−α)F(τ) +αB(τ). (3.16)
The overall contributions of these two signals are determined by a proportionality factorα.F(τ)is constructed by a convolution of the single exponential decay and the predetermined instrument response function (IRF), obtained from the scattered light of a water only sam- ple.
The comparison of data and model is done using the likelihood function Eq. 3.17 L(S|M(x)) = k
∏
i=1 ω(Si|Mi(x)). (3.17)Here,xis the set of model parameters forM,Sis the experimental decay histogram andωis the probability to detectSi photons in the ith out of kTCSPC channels. Throughout the fit the normalized fit quality parameter 2I∗is minimized Eq. 3.18
2I∗ =−2L(S|M(x))
3.2 Multi parameter fluorescence detection 33
where ζ is the number of model parameters. This fit quality pa- rameter is basically comparing the likelihood functionL(S|M(x))to the likelihoodL(S|S) =1. A derivation of the 2I∗equation for a MFD setup was done by [60] resulting in Eq. 3.19
2I∗ = 2 k−ζ−1 " Skln Sk Mk+S⊥ln S⊥ M⊥ ! +
∑
i Skiln S k i Mki∑
i S⊥i ln S ⊥ i Mi⊥ # . (3.19) The custom written fit algorithm used in this work calculates 10 values of the 2I∗ parameter on an evenly spaced grid ranging from 0 to 9 ns which is a sufficient range for all dyes in use. The two datapoints next to the one with the lowest 2I∗ are set as new outer boundaries and 10 new evenly spaced 2I∗values are calculated. This is iteratively done until the difference of the boundaries reaches 10 ps. At this stage the point with the lowest 2I∗is taken as result.In order to speed up the lifetime determination of huge measure- ments where a lifetime fit has to be performed for up to 1 million molecules some optimizations have been applied.
First, since only few photons are sorted to a huge amount of chan- nels (k=4096) this binning has been reduced tok=256 channels, reduc- ing the amount of data to be processed.
Second the background share α was set to a fixed value of 0% which is close to the actual value throughout the high countrates of a burst to avoid an additional fit parameter.
Third, instead of Eq. 3.19, the less computationally expensive Eq. 3.20 by [111] 2I∗= 2 k−ζ−1
∑
i Siln Si Mi . (3.20)was used for the 2I∗ calculation. Since this equation does not in- clude corrections for anisotropy related effects occurring in an MFD experiment (i.e. different decays in parallel and perpendicular chan- nels due to the polarized excitation) the signal had to be preprocessed prior to the lifetime fit. To this end the total fluorescence signal of
34 Experimental setup and methodology
each channelSxy,i(xy = GG, RR or GR) was calculated including cor-
rections for different detection effieciencies (β) and microscope objec- tive depolarization (l1,2 see chapter 3.2.4.3) which however has to be done only once prior to the fit iterations
Sxy,i= (1−3l2)βSkxy,i+ (2−3l1)S⊥xy,i. (3.21)
Since out of the two polarizations perpendicular to the excitation polarization only one can be detected, equation 3.21 counts perpen- dicular photonsSktwice compared to the parallel photonsS⊥ hence removing possible artifacts from high anisotropy molecules.
Results obtained using this simplification are comparable to the ones obtained using Eq: 3.19 and includingαas a free fit parameter (see chapter 4.3.5, Table 4.10) but can be calculated significantly faster. Since only a slight loss in precision and no differences for the separa- tion of populations could be detected for the simplified method it is used throughout this work if not stated otherwise.