### Abstract

- Top of page
- Abstract
- Theoretical Consideration
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- LITERATURE CITED
- Supporting Information

We calculate here analytically the performance of the polar approach (or phasor) in terms of signal-to-noise ratio and *F* values when performing time-domain Fluorescence Lifetime Imaging Microscopy (FLIM) to determine the minimal number of photons necessary for FLIM measurements (which is directly related to the *F* value), and compare them to those obtained from a well-known fitting strategy using the Least Square Method (LSM). The importance of the fluorescence background on the lifetime measurement precision is also investigated. We demonstrate here that the LSM does not provide the best estimator of the lifetime parameter for fluorophores exhibiting mono-exponential intensity decays as soon as fluorescence background is superior to 5%. The polar approach enables indeed to determine more precisely the lifetime values for a limited range corresponding to usually encountered fluorescence lifetime values. These theoretical results are corroborated with Monte Carlo simulations. We finally demonstrate experimentally that the polar approach allows distinguishing in living cells two fluorophores undetectable with usual time-domain LSM fitting software. © 2010 International Society for Advancement of Cytometry

Fluorescence Lifetime Imaging Microscopy (FLIM), which relies on the measurement of the fluorescence lifetime at each pixel in an image, is now routinely performed in many biological and biophysical laboratories. Since this fluorescence lifetime is sensitive to the local environment of the fluorophore [e.g. [Ca^{2+}], pH, temperature, viscosity, energy transfer (1)], a large number of biologically relevant questions can now be assessed without the need for ratiometric measurements. For example, it becomes possible to visualize and to quantify the dynamic interactions between proteins in vivo by detecting lifetime modifications associated with Förster Resonance Energy Transfer (FRET) occurring between two fluorescent probes (a donor and an acceptor) (2, 3).

Up to now, a large number of different techniques have been developed and have been used to measure the fluorescence lifetime. These techniques can be divided into two main groups: frequency domain methods (4–6) and time domain methods (7–9). In this article, we limit our study to this second group.

In time domain methods, a series of short pulses of light excites a fluorescent sample and the consecutive intensity *I*(*t*) emitted by this sample is measured. The intensity profile, which can be a single or multi-exponential decay, varies according to:

- (1)

In most cases, the determination of the different lifetimes τ_{i} and contributions *a*_{i} are achieved by fitting the collected data at each pixel with this equation with two or more unknowns. The major problems with this fitting method are that it requires computation time and a high level of expertise to obtain reliable results, due to the large number of existing minimization algorithms, the large number of unknown parameters, the correlation between lifetimes and species contributions and the low number of detected photons in biological samples.

To simplify the analysis of FLIM images and to make it accessible to the non-expert user, alternative strategies have been developed (10–12). Among all these techniques, the polar plot or phasor initially described by Jameson et al. (13) and then developed by different groups (14, 15) is a promising approach. In the time domain, it consists of converting the standard temporal FLIM image into frequency domain data by calculating the Fourier sine and cosine transforms for each experimental fluorescence decay. These values are then represented in a two-dimensional histogram which corresponds to the polar image. With this nonfitting approach, a fast and visual representation of fluorescence lifetimes is obtained which greatly facilitated the analysis of FLIM data. Compared to the standard well-known fitting method, the qualitative interest of this original approach is obvious and it has been widely reported in the literature (12, 14, 15). However, to the best of our knowledge, none of these numerous studies addresses the quantitative issues of sensitivity, minimum lifetime resolution or Signal-to-Noise Ratio (SNR).

In this article, we describe and perform a quantitative and complete comparison of the two distinct strategies: the fitting method and the polar analysis. For quantifying the performance of each approach, we employ the *F* value introduced by Gerritsen et al. (16) which is defined as *F = (σ*_{τ}/τ)/(σ_{N}/N) where σ_{τ} is the standard deviation in repeated measurements of the lifetime value τ and σ_{N} is the standard deviation of the number of detected photons *N*. In fluorescence microscopy, this number of collected photons *N* is Poisson distributed, which implies that the SNR is . The *F* value becomes then .

Ultimately, with an ideal lifetime determination procedure, the number of detected photons required to measure both fluorescence lifetime and intensity is identical. In this case, the SNR of the lifetime measurement σ_{τ}/τ corresponds exactly to the SNR in fluorescence microscopy and the *F* value is equal to unity. However, in real FLIM experiments, σ_{τ}/τ is always superior to and the *F* value is then always greater than unity. Consequently, the more *F* is close to unity, the better is the lifetime determination procedure.

In the first part of this manuscript, we present the theoretical *F* values for an idealized fluorophore exhibiting monoexponential intensity decay and theoretically study the impact of the fluorescence background on these values. We then confront these theoretical values to computed ones obtained from Monte Carlo simulations. We finally corroborate these simulations with experimental results acquired in the temporal domain with the time correlated single photon counting (TCSPC) technique.

### Theoretical Consideration

- Top of page
- Abstract
- Theoretical Consideration
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- LITERATURE CITED
- Supporting Information

We begin by considering an idealized fluorescent sample whose intensity decay is a monoexponential of lifetime τ. In temporal domain methods, the fluorescence emitted by this sample is caused by the excitation with a Dirac light pulse at time *t* = 0. This signal is then recorded by an idealized background-free lifetime acquisition system composed of *k* time channels of width *T/k* [the Instrumental Response Function (IRF) of this acquisition system is then identical to the excitation Dirac pulse]. In this case, the fluorescence intensity function *f(t*) is

- (2)

which is normalized so that its integral value over the finite width *T* of the measurement window equals unity.

To correctly estimate the unique unknown parameter τ from a given dataset, several statistical methods exist (17, 18). In this work, we focus our attention on the estimation procedure that gives the highest accuracy with the minimal number of data. From the exhaustive work performed by Hall and Sellinger (18), it has been demonstrated that the most precise strategy to determine the lifetime parameter is by fitting the collected data with either the Least Square Method (LSM) or the Maximum Likelihood Estimator (MLE). This fitting method has been commonly used and it has been adapted in most iteration-based commercial software. The variance of the lifetime determined with the fitting method is also provided in Refs.18 and19. From these calculations, we can deduce the *F* value as

- (3)

where *r = T*/τ. For an infinitely small temporal interval of measurement (or, in other words, for an infinite number of time channels *k*), we can simplify this equation as

- (4)

This expression corresponds to the minimal *F* value accessible with an ideal fluorescence lifetime acquisition system. For example, to measure a lifetime τ = 2.5 ns with an acquisition temporal window of *T* = 12.5 ns, this limit is equal to *F*_{fit} = 1.0981 and it is almost reached for *k* = 64 time channels (*F*_{fit} = 1.0985).

If we consider now that there is a constant background noted *b* underlying the intensity decay, the fluorescence intensity function *f*(*t*) is then defined as

- (5)

In this case, there is no analytical expression for the *F* value of the fitting method.

Of course, it can be calculated numerically according to Ref.19. For instance, for the previously described example with an added constant background *b* = 0.1, we obtain *F*_{fit} = 1.3596. As anticipated, when a background is added to the measurement data, the precision of the lifetime determination deteriorates and hence the *F* value is increased. Complete plots of *F* value of the fitting method as a function of the parameter *r* with or without an added background are illustrated in black in Figure 1.

To avoid these complex fitting algorithm strategies, a promising alternative approach has been developed and recently employed in the time domain. This is the polar analysis also called phasor. Technically, this new representation is obtained by calculating the Fourier sine and cosine transforms (also noted [*u; v*] coordinates) of all temporal decays *f(t*). With this mathematical operation, each point in the polar representation corresponds to a single intensity decay curve present in each FLIM image pixel. More details can be found in (12, 20). The *u*- and *v*- coordinates of a monoexponential temporal decay *f(t*) are then simply defined by

- (6)

- (7)

where ω is the laser repetition frequency. From these equations, we can deduce the phase and modulation lifetime values τ_{m} and τ_{φ} which are well known parameters in the frequency domain (1)

- (8)

- (9)

To determine the theoretical *F* value of the polar approach, we have applied for the temporal domain the method described by Philip and Carlsson in frequency domain lifetime imaging techniques (21). In this work, we consider a fluorescent sample exhibiting monoexponential intensity decay. For such a sample, the phase and modulation lifetimes are equivalent. We then define the mean lifetime τ = (τ_{m} + τ_{φ})/2 and calculate analytically the *F* value in this case (the detailed calculation is presented in the Supporting Information Appendix). We obtain

- (10)

with the same parameter *r = T*/τ. Note that this theoretical *F* value has been obtained with a continuous probability distribution (which corresponds to an infinite number of time channels *k*). It is then obvious that in this case, *F* is independent of *k*. Moreover, the numerical integration errors of the *u* and *v* coordinates that may occur in real experiments are not considered in these calculations.

By applying this equation to the same fluorescent sample of lifetime τ = 2.5 ns acquired on a temporal window of *T* = 12.5 ns, we found *F*_{polar} = 1.3617. The theoretical accuracy of the fluorescence lifetime measurement performed with the polar approach is then slightly worse (25%) than one obtained with the standard fitting method.

When a background intensity *b* is added to the idealized fluorescence intensity *f(t*), we can also calculate analytically the expression of *F* value deduced from the polar procedure (see Supporting Information Appendix for details). This expression is

- (11)

Note that we well retrieve Eq. (10) in the absence of fluorescence background (*b* = 0). If we consider the same previous example with an added background *b* = 10%, we have *F*_{polar} = 1.3991 which is almost identical to *F*_{fit} obtained with the fitting method. It is thus interesting to plot *F*_{polar} as a function of *r* (cf. Fig. 1) and to compare *F*_{polar} and *F*_{fit}. We remark in Figure 1 that in the presence of background, the theoretical gain of the polar analysis over the fitting procedure is improved in comparison with the background free case for *r* < 10 (or τ > 1.25 ns when *T* = 12.5 ns and *b* = 0.1). The lifetime precision accessible with the polar approach can even be better than that obtained with the well-known fitting method (or in other words *F*_{fit}/*F*_{polar}>1) for the lifetime range 1.25 < τ < 2.5 ns (with *T* = 12.5 ns) when the fluorescence background *b* is superior to 5% (Fig. 1b).

However, in all these calculations, in order to resolve *F* values analytically, we have considered a constant known background which is not true in typical experimental conditions. In a usual acquisition situation, this parameter is totally unknown and it also has to be estimated. As previously shown by Köllner and Wolfrum for the fitting method (19), *F*_{fit} becomes notably larger than the expected value when *b* is unknown and it can even be doubled. Concerning the polar approach, the effects of an unknown background on the lifetime measurement have never been studied. In the following sections of this manuscript, we study these effects by considering simulated data and we then confront the results to the theory.

### DISCUSSION

- Top of page
- Abstract
- Theoretical Consideration
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- LITERATURE CITED
- Supporting Information

We have demonstrated herein that the widely used standard fitting method is extremely dependent on the fluorescence background which is always present in time-domain FLIM image experiments. The signal-to-noise ratio as well as the sensitivity and the lifetime resolution are severely degraded when this parasite background has to be estimated (19) due to the fitting algorithm strategy. The situation is notably different when FLIM image analysis is performed with the polar approach. We have thus demonstrated that the signal-to-noise ratio as well as the sensitivity and lifetime resolution calculated with the polar approach are weakly affected by this fluorescence background. The benefit of using this nonfitting polar approach becomes then evident in biological samples.

As previously mentioned, the simulated *F* values achieved with the fitting method and presented in this work are significantly higher than anticipated by theory. This is largely due to the fact that the robustness of the minimization algorithm used in the fitting method is altered when the fluorescence background is an additional unknown fitting parameter (the minimization algorithm has probably found a local minimum). To improve the concordance between simulation and theory and to converge towards the global, rather than a local minimum, a simple solution would consist in using more robust fitting algorithms. However this will extend the fluorescence lifetime determination time. We have performed all our lifetime measurements by using widely used and commercially available fitting software called SPCImage (Becker & Hickl) using the Levenberg-Marquardt algorithm which has the advantage to be a good compromise between optimization speed and lifetime precision. We have also tested another FLIM image analysis software called Tri2 (23) from the Gray Cancer Institute for Radiation Oncology and Biology (University of Oxford, UK) and obtained comparable results when fitting pixel by pixel with Levenberg-Marquardt algorithm and all parameters free (data not shown). It would be interesting in a future work to compare these results with other fitting algorithms and estimation procedures (like the maximum likelihood estimation). To the best of our knowledge, such an exhaustive study has never been reported in the presence of fluorescence background.

Even if we use a fitting algorithm enabling to perfectly match simulated and theoretical *F* values (ideal case), we remind the reader that the polar approach allows reaching higher lifetime precision than the fitting method in presence of low background (*b* > 0.05) which is always present in time-domain FLIM experiments. The gain of the polar approach over the fitting method is low but non negligible (<10%) for a limited range (1.25 < τ < 2.5 ns when *T* = 12.5 ns). We emphasize the fact that this limited range coincides exactly with experimental lifetime values of fluorophores commonly used in FRET experiments (eGFP or CFP).

Note that, in this present work, we have supposed both theoretically and experimentally that the IRF of our FLIM acquisition system was a Dirac delta function. This assumption which is valid for our specific system since the full width half maximum of its measured IRF is 32 ps (9) is not generally correct. In this case, in order to obtain exact lifetime values, it is necessary to deconvolute all fluorescence intensity decays from the experimental IRF before determining fluorescence lifetime with the polar approach or the fitting method. If this IRF deconvolution is correctly performed, the intensity decays become equivalent to those obtained with Dirac delta function. Consequently, all the results presented in this work are still valid.

For the simulated resolution and sensitivity studies provided here, we have considered particular lifetime values τ = 2.5 ns and fluorescence background *N*_{b} (*N*_{b} = *N*/10). While this is a usual lifetime value and a typical fluorescence background encountered in FLIM experiments (9), our study is also easily transposable to more general experimental conditions thanks to our theoretical *F* values treatment.

Finally, with our theoretical treatment presented here [Eq. (11)], when fluorescence background is present, we demonstrate that the least square method does not provide the best estimator of the lifetime parameter for fluorophores exhibiting mono-exponential intensity decays. In this study, we demonstrate indeed that the polar approach allows determining more accurate lifetime values for a limited range (1.25 < τ < 2.5 ns when *T* = 12.5 ns and *b* > 0.05). This study could also be extended for samples emitting biexponential intensity decays (like in FRET experiments): . For such samples, it is possible to determine the lifetime components numerically by fixing the donor lifetime in the absence of acceptor τ_{2} and by solving the system of Eqs. (6) and (7) (as described in Ref.20). In this case, the polar approach continues to help, however with reduced benefits due to the necessity to fix one unknown parameter in order to determine other lifetime components. A theoretical treatment of these benefits in presence of background is under investigation.