A modified phasor approach for analyzing time-gated fluorescence lifetime images

Authors


Farzad Fereidouni, Debye Institute, Utrecht University, PO Box 80.000, NL 3508 TA, Utrecht, the Netherlands; Tel: + 313 02532823; Fax: +313 02532706; e-mail: f.fereidouni@uu.nl

Summary

Fluorescence lifetime imaging is a versatile tool that permits mapping the biochemical environment in the cell. Among various fluorescence lifetime imaging techniques, time-correlated single photon counting and time-gating methods have been demonstrated to be very efficient and robust for the imaging of biological specimens.

Recently, the phasor representation of lifetime images became popular because it provides an intuitive graphical view of the fluorescence lifetime content of the images and, when used for global analysis, significantly improves the overall S/N of lifetime analysis. Compared to time-correlated single photon counting, time gating methods can provide higher count rates (∼10 MHz) but at the cost of truncating and under sampling the decay curve due to the limited number of gates commonly used. These limitations also complicate the implementation of the phasor analysis for time-gated data. In this work, we propose and validate a theoretical framework that overcomes these problems. This modified approach is tested on both simulated lifetime images and on cells. We demonstrate that this method is able to retrieve two lifetimes from time gating data that cannot be resolved using standard (non-global) fitting techniques. The new approach increases the information that can be obtained from typical measurements and simplifies the analysis of fluorescence lifetime imaging data.

Introduction

Lifetime imaging has become an important tool in life sciences. The lifetime of the excited state of fluorophores ranges from picoseconds to microseconds and provides valuable information on the state of the fluorescent molecules and their immediate molecular environment including temperature, pH, viscosity, ionic concentration and oxygen saturation (Sanders et al., 1995; Agronskaia et al., 2004; Lakowicz, 2006). Lifetime imaging can be employed to probe the local environment of the fluorescent molecules (Schneckenburger & Koenig, 1992, Van Zandvoort et al., 2002), to discriminate different molecules (fluorophores; Clayton et al., 2004; Kremers et al., 2008) and to detect molecular interactions at the nanometre scale (Lu et al., 1982; Grailhe et al., 2006). The latter is accomplished by monitoring the lifetime reduction of the donor dye molecules caused by Förster Resonance Energy Transfer (FRET) to spectrally matched acceptor molecules (Clegg, 1996). FRET can not only be used to image molecular interactions but also to measure distances on the nanometre scale (dos Remedios & Moens, 1995; Hillisch et al., 2001; Stryer, 1978).

In many cases the measured fluorescence decay curve is modelled as a mono-exponential function. Often, however, multiple molecular species at different concentrations or fluorophores exhibiting bi-exponential decays are present in the specimen. Therefore, resolving lifetimes and concentrations of each species in the same volume requires multiexponential fitting of the experimental fluorescence decays.

High signal-to-noise (S/N) ratios are required for the reliable analysis of multiexponential decays. The dependence of the relative error and separability of lifetimes on the number of detected photons has been extensively studied (Bajzer et al., 1991; Köllner & Wolfrum, 1992; Gerritsen et al., 2002); for instance, it was found that separating two lifetime components that differ by a factor of two requires at least 1000 detected photons per pixel (Gratton et al., 2003). The collection of thousands of photons per pixel with biological samples is challenging because of photo bleaching, imaging speed and saturation of the detection system.

As an alternative to a per pixel analysis, global analysis methods can be employed to improve the accuracy of data analyses (Verveer et al., 2000). In global approaches, it is assumed that the fluorescence lifetimes in the whole image are spatially invariant and only the relative concentrations of the fluorescent species (amplitudes of the individual components) vary at each pixel location. It was shown that the accuracy of the analyses improves because the signal of a large number of pixels is used to resolve multiexponential decays (Pelet et al., 2004).

The phasor approach (also referred to as AB-plot) has been employed for the global analysis of frequency domain lifetime data (Verveer & Bastiaens, 2003; Clayton et al., 2004). This method simplifies the analysis of lifetime images and offers a graphical approach to analyze lifetime distributions and fractional concentrations of fluorophores. Briefly, phase and modulation values of each pixel are converted to real and imaginary parts and the result are presented in a two-dimensional plot. Here, the y-axis is the imaginary axis and x is the real axis. The phasor approach can also be employed for analyzing time domain data (Digman et al., 2008, Grecco et al., 2009). In this case, the imaginary and real parts of the Fourier transformation (FT) of the fluorescence decay curves of each pixel are mapped as points in the phasor plot. Again, each point in this two-dimensional histogram corresponds to the pixels of the lifetime image.

Implementing the phasor method for the analysis of fluorescence lifetime images recorded using Time Gating is potentially interesting. Photon counting based Time Gating electronics have a low dead time and can therefore in general handle higher count rates than time-correlated single photon counting (TCSPC) methods. As a result, time gating is well suited for rapid image acquisition (De Grauw & Gerritsen, 2001; Gadella, 2008). However, in Time Gating the gates are usually comparatively long and only a limited number of gates are employed (Buurman et al., 1992). Moreover, fluorescence is in general acquired in a limited time range; as a result part of the decay is not recorded. Importantly, due to the limited number of gates the sampling of the decay curve is not compatible with the standard implementation of the phasor method. This renders the original phasor method not adequate for the analysis of time-gated lifetime images.

In this paper, we present a modification of the phasor analysis method that takes into account the sampling limitations in time-gated detection and make it compatible with time-gated data. This modified approach is tested on both simulated lifetime images and on 3T3 2.2 cells transfected with glycosylphosphatidylinositol fused to green fluorescent protein (GPI-GFP) and incubated with cholera toxin subunit B labelled with Alexa-594 (CTB-Alexa). In addition we measured FRET efficiencies in cells transfected with EGFR-GFP and mCherry. We demonstrate that this method can be applied to retrieve two lifetimes from time gating data that cannot be resolved using standard (non-global) fitting techniques.

Measuring lifetimes

Here, we will restrict ourselves to the single photon counting based lifetime detection methods often used in laser scanning microscopes. Complete decay curves can be recorded using TCSPC systems. Due to the large number of narrow time channels, typically 128–1024, TCSPC affords the recording of complete fluorescence decay curves with high time resolution. TCSPC imaging generally requires comparatively long acquisition times; operating TCSPC systems at high-count rates, results in pulse pile-up, which distorts the recorded decay curve (Lakowicz, 2006).

In Time Gating, the fluorescence emission is detected in two or more time gates. In optimized schemes the gates open sequentially after excitation of the sample with a laser pulse (De Grauw & Gerritsen, 2001). Time-gated data can be recorded at high-count rates (∼10 MHz); in practice the maximum count rate is limited by the detector. The high maximum count rate in Time Gating is due to the low dead time of the detection electronics (<1 ns). The dead time of TCSPC electronics is usually on the order of 100–350 ns which results in pulse pile-up and loss of signal at high count rates (Gadella, 2008).

The measured decay curve is the convolution of the fluorescence decay function with the instrument response function (IRF). In TCSPC this is taken into account in the analyses by data fitting algorithms. In Time Gating the effect of the IRF is usually ignored by applying an offset between the laser pulse and the opening of the first gate. In this way the later part of the signal is measured where the IRF has dropped to zero. This also offers the possibility to suppress background signals like Raman scattering or auto fluorescence, which improves the signal to background ratio of the images.

In spite of the relatively long gate durations (usually > 500 ps) and the limited number of time channels (2–8) the lifetime of the single exponential decays can be accurately determined with proper gate settings (Gerritsen et al., 2002). Moreover, even multiexponential decays can be detected when the number of gates is large enough (Gerritsen et al., 2002).

Theory

Application of the phasor approach to analyze time domain data requires Fourier transformation from time zero to infinity. In practice the narrow time bins and long acquired time range make implementation of the phasor approach to analyze TCSPC data straightforward. However, in time gating techniques the decay is under sampled and the recorded time range is more limited. Therefore, a revised phasor analyses is required to analyze time-gated data.

The standard phasor analyses method has been described in detail previously (Digman et al., 2008; Grecco et al., 2009). Briefly, d(t) the fluorescence response of a multiexponential decay to pulsed laser excitation can be described by a sum of Q exponentials

image(1)

where inline imageis the lifetime of the q-th species, and aq is the pre-exponential coefficient. The Fourier transformation of d(t) at frequency inline image is given by

image(2)

The normalization integral in the denominator allows rewriting the pre-exponential coefficient aq into inline image, the fractional contribution of the q-th exponential to the average fluorescence intensity.

image(3)

For each pixel in an image with index i, Eq. (2) can be rewritten as

image(4)

where inline image is defined as

image(5)

For a bi-exponential decay inline image is in the range 0–1 which simplifies Eq. (4) to

image(6)

where inline image and inline image

A linear relationship exists between the imaginary and real components of Dthat can be written as (Grecco et al., 2009)

image(7)

with

image(8)

For single exponential decays with lifetimes in the range 0–∞ plotting the imaginary versus the real part of inline image for 0 <inline image < ∞ results in a semicircle. Mixtures of two single exponential decays fall on a straight line given by Eq. (7) which intersects the semicircle at two points corresponding to the two fluorescence lifetimes

image(9)

For measurements with a discrete number of time bins of equal width, like time gating and TCSPC, Eq. (5) can be rewritten as

image(10)

with K is the number of time bins, m the bin index, n the harmonic number, inline image and T the total time range of detection. The original continuous decay is described by inline image; here, it is replaced by inline image to take into account the width of the detection bins.where δ is the Dirac delta function. We note that in Eq. (10) although the summation runs from m =–∞ to +∞, the integration from 0 to T effectively restricts the summation to m= 0 to K–1. The summation from –∞ to +∞ is mathematically more convenient because of its compatibility with the Fourier Transform. Now we can rewrite Eq. (10) into

image

where inline image denotes the convolution operator and inline image the boxcar function which is equal to 1 for 0 < t < T and 0 otherwise. For inline imageEq. (11) can be approximated by

image(12)

Compared to Eq. (5), which is obtained without truncation and discrete sampling, this equation repeats every inline image. The periodicity is due to the discrete sampling of the decay curve. Examples of the Fourier transformation of decay curves for different numbers of gates and lifetime inline image are shown in Figure 1. The ‘fast’ ripples in the curves are the result of truncation effects and the periodicity is the consequence of the discrete sampling of the decay curve. As K increases the period increases and the curves get closer to the continuous Fourier transform at the harmonics. It is straightforward to show that if inline image then the inline image at the harmonics converges to inline image. Truncation does not alter the Fourier transform at the harmonics. It is, however, affected by sampling.

Figure 1.

The Fourier transform of decay curves recorded with different gate numbers (K = 4, 8, 16 and 32), the same total time window width T and a lifetime ofinline image.

The reference semicircle is generated by plotting the Imaginary part of inline imageversus the real part which can be done by using Eq. (5). For low number of gates and discrete measurements we need to use the modified equation which is given by Eq. (10). Figure 2 shows the modified reference semicircle at the first harmonic inline imagefor different gate numbers and the same total window for a lifetime range from inline image to inline image.

Figure 2.

Plot of the modified semi-circle inline image versus inline image. The curve shows all possible single component lifetimes from inline image to inline image at inline image.

Interestingly, even for a low number of gates the deviation of the modified semicircle at long lifetimes (small values of R) from the standard semicircle is small. By increasing the number of gates the modified semicircle gets closer to the standard semicircle in particular for shorter lifetimes.

We can rewrite Dfor discrete time measurements at the nth harmonic, inline image as

image(13)
image(14)

Equation (13) shows that in the case of a bi-exponential decay, Eqs (6) and (7) are still applicable and there is a linear relationship between imaginary and real parts of the Fourier transformation of the decay curve. This line will connect all noiseless mixtures of two components defined by Eq. (13) and the intersections with the adapted semicircle correspond to the two lifetimes which now can be evaluated by

image(15)

This approach is illustrated in Figure 3 for a bi-exponential decay curve with two lifetimes with inline image (R1), inline image (R2) and fractional intensityinline image. Here, the adapted semicircle for a system with four gates is employed.

Figure 3.

The standard and modified semicircle for a system with 4 gates. The phasor of a bi-exponential decay curve with lifetimes inline image, inline image and fractional intensity inline image(open circle) is shown. R1and R2 correspond to lifetimes of inline image and inline image filled circles.

Methods and materials

Simulations

Simulations were programmed in Compaq visual FORTRAN using the IMSL math library. Exponential decay curves were generated for a pre-set total photon count and binned to simulate truncation and sampling by time gating. Next, Poisson noise was added to each time gate. This procedure was repeated for different numbers of gates (K= 4, 8 and 16) using the same fixed total gate width.

Phasor diagrams of the simulated images were calculated using Eq. (16). One thousand images were simulated with size 100×100 pixels and two fixed lifetimes inline image andinline image. α, the fraction of the short lifetime fluorescence, was varied between 0 and 1 from the upper left to the lower right of the image. The inline image term in Eq. (13) was replaced byNi,m, the simulated number of counts in the ith pixel and mth gate. Next, the Fourier transformation of the ith pixel and nth harmonic was carried out using

image(16)

Applying this equation to all image points yields the data points for a phasor plot. A straight line was fitted through the clouds of points in the phasor diagrams and the intersection of the line with the (adapted) reference curves was calculated using Eq. (15). Here, the orthogonal least square method (Weisstein) was employed to fit the line. This provided significantly better results than the ordinary least square method; it is more accurate in the case that both dependent and independent variables contain errors.

To investigate the accuracy of the modified phasor approach for separating two lifetimes in simulated images we employed the index of separability S (Köllner & Wolfrum, 1992). This index is defined for a system with two lifetimes inline imageand inline image as

image(17)

where var is the variance of the estimated lifetime. A higher value of S indicates better separability of the two lifetimes. When S < 1 the two components are considered to be non-separable. In general an S value of 2 is required for separability.

Imaging experiments

A 473 nm solid-state diode laser (BDL-473-SMC, Becker & Hickl, GmbH, Berlin, Germany) with a pulse repetition rate of 50 MHz was coupled into a confocal microscope (Nikon C1, Nikon Instruments Europe BV, Badhoevedorp, The Netherlands). Imaging of solutions of Rhodamine-B and Fluorescein was performed using a 10× objective (NA = 0.3 Nikon, Japan); images of cells transfected with GPI-GFP and CTB-Alexa-594 were acquired with a 60× water immersion objective (NA = 1.2, Nikon, Japan). Fluorescence was detected by a fibre coupled PMT with a GaAsP photocathode (H7422P-40, Hamamatsu Japan). In the FRET measurements donor emission was selected by a 515/30 nm band pass filter. The output of the PMT was fed into a four channel time-gating module (LIMO, Nikon Europe BV, The Netherlands; De Grauw & Gerritsen, 2001). The time gates were all set to a width of 2 ns and the opening of the first gate was delayed 0.5 ns with respect to the excitation pulse. The IRF was accounted for by reference measurements on a solution of Fluorescein. TCSPC measurements on this solution yielded a monoeponential decay with a lifetime of 3.98 ns. The phasor of Fluorescein falls on the adapted semicircle and serves as a reference for the other measurements. The reference measurement was repeated using the LIMO system to calibrate the time gated measurements. All experiments were carried out at room temperature.

Sample preparation

COS-cells were transfected with EGFR-eGFP and mCherry and 3T3 2.2 cells were transfected with GPI-GFP using FugeneHD (Roche, Mannheim, Germany). 3T3 2.2 cells were further incubated with CTB-AlexaFluor-594 (Invitrogen, Breda, The Netherlands). Both samples were subsequently fixed using formaldehyde (Polysciences, Eppelheim, Germany), embedded in Mowiol (Polysciences) and stored at –20°C until further use.

Spectral un-mixing

As a comparison to the lifetime unmixing results, spectral unmixing experiments were performed based on measurements recorded in two different detection channels. GPI-GFP fluorescence was detected with the 515/30 nm band pass filter and the CTB-Alexa emission using a 660 nm long pass filter. The results of spectral unmixing are not corrected for bleed through of GPI-GFP in the CTB-Alexa channel and CTB-Alexa in the GFP channel. Correcting bleed through did not significantly improve the spectral unmixing results.

The Manders overlap coefficient (Manders et al., 1993) was used for a quantitative comparison of the results obtained by lifetime unmixing and by spectral unmixing. The value of this coefficient ranges from 0 to 1, with 1 indicating perfect overlap and zero indicating no overlapping of the two images.

Results and discussions

Numerical simulation

For different combinations of two normalized lifetimes, τ1/T and τ2/T, ranging from T/10 to T and different gate numbers (K= 4, 8, 16) the index of separability was calculated as a function of the average number of counts. Figure 4 shows the required number of counts per pixel to obtain an index of separability of 2. This corresponds to the lowest number of counts required to separate the two lifetimes. The smallest difference between the two lifetimes considered here amounts to T/20. As expected, the number of photons needed to separate two closely spaced lifetimes is higher than for two less closely spaced lifetimes. As the two lifetime values get closer, the number of required photons for separability increases rapidly. For example, a system with K= 4, T= 10 ns and lifetimes ofinline imagens and inline imagens requires 5000 counts per pixel for an S of 2. This number decreases to less than 500 counts for lifetimes ofinline imagens andinline imagens. This amount decreases further when the number of gates increases. Keeping the condition constant and changing the number of gates from 4 to 16 results in a decrease of the required photons to 200 counts per pixel for inline imagens and inline imagens. Under the same conditions a pixel-by-pixel analyses would require inline imagecounts per pixel to separate the two lifetimes. This calculation is done based on the estimations described by Köllner & Wolfrum (1992). Also at longer fluorescence lifetime values, separability of two component mixtures is not achieved. This is due to the truncation of the decay curves that, even at equal number of detected photons, causes a detrimental loss of information and makes the unmixing algorithms not usable in practical conditions.

Figure 4.

The number of counts per pixel required for separating (S= 2) the components of a bi-exponential decay as a function of the second normalized lifetimeinline image. Curves are shown for different normalized lifetimes inline imageand numbers of gates K= 4, 8, 16.

Experiments

Images of a binary mixture of Rhodamine-B and Fluorescein in buffer were recorded and analyzed using the adapted phasor approach. The concentrations of inline imagefor Fluorescein and inline image for Rhodamine-B yielded average intensity ratios of about 1: 1 at an excitation wavelength of 473 nm. Figure 5(a) shows the intensity image and (b) the phasor diagram at the first harmonic (125 MHz) which corresponds to a total width of the detection window of 8 ns. The straight line is obtained by orthogonal linear regression of points in the phasor plot and it intersects the phasor curve at lifetimes of 1.61 ns and 3.89 ns. These lifetimes are in agreement with reference average lifetimes of pure Rhodamine (1.7 ns) and Fluorescein (3.88 ns) measured independently by TCSPC. The comparatively short lifetime of Rhodamine is explained by a small amount of scattered light detected in the first gate. The shape of the distribution of points in the phasor plot confirms a gradient distribution of the species. The left side of the plot is dominated by Fluorescein and the right side mainly consists of Rhodamine-B.

Figure 5.

(a) The intensity image of a binary mixture of Rhodamine and Fluorescein. The dark area on the right side of the image is caused by a small air bubble in the sample. (b) The phasor diagram of the lifetime image of (a). The colour bar shows the density of points in the phasor cloud. Fractional intensities of Fluorescein (c) and Rhodamine (d) as obtained from the phasor analyses. (e) Overlay of the two distributions. (f) Average lifetime image.

The fractional intensities of Fluorescein and Rhodamine-B and their overlay are shown in Figure 5(c)–(e), respectively. The fractional intensities were calculated from the positions on the line in the phasor plot connecting the two reference lifetimes. Figure 5(f) shows the average lifetime image; a clear gradient in the lifetime is observed that qualitatively matches the intensity distributions.

The intensity image (54 × 54 μm2) of a single cell labelled with GPI-GFP and CTB-Alexa 594 is shown in Figure 6(a). Figure 6(b) shows the phasor plot which reveals fluorescence lifetimes of 1.94 ns for GPI-GFP and 3.48 ns for CTB-Alexa 594. Reference measurements on samples tagged exclusively with GPI-GFP or CTB-Alexa exhibited average lifetimes of 2.10 ns and 3.51 ns, respectively. The small discrepancy in the short lifetime is again due to the detection of a small amount of scattered light in the first gate. The two components were separated using the phasor analyses results; the fractional intensities of CTB-Alexa and GPI-GFP are depicted in Figures 6(c) and (d), respectively. GPI-GFP is mainly present at the plasma membrane and the Golgi apparatus while the CTB-Alexa 594 is predominantly localized at the plasma membrane.

Figure 6.

(a) Intensity image of a cell stained with GPI-GFP and CTB-Alexa 594. (b) Phasor diagram. The colour bar shows the density of points in the phasor cloud. (c) Fractional intensities of GPI-GFP and (d) CTB-Alexa as derived from the phasor plot. Spectrally unmixed images of: (e) GPI-GFP (515–30 nm) and (f) CTB-Alexa 594 (660 nm LP).

As a comparison, also spectral unmixing results are shown in Figures 6(e) and (f) for GPI-GFP and CTB–Alexa, respectively. The results from the lifetime measurements correspond well to the spectral unmixing results, especially considering the overlap of the two dyes at the plasma membrane. Quantitative comparison of the lifetime and spectral separation of GPI-GFP and CTB-Alexa 594 using the Manders overlap coefficient yielded coefficients of 0.94 and 0.92, respectively.

FRET experiments on COS-1 cells expressing EGFR-GFP (donor) and mCherry (acceptor) were carried out by recording fluorescence lifetime images of the donor. Figure 7(a) shows the fluorescence intensity image of the COS-1 cells measured through a 515/30 nm band pass filter and using a 60× water immersion objective. Figure 7(c) shows the (average) lifetime image and Figure 7(b) the phasor diagram of the lifetime image. Pixels that don't exhibit FRET show up in the phasor plot near the phasor of the unquenched donor (τdonor). Pixels without any donor fluorescence, including pixel where the donor is fully quenched by FRET, are located around the phasor of the autofluorescence background. Phasors of quenched donor lifetimes (τdonor-acceptor) corresponding with other FRET efficiencies inline image (Digman et al., 2008) lie on a curved trajectory connecting the unquenched and quenched positions, see Figure 7(b). Independent time gated measurements were carried out to measure the unquenched donor (2.10 ns) and autofluorescence (2.99 ns) average lifetimes. Next, their phasors were mapped into the phasor plot. The phasor representation can be used for the visual segmentation of FRET images (Wouters & Esposito, 2008). Selection of pixels on this trajectory can be employed to, for instance, identify regions in the image with ‘high’ or ‘low’ FRET efficiencies.

Figure 7.

(a) Intensity image of cells expressing GPI-GFP and mCherry. (b) Phasor diagram. The colour bar shows the density of points in the phasor cloud. (c) Segmented intensity images for FRET efficiencies <5%, (d) average lifetime image of the cells and (e) >5%.

The presence of FRET is clearly visible in the phasor plot of Figure 7 (c); the centre of gravity of the phasor points is clearly shifted to shorter lifetimes with respect to the unquenched donor coordinate. To discriminate regions in the image with ‘low’ and ‘high’ FRET efficiencies, two regions in the phasor plot were selected as indicated by the circles in Figure 7 (c). The circle includes about 95% of the pixels when placed in the middle of the cloud and by placing it in the selected regions the higher (>5% efficiency) and lower (<5% efficiency) FRET regions are selected. At the positions indicated in Figure 7 (c) each circle includes ∼49% of all pixels. Figures 7(d) and (e) show the corresponding, phasor segmented, ‘low’ and ‘high’ FRET images, respectively.

Conclusions

Computationally efficient and robust data analysis methods are crucial for fluorescence lifetime imaging microscopy. In this paper we demonstrated that, after modification the phasor representation and its use for global analysis is also applicable to time gated fluorescence lifetime images. This method provides a fast and intuitive representation of time-gated data sets without the need for time-consuming non-linear fitting procedures. In addition the phasor analyses can be also employed as a global approach to extract the lifetimes of two components in a mixture, to estimate fractional contributions of two fluorescent dyes in cells and for the detection of FRET.

Typically, fluorescence lifetime imaging based on time gating provides faster image acquisition rates than TCSPC, and requires similar data analyses. Furthermore, Time gating usually does not permit detection of multiexponential fluorescence decays because of the low number of gates typically used. The combination of phasor analysis with time gating, however, provides high imaging speeds in combination with fast data analysis and quantitative estimation of multiexponential decays. Regardless of the method used for recording the fluorescence decays, the phasor approach assumes that the fluorophores exhibit mono-exponential decays. When a specimen contains mixtures of fluorophores with multiexponential decays the estimation of lifetimes and fractional intensities is not straightforward; independent measurements of the two dyes are required to extract the phasors of each component.

We used the separability index to investigate the effect of gate widths on the accuracy of the estimated lifetimes. Longer gate widths reduce the accuracy compared to TCSPC, a finding that is in agreement with previous figure of merit calculations of lifetime data (Gerritsen et al., 2002). However, the average number of counts required to separate two lifetimes in the adapted phasor analyses is considerably lower than in conventional pixel-by-pixel analysis.

The global nature of phasor analysis permits estimating global parameters (here the two fluorescence lifetime components) with an accuracy determined by the total number of counts in the whole image. Local parameters, (here the fractional contribution of individual fluorophores), can be estimated with an accuracy determined by the number of counts per pixel. By effectively reducing the number of fit parameters, phasor analysis achieves superior accuracies compared to non-global approach at equal photon counts.

Finally, we note that multidimensional TCSPC systems are commercially available. The memory constrains of these acquisition boards may force investigators to reduce the number of time bins to those typically used in time gating. Moreover, the huge multidimensional lifetime images also call for fast and efficient data analysis methods. Here, the modified phasor approach may be of great value. Therefore, the theoretical framework we developed here may find broader usage than in time gating alone.

Finally, a versatile image J plugin for phasor analyses of lifetime images is available upon request from the authors.

Acknowledgements

The authors thank Jarno Voortman and Dr. Erik Hofman from the Cellular Architecture and Dynamics group at Utrecht University for providing the biological samples. F.F. was funded by the Iranian Ministry of Science Research and Technology.

Ancillary