Optimizing frequency-domain fluorescence lifetime ... - OSA Publishing

Esposito et al.

Vol. 24, No. 10 / October 2007 / J. Opt. Soc. Am. A

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Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed Alessandro Esposito,1,2,3,4,* Hans C. Gerritsen,3 and Fred S. Wouters1,2,5 1

Cell Biophysics Group, European Neuroscience Institute—Göttingen, Waldweg 33, 37073 Göttingen, Germany 2 DFG Center for Molecular Physiology of the Brain (CMPB), Göttingen, Germany 3 Debye Institute, Utrecht University, P.O. Box 80.000, NL 3508 TA, Utrecht, the Netherlands 4 Present address, Laser Analytics Group, Department of Chemical Engineering, University of Cambridge, Pembroke Street, CB2 3RA, Cambridge, UK 5 Current address, Laboratory for Molecular and Cellular Systems, Department of Neuro- and Sensory Physiology Centre II, Physiology and Pathophysiology, University Of Göttingen, Humboldtallee 23, 37073 Göttingen, Germany *Corresponding author: [email protected] Received February 2, 2007; revised July 19, 2007; accepted July 21, 2007; posted July 25, 2007 (Doc. ID 79498); published September 20, 2007 The signal-to-noise ratio of a measurement is determined by the photon economy of the detection technique and the available photons that are emitted by the sample. We investigate the efficiency of various frequencydomain lifetime detection techniques also in relation to time-domain detection. Nonlinear effects are discussed that are introduced by the use of image intensifiers and by fluorophore saturation. The efficiency of fluorescence lifetime imaging microscopy setups is connected to the speed of acquisition and thus to the imaging throughput. We report on the optimal conditions for balancing signal-to-noise ratio and acquisition speed in fluorescence lifetime sensing. © 2007 Optical Society of America OCIS codes: 030.5290, 110.0110, 170.3650, 180.2520.

1. INTRODUCTION Fluorescence lifetime detection techniques and their microscopy implementation (fluorescence lifetime imaging microscopy FLIM) are well established and characterized. The noise sensitivity, or photon economy, of a number of different implementations has been described in the recent past [1–9]. Lifetime detection can be performed in the time domain (TD) and in the frequency domain (FD). The former requires the transient excitation of the fluorophore and the time-resolved detection of its fluorescence emission. This is not usually performed during a single excitation event. Typically, the sample is illuminated with a train of short light pulses, and the emitted photons are collected in a number of different time windows. The main techniques adopted in the TD are time-correlated single-photon counting (TCSPC) and time gating (TG) [10,11]. Both TG and TCSPC produce histograms of arrival times that are fitted to (exponential) models to retrieve the decay time of the system. Detection in the FD relies on the excitation of a fluorophore with a periodic light pattern and the subsequent analysis of the harmonic response of the fluorescence emission. Importantly, fluorophores are considered to be linear and time-invariant systems; however, this only holds in the absence of saturation and photobleaching. Within the limits of this assumption, the harmonic content of the fluorescence is related to only the harmonic content of the excitation light. 1084-7529/07/103261-13/$15.00

Here, the phase delay and the demodulation of each harmonic are related to the lifetime of the fluorophore [12]. FD lifetime detection is accomplished by cross-correlation techniques. TD and FD techniques are both implemented in laser scanning and wide-field microscopy. Laser scanning microscopes offer higher resolution and image contrast at the cost of longer acquisition times. Furthermore, laser scanning microscopes commonly use point detectors and therefore comparatively simple detection electronics. This permits, for instance, the simultaneous recording of different time or phase gates or multiple harmonics. Widefield detectors potentially offer higher acquisition speeds at the detriment of spatial resolution and image contrast. Furthermore, images of time and phase stacks are usually collected sequentially, therefore decreasing the acquisition throughput. Recent methodological and technological advances [13–16] make parallel imaging with widefield detectors feasible, thus alleviating most of these problems. However, these techniques are still under development. When the excitation light consists of a train of Dirac pulses, both TD and FD detection perform equally well, reaching the maximally possible signal-to-noise ratio (SNR) [7]. FD detection allows the use of different excitation waveforms, e.g., rectangular and sine waves. However, correlations between measurements at different © 2007 Optical Society of America

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phase delays deteriorate the SNR. These can be prevented only by excitation with very short light pulses, since with nonpulsed excitation not all the photons carry new information about the fluorescence decay. FD detection is usually performed with sinusoidal excitation or other excitation patterns with a high duty cycle. FD detection exhibits a reduced SNR, and its photon economy is generally worse than TD techniques. Nonpulsed excitation in FD methods allows the use of cost-effective lasers and light-emitting diodes. Furthermore, depending on its particular technological implementation, FD may be able to handle higher photon fluxes than TD techniques [17]. The photon economy of TD techniques like TCSPC and TG has been extensively characterized [1–6,9], also with respect to the optimal number of time gates or bins. Recent work describes the photon economy of FD measurements by both lock-in and image-intensifier detection [8]. Image intensifiers and other wide-field detectors that operate in the FD directly cross correlate the optical signal with the gain modulation of the detector, which is phase locked to the excitation modulation. It should be noted that there is an intrinsic dc component in the optical signal that is not removed by these detectors. FD detection implemented by using the above detectors is commonly called lock-in imaging, but this should not be confused with the actual operation of a lock-in amplifier. In this work and its respective literature on photon economy, lock-in detection exclusively refers to the actual detection mechanism used by lock-in amplifiers, i.e., the cross correlation of an unbiased electrical signal with the excitation reference signal. A challenging development for FLIM is its use in highspeed and high-throughput applications where the integration time and photon economy need to be balanced. In this work, several parameters are studied that are important for the optimization of the rate of FLIM acquisition. The photon economies of the FD average-lifetime and the FD rapid lifetime determination (RLD) estimators are characterized. The dependence of photon economy on the excitation profile and on the number of acquired phase images (using parallel or sequential acquisition) and the effects of sine or gated gain modulation of the detectors are investigated. Potential artifacts related to aliasing, photobleaching, and fluorophore saturation are described. Finally, the photon economy is related to the acquisition speed by the analysis of the detection throughput by using a novel figure of merit.

2. METHODS Monte Carlo simulations were carried out by using custom-developed software in Matlab (Mathwork, Natick, Massachusetts, USA). Modulation frequencies were normalized 共x兲 to the sample lifetimes 共x = ␻␶兲. For each modulation frequency and lifetime value, photon emission probability density functions were synthesised in accordance with a particular excitation light profile: Dirac, sine, or rectangular shape. Different rectangular profiles were simulated with variable duty cycles, i.e., the ratio of “on” time to period. The photon emission probability density functions were binned, with the bin width set to avoid

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probabilities higher than 1% per bin. Each photon was generated by a random number generator. Subsequently, the photons were integrated in images with a phase offset relative to the excitation light. The detector gain characteristics were taken to be sinusoidal or square wave (50% duty cycle) for cross correlation with a sine wave or by gating, respectively. The normalized modulation frequency x was varied between 0.3 and 1.9 in 25 equidistant steps. Images of 30⫻ 30 pixels were synthesized, giving 900 data points for the computation of the standard deviation from which the photon economy is estimated. The noise level in the phase images was always monitored; in all cases Poissonian noise was observed at the expected level. The simulated number of photons per pixel was 2500. As expected, the photon economy did not depend on this number. However, at low modulation frequencies 共x ⬍ 0.4兲, the algorithms for the lifetime estimation may converge not to the right solution but to an erroneous limit value. Under these conditions, the relative error in the estimated lifetime can decrease significantly, resulting in apparently high photon economies. These systematic errors were monitored by computing the average relative difference between the simulated and the real lifetimes. Simulations for which the relative difference exceeded 5% were discarded and repeated with higher photon counts (10,000–25,000). Typically, higher-photon-count simulations were carried out when the simulated detector operated in gating mode with only 2–3 simulated phase images. Monte Carlo simulation results are shown with data fitting using rational functions. Saturation effects were studied by using the solved differential equations that govern the photophysics of a fluorophore. Here, only the transition between the ground state and the first excited state and a radiative deexcitation pathway whose rate constant was set to 1 / ␶ were considered. Normalized modulation frequencies of 0.5, 1.0, and 2.0 were used. The simulated rectangular excitation profiles had a variable duty cycle (0%, 5%, 10%, 20%, 30%, 40% and 50%)—the 0% duty cycle corresponding to Dirac excitation. The relative difference between the simulated and the real lifetime was computed. Subsequently, the saturation levels that cause a 5% relative error were mapped. The solutions for the differential equations and the simulations of saturation effects were obtained by software developed in Mathematica (Wolfram Research Europe Ltd., Long Hanborough, UK). Every simulation was carried out with 100% modulation depth of both the excitation and detector gain. In practice, excitation and detector gain modulations are usually less than 100%. The simulated figure of merit F values thus represent best-case values, and the performance in real experiments may be worse. Moreover, the simulations did not consider background signals that will also degrade the photon economy.

3. THEORY The photon economy [18] can be quantified by a figure-ofmerit F that is defined as F=

冉冊

⌬␶ ⌬N

␶

N

−1

=

␴␶

␶

冑N.

共1兲

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F is the ratio between the relative error in the lifetime 共␴␶␶−1兲 and the minimal possible noise limit when Poissonian noise is assumed 共N−1/2兲. F = 1 corresponds to the best possible photon economy; i.e., the lifetime estimation only exhibits Poissonian noise. Higher F values imply poorer photon economies. F2 represents the fold increase in the number of photons necessary to achieve the SNR of an ideal (efficient) system with an F value of 1. F−2 is thus the fraction of photons that an efficient system would require for obtaining a similar SNR. Therefore, F−2 can be defined as the photon economy of a system. Interestingly, TCSPC and TG in the TD and lock-in detection in the FD have equally efficient estimators 共F ⬃ 1兲 when Dirac excitation is used [7]. FD provides both a phase- and modulation-lifetime estimator [11,12]:

␶␾ = ␻−1 tan ␾ ;

␶m = ␻−1共m−2 − 1兲1/2 .

共2兲

The demodulation 共m兲 and relative phase 共␾兲 of the fluorescence emission are computed by Fourier analysis or by data fitting [10] of a phase stack composed of an ensemble of J images taken at different relative phase delays (usually ␸j = 2␲共j − 1兲J−1, j = 1 , . . . , J). Differences between the phase- and modulation-lifetime estimations are related to the lifetime heterogeneity. [19]. Neglecting lifetime heterogeneity, it is possible to retrieve a FD lifetime estimator from only two phase acquisitions [20]. These measurements (I0, I180) have to be collected at relative phases of 0° and 180°. I0 and I180 can be used to estimate the modulation of the fluorescence emission and therefore to estimate the fluorescence lifetime: m = 兵共I0 − I180兲/关2共I0 + I180兲兴其1/2 .

共3兲

As this estimation requires the collection of only two images 共J = 2兲, it is somewhat comparable with the RLD method in the TD [2]. The photon economy, the fluorescence emission intensity, and the maximum count rate of the detection system are relevant for the estimation of the potential throughput of a technique. If one neglects detector limitations, the limiting factor of fluorescence collection is fluorophore saturation. The maximum count rate would be n␶−1, where n is the number of molecules in the focal volume and ␶ is the fluorophore lifetime. As a figure of merit for the acquisition speed of a microscope, we can use the ratio of the effective photon emission rate 共k兲 to the maximum achievable rate 共n␶−1兲, multiplied by the photon efficiency of the system: E A=

k

F n␶−1 2

,

共4兲

where E is the efficiency of the system and F−2 the photon economy. E is a function of factors like the numerical aperture of the objective, the quantum efficiency, and the dead time of the detector; unless otherwise specified, we consider E to be equal to 1. For the fastest and most efficient techniques, A approaches 1 and for slow and less efficient techniques approaches 0. At decreasing values of A, A−1-fold longer acquisition times are required in order to obtain the SNR of

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the case where A = 1. A thus represents the relative throughput of a detection technique.

4. RESULTS A. Average-Lifetime Estimator The photon economy of both the ␶␾ and ␶m estimators has been characterized [8,18] for J = 3. However, the arithmetic average of the two estimates (␶a, average-lifetime estimator) is normally used when no information on lifetime heterogeneity or excited state reactions is required [19,21]. ␶a can be a convenient estimator, as it is often less sensitive than ␶␾ and ␶m to biases caused by photobleaching, heterogeneity, and fluorophore saturation. In the case where the bias of the phase- and modulation-lifetime estimators differ in sign (for instance, when photobleaching is not properly corrected for), the average lifetime is the least biased estimator. Figure 1 shows the results of Monte Carlo simulations of the F value for sine and Dirac excitation cases. The gray dashed lines mark the lowest achievable F value 共F = 1兲. With sinusoidal excitation light and detector gain (J = 3, parallel acquisition), the modulation- [Fig. 1(a), gray curve] and phase- [Fig. 1(a), black curve] lifetime estimators exhibit minimum F values of 3.8 共⫻ ⬃ 0.74兲 and 3.3 共⫻ ⬃ 1.3兲, respectively. When the phase images are acquired sequentially, the F values scale with a factor J1/2. In the previous case 共J = 3兲, the minimum F values would increase to 6.6 and 5.7, respectively. Although the optimal frequency is equal to that reported in the literature [8,18], the F values are slightly different. This is likely explained by the fact that we simulated the operation of an image intensifier that acquires the three different images at three different phases, i.e., the way the instrument is typically operated. In previous work, the photon economy was also simulated with three images, but including one with unmodulated excitation light [8,18,22]. The average-lifetime estimator [Fig. 1(a), dotted– dashed curves] exhibits an intermediate behavior with a minimum F value 共J = 3兲 of 2.6 (4.6 for sequential acquisition) at a modulation frequency of ⬃0.97. In general, the average-lifetime estimator exhibits a better maximum photon economy (14%) because the phase- (7%) and the modulation- (9%) estimators are correlated but carry different information. Interestingly, the three estimators exhibit higher efficiencies over three different modulation frequency regions, and the average-lifetime estimator is more efficient over a broader modulation frequency range. Analogous simulations with Dirac excitation yield similar results, but now all estimators are considerably more efficient (Fig. 1). In this situation, the minimum F values for phase, modulation, and average estimators 共J = 3兲 are ⬃1.1, ⬃1.2, and ⬃1.0, respectively (1.8, 2.1, and 1.7 with sequential acquisition). B. Rapid Lifetime Determination Estimator Lock-in imaging and the RLD estimator [Fig. 1(b), gray and black curves, respectively) provide F values that are equal to those of the phase and average estimators, respectively, but require the collection of only two phasedependent images 共J = 2兲. Analogously, TD-RLD allows

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gebraic strategies previously used to characterize the other estimators [8]. The minimum F value of ⬃1.1 is located around x ⬃ 0.5. Figure 1(b) shows the analytical solution (black curve), which is in perfect agreement with the data points from the Monte Carlo simulation (gray circles). Therefore, both lock-in detection and FD-RLD in combination with Dirac excitation exhibit the same efficient behavior (F ⬃ 1.1, F−2 ⬃ 83%) and a high photon economy, and also for sequential acquisition (F ⬃ 1.6, F−2 ⬃ 48%).

Fig. 1. Dependence of the F value of FD detection on modulation frequency for sine and Dirac excitation. Different detection techniques and estimators exhibit different photon economies. Shown are the fitted Monte Carlo simulations. (a) Phase- (dashdotted curve), modulation- (gray), and average- (solid black) lifetime estimators. (b) Phase lock in (gray), RLD algorithm (upper black curve and gray circles), and the analytical solution for the RLD (lower black curve). Results are shown for both sine and Dirac excitation. Note the better performance of Dirac excitation, the high efficiency of the RLD, and the different optimal modulation frequency for the various estimators.

C. Excitation, Cross-Correlation Signals, and Number of Collected Images The photon economy of a FLIM system highly depends on the temporal shape of the excitation light. A comparison between the minimum F value of the estimators with different excitation patterns is shown in Fig. 2. Here, rectangular-wave excitation with 50% and 20% duty cycles are simulated in addition to sinusoidal and Dirac excitation profiles. In both TD and FD measurements, the photon-economy may also depend on the number of time bins or acquired phase images. TG systems with two time bins, analyzed by the TD-RLD, exhibit a minimum F value of 1.5, whereas F values approach unity with increasing gate numbers. TCSPC is based on the detection of multiple time windows (usually J ⬎ 16) and therefore provides F values close to 1. Wide-field time-gated systems require the sequential acquisition of the time-gated images to the detriment of their photon economy. FD F values do not depend on J when the detector gain is sinusoidally modulated and parallel acquisition is performed (see Figs. 3 and 4); the dependence on J for sequential detection is trivial 共F ⬀ J1/2兲. However, wide-field detection is usually accomplished by using image intensifiers. Because of the nonlinear response of the intensifiers, their gain modulation can exhibit a high-frequency harmonic content [23], causing the photon economy of typical MCP-based FD systems to depend on the number

the lifetime determination from photons that are integrated in only two time windows [2]. The photon economy of FD-RLD was not previously characterized. Figure 1(b) shows that FD-RLD has a high photon economy. This unexpected result is confirmed by the analytical solution for the figure of merit F of the FD-RLD that, in the case of Dirac excitation and parallel acquisition, is Dirac FFD−RLD

1

= 共1 + x 兲 2 2

冑

5 + 2x2 1 + 4x2

,

共5兲

where x is the normalized modulation frequency. This analytical solution was obtained by the statistical and al-

Fig. 2. Comparison between different excitation methods: sine, rectangular, and Dirac excitation. Rectangular excitation was simulated with 50% and 20% duty cycles. With simultaneous collection, lock-in detection 共J = 2兲 exhibits a photon economy similar to phase detection 共J = 3兲. The optimal photon economy for the RLD estimator 共J = 2兲 and the average 共J = 3兲 estimator are similarly high.

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Fig. 3. Dependence of FD photon economy on the number of phase images. For gated detectors (gray bars), the photon economy depends on the number of phase images, even for simultaneous collection, shown here for sine (left column) and Dirac (right column) excitation. The photon economy of all of phase(top row), modulation- (middle row) and average- (bottom row) lifetime estimators generally converge to that observed with sine-modulated (white bars) gain.

of acquired images. In general, the presence of higher harmonics in the gain modulation is disadvantageous for the photon economy. Figures 3 and 4 show the F values for FD detection with Dirac, rectangular, and sine excitation profiles, varying numbers of phase images (J = 3, 4, 8, 16), and sine(white bars) or square- (gray bars) modulated detector gains. The F value decreases with increasing phase number 共J兲 when the detector is gated, whereas it is constant when the gain is sine modulated. At higher numbers of phase steps 共J ⬎ 4兲, the F values of the different lifetime estimators generally converge to values that are similar to those of sinusoidally modulated detector gain. The frequency at which the photon economy is the highest does not depend on J for J ⬎ 2 (data not shown). In some instances, a higher number of images can improve the photon economy. However, it deteriorates in the case of sequential acquisition. Figure 5 shows the F values at the optimal modulation frequency for the averagelifetime estimator when a detector is gated and used in combination with sine and Dirac excitation. Five or six phase images are a good trade-off between the minimization of photobleaching related artifacts [24], aliasing artifacts [23], and the photon-economy.


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Fig. 4. Dependence of FD photon economy on the number of phase images for rectangular excitation with 50% and 20% duty cycles (see also Fig. 3).

D. Relative Throughput The photon economy is one of the most important parameters that determine the throughput of a detection technique. Techniques exhibiting a poor photon economy require the acquisition of a (F2-fold) higher number of photons than an efficient system 共F = 1兲. This implies that longer acquisition times are needed for a less efficient technique to achieve the same SNR. As the excitation light profile may influence the average fluorescence emission rate of the sample, a more de-

Fig. 5. Minimal F values of the average-lifetime estimator versus number of phase images. The hatched and solid histograms show the photon economy for parallel and sequential acquisition, respectively. The beneficial effect of increasing the number of phase steps is outweighed by the consequent loss of photons for a number of phase steps higher than 5 or 6.

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When the light source is gated, e.g., with an electrooptical modulator, the average power at the specimen is reduced. Consequently, the maximum emitted photon rate 共kG兲 from n molecules is lower than n␶−1 (see Appendix A) by a fraction ␣: kG =

␬ ␬+1

␣共n␶−1兲,

共6兲

where ␬ is the excitation rate expressed in reciprocal fluorescence lifetime units and ␬ / 共␬ + 1兲 corresponds to the fraction of molecules in the excited state. On the other hand, when the average excitation power at the specimen is kept constant while the duty cycle is reduced for instance, when using light-emitting or laser diodes, the photon rate 共kc兲 is higher than kG (see Appendix A): kC =

␬ ␬+␣

␣共n␶−1兲.

共7兲

Because of the transcendent nature of the analytical solution for F共␣兲 [8], i.e., the minimum F value as a function of the duty cycle, only approximated solutions are given [see Fig. 6(a) and Table 1]. The Monte Carlo simulations were also used to estimate the quadratic interpolation curve for the modulation-lifetime estimator and the average-lifetime estimator. The relative throughput is therefore described [Eqs. (4), (6), and (7)] by simple relationships: AG ⬇

AC ⬇

Fig. 6. F values and relative throughput of the average-lifetime estimator versus excitation duty cycle. The photon economy of lifetime detection improves at shorter duty cycles (a) converging to the optimal value close to zero (i.e., Dirac excitation). A constant SNR is maintained by scaling the integration time to compensate for the reduction of emitted photons. This is shown for different excitation rates 共␬兲, with (b) a gated light source and (c) a pulsed source with constant average excitation intensity.

tailed analysis of the effect of the excitation profile on the acquisition time is important for the evaluation of the throughput of the lifetime techniques and for their adaptation to high-throughput and high-speed applications. For instance, reducing the duty cycle 共␣兲 in rectangularwave excitation can increase the photon economy, but can also decrease the fluorescence emission. Therefore, lower F values may not necessarily provide faster acquisition.

1 kG F n␶ 2

−1

1 kC F n␶ 2

−1

=

=

␣

␬

F共␣兲 ␬ + 1 2

␣

␬

F共␣兲 ␬ + ␣ 2

,

共8兲

.

共9兲

At high excitation rates, A also depends on the modulation frequency (see Appendix A). Figures 6(b) and 6(c) show the plot of the exact solutions for A for the average estimator, where the modulation frequency was set to its optimal value for each duty cycle. The maximum achievable relative throughput is ⬃25% 共␣ ⬃ 24% 兲, ⬃17% 共␣ ⬃ 27% 兲, and ⬃15% 共␣ ⬃ 16% 兲 for the average, modulation, and phase-lifetime estimators, respectively. The figure of merit A can also be conveniently used to compare different techniques. With Dirac excitation, the maximum emission rate is one photon per excitation pulse per molecule, leading to Table 1. Photon Economies of Rectangular-Wave Excitation for Different Duty Cycles Is Obtained by Quadratic Curve Fitting of Monte Carlo Simulation Data F Value versus Duty Cycle Phase lifetime Modulation lifetime Average lifetime

1.00+ 0.605 ␣ + 6.73␣2 1.23− 0.26 ␣ + 4.92␣2 1.03− 0.26 ␣ + 4.10␣2

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Table 2. Relative Throughput of FD Detection Excitation

A

a

1 / F共x兲2

Sine b

␣ / F共␣兲2

Rectangular b

x / 关2␲F共x兲2兴

Dirac a

Maximum values at highest fluorophore saturation level.

b

Limit set by fluorophore saturation.

c

␶a (RLD)

␶m

␶⌽ (Lock-in)

15% 共x = 0.97兲 1%–9% c 25% 共␣ = 24% 兲 ⬃16% 共⬃12% 兲 8% 共x = 0.50兲

9% 共x = 0.74兲 ⬍2% 共x ⬍ 2.0兲 17% 共␣ = 27% 兲 ⬃20% 共␣ ⬃ 20% 兲 7% 共x = 0.69兲

7% 共x = 1.03兲 ⬍0.4% 共x ⬍ 2.0兲 15% 共␣ = 16% 兲 ⬃13% 共␣ ⬃ 7 % 兲 5% 共x = 0.38兲

Values strongly depend on x.

␬

1 AD =

␶

F共x兲2 ␬ + 1 T

␬

1 =

x

F共x兲2 ␬ + 1 2␲

,

共10兲

where T is the modulation period, or the inverse of the repetition rate of the Dirac pulse train. The maximum relative throughput provided by the FDRLD and the average-lifetime estimators is therefore predicted [Eqs. (4) and (10)] to be equal to ⬃8%, at a normalized modulation frequency x = 0.50. Phase and modulation estimators provide A values equal to ⬃5% and ⬃7% at normalized modulation frequencies of 0.38 and 0.69, respectively. In the case of FD-FLIM with sine excitation, high photon fluxes can be afforded (see Appendix A), but the comparatively low photon economy of the technique and the fluorophore saturation will limit its relative throughput:

␬

1 AS =

F共x兲 ␬ + 1 2

.

共11兲

The highest relative throughput can therefore equal the system photon economy, i.e., ⬃15%, ⬃9%, and ⬃7%, for the average, modulation, and phase estimators, respectively. The maximum photon economy of TD-RLD is equal to ⬃44% 共F ⬃ 1.5兲 with two symmetric time gates of 2.5␶. This sets a maximum repetition rate of 共5␶兲−1 and a maximum throughput of ⬃9%. A time-gated system with J ⬎ 16 exhibits a photon economy of 100% 共F = 1兲 and a maximum relative throughput up to 20%. It was shown [3] that the repetition rate should be at least 5 times lower than 1 / ␶ in order to achieve F = 1 in a TCSPC experiment. This furthermore avoids the expoTable 3. Relative Throughput of TD Detection

System

Aa

Without Detector Rate Limit on Ab 9% 共T = 2 ⫻ 2.5␶兲

TG-RLD T / 关␶F共J兲2兴

20% 共T = 5␶ , F = 1兲

TG 共J ⬎ 16兲 TCSPC

1 / 共5n兲

20% 共n = 1兲

a

Maximum values at highest fluorophore saturation level.

b

For ␶ = 2 ns.

Detector Rate Limit on Ab 0.9n−1% (PMT) 0.02n−1% (MCP-PMT) 2n−1% (PMT) 0.05n−1% (MCP-PMT)

nential decays elicited by overlapping consecutive pulses. Therefore, the maximum relative throughput achievable by TCSPC equals 20%. However, TCSPC is inherently limited by the requirement to collect less than one photon per excitation pulse, i.e., En␬ / 共␬ + 1兲 1. Furthermore, the throughput of single-photon counting (SPC) systems is generally limited by the maximum counting rate of detector and electronics. This effect can be taken into account by introducing E = 共1 + ktd兲−1 in the parameter that describes the fraction of detected photons, where k is the incident count rate on the detector and td is the detector or electronics dead time. Photomultiplier tubes (PMTs) and microchannel plate (MCP)-PMTs grant maximal rates (i.e., the inverse of the detector dead time) of 10 MHz and 250 kHz, respectively. Also, common electronics limit the maximum count rate to a few megahertz, although the use of multiple TCSPC channels pushes this limit [25]. With a fluorescence lifetime of 2 ns, the limited detector count rate may reduce the relative throughput of SPC systems to (2% n−1 for a PMT) and (0.05% n−1 for a MCP-PMT), where n is the number of fluorophore molecules in the focal volume. If F ⫽ 1, these values are scaled by a factor of F2. These results are summarized in Tables 2 and 3.

E. Fluorophore Saturation Fluorophore saturation can introduce nonlinearities in the fluorescence emission, which alter its harmonic content. The inset in Fig. 7(a) shows that when the excitation (dashed curve) is sinusoidal, fluorophore saturation (here 90%) causes a nonsinusoidal fluorescence emission response (solid curve, lower graph) because of the generation of higher harmonics. FD detection is therefore prone to artifacts at high excitation rates. Figure 7(a) shows the modulation- (black) and phase- (gray) lifetime estimators plotted versus fluorophore saturation at different duty cycles (sine excitation, rectangular excitation with 50%, 40%, 30%, 20%, and 10% duty cycle and Dirac excitation). The estimators rapidly diverge from the simulated true normalized lifetime 共x = 1兲 toward opposite extremes. The average estimator is somewhat less prone to this artifact (not shown). Figure 7(b) shows the maximum fluorophore saturation level that can be used before exceeding a 5% error in the estimation for the phase- (gray curves) and modulation- (black curves) lifetime estimators at three normalized lifetimes of 0.5 (dashed–dotted curve), 1.0 (dashed curve) and 2.0 (solid curve). The limits on modulation and average estimators highly depend on the rela-

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F. Scanning versus Parallel Acquisition, Analog versus Single-Photon Counting Detection There is a substantial difference between sequential (scanning) and parallel (wide-field) imaging. The scanning process decreases the throughput of an x – y scanning system by a factor 共E兲 that is approximately equal to the inverse of the acquired pixel number:

E=

tdwell−time tframe

⬃

1 Npixels

.

共12兲

This already yields ⬃0.002% for a 256⫻ 256 image resolution. MCP-based detection is typically based on the sequential imaging of different phase- or time-dependent frames 共J兲 captured with specified exposure times and with interimage delays that are caused by the readout process and the operation of the electronics and optoelectronics. This implies photon losses during the acquisition of multiple images and correspondingly lower E values: E ⱕ 1/J.

Fig. 7. Artifacts caused by fluorophore saturation in FD detection. Inset of (a), significant saturation of the fluorophore (shown is 90%) introduces higher harmonics in the fluorescence emission (solid curve) even when a single harmonic is present in the excitation signal (dashed curve). (a) Modulation- (black curves) and phase- (gray curves) lifetime estimation as a function of fluorophore saturation level for x = 1. The saturation-dependent errors are reduced at lower excitation duty cycles. (b) Saturation levels that cause a 5% relative error in modulation- (black curves) and phase- (gray curves) lifetime estimators. Saturation levels are shown as a function of duty cycle for rectangular excitation. Dirac excitation (duty cycle →0%) exhibits no systematic errors in any of the estimators.

tive modulation frequency; ␶m appears to be less affected by fluorophore saturation. In scanning systems, where high saturation levels can be achieved, the maximum relative throughput of FD detection is therefore limited to ⬃20% 共␣ ⬃ 20% 兲 for the modulation-lifetime estimator and even lower values for the others (see Table 3).

共13兲

For example, a typical integration time of 100 ms and a delay of 60 ms will scale the relative throughput by a factor E ⬃ 8% 共J = 8兲. MCP-based time-gated systems have been implemented with parallel imaging [13,14] where the delay times caused by electronics and optomechanics operation are negligible, but the gating process still causes the loss of photons. For these systems, Eq. (13) shows an equals sign. In this case, E equals 50% or 13% for two or eight collected images, respectively. We recently [15,16] demonstrated the use of a parallel wide-field detector for FD that can provide E = 100%. The complete parallelization of the detection process could therefore result in faster systems. A comparison between different scanning and widefield implementations is presented in Tables 4 and 5, which illustrate the different throughputs of the techniques with specific numerical examples. Tables 4 and 5 give acquisition times that are estimated for about 100 fluorophore molecules exhibiting a 60% quantum yield and 2 ns lifetime, conditions that are representative for enhanced green fluorescent proteins. Further details are available in Appendix A. The physical limits of the different techniques are more generally represented by the figure of merit A shown in Tables 2 and 3. So far we have assumed that the dominant source of noise is Poissonian. However, when analog detection is used (e.g., MCP wide-field time-gated or FD systems), multiplicative noise is also present that decreases the photon economy [26]. The ranges of acquisition times shown in Tables 4 and 5 represent systems with analog detection, where the F values could be as high as 1.5 times the presented ones. The lower acquisition time limit is computed by considering the nominal photon economies in the presence of Poissonian noise; the upper limit takes into account a 1.5-fold higher F value.

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Table 4. Acquisition Speed of Lifetime Imaging–Scanning Systemsa Acquisition Time (s) for Collection Efficiencyb Equal to System TG [6] J = 4 TG [6] J = 8 TCSPC [25] TCSPC [25] TCSPC [25] and TG J ⬎ 16 FD average [17] FD lock-in [29] FD average [30] FD average [16]

Exc.

td

F (photons)

1%

10%

D D D D

100 ns 100 ns 4 ms 650 ns

1.3 (680) 1.2 (580) 1 (400) 1 (400)

30/24–54 26/21–46 120/— 28/—

8/2.5–5.5 7/2.1–4.7 110/— 18/—

D D S S R

100 ns — — — —

1 (400) ⬃1 (420) 3.8 (5800) 2.6 (2700) ⬃1.2 (540)

18/— —/31–70 —/25–57 —/12–27 —/10–22

5/— —/3.1–7.0 —/2.5–5.7 —/1.2-2.7 —/1–2.2

a

Abbreviations: Exc., excitation; D, Dirac; S, sinusoidal; R, rectangular ␣ = 24%; td, dead time of the detector.

b

Point estimations are for SPC, ranges are for analog detection.

Table 5. Acquisition Speed of Lifetime Imaging: Wide-Field Systemsa System TD [13] TD [14] TD [4] TD [4] TD [4] TCSPC [31] FD [32] FD [32] FD [20] FD [16]

Exc.

J

Acq.

D D D D D D S S S R

2 4 2 4 8 1 4 8 2 2

Par. Par. Seq. Seq. Seq. Seq. Seq. Seq. Seq. Par.

F (photons) 1.5 1.3 1.5 1.3 1.2 1.0 2.6 2.6 2.6 ⬃1.2

(900) (680) (900) (680) (580) (400) (2700) (2700) (2700) (540)

Acquisition Time 8–17 ms 11–25 ms 70–80 ms 190–200 ms 440–460 ms 110 s 190–200 ms 430–450 ms 64–70 ms 1–3 ms

a Abbreviations: Exc., excitation; D, Dirac; S, sinusoidal; R, rectangular ␣ = 24%; Acq., acquisition; Seq., sequential acquisition; Par., parallel acquisition. References are to the implementations.

5. DISCUSSION Theoretical knowledge of the photon economy and throughput of optical systems is necessary for the understanding of the limits of microspectroscopic techniques and for the rationalization of technical developments that could realize residual margins of improvement. FD lifetime detection is commonly considered to be less efficient but faster than TD techniques. However, this is a consequence of particular technological implementations and not a fundamental limitation. We showed that both detection in the TD and in the FD can exhibit high photon economies 共F−2 → 100% 兲 and similar relative throughputs 共A → 20% 兲. Furthermore, the photon economy of the average and RLD FD estimators were characterized for the first time, and a novel figure of merit 共A兲 was introduced that aids in the quantitative comparison of the acquisition speed of different systems. Moreover, the effects of the shape of the excitation light profile and the cross-correlation reference signal and the number of collected images on the photon economy were also investigated in relation to issues like photobleaching, aliasing, and fluorophore saturation.

Photobleaching causes artifacts that can be compensated by the acquisition of specific sequences 共J ⬎ 3兲 of phase-dependent images [24]. Most of the current implementations of FD lifetime detection suffer from higher harmonic content in the excitation or gain modulation. These harmonics cause aliasing [23] that may affect the measurements. To avoid aliasing, at least 5–6 phase images need to be acquired. The higher harmonics in the gain modulation cause the photon economy to depend on the number of acquired images. In general, the acquisition of eight phase-images is sufficient to optimize the photon economy. However, collection of only 5–6 images is advisable with sequential acquisition, where the photon economy decreases with increasing J. The photon economy is highly dependent on the excitation light profile. In general, decreasing the duty cycle of the excitation profile is beneficial. The reduction of duty cycle, however, can cause a decrease in detected photons that may result in longer acquisition times. A duty cycle of 10%–20% presents the best trade-off between photon economy and photon flux and provides the highest acquisition speed possible. For example, when a modulation frequency of 80 MHz is used, 3 ns broad pulses may provide the highest throughput for the average-lifetime estimator and a gated light source. If the reduction of duty cycle is accompanied by an increase in peak power, the fastest acquisition speed can be obtained with pulses of 1.2– 2.5 ns. The response of a fluorophore is not linear and time invariant when photobleaching or saturation occurs. Photobleaching can alter time- and phase-dependent data for techniques that rely on sequential acquisition, leading to possible artifacts that have to be compensated for. For all techniques, photobleaching can cause other artifacts, like the generation of fluorescent photoproducts and sensitized acceptor photobleaching in the case of Förster resonance energy transfer imaging. Photobleaching has been previously investigated [24,27,28]. The present work demonstrates that fluorophore saturation introduces higher harmonics in the fluorescence emission, which cause artifacts. Sine and rectangular excitation with high duty cycles may cause significant errors in the lifetime es-

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timation even at low (5%) fluorophore saturation levels. Therefore, high excitation duty cycles are advisable only with wide-field microscopes where fluorophore saturation is very low. TCSPC, TG, and lock-in imaging implemented on laser scanning systems offer high photon economy, spatial resolution, and image contrast. TD systems offer the simultaneous measurement of multiple time gates and therefore can provide high lifetime resolution and the separation of multiexponential decays. FD detection can exhibit similar advantages when a broad range of the fluorescence harmonic response is analyzed. Although the photon economy of FD is generally worse than TD, its analog detection offers a high throughput compared with SPC [17], and it commonly makes use of higher emission photon fluxes than provided by Dirac excitation. The limited detection count rates of PMTs, single-photon avalanche diodes, MCP-PMTs, and SPC electronics restrict the acquisition speed of SPC techniques. On the other hand, SPC is affected mainly by Poissonian noise, whereas analog detection is affected by additional sources of noise to the further detriment of photon economy (see Tables 4 and 5). Both wide-field and scanning systems, SPC and analogical detection, are modeled. The shortest acquisition times are obtained with optimized FD detection by using a rectangular excitation wave. TD (both TCSPC and TG) could also provide such high acquisition speeds if the count rate were not limited by detector and electronics [25]. This is, however, always the case for TG systems that are based on analog detection or when SPC instrumentation is used with fast detectors or with dim samples. All the other configurations provide longer acquisition times. A scanning TCSPC microscope equipped with a MCP-PMT provides the highest lifetime resolution but requires very long integration times. However, at low count rates, TCSPC could provide the fastest performances because, in agreement with recent work [17], it is not affected by detector and electronics limitations at low count rates. Time-gated systems are generally less demanding in terms of electronics and can provide a good balance among throughput, simplicity of the system, and lifetime resolution. Similar conclusions can be reached in more general terms by the analysis of the relative throughput 共A兲. Wide-field systems can be significantly faster than scanning systems, which exhibit relative throughputs significantly lower than 1% (see Table 4). A TG system that sequentially collects two images provides a relative throughput of 3.4%. For a proper comparison with scanning systems, we compare only maximally achievable rates. However, wide-field systems operate at low excitation rates, resulting in relative throughputs that are 1–2 orders of magnitude lower 共⬃0.1% – 1 % 兲. We recently demonstrated [13,15,16] that wide-field lifetime detection is possible with the simultaneous acquisition of two timeor phase-dependent images. Wide-field TD TG systems have proven capable of lifetime sensing with a single exposure, enabling high acquisition speeds of up to 100 Hz. These MCP-based systems require the fluorescence emission to be split into multiple images that are focused on the image-intensifier photocathode. Different images can be optically delayed [13] or photocathode quadrants can

Esposito et al.

be gated with different timings [14] to allow single-shot measurements. These two systems exhibit a relative throughput of 4.5% (TG, J = 2) and 3% (TG, J = 4), respectively. The parallelization of the acquisition process diminishes the dead times of acquisition, photobleaching, and motion artifacts, but a large number of the photons are lost during the gating process 共1 – J−1兲. Previously, we characterized a detector that is capable of lifetime sensing by the simultaneous accumulation of images at different relative phases on the same sensor [15,16]. Here, the system photon economy improves together with the acquisition speed. The FD-RLD algorithm can be conveniently used with this technology. For this reason, we characterized the FD-RLD and the averagelifetime estimators, whose photon economy was not previously investigated. Both estimators exhibit the same efficient behavior over a relatively broad modulation frequency range. What implementation of fluorescence lifetime imaging exhibits the highest throughput? This work shows that the highest throughput that can be achieved by lifetime imaging in the presence of Poissonian noise is ⬃20%. Our theoretical investigation suggests that this speed could be realized by a wide-field time-gated detector with at least eight gates and fully parallel acquisition. However, at this moment, wide-field FD technologies appear to be more advanced and could provide fast and cost-effective lifetime imaging. This technology is already available, although it requires optimization for its application to microscopy (e.g., more pixels, higher fill factor, cooling). This system could provide a relative throughput of ⬃16% with light pulses of ⬃13% duty cycle (1.6 ns at 80 MHz of modulation frequency) that can be conveniently provided by laser diodes and light-emitting diodes. Furthermore, with the implementation of a higher number of simultaneous phase acquisitions, FD could also achieve the highest throughput (20%–25%) by the use of broad pulses. These figures should be compared with the currently achieved speeds, i.e., 3%–4.5% for the fastest time-gated systems [13,14] and ⬍3% for a typical FD system (RLD, J = 2, sequential acquisition). It can be concluded that the time-gated systems with Dirac excitation and FD systems with nanosecond pulsed light sources currently appear to offer the most optimal throughput speed for laser-scanning and wide-field microscopy, respectively. FD-FLIM high-throughput applications will benefit from the new all-solid-state technologies that allow the implementation of the highest throughput possible for fluorescence lifetime imaging microscopy.

APPENDIX A: SUPPLEMENTARY MATERIAL 1. Emission Rates With spontaneous emission, the maximum rate is limited by fluorophore saturation and the finite fluorescence lifetime. The continuous excitation of an ensemble of fluorophores results in a continuous emission whose photon rate is given by Sn / ␶, where n is the number of molecules, S is the saturation level, and ␶ the fluorescence lifetime. Up to saturation levels that do not significantly distort the harmonic response of the fluorophore, both sine and

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Dirac excitation generally provides lower emission rates than sine and rectangular excitation. This is exacerbated by the comparatively low detection limit of detectors used for SPC. However, the maximum photon rate provided by the high duty cycles of sine and rectangular 共␣ → 50% 兲 excitation cannot be conveniently exploited given the accompanying reduction in photon economy and the low saturation values that can be tolerated. 2. Computation of Values for Tables 4 and 5 Essential parameters: N (number of molecules)⫽100, Q (fluorophore quantum efficiency)⫽60%, ␶ (fluorophore lifetime)⫽2 ns, T−1 (repetition rate)⫽80 MHz and T−1 = 30 MHz for TD and FD, respectively.

Fig. 8. Fluorescence emission (bold line and grayed area) and excitation (dashed line) time profiles for A, sine, B, rectangular, and, C, Dirac excitation. Here, low excitation rates are considered in order to allow an intuitive comparison between these three excitation regimes. The hatched areas in panels A and B show equivalent areas from which the emission rates [Eqs. (6)–(9)] can be directly appreciated.

rectangular excitation (see Fig. 8) exhibit the same photon rate, and S corresponds to the average saturation level. Equations (6) and (7) represent the actual emission rate of the fluorophore for rectangular-wave excitation. At higher saturation rates, the analytical solution derived by integration of the differential equation governing the fluorescence emission are required:

kG =

kC =

冉

␬ ␬+1

冉

n␶−1

␬ ␬+␣

冊

n␶−1

␣共1 + ␬兲 +

冊

x 2␲

␬兵1 − exp关− 2␲␣共1 + ␬兲x−1兴其 1+␬

␣共␣ + ␬兲 +

x 2␲

,

␬兵1 − exp关− 2␲共␣ + ␬兲x−1兴其 ␣+␬

.

For ␣ → 0, kc properly converges to the limit of Dirac excitation, while kG converges to 0. Note that the latter convergence cannot be clearly represented in Fig. 6.

F values are given considering only Poissonian noise. For SPC, the rate is decreased by a factor 共1 + ktd兲, where k is the photon rate and td is the dead time of the detector. This factor is ignored for analog detection. For the latter, F values are increased 1.5-fold to take non-Poissonian noise into account to realistically represent the degradation of the signal-to-noise ratio. Ranges of acquisition times, presented for analog detection, give the minimum and the maximum acquisition times corresponding to the cases of exclusive Poissonian noise and increased F value, respectively. Note that the optimized FD excitation with rectangular-wave excitation and duty cycle of 24% is computed for the case of a gated light source. Higher count rates are expected when a constant average light intensity is used. The repetition rate of the FD was limited to 30 MHz to achieve the optimal photon economy for a lifetime of 2 ns. Acquisition times corresponding to a relative error of 5%, i.e., when 400F2 photons per pixel are counted, were computed. Photon fluxes were computed for scanning systems: FD sine: NSQE / ␶ = 100⫻ 0.05⫻ 0.6⫻ 0.01/ 2 ns= 15 MHz, TD Dirac: NSQE / T = 100⫻ 0.05⫻ 0.6⫻ 0.01⫻ 80 MHz= 2.4 MHz, FD Dirac: NSQE / T = 100⫻ 0.05⫻ 0.6⫻ 0.01⫻ 30 MHz= 900 kHz, FD 共␣ = 24% 兲: ␣NSQE / ␶ = 0.24⫻ 100⫻ 0.05⫻ 0.6⫻ 0.01/ 2 ns= 3.6 MHz, where E, the photon efficiency of the microscope, is set to 1% to model a confocal system and S, the fluorophore saturation, is set to 5%. Table 4 also shows values for E = 10%, where the above computed photon rates should be increased by 1 order of magnitude. Example: TG with RLD measurement using a PMT with a dead-time of 100 ns. F = 1.5. The number of photons required in order to reach a relative error of 10% considering only Poissonian noise is 900. The exposure time is 共900/ 2.4 MHz兲共1 + 2.4 MHz ⫻ 100 ns兲−1 ⫻ 2562⬃ 30 s.

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For analog detection, we do not need to invoke the factor 共1 + 2.4 MHz⫻ 100 ns兲−1; this results in an exposure time of 24 s. However, the F value for analog detection can be higher, e.g., F = 2.25; required photons, ⬃2000; exposure time, ⬃54 s. With a higher count rate of 24 MHz, these values decrease to ⬃8 s (SPC) and 2.5– 5.5 s (analog). Example: FD with gated light source and averagelifetime estimator. The optimal photon economy is achieved for ␣ = 0.24, where F ⬃ 1.16 for the average estimator. This implies that the exposure time becomes 共540/ 3.6 MHz兲 ⫻ 2562⬃ 10 s (22 s for analog acquisition). Photon fluxes for wide-field systems are FD/sine: NSQE / ␶ = 100⫻ 0.001⫻ 0.6⫻ 0.05/ 2 ns= 1.5 MHz, TD/Dirac: NSQE / T = 100⫻ 0.001⫻ 0.6⫻ 0.05⫻ 80 MHz= 240 kHz, FD 共␣ = 24% 兲: ␣NSQE / ␶ = 0.24⫻ 100⫻ 0.001⫻ 0.6⫻ 0.05/ 2 ns= 360 kHz. Here, E is set to 5% to model the use of an S0 photocathode and the typical losses in a microscope, and S is set to 0.1% to model the lower saturation levels used in wide-field microscopy. For sequential acquisition, a time delay of 共J − 1兲 ⫻ 60 ms is added to the exposure time to represents the typical time lags that occur during the switching between time and phase delays. F2 is scaled by the number of acquired images if photons are lost due to sequential exposures or image splitting and gating. Example: TG with RLD measurement and images acquired by using an MCP and a dual-viewer image splitter.

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ACKNOWLEDGMENTS This investigation was supported by the Deutsche Forschungsgemeinschaft Research Center for Molecular Physiology of the Brain, Cluster of Excellence 171, the Federal Ministry of Education and Research (BMBF; Biophotonic III project “Fluorescence Lifetime Imaging Camera, FLI-CAM”), and the ENI-NET consortium. The European Neuroscience Institute Göttingen (ENI-G) is jointly funded by the Göttingen University Medical School, the Max-Planck-Society, and Schering AG.

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