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Front. Phys., 16 October 2020
Sec. Optics and Photonics
Volume 8 - 2020 |

Investigations on Average Fluorescence Lifetimes for Visualizing Multi-Exponential Decays

Yahui Li1,2,3, Sapermsap Natakorn4, Yu Chen4, Mohammed Safar1, Margaret Cunningham1, Jinshou Tian2,3 and David Day-Uei Li1*
  • 1Strathclyde Institute of Pharmacy and Biomedical Sciences, University of Strathclyde, Glasgow, United Kingdom
  • 2Key Laboratory of Ultra-fast Photoelectric Diagnostics Technology, Xi'an Institute of Optics and Precision Mechanics, Xi'an, China
  • 3University of Chinese Academy of Sciences, Beijing, China
  • 4Department of Physics, Scottish Universities Physics Alliance, University of Strathclyde, Glasgow, United Kingdom

Intensity- and amplitude-weighted average lifetimes, denoted as τI and τA hereafter, are useful indicators for revealing Förster resonance energy transfer (FRET) or fluorescence quenching behaviors. In this work, we discussed the differences between τI and τA and presented several model-free lifetime determination algorithms (LDA), including the center-of-mass, phasor, and integral equation methods for fast τI and τA estimations. For model-based LDAs, we discussed the model-mismatch problems, and the results suggest that a bi-exponential model can well approximate a signal following a multi-exponential model. Depending on the application requirements, suggestions about the LDAs to be used are given. The instrument responses of the imaging systems were included in the analysis. We explained why only using the τI model for FRET analysis can be misleading; both τI and τA models should be considered. We also proposed using τAI as a new indicator on two-photon fluorescence lifetime images, and the results show that τAI is an intuitive tool for visualizing multi-exponential decays.

1. Introduction

Fluorescence lifetime imaging (FLIM) is a crucial technique for assessing microenvironments of fluorescent molecules [1, 2], such as pH [3], Ca2+ [4, 5], O2 [6], viscosity [7], or temperature [8]. Combining with Förster Resonance Energy Transfer (FRET) techniques, FLIM can be a powerful “quantum ruler” to measure protein conformations and interactions [912]. Compared with fluorescence intensity imaging, FLIM is independent of the signal intensity and fluorophore concentrations, making FLIM a powerful quantitative imaging technique for applications in life sciences [13], medical diagnosis [1416], drug developments [1719], and flow diagnosis [2022]. FLIM techniques can build on time-correlated single-photon counting (TCSPC) [2325], time-gating [2628], or streak cameras [29]; they record time-resolved fluorescence intensity profiles to extract lifetimes with a lifetime determination algorithm (LDA) [1]. There is a rapid growth of real-time applications that fast analysis is sought after [12, 30]. Traditional LDAs usually use the least square method (LSM) or maximum likelihood estimation (MLE) [31] to analyze decay models chosen by users, and model-fitting analysis follows a reduced chi-squared criterion [1]. In reality, however, it is difficult to know the exact decay model as fluorescent molecules in biological systems can demonstrate complex multi-exponential decay profiles. For instance, a mixture of fluorophores, a multi-tryptophan protein, single fluorophores in varied environments, and single-tryptophan proteins in multiple conformational states [1] can show multi-exponential decays as

f(t)=Ai=1pqiexp(-t/τi),wherei=1pqi=1,    (1)

where A represents the amplitude, qi and τi (i = 1, …, p) denote the amplitude fractions and lifetimes, respectively, and p is the number of lifetime components. There are time-domain or frequency-domain [3235] FLIM systems to measure a fluorescence decay. In this work, we focus on time-domain approaches.

Suppose the instrument response function (IRF) of the measurement system is irf(t), the task performed by FLIM analysis tools is to extract f(t) from the measured decay h(t), as

h(t)=irf(t)*f(t).    (2)

The problems with traditional LSM or MLE are two-fold. (1) It is challenging to categorize a fluorescence emission into a specific exponential model described by Equation (1) in complex biological processes. An arbitrary choice of p in Equation (1) simply based on reduced chi-squared tests [36] would lead to totally different interpretations. As the fitting routine is not mathematically unique; a measured decay could be fitted equally well with a bi-exponential or a tri-exponential model. (2) To ensure the accuracy, it usually needs a high photon count (long acquisition time) when p ≥ 2 [37]. Instead of completely extracting qi and τi (i = 1, …, p), which is doubtful as mentioned above and time-consuming, in many applications, it is often useful to determine only the average lifetime which can be expressed in two forms [1]: the intensity-weighted average lifetime

τI=i=1pqiτi2/i=1pqiτi,    (3)

and the amplitude-weighted average lifetime

τA=i=1pqiτi/i=1pqi=i=1pqiτi.    (4)

The question about which average lifetime we should use according to the applications has been investigated in [38]. For instance, they suggested:

(a) τA can estimate the energy transfer efficiency in FRET [39],

E=1-IDAID=1-τDA,AτD,A,    (5)

where E is the energy transfer efficiency, IDA and ID are the fluorescence intensities of the donor in the presence and absence of energy transfer, respectively, and τDA, A and τD, A are τA of the donor in the presence and absence of energy transfer, respectively. E can further estimate the donor-acceptor distance.

(b) τA can also assess dynamic quenching behaviors, described by the Stern-Volmer equation [40],

I0I1=1+KD[Q]=τ0,Aτ1,A,    (6)

where I0 and I1 are fluorescence intensities, τ0, A and τ1, A are τA of the fluorophore in the absence and presence of the quencher, respectively, KD is the Stern-Volmer quenching constant, and [Q] is the concentration of the quencher. Additionally, the average radiative rate constant can be expressed as, kr = QE1, A, where QE is the quantum yield.

(c) τI can be used to estimate the average collisional constant kq from the Stern-Volmer constant KD.

Average lifetimes can either be calculated by extracting the lifetime components using model-based LDAs and then using Equations (3) and (4). Or they can be directly obtained with model-free LDAs, such as hardware-friendly center-of-mass methods (CMM) [4144], the phasor method (Phasor) [4547], the rapid lifetime determination method (RLD) [30, 4851], or the integral extraction method (IEM) [52, 53], without assuming any decay model.

In this work, we theoretically investigated two types of average lifetimes evaluated by model-free LDAs, examined the performances of τA and τI estimations using different LDAs, and suggested the choices of LDAs in terms of accuracy, precision, and estimation speeds according to the applications. We also described a multi-exponential decay visualization tool using the ratio τAI. Experimental results demonstrate the performance of τAI in comparison with Phasor.

2. Theory

In this section, we derived the average lifetimes determined by the model-free methods, CMM, Phasor, and IEM and described the general work flow of average lifetime estimations with the model-free and model-based LDAs.

As Equation (2), the measured signal h(t) is the convolution of f(t) with irf(t). Here we focus on the signal hm and irfm obtained from a TCSPC system, as shown in Figure 1,

    hm=k=0mirfk-m·fm,m=0,1,2,,M-1,irfm=mΔt(m+1)Δtirf(t)dt,    fm=mΔt(m+1)Δtf(t)dt=Ai=1pqiτie-mΔtτi[1-e-Δtτi],    (7)

where hm is the photon count collected in Bin m at tm = (m + 1/2)Δt, M is the number of bins, and Δt is the time resolution.


Figure 1. Illustration of hm and irfm obtained with a TCSPC system.

(a) CMM

The average lifetime evaluated with CMM is

τCMM=0t·h(t)dt0h(t)dt-0t·irf(t)dt0irf(t)dt=i=1pqiτi2i=1pqiτi             m=0M-1tm·hmm=0M-1hm-m=0M-1tm·irfmm=0M-1irfm,    (8)

which is equal to τI. The derivation of Equation (8) is shown in the Appendix.

(b) Phasor

The average lifetime evaluated with Phasor is

τP=sgω=i=1pqiτi2/(1+ω2τi2)i=1pqiτi/(1+ω2τi2),    (9)

where ω = 2π/T, T = MΔt is the measurement window, and g and s are the phasor components expressed as

g=0f(t)·cos(ωt)dt0f(t)dt=i=1pqiτi/(1+ω2τi2)i=1pqiτi       =Rh+s·IirfRirf,s=0f(t)·sin(ωt)dt0f(t)dt=i=1pωqiτi2/(1+ω2τi2)i=1pqiτi       =Ih·Rirf-Rh·IirfRirf2+Iirf2,


Rh=0h(t)·cos(ωt)dt0h(t)dtm=0M-1hm·cos(ωtm)m=0M-1hm,  Ih=0h(t)·sin(ωt)dt0h(t)dtm=0M-1hm·sin(ωtm)m=0M-1hm,
Rirf=0irf(t)·cos(ωt)dt0irf(t)dtm=0M-1irfm·cos(ωtm)m=0M-1irfm,  Iirf=0irf(t)·sin(ωt)dt0irf(t)dtm=0M-1irfm·sin(ωtm)m=0M-1irfm.

τP is a weighted average lifetime whose weights are qiτi/(1+ω2τi2). If τiT, then the weights are approximately equal to qiτi, i.e., τP is close to τI.

(c) IEM

For IEM, the underlying exponential decay should be extracted by a model-free deconvolution method. With the estimated exponential decay f^m, the average lifetime with IEM is

τIEM=-0g(t)dt0g(t)dt=i=1pqiτii=1pqi-Δtm=0M-1Sm·f^mf^M-1-f^0,   g(t)=Ai=1pqiτie-t/τi[1-e-Δt/τi],    (10)

where Sm = [1/3, 4/3, 2/3, 4/3, 1/3] are the coefficients for numerical integration based on Simpson's rule. τIEM is actually an estimator for τA.

Figure 2 summarizes the flow diagram for τI and τA estimations with different algorithms used in this study. The simulated signals hm and irfm are directly sent into CMM and Phasor blocks to estimate τI. The estimated fm (from hm and irfm with the Laguerre expansion deconvolution method with L = 16 and α = 0.912 [54, 55]) is sent to IEM to estimate τA and sent to the bi-decay center-of-mass method (BCMM; j = 2) [56], the variable projection method (VPM; j = 2) [57], or LSM with a j-exponential model (denoted as LSM-j), to estimate τI and τA. CMM and Phasor are fast as no deconvolution routine is needed, whereas IEM, BCMM, VPM, and LSM are direct or iterative estimation approaches once fm is extracted. Artificial neural network assisted analysis tools [58, 59] can be included in this diagram, but they are out of the scope of this work.


Figure 2. Flow diagram for τI and τA estimations.

3. Results

3.1. Simulations

In reality, it is difficult to characterize a real fluorescence profile with a suitable exponential model described in Equation (1). To demonstrate how model-free analysis can be beneficial, we examined two scenarios. Case A: we used exponential decay signals with p = 1 ~ 4 to assess the influence of the model mismatch between the signal and the algorithm on τI and τA estimations. This study is to investigate the scenario when users select a j-exponential model to analyze a p-exponential decay (p can be different from j). Case B: we generated synthetic bi-exponential (p = 2) decay signals to assess the performances of τI and τA estimations with the model-free and model-based LDAs.

The performances of lifetime estimations can be assessed in two aspects: (1) the accuracy Bn=|τ^n-τn|/τn and the precision Fn=Ntotστ^n/τ^n [60], where n = I or A for the intensity- or the amplitude-weighted lifetimes, τn and τ^n are actual and estimated values, στ^n is the standard deviation of τ^n, and Ntot is the total photon count. The lower the F, the higher the precision (F = 1 for the ideal case).

3.1.1. Case A: Model Mismatch

Ideally, a bi-exponential signal should be analyzed by a bi-exponential model. For instance, BCMM, VPM, and LSM-2 are used for bi-exponential decay models, and LSM-j for j-exponential models, j > 2. However, in realistic biological processes, it is difficult to know precisely how many lifetime components a decay profile contains. In traditional FLIM analysis tools, users usually need to select an exponential model to fit measured decays and use the reduced chi-squared to evaluate the goodness-of-fit. If the reduced chi-squared is not satisfactory, then a different exponential model is chosen. This process continues until the reduced chi-squared is acceptable. Often different exponential models can produce similar reduced chi-squared values, and the question is which fitting we should use? It is quite common that a j-exponential model might analyze a signal containing p lifetime components and jp. We would like to know if p is unknown to the user, whether using a different analysis model (jp) would lead to a different biological story.

We generated exponential decay signals hm (m = 0,…, M-1) to test the LDAs for τI and τA estimations. hm can be artificially generated with f(t)=Ai=1pqiexp(-t/τi), where p = [1, 2, 3, 4], qi = 1/p, and the IRF is approximated with a Poisson distribution irfm=exp(λ)λm/m! with λ = 500 ps, FWHM ≃ 300 ps, and M = 256. The measurement window T = 10 ns, and the total photon count Ntot = 103. τi, τI, and τA for each p are summarized in Table 1.


Table 1. τi, τI, and τA for p = 1 ~ 4 with qi = 1/p.

The performances of τI and τA estimations with the simulated exponential decays are shown in Figures 3A–D for BI, FI, BA, and FA, respectively. For model-free LDAs, BI and BA are below 10% and are independent of p. For LSM-1, when p = 1, BI, and BA are zero, whereas when p > 1, BI and BA increase especially for p = 2. For model-based LDAs where j > 1, BCMM, VPM, and LSM-j have similar performances even for pj, seemingly suggesting that a bi-exponential model can well approximate a signal following an arbitrary p-exponential model. We generated 100 signals with τi and qi chosen randomly in the ranges of 0.1 ~ 2.5 ns and 0.1 ~ 0.9, respectively, for each p. BCMM, VPM, and LSM-2 were used to fit the signals with bi-exponential decays. The goodness-of-fit is judged by the reduced chi-squared χ2=1Mm=0M-1(fm-fc,m)2/fm, where fm and fc, m are actual and fitted signals of Bin m. The box plots of χ2 for BCMM, VPM, and LSM-2 are shown in Figures 3E–G, respectively. The χ2 values are insensitive to p for the three LDAs so that we conclude that a bi-exponential decay is suitable to approximate an arbitrary p-exponential decay (p ≤ 4).


Figure 3. Performances of τI and τA estimations with exponential decay signals using different algorithms, (A) BI, (B) FI, (C) BA, and (D) FA. (E–G) Box plots of χ2 for BCMM, VPM, and LSM-2.

Therefore, if the decay model of the signal is inaccessible, model-free and model-based LDAs, BCMM, VPM, and LSM-2 are enough for τI and τA estimations.

In practice, users can choose an optimization algorithm and set initial conditions to analyze FLIM images when LSM-2 is used. We would like to know how they can affect τI and τA estimations. Four bi-exponential decays, Decays 1 ~ 4, with different parameters (q1, τ1, τ2) were analyzed using LSM-2 with different initial conditions (q10, τ10, τ20), denoted as Init. 1 ~ 4 listed in Table 2 with Ntot = 103. When either of the estimated τ1 and τ2 is larger than T (10 ns), we say that the estimation fails. The probabilities of producing a failed trial, P1 or τ2 > 10) and producing biased τI and τA with BI and BA > 0.3, i.e., P(Bn > 0.3), n = I or A, are shown in Figure 4. Figures 4A–F are the LSM-2 results with the unconstrained and constrained trust-region-reflective (TRR) algorithms, respectively. The constraints are 0 < q1 < 1 and 0 < τ1, τ2 < 10 ns. Figures 4G–I are the LSM-2 results using the Levenberg-Marquardt (LM) algorithm. For the unconstrained TRR, the performances are relatively sensitive to initial conditions. P1 or τ2 > 10) for Init. 4 is quite significant which results in large P(Bn > 0.3), n = I or A, for all four decays. Although Init. 3 leads to a low P for Decays 2 ~ 3, P(BA > 0.3) for Decay 1 rises to 0.7. Thus, if the initial conditions are not chosen properly, the quality of τI and τA images cannot be guaranteed. The constrained TRR and LM are insensitive to initial conditions. Although the LM has failed trials, they barely affect P(Bn > 0.3), n = I or A. Therefore, to ensure accurate τI and τA estimations, the constrained TRR and LM are recommended for LSM-2.


Table 2. Bi-exponential decays and initial conditions for τI and τA estimations with LSM-2.


Figure 4. (A,D,G) P1 or τ2 > 10), (B,E,H) P(BI > 0.3), and (C,F,I) P(BA > 0.3), for estimations of Decays 1 ~ 4 with Init. 1 ~ 4 using (A–C) the unconstrained TRR, (D–F) the constrained TRR, and (G–I) the LM algorithms.

3.1.2. Case B: Performances of Average Lifetime Estimations

As mentioned above, it might be challenging to use a proper exponential model to describe realistic biological processes; a bi-exponential model might well approximate them. Here we will use a bi-exponential model to explain why model-free LDAs have the benefits of higher photon efficiency and faster analysis than model-based LDAs for τI and τA estimations.

hm can be artificially generated with the same IRF used in Case A and f(t) = A[q1exp(−t1) + (1 − q1)exp(−t2)], where τ1 < τ2 and q1 is the amplitude fraction of τ1. Figure 5A shows the signal and IRF. In FRET and dynamic quenching applications, the fluorescence lifetime of the donor fluorophore is in general decreasing, and we assume τ2 = 2.5 ns and τ1 varying from 0.1 to 2.5 ns to emulate FRET or quenching. The theoretical τI, τA, and τP with q1 = 0.5 are shown in Figure 5B. τP has a negative bias from τI. With T2 increasing, τP approaches τI. Figure 5B that two different (τ1, τ2) sets can deliver the same τI, for instance, (0.32, 2.5) ns and (2.1, 2.5) ns have the same τI of 2.3 ns.


Figure 5. (A) Simulated signal (blue) and IRF (black), (B) theoretical τI (blue), τA (black), and τP (magenta) average lifetimes with q1 = 0.5.

Therefore, only estimating τI can be misleading. Figure 5B also shows that the dynamic range of τI is only 2.5–2.23 = 0.27 ns and within which the above problem persists. Whereas τA does not have this problem for this case. We conducted Monte Carlo simulations to estimate τI and τA with the simulated signals, including Poisson noise under different conditions q1 = 0.2, 0.5, and 0.8.

The performances of τI and τA estimations with bi-exponential decay signals are shown in Figures 6A–D for BI, FI, BA, and FA, respectively. For τ^I, BI, CMM, and BI, BCMM are roughly 10 and 8%, respectively determined by T2. The larger T2 is, the smaller BI becomes (with FI, CMM and FI, BCMM being closer to 1). Phasor has a lower accuracy when q1 becomes larger and τ1 smaller, and it is less precise than CMM. VPM and LSM-2 both have a smaller BI = 3% but higher FI (1.5 ~ 5) than CMM and BCMM. For τ^A, BA is 7% except for τ1 = 0.1 ns, and FA is around 5 for the four LDAs. Figures 6C,D show that if only τA is needed, there is no need to resort to slower model-based LDAs.


Figure 6. The performances of τI and τA estimations with bi-exponential decay signals with τ1 = 0.1 ~ 2.5 ns, τ2 = 2.5 ns, q1 = 0.2, 0.5, 0.8, (A) BI, (B) FI and (C) BA, (D) FA.

For τI estimations, LSM-2 and VPM are preferred when high accuracy is required. Still, they are slower and have lower photon efficiency than CMM and BCMM which means the photon count should be higher to have similar precision, for instance, a relative standard deviation of 5% can be reached with Ntot = 3,600 for LSM-2 and Ntot = 500 for CMM and BCMM. When the accuracy of CMM or BCMM (10% @ T2 = 4) is acceptable, CMM or BCMM should be employed for their high photon efficiency and estimation speeds. CMM is faster than BCMM as it can work without deconvolution. For τA estimations, since the performances of IEM, BCMM, VPM, LSM-2 are similar, IEM can be the right candidate for fast analysis. Notice that the τA method is less photon efficient than the τI method as FA is higher than FI.

3.2. Experimental Results

tSA201 cells, which are a transformed human kidney cell line, were co-transfected with hP2Y12-eCFP and hP2Y1-eYFP receptors. After 48 h of transfection, the cells on the coverslips were washed once gently with PBS followed by fixation with ice-cold methanol for 10 min at room temperature. After being washed three times with PBS, they were mounted on to glass microscope slides with Mowiol. The microscope slides were then stored in the dark at room temperature overnight to allow the coverslips to dry, then stored at 4°C for later use.

Cells were imaged on LSM510 (Carl Zeiss) equipped with a TCSPC module (SPC-830, Becker & Hickl GmbH), to determine the fluorescence lifetime and consequently the amount of FRET. The donor is CFP with the excitation wavelength range of 350 ~ 500 nm and the emission wavelength range of 450 ~ 600 nm. The acceptor is YFP. The sample is scanned pixel by pixel by a femtosecond Ti:Sapphire laser (Chameleon, Coherent) with an average output laser power of 3.8 W at 800 nm, as a two-photon excitation source to reduce cellular damage. The laser power is controlled with two polarizers. The repetition rate is 80 MHz with illuminating duration <200 fs. The emitted fluorescence signal from the donor is collected through a 63 × water-immersion objective lens (N.A. = 1.0), a 480 ~ 520 nm bandpass filter, and transferred into a photomultiplier tube (PMT) detector. The FLIM scanning was performed in a dark room containing the microscope. A set of experimental data (256 × 256 pixels, M = 256, T = 10 ns) was collected over an exposure period of up to 15 min. The IRF is obtained from the measurement of dried urea [(NH2)2CO] [61].

3.2.1. Average Lifetime Images With LSM, CMM, and IEM

Figures 7A,C show the τI and τA images of the data evaluated by LSM-2 with an execution time of 410 s. The lifetime images were evaluated on Matlab R2016a, 64-bit with the Intel(R) Celeron(R) CPU (2950M @ 2 GHz) with 20923 pixels above an intensity threshold. Figures 7B,D show the τI and τA images evaluated by CMM and IEM with execution times of 0.25 and 92.3 s, respectively. IEM can be further accelerated to 0.6 s per image with histogram classification methods (we will report the details soon), as shown in Figure 7E. Although Fast-IEM causes a small bias in some pixels, the mean square error is acceptable with 0.005 ns2. The color bar represents lifetimes and the pixel brightness represents photon counts. The Figures 7F,G are histograms of τI and τA, respectively. Although the histogram of τI with CMM deviates slightly from the one with LSM-2, CMM is 1,800-fold faster than LSM-2. If Ti > 4, the bias of τI with CMM would become smaller. The τA images are almost the same with IEM and LSM-2, whereas IEM and Fast-IEM are much faster than LSM-2.


Figure 7. (A,C) τI and τA images evaluated with LSM-2, (B,D) τI and τA images evaluated with CMM and IEM, respectively. (E) τA image with Fast-IEM. (F) Histograms of τI with LSM-2 (blue) and CMM (black). (G) Histograms of τA with LSM-2 (blue), IEM (magenta), and Fast-IEM (yellow). The color bar represents lifetimes and the pixel brightness represents photon counts. (A–E) Can also represent FRET efficiency (E) images evaluated with the corresponding lifetime images with the color bar representing the range of E, 0 ~ 100%. (F,G) Can also be used to show the histograms of E with the upper x label.

Since the FRET efficiency E has a linear relationship with the average lifetimes as shown in Equation (5), Figures 7A–E can also be used to represent E images with the color bar in the range of 0 ~ 100%. As we mentioned in Introduction, it is straightforward to obtain E images from τA images, so that Figures 7C–E are proper E images. If τI images are misused for E, the results would be different, as shown in Figures 7A,B, leading to a different biological story.

3.2.2. Visualization of Multi-Exponential Decays With τAI

τI and τA can not only access the essential parameters in FRET and dynamic quenching processes but also indicate the positions where multi-exponential decays occur. As mentioned previously, a fluorescence signal can be approximated by a bi-exponential decay, so that the ratio of τI and τA can be expressed as

τAτI=[1+q1(R-1)]21+q1(R2-1),    (11)

where R = τ12. The distribution of τAI (Figure 8) shows that when R ≃ 1 or q1 ≃ 0 or 1, τAI ≃ 1. With a decrease of R or an increase of q1, τAI decreases. Therefore, the ranges of q1 and R of a pixel can be determined by τAI.


Figure 8. Distribution of τAI with q1 = 0 ~ 1 and R = 0 ~ 1.

To present the multi-exponential decay visualization performance of τAI, the τI and τA images evaluated by LSM-2, Figures 9A,B, were used to generate the τAI image as shown in Figure 9C. The histograms of τI and τA and the phasor plot are shown in Figures 9D,E. Figure 9F shows the possible range of q1 and R of the selected pixels in Figure 9C. Figures 9C,F share the same color bar. Figure 9D shows that τA has a broader lifetime dynamic range than τI, which is consistent with the theoretical lines shown in Figure 5B. The τA histogram shows two clusters with different peaks, whereas the τI histogram only indicates a single merged group, meaning that there is no way to differentiate these two clusters. It is why using τI to analyze samples with a strong FRET can be misleading.


Figure 9. (A) τI-intensity image, (B) τA-intensity image evaluated by LSM-2, (C) τAI ratio image, (D) histograms of τI (yellow) and τA (blue), (E) phasor plot, and (F) distribution of τAI of the selected pixels in (C).

The results of the selected pixels within different τAI ranges are shown in Figure 10, τAI = 0.2 ~ 0.5, and Figure 11, τAI = 0.5 ~ 1. For the pixels with τAI = 0.2 ~ 0.5, the histograms clearly show that τA is much smaller than τI, which means the difference between τ1 and τ2 is significant. Figure 10F shows that the ranges of q1 and R are approximately 0.5 ~ 1 and 0 ~ 0.2, respectively. For the pixels with τAI = 0.5 ~ 1, τA is closer to τI, meaning the pixels have decays close to mono-exponential. Separating the average lifetime images with τAI is easier than phasor plots because τAI is one dimensional and phasors are two dimensional. Furthermore, τAI can show the q1 and R ranges more intuitively than phasor plots. τAI can be a useful tool to visualize the properties of the fluorescence decays within a lifetime image.


Figure 10. (A) τI-intensity image, (B) τA-intensity image evaluated by LSM-2, (C) τAI ratio image, (D) histograms of τI (yellow) and τA (blue), (E) phasor plot, and (F) distribution of τAI of the selected pixels in (C) with τAI = 0.2 ~ 0.5.


Figure 11. (A) τI-intensity image, (B) τA-intensity image evaluated by LSM-2, (C) τAI ratio image, (D) histograms of τI (yellow) and τA (blue), (E) phasor plot, and (F) distribution of τAI of the selected pixels in (C) with τAI = 0.5 ~ 1.

4. Discussion

In realistic samples, fluorescence signals always follow multi-exponential decay models. However, extracting lifetime components with a traditional fitting method is a time-consuming process. For some applications that require calculating FRET efficiency and accessing dynamic quenching behaviors, average lifetimes are satisfactory. Model-free lifetime determination algorithms can be used to evaluate average lifetimes directly, for instance, CMM and Phasor for intensity-weighted average lifetimes τI and IEM for amplitude-weighted average lifetimes τA. Discussions of the influence of the model mismatch between the real signal and the model-based LDAs on τI and τA estimations suggest that a bi-exponential model can well-approximate a signal following a multiple-exponential model. The results of the Monte-Carlo simulations suggest that VPM and LSM based on a bi-exponential model can be used for applications requiring high accuracy. The constrained TRR and LM algorithms with proper initial conditions are supported for LSM to guarantee accuracy. In contrast, CMM and IEM are recommended for applications requiring high estimation speeds. We also explained why τI models can be misleading, and τI and τA models should be considered. Experimental data were used to compare the performances of LSM-2, CMM, and IEM for evaluating τI and τA images. Similar τI and τA images were generated, whereas CMM and IEM are much faster than LSM-2. The data were further analyzed with τAI, which is capable of indicating the possible ranges of the amplitude proportion of the short lifetime and the ratio of the short and long lifetimes. We believe τAI is a useful and intuitive tool for visualizing multi-exponential decays in a lifetime image.

Data Availability Statement

FLIM image raw data and the instrument response are available at

Author Contributions

YL conducted theoretical and experimental analysis and developed analysis tools. MS and MC conceived FRET experiments and prepared samples. SN and YC contributed to FRET-FLIM experiments. JT contributed to tool developments. DL initiated the research concept and supervised the project. All authors wrote and revised the paper.


We would like to acknowledge support from Medical Research Scotland, China Scholarship Council, and Engineering and Physical Sciences Research Council (EP/M506643/1).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


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Derivation of τCMM. Take the integration of t·h(t) and h(t),

0t·h(t)dt=0t0irf(t-t)·f(t)dtdt                           =00(t-t)·irf(t-t)·f(t)dtdt                            +00irf(t-t)·t·f(t)dtdt                            =0[t·irf(t)]*f(t)dt+0irf(t)*[t·f(t)]dt                            =0t·irf(t)dt0f(t)dt                            +0irf(t)dt0t·f(t)dt,    (A1)
0h(t)dt=0irf(t)dt0f(t)dt.    (A2)

Dividing Equation (1) by Equation (2) gives

0t·h(t)dt0h(t)dt=0t·irf(t)dt0irf(t)dt+0t·f(t)dt0f(t)dt.    (A3)


τCMM=0t·f(t)dt0f(t)dt=i=1pqiτi2i=1pqiτi              =0t·h(t)dt0h(t)dt-0t·irf(t)dt0irf(t)dt              m=0M-1tm·hmm=0M-1hm-m=0M-1tm·irfmm=0M-1irfm.    (A4)

Keywords: fluorescence lifetime imaging, lifetime determination algorithm, average lifetimes, multi-exponential decays, lifetime image visualization, FRET—fluorescence resonance energy transfer

Citation: Li Y, Natakorn S, Chen Y, Safar M, Cunningham M, Tian J and Li DD-U (2020) Investigations on Average Fluorescence Lifetimes for Visualizing Multi-Exponential Decays. Front. Phys. 8:576862. doi: 10.3389/fphy.2020.576862

Received: 27 July 2020; Accepted: 09 September 2020;
Published: 16 October 2020.

Edited by:

Klaus Suhling, King's College London, United Kingdom

Reviewed by:

Nirmal Mazumder, Manipal Academy of Higher Education, India
Ilia L. Rasskazov, University of Rochester, United States

Copyright © 2020 Li, Natakorn, Chen, Safar, Cunningham, Tian and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: David Day-Uei Li,