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The authors have declared that no competing interests exist.

Conceived and designed the experiments: PRB ACCC BV. Performed the experiments: MIR PRB. Analyzed the data: MIR PRB. Wrote the paper: MIR ACCC BV PRB. Implemented the algorithms in software: MIR PRB.

We present novel Bayesian methods for the analysis of exponential decay data that exploit the evidence carried by every detected decay event and enables robust extension to advanced processing. Our algorithms are presented in the context of fluorescence lifetime imaging microscopy (FLIM) and particular attention has been paid to model the time-domain system (based on time-correlated single photon counting) with unprecedented accuracy. We present estimates of decay parameters for mono- and bi-exponential systems, offering up to a factor of two improvement in accuracy compared to previous popular techniques. Results of the analysis of synthetic and experimental data are presented, and areas where the superior precision of our techniques can be exploited in Förster Resonance Energy Transfer (FRET) experiments are described. Furthermore, we demonstrate two advanced processing methods: decay model selection to choose between differing models such as mono- and bi-exponential, and the simultaneous estimation of instrument and decay parameters.

Optical microscopy methods are extensively and increasingly used in biomedicine. In particular, quantification of the acquired images has become essential, both in terms of morphology and in terms of intensity. More advanced fluorescence imaging techniques (e.g. confocal [

The finite probability of fluorescence light emission following fluorophore excitation results in a decaying profile of fluorescence intensity from a given ensemble of molecules. Since the probability per unit time is usually constant over the time period of the decay (typically nanoseconds for organic fluorescent molecules), the profile is a decaying exponential function that can be characterised by the

When linked with single or two-photon laser scanning techniques to form Fluorescence Lifetime Imaging Microscopy (FLIM), these techniques provide powerful tools for biological investigation. For example, molecular interactions between specific proteins in a cell can be robustly detected in living and fixed cells and tissues using FLIM by exploiting Förster Resonance Energy Transfer (FRET) [

Our previous analysis [

As a preliminary example of what can be achieved with bi-exponential analysis in the context of a FRET experiment,

Lifetime analysis of FLIM data from a cell pellet sample. In (a) an intensity image having pixels with a total photon count of between about 20 and 400 were analysed (having invoked 9 × 9 spatial binning to provide sufficient photon counts for a bi-exponential analysis) using the bi-exponential Bayesian algorithm. In (b) and (c) the interacting fraction and the FRET efficiency respectively (as computed using the Bayesian estimates of the decay parameters). The size of a 9 × 9 spatial bin is indicated in (a) by a red square at the centre of the image and in (d) the decay data from from the spatial bin is shown. All of the images correspond to a 334 × 334

The data for this kind of time-domain FLIM is usually acquired using Time Correlated Single Photon Counting (TCSPC): this involves the collection of fluorescence decay photon emission times to yield a histogram of photon count (fluorescence intensity) against time, and Poisson statistics dictates that the more photons that are counted the more accurately the histogram represents the fluorescence decay. The most commonly applied analysis methods for this type of data involve the ‘direct fitting’ of a fluorescence decay model to the measured histogram. In the direct fitting approach a decay model, typically of the form
_{ℓ} and a decay lifetime _{ℓ}. The optimal fit is determined by iteratively minimising a distance metric between the fit and the measured photon counting histogram. In performing such an analysis, the necessary convolution of

The effectiveness of the ‘direct fitting’ approach depends crucially on the choice of the distance metric; different distance metrics introduce into the analysis different assumptions regarding the statistical noise in the counted photon data.

In the least-squares (LS) fitting approach, the distance metric is based on the squared difference between the measurements and the proposed fit; a choice which implicitly imposes a Gaussian ‘noise model’ that does not correspond to counting discrete events and approximates experimental reality only when the number of photons counted is large (where Gaussian statistics approximate Poisson statistics). Different adaptive and re-binning schemes have been investigated as a means of counteracting the problem of histogram bins having either zero or very low photon counts and have been found to be effective in improving the accuracy of LS estimates [

The maximum-likelihood (ML) approach to direct fitting [

Global analysis algorithms have been applied to time-domain FLIM [

Of course, in order to obtain accurate decay parameter estimates it is necessary that the decay underlying the data be faithfully represented by the analysis model; the analysis of bi-exponential decay data with a mono-exponential model may not offer sufficient insight into the underlying fluorescence or interaction process. Similarly any significant peculiarities of the data acquisition process should also be modelled and although the influence of repetitive excitation is often acknowledged, it has only been included formally in the analysis in a small number of publications [

A number of time domain FLIM analysis methods which are not based on direct histogram fitting have also been used. A phasor analysis has been applied to time-domain FLIM [

In time-domain FLIM, a pulsed laser is often used to periodically excite the sample, causing fluorescence emission. Fluorescence decay photons are subsequently detected by a photon-counting detector; their arrival times (relative to the pulsed excitation) being recorded electronically with a high accuracy. Over time, a set of photon arrival times that represent the fluorescence emission accumulates; it is this set that form the acquired data in our analysis. In parameterizing the system for use with this analysis, a model that captures the relevant characteristics of the system as accurately as possible, and relates a particular photon arrival time to a set of fluorescence decay and other model parameters, is developed. In the interest of developing a model that remains generally applicable, those elements of the system that exert little influence on the behaviour of the typical system (and are easily controlled experimentally through appropriate configuration of the hardware) can be safely left out. Nevertheless, it should be remembered that emitting species located at different physical sites within the microscope’s point spread function (PSF) emit with exponential decay times characteristic of each type of site, and the resulting overall decay profile is a superposition of all these emissions. When the number of sites is sufficiently large, the system can be viewed as a continuum of environments, characterized by a multi-exponential profile. In practice the photon count is far too low to be able to separate these individual components, with very similar lifetimes and the best that can be used, in the context of FLIM performed on biological material, is an approximated single exponential decay. Super resolution methods e.g. [

Considering time-resolved FLIM in fairly simple terms, as illustrated in _{m}) and emits fluorescence photons, some of which traverse a path through the experimental apparatus before they are detected. It is neither desirable nor likely to be advantageous to attempt to model independently the influence of all of the individual optical components of the time-domain FLIM system; instead, these effects are modelled by an overall delay

An illustration of the key components of the FLIM system model having repetition period _{m} and a measurement window of duration _{m}/6 as a consequence of a sample being subjected to repetitive excitation at discrete times _{m} (_{m}]), and typical data for such a decay having about 1,000 total photons counted into 64 bins of equal width subdividing the measurement interval. The bottom half of the figure repeats the top for a mono-exponential decay of lifetime _{m}/2, and again shows data having about 1000 photon counts.

TCSPC involves determining the arrival time Δ

In the interest of readability, wherever possible only the key assumptions that guide the model development and significant results along the way are presented here; intermediate steps, technical and mathematical details can be found in Appendix A1 Mathematical Details.

In this analysis, the events that are analysed are photon arrival times that have been

_{m}); or could result from a photon emitted a number of repetition periods earlier than the most recent excitation pulse but delayed sufficiently as to be seen in the latest window (_{m})—the likelihood of such situations depends on the decay distribution _{m} than is Γ(

Arrival times are recorded only during a measurement interval of duration _{m}, and therefore Δ_{0} ∈ [0, 1] represents the contribution of a uniform background. The step function is denoted by _{m}) is given by

Observe that _{0}, the effects of repetitive excitation, an arbitrary instrument response Γ(

^{L}, ^{H}] ⊆ [0, ^{H} − ^{L} denotes the width of the interval ^{L}, ^{H}], and

A multi-exponential decay signal _{K} of permitted weight and lifetime values
_{1},…,_{K}_{1},…,_{K}_{k} weights the contribution of an exponential decay of lifetime _{k} to the overall signal, the fluorescence bin-likelihood ^{L}, ^{H}] due to a mono-exponential decay with lifetime

_{i} ∈ [0, 1] weights the contribution of a truncated Gaussian distribution of width _{i} ≥ 0, centered about a delay parameter _{i} ≥ 0 and having a lower cut-off _{i} ≥ 0. The set

The introduction of

The Bayesian framework is now applied to the system model developed above with the purpose of quantifying probabilistically the model parameters of some fluorescence decay data and also to provide answers to the often unasked questions regarding the nature of both the fluorescence decay and of the FLIM instrument. In particular, the following are explored:

What is the probability of a set of fluorescence decay model parameter values given the detected data, the decay model, and the parameterized instrument response approximation?

How many exponential decay components,

What is the most likely form of the IRF given the

The following notation is introduced and shall be used throughout the remainder of this document:

The data _{j}, _{j})|_{j}, _{j}) where _{j} represents the _{j} is the number of photons recorded as having been counted into that bin. It is required that none of the bins overlap (i.e. _{j} ∩ _{k} = ∅, ∀

The fluorescence decay model is denoted by _{k} contribute to the overall decay according to weight _{k} as defined by _{K}_{1},…,_{K}) and _{K}_{1},…,_{K}) shall also be used.

The characteristics of the FLIM equipment are denoted by

The Bayesian analysis developed in this section intentionally does not explicitly incorporate any particular form for the required prior distributions, in order that the expressions developed remain general and can be used as a starting point and be suitably updated on making specific the choice of prior distribution.

_{K}_{K}_{K}

Of course, as any parameter estimates, (_{K}_{K}_{K}

_{K}_{K}

Alternatively, the most probable decay model and its hyperparameters conditioned on the data can be sought by finding the maximum of _{K}

Of course, should the data be acquired using a fluorophore known to exhibit a purely

Additionally, it is also possible to estimate the instrument response parameters at the same time as estimating the fluorescence decay parameters. Again, under the assumption of a purely

The Bayesian analysis algorithms were implemented in the C programming language and incorporated into the TRI2 (Time Resolved Imaging 2) image processing software package [

The optimal mono- and bi-exponential parameter estimates are those which maximise the posterior distribution as given by

A Gaussian approximation, as developed in e.g. [_{K}_{K}_{K}_{K}_{K}_{K}

Determination of the optimal IRF approximation has been implemented using a simulated annealing algorithm [

The performance of the Bayesian algorithms was compared to that of the direct-fitting approach using ML and LS for the analysis of synthetic data simulating various mono- and bi-exponential decay conditions and a typical TCSPC system (see

The Bayesian mono- and bi-exponential decay analysis algorithms were tested against ML and LS, all approaches operating directly on the accumulated histogram. The ML estimation routines were implemented as described in [

The Bayesian-determined optimal single Gaussian IRF approximation, having a FWHM width of 0.129 ns (i.e. a standard deviation of 0.055 ns) centered about a delay of 2.067 ns, was determined from a single high-count data IRF measurement (about 5 million photon counts) and was used for the analysis of synthetic data presented below.

The mono-exponential parameter estimates obtained with this model are in close agreement with those reported in [

The enhanced model described in this paper offers improvements over the model of [

In bi-exponential decay analysis, the lifetimes, _{1} and _{2}, and initial amplitudes, _{1} and _{2}, of the two decay components are estimated from the fluorescence decay data.

A comparison of the performance of the Bayesian bi-exponential algorithm with LS and ML is shown in

The uncertainty, as measured by the standard deviation of the estimated parameter distribution, in the bi-exponential decay parameter estimates obtained using ML, LS, and Bayesian analysis for the analysis of synthetic data simulating a bi-exponential decay (^{3} and 10^{4} photon counts. In (a) and (c) the fractional errors in the estimated decay lifetimes versus photon count; in (b) and (d) the fractional errors in the initial amplitudes of the two decay components. In all cases, the normalised width is displayed only when the respective estimates are not biased by more than 5% of the true value.

The Bayesian bi-exponential algorithm provides parameter estimates to a greater precision, as measured by the standard deviation of the estimated parameter distributions, than is achieved using ML or LS analysis, as is evident on visual inspection of ^{3} and 10^{4} photon counts. At an intensity of about 5×10^{3} photon counts, the lifetime of the slower decay component, _{1}, is estimated to a precision of about 4.2% by our Bayesian algorithm; at the same intensity ML and LS achieve a precision of about 5.2% and 6.1% respectively. Indeed, to achieve the precision that is offered by the Bayesian algorithm at 5×10^{3} photon counts, ML requires an intensity of almost 6.5×10^{3} photon counts and LS requires about 9×10^{3} photon counts. A similar improvement in precision is achieved for the estimates of the corresponding initial amplitude, _{1}, at 5×10^{3} photon counts; the Bayesian algorithm offers estimates within a precision of 8.9%, whereas ML and LS achieve a precision of 10.3% and 11.7% respectively. The precision of the estimated lifetime and initial amplitude of the faster decay component is expected to be inferior to that of the slower component, as for such a decay the slower component contributes roughly four times as many of the counted photons to the intensity than the slower decay component. However, it is for the characterisation of the faster decay component that the Bayesian algorithm provides greatest advantage over ML and LS. The faster decay lifetime, _{2}, is estimated to a precision of 16.4% by the Bayesian algorithm at 5×10^{3} photon counts; ML and LS do not offer such precision below intensities of about 8.5×10^{3} and 9.5×10^{3} photon counts respectively. The initial amplitude of the faster decay component, _{2}, is also estimated with greater precision by the Bayesian algorithm; at 5×10^{3} photon counts the Bayesian algorithm achieves a precision of 10.0%, as compared to a precision of 13.5% offered by ML and 13.8% offered by LS analysis.

In a biological example where levels of protein-protein interactions can be determined by FRET (via FLIM), we can demonstrate the use of the bi-exponential model. We are thus making the assumptions that the donor has a single dominant decay path, resulting in mono-exponential kinetics, and that the target proteins (for example, HER2 and HER3 for the cell pellet data (_{1} and _{2} and the initial amplitudes _{1} and _{2} respectively, as follows,

FRET efficiency: _{2}/_{1}, _{1} > _{2},

Interacting fraction: _{2} = _{2}/(_{1} + _{2}).

The FRET efficiency _{2} provides a measure of the proportion of molecules undergoing FRET.

Of course, in such experiments, superior bi-exponential parameter estimates lead to more precise estimates of derived quantities such as the FRET efficiency and interacting fraction. Indeed, the FRET efficiency for the bi-exponential decay data analysed for ^{3} photon counts; the ML and LS estimates resulted in a precision of 6.2% and 6.3% respectively. Similarly, a precision of 8.2% was achieved for the interacting fraction estimated from the Bayesian bi-exponential initial amplitudes; ML and LS offered a precision of 9.5% and 10.3% respectively.

A comparison of the performance of the Bayesian bi-exponential algorithm with LS and ML is shown in

The uncertainty, as measured by the standard deviation of the estimated parameter distribution, in the bi-exponential decay parameter estimates obtained using ML, LS, and Bayesian analysis for the analysis of synthetic data simulating a bi-exponential decay. In (a) and (d) are the fractional errors in FRET efficiency and interacting fraction respectively versus photon count, in (b) and (e) the same for different FRET efficiencies ^{⋆}, and in (c) and (f) for different interacting fractions

The fractional errors in FRET efficiency and interacting fraction versus photon count (^{⋆} = 0.75 (i.e. lifetimes of ^{3} and 10^{4} photon counts. The improvement in the estimated FRET efficiency offered by Bayesian analysis is a consequence of superior lifetime estimates compared to those from ML and LS analysis. At an intensity of about 5×10^{3} photon counts the FRET efficiency derived from Bayesian lifetime estimates has an uncertainty of about 4.7%, whereas the estimates due to ML and LS analysis have an uncertainty of about 6.2% and 6.4% respectively. Bayesian analysis also offers improved estimates of the interacting fraction; at an intensity of about 5×10^{3} photon counts the interacting fraction determined from the Bayesian bi-exponential parameter estimates has an uncertainty of about 8.3% as compared to about 9.6% and 10.3% for ML and LS respectively.

The performance for different FRET efficiencies is shown in (^{4} photon counts. Bayesian analysis provides a slight improvement in the estimation of the FRET efficiency and the interacting fraction and could therefore offer more precise estimates at a given FRET efficiency or estimates within a given precision for an increased range of FRET efficiencies. However, it is evident that at such an intensity the uncertainty in the estimated FRET efficiency exceeds 10% and increases rapidly for all of the analysis techniques for actual FRET efficiencies of less than about 0.55 and that an equally rapid deterioration in the estimated interacting fraction occurs at FRET efficiencies lower than about 0.65. The estimated interacting fraction demonstrates a bias of greater than 5% for FRET efficiencies greater than about 0.8 for ML and LS analysis, and at about 0.9 for Bayesian analysis (data where the bias is greater than 5% has not been plotted).

The sensitivity to different interacting fractions is shown in (^{⋆} = 0.75 (i.e. ^{4} total counts. Bayesian analysis offers a modest improvement in precision over ML and LS analysis in this instance.

Of course, some bi-exponential decays are more amenable to accurate analysis than others; it would be reasonable to expect it to be more difficult to resolve the two decay components if they have similar lifetimes (i.e. a low FRET efficiency) and impossible at the point where they are equal (FRET efficiency equal to zero) or if one of the components dominates the decay (i.e. a very high or very low interacting fraction). It is these competing effects that give rise to the minima in (

Bayesian analysis offers superior estimation of biologically relevant quantities in many situations such as the FRET efficiency and interacting fraction in comparison to ML and LS analysis. Although the improvement in precision using Bayesian analysis is modest compared to ML and LS analysis, in some cases this improvement does mean that fewer total photon counts are required for analysis at a given precision or that a wider range of FRET efficiency or interacting fractions can be studied. It may be that a modest improvement in the precision of the estimated FRET efficiency could be the difference between observing a statistically significant difference in an experiment in which only a small change in the FRET efficiency is expected. It is also worth noting that Bayesian analysis in this form is very rarely worse than the other methods, and this is true at higher photon counts, so the only possible penalty from exclusively employing the Bayesian analysis is increased analysis time.

Typically, a combination of factors influence which decay model is chosen to analyse the experimental data. In choosing the decay model the expectation of what form the decay is

The Bayesian decay model selection analysis algorithm (

The results of model selection are shown in ^{2} model selection of [

In (a) an intensity image having pixels with intensity of about 750 photon counts, those on the left half of the image simulating a mono-exponential decay and those on the right half of the image simulating a bi-exponential decay; in (b) and (c) the Bayesian determined probability of the decay model being bi-exponential, ^{2} model selection algorithm of [

The Bayesian model selection algorithm is able to successfully distinguish between mono-exponential and bi-exponential decay data at very low intensities of about 750 photon counts, as shown in (^{2} based model selection algorithm predicted only 89% of the mono-exponential decays correctly, and the performance on bi-exponential data is visibly poorer.

The results of application of Bayesian and ^{2}-ML decay model selection to the GFP-expressing human carcinoma cell data (see ^{2}-ML decay model selection algorithm, as evident on inspection of (^{2}-ML decay model selection may be biased towards selection of the bi-exponential decay model as a consequence of poor decay estimates compensating for an underestimation of the background level by ML analysis at these pixels.

In (

It is important to appreciate that in interpreting the results of Bayesian model selection, inferences can only be made concerning those models that are present in the ensemble. The algorithms as applied in this work operate over an ensemble consisting only of mono- and bi-exponential decays; and it is therefore not possible to make any inference as to whether the data may actually be more likely to be due to, for example, a tri-exponential or even some non-exponential decay process. However, the Bayesian algorithm could be extended to offer the relative likelihoods of mono-, bi-, and tri-exponential decays. Similarly, a background noise only model could be incorporated to permit determination of the presence or absence of a decay signal in some data, along the lines of the method realised in [

The Bayesian IRF determination algorithm has been developed according to

Typically, the Bayesian IRF determination algorithm will be run over a data set having very high photon counts in order to estimate the optimal IRF approximation to be used in subsequent decay analysis; in this instance, the decay parameter estimates would be of little interest. An alternative application of the Bayesian IRF determination algorithm, however, could be to provide

The effectiveness of the Bayesian IRF determination algorithm to provide accurate decay and IRF approximation parameter estimates simultaneously is demonstrated in

In (a) the measured and optimal single- and double-Gaussian IRF approximations as determined on application of the Bayesian SID algorithm to a single data set having over 10^{7} counts obtained by binning the data from all image pixels from a single image of the human carcinoma cell data (see

The optimal single- and double-Gaussian IRF approximations for these data are shown along with the measured IRF in (^{7} counts, obtained by binning the data from all image pixels from an image having intensities between about 350 and 3500 counts. The optimal single-Gaussian IRF approximation is defined by the estimates

The single-Gaussian IRF approximation width and delay estimates obtained on independent application of the Bayesian SID algorithm, under the assumption of a mono-exponential decay, to each pixel are shown in (

The distribution of the delay estimates has an average of 2.342 ns and a standard deviation of 0.019 ns, and the width parameter estimates are distributed around an average value of 0.086 ns with a standard deviation of 0.022 ns; both are in close agreement with the optimal values determined using the high count data set. The mono-exponential lifetime estimates obtained on application of the Bayesian SID are inferior to those obtained on applying Bayesian decay analysis with the optimal IRF approximation, as might be expected given that the IRF parameters are additionally estimated from the same data. However, for such decay data and for such an instrument for which a single Gaussian serves as a reasonable approximation of the IRF, it is noteworthy that the lifetime estimates are only slightly degraded in comparison to those obtained from Bayesian decay analysis alone using the optimal approximated IRF, so there remains a significant improvement over the estimates offered by ML (which uses the experimentally measured IRF). The Bayesian SID lifetime distribution is centered about an average of 2.18 ns with standard deviation 0.17 ns. The estimates derived from Bayesian decay analysis using the optimal IRF approximation have an average of 2.16 ns and a standard deviation of 0.15 ns. The ML estimates (lifetime image not shown) were centered about an average lifetime of 2.17 ns with standard deviation 0.25 ns.

In this paper we have presented the novel analysis of exponential fluorescence decay data using Baysian inference that is based on the concept that each single photon carries some evidence about the photo-physical system from which it was emitted. In particular, algorithms that relate to the analysis of time-domain FLIM data from microscopes have been presented which may be used in the microscopical studies of cellular protein interactions employing FRET. We have extended our previous work on mono-exponential analysis [

complex decay modelling, demonstrated here by bi-exponential analysis

simultaneous estimation of additional system parameters such as the IRF

decay model selection

rigorous treatment of repetitive pulse excitation

complex IRF modelling with multiple Gaussians

We have shown that this novel analysis performs better than the two most popular methods of ‘direct fitting’, LS and ML, in terms of accuracy and bias, and offers a distinct advantage when photon counts are low and the TCSPC-accumulated histogram is not a good representation of the underlying fluorescence decays. The greatest improvement over previous techniques was seen when performing mono-exponential analysis, where a factor of two improvement in accuracy was observed. Since time-domain FLIM using TCSPC is a relatively slow method of acquisition, this factor of two improvement can be exploited to image significantly faster for the same accuracy as previously. This may allow time-lapse experiments where imaging speed is important. The improvement due to the new algorithms when performing bi-exponential analysis was less impressive, but still offers a significantly better accuracy in determined parameters, extending the range of FRET parameters that can be accommodated and therefore the biological protein interactions that can be studied.

The analysis has also been extended to include model selection to infer robustly whether underlying photo-physical system(s) have a single fluorescence lifetime or would be better represented by a bi-exponential model. This type of analysis is not common in the FLIM field and as well as indicating which type of model to use, it can also be used to indicate to the user whether there data is of sufficient quality (e.g. contains sufficient photon counts) to provide robust parameter estimates of a high-order model; a common pitfall for the novice FLIM user.

We also presented the first published simultaneous estimate of the IRF and decay parameters based only on the assumption of the IRF being approximated by a mixture of Gaussian distributions. This represents a step forward in determining dynamically the response of the instrument as well as the sample. Although it is not impossible to obtain a reasonable estimate of the IRF for many FLIM systems experimentally, obtaining parameter estimates algorithmically has advantages. One is that it enables us to use analytical expressions that aid implementation and execution speed. A second and more generally applicable advantage is relevant to the use of other types of fluorescence lifetime equipment (or other ‘time-of-flight’ systems [

All the algorithms are available for use in the TRI2 image processing program. This is currently available as compiled ‘freeware’ for Windows from the Oxford Group’s web site (

The development of the signal photon-likelihood, from the integral form defined by _{i,j} is the Kronecker delta function which exists only when _{m} (i.e. Δ_{m}), consequently _{ℓ,⌊ℓ+Δt/Tm⌋} = _{ℓ,ℓ} = 1 and the integral simplifies further to give

Since the instrument response approximation Γ(

Incorporating the discrete-time nature of these time-domain FLIM data into our model to give the photon and signal photon likelihood in discrete time, ^{L}, ^{H}]. The introduction of a multi-exponential fluorescence decay signal of the form defined by ^{L}, ^{H}] due to a mono-exponential decay with lifetime

The introduction of an approximation to the IRF,

Determining now the convolution of a component of the multi-exponential decay signal, having a decay lifetime

Notice that the term _{m} + Δ_{i}] ensures that the integral is positive. The developed expression describes (without normalisation) the likelihood of a fluorescence decay photon at time Δ

Observe that the term _{m} + Δ_{i}] ensures that the fluorescence photon likelihood is zero until the decay time _{m} + Δ_{i}. In determining the remaining integral, which accounts for the discrete time nature of TCSPC data, it is convenient to incorporate the bin boundaries directly into the developed expression, to give

It is evident that if the time bin lies entirely before the cutoff there is no likelihood of a fluorescence decay photon being counted into it, the likelihood of photon being counted into a time bin which straddles the cutoff is determined by integrating between the cutoff and the upper bin boundary, and if the time bin is entirely beyond the cutoff then the likelihood of a photon arrival time in the bin is determined by integrating between the bin boundary values. These conditions are encapsulated by the following expression,

The remaining integral is determined analytically, by parts, giving

Defining the quantity,

Consequently, the fluorescence decay bin-likelihood due to the _{i}^{L}^{H}_{m}_{i}_{i}_{i}

Incorporating this into the model yields the following expression for the bin-likelihood (which includes the contribution of both background and a fluorescence decay signal):

Synthetic data were generated to simulate a variety of mono- and bi-exponential decay conditions; all generated data reflected a modelled TCSPC system having a repetition rate of 40 MHz, a measurement interval of 20.0 ns partitioned into 256 bins of equal width, the effects of a Gaussian IRF having a FWHM width of 0.15 ns, and incorporated Poisson noise at each bin.

Human epithelial

Cell pellet preparations offer the opportunity to have the controllability of a cell experiment but in a medium that is similar to a tissue section. MCF7 cells (ATTC, UK) were cultured in DMEM supplemented with 10% FCS. Cells were transfected with HER2-GFP and HER3-GFP plasmids at ratio 3:1 using FuGene6 (Promega) according to the manufacturer’s protocol, and cultured for 24 hours. Cells were fixed with 10% formalin for 5 hours and then processed overnight on Leica ASP300S and embedded into paraffin on Leica EG11508C. For staining, cell pellets or tissue were cut (3

FLIM data was obtained from an ongoing study of protein interactions in human tissue by our collaborators. Tissue microarrays were created from 218 primary breast cancers from patients included in the METABRIC (Molecular Taxonomy of Breast Cancer International Consortium) study [

Time-domain FLIM was performed with an in-house Open Microscope system [

To avoid pulse pile-up, peak photon counting rates were adjusted to be well below the maximum counting rate offered by the TCSPC electronics [^{4} − 10^{5} photons/second. The photon arrival times, with respect to the 40 MHz repetitive laser pulses, were binned into 256 time windows over a total measurement period of 15 ns. Images were captured with a 0.75 NA objective lens (S Fluor 20x/0.75 air, Nikon, UK) at 256 × 256 pixels corresponding to 334 × 334

IRF measurement was performed by replacing the sample with an Aluminium-coated reflective slide and removing the emission filter such that reflected excitation light reaches the detector. In order to replicate experimental conditions in terms of laser power and photon detection rate, neutral density filters (ND10A and ND40A, Thor Labs, UK) were placed in the excitation light path.

We thank Prof T. Ng and Dr G. Weitsman for cell and tissue samples and other supporting staff at the Randall Division of Cell and Molecular Biophysics, King’s College London and at the Cancer Research UK and Medical Research Council Oxford Institute for Radiation Oncology, University of Oxford.