Optimal parameters in variable‐velocity scanning luminescence lifetime microscopy

Abstract We determine the optimal parameters (scan velocities) for measuring the luminescence lifetime on the microsecond scale using the recently introduced method based on scanning the excitation beam across the sample. Using simulations, we evaluate the standard deviation and bias of the luminescence decay rate determined by scanning with two different velocities. The analysis is performed for Poisson‐ and normal‐distributed signals, representing different types of detection techniques. We also show that a weak uncorrected background induces a bias in the obtained decay rate, and take this effect into account when choosing optimal measurement parameters. For comparison, the analysis is additionally performed for two conventional gating schemes for lifetime measurement. The variable‐velocity scanning method is found to be more robust to the effect of the background signal than the gating schemes.


| INTRODUCTION
The luminescence lifetime is an important parameter often employed for sensing the environment and the interactions of the emitting molecules, or simply for distinguishing between dyes of similar spectral properties (Lakowicz, 2006;Meyer-Almes, 2017;Suhling, French, & Phillips, 2005;Valeur, 2001). The independence of lifetime on the excitation intensity and on the dye concentration within a broad experimental range is a major advantage over measurements based only on emission intensity, and a reason for the many applications of time-resolved fluorescence (Bolivar, Consolati, Mayr, & Nidetzky, 2013;Gruber, Marques, Szita, & Mayr, 2017;Wang & Wolfbeis, 2014). However, the experimental determination of luminescence lifetime typically requires complex equipment, involving pulsed excitation and time-resolved detection. Furthermore, analytics based on lifetime generally necessitates a higher signal-to-noise ratio than methods using solely emission intensity. This is especially relevant in microscopy applications, where the single volumes within which the measurements are performed (pixels, voxels) are small, and the signals are limited. The particular choice of experimental parameters can, therefore, have profound effects on the quality of the results, quantified by the standard deviation and bias of the determined lifetime or decay rate.
We have recently introduced an alternative method for the measurement of luminescence lifetimes on the microsecond scale based on scanning the beam across the sample with different velocities and simultaneously detecting the emitted signal (Petrášek, Bolivar, & Nidetzky, 2016 (Petrášek et al., 2016). This time is determined predominantly by the luminescence decay rate. The scan velocity of the excitation beam determines the duration of excitation at any point within the sample. At fast scan speeds, resulting in an excitation period that is short compared to the luminescence lifetime, the excited state will not have reached its steady-state population and the emitted signal will be relatively low. At slower scan speeds, where the population of the excited state gets closer to the steady-state, the emitted signal will be on average higher (Figure 1). The luminescence lifetime can be determined from the intensities measured at least two different scan speeds.
We have previously shown that the dependence on the scan velocity v of the luminescence intensity, f, can be expressed as (Petrášek et al., 2016): where k is the decay rate (the inverse of the luminescence lifetime τ: τ = 1/k), and α is a factor nominally equal to the linear size of the illuminated area, typically a focused laser beam in a confocal laser scanning microscope. In practice, α is determined by calibration with a dye of a known luminescence lifetime. The emission intensity increases with decreasing scan velocity, and in the limit of very low velocity approaches the steady-state value f 0 .
When scanning with velocity v, any position within the sample is illuminated for time t = α/v. Employing two different scan speeds gives two illumination times t 1 and t 2 . In the Results section, we determine the optimal times t 1 and t 2 , from which the scan velocities v 1 and v 2 can be calculated ( Figure 2).
In order to obtain the signals F 1 and F 2 , the luminescence intensity is integrated for times t 1 0 and t 2 0 . In a confocal laser scanning microscope, these times correspond to the time per pixel where d is the pixel size. It follows from Equation (1) (substituting The integration times t 1 0 and t 2 0 are generally different from the illumination times t 1 and t 2 , in the same way as the size d of an image pixel is usually not the same as the size α of the focused laser spot in the sample. Substituting t i 0 = (d/α)t i in Equation (2) introduces a constant factor d/α, which, however, cancels out in the calculation of k because only the ratios of intensities F i are used, as explained in detail The principle of the variable-velocity scanning method: the detection area of the size α where the molecules are excited moves across the sample (dimension x). The detected signal F i (integrated over the area α) depends on the scan velocity v i ; this dependence is used to determine the luminescence lifetime. The vertical axis represents the emission intensity below. Therefore, without a loss of generality, we will assume in the following that these times are the same: t 1 0 = t 1 , t 2 0 = t 2 . The two equations above (Equation (2)) can then be used to calculate the decay rate k from the measured signals F 1 and F 2 and the known times t 1 and t 2 . In the following text, this measurement method will be denoted as the Scheme S ("scanning").

| Conventional gating schemes
We compare the scanning method (Scheme S) with the commonly employed time-gating method, which relies on pulsed excitation and time-resolved detection of the subsequent light emission in several time windows (Suhling et al., 2005;Webb et al., 2002).
Two gating schemes are considered here, each with two-time channels ( Figure 2). In both cases, the first gate starts at time zero, immediately after the pulsed excitation. In the scheme denoted G1 ('gating') the second gate starts immediately after the end of the first gate; the gates generally have different widths: t 1 and t 2 − t 1 . In the second scheme, denoted G2, the second gate also starts at time zero, so both gates are fully overlapped, and the unequal gate widths in this case are t 1 and t 2 .
Assuming a mono-exponential luminescence decay f(t): the intensities F 1 and F 2 detected in the two-time windows in the G1 scheme are: and in the G2 scheme: For both distributions the maximum-likelihood criterion (Bajzer, Therneau, Sharp, & Prendergast, 1991;Tellinghuisen & Wilkerson, 1993) for parameter estimation leads to the following equations, from which the unknown parameters f 0 and k can be determined: Since we are interested only in the decay rate k, it is practical to calculate the following ratio r from the experimental data: Using Equation (6) for all three measurement schemes, we obtain the equation, from which the decay rate k can be calculated numerically: the Scheme S: F I G U R E 2 The three measurement schemes analyzed in this work. S: the scanning method employing two different scan velocities; G1, G2: time-gating methods using pulsed excitation at time t = 0 and detection in twotime windows. In the Scheme G1, the second time window begins immediately after the first one (no overlap); in the Scheme G2 both windows start at the time t = 0 (full overlap) the Scheme G1: and the Scheme G2: The signal strength in the three measurement schemes is expressed by the parameter f 0 (Equations (1) and (3)). It is convenient to substitute f 0 = nk in the equations defining F 1 and F 2 (Equations (2), (4), and (5)), where the new parameter n replaces f 0 , and can be interpreted as the number of photon counts if photon-counting detection is used, or as a time-integrated signal in general. In Scheme S, the parameter n represents the signal detected by integration over the time t 0 = 1/k equal to the luminescence lifetime in the steady-state regime (kt i ) 1, Equation (2)). In the Schemes G1 and G2, n is the total signal detected after one excitation pulse.
The numerical values of the times t 1 and t 2 in the simulation results, in figures, and in-text are given in relative units of 1/k.

| The effect of the background
The presence of a constant background signal influences the measured decay rate k. If the background is not taken into account during the analysis, it biases the calculated value of k. Even if included in the fitting model as an unknown parameter, the background affects the precision with which k is determined, and also influences the optimal experimental parameters (t 1 , t 2 ). It has been shown that even if the correct background is subtracted, the noise associated with the background negatively influences the precision with which the decay rate can be determined (Heeg, 2013;Köllner & Wolfrum, 1992).
The background effect has been described for the gating Schemes (G1) several times in the past (Ballew & Demas, 1991;Heeg, 2013Heeg, , 2014Köllner & Wolfrum, 1992;Moore et al., 2004). The best practice is to determine the background independently, and include it in the analysis as a known parameter. This is, however, not always possible or cannot be done exactly. Consequently, a small contribution of the background still remains in the signal. Here, we analyze the bias in k when the background is not corrected for, or when the residual background remains, and show that the impact of background differs widely among the considered measurement schemes. Even at low background levels, its effect has to be taken into account when choosing t 1 and t 2 .
The background intensity B is assumed to be constant in time, and we express it relatively to the luminescence amplitude f 0 in Equations (1) and (3) for the Scheme S (where F 1 and F 2 are taken from Equations (2)), and: for the Scheme G1 (where F 1 and F 2 are taken from Equations (4)).
The bias of k as a result of background in the measurement Scheme G2 is the same as in the Scheme G1, therefore, it is not shown it explicitly here.
In order to calculate the biased decay rate k 0 , the ratio r 0 is calculated from the equations above (Equations (12) and (14)), substituted to the equation for unbiased r (r = r 0 ) without any background (Equations (8) and (9)), and subsequently the biased k 0 is calculated from r.

| METHODS
For all the measurement schemes ( Figure 2) the noisy experimental signals N 1 and N 2 were simulated (Monte Carlo) and analyzed to obtain the decay rate k, in the following way. The signals N 1 and N 2 were generated as random numbers with Poisson or normal (Gaussian) distribution, with the mean given by F 1 and F 2 in Equations (2), (4), and (5), and the variance (normal distribution) equal to nΔt, where Δt is the width of the time window (Δt = t 1 , Δt = t 2 or Δt = t 2 − t 1 depending on the measurement scheme; see Figure 2). Then, the ratio r (Equation (7)) was calculated from N 1 and N 2 , and subsequently the estimate of the decay rate k was calculated from r by numerically inverting Equations (8), (9), or (10), depending on the measurement scheme.
The simulations were performed for ten different signal levels n, ranging from 100 to 10 5 . The simulation was repeated 10,000 times for every set of parameters, the obtained rates k were averaged and their standard deviation σ k and bias relative to the true value k 0 were calculated. All simulations were done in Matlab (The MathWorks, Natick, MA).
The standard deviation σ k of the decay rate k was also estimated analytically using the standard error propagation method. The σ k is related to the standard deviation σ Fi of the signals F i as follows: The derivatives in Equation (15) were determined from Equations (6) to (10). The analytical calculations of error propagation were done in Mathematica (Wolfram Research Inc., Champaign, IL).

| The standard deviation of the decay rate
In order to determine the optimal time windows t 1 and t 2 for all three measurement schemes, we simulated the measurement signals N 1 and N 2 for a range of values of t 1 and t 2 and calculated the decay rate k using Equations (8)-(10). The presence of noise, the level of which is determined by the parameter n, means that the calculated rate k was distributed around the true value k 0 . By repeating the simulations many times, we could evaluate the mean value of k, its bias from the true value k 0 , and the standard deviation σ k .
The standard deviation σ k of the decay rate k depends on the number of counts n as follows ( Figure 3): where the parameter σ * differs among the considered measurement schemes, and is a measure of the precision of the method. The signal strength n was found to have no influence on σ * , meaning that the optimal parameters determined below are independent of the signal intensity.
The position of the minimum of σ * indicates the optimal times t 1 and t 2 (Figure 4). The value of σ * for the scanning method (Scheme S) decreases steadily with increasing time t 2 , however, from t 2 $ 8, this decrease is not significant. The optimal time t 1 then lies between 2 and 2.5 for the Poisson noise, depending on the particular value of t 2 , but with a rather weak sensitivity to its exact value. The situation is very similar in the case of the Gaussian noise, with the optimal value of t 1 between 2 and 3.2.
Similarly, in the two gating Schemes G1 and G2 with Poisson noise the relative standard deviation σ * steadily decreases with t 2 , with little significant decrease above t 2 = 4. The optimal value of t 1 is different for each scheme: it lies between 1.3 and 1.6 for the G1 scheme, and is approximately equal to 0.5 for the G2 scheme.
Contrary to the other cases, the gating Schemes G1 and G2 with normally-distributed signals exhibit a well-defined minimum of σ * : at t 2 = 3.80, t 1 = 0.44 for the G1 scheme, and at t 2 = 4.46, t 1 = 0.31 for the G2 scheme. The existence of the minimum can be understood by realizing that extending the gate width beyond the times when the luminescence has decayed only adds more noise but no signal, thus decreasing the signal-to-noise ratio.
For comparison with previously published results, we also looked for the optimum gate width in the Scheme G1 in the situation where the widths of both gates are equal (t 2 = 2t 1 ). This is equivalent to finding the minimum of σ * along the line t 2 = 2t 1 in the corresponding plot in Figure 4 (dash-dotted line). The optimal gate width in this situation is t 1 = 2.43, in agreement with previous results (Ballew & Demas, 1989;Chan et al., 2001;Köllner & Wolfrum, 1992).
The standard deviation obtained from simulations ( Figure 4) agrees with analytical calculations using standard error propagation method. The minimum of σ * (Equation (16) Table 1.
In addition to the standard deviation, the simulations allowed us to evaluate the bias of the calculated decay rate k. For this purpose, we define the relative bias Δ r k as a difference between the calculated (k) and true (k 0 ) decay rate relative to the true decay rate: The bias was found to decrease with the number of counts n in the following way ( Figure 3): The bias parameter γ (Equation (18)) was determined from simulations with a range of n at the position of the minimum of standard deviation within the range of tested times t 1 and t 2 , and is listed in Table 1.
The bias turns out to be positive and rather small compared to the expected error expressed by the standard deviation (Figure 3).
Even at the smallest considered signal n (n = 100), where the bias is strongest, it is about one order of magnitude smaller than the standard deviation. This means that this bias can be usually ignored, or, at F I G U R E 3 The dependence of the relative standard deviation σ k / k 0 and the relative bias Δ r k on the signal strength n as determined from simulations. The plotted values are taken at t 1 and t 2 , for which the minimum of σ k was found within the tested range (Figure 4). The results are shown for the three measurement Schemes (S, G1, and G2) with either Poisson (p) or normal-distributed (n) signal the weakest signal levels on the order of 100 counts per decay, corrected for by using the curves in Figure 3. Table 1 summarizes the position (t 1 and t 2 ) of the minimum of the standard deviation σ * for the considered measurement schemes, together with the standard deviation (σ * ) and bias (γ) parameters at this minimum. As mentioned above, in the absence of a localized minimum for finite t 2 , there is a value of t 2 , beyond which the standard deviation does not significantly decrease, and further extension of t 2 does not make a practical difference. These results, however, apply to an idealized situation, and as we show in the next section, the presence of even a small background influences the choice of the optimal t 1 and t 2 in a more realistic setting.
The value of the parameter σ * in Table 1, together with Equation (16) can be used to estimate the signal needed to reach a required measurement error. For example, if a relative error of 10% is tolerable (σ k /k 0 = 0.1), the signal n detectable over the lifetime 1/k should be at

| The effect of uncorrected background
When the detected signal contains a constant background in addition to the luminescence, failure to fully account for the background in analysis leads to an additional bias in the measured luminescence decay rate.
The sensitivity of the relative bias Δ r k (Equation (17)) to the background depends on the chosen values of t 1 , t 2 . Figure 5a shows how the bias depends on the background level for the two measurement Schemes S and G1, with t 1 , t 2 chosen so that the standard deviation of k is near its minimum for each measurement scheme. The presence of background leads to overestimated values of k in the measurement Scheme S (positive bias), and to lower k (longer lifetime) in the measurement Scheme G1 (negative bias).
Since we are interested in the effects of a weak background (small b), we can approximate the bias of the rate k for a low background b by a linear function with a slope β: The positions and values of the minima of normalized standard deviation and bias of the decay rate k expressed by σ * (Equation (16)) and γ (Equation (18) The dependence of the parameter σ * , describing the relative standard deviation σ k /k 0 (Equation (16) The parameter β is then a dimensionless number expressing the sensitivity of k to the background that has not been accounted for. A comparison of β values (the ratio of their absolute values) of the two schemes shows that, for the t 1 and t 2 values chosen in Figure 5a, the gating Scheme G1 is about 3.4× more sensitive to the low background than the Scheme S.
The bias of k, as expressed by the sensitivity to background β, depends on the times t 1 and t 2 . We have calculated β as explained above for the same range of t 1 and t 2 as the standard deviation σ * (Figure 4), and show its absolute value in Figure 5b,c. These results reveal a substantial difference in the sensitivity to the background between the two considered schemes.
In the Scheme S, β decreases with increasing t 1 and t 2 (Figure 5b).
This means that in order to minimize the background effects, it is favorable to choose a rather long time t 2 . This is the same conclusion as that reached by observing the dependence of the standard deviation σ * of the decay rate on t 2 ( Figure 4, Table 1). For t 2 larger than approximately eight, the decrease in σ * and β is not particularly strong, therefore, any value above eight is close to optimal.
Contrary to this, in the Scheme G1, jβj increases with increasing t 1 and t 2 (Figure 5c). Comparing this with the dependence of the standard deviation σ * on t 2 (Figure 4) means that a compromise has to be found, where t 2 is high enough to keep σ * small and at the same time sufficiently low to minimize the bias due to the background. The final choice of t 1 and t 2 will depend on the particular circumstances of the experiment: the tolerable standard deviation of k, the actual expected background level, and the maximum acceptable bias due to the background.
These observations allow us to make the following suggestions for the choice of the times t 1 and t 2 (all expressed in relative units of 1/k). The optimal t 2 for the Scheme S is any value larger than approximately eight, as there is no significant decrease of σ * beyond this value. The optimal t 1 lies between 2 and 3.2, depending on the chosen t 2 , and on whether the noise character is closer to the Poisson or the normal distribution ( Figure 4, Table 1).
The optimal t 2 for the Schemes G1 and G2 will typically be around t 2 = 4, since higher t 2 leads to a minimal decrease of σ * but a considerable increase of bias due to a possible background. The presence of the background will eventually influence the choice of t 2 , as discussed above. The optimal t 1 is then determined by the Scheme (G1 or G2) and the type of noise, as shown in Table 1. It is minimally dependent on the chosen t 2 , with the exception of the G1 scheme with Poisson noise, where it varies between 1 and 1.6 depending on t 2 , as can be seen in Figure 4.

| CONCLUSIONS
The presented analysis allowed us to find the optimal experimental parameters t 1 and t 2 for the recently introduced variable-velocity scanning method for lifetime determination. The optimal parameter values were found to be independent of the signal intensity and were determined by the decay rate k. The range of optimal t 2 is relatively broad, with t 2 being sufficiently high so that the conditions are close to the steady-state regime. The value of t 1 is chosen accordingly, but the precise choice is not critical, as the standard variation of the decay rate is only weakly sensitive to t 1 variation around its minimum ( Figure 4). If a broad range of decay rates is measured, as may often be the case in lifetime imaging with a considerable spatial lifetime variation, the values at the lower end of the expected range of k (longer lifetimes) should be used to determine the optimal times t 1 and t 2 .
Importantly, in the scanning scheme, high t 2 assures both minimal standard deviation and minimal bias due to background, simplifying the choice of t 1 and t 2 .
(a) (b) (c) F I G U R E 5 A: the bias Δ r k of the decay rate k in dependence of the background level b for the two measurement schemes: S (solid line) and G1 (dashed line). The bias is calculated for the following choice of the times t 1 and t 2 : Scheme S: t 1 = 2.0, t 2 = 8.0; Scheme G1: t 1 = 1.3, t 2 = 4.0. The gray lines indicate the linear slopes in the limit of low background (b ! 0). B, C: The dependence of the absolute value of the bias slope jβj (Equation (19)) on the times t 1 and t 2 for the Schemes S (B) and G1 (C). The solid line indicates the minimum of the standard deviation of k for a given value of t 2 [Color figure can be viewed at wileyonlinelibrary.com] Contrary to the scanning Scheme S, compromise has to be sought for the gating Schemes G1 and G2 when considering both standard deviation and bias due to background: while the criterion of minimum standard deviation favors larger t 2 , minimizing the effects of background on the bias of k calls for smaller t 2 .
Furthermore, the sensitivity to the background-induced bias, quantified by jβj, is stronger for the gating Schemes G1, G2 than for S. The higher robustness of the scanning Scheme S compared to the gating schemes constitutes a clear advantage in practical applications.
In this work, we have considered measurements using two scan velocities, which is the minimum required to determine the luminescence lifetime. Employing more than two scan velocities is expected to expand the possibilities of the scanning method, similarly to using more gates in the time-gating approaches. More scan velocities would mean that a broader range of lifetimes could be optimally measured within one experiment. Furthermore, non-exponential kinetics could be detected, and more advanced analysis methods analogous to those used with conventional lifetime imaging, either on a pixel-by-pixel basis or as global image analysis, could be applied.