1. Introduction

Fluorescence lifetime imaging microscopy (FLIM) is an increasingly widely used tool for biological and medical applications. It provides information about specific fluorophores and their environment such as ion concentration, dissolved oxygen concentration, pH, and refractive index by measuring the fluorescence decay lifetime of the excited fluorophores [1–3]. FLIM becomes especially powerful when combined with multiphoton microscopy (MPM) [4–7] which provides 3D resolution, deep penetration, and minimal phototoxicity [8–10]. FLIM can be performed using time-domain methods such as time-correlated single photon counting [11] or frequency-domain (FD) methods as performed here. FD methods, specifically, are attractive for their rapid acquisition speed, easy implementation, and reduced system bandwidth requirements [1].

A high imaging sensitivity, or equivalently a sufficient signal-to-noise ratio (SNR), is essential in FD-FLIM in order to study kinetic processes and physiologically relevant structures in biological systems. Although the image SNR can be improved by increasing the excitation power, this solution can produce photodamage and fluorophore saturation [12]. Recently, we theoretically described and modeled the SNR performance of MPM-FD-FLIM systems under various conditions [13]. We showed theoretically that the MPM based FLIM required 50% fewer photons to achieve the same SNR as a one-photon FLIM system. More importantly, we theoretically proposed a method to increase the SNR of a conventional MPM-FD-FLIM by a factor of 2 without increasing the excitation power. Although the theory was verified by Monte Carlo simulations, this paper describes an experimental validation of these methods by comparing the measured lifetime sensitivity with the theoretical performance to validate the theory. A sensitivity comparison among the 1ω, 2ω, and DC&1ω methods is also performed in these experiments, confirming the sensitivity improvement enabled by the DC&1ω method. Furthermore, we demonstrate experimentally that the proposed super-sensitivity MPM-FD-FLIM using the DC&1ω method can produce unprecedented sensitivities over a wide lifetime range.

2. Methods

Conventional MPM-FD-FLIM calculates fluorophore lifetime from phase or magnitude measurements of an intensity modulated excitation and emission signal. Because of the nonlinearity of 2PE fluorescence, the emission is quadratically dependent on the excitation intensity; therefore, the second harmonic (2ω) of the modulation frequency is also present in the emission and can be employed to extract lifetime [14]. In this paper, we similarly analyze the nonlinear harmonics of MPM-FD-FLIM.

The system is modeled in the same way as in [13]. The fluorescence sample is excited by intensity-modulated excitation light, e(t), at an angular frequency of ω. The sample generates 2PE fluorescence, p(t). Based on the quadratic nature of the 2PE, p(t) is the convolution of e²(t) and f(t), where f(t) = exp(−t/τ)/τ is the impulse response of the fluorophore, τ is the lifetime, and f(t) is normalized such that its integral on the time domain (t ⩾ 0) is unity. We assume a mono-exponential fluorescence decay model, which is useful for many fluorophores; it can be extended to multi-exponential decay with multiple frequencies of excitation [15]. Because e(t), e²(t), and p(t) are all periodic signals with period T = 2π/ω, the system can be described with Fourier series. Define Fourier coefficients

Although a variety of waveforms can be used to modulate the excitation light, in our experimental validation, we will only consider sinusoidal modulation, i.e., e(t) = 1 + msin(ωt), where m is the degree of modulation 0 < m ⩽ 1. Therefore, we have a₀ = (m² + 2)/2, a₁ = −mi, a₂ = −m²/4, and correspondingly,

Though the three methods above are all capable of measuring lifetime, their measurement sensitivities are different [13]. To quantify the sensitivity (SNR), we use the photon economy (F-value) as a figure-of-merit. It is defined as the ratio of the uncertainty in lifetime (τ) measurement to the one in intensity (I) measurement with the same amount of detected photons. $F=({\sigma }_{\tau }/\tau )/({\sigma }_I/I)=\sqrt{N_{\mathit{det}}}({\sigma }_{\tau }/\tau )$, where σ_τ and σ_I are the standard deviations of the measured lifetime and intensity, respectively; N_det, the number of detected photons, is proportional to I, and we have used the relation ${\sigma }_I/I=\sqrt{N_{\mathit{det}}}/N_{\mathit{det}}$ based on the Poisson distribution of N_det [1]. Smaller F-values indicate improved sensitivity with a shot noise limited minimum of 1. We have analytically found minimum F-values for the conventional 1ω method, the 2ω method, and the proposed DC&1ω method [Eq. (3)] to be 2.62, 11.02, and 1.87, respectively in [13].

The MPM-FD-FLIM sensitivity theory validation experiment is depicted in Fig. 1. The intensity of a mode-locked Ti:sapphire laser (Spectra Physics Mai Tai BB, 800 nm, 100 fs, 80 MHz) was modulated by an electro-optic modulator (EOM) (Thorlabs EO-AM-NR-C1) controlled by a function generator. The excitation light was filtered through a longpass filter to block ambient light from entering the microscope. The excitation beam was expanded by a telescope (L1 and L2) to overfill the back aperture of an objective lens (Zeiss Plan Neofluar, 10×, 0.3 NA), which created a diffraction-limited spot inside a cuvette. We used [Ru(dpp)₃]²⁺ nanomicelle probe solution in deionized water as our sample in the cuvette according to the protocol described in [3] due to the easy preparation, large two-photon cross-section, and the ability to tune lifetime with dissolved oxygen. The 2PE fluorescence p(t) was epi-collected by the objective lens, reflected by a dichroic mirror, filtered through a set of bandpass and shortpass filters to eliminate residual excitation, and detected by a PMT (Hamamatsu H7422PA-40). The excitation e(t) was monitored by a photodetector (Thorlabs PM100D). Both excitation and emission signals were digitized by a data acquisition card (National Instruments PCI-6110). Lifetime calculations were performed in real time in LabView (National Instruments). The system phase offset was calibrated by measuring and subtracting the phase difference between e(t) and p(t) in a near zero lifetime sample (Rhodamine B, 1000× shorter in lifetime than [Ru(dpp)₃]²⁺), or alternatively a reflective surface; the low-frequency system response was calibrated by scaling the DC Fourier component such that the computed DC&1ω and 1ω lifetimes were consistent. Additionally, a photon counter (Stanford Research Systems SR400) was used to extract the photon number information from the PMT signal in order to calculate the experimental F-values.

 

Fig. 1 MPM-FD-FLIM experimental setup.

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It should be noted that while this technique can be applied generally, technical requirements need to be considered for short lifetime fluorophores where required modulation bandwidth exceeds that of an EOM (e.g., NADH, FAD). In such cases, an acousto-optic modulator (AOM) may be preferred. Additionally, when fluorophore lifetime < 2 ns, the Dirac pulse train of a mode-locked laser (f ≈ 80 MHz) provides an even superior modulation source [13].

3. Results

3.1. Experimental validation of the analytical SNR model

We used the setup in Fig. 1 to experimentally validate the analytical SNR result in [13] by finding the F-values of each method as a function of normalized modulation frequencies. The same excitation and emission signal dataset was analyzed using the 1ω, 2ω, and DC&1ω methods. Lifetime was determined from 10 ms of data, and 1000 measurements were obtained for each modulation frequency to obtain means (τ) and standard deviations (σ_τ) for each method. Experimental F-values $\left[F=\sqrt{N_{\mathit{det}}}({\sigma }_{\tau }/\tau )\right]$ were obtained from the means (τ), standard deviations (σ_τ), and photon counts during the integration time (N_det). The [Ru(dpp)₃]²⁺ nanomicelle sample was held at a lifetime of 1.6 μs by air saturating the solution and sealing the cuvette. While the modulation frequency varied, the degree of modulation m was kept constant at 0.75. Representative experimental results are shown in Fig. 2(a) for measurements at a modulation frequency of 131.25 kHz (= 0.21/τ). Calculated lifetimes and the corresponding histograms show that the DC&1ω method has the best sensitivity with a measurement standard deviation of 15 ns, which is clearly smaller than the 1ω method (24 ns) and significantly smaller than the 2ω method (850 ns). This result is in accordance with the analytical model in [13], where a 0.21/τ modulation produces a SNR relation that DC&1ω > 1ω >> 2ω.

 

Fig. 2 (a) Representative experimental lifetime temporal plots (left) and histograms (right) for the 1ω, 2ω, and DC&1ω methods. (b) Experimental (symbols) and analytical (curves, from [13]) values of F versus modulation frequency for the 1.6 μs [Ru(dpp)₃]²⁺ nanomi-celle sample. Inset: zoomed scale.

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Photon economy (F-value) versus modulation frequency is presented in Fig. 2(b), where the symbols and curves represent the experimental and analytical theoretical F-values, respectively. For the 1ω and DC&1ω methods, the experiment matches the theory extremely well, considering the noise sources other than shot noise are not included in the theoretical model but exist in real measurements. Although the 2ω method’s experimental results obviously depart from the theory due to the low SNR of 2PE fluorescence’s 2ω components, they show a very similar trend that both have a minimal F-value around a frequency of 0.05/τ. The results suggest that the analytical SNR model in [13] accurately describes MPM-FD-FLIM and serves as a guideline for experimentalists choosing optimal conditions for their MPM-FD-FLIM experiments. Additionally, the DC&1ω method is proven to be superior to the conventional methods when the modulation frequency is larger than 0.11/τ, and the advantage increases with frequency.

3.2. Sensitivity comparison among the 1ω, 2ω, and DC&1ω methods

From the experimental results in Fig. 2(b), we know that the modulation frequency of 0.11/τ is the optimal condition for the 1ω method (F = 3.62); 0.05/τ for the 2ω method (F = 38.69); and 0.17/τ for the DC&1ω method (F = 2.81). Thus, each method must be analyzed at its optimal performance condition for a fair comparison. For a [Ru(dpp)₃]²⁺ nanomicelle sample with lifetime τ = 1.6 μs, we therefore chose modulation frequencies of 0.11/τ = 68.75 kHz for the 1ω method, 0.05/τ = 31.25 kHz for the 2ω, and 0.17/τ = 106.25 kHz for the DC&1ω. For each of these three experiments, the lifetime measurement relative error (σ_τ/τ) was obtained from 1000 individual measurements as a function of integration time.

The symbols in Fig. 3(a) plot experimental results; the experimental data are fit to ${\sigma }_{\tau }/\tau \propto N_{\mathit{det}}^{-1/2}$ using the nonlinear least squares method. These data illustrate two important results. First, even when measurements are performed at each method’s optimal condition, the DC&1ω method is superior. For example, for a desired lifetime measurement relative error of 1%, the 1ω method requires 10.48 ms, the 2ω needs 668 ms, while the DC&1ω only requires 5.28 ms. Based on the fits, for any given lifetime relative error, the DC&1ω method is 2.0 times as fast as the 1ω method, and is 126.5 times as fast as the 2ω method. Since the sensitivity of the DC&1ω exceeds the optimal sensitivity produced by the conventional 1ω MPM-FD-FLIM (i.e., fundamental limit), a super-sensitivity MPM-FD-FLIM can be built [Eq. (3)]. Second, the results show that the photon economies of both the 1ω and DC&1ω two-photon FD-FLIM exceed the theoretically optimal one-photon FD measurement, as expected from the analytical SNR model in [13]. This is seen by comparing the experimental results to the theoretically optimal (m = 1) one-photon FD-FLIM performance [dashed green line in Fig. 3(a)].

 

Fig. 3 (a) Measured (symbols) and linear fit (lines) lifetime relative error versus integration time. Dashed green line: theoretical limit for one-photon sinusoidal FD-FLIM. (b) Lifetime standard deviation versus fluorescence lifetime for a constant modulation frequency of 62.5 kHz. Inset: zoomed view, clearly showing the expanded DC&1ω frequency range.

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3.3. Enhanced MPM-FD-FLIM lifetime range

Typically, FD-FLIM measurements of fluorophore lifetimes are limited to narrow range of values since systems have a fixed excitation intensity modulation frequency, f. For example, to maintain F < 4, Fig. 2(b) shows that the 1ω method is limited to lifetimes of 0.09/f < τ < 0.17/f while the DC&1ω MPM-FD-FLIM expands this range to 0.11/f < τ < 0.31/f. We experimentally validated this point by varying the lifetime of the [Ru(dpp)₃]²⁺ nanomicelle probe by slowly pumping nitrogen into the air-saturated solution to change the dissolved oxygen concentration. The modulation frequency was held at 62.5 kHz. The lifetime τ increased from 1.6 μs to 4.2 μs, corresponding to a normalized modulation frequency of 0.1/τ to 0.26/τ. For each τ, the 10 ms measurements were repeated for 1000 times and their standard deviations (σ_τ) were recorded.

The experimental sensitivity results with respect to different lifetimes are presented in Fig. 3(b). As the τ varies from 1.6 μs to 4.2 μs, the σ_τ of the 1ω method degrades from 19.7 ns to 53.1 ns, while the σ_τ of the DC&1ω method stays at a stable level (22.1 ns to 32.8 ns). These results match well with expected F-values in Fig. 2(b). When the normalized modulation frequencies range from 0.1/τ to 0.26/τ, the F-value of the DC&1ω method is flatter compared to the 1ω method. This enhanced lifetime range is important since 2D or 3D lifetime images usually span a wide range. The super-sensitivity MPM-FD-FLIM using the DC&1ω method is capable to keep a universally high SNR over all pixels, even if there are large lifetime variations.

4. Conclusions

We have experimentally validated the analytical SNR model in [13] that the 1ω, 2ω, and DC&1ω MPM-FD-FLIM methods have distinct sensitivity performances which can be precisely described by their analytical F-value curves. The theoretical model can act as a guideline for experimentalists using MPM-FD-FLIM to obtain the best imaging sensitivity. Based on the experimental results, the presented super-sensitivity MPM-FD-FLIM using the DC&1ω method is able to produce high-sensitivity lifetime measurements beating the conventional 1ω MPM-FD-FLIM and the 2ω one. In this work, all experiments were performed on the same hardware setup as well as obtained the same raw dataset; the only difference for the 1ω, 2ω, and DC&1ω methods was in the data analysis implementation. Therefore, it is easy and straightforward to implement the super-sensitivity MPM-FD-FLIM by simply programmatically modifying a conventional MPM-FD-FLIM setup, and the benefits will be approximately 2 times sensitivity improvement over a wide lifetime range compared to conventional MPM-FD-FLIM.