Representation Method for Spectrally Overlapping Signals in Flow Cytometry Based on Fluorescence Pulse Time-Delay Estimation

Abstract

Flow cytometry is being applied more extensively because of the outstanding advantages of multicolor fluorescence analysis. However, the intensity measurement is susceptible to the nonlinearity of the detection method. Moreover, in multicolor analysis, it is impossible to discriminate between fluorophores that spectrally overlap; this influences the accuracy of the fluorescence pulse signal representation. Here, we focus on spectral overlap in two-color analysis, and assume that the fluorescence follows the single exponential decay model. We overcome these problems by analyzing the influence of the spectral overlap quantitatively, which enables us to propose a method of fluorescence pulse signal representation based on time-delay estimation (between fluorescence and scattered pulse signals). First, the time delays are estimated using a modified chirp Z-transform (MCZT) algorithm and a fine interpolation of the correlation peak (FICP) algorithm. Second, the influence of hardware is removed via calibration, in order to acquire the original fluorescence lifetimes. Finally, modulated signals containing phase shifts associated with these lifetimes are created artificially, using a digital signal processing method, and reference signals are introduced in order to eliminate the influence of spectral overlap. Time-delay estimation simulation and fluorescence signal representation experiments are conducted on fluorescently labeled cells. With taking the potentially overlap of autofluorescence as part of the observed fluorescence spectrum, rather than distinguishing the individual influence, the results show that the calculated lifetimes with spectral overlap can be rectified from 8.28 and 4.86 ns to 8.51 and 4.63 ns, respectively, using the comprehensive approach presented in this work. These values agree well with the lifetimes (8.48 and 4.67 ns) acquired for cells stained with single-color fluorochrome. Further, these results indicate that the influence of spectral overlap can be eliminated effectively. Moreover, modulation, mixing with reference signals, and low-pass filtering are performed with a digital signal processing method, thereby obviating the need for a high-speed analog device and complex circuit system. Finally, the flexibility of the comprehensive method presented in this work is significantly higher than that of existing methods.

1. Introduction

Flow cytometry is a cell biology technique that is widely applied in life science research and in the clinic. It is a well-established and powerful high-throughput fluorescence measurement tool that also allows for the sorting and enrichment of cell subpopulations expressing unique fluorescence signatures. The most common readouts in flow cytometry are the forward- and side-scattering properties of particles and the fluorescence intensities of attached probes measured in multiple spectral channels. Owing to the breadth of measurements conducted in both research and clinical settings, flow cytometers that detect up to 18 colors simultaneously are now available. Yet, the emission spectra of many common fluorophores are broad and the larger the number of colors used, the more prevalent the problem of fluorescence spillover. Because of its reliance on intensity-only signals, flow cytometry cannot easily discriminate between fluorophores that spectrally overlap [1,2].

Detecting the fluorescence lifetime, i.e., the average time for which a fluorophore resides in an excited state, is a possible alternative to intensity-based measurements. The fluorescence lifetime is a central quantity of fluorescence spectroscopy and reflects the fluorophore heterogeneity and the dynamics of the interactions in the fluorophore environment. Early demonstration of the time-resolved flow cytometry technique involved frequency-domain measurements performed with complex analog devices [3,4,5]. In the phase method, harmonic excitation is used when the fluorescence is also harmonic, but with a phase delay and demodulation relative to the excitation, from which the lifetime can be computed. The alternative method, i.e., the time-domain or pulsed technique, directly samples the fluorescence decay curves elicited by very short light flashes via single-photon counting or stroboscopic detection [6,7,8,9]. However, both the time- and frequency-domain approaches increase the cost and complexity of a traditional flow cytometer significantly. Moreover, several high-frequency hardware components must be added to a simple flow system for phase-sensitive flow cytometry to be performed. These modifications include high-frequency mixing and filtering steps, when the employed data acquisition system is unable to sample the high-frequency signals [10,11].

Focusing on the case of spectral overlap occurring for two kinds of fluorochromes, in this paper, we demonstrate a new method for representing spectrally overlapping signals using fluorescence lifetime-dependent methodology. In addition, the autofluorescence overlap are fully handled with the emission fluorescence, without distinguishing the separating influence. This approach, referred to herein as “pulse time-delay estimation” (PTDE), employs a method of digital modulation, rather than a complex and high-speed analog circuit, to separate the fluorescence signals observed from fluorophores emitting fluorescence with differing average lifetimes. PTDE is a practical approach to separating cell populations that are stained with spectrally overlapping fluorophores.

2. Theory

2.1. Cytometric Pulse Generation and Fluorescence Spectral Overlap

In flow cytometry, fluorescence signals are often referred to as “cytometric pulses” or “waveforms”, because they are the result of the rapid transit of a fluorescently labeled cell through a tightly focused laser beam with a Gaussian profile. As shown in Figure 1, the microsphere moves along the flow direction through the excitation area (blue ellipsoid, Figure 1). A Gaussian-like trace reflects the resulting signal, where the forward-scattered emission begins to increase at time Ι (the microsphere flows across the upper limb L₀ into the excitation area), peaks at time Π, and decreases as the microsphere moves from time Π to time Ш (the microsphere flows across the lower limb L₁).

Therefore, the original signals fl₁ and fl₂ are mathematically expressed as:

$$\{\begin{array}{c}fs=K\exp [-{(t-t_0)}^2/(2{\mathsf{\sigma }}^2)]\\ f{l}_1=A\exp [-{(t-t_0+\Delta t_1)}^2/(2{\mathsf{\sigma }}^2)]\\ f{l}_2=B\exp [-{(t-t_0+\Delta t_2)}^2/(2{\mathsf{\sigma }}^2)]\end{array}$$

(1)

where fs is the forward-scattered pulse; K, A, and B are constants representing the heights of the curve peaks; t₀, t₀ − Δt₁, and t₀ − Δt₂ represent the peak positions; and σ represents the curve width.

Spectral overlap, or “spillover”, results from the use of fluorescent dyes that are measurable in more than one detector. This spillover is correlated by a constant known as the spillover coefficient. Spectral overlap is illustrated schematically in Figure 2. Two fluorochromes with different lifetimes are excited simultaneously and then detected using detector channels having fixed bandwidths. The time-intensity Gaussian-like cytometric pulses (fs, fl₁, and fl₂) are generated as the microspheres flow through the extinction area. Then, the spectrum of each fluorochrome is detected by a different detector channel having a fixed bandwidth. Both the peak value and the peak location of the detected signal (time-intensity cytometric pulse) of each detector channel are influenced by the spectral overlap due to the different fluorochrome lifetimes. That is, the time delays between the observed fluorescence signals (fl_a, fl_b) and fs are influenced by the spectral overlap [12,13,14,15].

In addition, considering the influence of autofluorescence, the dim expression of the exogenous fluorophores adds a number of spectral overlap. Autofluorescence is due to many cellular constituents, rather than a single fluorochrome with a single exponential decay, and it is difficult to distinguish the influence of this behavior from the observed fluorescence spectrum. However, the autofluorescence is dependent on certain cell constituents, it is relatively stable, and its dim intensity barely shifts the peak locations of the fluorescence pulse signals, even with multi-exponential decay. Therefore, the autofluorescence can be fully handled with the emission fluorescence. In what follows, the autofluorescence overlap are regarded as elements of the emission fluorescence for the lifetime calculations of cells labeled with single- or two-color fluorochromes.

The signals after crossover (fl_a, fl_b) are shown in Figure 3 and described by Equation (2) below. The expected results of the detector channels are k₁₁fl₁ and k₂₂fl₂. However, the peak values and peak locations of the observed signals are influenced by signal crossover. The peak values of the observed signals increase with the addition of parts of the crossover signals. The peak-value errors (Ep_a, Ep_b) introduced through crossover can be calculated using Equation (3). Meanwhile, the peak locations of fl₁ and fl₂ differ, and the fl_a and fl_b peak locations move to the left or right when parts of the crossover signals are added. The peak-location errors (Et_a, Et_b) introduced through crossover can be calculated using Equation (4):

$$\{\begin{array}{c}f{l}_a=k_{11}f{l}_1+k_{21}f{l}_2\\ f{l}_b=k_{12}f{l}_1+k_{22}f{l}_2\end{array}$$

(2)

$$\{\begin{array}{c}E{p}_a=(P_a-k_{11}{P}_1)/P_a\\ E{p}_b=(P_b-k_{22}{P}_2)/P_b\end{array}$$

(3)

Here, P_a, P₁, P_b, and P₂ are the peak values of fl_a, fl₁, fl_b, and fl₂, respectively.

$$\{\begin{array}{c}E{t}_a=(\Delta t_a-\Delta t_1)/\Delta t_a\\ E{t}_b=(\Delta t_b-\Delta t_2)/\Delta t_b\end{array}$$

(4)

The peak-value errors and the pulse time delay errors (Δt_a − Δt₁, Δt_b − Δt₂: the time delays of the peak locations between fs and the fluorescence pulses (fl₁, fl₂, fl_a and fl_b)) introduced by the signal crossover are shown in Figure 3. It is apparent that ΔV₁ and ΔV₂ are significantly larger than Δt_a − Δt₁ and Δt_b − Δt₂. Note that the peak values of different fluorescence signals can vary widely and fluorescence intensity measurements are impacted by nonlinearity. Therefore, the peak-value errors introduced by crossover are typically large and are difficult to amend. On the other hand, the differences in the pulse time delays between the different fluorescence pulses are significantly smaller. The pulse time delays vary little with signal crossover. Therefore, the influence of crossover can be decreased be using the pulse time delay as a parameter when representing the fluorescence pulse.

2.2. Fluorescence Pulse Time-Delay Estimation

The time delay between the observed fluorescence signal and the forward-scattered signal includes the lifetime and delay introduced by the hardware [3]. However, only the fluorescence lifetime is relevant for removing the crossover signal from the observed signal in the following sections. Therefore, it is necessary to determine the relationship between the time delay and lifetime. We assume that the fluorescent light is emitted in accordance with the single exponential decay model. The fluorescent-light signal can be expressed as a convolution of the forward-scattered light and the single exponential decay model, such that:

$$L\_fl=K_0\left(L\_fs\times e^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\tau }$}\right.}\right)$$

(5)

where L_fl is the fluorescent-light pulse; L_fs is the forward-scattered light pulse; K₀ is the signal intensity; and τ is the fluorescence lifetime. Then, we can conclude that the phase shift between L_fl and L_fs is the phase of $e^{-\raisebox{1ex}{$\mathrm{t}$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\tau }$}\right.}$: arctan(ωτ). Therefore, the time delay between the fluorescent-light pulse and the forward-scattered light pulse is equal to τ.

To estimate the time delay of output pulse signals from circuit systems, the cross-correlation function algorithm is applied to restrain the influence of the noise. However, the estimation accuracy with a traditional cross-correlation function is limited by the time interval between adjacent points in the pulse signal, i.e., by the sampling frequency of the analog-to-digital converter (ADC). In this work, zoom analysis of the frequency spectrum within the scope of a certain frequency is executed using the modified chirp Z-transform (MCZT) method, so as to acquire the frequency spectral information with high resolution. The frequency resolution is higher with larger N₁, however, more time would be taken for computing. The frequency spectral information is neither limited by the pulse signal length nor the ADC sampling frequency. The fine interpolation of the correlation peak (FICP) algorithm is then used to improve the resolution of the time-domain cross-correlative function. This method avoids modulating the laser at different frequencies and effectively presents a platform for lifetime detection that is easier to practically implement on any standard flow cytometer.

Assuming that the forward-scattered pulse signal fs(n) and the fluorescence pulse signal fl_m(n) are data sequences containing N points, the corresponding zoomed frequency spectra (FS(k) and FL_m(k), respectively) are described by the transform:

$$\{\begin{array}{c}FS(k)=MCZT(fs(n))={\displaystyle \sum _{n=0}^{N-1}fs(n)\exp (-\frac{j2\mathsf{\pi }}{N_1}kn)}\\ F{L}_m(k)=MCZT(f{l}_m(n))={\displaystyle \sum _{n=0}^{N-1}f{l}_m(n)\exp (-\frac{j2\mathsf{\pi }}{N_1}kn)}\end{array}$$

(6)

where m = 1, 2, a, b and k = 0, 1, …, N − 1.

Based on the sampling theorem, FS(k) and FL_m(k) are periodic functions and the cycle is N₁, while the first and last parts of the functions are conjugate symmetric. With the FICP analysis, the sampling frequency is raised to N₂/N₁ times the initial sampling frequency, when the frequency spectrum cycle is extended from N₁ to N₂ with the insertion of zero. That is, the resolution of the time delay between fs(n) and fl_m(n) in the time domain is increased to N₂/N₁ times for the same number of calculations. The complete cross-correlation frequency spectrum is constructed as:

$$R_m(k)=\{\begin{array}{c}{R}_1(k)=FS(k)F{L}_m^{*}(k),\\ 0\\ R_1^{*}(N_2-k)\end{array}\begin{array}{c}k=0,1,\cdots ,N-1\\ k=N,\cdots ,N_2-N\\ k=N_2-N+1,\cdots ,N_2-1\end{array}$$

(7)

The time delays are distributed within a certain limited range and, similarly, the time delays between fs(n) and fl_m(n) fall into a certain limited range. In this case, only a finite number of the first and last parts of the points are required for the calculation. The first and last parts of the cross-correlation function in the time domain, r₁(n) and r₂(n), respectively, are calculated from Equations (8) and (9), respectively, as follows:

$$r_{1m}(n)=[{\displaystyle \sum _{k=0}^{N-1}{R}_{1m}(k)\exp (j\frac{2\mathsf{\pi }}{N_2}kn)}+\exp (-j\frac{2\mathsf{\pi }}{N_2}Nn){\displaystyle \sum _{k=0}^{N-1}{R}_{1m}^{*}(N-k)\exp (j\frac{2\mathsf{\pi }}{N_2}kn)}]/N_2$$

(8)

$$\begin{array}{l}{r}_{2m}(n)=\exp (j\frac{2\mathsf{\pi }}{N_2}Nn){\displaystyle \sum _{k=0}^{N-1}[R_{1m}(k)\exp (-j\frac{2\mathsf{\pi }}{N_2}kn)}]/N_2\\ +\exp [j\frac{2\mathsf{\pi }}{N_2}(N^2-nN)]{\displaystyle \sum _{k=0}^{N-1}[R_{1m}^{*}(N-k)\exp (-j\frac{2\mathsf{\pi }}{N_2}kn)}]\exp (j\frac{2\mathsf{\pi }}{N_2}kn)/N_2\end{array}$$

(9)

where m = 1, 2, a, b and k = 0, 1, …, N − 1. N₂ is the length of both FS(k) and FL_m(k) after the zeros are inserted.

The value of N₂ is set arbitrarily, and the resolution of r₁(n) and r₂(n) is increased by N₂/N₁. Meanwhile, the integrated cross-correlation function in the time domain r(n) is constructed with r₁(n) and r₂(n), such that:

$$r_m(n)=\{\begin{array}{cc}{r}_{1m}(n),& n=0,1,\cdots ,N-1\\ r_{2m}(2N+n)& n=-N,\cdots ,-1\end{array}$$

(10)

The resolution of the cross-correlation peak is improved by increasing the interpolation multiple of the frequency spectrum. That is, the computational accuracy of the time-delay estimation is higher with FICP, particularly for pulse signals of low sampling frequency.

It is necessary to correct for the inherent time delay introduced by the hardware variables, such as the differences in the photoelectric detector, circuitry, and cable. The time delay introduced by the hardware can be eliminated via calibration, which is performed as shown in Figure 4. Two light-emitting diodes (LEDs) are used as light sources and are pulsed synchronously by a standard function generator with Gaussian waveforms. As shown in Figure 4a, the Gaussian-shaped light pulses L_fs and L_fl₁ are used to represent the forward-scattered light pulse and the fluorescent-light pulse of a flow cytometer, respectively. L_fs and L_ fl₁ are synchronous with the same pulsed signal. Figure 4b shows the photovoltaic conversion and the electric systems. The corresponding electrical pulse signal is presented in terms of V_fs and V_fl₁, and then converted to digital signals using ADCs labeled ADC0 and ADC1, respectively, as illustrated in Figure 4c. The ADC0 and ADC1 clock signals are provided by one crystal oscillator, to ensure that V_fs and V_ fl₁ are converted simultaneously. The time delay between V_fs and V_fl₁ can be acquired and taken as the time delay introduced by the photovoltaic conversion and electric systems.

2.3. Representation of Spectrally Overlapping Fluorescence Signals

Spectrally overlapping fluorescence signals are represented using the process shown in Figure 5. The cells are analyzed as they intersect a laser excitation beam. The fluorescence is measured orthogonally to the laser beam-cell stream intersection point in the flow chamber, using a dichroscope, a filter, a signal detector, and an electric system [16]. The time delays (Δt_a, Δt_b) of the fluorescence pulses (fl_a, fl_b) are estimated using the above-mentioned FICP arithmetic. As mentioned above, the forward-scattered signal and the fluorescence signal are Gaussian in shape. For a generated signal, fs, fl_a, and fl_b can be expressed as shown in Equation (11) [16,17,18].

$$\{\begin{array}{c}fs=K_0{\mathrm{e}}^{-\frac{{(t-t_0)}^2}{2{\mathsf{\sigma }}^2}}=K_1{\mathrm{e}}^{-\frac{{(t-t_0^{\text{'}})}^2}{2{\mathsf{\sigma }}^2}}\times h_0(t)\\ f{l}_a=C_0{\mathrm{e}}^{-\frac{{(t-t_0-\Delta t_a)}^2}{2{\mathsf{\sigma }}^2}}=C_1{\mathrm{e}}^{-\frac{{(t-t_0^{\text{'}})}^2}{2{\mathsf{\sigma }}^2}}\times {\mathrm{e}}^{-\frac{t}{{\mathsf{\tau }}_a}}\times h_1(t)\\ f{l}_b=D_0{\mathrm{e}}^{-\frac{{(t-t_0-\Delta t_b)}^2}{2{\mathsf{\sigma }}^2}}=D_1{\mathrm{e}}^{-\frac{{(t-t_0^{\text{'}})}^2}{2{\mathsf{\sigma }}^2}}\times {\mathrm{e}}^{-\frac{t}{{\mathsf{\tau }}_b}}\times h_2(t)\end{array}$$

(11)

where, K₁, C₁, and D₁ are the signal intensity, t₀' is the center moment of the scattered signal; τ_a and τ_b are the calculated lifetimes of fl_a and fl_b, respectively; and h₀(t), h₁(t), and h₂(t) are the transfer functions of electric systems 0, 1, and 2, respectively.

The individual lifetimes τ₁ and τ₂ are acquired so as to express the phase shift introduced by the lifetime. In order to acquire the individual lifetime components of each fluorochrome, the cells are labeled with one kind of fluorochrome only and tested separately. Then, fl₁ and fl₂ are detected, and τ₁ and τ₂ are acquired, respectively, using the MCZT and FICP techniques.

The signals after modulation (fs_mod, fl_a_mod, and fl_b_mod) are artificially created using a digital signal processing method, as shown in Equation (12). The modulating signal is a cosine function, and the phase shifts of fl_a and fl_b introduced by τ_a and τ_b are ϕ_a = arctan(ω₀τ_a) and ϕ_b = arctan(ω₀τ_b), respectively. Considering the relationship between the observed (fl_a, fl_b) and original signals (fl₁, fl₂), the expressions given in Equation (13) can be acquired. Hence, the phase shifts introduced by τ₁ and τ₂ are ϕ₁ = arctan(ω₀τ₁) and ϕ₂ = arctan(ω₀τ₂), respectively.

$$\{\begin{array}{c}fs\_\mathrm{mod}=\cos ({\mathsf{\omega }}_{0}t)K_2\exp [-{(t-t_0)}^2/(2{\mathsf{\sigma }}^2)]\\ f{l}_a\_\mathrm{mod}=C_2\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_a)\exp [-{(t-t_0+{\mathsf{\tau }}_a)}^2/(2{\mathsf{\sigma }}^2)]\\ f{l}_b\_\mathrm{mod}=D_2\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_b)\exp [-{(t-t_0+{\mathsf{\tau }}_b)}^2/(2{\mathsf{\sigma }}^2)]\end{array}$$

(12)

$$\{\begin{array}{c}\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_a)=C_3\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_1)+C_4\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_2)\\ \cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_b)=D_3\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_1)+D_4\cos ({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_2)\end{array}$$

(13)

Here, K₂, C₂, C₃, C₄, D₂, D₃, and D₄ are the signal intensities; t₀ is the center moment of fs_mod; and ω₀ is the angular frequency of the modulation.

The aim of PTDE is to transform the observed summation signal, which is measured experimentally, into a signal that is dependent on one fluorophore only. Nullifying one of the fluorescence signals by exploiting the fluorophore lifetime is necessary for this to occur. Nullification is accomplished by mixing a cosine reference signal, which is performed by altering the reference signal phase. The reference signal has the same angular frequency as the modulating signal and the reference phase is shifted by an amount ϕ_ref1 = ϕ₂ or ϕ_ref2 = ϕ₁. The mixing results (fl_a_mix, fl_b_mix) are expressed as:

$$\{\begin{array}{c}f{l}_a\_mix=f{l}_a\_\mathrm{mod}\cdot \cos \left({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_{ref1}\right)\\ f{l}_b\_mix=f{l}_b\_\mathrm{mod}\cdot \cos \left({\mathsf{\omega }}_{0}t-{\mathsf{\phi }}_{ref2}\right)\end{array}$$

(14)

Finally, the original fluorescence signal (fl₁, fl₂) can be obtained via low-pass filtering, as expressed in Equation (15). Both signals are resolved, but with a loss in amplitude [1/2cos(ϕ₁ − ϕ₂)]:

$$\{\begin{array}{c}f{l}_1=\frac{1}{2}\cos ({\mathsf{\phi }}_1-{\mathsf{\phi }}_2)C_3\exp [-{(t-t_0+{\mathsf{\tau }}_1)}^2/(2{\mathsf{\sigma }}^2)]\\ f{l}_2=\frac{1}{2}\cos ({\mathsf{\phi }}_2-{\mathsf{\phi }}_1)D_3\exp [-{(t-t_0+{\mathsf{\tau }}_2)}^2/(2{\mathsf{\sigma }}^2)]\end{array}$$

(15)

3. Materials and Method

In this study, three steps are employed to evaluate and demonstrate PTDE. Mathematical modeling is first performed using MATLAB^®, to simulate the PTDE theory and to verify the methodology. A variety of fictional fluorescence lifetime, peak, and crossover coefficients are employed. Secondly, fixed mammalian cells are stained with single-color reagents, so that the individual lifetime components of each fluorochrome can be acquired. Finally, experiments are conducted using fixed mammalian cells to test the fluorescence signal representation by discriminating between the fluorescence lifetimes of all samples.

3.1. Simulation of Time-Delay Estimation Fluorescence Measurements

The peak values and peak locations of the observed fluorescence pulse waveforms are influenced by A, B, Δt₁, Δt₂, and the crossover coefficients.

Table 1. Example parameters used in simulation.

No.	A (V)	B (V)	Δt1 (μs)	Δt2 (μs)	k11	k12	k21	k22
1	1.40	1.20	0.33	0.35	0.80	0.15	0.15	0.80
2	1.10	1.20	0.33	0.35	0.80	0.15	0.15	0.80
3	1.40	1.20	0.30	0.40	0.80	0.15	0.15	0.80
4	1.10	1.20	0.30	0.40	0.80	0.15	0.15	0.80
5	1.40	1.20	0.33	0.35	0.80	0.10	0.10	0.80
6	1.40	1.20	0.33	0.35	0.75	0.15	0.10	0.75

lists the different specifications and sample parameters used for the simulation.

3.2. Experiments Using Mammalian Cells

The samples used in this study were the Cyto-Trol Control Cells Kit (Ref. 6604248, Beckman Coulter Inc., Brea, CA, USA) and the fluorochrome-conjugated antibodies, CD3-FITC (Ref. A07746, Beckman Coulter Inc., Brea, CA, USA) and CD4-PC5 (Ref. A07752, Beckman Coulter Inc., Brea, CA, USA).

Cyto-Trol Control Cells are used for activity assessments of monoclonal antibodies via flow cytometry. These cells are a lyophilized preparation of human lymphocytes that exhibit surface antigens detectable with monoclonal antibodies. The cells are isolated from peripheral blood and express antigens representative of those found on normal lymphocytes. They are used with direct conjugation on monoclonal antibodies. The fluorochrome-conjugated antibodies facilitate the identification and numeration of cell populations expressing the CD3 and CD4 antigens present in human biological samples using flow cytometry. All fluorochrome-conjugated antibodies are excited at 488 nm with broad emission spectra that overlap from ~500 to 700 nm. The emission peaks of CD3-FITC and CD4-PC5 occur at 525 and 670 nm, respectively. Here, the flow cytometry measurements were acquired with a Gallios Flow Cytometer (Beckman Coulter Inc., Brea, CA, USA).

• The compensation tubes were stained with single-color reagents (CD3-FITC, CD4-PC5) in each fluorochrome and used to acquire the average fluorescence lifetime, with all these tubes being run in the same manner. The average fluorescence lifetime was taken as the individual lifetime component of FITC and PC5. The fluorescence pulse signals were detected, and the individual lifetime component was acquired using the MCZT and FICP techniques mentioned above.
• The verification tube was stained with two-color reagents simultaneously and used to ensure that the representation method based on the fluorescence PTDE is satisfactory for the samples and antibodies used in this work.

4. Results and Discussion

4.1. PTDE Simulation

Sample results of the simulation output, for which the mathematical functions of each stage were modeled, are plotted in Figure 6a–c. For this example, the two fluorescence signals had simulated time delays of 0.33 and 0.35 μs, and the obtained peak intensities are 1.40 and 1.20 V, respectively. Figure 6a shows the original and observed fluorescence signals after crossover. The fluorescence-signal waveform peak values and peak locations are influenced by crossover.

As shown in Figure 6a, fl₁ is ahead of fl₂. The crossover causes the location of the fl_a peak to shift to the right, which increases the time delay between fl_a and fs slightly. Further, fl₂ lags behind fl₁, and the location of the fl_b peak shifts to the left as a result of the crossover. The time delay between fl_b and fs decreases slightly. Figure 6b shows plots of the zoomed frequency spectra of the pulse signals (fs, fl_a, and fl_b). A zoom analysis of the frequency spectrum was then executed within the scope of a certain frequency and using the MCZT method, so as to acquire high-resolution frequency spectral information. Recall that this information was not limited by the pulse signal length or by the ADC sampling frequency. With considering the characteristic of frequency spectrum and the amount of calculations synthetically in this work, the values of N₁ and N₂ are set to be N₁/N = 10 and N₂/N₁ = 10. Compared with the fast Fourier transform (FFT) result, less detailed information is lost and the frequency resolution of the spectrum obtained using the MCZT method is significantly higher; therefore, the corresponding cross-correlation frequency spectrum is smoother. Figure 6c shows plots of the cross-correlation functions calculated using the FICP algorithm. The time-delay resolution between fs and fl_m in the time domain is increased to N₂/N₁ times with the same number of calculations.

Finally, the simulation results (see Table 1 for each parameter) are listed in

Table 2. Example simulation results.

No.		fla		flb
		Peak	Time-Delay	Peak	Time-Delay
1	Expected values	1.12 V	0.33 μs	0.96 V	0.35 μs
	Errors (%)	20.95	1.70	39.01	1.42
2	Expected values	0.88 V	0.33 μs	0.96 V	0.35 μs
	Errors (%)	28.37	3.46	38.99	2.32
3	Expected values	1.12 V	0.30 μs	0.96 V	0.40 μs
	Errors (%)	20.79	1.84	38.82	−0.93
4	Expected values	0.88 V	0.30 μs	0.96 V	0.40 μs
	Errors (%)	28.15	2.92	38.82	1.12
5	Expected values	1.12 V	0.33 μs	0.96 V	0.35 μs
	Errors (%)	20.27	1.12	25.88	−0.78
6	Expected values	1.05 V	0.33 μs	0.90 V	0.35 μs
	Errors (%)	27.32	1.58	33.68	0.31

. The expected peak values correspond to the signal intensities of the observed fluorescence, which form the corresponding original signals; that is, the expected peak values are k₁₁ × A and k₂₂ × B. The expected time delay values are ΔT₁ and ΔT₂. The peak-value errors and time delays were calculated using Equations (3) and (4). In this simulation, the crossover-induced peak-value errors ranged from 20.27% to 39.01%, whereas the crossover-induced time-delay estimation errors ranged from 0.31% to 3.46%. Thus, the influence of crossover on the peak values is significantly larger than that on the time-delay estimation. Further, the fluorescence lifetime, which is obtained from the time-delay estimation, is more suitable for the representation of spectrally overlapping fluorescence signals.

4.2. Time-Delay Estimation of Fluorescently Labeled Cells

The time delay between the fl_a, fl_b, and fs of the fluorescently labeled microspheres was estimated using the FICP algorithm, and the peak locations of these signals were acquired using a Gauss fitting algorithm, for comparison with the results of the proposed method.

One series of observed fs, fl_a, and fl_b signals was taken as an example for the FICP feasibility analysis. The curves obtained after Gauss fitting are shown in Figure 7, where the colored curves are the observed signals and the black curves are the Gauss fitting results.

The Gauss fitting was modeled by $a{e}^{-{(\frac{t-b}{c})}^2}$ and the results were evaluated by calculating the root mean square error (RMSE) and the R-squared values. The RMSE is an estimate of the standard deviation of the random component in a data set, and was used to evaluate the deviation between the curve fitting result and the observed signal in this study. The RMSE is defined as:

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(X_{obs,i}-X_{fit,i})}^2}}$$

(16)

where X_obs is the observed value, X_fit is the curve fitting result, and n is the length of the observed signal, which is equal to that of the fitting result.

R-squared is the square of the correlation between the observed and predicted values. It is also known as the “square of the multiple correlation coefficient” and the “coefficient of multiple determination”. R-squared is defined as the ratio of the sum of the squares of the regression (SSR) and the total sum of squares (SST):

$$R-squared=\frac{SSR}{SST}$$

(17)

SSR and SST are defined as:

$$SSR=\frac{1}{n}{\displaystyle \sum _{i=1}^n{(X_{fit,i}-{\overline{X}}_{obs,i})}^2}$$

(18)

$$SST={\displaystyle \sum _{i=1}^n{(X_{obs,i}-{\overline{X}}_{obs,i})}^2}$$

(19)

where ${\overline{X}}_{obs}$ is the mean value of the observed signal. R-squared can take any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted for by the model.

Table 3. Gauss fitting and FICP results (N₁/N = 10, N₂/N₁ = 10).

	Gauss Fitting				FICP
	Peak Location (ns)	RMSE	R-Square	Time-Delay (ns)	Time-Delay (ns)
fs	5870	0.0186	0.9937	-	-
fla	5108	0.0162	0.9986	762	758.6
flb	5120	0.0295	0.9991	750	748.4

presents the Gauss fitting and FICP results. The RMSE of the Gauss fitting is less than 0.03 and the R-squared values are larger than 0.99. Therefore, the observed signals were represented perfectly by the Gauss fitting results. Meanwhile, the deviations in the time delays between fl_a and fs and between fl_b and fs are 3.4 ns and 1.6 ns, respectively.

Note that the time delays calculated above contain the time delays introduced by both the fluorochrome lifetimes and the electrical system. In the calibration process, Gaussian-shaped pulses having an average peak voltage of 2 V and an average pulse width ranging from 1 to 5 μs, in increments of 0.5 μs, were artificially created. Photoelectric detectors were configured to collect light signals from the LEDs. The processing methods were verified by introducing artificial fluorescence delays, which was achieved by setting the phase-shift values between the LED pulses to 0 ns. For each group of delays, 1000 events were collected and processed, and the mean values were taken as the calibration results.

Thus, 1000 serial pulse signals of fluorescently labeled microspheres were detected, and the time delays between fl_a, fl_b, and fs were calculated using Gauss fitting and the FICP method.

Table 4. Statistical results of time-delay estimation after calibration.

	Gauss Fitting			FICP
	Mean (ns)	SD (ns)	RSD (%)	Mean (ns)	SD (ns)	RSD (%)
τa	8.35	0.38	4.44	8.28	0.32	3.77
τb	4.90	0.15	3.26	4.86	0.11	2.36
τ1	8.56	0.76	8.50	8.48	0.58	6.55
τ2	4.58	0.24	5.36	4.67	0.17	3.84

lists the statistical results of the time-delay estimation after calibration of the 1000 events. The mean values, standard deviations (SDs, $\mathsf{\sigma }=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}$) and relative standard deviations (RSDs, $\frac{\mathsf{\sigma }}{\overline{x}}\times 100\%$) were taken as indexes to evaluate the statistical properties of the fluorescence pulse signals obtained from the same type of fluorochrome.

As we can see, the mean values of each channel obtained via Gauss fitting and FICP are very close, and the SDs and RSDs determined for the FICP method values are significantly smaller than those for the Gauss fitting results. Therefore, the time delay can be estimated correctly using the FICP method, and the lifetimes (τ_a and τ_b) can be acquired with calibration. Meanwhile, the individual lifetime (τ₁, τ₂) components of each fluorochrome, which were acquired for cells labeled with one kind of fluorochrome and tested separately, are also listed in Table 4. It is apparent that τ_a is smaller than τ₁, whereas τ_b is larger than τ₂. Note that the deviations between τ_a, τ_b and τ₁, τ₂ are introduced by spectral overlap. Further, τ₁ is longer than τ₂, and the peak location of fl_b moves right while that of fl_a moves left, because of the influence of the spectral overlap. Therefore, the calculated lifetime of fl_a is significantly smaller than that of fl₁, whereas the calculated lifetime of fl_b is considerably larger than that of fl₂.

4.3. Fluorescence Signal Representation of Fluorescently Labeled Cells Based on PTDE

The width of observed cytometric pulses and the differences between lifetimes τ₁ and τ₂, together with the amount of calculations are considered collectively, the modulating frequency is set to be 10 MHz (ω₀ = 10 MHz) for the analysis of crossover influence eliminating. Figure 8a–d is a panel of waveforms that exemplify the step-wise fluorescence signal representation of fluorescently labeled cells based on PTDE. The τ_a and τ_b lifetimes were used to artificially configure the signals after modulation with Equation (12), as shown in Figure 8b. In the figures, each lifetime is reflected and represented with the phase shift introduced by the lifetime. The signals after modulation are mixed with reference signals so as to eliminate the crossover influence, as shown in Figure 8c. Finally, the original fluorescence signals are acquired using a low-pass filter.

As Figure 8 reveals, the PTDE-based fluorescence signal representation method for the fluorescently labeled cells successfully and effectively eliminates the influence of the spectral overlap. Moreover, the modulation, mixing with reference signals, and low-pass filtering are performed using a digital signal processing method, thereby obviating the need for a high-speed analog device and complex circuit system. Further, the flexibility of the comprehensive method developed in this work is significantly higher than that of existing methods. For example, the modulating frequency can be significantly higher than that achievable for the analog method. Selected sample results for the mean phase shift at different modulating frequencies are listed in

Table 5. Mean values of phase shifts at different modulating frequencies.

	ω0 (MHz)
	1	2	5	10	20	50
ϕa (°)	83.13	86.54	88.61	89.29	89.65	89.86
ϕb (°)	78.95	84.37	87.70	88.84	89.41	89.77

.

Finally, the lifetime histograms of the fluorescence pulse signal (ω₀ = 10 MHz) are provided in Figure 9. Limited to the experimental conditions employed in this work, the mean values of the calculated individual lifetimes of each fluorochrome are 8.51 ns (SD = 0.49 ns) and 4.63 ns (SD = 0.13 ns). The calculated individual lifetimes after the influence of spectral overlap is eliminated are close to the individual lifetimes of each fluorochrome (for the examined cells stained with single-color fluorochrome). Thus, the influence of spectral overlap can be eliminated effectively using the comprehensive method presented in this work.

5. Conclusions

In this paper, a method through which fluorescence pulse signals with spectral overlapping can be represented was proposed. The MZCT algorithm was introduced for zoom analysis of the frequency spectra of forward-scattered and multicolor fluorescence pulse signals, to prevent information loss through the fence effect. Then, the cross-correlation function was calculated using FICP, so as to increase the time-domain resolution of the fluorescence pulse time-delay estimation (PTDE). Subsequently, after modulation, signals were created with a digital signal processing method to convert the lifetimes to phase shifts, and reference signals with specific phase shifts were introduced to eliminate spectral overlap. The proposed method using digital signal processing offers a more flexible approach than existing methods and, also, complex analog circuits are not required. In conclusion, a PTDE-based method for representing fluorescently labeled cells is recommended for fluorescence lifetime analysis via flow cytometry, especially for two-color analysis scenarios adversely affected by extensive spectral overlap. On the other hand, in the proposed method, the autofluorescence overlap are simplified by regarding them as part of the emission fluorescence. Thus, the accuracy with which the spectral overlap influence is eliminated will be improved when the autofluorescence can be distinguished and eliminated. Further, the method proposed in this paper is not suitable for multi-exponential decay or cases in which more than two colors are stained simultaneously. Thus, the focus of future work could be the representation of spectrally overlapping signals in both cases using simple algorithms and hardware.