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.

Keywords: digital signal processing; flow cytometry; fluorescence lifetime; spectrally overlapping signals; time-delay estimation.

Figure 1

Generation of forward-scattered and fluorescence pulses while microsphere flows through excitation area (fs: forward-scattered pulse, fl₁ and fl₂: fluorescence pulses of fluorochromes 1 and 2, respectively; L₀ and L₁: upper and lower limbs of excitation area, respectively).

Figure 2

Schematic of spectral overlap of two fluorescence signals. (a) The green and red areas represent the spectra of the fluorescence light emitted from fluorochromes 1 and 2, respectively. The spectrum of each fluorochrome is detected by a different detector channel having a fixed bandwidth, and the spectra within the bandwidth of each detector channel are detected as one sample point of the time-intensity cytometric pulse; (b) fl₁ and fl₂ represent the original fluorescence signals (time-intensity pulses); fl_a and fl_b represent the observed fluorescence signals after signal crossover; k₁₁ and k₁₂ are components of signal fl₁, which contribute to fl_a and fl_b; k₂₁ and k₂₂ are components of signal fl₂, which contribute to fl_a and fl_b; (c) The green and red pulse (time-intensity cytometric pulse) curves are the respective fl₁ and fl₂ contributions to each detector channel ((k₁₁ × fl₁, k₁₂ × fl₁) and (k₂₁ × fl₂, k₂₂ × fl₂), respectively).

Figure 3

Peak-value errors and pulse time delays introduced by three-signal crossover. ΔV₁ and ΔV₂ represent the differences in the peak values between fl_a and fl₁, and between fl_b and fl₂, respectively. The pulse time-delay details are provided in the yellow region. Δt_a and Δt_b are the pulse time delays of fl_a and fl_b (time delays of peak locations between fs and fl_a, fl_b) respectively; Δt₁ and Δt₂ are the pulse time delays of fl₁ and fl₂ (time delays of peak locations between fs and fl₁, fl₂), respectively.

Figure 4

Time-delay calibration. (a) Light-pulse generation. The LED1 and LED2 emission lights are blue and red, respectively, and L_fs and L_fl₁ are the respective forward-scattered light pulses; (b) L_fs is detected by a photodiode and processed with electric system 0. H₀(ω) is the transfer function of electric system 0 in the frequency domain. L_fl₁ is detected by a photomultiplier tube (PMT) and processed with electric system 1. H₁(ω) is the transfer function of electric system 1 in the frequency domain. Some light intensity attenuation modules are omitted in the detection of L_fl₁ with a PMT; (c) Data acquisition. The analog electrical signals (V_fs, V_fl₁) are simultaneously converted to digital signals by ADC0 and ADC1, respectively.

Figure 5

Conceptual diagram of signal detection and processing for PTDE-based fluorescence signal representation. The stages against the dark background are processed using the digital signal processing method. The cell stream is excited by the laser beam in the flow chamber, and the forward-scattered (blue arrows) and fluorescence (green and red arrows) signals from the two fluorochromes are separated with a dichroscope and filter in series. fs (blue line) is the forward scattered pulse signal; fl_a and fl_b are the observed summation signals from the first and second channels, respectively; Δt_a and Δt_b are the pulse time delays of fl_a and fl_b, respectively; τ₁ and τ₂ are the individual lifetime components of fluorochromes 1 and 2, respectively; ϕ₁ and ϕ₂ are the phase shifts introduced by τ₁ and τ₂, respectively; τ_a and τ_b are the calculated lifetimes of fl_a and fl_b, respectively; and ϕ_a and ϕ_b are the respective phase shifts introduced by τ_a and τ_b.

Figure 6

Simulation results at each stage during PTDE. (a) Waveforms of fluorescence and forward-scattered pulse signals. The colorized solid and dashed lines are the observed fluorescence signals after crossover and the original fluorescence signals, respectively; (b) The frequency spectra are zoom-analyzed 10 times (N₁/N = 10 in Equation (6)) using MCZT; (c) Time domain cross-correlation functions calculated using FICP (N₂/N₁ = 10 in Equations (8) and (9)).

Figure 7

Gauss fitting results for pulse signals (fs, fl_a, and fl_b). (a) The blue and black curves are the pulse signal fs and the Gauss fitting result, respectively; (b) The green and black curves are the observed fluorescence pulse signal fl_a and the Gauss fitting result, respectively; (c) The red and black curves are the observed fluorescence pulse signal fl_b and the Gauss fitting result, respectively.

Figure 8

Results of PTDE-based fluorescence signal representation at each stage (see also Figure 3 for each stage of the digital signal processing). (a) Waveforms of forward-scattered pulse signal (fs) and observed fluorescence pulse signals (fl_a, fl_b) from a mixture of two spectrally overlapping signals; (b) fl_a and fl_b are modulated by a cosine function with different phase shifts (ϕ_a, ϕ_b); (c) The modulated fluorescence pulse signal is mixed with a reference signal having the same angular frequency as the modulating signal, after which low-pass filtering of the signals in (c) results in (d).

Figure 9

Lifetime histograms of fluorescence pulse signals at 10-MHz modulating frequency. (a) Histograms of (a) fl₁ and (b) fl₂ lifetimes (SD: Standard deviation).