Luminescence lifetime encoding in time-domain flow cytometry

Time-resolved flow cytometry represents an alternative to commonly applied spectral or intensity multiplexing in bioanalytics. At present, the vast majority of the reports on this topic focuses on phase-domain techniques and specific applications. In this report, we present a flow cytometry platform with time-resolved detection based on a compact setup and straightforward time-domain measurements utilizing lifetime-encoded beads with lifetimes in the nanosecond range. We provide general assessment of time-domain flow cytometry and discuss the concept of this platform to address achievable resolution limits, data analysis, and requirements on suitable encoding dyes. Experimental data are complemented by numerical calculations on photon count numbers and impact of noise and measurement time on the obtained lifetime values.

. Spectroscopic ensemble measurements. (a) Photoluminescence emission spectra of lifetime codes. λ ex was 488 nm (organic dyes) or 425 nm (quantum dots, for higher signal intensity only). The emission band pass was 4-6 nm. (b) Photoluminescence intensity decay curves of lifetime codes measurements for excitation at 488 nm (16 nm band pass; 5 MHz repetition rate for code E, 10 MHz for all others) and detection with a 645 or 590 nm (sample C, for improved signal intensity only) long pass filter.

Numerical generation of synthetic decay curves
The study of synthetic decay curves is based on pseudo-random numbers following the respective probability distributions. The numerical generation of such distributions will be described in the following based on textbook procedures 1 .

Analytical approach for mono-exponential decay
In case of mono-exponential fluorescence decay with lifetime τ, the intensity distribution in time t follows Eq. (S1).
Standard pseudo random number generators (PRNG), however, usually provide uniformly distributed random numbers x on the interval [0, 1]. Thus, we need to map these numbers on a probability distribution that would produce intensity decays according to Eq. (S1). Such a probability distribution is given by Eq. (S2).
We now define Eq. (S3), which maps x ∈ [0, ∞[→ y ∈ [0, 1]. If Eq. (S3) can be solved for x, we can transform a uniform distribution of values of y to the required distribution of values of x. In case of a mono-exponential decay, cf. Eq. (S1), we have which can be easily inverted to give Eq. (S4).
Thus, for mono-exponential decays we can generate uniform distributions of y and transform them to the desired monoexponential distributions of x, i.e. time t in the physical process.

Von-Neumann rejection for multi-exponential decays
For multiple decay components or non-exponential decays, Eq. (S3) might not be invertible with respect to x. Thus, another solution for the problem must be found. In addition to the now modified w(x) we now need a second distribution w (x) with w (x) > w(x) for all x in the required interval. The only requirement is that w (x) allows for generation of a random number distribution as described above. One then generates pairs of values x i (distributed according to w ) and ξ (distributed uniformly on [0, 1]). For each pair, the value of x i is only accepted when and discarded otherwise. The process is repeated until the desired number of random values has been generated. For multi-exponential decay and a suitable choice of w could be where τ m = max(τ j ). Practically, a loop generates pairs of x i and ξ and checks the condition of Eq. (S5) each time. The loop terminates when the required number of accepted x i has been generated.

Numerical simulation of lifetime determination in flow cytometry
Synthetic decay curves generated from pseudo-random numbers as described above can be used as starting point to deduce theoretical dependencies of the measured luminescence lifetime in flow cytometry on various parameters such as integration time range, bin width, signal intensity, or signal-to-background ratio. Therefore, a set of decay curves with certain decay kinetics and given overall number of counts is generated. The data are binned as desired and background counts can be added. Lifetime determination on the simulated data is done by means of where t j is the time value of bin j and I j is the respective number of counts in that bin. The integration time range is determined by j max . The number of generated decay curves resembles the number of objects in a flow cytometry experiment and provides statistical relevance of the simulation results. Simulations were performed with custom-made Octave 2 scripts.

3/7 Complementary simulation results
In addition to the investigation of the impact of the photon count number on the calculated lifetime for mono-exponential decays without background signal, Fig. S2 shows calculations for bi-exponential decay and background counts. The relative deviation from the notional lifetime approaches a constant value for higher photon count numbers. The offset from zero deviation is caused by the background counts. The step-like changes in the relative deviation stem from the discrete nature of the counts. Therefore, a background count is, on average, only added when the photon count number exceeds certain values which results in a stepwise change of the simulated SBR at small photon count numbers. The discretisation effects vanish for increasing photon count numbers as the background count increase in a more continuous manner. Apart from these artefacts, the relative deviation is independent of the photon count number and therefore also luminophores with bi-exponential (or multi-exponential) decays are expected to be suitable for lifetime encoding.
It was shown in the main manuscript that the SBR affects the calculated lifetime. This effect is displayed again for several different notional lifetimes in Fig. S3. Obviously, the relative deviation increases drastically with decreasing notional lifetimes. . As the calculated lifetime approaches θ /2 for very low SBR independent of the notional lifetime, the relative deviation is larger for small notional lifetimes lifetimes. The course of the standard deviation is independent of the notional lifetime.

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This means, shorter lifetimes undergo a stronger relative distortion due to background counts. Anyway, all measured lifetimes will approach θ /2 with decreasing SBR. Consequently, no distinction will be possible beyond a certain background level and lifetime codes may mix upon different SBR per code carrier. The standard deviation, however, is basically independent of the notional lifetime for small SBR and it is slightly larger for longer notional lifetimes for increasing SBR.
As discussed in the main manuscript, the bin width plays an important role for the correct determination of lifetimes. On the contrary, the bin width does not affect the standard deviation, i.e. the precision, of the determined lifetimes as shown in Fig. S4.  Figure S4. Dependence of the standard deviation of the calculated lifetime on the photon count number for different bin widths. Time range θ was 60 ns, 1000 repetitions were performed (equivalent to 1000 objects in an experiment). Clearly, the bin width does not influence the lifetime distribution standard deviation and is therefore less critical for improved precision.

Analytical expressions for lifetime standard deviation
The analytical expressions for the standard deviation in lifetime estimation procedures used for comparison with our numerical results shall be given here for the sake of completeness. When background is absent, the variance of the lifetime is 3 where N is the number of photon counts, τ the actual lifetime, θ the integration time range, k the number of bins, and r = θ /τ. In case of background signal being present, the expression changes to 3

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Technical characteristics of the lifetime flow cytometer

Description of optical components
A detailed list of the optical filters and mirrors of the lifetime flow cytometer (LT-FCM) is given in Tab. S1.

Linearity of the detection system
Operating a detection system outside its linear range leads to signal distortion. Thus, we assessed the response of our detection system in terms of photon counts as a function of the transmittance of the used neutral density filter transmission, see Fig. S5. Even though the number of acquired photons is low on the scale of typical time-resolved measurements, the detector is already