Algebraic Multicomponent Fluorescence Lifetime Estimation from Harmonic Phase Measurements under Two-Photon Excitation.

We formulate and numerically evaluate a direct, non-iterative algebraic estimator for multicomponent fluorescence lifetimes using harmonic phase measurements under two-photon excitation. Complex harmonic responses at the repetition frequency of a femtosecond pulse train and its harmonics are mapped through linear coefficient reconstruction to the denominator polynomial of a rational representation of the multiexponential decay, and the decay constants are obtained from its roots. The method requires neither initial lifetime estimates nor iterative nonlinear minimization.

Photon-limited performance was evaluated for a three-component system using Poisson statistics. For closely spaced lifetimes of 0.5, 1.5, and 2.5 ns, the mean relative lifetime error decreased from 14.49% at 100 peak expected photons per time bin to 5.94% at 1000 photons and 1.31% at 10000 photons after averaging 500 statistically independent acquisitions. In a matched 500-trial Monte Carlo comparison at (1 x 107) expected photons per pixel, success probabilities were 0.484, 0.854, and 0.898 for the unconstrained algebraic, DC-constrained algebraic, and single-start nonlinear least-squares (NLLS) estimators, respectively. Thus, the DC constraint substantially improved photon-noise robustness, bringing the algebraic estimator close to NLLS performance.

In a separate matched computational benchmark at 1 x 108photons per pixel, median estimation times were 0.138, 0.095, and 5.552 ms per pixel, respectively. These results demonstrate a constraint-induced robustness improvement and a robustness-computation trade-off relative to iterative fitting. The present work is a theoretical and numerical implementation within the established multifrequency frequency-domain framework; experimental validation and harmonic-dependent instrument-response calibration remain necessary for practical two-photon FLIM.