spechronImage Analysis tools

Introduction

Phasor approach is a graphical method which has been used for representation of alternative currents and electromagnetic waves which simplifies their analysis by converting them into a vector. In Fluorescence lifetime imaging, the decay curves are transformed into a vector where the magnitude and the phase of the vector correspond to the lifetime of the decay curve.


                                   



















 Assuming an exponential decay curve with lifetime of τ :
                                                                        



The Fourier transformation is represented by:

                         





This is a complex function where plotting the imaginary part versus real part for lifetimes ranging from zero to infinity will make a semicircle :

                                                             
















As demonstrated in the figure above, by varying the lifetime from zero to infinity the phasor point moves along a semicircle in the counter clock wise direction from coordinate (1,0) to (0,0). The lifetimes can be estimated by measuring the phase or the amplitude of the phasor shown by pink vector. For measurements with a discrete number of time bins of equal width, like time gating and TCSPC, equation for reference semicircle R can be rewritten as(Fereidouni et al.):

                                                                    








Where T is the total time window and K is the number of time gates. ωis the first harmonic frequency and is defined by: 2π/T. Figure below shows the modified semicircle for different number of time gates.