Phasor Approach to Fluorescence Lifetime Imaging Microscopy
Enrico Gratton[math]^{1}[/math] and David M. Jameson[math]^{2}[/math]
[math]^{1}[/math]Laboratory for Fluorescence Dynamics, Professor of Biomedical Engineering, University of California, Irvine. Natural Sciences II, Irvine, CA 92697, USA.
[math]^{2}[/math]Professor of Cell and Molecular Biology, University of Hawai’i at Manoa, John A. Burns School of Medicine 651 Ilalo Street, BSB 222, Honolulu, HI 96813 USA.
1 Brief history
The concept of phasor to describe the fluorescence decay was first introduced by Jameson, Gratton and Hall in 1984.[1] Several other equivalent derivations were later developed by other authors using different names.[2, 3] The term phasor is commonly used in other fields of science and predates its use in the field of fluorescence decay analysis. For example phasors have been used for the analysis of dielectric relaxations and mechanical relaxations among others.[4] Traditionally, the concept of phasor is associated with sinusoidal excitation and the “phasor: describes the change of phase and amplitude of the response of the system to the sinusoidal excitation. More generally, a phasor can be used to describe the response of a system for each of the harmonics of a periodic excitation. In the fluorescence field, given that the concept of phasor was originally related to periodic excitation it was naturally to use phasors to describe the response of a fluorescent sample in response to periodic excitation. For this reason, the phasor analysis of fluorescence decay was developed in connection with the frequency-domain methods in the fluorescence lifetime field. Historically, in the fluorescence time-domain field, the excitation source was not periodic, or the period was very longand the methods of decay analysis based on curve fitting were universally used. Since the origin of frequency-domain methods, the community using this approach was attracted by the simplicity of the data analysis in the frequency domain with respect to the time domain. With the advent of high repetition periodic pulsed laser sources in the ’90s, the two traditional areas of data collection and analysis, frequency-domain and time-domain, started to converge. This field further developed for microscopy in the laser scanning or wide field microscopes where the fluorescence decay is measured at a very large number of pixels. For applications to image analysis, the differences between the simplicity and speed of the frequency-domain methods for the analysis of fluorescence decay compared to the time-domain analysis made a difference. It became clear that in the biological samples commonly used in microscopy, the identification of molecular species with complex fluorescence decays was impractical since it required the resolution of several exponential components at each pixel of an image and the simple phasor analyses could convey the information contained in images in a straightforward visualization without resolving exponential components.
2 The phasor transformation of the intensity decay and the phasor plot
Given a periodic excitation source with period [math]T[/math] and a decay of the fluorescence intensity given by [math]I(t)[/math], we construct a phasor by defining the following phasor components indicated with the letters [math]G[/math] and [math]S[/math]
[math]\left\{\begin{matrix}G_{i}(n\omega)=\int_{0}^{T}I_{i}(t)\cos (n\omega t)dt/)\int_{0}^{T}I_{i}(t) \\S_{i}(n\omega)=\int_{0}^{T}I_{i}(t)\sin (n\omega t)dt/)\int_{0}^{T}I_{i}(t) \end{matrix}\right.\tag {1}[/math]
The integrals (or sums) are done over a period of the excitation source, [math]n\omega[/math] represents the [math]n-th[/math] harmonic of the periodic function describing the excitation frequency,[math]\omega=2\pi f[/math] is the angular excitation frequency and [math]f[/math] is the frequency of the periodic source. The index i indicates the [math]i-th[/math] pixel of an image. In the phasor approach, the coordinates G and S are plotted in a polar plot called the phasor plot. There is a phasor point associated with each pixel of an image. An example of a phasor plot is shown in Figure 1.
