The Ultimate Phasor Plot and beyond
The Ultimate Phasor Plot and beyond
Yuansheng Sun and Shih-Chu Liao ISS
1Introduction to the phasor plot
Fluorescence Lifetime Imaging Microscopy (FLIM) is a useful technique that provides, with its peculiar selectivity of fluorophores, a higher contrast of confocal images; moreover, it is used for providing quantitative information of the cell environment (ions, pH, oxygen content, electrical signals, index of refraction). When used in Förster Resonance Energy Transfer (FRET) applications, the measurement of the decay times provide indirect information about the proximity of target fluorophores up to a distance of about 10 nm.
The data analysis approach utilized to extract the decay times information hampers the applications of FLIM and, in several cases, provides information that are just a rough approximation of the phenomena occurring on a molecular scale. The typical approach consists in separating the region of interest (ROI) in the image and, by using a chi- square minimization technique, extracting a 1-, 2- or 3-decay times. Yet, the limitation to one or three decay rates of fluorophores is in many cases an arbitrary one and in most cases limited as, in a cellular environment, several decay rates are present and active at the same time. Additionally, the process is fairly complicated. It has always been a challenge for many users to apply the traditional non-linear least square fitting for FLIM data analysis: questions like which decay model should be chosen–single, double, or tripleexponential?; and how to make initial guesses for the fitting parameters to get are liable convergence?; and after the analysis, how to evaluate the goodness of the fit and is my “Chi Square” good enough? The answers to these questions require a deep knowledge of the statistical analysis and the minimization technique algorithm functionality. The phasor plot leaves all of these questions behind and presents an intuitive simple interface for you to get instantaneous and quantitative results. Does it sounds too good to betrue?
What is the phasor plot? The phasor plot is a graphical representation of all the raw fluorescence lifetime imaging microscopy (FLIM) data in a vector space; that is, each pixel in a FLIM image is transformed to a point in the phasor plot. No assumption is made on the number of decay rates present in the environment as well as on the specific modeling of the decay (exponential, non-exponential). This transformation of the pixels of a FLIM image into the phasor plot can be equally applied to both time- and frequency-domain FLIM data. Since ISS provides both time- and frequency-domain FLIM instrumentation solutions, the phasor plot analysis routines for both are included in the VistaVision software by ISS.
An in-depth mathematical description of the phasor plot (the phasor plot for FLIM data analysis) is available on the ISS website – www.iss.com/resources/research/technical_notes. In this Note we only present the relevant results as applied to FLIM.
The phasor space is constructed by using two phasor vectors (G, S), where each component is represented as shown in Eq. [1].
|
gx,y(w) = mx,ycos(jx,y) |
|
sx,y(w) = mx,ysin (jx,y) |
[1] |
n a frequency-domain FLIM measurement, mx,y and φx,y are the modulation ratio and the phase delay measured given a particular modulation frequency (ω) at a pixel location (x,y). In a time-domain FLIM measurement (such as TCSPC - time correlated single photon counting), the two phasors can be obtained through the sine and cosine transforms of the raw decay data, given by Eq. [2], where Ix,y(t) is the photon count of each time bin “t” at a pixel location (x,y), and ω is the angular frequency of harmonic content of the excitation light and is equal to “2πf”, where f is the basis repetition rate of the pulsed excitation light.