Instrument Response Function¶
The concept of the instrument response function (IRF) is central to the correct analysis of FLIM measurements. The IRF describes the spread of photon arrival times due to purely instrumental effects.
textbox to shift the IRF in time by the value specified. This shift is applied using cubic interpolation so sub-time bin values may be specified. This value may be estimated by using the Estimate IRF Shift menu option (see below).
The IRF can sometimes be measured directly in the imaging system used. In some cases, it is instead convenient to measure a fluorophore with a monoexponential decay, which is referred to as a reference IRF \(g_R(t)\) to obtain information about the IRF. The Reference IRF is typically a high-quality experimental acquisition of a mono-exponential fluorescence decay with known fluorescence lifetimeτR obtained from a suitable sample in the spectral window of interest. The IRF can be pre-processed by baseline-subtraction and it is then normalized to have unit area in order to represent the instruments’ photon arrival time probability density. When using a reference decay that sufficiently short that it can be considered to be practically instantaneous (τR =0), the reference IRF becomes an impulse response function, or delta-function IRF gδ(t).
- Consider the fluorescence decay caused by the arrival of fast laser pulses at times t = −nT (n=0,1,2,… ∞), and recorded in the time window [0,T], where T is the pulse repetition period. One could derive the following model expression to fit the measured decay intensity to the Nexp exponential components, basing on the delta-function IRF gδ(t):
where Itot is the total number of photons in the decay (“total intensity”), τk are fluorescence lifetimes, and fk are fractional contributions of components to the total decay (fk are summing to 1). Formula (1) takes into account fluorescence intensities originating from the current laser pulse at t=0, and from all previous pulses. The term in brackets is the photon arrival time probability density for the model decay, as it integrates to 1 in the range [0,T]. The convolution of this term with the IRF then gives the joint probability density of the photon arrival at time t∈[0,T] due to independent delays of photons in the sample and in the instrument. The tilde is used to indicate that a decay model has been convolved with the IRF. In the case when reference IRF gR(t) is used instead of gδ(t), the decay model is modified to compensate for the additional convolution of the IRF with the reference decay. The model becomes
where δ(t) is Dirac delta. As expected, the expression (2) reduces to (1) in the limit τR →0, i.e. when gR(t)→gδ(t). For fluorescence decays, the fractional contributions of the decay components fk are proportional to their lifetime τk and to the molecule fraction βk of the corresponding fluorescent emitters, i.e. fk∝τk⋅βk. Substituting this into the models (1) and (2) and introducing the intensities of the scatter light S and the background B, one can rewrite these expressions in the form convenient for fitting purposes, as
The advantage of using intensity amplitudes Ak in the fitting model is that they are directly proportional to the above fractions βk of fluorescent molecules participating in corresponding decays (not to numbers of photons they emit). Molecule fractions are then calculated as , and reported by the software. The above description is suitable for TCSPC and widefield time-gated FLIM modalities.
Dealing with backgrounds¶
- In some cases, the background intensity B might be present in the measured FLIM data, i.e. a background decay that is due to the sample preparation or instrument, and therefore should be included as an additional component in the fitting model. The expression for B becomes
where b(t) is a spatially invariant time-varying background (TVB) function. This TVB can be determined by measuring a control sample without fluorescent objects of interest. The contribution V of this TVB can be fitted but it is preferred if it can be measured experimentally where possible. The term Z represents a constant offset to the data, e.g. due to detector dark/thermal current, CCD readout noise or a background of stray room light. In the case of reference reconvolution, i.e. using equation (4), but when it is necessary to fit a scattered contribution, the true IRF gδ (t) can be estimated from the reference IRF gR(t) through the use of a Fourier deconvolution method.
However, it is currently not possible to add a spatially and time varying background to the model, and so this option must be used if you wish to account for a spatially and temporally varying background.
- The integrated intensity for pixel s is calculated by summing the data over time bins or gates, accounting for any provided background and scatter and therefore represents an estimation of the total signal of interest. Using (5), one can write:
where ti is the photon arrival time assigned to the i-th bin or gate, ys(ti) is the measured photon count of the pixel s in the i-th time bin (gate). The time-independent contribution to background is split into the spatial varying (ZSV,s) and constant (Zconst) components for convenience. Note that, depending on software settings, any of the parameters except ys(ti), b(ti) and gδ,s(ti) might be fitted parameters and therefore the integrated intensity may change when the data is refitted using a different model.
In the software, the parameters of the expressions (3), (4) and (5) might be fitted locally (i.e. separately for every pixel in the FLIM image), globally (when their values are the same for groups of fitted pixels such as an image or a group of images), or fixed (set up by the user to a constant, not fitted). As the molecule fractions are of primary interest, they are often fitted locally. The parameters Nexp and T are always fixed. The IRFs and TVB functions are also considered fixed. However, the IRFs (as well as scatter light terms ) might be time-shifted, or different for different pixels in the image, or both. These shifts are not shown in the formulas in order to keep them observable.
timebins from the IRF which are excluded from the fit using Time Min and Time Max are set to the background value and no background is subtracted from the IRF. This correctly compensates for the presence of afterpulsing, which is common in TCSPC data acquired using a photomultiplier tube.