Fig 1.
Bayesian bi-exponential analysis of cell pellet data.
Lifetime analysis of FLIM data from a cell pellet sample. In (a) an intensity image having pixels with a total photon count of between about 20 and 400 were analysed (having invoked 9 × 9 spatial binning to provide sufficient photon counts for a bi-exponential analysis) using the bi-exponential Bayesian algorithm. In (b) and (c) the interacting fraction and the FRET efficiency respectively (as computed using the Bayesian estimates of the decay parameters). The size of a 9 × 9 spatial bin is indicated in (a) by a red square at the centre of the image and in (d) the decay data from from the spatial bin is shown. All of the images correspond to a 334 × 334 μm field of view, and are 256 × 256 pixels. See Appendix A4 Cell Pellet Preparation for details about the sample.
Fig 2.
Repetitive excitation and photon counting.
An illustration of the key components of the FLIM system model having repetition period Tm and a measurement window of duration T. In the upper left panel, a mono-exponential decay of lifetime τ = Tm/6 as a consequence of a sample being subjected to repetitive excitation at discrete times nTm (n = 1, 2, 3,…), the fluorescence signal being retarded by a typical IRF on progressing through the system apparatus. In the upper right panel, the normalised fluorescence signal likelihood within the measurement interval [0, T] (i.e. there is no likelihood of detecting a photon outside of the measurement interval (T, Tm]), and typical data for such a decay having about 1,000 total photons counted into 64 bins of equal width subdividing the measurement interval. The bottom half of the figure repeats the top for a mono-exponential decay of lifetime τ = Tm/2, and again shows data having about 1000 photon counts.
Fig 3.
Bi-exponential parameter estimation at low photon counts.
The uncertainty, as measured by the standard deviation of the estimated parameter distribution, in the bi-exponential decay parameter estimates obtained using ML, LS, and Bayesian analysis for the analysis of synthetic data simulating a bi-exponential decay (,
,
) for a range of intensities between 103 and 104 photon counts. In (a) and (c) the fractional errors in the estimated decay lifetimes versus photon count; in (b) and (d) the fractional errors in the initial amplitudes of the two decay components. In all cases, the normalised width is displayed only when the respective estimates are not biased by more than 5% of the true value.
Fig 4.
FRET efficiency and interacting fraction estimation at low photon counts.
The uncertainty, as measured by the standard deviation of the estimated parameter distribution, in the bi-exponential decay parameter estimates obtained using ML, LS, and Bayesian analysis for the analysis of synthetic data simulating a bi-exponential decay. In (a) and (d) are the fractional errors in FRET efficiency and interacting fraction respectively versus photon count, in (b) and (e) the same for different FRET efficiencies E⋆, and in (c) and (f) for different interacting fractions . In all cases, the normalised width is displayed only when the respective estimates are not biased by more than 5% of the true value. See main text for details.
Fig 5.
Bayesian decay model selection.
In (a) an intensity image having pixels with intensity of about 750 photon counts, those on the left half of the image simulating a mono-exponential decay and those on the right half of the image simulating a bi-exponential decay; in (b) and (c) the Bayesian determined probability of the decay model being bi-exponential, , and the optimal decay model,
, respectively, and in (d) the optimal model as determined by the χ2 model selection algorithm of [38] using the ML parameter estimates. Similar images are shown for human cancer cells expressing GFP (e-h), showing largely mono-exponential characteristics, and for human breast cancer tissue (i-l) which has many ‘contaminants’ from heterogeneous tissue types giving rise to bi-exponential, or higher order, responses. See main text for details.
Fig 6.
Bayesian simultaneous decay and IRF analysis.
In (a) the measured and optimal single- and double-Gaussian IRF approximations as determined on application of the Bayesian SID algorithm to a single data set having over 107 counts obtained by binning the data from all image pixels from a single image of the human carcinoma cell data (see Appendix A3 Human Epithelial Carcinoma Cell Preparation). In (b) and (c), the width and delay parameter estimates obtained on independent analysis of each pixel of the same human carcinoma cell data image using the Bayesian SID algorithm, assuming a mono-exponential decay and a single-Gaussian approximation, for pixels having intensities between about 350 counts and 3500 counts; in (d) the corresponding lifetime estimates. In (b), (c), and (d), the corresponding optimal single-Gaussian IRF approximation estimates are indicated in the histograms by a dotted black line.