A hybrid global fitting algorithm for decay-associated images from fluorescence lifetime image microscopy data Aleksandr V. Smirnov*a, Christian A. Combsa, Robert S. Balaban a, Kevin Tangb, Jay R. Knutsona a National Heart, Lung and Blood Institute, National Institutes of Health, 10 Center Dr., Bethesda MD USA 20892 b Dept. of Biomedical Engineering, Yale University, New Haven, CT USA 06520 ABSTRACT Fluorescence lifetime imaging microscopy (“FLIM”) is a technique for measuring fluorescence lifetime(s) at each pixel of a microscope image. Imaging of fluorescence lifetimes enables biochemical reactions to be followed at each microscopically resolvable location within the cell. FLIM has thus become very useful for biomedical tissue imaging. Global analysis [1] is a method of recovering fluorescence decay parameters from either time-resolved emission spectra to yield Decay-Associated Spectra [2], or equivalently, from FLIM datasets to yield Decay-Associated Images (“DAI”). Global analysis enables a sensitive and non-invasive probe of metabolic state of intracellular molecules such as NADH. Using linkage information, such as the spatial invariance of the lifetime of each fluorescent species in the image, global analysis can recover lifetimes and amplitudes more accurately than traditional pixel-by-pixel analysis. Here, we explain a method to analyze FLIM data so that more accurate lifetimes and DAIs can be computed in a reasonable time. This approach involves coupling an iterative global analysis with linear algebraic operations. Keywords: deconvolution, global fitting, FLIM, DAS, DAI, NADH, free-to-bound ratio. * E-mail:
[email protected] Fax: 1 301 451-5463 http://dir.nhlbi.nih.gov/labs/lmb/os/
1. INTRODUCTION Time-resolved fluorescence spectroscopy is a well-established technique for studying the excited state dynamics of fluorescent molecules i.e. the distribution of times between the electronic excitation of a fluorophore and the emitted photon. The characteristic decay time of this distribution is referred to as the fluorescence lifetime of the molecule. The fluorescence of a pure fluorophore in a homogeneous environment has both a unique emission spectrum and a characteristic lifetime. Interactions between an excited molecule and a heterogeneous environment often change both the fluorescence lifetime and spectrum. Lifetime measurements thus yield information on the molecular microenvironment of a fluorescent molecule. Factors such as ionic strength, hydrophobicity, oxygen concentration, binding to macromolecules and the proximity of acceptor molecules (that can deplete the excited state by resonance energy transfer) can all modify the lifetime of a fluorophore. Lifetime measurements of particular microenvironments are generally absolute, i.e. independent of the concentration of the fluorophore. This can have considerable practical advantages. For example, NADH molecules in mitochondria show changes in fluorescence intensity depending upon whether it is enzyme-bound or free [3]. The intensity-based calibration of NADH binding in a complex system is difficult and prone to errors. NADH, however, exhibits large lifetime changes upon enzyme binding. This means that measurements of relative concentration can be made without the more elaborate calibration procedures required with intensity data. The role of the intrinsic fluorophore NADH as the principal electron donor in glycolytic and oxidative energy metabolism makes it a convenient non-invasive fluorescent probe of metabolic state. Fluorescence spectroscopic techniques are well suited for studying intracellular NADH dynamics, but are hampered by the large differences between free and bound NADH fluorescence yield [3]. Fluorescence lifetime may be a more sensitive probe of NADH binding, although the fluorescence decay of bound NADH might be multi exponential with shorter components comparable with the decay time of free NADH. This makes it difficult to attribute a fast fluorescence decay component in tissue solely to
free NADH. Since intracellular NADH could either be free or bound to many different enzymes, the integrated cellular spectra are an unknown combination of many different spectral influences. Fluorescence decay F(t) can often be satisfactorily modeled as a sum of first order kinetic processes and is thus mathematically represented as a sum of exponentials: F (t ) = ∑ αi exp⎛⎜ − t ⎞⎟ ⎝ τi ⎠ i
where τi is the lifetime of component i and αi its amplitude contribution to the fluorescence decay. Note that the intensity of each lifetime component is not αi, but αiτi, the integral under each curve. A consideration in the analysis of time domain lifetime measurements is that this sum of exponential is necessarily convolved with the instrument response G(t). Thus the total intensity measured I(t) is: t
I (t ) = ∫ G (t − T ) F (T ) d T 0
The fitting of experimental data is traditionally done by iterative reconvolution. A guess of the coefficients αi and lifetimes τi, is used to calculate F(t), which is convolved with the known instrument response G(t) to obtain Imodel. Comparison between the model and the data allows the coefficients to be refined in the next iteration. Commercial imaging software employs these standard iterative reconvolution routines, fitting each pixel independently. This process requires the successive calculation of an enormous number of convolutions, which is a time-consuming process. Moreover, these software algorithms become inaccurate for very noisy decay curves (i.e. those with low photon counts per pixel). Further, at very low counts, the noise has Poisson rather than Gaussian character, leading to systematic underestimation of lifetimes [4]. Global analysis makes use of external information, such as the spatial invariance of the lifetimes in each pixel in the image, to significantly reduce the degrees of freedom in the fitting algorithm resulting in a better measure of the relevant parameters [5]. It has been shown in multi-exponential decay systems that global fitting, assuming spatially invariant lifetimes, can accurately model parameters in cases where pixel-by-pixel analysis can usually only extract a single average lifetime. To further improve lifetime image analysis under noisy or low photon count per pixel conditions, we have developed a technique that utilizes a combination of global iterative non-linear least squares fitting on just a few precise decays gleaned from binned sub-images with linear algebra operations upon every pixel to extract relative contributions of each fluorescence lifetime component. In addition, an adaptive weighting of residuals is implemented to further increase robustness in the Poisson regime. The resulting Decay Associated Images (DAI) allow us to estimate, e.g., relative concentrations of free and enzyme-bound NADH in vitro. The advantage of this hybrid global fitting algorithm over standard global analysis is that it greatly decreases overall FLIM data processing time.
2. METHODS 2.1 Time-resolved two-photon fluorescence spectroscopy Time domain fluorescence lifetime images were acquired with a Zeiss LSM 510 galvo-scanning microscope equipped with a wide-band tunable “Mai Tai” Ti-Sapphire laser (700-950 nm) as its two-photon excitation source. NADH two-photon fluorescence was excited with attenuated 300 fs laser pulses tuned to 710 nm and imaged through a short-pass 680 nm dichroic mirror by means of a non-descanned detection with a 1.3 NA Plan-Neofluar 40x objective. Chroma 700 nm short-pass and Semrock 443/60 nm bandpass filters were used in front of a Hamamatsu Peltier-cooled fast-rise PMT module to eliminate scatter, second harmonic generation and cells autofluorescence. A fast-response silicon photodiode was used for laser synchronization and photon counting events were discriminated, registered and assigned to pixel locations by Becker&Hickl SPC-830 PCI plug-in module for TCSPC. Preliminary data analysis was performed with SPCImage 2.5 software by Becker&Hickl. Typical data collection times varied from 60 s to 15 min. 2.2 Time-correlated single photon counting (TCSPC) TCSPC is an arrival histogramming technique based on the detection of first single photons from a periodic light signal, [6]. A pulsed laser with a repetition rate of typically several MHz excites the fluorescence. A fast PMT or a single
photon avalanche photodiode detects the fluorescence photons. For pileup avoidance these detectors allow count rates on the order of 1MHz, i.e. much less than the laser repetition rate. Thus the arrival probability for multiple photons in one laser period is small enough to be neglected. When a photon is detected A “picosecond stopwatch” (implemented as time-to-amplitude converter module) is started, soon to be stopped by a (cable-delayed) sync pulse from the laser pulse train. The measured times are transferred to a histogramming memory. The benefits of this method are a near-ideal counting efficiency and an ultra-high time resolution.
The instrument response function (FWHMد300 ps) in our case, limited by the PMT detector, was quite sufficient for measurement of the differing lifetime components of NADH (0.3 to 6 ns) [2,3].
3. DATA ANALYSIS Preliminary fluorescence lifetime images were obtained using standard iterative reconvolution routines fitting each pixel independently using the commercial Becker&Hickl SPCImage 2.5 program for FLIM analysis. Since this straightforward approach suffered from noticeable artifacts such as punctate errors found at regions with low counts per pixel, a custom-designed program titled Global FLIM was written in IDL (Interactive Data Language by RSI) that utilizes calls to a dynamic-linked library (DLL) compiled from FORTRAN code for non-linear least squares fitting algorithms to fit the fluorescence decay. Global FLIM utilizes a set of IDL commands that automate calls to the global fitting routine called from DLL to extract lifetime components τi from the measured (user selected binning) subsets of FLIM data. It then solves a system of linear equations to extract relative intensity contributions αi of each fluorescence lifetime component. The iterative global fitting are written in Fortran and compiled into a dynamic link library (DLL) using Microsoft’s Visual Studio C++ compiler and integrated into IDL as a custom function to be called by Global FLIM. When the curve fit approaches the minimum value of chi squared, the weighting method switches to adaptive weighting, which weighs the residuals with the model data instead of the experimental data and continues only a few steps until convergence is achieved [4]. The switch to adaptive weighting prevents Poisson noise distortion [4].
4. RESULTS Using global analysis, we fit each selected region of interest to, e.g. the bi-exponential equation
F (t ) = α1 exp(−t / τ 1 ) + α 2 exp(−t / τ 2 ) . After giving initial guesses for the values of αi and τi, global analysis fits the data (assuming the same τ1 and τ2 across all decay curves) according to the non-linear least squares algorithm and returns τi values. Since global analysis employs the concept of overdetermination to find τi (i.e. using more than two regions to fit two parameters, τ1 and τ2 ), these values of τi are more accurate. We then use these lifetimes to calculate accurate values for pre-exponential factors αi through linear algebra for each pixel element (as follows). Once we have two very accurate lifetimes, we can now create unit convolution curves, here shown as D1 and D2, that predict how much each species emits in a given time window. Now we take areas under the curves (ROIs) for arbitrary time windows and obtain a set of mixing coefficients Ci that tells us how much each lifetime component contributes in each time window. In an early time window, both species are still emitting, but in a late window the short-lived species, e.g. free NADH, is almost extinguished and won’t contribute many photons. From the lifetimes we get a set of mixing coefficients Ci that allow us to predict how much each species contributes. Using linear algebra, we set up our matrix of mixing coefficients, which were obtained by integrating each lifetime component over the time windows, invert the matrix and multiply it with the vector of total intensities F(t) obtained by integrating the original data set over the time windows, and obtain relative concentrations of each species [2]. −1
⎡α1 ⎤ ⎡C1 (a → b) C2 (a → b) ⎤ ⎡ F (t , a → b) ⎤ ⎢α ⎥ = ⎢C (c → d ) C (c → d )⎥ ⎢ F (t , c → d )⎥ ⎦ 2 ⎣ 2⎦ ⎣ 1 ⎦ ⎣
As an aside, we note that multiexponential curves for species could be used to generate D1 and D2, so SAI (species associated images) could be retrieved in the same way for, e.g. FRET or other excited state reactions. Our final product after unmixing time slices are concentration (actually amplitude, which is proportional to concentration times the nearly invariant factors extinction and radiative rate) maps of each emitting species that we call decay-associated images or DAI (Fig. 2C), fully analogous to DAS [2]. Our improved procedure reveals that the relative amounts of free and bound NADH in isolated cardiac myocytes are quite uniform, not punctate. It was previously presumed (from blueshifted spectra) that the proportion of free NADH in mitochondria was small. These results, along with our studies of isolated mitochondria [7] show that free NADH usually exceeds the amount of bound NADH, although we note again that not all of the shortlived component seen is necessarily free. Individual Fluorescence Lifetime Components 1000
D1
800
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600
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400
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0
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100 120 140 160 180 200 220 240 260 280
Time Short lifetime decay Long lifetime decay
Fig. 1. Individual fluorescence lifetime components. Curves D1 and D2 are the convolved decays due to components 1 and 2 respectively.
5. DISCUSSION AND CONCLUSION The combination of global regional fitting with linear algebra shows promise in providing more precise values for lifetimes and pre-exponential factors (DAI, Decay-Associated Images) for the noisy data that are common in FLIM techniques. Using experimental TCSPC images, we have found that the global iterative non-linear least squares fitting provides an accurate way to reduce confidence intervals for lifetimes gleaned from low-count data common to biological images. The linear algebra approach accurately and rapidly extracts relative amplitude contributions ai of each fluorescence lifetime component for each pixel. Although IDL faces many limitations for graphical user interface development, it affords a reasonable pathway for integration with previously written routines in other programming languages such as FORTRAN and C++. As a result, we can create a program that is both fast and readily adaptable to any type of fluorescence microscopy data analysis. We are presently extending the program to analyze data from time-resolved emission anisotropy to yield Anisotropy DAI or ADAI. Emission anisotropy measures not only how long a fluorophore remains in the excited state, but also how quickly it tumbles in solution (Brownian motion) [8]. ADAI of NADH (using associative decay terms) should be better able to discriminate and monitor the effects of changes in concentration, conformation, and the environment of the molecule, enabling a more precise measurement of NADH partitioning into free and multiple bound pools.
(A)
(B)
(C) Fig. 2 (A) Total intensity image. (B) Ratio of apparent free to bound ratio with standard software. The reconvolution analysis has difficulty fitting individual pixels with noisy data. The recovery of amplitude ratio is unreliable, yielding punctate distortions. (C) Linear algebra unmixing process is more accurate than fitting individual pixels; therefore we recover amplitudes of each component with greater accuracy. Long-lived ‘bound’ (right) and short-lived, mostly‘free’ (left) components are seen to be relatively uniform (without punctate features) across the isolated myocyte under these conditions.
6. ACKNOWLEDGMENTS This work was supported by the intramural program of National Heart, Lung, and Blood Institute (NIH) and the Biomedical Summer Internship Program (BESIP) sponsored by the National Institute for Biomedical Imaging and Bioengineering, NIH. Many thanks to John McManigle, now a student at Penn State University, who contributed to the early version of IDL image processing routines.
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