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FLIM

Single-Image Fluorescence Lifetime Imaging Microscopy

Toggel features a unique image sensor that was designed and optimized specifically for fluorescence lifetime imaging applications. It enables lifetime imaging at unprecedented frame rates with the single-image fluorescence lifetime imaging microscopy (siFLIM) method [1].

"We used a prototype of the Toggel camera to develop a method for acquiring quantitative lifetime images from a single exposure." says professor Kees Jalink of the Netherlands Cancer Institute. "siFLIM takes advantage of the technical capabilities of the Toggel camera and simultaneously records two 180°-phase-shifted images. This allows for video-rate lifetime imaging with minimal phototoxicity and bleaching."

During a single exposure, the Toggel camera records two images. The electrons in each pixel of the sensor are toggled between two storage areas at the same frequency as the modulation frequency of the light source. This results in two images that are shifted 180° in phase with respect to each other.

Because both images are recorded simultaneously, there are no artifacts of cellular movements and there is a significant improvement in the photon efficiency of the image acquisition.

These two images of convallaria were recorded simultaneously, and because of the 180° phase shift between the two images, the recorded light intensity differs. The difference in light intensity between these two images depends on the phase shift and demodulation of the fluorescence light and can be used to calculate changes in the fluorescence lifetime of a sample [1].

 

[1] siFLIM: single-image frequency-domain FLIM provides fast and photon-efficient lifetime data, M. Raspe et al, Nature Methods 13, 501–504 (2016)

 Schematic representation of one pixel in the image sensor of the Toggel. The electric field (rotating black line) in the pixel directs the photo-electrons (green) to one of the storage areas (gray).

Schematic representation of one pixel in the image sensor of the Toggel. The electric field (rotating black line) in the pixel directs the photo-electrons (green) to one of the storage areas (gray).

TIRF FLIM

TIRF-evanescent-wave.jpg

Total Internal Reflection Fluorescence (TIRF) microscopy is a super-resolution technique with high sensitivity of fluorescence near the cover glass. TIRF does not disturb cellular activity, and enables tracking of biomolecules, and the study of their dynamic activity and interactions at the molecular level. TIRF enables the selective visualisation of processes and structures of the cell membrane and pre-membrane space such as vesicle release and transport, cell adhesion, secretion, membrane protein dynamics and distribution or receptor-ligand interactions. The  combination of TIRF and frequency domain FLIM makes it possible to measure fluorescence lifetimes of for instance small focal adhesions near the cover glass.

High NA TIRF objectives (up to 1.49) make it possible to introduce illumination at incident angles greater than the critical angle (\(\theta\)) resulting in TIR (Total Internal Reflection) accompanied by the formation of an evanescent wave immediately adjacent to the coverglass-specimen interface. The evanescent wave energy drops off exponentially with distance from the coverglass and reaches about a hundred nanometers into the specimen. For TIR to occur, the refractive index of the coverglass should be higher than the refractive index of the specimen (which is e.g. the case when using buffered saline solution).

The white-TIRF as well as the laser-TIRF system utilises this evanescent wave to excite fluorescent molecules in a very thin section in contact with the coverglass (here: green dots). Because the specimen is not excited beyond the evanescent wave (here: white dots), this imaging system can produce fluorescence images with an extremely high signal-to-noise (S/N) ratio and z-resolution.

For more information: Wikipedia, Olympus FluoView.

Confocal FLIM

Confocal microscopy is a technique for high-resolution three-dimensional imaging which uses a pinhole to increase resolution in the image plane and eliminate out-of-focus light in thick specimens. The thickness of the focal plane is generally defined mostly by the objective lens and also by the optical properties of the specimen and the ambient conditions. With an imaging confocal only the light within the focal plane is detected, so that the resulting confocal images appear crisper than widefield images [read more at the cell biology wiki]. Typical applications occur within the life sciences, e.g. in cell biology. In essence there are two classes of confocal systems: single beam and multi-beam.

Confocal Scanning with CSU Spinning Disk

A Nipkow spinning disk is a multi-beam confocal scanner. The main advantage of this type of confocal imaging is the relatively fast imaging acquisition making it useful for live cell imaging applications.

The operating principle of the Yokogawa CSU spinning disk is explained here. Briefly, the disk has a spiral pattern of pinholes that is illuminated by an expanded laser beam. This generates a multi-beam illumination pattern which which the sample is illuminated. By rapidly rotating the disk this multi-beam visits all positions in the sample plane near simultaneously. The part of the fluorescence that travels back through the pinholes generates a full field confocal image at the camera detector. 

   Lifetime images of mixed polled grains, at three different z-positions.

Lifetime images of mixed polled grains, at three different z-positions.

Being a camera-based system, the Lambert Instruments LIFA system for frequency domain FLIM is compatible with multi-beam confocal microscopy techniques, most notably the Yokogawa CSU spinning disk series (based on the Nipkow disk scanner), and the VTInfinity series by Visitech International Ltd.

Frequency-Domain FLIM for Beginners

Fluorescence lifetime imaging microscopy (FLIM) can be performed in the time domain and in the frequency domain. Scanning single point lifetime detection units on confocal laser scanning microscopes mainly operate in the time domain. Camera-based lifetime detection on widefield, multi-beam confocal and total internal reflection fluorescence (TIRF) microscopes operate both in time domain and frequency domain. The Lambert Instruments LIFA for example is a fast frequency-domain system, whereas the Lambert Instruments TRiCAM can be operated both in the frequency and time domains.
 

Time Domain

In the time domain the fluorescence decay can be measured by using time-correlated single photon counting (TCSPC) or fast-gated image intensifiers. A measurement requires short excitation pulses of high intensity and fast detection circuits. Each point in the sample is excited sequentially. TCSPC records a histogram of photon arrival times at each spatial location using Photo Multiplier Tubes (PMTs) or comparable single photon counting detectors. Fast-gated image intensifiers measure fluorescence intensity in a series of different time windows. With both time domain techniques lifetimes are derived from exponential fits to the decay data. When sufficient channels (time windows) are used, multi-exponential lifetimes can be extracted.
 

Frequency Domain

The frequency-domain FLIM technique requires a modulated light source and a modulated detector. The excitation light is modulated or pulsed in intensity at a certain radio frequency (the blue curve in the figure below). The induced fluorescence emission will mirror this modulation pattern and show, due to the fluorescence decay, a delay in time in the form of a phase-shift (the red curve). In addition, the modulation depth will decrease with respect to the excitation light, while the average intensity remains the same. The phase-shift and modulation-depth directly depend on the fluorescence lifetime and the known modulation frequency (see figure).

To extract the phase shift and modulation depth from the fluorescence emission signal, a homodyne detection method is often used. In this method the sensitivity of the detector - often an intensified camera - is modulated (or gated) with the same radio frequency as the light source (the green curve in the figure on the right). For a camera detector the result is an intensity image with a fixed brightness. By shifting the phase of the image intensifier with respect to the light source in a series of fixed steps a low-pass signal is generated for each pixel: the output image will be brighter or dimmer depending on whether the detector sensitivity is in or out of phase with the fluorescence emission. The result is a frequency-domain FLIM signal as a function of the phase difference between light source and camera (purple curve in the figure on the right) for each pixel in the image.

The key is that this frequency-domain signal (purple curve) exactly mirrors the phase shift and demodulation in the time domain. The phase and modulation depth can be directly extracted from the measurements and are the fundamental data in a homodyne FD FLIM measurement.

From the acquired modulation depth and phase shifts, two independent determinations of the fluorescence lifetime can be calculated. For an absolute determination the system needs to be calibrated at the pixel level with a reference fluorophore of known lifetime. For this calibration the only requirement is a FLIM acquisition of the reference fluorophore with known lifetime (figure on the right).
 

Multi-Exponential Decay

Some fluorophores have a multi-exponential decay, consisting of two or more lifetime components. For example, the decay of CFP is bi-exponential. These multiple lifetime components can be separated and extracted using multiple frequency measurements and the polar (or phasor) plot.
 

 

Advantages of Frequency-Domain FLIM

The key advantage of frequency-domain FLIM is its fast lifetime image acquisition making it suitable for dynamic applications such as live cell research: the entire field of view is excited semi-continously - using relatively broad excitation pulses - and read out simultaneously. Hence frequency domain lifetime imaging can be near instantaneous. Another advantage of a camera-based FLIM setup, such as the Lambert Instruments LIFA, is its ease of use and its low maintenance requirements. For more information about our products, please visit the FLIM product pages and our FLIM software page.
 

 Time-lapse of EPAC sensor courtesy of Netherlands Cancer Institute

Time-lapse of EPAC sensor courtesy of Netherlands Cancer Institute


More information

Principles of Fluorescence Spectroscopy, Springer 3rd ed., J.R. Lakowicz (2006)

Advances in Biochemical Engineering/Biotechnology, chapter Fluorescence Lifetime Imaging Microscopy (FLIM) by E.B. van Munster & T.W.J. Gadella (2005) 95:143-175

FLIM microscopy in Biology and Medicine, CRC Press, by A. Periasamy and R.M. Clegg, editors (2010)

FRET & FLIM Techniques, Elsevier, by T.W.J. Gadella, editor (2009)

Fluorescence Lifetime Imaging Microscopy

What is the fluorescence lifetime?

The fluorescence lifetime - the average decay time of a fluorescence molecule's excited state - is a quantitative signature which can be used to probe structure and dynamics at micro- and nano scales. FLIM (Fluorescence Lifetime Imaging Microscopy) is used as a routine technique in cell biology to map the lifetime within living cells, tissues and whole organisms. The fluorescence lifetime is affected by a range of biophysical phenomena and hence the applications of FLIM are many: from ion imaging and oxygen imaging to studying cell function and cell disease in quantitative cell biology using FRET.

For fluorescent molecules the temporal decay can be assumed as an exponential decay probability function:

\[ P_{decay}(t)={1/\tau} e^{-{t/\tau}},\quad t\gt 0 \]

where \(t\) is time and \(\tau\) is the excited state lifetime.

More complex fluorophores can be described using a multiple exponential probability density function:

\[ P_{multiple-decay}(t)={\sum_{i=1}^{N}}\alpha_i\cdot P_{\tau_i}(t),\quad t\gt 0 \]

where \(t\) is time, \(\tau_i\) is the lifetime of each component and \(\alpha_i\) is the relative contribution of each component.

Why measure fluorescence lifetime?

A key advantage of the fluorescence lifetime is that it is a basic physical parameter that does not change with variations in local fluorophore concentration and is independent of the fluorescence excitation. Hence the lifetime is a direct quantitative measure, and its measurement - in contrast to e.g. the recorded fluorescence intensity - does not require detailed calibrations. Excited state lifetimes are also independent of the optical path of the microscope, photobleaching (at least to first order), and the local fluorescence detection efficiency.

The fluorescence lifetime does change when the molecules undergo de-excitation through other processes than fluorescence such as dynamic quenching through molecular collisions with small soluble molecules like ions or oxygen (Stern-Volmer quenching) or energy transfer to a nearby molecule through FRET. As a result the fluorophores (in the excited state) lose their energy at a higher rate, causing a distinct decrease in the fluorescence lifetime. The measured rate of fluorescence is actually a summation of all of the rates of de-excitation. In this way the fluorescent lifetime mirrors any process in the micro-environment that quenches the fluorophores; and spatial differences in the amount of quenching reveals itself as contrast in a lifetime image.


Related Posts

FLIM Papers and Reviews

A selection of papers papers based on Lambert Instruments FLIM systems is maintained here.

The following is a selection of books and papers on FLIM technology:

BOOKS / REVIEWS:

  • Gadella TW Jr., FRET and FLIM techniques, 33. Elsevier, ISBN-13: 978-0080549583. (Dec 2008) 560 pages. Elsevier link
  • Periasamy, A & Clegg RM, FLIM Microscopy in Biology and Medicine. Chapman & Hall/CRC, 1st edition, ISBN-13: 978-1420078909. (Jul 2009) 368 pages. Amazon link
  • Lakowicz JR. Principles of fluorescence spectroscopy, 3rd edition, ISBN-13: 978-0387312781 . Springer, 3rd edition (Sep 2006) 954 pages. Amazon link
  • Van Munster EB, Gadella TW Jr. Fluorescence lifetime imaging microscopy (FLIM). Review. Adv Biochem Eng Biotechnol. (2005) 95:143-75. Pubmed link

BOOKS; FLUORESCENT PROTEINS / CELL BIOLOGY:

  • Sullivan KF, Fluorescent Proteins, 2nd Edition Volume 85. Academic Press, ISBN-13: 978-0123725585. (Dec 2007) 660 pages. Elsevier link
  • Sullivan KF, Kay SA, Wilson L, Matsudaira PT,Green Fluorescent Proteins, Volume 58. Academic Press, ISBN-13: 978-0125441605. (1998) 386 pages. Amazon link

PAPER MULTIFREQUENCY:

  • Squire A, Verveer PJ, Bastiaens PIH, Multiple frequency Fluorescence lifetime imaging microscopy. Journal of Microscopy, (2000) 197(2):136-149. Pubmed link

PAPER PHASE STEP ORDER:

  • Van Munster EB, Gadella TW Jr, Suppression of photobleaching-induced artifacts in frequency-domain FLIM by permutation of the recording order. Cytometry A. (2004) 58(2):185-94. Pubmed link

PAPERS POLAR PLOT:

  • Redford GI, Clegg RM, Polar plot representation for frequency-domain analysis of fluorescence lifetimes. Journal of Fluorescence (2005) 15(5):805-815. Pubmed link
  • Clayton AHA, Hanley QS, Verveer PJ. Graphical representation and multicomponent analysis of single-frequency fluorescence lifetime imaging microscopy data. Journal of Microscopy (2004) 213(1):1-5. Pubmed link

PAPERS TIRF-FLIM:

  • Valdembri D, Caswell PT, Anderson KI, Schwarz JP, König I, Astanina E, Caccavari F, Norman JC, Humphries MJ, Bussolino F, Serini G, Neuropilin-1/GIPC1 signaling regulates alpha5beta1 integrin traffic and function in endothelial cells. PLoS Biol. (2009) 27:7(1):e25. Full text

Modulated Intensifiers for Lifetime Imaging

As part of its portfolio, Lambert Instruments designs and manufactures ICCD cameras based on modulated image intensifiers for frequency domain imaging techniques such as FLIM. Standard products such as the new TRiCAM M ICCD camera, the TRiCATT M modulated intensifier attachment, and the LIFA system are all based on a proximity-focused modulated Gen II or Gen III image intensifier.

 Figure 1. Schematic representation of a proximity-focused image intensifier, showing its photocathode, micro-channel plate and anode, fiber-optically coupled to a sensor.

Figure 1. Schematic representation of a proximity-focused image intensifier, showing its photocathode, micro-channel plate and anode, fiber-optically coupled to a sensor.

 Figure 2. Modulated detector gain (green), pulsed excitation (blue) and fluorescence emission (red) in a homodyne FLIM system. The phase and duty cycle of the detector gain is controllable.

Figure 2. Modulated detector gain (green), pulsed excitation (blue) and fluorescence emission (red) in a homodyne FLIM system. The phase and duty cycle of the detector gain is controllable.

The modulated sensitivity of an image intensifier is produced by high-frequency switching of its photocathode voltage. In this modulation mode the temporal and also the spatial characteristics of the image intensifier are different from the nominal characteristics under continuous operation.

The following table lists the nominal characteristics for the Gen II and Gen III intensifiers provided.

 

Gen II S20 & S25

Gen III GaAs

Gen III GaAsP

Spatial resolution
Photocathode QE at 550 nm

28 lp/mm
14%

17 lp/mm
28%

7 lp/mm
48%

The increased sensitivity of the Gen III image intensifier results in a better lifetime accuracy, although the amount further depends on the model and the modulated light source used. For a LIFA Gen III GaAs compared to a LIFA Gen II S25 at 550nm, both using the same Multi-LED light source, the increased sensitivity results in an increased lifetime accuracy of 40% at the same acquisition speed and fluorescence intensity.

For phosphorescence lifetime imaging with the TRiCAM GM and the LIFA-P the image intensifier is operated at lower frequencies spatial resolution is higher, reaching values in excess of 35 lp/mm for a Gen III GaAs image intensifier.

Please note that the characteristics of an image intensifier are unique and are known to vary between units.

 Figure 3. Image intensifier spectral response.

Figure 3. Image intensifier spectral response.

Frequency-Domain FLIM: Basic Equations

In this article the first principles of frequency domain (FD) fluorescence lifetime imaging microscopy (FLIM) are further explained through use of equations. Although these principles not necessary for the execution of basic lifetime measurements, a thorough understanding provides the groundwork that enables deeper insight into your results and into the possibilities of FD lifetime imaging.

Fluorescence Lifetime

For an ensemble of fluorescent molecules the rate at which molecules decay from the excited state to the ground state (cf. Fig.1) is proportional to the number of excited state molecules \(N\):

\begin{equation}
\frac{d N}{d t} = - k_{F} N
\end{equation}

where \(k_{F}\) is the fluorescence decay rate (in units of \(s^{-1}\)).

  Figure 1. Jablonski diagram and spectra showing the fundamental photophysical processes in organic molecules: absorption of a photon (      S  0  >  S  1  ,  S  2      ), internal conversion (      S  2  >  S  1      , non-radiative), fluorescence (      S  1  >  S  0      ), intersystem crossing (      S  1  >  T  1      ) and phosphorescence (      T  1  >  S  0      ).

Figure 1. Jablonski diagram and spectra showing the fundamental photophysical processes in organic molecules: absorption of a photon (S0>S1,S2), internal conversion (S2>S1, non-radiative), fluorescence (S1>S0), intersystem crossing (S1>T1) and phosphorescence (T1>S0).

In case the number of excited state molecules is not re-supplied and there are no quenching mechanisms other than fluorescence, the number of molecules that drop to the ground state at time \(t\) is given by the solution to equation [1]:
\begin{equation}
N(t) = N_{0}\ e^{-t/\tau}
\end{equation}

\noindent where \(\tau = k_{F}^{-1}\) is the fluorescence lifetime, or the average time the fluorescent molecules spend in the excited state. It is this characteristic timescale which is measured with Fluoresence Lifetime Imaging Microscopy (FLIM) technology.

Modulated excitation and fluorescence emission

The exponential decay is the impulse or fundamental response of an ensemble of fluorescent molecules and only follows after excitation by an infinitely short excitation pulse. In practice, the excitation light is not an infinitely short pulse, and the fluorescence emission \(F(t)\) is the convolution of the excitation light waveform \(E(t)\) with the impulse response:
\begin{equation}
F(t) = \int_{0}^{t} E(t') F_{\delta}(t-t') dt'
\end{equation}

For FLIM the excitation signal \(E(t)\) can be a train of pulses or any repetitive waveform:
\begin{equation}
E(t)=E_0+\sum_{\substack{n=-\infty\\n\ne0}}^{+\infty}\left|E_{\omega,n}\right|e^{i(\omega_nt + \phi_n^E)}
\end{equation}

  Figure 2. Typical excitation signal used in a frequency-domain FLIM system, the fundamental frequency adopted in this example is 50 MHz.

Figure 2. Typical excitation signal used in a frequency-domain FLIM system, the fundamental frequency adopted in this example is 50 MHz.

which is a Fourier series with \(|E_{w,n}|\) the amplitude of the \(n\)th frequency component which has frequency \(w_{n}^{E}\) and phase \(\phi_{n}^{E}\). The fundamental frequency of the excitation light is chosen to best resolve particular fluorescence lifetime components. For example, nanosecond fluorescence decays can be probed using MHz frequencies.

For a single fluorescence lifetime species (i.e. equation 2) the fluorescence emission \(F(t)\) following excitation by this repetitive waveform is then:

\begin{equation}
F(t)=E_0\tau+\sum_{\substack{n=-\infty\\n\ne0}}^{+\infty}\left|E_{\omega,n}\right|\frac{\tau+\omega_n\tau^2}{1+(\omega_n\tau)^2}e^{i(\omega_nt + \phi_n^E - \phi_n)}
\end{equation}

  Figure 3. Fluorescence emission (red), showing a phase lag and reduced amplitude with respect to the excitation signal (blue) due to the fluorescence lifetime.

Figure 3. Fluorescence emission (red), showing a phase lag and reduced amplitude with respect to the excitation signal (blue) due to the fluorescence lifetime.

  Figure 4. Modulated detector gain (green), pulsed excitation (blue) and fluorescence emission (red) in a homodyne fluorescence lifetime imaging system. The phase of the detector gain is controllable.

Figure 4. Modulated detector gain (green), pulsed excitation (blue) and fluorescence emission (red) in a homodyne fluorescence lifetime imaging system. The phase of the detector gain is controllable.

This waveform can be measured in the time domain or the frequency domain. It oscillates at the same high-frequencies as the corresponding Fourier components of the excitation light \(E(t)\), but exhibits a time delay or phase lag \(\phi_{n}\) at each frequency.

\begin{equation}
\phi_{n} = \arctan(\omega_{n} \tau)
\end{equation}

In addition to this phase lag the fluorescence emission is demodulated, i.e. the modulation depth or relative amplitude of the emission is attenuated relative to the pure excitation waveform.

For a single lifetime fluorescence species the modulation depth is:

\begin{equation}
M_{\omega, n} = 1 / \sqrt{1 + (\omega_{n} \tau)^2}
\end{equation}

Homodyne detection

In frequency-domain (FD) FLIM systems the fluorescence lifetime can be obtained from measurements of the phase lag and demodulation of the emission as compared to the excitation light. For convenience, simplicity of the instrumentation and to avoid high frequency noise, the high-frequency (HF) fluorescence signal \(F(t)\) is not measured directly in the time domain but instead converted to a low-frequency (LF) signal. This is accomplished using a detector of which the gain is shuttered just as the excitation light source. This frequency mixing phenomenon, the conversion of HF signals to LF signals, is well-known and the basis of radio technology. In the homodyne detection method, the excitation light and detector are modulated at the same frequency (\(\omega_{n}^{G} = \omega_{n}^{E} = \omega_{n}\)). The detector gain is then:

\begin{equation}
G(t)=G_0+\sum_{\substack{n=-\infty\\n\ne0}}^{+\infty}\left|G_{\omega,n}\right|e^{i(\omega_nt + \phi_n^G)}
\end{equation}

with \(|G_{w,n}|\) the amplitude of the \(n\)th frequency component, \(\omega_{m}\).

At a certain phase difference between the detector gain curve and the modulated excitation, the measured signal in an FD system \(S\) is the real-time product of the fluorescence emission and detector gain:

\begin{equation}
S(t)=\left\{F(t) \cdot G(t)\right\}_\textrm{LF} \propto F_0\left(1 +M_{\omega,n}\cos(\phi_n)\right)
\end{equation}

In a homodyne system, \(S\) is measured at a series of phase steps covering \(360\) degrees, and at each phase setting the detector signal is integrated for a time period much longer than the period of the HF modulation, averaging the signal. The resulting homodyne signal or phase-modulation diagram (an integral of Eqn.~9 over time \(t\)) exactly preserves the phase lag and the demodulation of the high frequency fluorescence emission, and can be directly translated to a fluorescence lifetime (through e.g. Eqns.6 & 7 for  a single lifetime species).

  Figure 5. FD homodyne signal of the fluorescence emission (red), showing a phase lag phi and demodulation M with respect to the excitation light (blue), exactly as in the time domain (figure 2).

Figure 5. FD homodyne signal of the fluorescence emission (red), showing a phase lag phi and demodulation M with respect to the excitation light (blue), exactly as in the time domain (figure 2).

For multiple fluorescence lifetime species the impulse response to an ultra-short excitation pulse is \(F_{\delta}(t) = \Sigma_{s=1}^{S} a_{s} e^{-t/\tau_{s}}\). It can be shown that the fluorescence emission following modulated excitation has a generalized form of Eqn.5, with corresponding generalized expressions for the phase and modulation (e.g. Spring & Clegg, 2009). The fluorescence lifetime components and their relative contributions can be identified using multi-frequency FLIM instrumentation.

The FD FLIM approach is implemented in the gain-modulated image intensified CCD camera of the Lambert Instruments LIFA for fast full-field lifetime imaging. The videorate imaging speeds attainable with full-field FD FLIM and its model free, rapid data analysis using the measured phase lag and demodulation make it applicable to real-time FRET imaging and promising for endoscopy and clinical applications. Recent advances in full-field FLIM include implementations of video-rate 3-D confocal FLIM with a multi-beam confocal spinning disk and hyperspectral FLIM.

  Figure 6. Fluorescence intensity (left, grayscale) and fluorescence lifetime (right, pseudo-color) images obtained with a Lambert Instruments LIFA.

Figure 6. Fluorescence intensity (left, grayscale) and fluorescence lifetime (right, pseudo-color) images obtained with a Lambert Instruments LIFA.

 

References

Robert M . Clegg and Bryan Q . Spring, in FLIM Microscopy in Biology and Medicine', eds. Robert M. Clegg and Ammasi Periasamy, Chapman and Hall/CRC 2009, p.115-142