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 \(s^{-1}\)).

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}

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}

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).

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.

## 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

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