Multi-exponential

After intravenous drug administration, a log-convex concentration decline points to a multi-exponential function. For the most frequent case, a bi-exponential equation with the inter-compartment rate constants (k 2) and (k21) can be fitted. 

The mean residence time (MRT) gives one parameter for the multi-exponential elimination kinetics with more than one half-life. 

The Laplace inversion (LI) is the key mathematical tool of the DDIF experiment. The ability to convert the measured multi-exponential decay into a distribution of decay times is crucial to the DDIF pore size distribution application. However, unlike other mathematical operations, the Laplace inversion is an ill-conditioned problem in that its solution is not unique, and is fairly sensitive to the noise in the input data. In this light, significant research effort has been devoted to optimizing the transform and understanding its boundaries [17, 53, 54],... 

Kremers, G. J., Van Munster, E. B., Goedhart, J. and Gadella, T. W. (2008). Quantitative lifetime unmixing of multi-exponentially decaying fluorophores using single-frequency FLIM. Biophys. J. 95, 378-89. 

The analysis of the histograms of photon arrival times is equivalent in both cases and relies on fitting appropriate model functions to the measured decay. The selection of the fitting model depends on the investigated system and on practical considerations such as noise. For instance, when a cyan fluorescent protein (CFP) is used, a multi-exponential decay is expected furthermore, when CFP is used in FRET experiments more components should be considered for molecules exhibiting FRET. Several thousands of photons per pixel would be required to separate just two unknown fluorescent... 

In case of slow exchange between phases, the relaxation decay becomes the sum of relaxation of the various phases and thereby multi-exponential ... 

Brownstein and Tarr35 considered if the size of a muscle cell could give rise to non-mono-exponential transverse relaxation by assuming simple diffusion and planar geometry, and found that under these conditions and in the slow diffusion range, multi-exponential transverse relaxation could be expected when the sample size is between 1 and 30 pm. Accordingly, as muscle cells have a diameter of 10-100 pm,36 the calculations performed by Brownstein and Tarr35 indicate that the anatomical features of muscle cells are consistent with the expectation of multi-exponential transverse relaxation. 

Attractive for the use of QDs are their long lifetimes (typically 5 ns to hundreds of nanoseconds), compared to organic dyes, that are typically insensitive to the presence of oxygen. In conjunction with time-gated measurements, this provides the basis for enhanced sensitivity [69]. This property can be also favorable for time-resolved applications of FRET. The complicated size-, surface-, and wavelength-dependent, bi- or multi-exponential QD decay behavior (Fig. 2) can complicate... 

General relations for single exponential and multi-exponential decays For a single exponential decay, the b-pulse response is... 

For a multi-exponential decay with n components, the (5-pulse response is... 

When the fluorescence decay of a fluorophore is multi-exponential, the natural way of defining an average decay time (or lifetime) is ... 

In practice, initial guesses of the fitting parameters (e.g. pre-exponential factors and decay times in the case of a multi-exponential decay) are used to calculate the decay curve the latter is reconvoluted with the instrument response for comparison with the experimental curve. Then, a minimization algorithm (e.g. Marquardt method) is employed to search the parameters giving the best fit. At each step of the iteration procedure, the calculated decay is reconvoluted with the instrument response. Several softwares are commercially available. 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The maximum entropy method has been successfully applied to pulse fluorometry and phase-modulation fluorometry3- . Let us first consider pulse fluorometry. For a multi-exponential decay with n components whose fractional amplitudes are a , the d-pulse response is... 

This can easily be extended to multi-exponential decays of I(t) and r(t). 

In phase fluorometry, the phase (and modulation) data are recorded at a given wavelength and analyzed in terms of a multi-exponential decay (without a priori assumption of the shape of the decay). The fitting parameters are then used to calculate the fluorescence intensities at various times, 2 > 3 > The procedure is repeated for each observation wavelength X, X2, A3,... It is then easy to reconstruct the spectra at various times. 

The most commonly used method for the evaluation of a distribution of distances is based on the measurement of the donor fluorescence decay3. If the <5-response of the donor fluorescence is a single exponential in the absence of acceptor, it becomes multi-exponential in the presence of acceptor ... 

Transverse relaxation of musculature is relatively fast compared with many other tissues. Measurements in our volunteers resulted in T2 values of approximately 40 ms, when mono-exponential fits were applied on signal intensities from images recorded with variable TE. More sophisticated approaches for relaxometry revealed a multi-exponential decay of musculature with several T2 values." Normal muscle tissue usually shows lower signal intensity than fat or free water as shown in Fig. 5c. Fatty structures inside the musculature, but also water in the intermuscular septa (Fig. 5f) appear with bright signal in T2-weighted images. 

Most methods assume an exponential decay for the resonances in the time domain giving rise to Lorentzian lineshapes in the frequency domain. This assumption is only valid for ideal experimental conditions. Under real experimental circumstances multi-exponential relaxation, imperfect shimming, susceptibility variations and residual eddy current usually lead to non-ideal... 

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

The type of the distribution of T-values is a much discussed topic. Experience shows that, in a mono-exponential case, the values should spread over an interval of more than 3 x Ti but not much over 4 x Ti, and a linear distribution appears to be slightly better than a logarithmic one. This is probably due to the fact that in a three-parameter exponential fit, the points with large x-values play as crucial a role in determining the relaxation rate as the slope at small x-values, and one needs both to determine R. On the other hand, it is evident that in multi-exponential cases, logarithmic distribution is often better suited for the task, especially when the relaxation rates of different sample components differ by an order of magnitude or more. 

The phasor method associates the decay dynamics with a vector in a so-called phasor space. In particular, purely exponential decay corresponds to a phasor with its end point on a semicircle of radius 1/2 and centered at (1/2, 0). Tuning of the decay time from zero to infinity results in a counterclockwise displacement of the end point from (1,0) to (0,0) along the semicircle. Multi-exponential decay is equivalent to a point inside the semicircle, but its dependence on the weight-averaged decay... 

E.g. tryptophane residues of proteins excite at 290-295 mn but they emit photons somewhere between 310 and 350 mn. The missing energy is deposited in the tryptophane molecular enviromuent in the form of vibrational states. While the excitation process is complete in pico-seconds, the relaxation back to the initial state may take nano-seconds. While this period may appear very short, it is actually an extremely relevant time scale for proteins. Due to the inherent thermal energy, proteins move in their (aqueous) solution, they display both translational and rotational diffusion, and for both of these the characteristic time scale is nano-seconds for normal proteins. Thus we may excite the protein at time 0 and recollect some photons some nano seconds later. With the invention of lasers, as well as of very fast detectors, it is completely feasible to follow the protein relax back to its ground state with sub-nano second resolution. The relaxation process may be a simple exponential decay, although tryptophane of reasons we will not dwell on here display a multi-exponential decay. 

Figure 2a shows a few fluorescence upconversion transients as measured for 1 dissolved in n-heptane, under magic angle conditions. The transients show multi-exponential decay behavior... 

The initial steep change in the extinction is ascribed to the temperature equilibration of the sample. Following this, a multi-exponential decay in the extinction is observed, indicating the complex relaxation kinetics of the homogeneous reaction between the various cationic SE s. Finally, the long-term single exponential relaxation... 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

This is simply a multi exponential decay with the ampHtude of the k-th mode given by ak Tk (1 - e tplTk). In the short and in the long exposure limit, this reduces to... 

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