# Data analysis in TCSPC for fluorescene decay (reconvolution with measured IRF)

I am trying to understand the data analysis of fluorescence decay counts measured by TCSPC technique, particularly with reconvolution with measured IRF.

I am able to get the fitted counts (given by the software according to the model only in the case of tail fitting and not in the case of reconvolution.

### Tail fit

The basic model is a multi-exponential decay with y-offset(background-offset).

$$I(t) = A + \sum_{i}^{N}B_{i}\textrm{e}^{\frac{-t}{\tau_{i}}}$$

Here $$B_{i}$$ and $$\tau_{i}$$ are the pre-exponential and lifetime due to the $$i$$th component respectively and $$A$$ is the y-offset.

The fitted parameters for a 2 exponential model ($$I(t) = A + B_{1}\textrm{e}^{\frac{-t}{\tau_{1}}} + B_{2}\textrm{e}^{\frac{-t}{\tau_{2}}}$$) are.

A   =   11.63431
T1 (ns) =   0.221457836
T2 (ns) =   2.69572
B1  =   6618.36
B2  =   644.1506

Time per channel (ns) = 0.026128421
Data range = 402:1289

Peak count  =   10000
Total count =   190155


Here I am able to get the fitted counts in the following example by using these parameters. For example in case of bin no. 405.

|    | Bin | Time (ns) | DecayCount | FittedCount |
|----|-----|-----------|------------|-------------|
| 0  | 401 | 10.47750  | 9464       | -           |
| 1  | 402 | 10.50363  | 8462       | 6531.378    |
| 2  | 403 | 10.52975  | 7298       | 5870.6414   |
| 3  | 404 | 10.55588  | 5758       | 5282.8124   |
| 4  | 405 | 10.58201  | 4800       | 4759.7832   |
| 5  | 406 | 10.60814  | 4069       | 4294.3481   |
| 6  | 407 | 10.63427  | 3396       | 3880.1036   |
| 7  | 408 | 10.66040  | 2908       | 3511.3585   |
| 8  | 409 | 10.68652  | 2649       | 3183.0549   |
| 9  | 410 | 10.71265  | 2315       | 2890.698    |
| 10 | 411 | 10.73878  | 2171       | 2630.2929   |


I am able to plug in the fitted values of the model parameters in the formula to get the fitted count values for any time $$t$$.

t =  10.58201035 - 10.47749666 = 0.104513682

FittedCount = 11.63431 +
(6618.36 * exp(-0.104513682/0.221457836)) +
(644.1506 * exp(-0.104513682/2.69572)) = 4759.783


### Reconvolution fit

Here the model is. $$I(t) = L(t)\otimes \sum_{i}^{N}B_{i}\textrm{e}^{\frac{-t}{\tau_{i}}}$$

$$F(t) = A + B.I(t+\Delta)$$

Here $$B_{i}$$ and $$\tau_{i}$$ are the pre-exponential and lifetime due to the $$i$$th component respectively, $$L(t)$$ the instrument response function (IRF), $$B$$ is the amplitude scaling factor, $$\Delta$$ is the shift parameter and $$A$$ is the y-offset.

A   =   11.41109
T1 (ns) =   0.191460122
T2 (ns) =   2.626401242
B1  =   0.2411146
B2  =   0.007343028
Shift (ns)  =   -0.158856098

Time per channel (ns) = 0.026128421
Data range = 398:1289

Peak count (Decay)  =   10000
Total count  (Decay)    =   190155
Peak count (IRF)    =   10000
Total count  (IRF)  =   101269


How to get the fitted counts in the case of reconvolution fit ?

| -  | Bin | Time (ns) | IRFCount | DecayCount | FittedCount |
|----|-----|-----------|----------|------------|-------------|
| 0  | 397 | 10.37298  | 9743     | 5665       | -           |
| 1  | 398 | 10.39911  | 10000    | 7982       | 9757.034    |
| 2  | 399 | 10.42524  | 9776     | 9420       | 9106.409    |
| 3  | 400 | 10.45137  | 8918     | 10000      | 8363.679    |
| 4  | 401 | 10.47750  | 7445     | 9464       | 7626.371    |
| 5  | 402 | 10.50363  | 5922     | 8462       | 6899.12     |
| 6  | 403 | 10.52975  | 4538     | 7298       | 6221.999    |
| 7  | 404 | 10.55588  | 3327     | 5758       | 5582.207    |
| 8  | 405 | 10.58201  | 2450     | 4800       | 5004.482    |
| 9  | 406 | 10.60814  | 1605     | 4069       | 4481.801    |
| 10 | 407 | 10.63427  | 1191     | 3396       | 4012.029    |


Specifically, in the calculation, How to

• get the amplitude scaling factor (in addition to the amplitudes) and
• shift the measured IRF using the shift parameter.

I have tried multiplying by the total IRF count value to get the $$B_{i}$$s corresponding to a tail fit, but still not getting the fitted counts as in the output.

### Update

Here are the comments regarding scaling of $$B_{i}$$s by the scaling factor $$B$$ in few fitting software program documentations.

http://www.nanoer.net/d/img/1-FASTissue6.pdf

If the fit was performed in the Reconvolution mode, $$B_{i}$$ values are scaled by the integrated IRF, i.e. if the same decay would be analysed with a 10 times bigger IRF, the numerical values of the $$B_{i}$$ values would be reduced by a factor of 10. If one determines the total number of counts of the IRF, and then multiplies the $$B_{i}$$ value with this number, this will result in a value that will be similar to the $$B_{i}$$ value for a tail fit, extrapolated to the channel of the peak of the IRF.

The magnitude of the $$B_{i}$$ values returned in the fit are dependent on whether or not reconvolution is used in fitting the data. Without reconvolution the values are in counts and can easily be linked to the peak number of counts (ie for a single exponential its magnitude should be ~ the count value in the start channel). The value after reconvolution is not in counts, but still reflects the amount of a particular emitting species.
• The shift parameter is part of the fitting and is used because different filters/wavelength etc may be used when measuring the irf vs fluorescence so this time delay is adjusted by the convolution process to get a best fit. It has no physical meaning otherwise. The amplitudes $B$ are relative to one another; the absolute values usually do not matter (as fluorescence yields are not often measured via tcspc), but you can normalise if you wish to get x% B1 and y% B2 etc. Jul 22 '20 at 12:04
• @porphyrin I understand why shift is used fitted for a reconvolution fit and how $B_i$s are different in case of tail fit vs reconvolution fit. I want to know how to get the fitted counts at a time $t$ by using the fitted parameter values in the model, as I can do with tail fit. Jul 22 '20 at 15:18
• @porphyrin Yes, I should be able to get that. But the problem is with the $Bi$s. They are scaled with the amplitude scaling factor $B$, which I am not able to figure out. When I try to fit with other software, I am getting the values for the amplitudes B1 = 16781.47 and B2 = 580.40. The relative amplitude from above are B1 = 0.2411146 and B2 = 0.007343028. But also in both cases the normalised pre-exponential values are identical (B1 = 0.97 and B2 = 0.03). Jul 27 '20 at 16:46