# What is the true way to determine time parameters from semilog graph?

I have experimental nuclear magnetic resonance data that describe T2-relaxation of the nuclei in the sample of interest. The experimental points obey bi-exponential dependence: I = I1*exp(-t/T2_1) + I2*exp(-t/T2_2), where I is current intensity, I1 and I2 are intensities which represent fractions of two T2 components - T2_1 and T2_2. The purpose is to calculate T2_1 and T2_2. To calculate T2_1 and T2_2, I fit bi-exponential equation mentioned above to experimental data points. But in order to ensure that calculated T2 values are true, I'd like to build semilog graph. For this, I take natural logarithm of experimental intensities, and thus the vertical axis in the graph is now in ln(I). Then, in obtained semilog graph, I need to divide points into two parts: first part of points will be fitted by linear equation y1 = C1 - t/T2_1, and the second part of points will be fitted by linear equation y2 = C2 - t/T2_2. And here I have two variants of how to do this fitting. The first variant is just to fit each part of data points with linear equation with corresponding initial guesses. And the second variant is the following: first, I fit the second part of points with equation y2 = C2 - t/T2_2, i.e. 7 points as you can see in the attached images. Thus I found C2 and T2_2 from fit. Then, using equation y2 = C2 - t/T2_2, I find ln(I) values for time values corresponding to the first part of points (i.e. first 3 points as you can see in the attached images). After that, I subtract these extrapolated ln(I) values from first three experimental ln(I) values. Let's call these new values as ln(I)1, ln(I)2, ln(I)3. Finally, I fit these 3 points with equation y1 = C1 - t/T2_1 and find the parameters C1 and T2_1. Could you please tell me, which variant of calculating T2 values is true? Please find attached two images which show the difference between two mentioned methods of calculating T2 values.

I would say that neither of your two methods is strictly correct, but your 2nd method would result in a more correct result than your 1st method. The problem with your 1st method is that you can't say that the first part of the signal decay is strictly associated with only the $$T_{2 1}$$-type decay process. In truth, both decay processes, $$T_{2 1}$$ and $$T_{2 2}$$, are simultaneously occurring both before and after the "kink" in the data. You seem to be tacitly assuming that the observed decay in the data before the kink is only associated with type $$T_{2 1}$$ decay, and that the data after the kink is only associated with type $$T_{2 2}$$ decay, but that's not so. Both decays are occurring throughout the displayed time interval, as indicated by the equation that you yourself wrote down of I = I1*exp(-t/$$T_{2 1}$$) + I2*exp(-t/$$T_{2 2}$$).
Your 2nd method at least recognizes the fact that type $$T_{2 2}$$ decay is occurring in the first time interval and tries to subtract out that contribution, but the problem is that type $$T_{2 1}$$ is also occurring in the second time interval, so strictly speaking it isn't correct to assume that red lines shown in your fits reflect pure type $$T_{2 2}$$ decay. There is also a little bit of type $$T_{2 1}$$ decay in the 2nd time interval. In practice, though, the type $$T_{2 1}$$ decay shown in your plots is so rapid that there is probably very little error in just fitting the 2nd time interval and assuming that you've measured $$T_{2 2}$$ from that, so that's why I say that the 2nd method is probably much more accurate, even though it's not rigorously correct, either.
If you want to be rigorously correct, you would have to do a least-squares fit of the entire set of data points shown to the function I(t; I1, $$T_{2 1}$$, I2, $$T_{2 2}$$) = I1*exp(-t/$$T_{2 1}$$) + I2*exp(-t/$$T_{2 2}$$), where I1, $$T_{2 1}$$, I2, and $$T_{2 2}$$ are the fitting parameters.