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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. Graph1 Graph2

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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.

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  • $\begingroup$ Thank you! Initially I fit the function I(t; I1, T2_1, I2, T2_2) = I1*exp(-t/T2_1) + I2*exp(-t/T2_2) to the entire ser of data, and thus I obtained values for fitting parameters I1, T2_1, I2, and T2_2. But my supervisor told me that I should fit y2 = C2 - t/T2_2 to the second part of points, then extrapolate this fitting line for the time values corresponding to the first part of points, then subtract extrapolated values from the first part of points, and finally fit y1 = C1 - t/T2_1 to these new points. $\endgroup$ – Leonid May 14 at 11:55
  • $\begingroup$ But I have doubt regarding this method, because T2_1 value obtained using this method (when subtracting extrapolated values from the first part of experimental points) is further from the truth than the T2_1 value obtained using the method when subtraction is not applied (Top graph in my main post). I.e. if I just fit first part of data points with linear equation, I get T2_1 value that is close to the truth. $\endgroup$ – Leonid May 14 at 12:03
  • $\begingroup$ In the first comment: when I wrote "But my supervisor told me that I should fit y2 = C2 - t/T2_2 to the second part of points, then extrapolate this fitting line for the time values corresponding to the first part of points, then subtract extrapolated values from the first part of points, and finally fit y1 = C1 - t/T2_1 to these new points.", I meant that I should do these steps in semilog graph, i.e. taking natural logarithm of intensity values. $\endgroup$ – Leonid May 14 at 12:05
  • $\begingroup$ @Leonid - I think that the method that your supervisor is suggesting should give a fairly accurate result because the T2_1 decay is so rapid that its contribution to the signal is nil after the first five points. So you can regard the following points as being pure T2_2 decay. The results you get for the decays by this method should agree very well with the full entire data fit method. You wrote that you used the entire data fit method that I suggested. So how well did the T2_1 and T2_2 decays agree with the decays that you obtained using your supervisor's suggested method? $\endgroup$ – Samuel Weir May 14 at 17:35
  • $\begingroup$ The 1st method which you described (no subtraction used) is incorrect because the slope of the data points in the first part of the decay before the kink is a function of BOTH the T2_1 decay AND the T2_2 decay. That's the point of your supervisor's suggestion to do the subtraction. By subtracting out an appropriate amount of signal from the data points of the first amount of decay, you take out the influence of the T2_2 decay and are left with new data points which more accurately reflect pure T2_1 decay. $\endgroup$ – Samuel Weir May 14 at 17:40

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