Mono-Exponential Fitting in T2-Relaxometry: Relevance of Offset and First Echo

Abstract

Introduction: T2 relaxometry has become an important tool in quantitative MRI. Little focus has been put on the effect of the refocusing flip angle upon the offset parameter, which was introduced to account for a signal floor due to noise or to long T2 components. The aim of this study was to show that B1 imperfections contribute significantly to the offset. We further introduce a simple method to reduce the systematic error in T2 by discarding the first echo and using the offset fitting approach.

Materials and methods: Signal curves of T2 relaxometry were simulated based on extended phase graph theory and evaluated for 4 different methods (inclusion and exclusion of the first echo, while fitting with and without the offset). We further performed T2 relaxometry in a phantom at 9.4T magnetic resonance imaging scanner and used the same methods for post-processing as in the extended phase graph simulated data. Single spin echo sequences were used to determine the correct T2 time.

Results: The simulation data showed that the systematic error in T2 and the offset depends on the refocusing pulse, the echo spacing and the echo train length. The systematic error could be reduced by discarding the first echo. Further reduction of the systematic T2 error was reached by using the offset as fitting parameter. The phantom experiments confirmed these findings.

Conclusion: The fitted offset parameter in T2 relaxometry is influenced by imperfect refocusing pulses. Using the offset as a fitting parameter and discarding the first echo is a fast and easy method to minimize the error in T2, particularly for low to intermediate echo train length.

Fig 1. Illustration of a EPG derived curve with an FA of 120° (x) and correct 180° FA (line) for a T2 = 100ms, T1 = 3000ms, ESP = 20ms and ETL of 24.

It can be seen that due to the incorrect FA the first echo point is lower than that of the second and the signal seems to oscillate between odd and even echoes.

Fig 2. EPG simulated curve for T2 = 50ms, T1 = 3000ms, ESP = 20, ETL = 24, and FA = 120°.

Fig 2A illustrates the difference (blue solid line) between the curve simulated at 120° (black dashed line) an optimal 180° pulse (red solid line). Fig 2B illustrates the envelope (black lines) calculated from the odd and even echoes of the signal (blue line).

Fig 3. Relative T2 deviation, dT2 (in %), for method 1 (first row), method 2 (second row), method 3 (third row) and method 4 (forth row).

These results are presented for 3 different FA. It is seen that T2 becomes longer as the FA decreases. Closer approximation to the actual T2 are seen when Eq 2 is used and the first point is excluded. Please note that the scales of dT2 are not uniform provide maximum dynamic range for the different ETL and ESP.

Fig 4. Relative S₀ deviation, dS₀ (in %), for methods 1–4 corresponding to rows 1–4 respectively.

Closer approximations with little difference are seen for both equations with the first point excluded from the fit. Please note that the scales of dS₀ are not uniform and therefore provide maximum dynamic range for the different ETL and ESP.

Fig 5. Determined offset values for different FAs for method 2 (top row) and method 4 (bottom row).

Distributions remain relatively similar with decreasing values as FA tends to 180°. Note the decreasing offset with increasing FA.

Fig 6. Example of the signal decay curves of the phantom measurements for a defined pixel (shown by the cross section of the lines): Curves are shown for the single spin echo sequence (top right) and the MSE sequences for different FAs (bottom row).

The X-axis represents TE and y-axis the signal (x 10⁶ a.u). It can be noticed that as the FA reduces the first point, in particular, deviates from the expected exponential decay curve. The variation of the refocusing FA was performed by variation of the FA in the sequence protocol. The actual FA at the respective position might even differ from this value due to B1 inhomogeneities and imperfect slice profiles.