# Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging

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## Abstract

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## Featured Application

**Computation of Relaxation Time Maps in Magnetic Resonance Imaging.**

## Abstract

## 1. Introduction

_{rel}). The type of pulse sequence and the technical parameters used during the acquisition determine how each time factor affects the final signal intensity. For a detailed description of MR signal formation, the reader is referred to the textbooks on this topic [11,12,13].

## 2. Materials and Methods

**Model I.**Inversion Recovery: In this case, ${T}_{rel}=T1$, $T={T}_{I}$ and $S=A+B\cdot \left|1-2\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)\right|$ where $A\ge 0$. To fit this model, let $S=y$ and $x=\left|1-2\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)\right|$. The linear fit is carried out according to: $y=A+B\cdot x$. If the result is $A<0$, the algorithm lets $A=0$ and repeats the fit according to: $y=B\cdot x$.

**Model II.**Inversion Recovery [11,13] or MOLLI [25]: In this case, ${T}_{rel}=T1$ (Inversion Recovery) or ${T}_{rel}={T}_{1}^{\prime}$ (MOLLI), $T={T}_{I}$ and $S=\left|A-B\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)\right|$. To fit this model, let $S=\left|y\right|$ and $x=-\mathrm{exp}\left(-T/{T}_{rel}\right)$. The linear fit is performed according to $y=A+B\cdot x$. This case is more difficult since the absolute value of $y$ is known but its sign is unknown. In principle, with $n$ data points, ${2}^{n}$ sign combinations are possible. However, since the signal is recovering from inversion, $y$ increases as $T$ increases, so only $n$ sign combinations need to be considered [24,26]. Therefore, $n$ linear fits are considered corresponding to the sign of $y$ and the one for which the error is lowest is taken.

**Model III.**Simplified Spin Echo (T2) or Simplified Gradient Echo (T2

^{*}): In this case, $T={T}_{E}$ and ${T}_{rel}=T2$ for spin-echo or ${T}_{rel}={T2}^{*}$ for gradient echo. To fit this model, let $S=y$ and $x=\mathrm{exp}\left(-T/{T}_{rel}\right)$. The linear fit is performed according to:$y=A\cdot x$.

- Inversion recovery (IR) images from a healthy mouse.
- A series of MRI human brain images, simulated using a web-based software available at http://brainweb.bic.mni.mcgill.ca/brainweb (accessed on 10 March 2018) [27,28].
- Multi-slice multi-echo spin-echo images of a phantom consisting of scaffolds in which cells were grown, filled with phosphate-buffered saline.

**Imaging:**Healthy mouse images were taken using an IR sequence in a Bruker 1T benchtop MRI scanner (ICON 1-T MRI; Burker BioSpin GmbH, Ettlingen, Germany). The main sequence parameters were as follows: T

_{R}= 7500 ms; Acquisition matrix = 80 × 80; FOV = 20 × 20 mm

^{2}, giving a pixel resolution of 0.25 × 0.25 mm

^{2}; Number of Slices = 3; Slice Thickness = 1.25 mm; Slice Gap = 1 mm; T

_{E}= 5 ms; T

_{I}= 35, 50, 75, 100, 200, 250, 350, 500, 650, 800, 1000, 1500, 2000, 3000, 5000, 7000 ms; flip angle = 90°.

^{3}and a 181 × 217 × 181 acquisition matrix, giving a voxel resolution of 1 × 1 × 1 mm

^{3}. An IR sequence with T

_{R}= 10 s was used. Echo time was T

_{E}= 10 ms, and inversion times were T

_{I}= 35, 50, 75, 100, 200, 250, 350, 500, 650, 800, 1000, 1500, 2000, 3000, 5000, and 7000 ms. Other simulation parameters were flip angle = 90°, INU field = “Field A”, INU level = 20%, and noise level = 10%. For simplicity, only 11 slices in the center were selected for fitting.

_{R}= 4000 ms; Acquisition matrix = 256 × 256; FOV = 32 × 32 mm

^{2}, giving a pixel resolution of 0.125 × 0.125 mm

^{2}; Number of Slices = 5; Number of echoes = 64; Slice Thickness = 1 mm; Slice Gap = 0.25 mm; T

_{E}= 7 ms; flip angle = 90°.

**Processing:**Before fitting, a region of interest (ROI) was selected in the image. Only pixels inside the ROI were fitted. Furthermore, the pixels in which the T

_{rel}value obtained in the fit was not in a suitable range were discarded. The range chosen was 100–3000 ms for T1 and 10–500 ms for T2. This range is arbitrary, but it seems appropriate to contain meaningful values.

_{a}refers to the error of the RD-NLS algorithm and ε

_{b}to which it is compared.

- Multiple minima: number of pixels (as a percentage of total pixels) for which the fitting error as a function of T
_{rel}after linear parameters have been fitted shows more than one local minimum. - Pixels: The number of pixels (as a percentage of total pixels), for which the T
_{rel}difference between the two fits is above a threshold and ε_{a}< ε_{b}. No pixels for which the T_{rel}difference between the two fits is above the threshold and ε_{a}> ε_{b}or ε_{a}= ε_{b}were found. Therefore, the remaining pixels (up to 100%) are pixels for which the T_{rel}difference between the two fits is below the threshold. The threshold exists because if it did not exist, it could be possible to conclude that the same solution is two different solutions due only to rounding errors. The threshold must be small enough not to allow two different minima to be confused. For these reasons, it was set arbitrarily as 1 ms. - Relaxation time: Average T
_{rel}value given by fit a (the RD-NLS algorithm) and fit b (the algorithm it is compared to).

_{rel}, given by fit a, and T

_{rel}, provided by fit b. Comparison is (T

_{rel}fit b—T

_{rel}fit a) vs. average. Each point in the plot represents the average T

_{rel}of 100 randomly selected pixels. Averaging was performed to decrease plot dispersion. In addition, it needs to be noticed that since the plotted values are a small random sample, the plots could vary if they are generated multiple times.

## 3. Results

- The T1 fits (Tests 1–10) show many pixels in which error vs. T1 has more than one local minimum, in the 9–70% range. Having more than one local minimum means that a local search algorithm could get trapped in a local minimum which is not a global minimum (see Figure 3). This is not the case for the T2 fits (Tests 11–14), where error vs. T2 has only one local minimum.
- The T
_{rel}value obtained by the RD-NLS algorithm always corresponds to the global optimum. - Comparing Bruker and Matlab fit versus RD-NLS for the T1 maps (Tests 1–10), the fit is different in a significant number of pixels, in the range of 5–97%. However, this is not true for the T2 maps (Tests 11–14).

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bland-Altman plot corresponding to Test 1 (inversion recovery, mouse data). It is evident that the T1 obtained by Bruker’s fit is greater than that derived from RD-NLS.

**Figure 2.**Bland-Altman plot corresponding to Test 8 (inversion recovery, simulated human data). T1 values show a significant dispersion. Moreover, T1 obtained by Matlab’s fit (Lebenberg-Marquardt) is greater than the one attained by RD-NLS.

**Figure 3.**Fitting error (y) as a T1 (x) function after parameters A and B have been fitted. The T1 value obtained in fit (RD-NLS) was T1 = 696 ms, whereas the T1 value obtained in fit b (Matlab, Levenberg-Marquardt) was T1 = 740 ms. It corresponds to Test 2 (mouse data).

**Table 1.**Some MRI signal models can be fitted using the algorithm described in this manuscript. The fit can be partially linearized in all cases, so only one parameter (relaxation time) needs to be varied to find the lowest error.

Name | Equation |
---|---|

I. Inversion Recovery | $S=A+B\cdot \left|1-2\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)\right|\hspace{1em}A\ge 0$ |

II. Inversion Recovery or MOLLI | $S=\left|A-B\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)\right|$ |

III. Simplified Spin Echo or Simplified Gradient Echo | $S=A\cdot \mathrm{exp}\left(-T/{T}_{rel}\right)$ |

**Table 2.**Summary of tests performed in this article. The RD-NLS algorithm has been compared to the reported algorithm in all cases. Inversion-recovery mouse images, inversion-recovery simulated human images, and multi-slice multi-echo images were used to fit T1 and T2 maps.

Test Number | Test Model | Algorithm to Compare | Software to Compare | Data |
---|---|---|---|---|

1 | I | Levenberg-Marquardt | ISA | mouse |

2 | I | Levenberg-Marquardt | Matlab | mouse |

3 | I | Nelder-Mead | Matlab | mouse |

4 | I | Split | Matlab | mouse |

5 | II | Levenberg-Marquardt | Matlab | mouse |

6 | II | Nelder-Mead | Matlab | mouse |

7 | II | Split | Matlab | mouse |

8 | II | Levenberg-Marquardt | Matlab | simulated |

9 | II | Nelder-Mead | Matlab | simulated |

10 | II | Split | Matlab | simulated |

11 | III | Levenberg-Marquardt | ISA | phantom |

12 | III | Levenberg-Marquardt | Matlab | phantom |

13 | III | Nelder-Mead | Matlab | phantom |

14 | III | Split | Matlab | phantom |

**Table 3.**Results from the tests. Notice that the T2 fits (Tests 11–14) are much more similar for the compared algorithms than the T1 fits (Tests 1–10). Also, the T2 fits (Tests 11–14) show no multiple minima, whereas multiple minima are common in the T1 fits (Tests 1–10).

Test Number | Minima (%) | Pixels | Relaxation Time (ms) |
---|---|---|---|

1 | 9.2% | 97.0% | 622/636 |

2 | 9.2% | 4.1% | 622/623 |

3 | 9.2% | 0.6% | 622/622 |

4 | 9.2% | 4.1% | 622/623 |

5 | 18.6% | 4.8% | 807/805 |

6 | 18.6% | 2.0% | 807/807 |

7 | 18.6% | 2.7% | 807/806 |

8 | 68.1% | 15.7% | 823/842 |

9 | 68.1% | 10.3% | 823/841 |

10 | 68.1% | 10.8% | 823/823 |

11 | 0.0% | 0.2% | 113/113 |

12 | 0.0% | 0.0% | 113/113 |

13 | 0.0% | 0.0% | 113/113 |

14 | 0.0% | 0.2% | 113/113 |

**Table 4.**Bias and limits of agreement from the tests. Bland-Altman plots corresponding to this table are given in Figure 1 and Figure 2, and in Figures S1–S14 in the supplementary materials.

Test Number | Bias (ms) | Limits of Agreement (ms) |
---|---|---|

1 | 14.07 | −1.63 to 29.76 |

2 | 0.88 | −13.11 to 14.86 |

3 | 0.07 | −5.22 to 5.36 |

4 | 0.92 | −13.19 to 15.03 |

5 | −1.87 | −27.21 to 23.47 |

6 | −0.61 | −19.67 to 18.45 |

7 | −1.32 | −22.08 to 19.45 |

8 | 18.20 | −497.54 to 533.95 |

9 | 17.34 | −472.89 to 507.57 |

10 | −0.63 | −449.79 to 448.53 |

11 | −0.01 | −3.32 to 3.31 |

12 | 0.01 | −2.76 to 2.78 |

13 | 0.01 | −2.75 to 2.78 |

14 | 0.02 | −2.82 to 2.85 |

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**MDPI and ACS Style**

Rodriguez, I.; Izquierdo-Garcia, J.L.; Yazdanparast, E.; Castejón, D.; Ruiz-Cabello, J.
Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging. *Appl. Sci.* **2023**, *13*, 4083.
https://doi.org/10.3390/app13074083

**AMA Style**

Rodriguez I, Izquierdo-Garcia JL, Yazdanparast E, Castejón D, Ruiz-Cabello J.
Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging. *Applied Sciences*. 2023; 13(7):4083.
https://doi.org/10.3390/app13074083

**Chicago/Turabian Style**

Rodriguez, Ignacio, Jose Luis Izquierdo-Garcia, Ehsan Yazdanparast, David Castejón, and Jesús Ruiz-Cabello.
2023. "Comparison of Algorithms to Compute Relaxation Time Maps in Magnetic Resonance Imaging" *Applied Sciences* 13, no. 7: 4083.
https://doi.org/10.3390/app13074083