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mathieuboudreau committed Oct 3, 2024
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Several factors impact the choice of the inversion recovery fitting algorithm. If only magnitude images are available, then a polarity-inversion is often implemented to restore the non-exponential magnitude curves ([](#irPlot2)) into the exponential form ([](#irPlot1)). This process is sensitive to noise due to the Rician noise creating a non-zero level at the signal null. If phase data is also available, then a phase term must be added to the fitting equation {cite:p}`Barral2010-qm`. [Equation 1.3](#irEq3) must only be used to fit data for the long TR regime (TR > 5T<sub>1</sub>), which in practice is rarely satisfied for all tissues in subjects.

Early implementations of inversion recovery fitting algorithms were designed around the computational power available at the time. These included the “null method” {cite:p}`Pykett1983`, assuming that each T<sub>1</sub> value has unique zero-crossings (see Figure 2), and linear fitting of a rearranged version of [Equation 1.3](#irEq3) on a semi-log plot {cite:p}`Fukushima1981`. Nowadays, a non-linear least-squares fitting algorithm (e.g. Levenberg-Marquardt) is more appropriate, and can be applied to either approximate or general forms of the signal model ([Equation 1.3](#irEq3) or [Equation 1.1](#irEq1)). More recent work {cite:p}`Barral2010-qm` demonstrated that T<sub>1</sub> maps can also be fitted much faster (up to 75 times compared to Levenberg-Marquardt) to fit [Equation 1.1](#irEq1) – without a precision penalty – by using a reduced-dimension non-linear least squares (RD-NLS) algorithm. It was demonstrated that the following simplified 5-parameter equation can be sufficient for accurate T<sub>1</sub> mapping:
Early implementations of inversion recovery fitting algorithms were designed around the computational power available at the time. These included the “null method” {cite:p}`Pykett1983`, assuming that each T<sub>1</sub> value has unique zero-crossings (see [](#irPlot1)), and linear fitting of a rearranged version of [Equation 1.3](#irEq3) on a semi-log plot {cite:p}`Fukushima1981`. Nowadays, a non-linear least-squares fitting algorithm (e.g. Levenberg-Marquardt) is more appropriate, and can be applied to either approximate or general forms of the signal model ([Equation 1.3](#irEq3) or [Equation 1.1](#irEq1)). More recent work {cite:p}`Barral2010-qm` demonstrated that T<sub>1</sub> maps can also be fitted much faster (up to 75 times compared to Levenberg-Marquardt) to fit [Equation 1.1](#irEq1) – without a precision penalty – by using a reduced-dimension non-linear least squares (RD-NLS) algorithm. It was demonstrated that the following simplified 5-parameter equation can be sufficient for accurate T<sub>1</sub> mapping:

```{math}
:label: irEq4
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* It offers a wide dynamic range of signals ([up to `-kM0`, `kM0`]), allowing a number of inversion times where high SNR is available to sample the signal recovery curve {cite:p}`Fukushima1981`.
* T<sub>1</sub> maps produced by inversion recovery are largely insensitive to inaccuracies in excitation flip angles and imperfect spoiling {cite:p}`Stikov2015`, as all parameters except TI are constant for each measurement and only a single acquisition is performed (at TI) during each TR.

One important protocol design consideration is to avoid acquiring at inversion times where the signal for T<sub>1</sub> values of the tissue-of-interest is nulled, as the magnitude images at this TI time will be dominated by Rician noise which can negatively impact the fit under low SNR circumstances (Figure 6). Inversion recovery can also often be acquired using commonly available standard pulse sequences available on most MRI scanners by setting up a customized acquisition protocol, and does not require any additional calibration measurements. For an example, please visit the interactive preprint of the ISMRM Reproducible Research Group 2020 Challenge on inversion recovery T1 mapping {cite:p}`Boudreau2023`.
One important protocol design consideration is to avoid acquiring at inversion times where the signal for T<sub>1</sub> values of the tissue-of-interest is nulled, as the magnitude images at this TI time will be dominated by Rician noise which can negatively impact the fit under low SNR circumstances ([](#irPlot5)). Inversion recovery can also often be acquired using commonly available standard pulse sequences available on most MRI scanners by setting up a customized acquisition protocol, and does not require any additional calibration measurements. For an example, please visit the interactive preprint of the ISMRM Reproducible Research Group 2020 Challenge on inversion recovery T1 mapping {cite:p}`Boudreau2023`.


:::{figure} #fig2p6cell
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