Data Fitting

In qMT imaging, the biophysical model relates the parameters observed in the two-pool tissue model to physical quantities such as the fractional size of the pools, relaxation times and magnetization exchange rates of the free and restricted pool Sled & Pike, 2001Sled, 2018. However, qMT experiments usually consist of long acquisition imaging protocols accompanied by complex data fitting. To this end, some software solutions have been proposed Karakuzu et al., 2020Tabelow et al., 2019C Wood, 2018. qMRLab is an open-source project for quantitative MR analysis that is an extension of qMTLab, a software for data simulation and analysis of three MT models: qMT-SPGR, qMT-bSSFP and qMT-SIRFSE. In addition to the quantitative MT methods, qMRLab also contains semi-quantitative MT models including the magnetization transfer ratio (MTR) and magnetization transfer saturation (MTsat).

The qMT-SPGR method in qMRLab contains three fitting models: Sled and Pike, Ramani, and Yarnykh and Yuan Sled & Pike, 2001Ramani et al., 2002Yarnykh, 2002. For the Sled and Pike model, the saturation fraction effect of the MT pulse on the free pool is pre-computed to accelerate the processing times. The MT effect of the pulse is approximated as an instantaneous fractional saturation of the longitudinal magnetization of the free pool, assuming the absence of chemical exchange processes Pike, 1996Sled & Pike, 2001. To fit the model, additional parameters related to the pulse sequence are required, namely timing parameters, the absorption lineshape, and the characteristics of the MT pulse, such as the shape and the bandwidth or the time-bandwidth product. In Figure 6.5, the qMT-SPGR method is used to show a single voxel curve simulation for the same MT data fitted by three different models. The fitted parameters were the pool size ratio F, the magnetization transfer rate from the restricted to the free-water pool (k_r), and the transverse relaxation time of the free-water (T_2,f) and restricted (T_2,r) pool.

In addition to acquiring the MT data, three more quantitative measurements are typically required for model correction purposes. The magnetic field B₀ inhomogeneity affects the actual off-resonance frequency experienced by the tissue at each voxel. In MT, B₀ maps are computed to correct for off-resonance frequency values of the MT pulse in the presence of magnetic field non-uniformity. Radiofrequency field B₁ inhomogeneity is another source of inaccuracies that depend on the operating frequency of the scanner, the pulse sequence, and the shape and electrical properties of the sample Sled & Pike, 1998. Therefore, B₁ maps are typically needed to correct the radiofrequency amplitude variations that affect the nominal values of the MT pulse flip angle and the excitation flip angle Boudreau et al., 2018. Longitudinal relaxation T₁ values vary naturally in biological tissue, but the choice of the T₁ mapping method, has also been shown to influence the variability of T₁ measurements Stikov et al., 2015. In the context of a qMT experiment, T₁ maps are acquired with an independent measurement of the apparent relaxation time T₁ (T_{1, meas}). The T₁ map is related to the relaxation rate of the free pool (R_1,f) as described by Eq. 6.6, expressed in terms of 𝐹, k_f, R_1,meas and R_1,r, where the relaxation rate of the restricted pool is arbitrarily set to 1 s^-1 because it is insensitive to this kind of MT experiments Sled & Pike, 2001. Multiple qMRI maps with a range of B₀ and B₁ inaccuracies, as well as T₁ maps with a variety of relaxation times, have been simulated to show the effect of the quality of these input maps on the qMT fitted parameters as shown in Figure 6.6.

As described above in Figure 6.6, inaccurate MT pulse flip angles and excitation flip angles affect the fitted MT parameters, and there is an additional error source related to the T₁ mapping measurement. As shown in Boudreau et al., 2018, using specific acquisitions protocols, T₁ values can be affected by B₁ field non-uniformities, such as the variable flip angle method, while the inversion recovery method is insensitive to these field inhomogeneities Stikov et al., 2015.

Figure 6.7 displays an example human dataset with the input qMRI maps used to fit the qMT parameters F, k_f, T_2,f, T_2,r.