Signal Modelling

Prior to considering the full signal equations, we will first introduce the equation for the MP2RAGE parameter (S_MP2RAGE) that is calculated in addition to the T₁ map. For complex data (magnitude and phase, or real and imaginary), the MP2RAGE signal (S_MP2RAGE) is calculated from the images acquired at two TIs (S_GRE,TI1 and S_GRE,TI2) using the following expression Marques et al., 2010:

This value is bounded between [-0.5, 0.5], and helps reduce some B₀ inhomogeneity effects using the phase data. For real data, or magnitude data with polarity restoration, this metric is instead calculated as:

Because MP2RAGE is a hybrid of pulse sequences used for inversion recovery and VFA, the resulting signal equations are more complex. Typically, a steady state is not achieved during the short train of GRE imaging blocks, so the signal at the center of k-space for each readout (which defines the contrast weighting) will depend on the number of phase-encoding steps. For simplicity, the equations presented here assume that the 3D phase-encoding dimension is fully sampled (no partial Fourier or parallel imaging acceleration). For this case (see appendix of Marques et al., 2010 for derivation details), the signal equations are:

where B₁^- is the receive field sensitivity, “eff” is the adiabatic inversion pulse efficiency, $\text{ER} = \text{exp}(-\text{TR}/T_{1})$, $\text{EA} = \text{exp}(-\text{TA}/T_{1})$ , $\text{EB} = \text{exp}(-\text{TB}/T_{1})$, $\text{EC} = \text{exp}(-\text{TC}/T_{1})$. The variables TA, TB, and TC are the three different delay times (TA: time between inversion pulse and beginning of the GRE₁ block, TB: time between the end of GRE₁ and beginning of GRE₂, TC: time between the end of GRE₂ and the end of the TR). If no k-space acceleration is used (e.g. no partial Fourier or parallel imaging acceleration), then these values are TA = TI₁ - (n/2)TR, TB = TI₂ - (TI₁ + nTR), and TC = TR_MP2RAGE - (TI₁ + (n/2)TR), where n is the number of voxels acquired in the 3D phase encode direction varied within each GRE block. The value m{sub}`1z,ss is the steady-state longitudinal magnetization prior to the inversion pulse, and is given by:

From Equations 2.13, 2.14, 2.15, and 2.13, it is evident that the MP2RAGE parameter S_MP2RAGE (Equations 2.11, 2.12) cancels out the effects of receive field sensitivity, T₂^*, and M₀. The signal sensitivity related to the transmit field (B₁⁺), hidden in Equations 2.13, 2.14, 2.15, and 2.16 within the flip angle values $\theta_{1}$ and $\theta_{2}$, can also be reduced by careful pulse sequence protocol design Marques et al., 2010, but not entirely eliminated Marques & Gruetter, 2013.