Danielle Kara^{1}, Mingdong Fan^{1}, Jesse I. Hamilton^{2}, Nicole Seiberlich^{2,3}, Mark Griswold^{2,3}, and Robert Brown^{1}

With the invention of MRF imaging, there is considerable freedom in input parameter selection, but it is difficult to determine how each choice affects the resulting parameter maps. Quality factors are introduced as a means of comparing MRF sequences with various input parameters (FA, TR, TE, N) on their abilities to precisely quantify T1 and T2. Simulations, fully sampled, and undersampled experiments verified that sequences with higher quality factors result in lower standard deviations in R1 and R2. With quality factor analysis, researchers and clinicians can readily determine the appropriate MRF input parameters to image more efficiently and precisely.

The goal of this work is to develop a mathematical framework with which the error in measuring relaxation times using various MRF sequences can be quantitatively assessed. MRF experiments can then be performed with assurance that chosen imaging parameters (FA, TR, TE, and N) will successfully produce reliable T1 and T2 maps.

The standard deviations of R1=1/T1 and R2=1/T2 are related to the dictionary signals through a multivariate Taylor series expansion of transverse magnetization about $$$R1_0$$$ and $$$R2_0$$$ at each time step. Then the first order corrections to the N-dimensional dictionary vector $$$\overrightarrow{D_{0}}=\overrightarrow{D}\left(R1_0,R2_0\right)$$$ yield $$\overrightarrow{D}\left(R1,R2\right)\cong\overrightarrow{D_{0}}+\overrightarrow{C1}\left(R1-R1_0\right)+\overrightarrow{C2}\left(R2-R2_0\right)$$ with N-dimensional coefficients $$$\overrightarrow{C1}$$$ and $$$\overrightarrow{C2}$$$, containing the first derivative of the transverse magnetization with respect to R1 and R2 respectively at each time step. Thus, the acquired signal with noise $$$\eta$$$ from a tissue with $$$R1_{0}$$$ and $$$R2_{0}$$$ will be $$$\overrightarrow{S}=\overrightarrow{D_0}+\overrightarrow{\eta}$$$.

The extremum calculation of the dot product $$$Re\left(\overrightarrow{S}\cdot\overrightarrow{D}^*\right)$$$ leads to the standard deviations $$$\sigma_{R1}=\frac{\eta}{\sqrt{Q1}}$$$ and $$$\sigma_{R2}=\frac{\eta}{\sqrt{Q2}}$$$ with quality factors $$Q1=C1^{2}\frac{1+2\;\widehat{D_{0}}\cdot\widehat{C2}^{\star}\;\widehat{D_{0}}\cdot\widehat{C1}^{\star}\;\widehat{C1}\cdot\widehat{C2}^{\star}-\left(\widehat{D_{0}}\cdot\widehat{C2}^{\star}\right)^{2}-\left(\widehat{D_{0}}\cdot\widehat{C1}^{\star}\right)^{2}-\left(\widehat{C1}\cdot\widehat{C2}^{\star}\right)^{2}}{1-\widehat{D_{0}}\cdot\widehat{C2}^{\star}}$$ and $$Q2=C2^{2}\frac{1+2\;\widehat{D_{0}}\cdot\widehat{C2}^{\star}\;\widehat{D_{0}}\cdot\widehat{C1}^{\star}\;\widehat{C1}\cdot\widehat{C2}^{\star}-\left(\widehat{D_{0}}\cdot\widehat{C2}^{\star}\right)^{2}-\left(\widehat{D_{0}}\cdot\widehat{C1}^{\star}\right)^{2}-\left(\widehat{C1}\cdot\widehat{C2}^{\star}\right)^{2}}{1-\widehat{D_{0}}\cdot\widehat{C1}^{\star}}$$ where all individual products are understood to be $$$Re\left(\overrightarrow{A}\cdot\overrightarrow{B}^\star\right)$$$, and $$$\overrightarrow{C1}$$$ and $$$\overrightarrow{C2}$$$ are obtained through first order parametric fitting.The quality factors were initially tested with Bloch equation simulations in MATLAB, where error analysis was performed by repeatedly matching a simulated signal containing Gaussian white noise to a pre-calculated dictionary by maximizing $$$Re\left(\overrightarrow{S}\cdot\overrightarrow{D}^*\right)$$$. The matching was repeated for various FAs, TRs, TEs, Ns, and preparation pulses for both FISP-based and bSSFP-based MRF sequences [3][4]. A FISP-MRF experiment was then performed (3T Siemens Skyra and an 18-channel brain array coil) with a variable-density spiral (48 projections for full k-space coverage, 192x192, 300mm^{2} FOV) and a phantom with ten separate compartments each with a different T1 and T2 combination. Three FA distributions of 1000 steps were tested (see figure 1) with TE=0.75ms and TR=6.98ms [3][4]. Matching for N=1000 and N=500 produced T1 and T2 maps from the fully sampled data, from which the standard deviations of matched R1 and R2 were calculated for each gel. Finally, matching was performed over 1000 and 500 steps to produce T1 and T2 maps for the FA distributions in Figure 1, with undersampled experimental data (acceleration factor R=48).

The results of matching simulated and experimental data to a dictionary are presented for six values of T1 and T2 with ranges [370, 1710] ms and [40,105] ms respectively. Shown in Figure 2, simulated results demonstrate exemplary linearity between Q1 and 1/$$$\left(\sigma_R1 \right)^2$$$ and Q2 and 1/$$$\left(\sigma_R1 \right)^2$$$, where $$$\sigma_R1$$$, and $$$\sigma_R2$$$ were calculated by matching simulated noisy data to the dictionary. Figure 3 shows the results of matching the fully sampled, experimental data and Figure 4 shows result of matching the undersampled, experimental data. The fully sampled data contains significant scatter from the expected linear result, likely due to local and non-additive experimental uncertainties, which is overwhelmed by global and additive undersampling noise resulting in improved linearity in analysis the undersampled experimental data.

Figure 1: Flip angle distributions tested with simulation and experiment where a) contains an initial inversion and is based on [3] b) contains inversions and pi/2 pulses intermittently as introduced in [4], and c) contains no preparation pulses and is based on [4].

Figure 2: Simulated results showing the linear relationship between quality factors and error.

Figure 3: Fully sampled experimental results showing the linear relationship between quality factors and error.

Figure 4: Undersampled experimental results showing the linear relationship between quality factors and error.