MR fingerprinting has been used for estimating the intra-voxel tissue component fractions ^{1, 2}. The previous method ² could lead to over-fitting error, which hinders reliable mapping of tissue fractions. In this work, we propose to use template matching with a dictionary whose entry is dependent on composition fraction, in solving the MRF signal equation.

The previous method resolves the MRF signal equation S_voxel=Dw by pseudo-inverse, namely (D^HD)^-1D^HS_voxel=w, where S_voxel is the measured signal evolution, D is pre-calculated signal evolution of interested components and w is the group of component fractions. However, the results often exhibit abnormal fraction values (i.e. negative or above 1), so normalization after fractions obtained are needed, leading to over-fitting error. Besides, no consensus solution for the necessary normalization is reached yet in the field. One approach often used is to take the absolute value of results and then normalize them to summation of 1 ².

In this study, instead of the dictionary based on T1 or T2 for MRF, potential fractions of interested components were used. Only the signal evolutions of interested components, D, was pre-calculated by extended phase graph algorithm ³, and then multiplied by W (W=[w₁,w₂,…w_M], where M is the number of potential fractions groups w_i,), to build the dictionary. By using template matching between each column vector of DW and S_voxel, the component fraction is obtained from the best matched one. The solution of S_voxel=Dw actually is therefore turned into an optimization problem:

$$ \hat{\bf{w}}=argmin\parallel \bf {S}_{\it {voxel}}-\bf{Dw}\parallel^{2}$$

$$s.t.{\sum_{{\it{n}=1}}^{{\it{N}}}{\bf{w}}({\it{n}})=1, {\bf{w}}({\it{n}})\in1}.$$

where w is the column vector of fractions with N the interested components. S_voxel is L×1 vector with L the number of time points for MRF acquisition and w is a N×1 vector. The matrix size is L×N for D and N×M for W.

To demonstrate the effectiveness of the proposed method, two types of MRF measurements were performed on a Siemens 3T Prisma scanner based on an inversion-prepared FISP MRF sequence ⁴ with TR varying from 10 to 12ms, flip angle varying from 5 to 80 degrees. The total scanning time was about 10s. A Polyvinylpyrrolidone (PVP) phantom with concentration of, 5%, 10%, 15%, 20%, 30%, and pure water was made in separate compartment respectively. Since PVP solution has a good linear relationship between T1, T2 and its concentration especially when the concentration is smaller than 30% ^{5, 6}, the signal of dilute PVP solution could be regarded as a mixture of water and thick PVP solution. In the dictionary, the interested components were water (T1/T2=3200/3000ms) and 30% PVP solution (T1/T2=1100/800ms). Therefore, the theoretical fraction values of PVP solution with concentration from 5% to 20% should be regarded as the linear combination of water and 30% PVP solution. The in-vivo experiment was performed using the same imaging parameters. The interested tissue components included CSF (T1/T2=4000/1500ms), gray matter (T1/T2=1300/120ms) and white matter (T1/T2=800/80ms). To test whether the methods can separate white matters with long T1/T2 (800/80ms, WM II) from that with comparatively short T1/T2 (660/70ms, WM II), these plus CSF and GM were introduced to form a 4-components tissue fractional mapping.

Figure 1 shows the fractions of interested component (30% PVP solution) with different concentration from 0% to 30%. Results using proposed method are closer to the theoretical values and have a smaller RMSE, demonstrating better precision for estimating components. The results from in-vivo experiment, as shown in Figure 2, exhibit that the proposed method performs better, especially in separating the white matter from gray matter (red arrow). Furthermore, when short WM components was introduced (Figure 3), the performance of previous method deteriorated for all components estimating. The proposed method could distinguish white matters of long T1/T2 from short T1/T2 and did not influence the estimation of CSF and grey matters.With the regularization introduced to the calculation of component fractions, the results are more precise and robust since the irrational over-fitting error is alleviated. Therefore, the proposed method could provide more robust quantification of tissue composition and estimation of partial volume effects. It also provides a potential for refined classification of tissue by introducing more relevant components.