MR fingerprinting, or MRF, is a promising method that offers an efficient way to acquire multiple quantitative maps within a single sequence, which uses a pseudorandomized acquisition instead of a repeated one. Currently in the reconstruction of MRF, parameters such as T1, T2 and off-resonance are determined by matching the signal with a dictionary.¹ However, the dictionary usually has a tremendous size, which requires considerable generation time and large memory and relies on the signal model heavily. Here we introduce an alternative reconstruction method, based on the use of Kalman filter and Bloch equation to recursively derive the MR parameters from acquired MRF data. In this way, we can derive the quantitative maps without generating the dictionary. Furthermore, the value calculated by Kalman filter is more accurate than dictionary reconstruction because there is no discrete parameter space assumption.

Theory: Kalman filtering involves the recursive estimation of the mean and covariance of the state, and will gradually converge. It includes two steps: predicting and updating. In the predicting step, we derive a system dynamic function based on Bloch equation, and then calculate the next state of a joint-state vector, ² S= [Mx, My, Mz, T1, T2, df], using this function. In the updating step, we uses the observation value, y= [Mx, My], to modify the prediction by calculating Kalman gain and determining the weight between prediction and observation.³ When the predicting-updating cycle is repeated, the covariance of joint-state vector gradually decreases.

Experiment: Simulated signal evolution was generated with Bloch equation and additive Gaussian noise. The parameter maps (T1, T2, off-resonance) were calculated with Kalman filtering and dictionary-matching, respectively. The dictionary contained 132,651 entries, with T1 between 100 and 2,100 ms in increments of 40 ms, T2 between 20 and 520 ms in increments of 10 ms, off-resonance between -50 and 50 Hz in increments of 2 Hz. The signal evolution contained 500 time frames. The temporal SNR was 3.0. The mean relative error was calculated by taking average of absolute relative error for each point to evaluate the accuracy of the algorithm. All experiments were performed with MATLAB (the MathWorks).

As shown in Figure 1, the T1, T2 and off-resonance estimation converged to their actual values after 3,000 iterations. Comparison with dictionary reconstruction is shown in Figure 2, 3 and 4. As can be seen in Table 1 which lists the STDs and mean relative errors, T1, T2 and off-resonance estimations are more accurate with Kalman filter reconstruction and. In off-resonance estimation, we can clearly see that there is a staircase in the result of dictionary method, while the Kalman filter can estimate the off-resonance more precisely.Compared to the original dictionary-matching method, the Kalman filter has several advantages for MRF. First, it does not need the definition of the dictionary and overcomes the problem in the time and memory required by dictionary. Second, using the same data, Kalman filter can achieve more accurate result because it employs recursive calculation not discrete dictionary entries. Third, since it does not need dictionary, the pulse sequence design can be more flexible than dictionary algorithm. We can flexibly change the parameters such as TR and Flip Angle, while for dictionary-matching method, a change in TR and Flip Angle means the need of recalculating the whole dictionary, which is quite time-consuming. Finally, the Kalman filtering approach is very flexible so that it might be used to accommodate deviations from simple signal models currently used to generate the dictionary.In this paper, we describe an alternative parameter estimation method in MRF based on Kalman filtering. The result shows that this approach not only enables reconstruction in MRF free of the use of dictionary, but also provides better results than dictionary-matching.