The Partial Volume Problem in MR Fingerprinting from a Bayesian Perspective
Debra F. McGivney1, Anagha Deshmane2, Yun Jiang2, Dan Ma1, and Mark A. Griswold1

1Radiology, Case Western Reserve University, Cleveland, OH, United States, 2Biomedical Engineering, Case Western Reserve University, Cleveland, OH, United States

### Synopsis

Magnetic resonance fingerprinting (MRF) is a technique that allows us to produce quantitative maps of tissue parameters such as T1 and T2 relaxation times, however it is susceptible to artifacts due to the partial volume effect. The aim of this work is to provide a blind solution to the partial volume problem in MRF using the Bayesian statistical framework. A complete description of the algorithm is presented as well as applications to in vivo data.

### Purpose

Magnetic resonance fingerprinting1 (MRF) is a technique used to produce quantitative maps of tissue parameters such as T1 andT2 relaxation times on a pixel-wise basis. In MRF, the signal at each pixel is matched to a large dictionary of simulated signal evolutions for the assignment of one T1 and one T2 value. Due to this type of matching, pixels that contain signal from multiple tissues may be incorrectly assigned. Previously, an algorithm was proposed to solve this partial volume problem in MRF (PVMRF) using the Bayesian statistical framework applied in simulation.2 We have refined the algorithm and applied it to volunteer brain data using a FISP MRF3 sequence to validate the method in vivo.

### Methods

We assume that the MRF signal $y$ for one pixel is modeled as a weighted sum of dictionary elements $y = \mathsf{A}x + \xi$, where $\mathsf{A}$ is the MRF dictionary of simulated signal evolutions, $x$ is a vector of weights determining the contribution of each dictionary entry to the signal, and $\xi$ is additive zero mean Gaussian noise.

Assuming that the signal at each pixel is comprised of only a few dictionary elements, this corresponds to imposing a sparsity prior on the weight vector $x$. To this end, we model each weight $x_j$ as a Gaussian random variable, $x_j \sim \mathcal{N}(0,\theta_j)$ with unknown variance $\theta_j$.4 Noting that the variance controls the width of the distribution, we impose an inverse gamma hyperprior density on each $\theta_j$ with shape and scale parameters $\alpha$ and $\beta$.

Applying Bayes' formula leaves us with the minimization problem in $x$ and $\theta$,

$$\min_{x,\theta} \left\{\frac{1}{2\sigma^2} \| y - \mathsf{A} x \|^2 + \frac{1}{2} \| \mathsf{R}_\theta^{-1/2} x\|^2 + \beta \sum_{j = 1}^n \frac{1}{\theta_j} + \left( \alpha + \frac{3}{2} \right) \sum_{j = 1}^n \log \theta_j \right\}$$

where $\mathsf{R}_\theta$ is a diagonal variance matrix, and the problem is solved iteratively, using the conjugate gradient method to update $x$, while minimization in $\theta$ yields an analytic solution. To ease the computational burden and to encourage more sparsity, at each iteration the dictionary is pruned keeping only the entries that are most likely to contribute to the mixed signal $y$. A conservative pruning approach is taken, to reduce the likelihood that the true components will be omitted in the earlier iterations. In addition, the sparsity constraint is eased as the iterations proceed, resulting in a more evenly spread distribution of weights at the final iterations.

### Results and Discussion

The algorithm was applied to selected pixels from a data set acquired using a FISP MRF acquisition. The dictionary size is $5790 \times 3000$ and the image matrix is $400\times 400$. The outer iterations of the algorithm were stopped using an L-curve analysis on the subproblem of minimizing in $x$, and at the final iteration, the dictionary entries that remain were visualized as scatter plots of their T1, T2 values to determine if the partial volume effect was present. Three examples are shown. Two locations, indicated in yellow and blue in Figure 1, converge to their respective MRF results, shown in Figures 2 and 3, implying the presence of one tissue within each pixel. The third pixel considered, highlighted in magenta in Figure 1, exhibits partial volume, as shown by the scatter plot after 60 iterations of the algorithm. See Figure 4.

The result of this mixed pixel was analyzed using a Gaussian mixture model to cluster the data points with Gaussian densities. This method requires a predefined number of clusters, which we chose based on the scatter plot in Figure 4. The 95% credibility ellipses of these densities, along with their means and random draws are shown in Figure 5. Using the mean T1, T2 values, we can calculate weights using, for example, the pseudoinverse model previously presented.5

### Conclusion

Having a method to blindly solve the partial volume problem in MRF is a large step forward in diagnosing pathologies that may be present at the sub-voxel level. The Bayesian framework is a natural setting for this problem, as it allows us to build into the model prior information at hierachically different levels so that the true components in the mixed signal can be observed. Further refinements to the algorithm, in particular a more efficient coding scheme and other methods of soft-clustering will be added in the future.

### Acknowledgements

The authors would like to acknowledge funding from Siemens Healthcare and NIH grants 1R01EB016728-01A1 and 5R01EB017219-02.

### References

1. Ma D, et al. (2013), Nature 495, 187-92.

2. McGivney D, et al. (2015), Proc. ISMRM 23, 3381.

3. Jiang Y, et al. (2014), Magn. Reson. Med. 10.1002/mrm.25559.

4. Calvetti D and Somersalo E (2008), Inverse Problems 24, 034013.

5. Deshmane A, et al. (2014), Proc. ISMRM 22, 94.

### Figures

Locations of three pixels analyzed.

Result after 95 iterations of the Bayesian PVMRF algorithm for a pure CSF pixel (located in yellow in Figure 1). The result from MRF is also shown.

Result after 85 iterations of the Bayesian PVMRF algorithm for a pure CSF pixel (located in blue in Figure 1). The result from MRF is also shown.

Result after 60 iterations of the Bayesian PVMRF algorithm for a pixel exhibiting the partial volume effect (located in magenta in Figure 1). The result from MRF is also shown.

Random draws and credibility ellipses from the Gaussian densities that approximate the result corresponding to the mixed pixel.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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