The Partial Volume Problem in MR Fingerprinting from a Bayesian Perspective

Debra F. McGivney^{1}, Anagha Deshmane^{2}, Yun Jiang^{2}, Dan Ma^{1}, and Mark A. Griswold^{1}

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.

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 T_{1}, T_{2} 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 T_{1}, T_{2} values, we can calculate weights using, for example, the pseudoinverse model previously presented.^{5}

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.

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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