Antal Horvath^{1}, Christoph Jud^{1}, Simon Pezold^{1}, Matthias Weigel^{1,2}, Charidimos Tsagkas^{3}, Katrin Parmar^{3}, Oliver Bieri^{1,2}, and Philippe Cattin^{1}

The averaged magnetization inversion recovery acquisitions (AMIRA) spinal cord imaging sequence acquires images of different inversion contrasts. Despite the different contrasts the images can be combined to even enhance tissue contrast. We give a principled justification for such averaging. Using energy optimization, we describe how to automatically optimize the contrast-to-noise ratio between different tissues using a compressed sensing inspired approach. We show that the uniform weights in the recently proposed AMIRA sequence are close to the optimum but can nevertheless still be improved. As an example we optimize the contrast-to-noise ratio between different compartments in the spinal cord.

We present an energy functional, which is based on the idea of compressed sensing [2]. Given some images $$$I_1,\ldots,I_n$$$, with values between 0 and 1, we search for coefficients $$$c = (c_1,\ldots,c_n)$$$ where the linear combination $$$I(c) := c_1 \cdot I_1 + \ldots + c_n \cdot I_n$$$ yields an optimal contrast-to-noise ratio. Suppose we have a manual segmentation of tissue $$$A$$$ and $$$B$$$ (subsets of the image domain) and suppose tissue $$$A$$$ has bright intensities close to 1 and tissue $$$B$$$ has dark intensities close to 0 (or at least darker than tissue $$$A$$$). We define the energy $$\begin{align}\mathscr{E}(c) := &\,\, \lambda_1 \frac{\sum_{x\in A} (I(c)(x)\, – 1)^2 }{|A|} + \lambda_2 \frac{\sum_{x\in B} (I(c)(x))^2 }{|B|} \,\,+ \\\\ & - \lambda_3 \big| E[I(c)(A)]-E[I(c)(B)] \big| + \lambda_4 V[I(c)(A)] + \lambda_5 V[I(c)(B)] \,\,+ \\\\ &- \lambda_6 \sum_{k=1}^n c_k \big| E[ I_k(A) ]-E[ I_k(B)] \big|+ \lambda_7 \sum_{k=1}^n c_k V[I_k(A)] + \lambda_8 \sum_{k=1}^n c_k V[I_k(B)] \,\,+ \\\\ &+ \lambda_9 \mathcal{R}(c),\end{align}$$ where $$$|A|$$$ is the cardinality of $$$A$$$, $$$I(A)$$$ is the set of all intensities of tissue $$$A$$$ on an image $$$I$$$, and $$$E[I(A)]$$$ and $$$V[I(A)]$$$ is the mean and variance of the intensities of tissue $$$A$$$ on an image $$$I$$$, respectively. The first two summands in $$$\mathscr{E}$$$ encourage the linear combination $$$I(c)$$$ to be as close to the segmentation as possible, the third negative term maximizes the contrast, and the fourth and fifth terms minimize noise. Terms 6, 7 and 8 in the third line also maximize contrast and minimize noise, however on the level of the individual inversion images. With the regularization $$\mathcal{R}(c) := \left( 1-\sum_{k=1}^n |c_k|\right)^2,$$ we weakly constrain the coefficients’ absolute values to sum up to 1. This way we omit arbitrarily large coefficients and we enable meaningful negative coefficients. A negative coefficient $$$c_k$$$ flips the contrast of an inversion image $$$I_k$$$ between tissue $$$A$$$ and $$$B$$$. $$$\mathscr{E}$$$ is minimized using BFGS.

We optimized the coefficients for a total of 68 slices on 4 different subjects at different vertebral heights (C2-C5) acquired with the AMIRA sequence. For each slice, we calculated linear coefficients $$$c_{\text{CSF/WM}}$$$ for optimal CSF/WM contrast and another set of linear coefficients $$$c_{\text{GM/WM}}$$$ for optimal GM/WM contrast, see Figure 2 and 3.

Figure 1 shows a representative series of the 8 images acquired by the AMIRA sequence and the suggested optimized CSF/WM and GM/WM contrast averages of an exemplary single axial slice located at level C2-C3. The optimized coefficients and their statistics are shown in Figure 2 and 3. The calculated optimal coefficients for GM/WM and for CSF/WM contrast are similar to uniform averaging, but they also address the weaker signal of inversion images at later inversion times, e.g. the drop of $$$c_8$$$ in Figure 2. For the calculated CSF/WM coefficients, we may now include the first inversion image, which has darker CSF and brighter WM, by allowing negative coefficients $$$c_k$$$. A flipped image $$$c_k \cdot I_k$$$ can be combined with the inversion images $$$I_5$$$ to $$$I_8$$$, where the CSF is brighter than WM. We empirically have chosen $$$\lambda_1,\ldots,\lambda_9$$$ to be 100, 100, 1, 1, 1, 10, 10, 10, and 1000, respectively.

For quantitative comparison we calculated contrast to noise ratios $$$\text{CNR}_{A/B}=\text{SNR}_{A}-\text{SNR}_{B}$$$ and signal to noise ratios $$$\text{SNR}_{A}=E[I(A)] \big/ SD[I(C)]$$$. We estimated the standard deviation of noise $$$SD[I(C)]$$$ on a homogeneous part $$$C$$$ of the background. Figure 4 shows a quantitative comparison of CNR between uniform and proposed averaging. Leave-one-subject-out results are shown in Figure 5.

With the proposed method we analyze the uniform averaging technique of the inversion images of the AMIRA sequence. The found calculated coefficients are close to the uniform coefficients and the contrast-to-noise ratios can slightly be improved. For CSF/WM contrast a notable improvement was possible. With the developed approach, we could justify the decisions made in [1] on a quantitative basis. The optimization coefficients can be chosen for the needs, e.g. prioritizing less noise or better contrast.

[1] Weigel M, Bieri O. Spinal cord imaging using averaged magnetization inversion recovery acquisitions. Magn Reson Med 2017 Jul 16. doi: 10.1002/mrm.26833.

[2] D. L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006.

Figure 1: *Top and middle row: *series of the 8 acquired inversion images of the AMIRA sequence in chronological acquisition order from top left to bottom right (zoomed in and upsampled). *Bottom row:* resized CSF/WM and GM/WM averages of the AMIRA images (combined with the mean coefficient values of Figure 2 and Figure 3, respectively).

Figure 2: Linear coefficients $$$c_{\text{CSF/WM}}$$$ for optimal CSF/WM contrast. In red, the uniform 6:8-averaging; in black, box plots of the optimized coefficients of all 68 slices with median, lower and upper quartile box, and 10/90th percentile whiskers; and in blue, the mean values are shown.

Figure 3: Linear coefficients $$$c_{\text{GM/WM}}$$$ for optimal GM/WM
contrast. In red, the uniform 1:5-averaging; in black, box plots of the optimized coefficients of all 68 slices
with median, lower and upper quartile box, and 10/90th percentile
whiskers; and
in blue, the mean values are shown.

Figure 4: Contrast-to-noise ratios, comparing the uniform averaging (unif) and the optimized coefficients (opt). The optimized coefficients are evaluated for two cases: the optimal case (s), where the optimal coefficients $$$c$$$ of each slice were used to calculate the linear combination $$$I(c)$$$ for each slice; and the global case (g), where the blue mean value coefficients of Figure 2 and 3, respectively, were used for all slices. *Left*: CNR comparison for CSF/WM contrast enhancement; *right: comparison for GM/WM contrast enhancement.*

Mean values from* left *to *right*: 83, 139, 97, 22, 24, 23.

Figure 5: Leave-one-subject-out cross-validation: CNR mean values over all slices of the subjects that were not left out are shown for uniform averaging and optimized averaging. Each cross-validation uses the coefficients $$$c_{\text{CSF/WM}}$$$ and $$$c_{\text{GM/WM}}$$$ averaged over the optimal coefficients of the slices of the left out subject. Positive improvements with the proposed method are possible for both contrasts.