Simultaneous Estimation of Proton Densities and Receiver Coil Sensitivities using Optimized Basis Functions
Dietmar Cordes1,2, Zhengshi Yang1, Xiaowei Zhuang1, Karthik Sreenivasan1, and Le Hanh Hua1

1Cleveland Clinic Lou Ruvo Center for Brain Health, Las Vegas, NV, United States, 2Department of Psychology and Neuroscience, University of Colorado, Boulder, CO, United States

### Synopsis

In this study, a new algorithm to better model the receiver coil sensitivities with the purpose of obtaining unbiased proton density maps is proposed. Using optimized orthonormal basis functions for the modeling produces an accurate fit of potential inhomogeneities of the signal due to receiver coil bias. The obtained final image of the proton density has low variance, suitable for quantitative diagnostic information of brain tissue. Results are shown for nine MS patients and one control subject.

### Purpose

A difficult problem in quantitative MRI is the accurate determination of the proton density, which is an important quantity in measuring brain tissue organization in multiple sclerosis (MS) 1,2. After determining T1 and correcting for transmission sensitivity, we solve for the proton density and receiver coil sensitivity by constructing an optimized orthonormal set of basis functions using a family of functions where the proton density has an inverse linear relationship to the T1 value. We provide efficient and accurate algorithms to solve the entire problem of determining T1 estimates, transmission coil sensitivies, receiver coil sensitivities, proton densities and final normalization of cubes to arrive at a composed proton density image in less than 20 min on a standard PC running MATLAB.

### Methods

Imaging: 3T MR data for 9 MS patients and one healthy subject using a 32-channel head coil were obtained using a 2D SE-IR EPI sequence with different inversion times (TI 50ms, 400ms, 1000ms, 2400ms, TR 3s, resolution 1.9mm x 1.9mm x 4mm, TE 12ms, parallel imaging factor 2) and a 3D SPGR sequence with different flip angles (FA 4deg, 10deg, 20deg, 30deg, TR14ms, resolution 0.94mm x 0.94 mm x 1mm, TE 2ms). Data analysis: Voxel-specific T1 values were calculated from the SE-IR images using a method previously reported 3 and corrected for the transmission coil sensitivity. Then, using the signal equation for the SPGR, the magnetization $M_0^{(i)}$ can be computed for each coil i (1 to 32). According to MR physics, this magnetization is related to the receiver coil sensitivity $g^{(i)}$ and proton density $\rho$ by $M_q^{(i)}=g_q^{(i)}\rho_{q}$ (Eq.(1), where q labels each voxel. Optimized basis functions: The relationship between T1 and proton density $\rho$ in gray and white matter can approximately be expressed by an inverse linear relationship according to literature 1. This relationship is expressed by $\frac{1}{\rho}=A+\frac{B}{T_{1}}$ (Eq.2) where A and B are constants which are slightly different for gray and white matter. Typical values are A=0.879 and B=503ms 4 and A=0.858 and B=522ms 1 for gray and white matter combined. Since the basis functions that are needed to model the individual coil images are unknown, Eq.(1) offers an optimal solution to generate optimized basis functions. Using Eqs.(1,2) we obtain for the i-th coil sensitivities the expression $g^{(i)}(\overrightarrow{r})=const. \overline{M_0^{(i)}}(\overrightarrow{r})(1+\frac{\widetilde{B}}{T_{1}(\overrightarrow{r})})$ where $\widetilde{B}=\frac{B}{A}$ . Since $\widetilde{B}$ is of magnitude 572ms or 608ms, we define a uniformly random variable $\widetilde{B}\thinspace \epsilon[500,700]ms$ and create a large family of functions $g^{(i)}(\overrightarrow{r})$ from which we generate orthonormal basis functions using principal component analysis (PCA). We retain the most dominant PCA components belonging to the largest eigenvalues of the PCA decomposition. By partitioning the entire volume into smaller cubes of size 3cm x 3cm x3cm, we obtain cube-optimized basis functions using above method. Using the notation $R_{q}=\frac{1}{\rho_{q}}$, we then solve the resulting equation $R_{q}M_q^{(i)}=\sum_{j=1}^pf_{jq}^{(i)}A_j^{(i)}$ (Eq.2) for the unknown coefficients $A_j^{(i)}$ and $R_{q}$ simultaneously using a weighted least square algorithm for each cube and repeat this analysis for a different 3D gridding where each cube is shifted by half the edge length in x, y and z. Then, each voxel provides two values for the proton density, and we can calculate the best multiplication factor of each cube by minimizing the overall variance in proton density.

### Results

Images in Fig.1 on the left and middle show the results obtained for $R_{q}$ by solving Eq.(2) for each cube belonging to the two different 3D gridding approaches. On the right of Fig.1 the final result is shown for the proton density $\rho_{q}$ without any artifacts. Table 1 shows mean proton density values for all subjects in white and gray matter. We also carried out simulations with assumed proton density values from real data. Our approach gave a RMSQ error less than 0.4%.

### Discussion

We have developed optimized basis functions based on the known relationship between T1 and proton density. Since we are not limiting this relationship by a discrete value of the unknown constants but chose a larger interval range to create a large family of orthonormal functions, the proton density vs. receiver coil sensitivity problem can be elegantly and accurately solved.

### Conclusion

We propose a new algorithm to better model the receiver coil sensitivities with the purpose of obtaining unbiased proton density maps. Using optimized basis functions for the modeling produces an accurate fit of potential inhomogeneities of the signal due to receiver coil bias. The obtained final image of the proton density has low variance, suitable for quantitative diagnostic information of brain tissue.

### Acknowledgements

No acknowledgement found.

### References

1. Volz S., et al. 2012. Correction of Systematic Errors in Quantitative Proton Density Mapping. Magn Reson Med 68:74-85.

2. Mezer, A., et al. 2013. Quantifying the local tissue volume and composition in individul brains with magnetic resonance imaging. Nat Med.19:1667-72.

3. Barrel J.K. et al. 2010. A Robust Methodology for in Vivo T1 Mapping. Magn Reson Med 64:1057-1067.

4. Gelman, et al. 2001. Interregional variation of longitudinal relaxation rates in human brain at 3.0T: relation to estimated iron and water contents. Magn. Reson. Med. 45, 71-79.

### Figures

Fig.1. Inverse proton density image (left and middle) obtained as solutions of the first stage for the two different 3D gridding approaches (note the visibility of cubes with edge length 3cm). On the right is the final proton density map without any artifacts as obtained by solving an optimization problem to determine the proportionality factors of proton density in each cube of the left two images.

Table 1: Mean Proton density values in non-lesion white matter (WM) and Gray matter (GM) for all 9 MS subjects and 1 healthy control. The MS lesions in WM were segmented before calculating the mean proton density.

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