Dietmar Cordes^{1,2}, Zhengshi Yang^{1}, Xiaowei Zhuang^{1}, Karthik Sreenivasan^{1}, and Le Hanh Hua^{1}

^{1}Cleveland Clinic Lou Ruvo Center for Brain Health, Las Vegas, NV, United States, ^{2}Department 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.