Jinhu Xiong^{1}, William R Kearney^{1}, Mathews Jacobs^{1}, Rolf Schulte^{2}, Baolian Yang^{2}, and Vincent Magnotta^{1}

We have developed two models (anisotropic-anisotropic (AAS) vs. anisotropic-isotropic (AIS) models) to correlate the measured fast/slow decay components of tissue sodium to the intra-/extra-cellular sodium concentrations. The models were evaluated based on theoretical and experimental results. Our results indicate that AAS model fits experimental data much better than AIS model does.

**Introduction**

**Methods**

Data
collection: Images were collected using a 1H/23Na
dual tuned head coil (Rapid Biomedical, Germany)
on a 7T human scanner (GE MR950, USA) from ten subjects. Data were acquired using a fast 3D multi-echo ultrashort
TE radial free induction decay sequence with TR = 50ms, the minimum TE = 455µs, 8 echoes with a 2.176ms
echo spacing, tip = 15^{0}, 200mm isotropic field-of-view, 7333 X 33
matrix, and total acquisition time of 6.07 minutes. The k-space data was re-gridded to a 48 x 48 x
48 isotropic matrix. Image reconstructions were performed off-line
using Tikhonov based regularization [1, 2]. Two NaCl reference standards (51.3mM
and 154.0mM saline solution) were imaged along with the subjects and used for
normalization.

Image processing: A bi-exponential decay model was fit to the multi-echo
23Na signal, *c(t)*, for each voxel to estimate sodium concentrations and R2*
values for both fast decay component (FDC) and slow decay component (SDC).

$$c(t)=C_{f}e^{-R2ft}+C_{s}e^{-R2st } $$ 1

where C_{f} and C_{s}
correspond to FDC and SDC sodium signal, respectively; R_{2f}* and R_{2s}*
are transverse relaxation
rates of the FDC and SDC. A least square estimation was used for estimating sodium relaxation times and
compartmental concentrations. Region
of interest measures were obtained for the following tissue types: white
matter, gray matter, and cerebral spinal fluid.

Modeling: In our anisotropic and isotropic systems (AIS) model, intra-cellular sodium ions have restricted motions, are characterized by a bi-exponential decay, and contribute to both FDC and SDC; extra-cellular sodium ions have a free motion, are characterized by a mono-exponential decay, and contribute to SDC only. The measured FDC and SDC should be related to ISC and ESC by:

$$C_{f}=ηα_{i}C_{i} $$ 2

$$C_{s}=(1-η)α_{i}C_{i}+α_{e}C_{e} $$ 3

Where *C _{i}* and

In our anisotropic-anisotropic systems (AAS) model, macromolecular anions are actually present in both intra- and extra-cellular compartments [3, 4, 5]. Sodium ions in both compartments experience restricted motions and are characterized by bi-exponential T2 decay. The measured FDC and SDC should be related to ISC and ESC by:

$$C_{f}=η(α_{i}C_{i}+δα_{e}C_{e}) $$ 4

$$C_{s}=(1-η)(α_{i}C_{i}+δα_{e}C_{e})+(1-δ)α_{e}C_{e} $$ 5

where δ is the fraction of extra-cellular sodium ions experiencing restricted motion. In our compartmental model, extra-cellular sodium compartment includes the interstitial and vascular spaces. The value of δ should vary according to fraction of spins restricted by macromolecules and range from 0 to 1.

Both AIS and AAS models were evaluated based on theoretical and
experimental results. First, ranges of
the theoretical sodium concentrations of FDC and SDC were computed based on AIS
and AAS models as well as theoretical values of ISC and ESC. The theoretical ranges were then compared to
the measured C_{f} and C_{s}.
Chi-square goodness of fit was performed to test which model fits our
results better. Second, theoretical T2*
decay curves were computed based on AIS and AAS model. The theoretical T2* decay curves
were compared to experimental T2* decay curves to evaluate which
model fits the curves better.

**Results**

Table 1 summarizes the modeling results of AIS and AAS models, as well
as experimental results. AIS predicts
very low ^{23}Na concentrations of FDC.
AAS predicts higher FDC and lower SDC when compared to AIS. The predicted FDC and SDC ^{23}Na
concentrations by AAS partially overlap with the measured concentrations, as
depicted in Figure 1. Based on our Chi-square goodness of
fit test, the AAS model fit our
experimental data better than the AIS model when d≥0.7.

Theoretical T

**Discussion and Conclusions**

[1] A. Tikhonov and A. VI, Solutions of ill-posed problems, Washington/New York: Winston, distributed by Halsted Press, 1977.

[2] L. Ying, D. Xu and Z. Liang, "On tikhonov regularization for image reconstruction in parallel mri," Conf Proc IEEE Eng Med Biol Soc, pp. 2:1056-9, 2004.

[3] P. M. Winter and N. Bansal, "TmDOTP5– as a 23Na Shift Reagent for the Subcutaneously Implanted 9L Gliosarcoma in Rats," Magnetic Resonance in Medicine, p. 45:436–442, 2001.

[4] H. Naritomi, M. Kanashiro, M. Sasaki, Y. Kuribayashi and T. Sawada, "In vivo measurements of intra- and extracellular Na+ and water in the brain and muscle by nuclear magnetic resonance spectroscopy with shift reagent," Biophys. J, p. 52 (4):611–616, 1987.

[5] G. Navon, "Complete elimination of the extracellular Na-23 Nmr signal in triple quantum filtered spectra of rat hearts in the presence of shift-reagents," Magn. Reson Med, p. 30 (4): 503–506, 1993.

Figure 1.
Predicted and measured 23Na concentrations of FDC and
SDC. The solid line represents the range
of the measured 23Na concentrations (mean ± standard
deviation). The dashed line is the range
of the predicted 23Na concentrations using AIS model. The dotted line is the range of the predicted
23Na concentrations using AAS model (δ=0.9).
The measured 23Na concentrations partially overlap with those
predicted by AAS model for both FDC and SDC.

Figure 2.
Predicted and measured T2* decay curves, averaged across GM and WM VOIs.
The solid lines represent the average T2* decay curves predicted by
AAS (Red) and AIS (Green) models. The dashed lines represent the Upper-limit
(UL) and Lower-limit (LL) of the predicted decay curves. The ¨
represents the measured 23Na concentrations (mean ± standard
deviation). Sodium signals are
normalized to be 1 at t=0.