Introduction

Slow modulations of the electric field gradient and the quadrupole interactions are causing biexponential ²³Na T₁ and T₂ MR relaxation. Recently, we presented a sequence to determine reliably¹ the T_1s and T_1f components of the relaxation times.

It was previously shown that the Debye model does not adequately describe sodium interactions^2-4. The T₁ differences are much smaller than expected from the Debye model. However, the model is still widely used^4-6, since alternative models^2-4,7 usually use too many parameters or are limited to specific situations like anisotropic environment.

This study investigate an alternative model of sodium interactions based on the Debye model that is simple, yet providing more compatibility with the experimental T₁ and T₂ relaxation times. We compare and discuss the parameters of this model for several samples at 9.4T and 21.1T.

Materials and Methods

The 23Na relaxation times are calculated by⁴
$$T_{1s}=\frac{1}{2J_1},\;T_{1f}=\frac{1}{2J_2}\\T_{2s}=\frac{1}{J_1+J_2},\;T_{2f}=\frac{1}{J_0+J_1},$$
where J_m are the spectral densities that carry the information about interactions of the sodium nucleus with its environment.

The most general model for describing the spectral densities uses a distribution of correlation times $$$p(\tau_c)$$$ and time dependent interaction strength $$$\omega_Q(\tau_c)$$$:
$$J_m=\int_{0}^{\infty}p(\tau_c)\cdot\omega_Q^2(\tau_c)\frac{\tau_c}{1+(m\omega_0\tau_c)^2}d\tau_c$$
Assumptions about $$$p(\tau_c)$$$ and $$$\omega_Q(\tau_c)$$$ are necessary to simplify $$$J_m$$$ for application in experiments.

The Debye model uses a Delta distribution for $$$p(\tau_c)$$$ at single correlation time $$$\tau_c$$$ and interaction strength $$$\omega_Q$$$ to describe the spectral densities:
$$J_m=\omega_Q^2\frac{\tau_c}{1+(m\omega_0\tau_c)^2}$$

There are two relevant motional regimes, the extreme narrowing regime with $$$\omega_0\tau_c\ll1$$$ that causes mono-exponential decay and the slow-motion regime that causes biexponential relaxation. Both regimes can occur simultaneously, e.g. by Bull exchange⁸. Therefore, a model of sodium interactions should consider the influence of both regimes.

We propose a $$$p(\tau_c)$$$ consisting of two delta distributions, one for each motional regime leading to spectral densities of the form
$$J_m=\omega_{Q,en}^2\tau_{c,en}+\frac{\tau_c}{1+(m\omega_0\tau_c)^2}=J_{en}+\frac{\tau_c}{1+(m\omega_0\tau_c)^2}$$

where $$$J_{en}=\omega_Q^2\tau_{c,en}$$$ is the spectral density in the extreme narrowing regime and $$$\tau_c$$$ and $$$\omega_Q$$$ are the effective correlation time and interactions strength, respectively.

To decouple T₁ and T₂, we introduce B₀-inhomogeneities in the model. Local frequency shifts of the Lamor frequency $$$\omega_{shift}$$$ change the transfer functions according to⁹
$$f_{ij,shift}^q(t)=f_{ij}^q(t)\exp(-iq\omega_{shift}t)$$

Typically, B₀ deviations follow a Lorentz distribution^9,10, such that the transfer functions become

$$f_{ij,B0}^q(t)=f_{ij}^q(t)\int_{-\infty}^{\infty}p(\omega_{shift})\exp(-iq\omega_{shift}t)d\omega_{shift}=f_{ij}^q(t)\exp(-qR_{B0}t)$$
The last step uses that the integral has the form of a Fourier transform (FT) and the FT of the Lorentzian is an exponential function. Therefore, the relaxation rates are effectively
$$R_{i,B0}=R_i+q\cdot{}R_{B0}$$

With q=0 for T₁, q=1 for T₂ and $$$R_{B0}$$$ being the influence of B₀-inhomogeneities on the relaxation times.
In total, the model has four intrinsic parameters, $$$J_{en}$$$, $$$\tau_c$$$, $$$\omega_Q$$$ and $$$R_{B0}$$$ and four measurement parameters, $$$T_{1s}$$$, $$$T_{1f}$$$, $$$T_{2s}$$$ and $$$T_{2f}$$$. To calculate the model intrinsic parameters from the measurement parameters, we use an optimization algorithm that minimizes the difference between the measurement parameters and the relaxation times calculated from the model parameters.

Measurement data was acquired at a 9.4T preclinical MRI (Bruker Biospec 94/20) using a linear ¹H/²³Na Bruker volume coil and at the 21.1T preclinical MRI scanner at the NHML (Tallahassee, Florida, USA)¹¹ equipped with a custom-built ¹H/²³Na volume coil¹². The samples consisted of [2,4,6]% w/w agarose with 154mM. The T₂ and T₁ relaxation times were obtained using the TQTPPI¹³ and the IRTQTPPI sequences¹, respectively. T_1f is determined using T_1s from a biexponential fit and T_1m from a mono-exponential fit of the FID.

Results/Discussion

Fig.1 shows the relaxation times that were used to calculate the Debye and 2xDebye model parameters. All input and model parameters are summarized in Tab.1 for 9.4T and in Tab.2 for 21.1T.

Fig.2 shows the Debye model parameters determined with T₁ and T₂ at 9.4T and 21.1T. The parameters substantially deviate from each other. This indicates that the model does not adequately describe the sodium-environment interaction and provides space for a more sophisticated model.

The model parameters for the new model proposed in this study for 9.4 T and 21.1T are also presented in Fig.2. By design, the model parameters are consistent with the measured relaxation times. However, the parameters differ between 9.4T and 21.1T since this model is still an approximation. The model parameters vary as expected when considering a more sophisticated $$$p(\tau_c)$$$ as indicated in the results of others³. Since $$$\omega_0\tau_c\sim1$$$ for smaller $$$\tau_c$$$, the slow motion regime includes faster interactions. This causes the effective $$$\tau_c$$$ to decrease and the average $$$\omega_Q$$$ to increase, since the fraction in the slow motion regime increases. $$$J_{en}$$$ decreases as the fraction in the extreme narrowing regime decreases.

The value of R_B0 (the inhomogeneity parameter) is negative for all samples, which is unphysical. This indicates that T_2s is too large relative to T_1s and T_1f. The relaxation times depend on the temperature and sometimes change in the 1-3ms range during measurement. Such small difference can shift to positive values. Moreover, T_1f makes only 20% of the signal, which may vause inaccurate fit values.

Conclusion

This study presents a simple model of sodium interactions with environment that includes contribution of both the extreme narrowing as well as the slow motion regimes in presence of B₀ inhomogeineities. The model parameters are easy-to-compute and inherently provide relaxation times closer to the experiment and even works with almost mono-exponential T₁.

Acknowledgements

A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida.