Introduction:

Interactions of sodium ions with proteins yield a sodium triple-quantum (TQ) signal, which can be a valuable biomarker for cell viability due to its intracellular sensitivity. To leverage its full potential, a deeper understanding of the sodium TQ signal using model solutions and cell experiments is necessary. To do so, the TQ time proportional phase incrementation (TQTPPI) pulse sequence represents an elegant way to sample the SQ and TQ signals using simultaneous evolution time and phase increments¹. This unique feature of the TQTPPI pulse sequence allows extracting the maximum TQ signal despite possible changes in transverse relaxation times which is in contrast to a fixed-delay TQ pulse sequence. The quantification of the TQ signal using the TQ/SQ ratio has been shown to provide valuable insights into sodium interactions with proteins^1-4. However, the uniform sampling of the TQTPPI FID in a second dimension increases the measurement time compared to a fixed-delay TQ pulse sequence. However, the sparsity of the corresponding TQTPPI spectrum can be exploited using compressed sensing to achieve a higher time efficiency.

In this study, we investigated the feasibility of low rank compressed sensing (CS) reconstruction^5,6 to speed up sodium TQTPPI measurements. Despite the sparsity of the TQTPPI spectrum, the data analysis of the TQTPPI FID using non-linear fitting requires a high accuracy of the reconstructed TQTPPI FID. Thus, we verified CS based reconstruction of undersampled TQTPPI measurements using simulations and measurements.

Material and Methods:

Measurement data was acquired at a 9.4T preclinical MRI (Bruker Biospec 94/20). A ¹H/²³Na Rapid volume coil and a ²³Na Rapid surface receiver coil was used for the protein samples consisting of either 30%w/v bovine serum albumin or 10%w/v Hemoglobin with 154mM NaCl. A ¹H/²³Na/³¹K Bruker volume coil was used for 2%w/v agarose sample with 134.75mM NaCl. To quantify the SQ and the TQ amplitudes and the transverse relaxation times the TQTPPI FID (Fig.1) was non-linearly fitted using:
$$ Y(t)=\sin(\omega t)\cdot\left(A_{SQ,1}e^{-\frac{t}{T_{2F}}}+A_{SQ,2}e^{-\frac{t}{T_{2S}}} \right )+A_{TQ}\sin(3\omega t)\left(e^{-\frac{t}{T_{2F}}}-e^{-\frac{t}{T_{2S}}} \right ) ,\ \ (1)$$
where Y(t) is the TQTPPI FID amplitude, A_SQ,i and A_TQ are the SQ and TQ amplitudes, respectively. T_2S and T_2F are the slow and fast transverse relaxation time.

Simulation of TQTPPI FIDs was performed using equation (1) with addition of Gaussian white noise. To cover the possible range of tissue parameters^1,4,7, we varied the TQ amplitude A_TQ/A_SQ=5-25%, the fast relaxation time T_2F=5-30ms and the signal-to-noise ratio SNR=40-200. The range of SNR was chosen based on the measurements results, which yielded a SNR≥65. Each parameter was varied individually, while all other values remained at the standard values SNR=70, A_TQ/A_SQ=10%, T_2S=39ms, T_2F=10ms.
The TQTPPI FID was retrospectively non-uniformly undersampled (NUS). The undersampling pattern was based on sinusoidal Poisson-gap sampling⁸, which favors small evolution times with a large signal. Moreover, it avoids large gaps between data points at higher t_evo. The undersampling factor was varied in the range of 2 to 16.

The CS algorithm low rank reconstruction (LR)^5,6 was adapted to the TQTPPI pulse sequence using MATLAB (MathWorks). Model parameters were optimized for a high data consistency leading to optimal performance. In contrast to iterative thresholding algorithms, LR can recover the Lorentzian shape of the SQ and TQ peaks in the TQTPPI spectrum (Fig.1). This is essential to obtain the transverse relaxation time and the maximum TQ amplitude. Additionally to the CS reconstruction, the undersampled TQTPPI FID was non-linearly fitted without reconstruction of the missing data points (NUSF).
The ground truth for the measurement data represents the fit result of the fully sampled FID (FSF).

Results/Discussion:

Fig.2 shows the accuracy of the A_TQ/A_SQ, T_2S and T_2F for different undersampling factors. Up to undersampling factors of 5, both algorithms yielded less than 5% deviation from ground truth for all parameters.

Decreasing TQ amplitude and SNR led to larger deviations (Fig.3). NUSF and LR did not yield reliable results for a SNR below 40 and 60, respectively. For increasing SNR, LR yielded the same results as FSF, while NUSF resulted in a small offset for A_TQ/A_SQ and the transverse relaxation times. For a decreasing difference in transverse relaxation times $$$\Delta T_2 = T_{2S}-T_{2F}$$$, LR resulted in a small deviation for all values, while NUSF substantially deviated at small differences (Fig.4). This indicates high stability and reliability for LR while NUSF shows some outliers.

In summary, NUSF and LR achieved accurate results up to undersampling factors of 5 using simulated data. These results were confirmed with measurement data of protein and agarose samples, where only small deviations for all parameters were observed for undersampling factors up to a factor of 5 (Fig.5). This can potentially lead to measurement time reductions of up to 80%.

Conclusion:

NUS with and without CS reconstruction resulted in small deviations of less than 10% for undersampling factors up to 5 using simulated and measurement TQTPPI data. A potential measurement time reduction of up to 80% can be achieved. This could be beneficial for TQTPPI applications which require a high temporal resolution such as dynamic studies of perfused organs or bioreactor systems.

Acknowledgements

No acknowledgement found.