Modelling and correcting the impact of RF pulses for continuous monitoring of hyperpolarized NMR

Abstract Monitoring the build-up or decay of hyperpolarization in nuclear magnetic resonance requires radio-frequency (RF) pulses to generate observable nuclear magnetization. However, the pulses also lead to a depletion of the polarization and, thus, alter the spin dynamics. To simulate the effects of RF pulses on the polarization build-up and decay, we propose a first-order rate-equation model describing the dynamics of the hyperpolarization process through a single source and a relaxation term. The model offers a direct interpretation of the measured steady-state polarization and build-up time constant. Furthermore, the rate-equation model is used to study three different methods to correct the errors introduced by RF pulses: (i) a 1/cos⁡n-1θ correction ( θ denoting the RF pulse flip angle), which is only applicable to decays; (ii) an analytical model introduced previously in the literature; and (iii) an iterative correction approach proposed here. The three correction methods are compared using simulated data for a range of RF flip angles and RF repetition times. The correction methods are also tested on experimental data obtained with dynamic nuclear polarization (DNP) using 4-oxo-TEMPO in 1 H glassy matrices. It is demonstrated that the analytical and iterative corrections allow us to obtain accurate build-up times and steady-state polarizations (enhancements) for RF flip angles of up to 25 ∘ during the polarization build-up process within ±10  % error when compared to data acquired with small RF flip angles ( <3 ∘ ). For polarization decay experiments, corrections are shown to be accurate for RF flip angles of up to 12 ∘ . In conclusion, the proposed iterative correction allows us to compensate for the impact of RF pulses offering an accurate estimation of polarization levels, build-up and decay time constants in hyperpolarization experiments.

This document is structured into three sections: The first two deal with the simulated performance of the three discussed methods to correct for the readout RF pulses.The last section shows experimental (uncorrected) FIDs of both samples to support the signal-to-noise ratio (SNR) discussion of the main text.
For the assessment of the three corrections (1/ cos n , CC-model as introduced in [1] and extended to the steady-state polarization (enhancement) in our work as well as the introduced iterative correction involving the CC-correction) in the simulations, we used a time slicing approach to numerically integrate Eq. 1 of the main paper with RF pulses applied at a fixed repetition time T R .The simulations included a large noise (compare SNR in the simulations for 2.5°pulses with the experimental FID shown in Figs.S9 and S10 with an SNR of 1000 or more based on the first data point of the FID).The build-ups and decay observed with noise and RF pulses can show very distinct apparent hyperpolarization dynamics compared to the assumed experimental parameters (compare Figs. 1a and S5 as well as Tab. 1 and S1).The discussed corrections were then applied to these apparent build-ups and decays and the results of the corrections can be compared with the expectations based on the assumed experimental parameters.

S1 Simulations of noisy build-ups with RF pulses
The build-ups were simulated based on a time slicing approach of Eq. 1 of the main text.The input parameters (assumed thermal electron polarization, time constant and steady-state polarization) are inspired by experimental values.Application of RF pulses at a fixed repetition time T R alters the polarization build-up.We applied the extended CC-correction and the iterative correction, involving the time constant estimate of the CC-model, to correct for the impact of the RF pulses.To quantify the performance, we focus on the steady-state polarization and time constant characterizing the build-up.As random noise is applied in the simulation, thousands of random noise configurations are used to estimate the accuracy and precision of the corrections.The resulting histograms for the corrected build-up time together with the expected value are shown in Figs.

S2 Simulations of noisy decays with RF pulses
We performed similar simulations for the decay as for the build-up.In particular, the impact of RF pulses in noise-free simulations is studied in Fig. S5 and Tab.S1 -in analogy to Fig. 1 and Tab. 1 of the main text.Subsequently, the effect of noise on the accuracy and precision of the corrections (iterative, CC and 1/ cos θ n−1 ) is studied over thousands of configurations.The resulting histograms are shown in Figs.S6, S7 and S8.
S1 and S2.The respective results for the corrected SNR are shown in Figs.S3 and S4.
ig. S1.Results of the CC-correction for the build-up time constant for different flip angles θ and repetition times T R .The vertical black line refers to the theoretically expected value.Assumed experimental parameters without pulses: P 0 = 0.3, τ bup = 50, A = 1, noise = 3.2 • 10 −4 , 5000 configurations.Results of the iterative correction for the build-up time constant for different flip angles θ and repetition times T R .The vertical black line refers to the theoretically expected value.Assumed experimental parameters without pulses: P 0 = 0.3, τ bup = 50, A = 1, noise = 3.2 • 10 −4 , 5000 configurations.ig.S3.Results of the CC-correction for the SNR for different flip angles θ and repetition times T R .The vertical black line refers to the theoretically expected value.Assumed experimental parameters without pulses: P 0 = 0.3, τ bup = 50, A = 1, noise = 3.2 • 10 −4 , 5000 configurations.Results of the iterative correction for the SNR for different flip angles θ and repetition times T R .The vertical black line refers to the theoretically expected value.Assumed experimental parameters without pulses: P 0 = 0.3, τ bup = 50, A = 1, noise = 3.2 • 10 −4 , 5000 configurations.