Background and Purpose

CEST spectra are commonly described using the Bloch-McConnell equations, and numerical solutions to these equations have been developed [1-3]. While numerical computations are valuable for simulations, they tend to be time-consuming and require extensive experience for Z-spectral fittings [4-6]. In response to the challenges of Z-spectral analysis, various approximations have been proposed [7-15]. The most prevalent approach involves asymmetry analysis of CEST spectra with respect to the water proton resonance. This method assumes that contaminating effects on the CEST signal are symmetric, making it inadequate for addressing contaminations caused by intrinsic asymmetric effects like semi-solid magnetization transfer (MT). Another commonly used quantification method is the multicomponent CEST Z-spectral fitting, which models the Z-spectrum as a sum of Lorentzian functions corresponding to the exchanging components [16-18].
In this study, we have derived an analytical solution for the Bloch-McConnell equations pertaining to steady-state Z-spectra. With this precise mathematical expression for Z-spectra in two-pool exchange systems, we will demonstrate direct Z-spectral fitting and compare it with numerical solutions.

Theoretical considerations and Methods

The analytical solution for the Bloch–McConnell equations in a steady-state two-pool exchange system is obtained by directly solving the set of differential equations in matrix form for the inverse matrices' components. We derive this analytical solution for steady-state CEST Z-spectra by directly solving for the steady-state magnetization of water (pool a), denoted as $$$ \frac{M_{z,ss}^a}{M_0^a}=\frac{M_z^a (t_{sat}=∞)}{M_0^a} $$$ as a function of the saturation offset, denoted as $$$ω≡X$$$.

Results

The analytic form of the steady-state Z-spectrum in a two-pool exchange system is obtained as the equation: $$ 1-Z(X)=\frac{ω_1^2 (AX^2-Bω_b X+C)} {DX^2 (X-ω_b )^2+(E+Fω_1^2 ) X^2-(Gω_1^2+H) ω_b X+I} $$ Here, ω₁ is the intensity of the saturation pulse, ω_b is the Larmor frequency of the exchangeable protons (all terms involving ω are in units of rad/s), and the capital letters denote constants that encompass various contributions from the physical parameters of the pools, including exchange rates, proton concentrations, saturation parameters, and relaxation rates. This analytical solution incorporates both exchange and saturation parameters.
We compared and validated the analytic solution for steady-state Z-spectra with the numerical solutions derived from the Bloch–McConnell equations under prolonged saturation. The analytical solution accurately reproduces the spectra obtained through numerical methods for the steady-state magnetization of two exchanging pool systems (Fig. 1-2). Through direct fitting of the simulated CEST spectra with this analytical solution, we demonstrated the determination of the physical parameters of the exchange system (Fig. 3). Furthermore, the analytical solution allows the fitting of sparsely sampled Z-spectra to reconstruct complete spectra.
Additionally, we confirmed that the exact solution of a Z-spectrum can be approximated by the summation of multi-pool components in the form of Lorentzian functions. We also derived and discussed the analytical solution for three-pool Z-spectra in a similar manner.

Discussion and conclusion

Previously, there was no straightforward analytical solution available for the Bloch-McConnell equations to accurately describe and fit entire Z-spectra. Instead, fitting methods based on numerical solutions were commonly used. These methods involve calculating pool magnetization under CEST conditions using ordinary differential equations [2, 19, 20]. However, these numerical methods had a major drawback in terms of processing time. On a typical desktop computer, fitting a single Z-spectrum could take up to an hour [5, 6]. The primary reason for this extended duration is the computational cost associated with repeatedly computing matrix inversions for every saturation frequency offset during each iteration of the Z-spectrum fitting algorithm. In contrast, the analytic fitting method presented in this paper involves computing the matrix inversion only once to obtain the analytical solution for the Z-spectrum. With this analytical solution, the fitting algorithm requires rapid iterations using straightforward arithmetic computations. As a result, in contrast to numerical solution-based fitting, which can take up to an hour [5, 6], the analytical solution significantly reduces the fitting time to seconds.
In summary, our derived analytic solution for steady-state Z-spectra can accurately and rapidly fit Z-spectrum data. The exact solution of a Z-spectrum can be approximated as the sum of multi-pool components in the form of Lorentzian functions, supporting the Lorentzian line shape for CEST peaks in Z-spectra. This analytical solution provides a valuable tool for Z-spectral analysis and the direct assessment of physical parameters, including the exchange rate and the exchangeable proton concentration. It also facilitates the reconstruction of the full CEST spectrum from limited data points, potentially reducing acquisition time.

Acknowledgements

This work is supported by the NIH grants R01DK135772 and R01CA283548.

References

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