Analytical solution of the Bloch-McConnell equations for steady-state CEST Z-spectra

Purpose: To derive an analytic expression for the steady-state Chemical Exchange Saturation Transfer (CEST) Zspectra of a two-pool proton-exchanging system, facilitating simulations and expedited fitting of steady-state Zspectra. Method: The analytical expression is derived by directly solving the set of Bloch-McConnell differential equations in matrix form for a two-pool exchanging system, determining water magnetization under steady-state saturation across the entire Z-spectrum. The analytic solution is compared and validated against the numerical solution of Bloch-McConnell equations under prolonged saturation. The study also explores the line shape of a CEST peak, interpolating under-sampled Z-spectra, and Z-spectral fitting in the presence of noise. Results: The derived analytic solution accurately reproduces spectra obtained through numerical solutions. Direct fitting of simulated CEST spectra with the analytical solution yields the physical parameters of the exchanging system. The study shows that the analytical solution enables the reproduction of fully sampled spectra from sparsely sampled Z-spectra. Additionally, it confirms the approximation of the CEST spectrum of a single exchanging proton species with a Lorentzian function. Monte Carlo simulations reveal that the accuracy and precision of Z-spectral fittings for physical parameters are significantly influenced by data noise. The study also derives and discusses the analytical solution for three-pool Z-spectra. Conclusion: The derived analytic solution for steady state Z-spectra can be utilized for simulations and Z-spectrum fitting, significantly reducing fitting times compared to numerical methods employed for fitting CEST Z-spectra.