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

Motivation: Fitting CEST-MRI spectra using numerical methods is currently a time-consuming process, involving various approximations for initial parameters to expedite the process. Goal(s): We aimed to derive the exact analytical expression for CEST Z-spectra of a two-pool exchange system. Approach: We directly solved the Bloch-McConnell differential equations in matrix form for a two-pool exchange system to determine water magnetization under a steady-state saturation over the entire Z-spectrum. Results: The analytical solution accurately reproduces spectra obtained through numerical methods. It allows fitting for physical parameters of the exchange system (like the exchange rate) as demonstrated by fitting simulated CEST spectra. Impact: The analytical solution significantly reduces fitting time compared to the numerical methods used for fitting CEST Z-spectra. This solution has been demonstrated for the determination of physical parameters in the exchange system with fewer assumptions.