摘要
The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch-McConnell equations describing the behavior of magnetization in magnetic resonance imaging (MRI) in the presence of multiple chemical exchange pools. First, the time-dependent Bloch- McConnell equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation and Kronecker tensor product. From these solutions, the longitudinal relaxation rate (R1ρ) and transverse relaxation rate in the rotating frame (R2ρ) and Z-spectra were obtained. As illustrative examples, the numerical solutions for linear and star-type three-pool chemical exchange models and linear, star- type, and kite-type four-pool chemical exchange models were presented. The effects of saturation time (ST) and radiofrequency irradiation power (ω1) on the chemical exchange saturation transfer (CEST) effect in these models were also investigated. Although R1ρ and R2ρ were not affected by the ST, the CEST effect observed in the Z-spectra increased and saturated with increasing ST. When ω1 was varied, the CEST effect increased with increasing ω1 in R1ρ, R2ρ, and Z-spectra. When ω1 was large, however, the spillover effect due to the direct saturation of bulk water protons also increased, suggesting that these parameters must be determined in consideration of both the CEST and spillover effects. Our method will be useful for analyzing the complex CEST contrast mechanism and for investigating the optimal conditions for CEST MRI in the presence of multiple chemical exchange pools.
The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch-McConnell equations describing the behavior of magnetization in magnetic resonance imaging (MRI) in the presence of multiple chemical exchange pools. First, the time-dependent Bloch- McConnell equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation and Kronecker tensor product. From these solutions, the longitudinal relaxation rate (R1ρ) and transverse relaxation rate in the rotating frame (R2ρ) and Z-spectra were obtained. As illustrative examples, the numerical solutions for linear and star-type three-pool chemical exchange models and linear, star- type, and kite-type four-pool chemical exchange models were presented. The effects of saturation time (ST) and radiofrequency irradiation power (ω1) on the chemical exchange saturation transfer (CEST) effect in these models were also investigated. Although R1ρ and R2ρ were not affected by the ST, the CEST effect observed in the Z-spectra increased and saturated with increasing ST. When ω1 was varied, the CEST effect increased with increasing ω1 in R1ρ, R2ρ, and Z-spectra. When ω1 was large, however, the spillover effect due to the direct saturation of bulk water protons also increased, suggesting that these parameters must be determined in consideration of both the CEST and spillover effects. Our method will be useful for analyzing the complex CEST contrast mechanism and for investigating the optimal conditions for CEST MRI in the presence of multiple chemical exchange pools.
