The “Spin-3/2 Bloch Equation”: System matrix formalism of excitation, relaxation, and off-resonance effects in biological tissue
Chengchuan Wu, Yasmin Blunck, Leigh A. Johnston
Abstract
Purpose
This work proposes “Spin-3/2 Bloch Equation” (SBE), a consolidated formalism for spin-3/2 dynamics in biological environments. The formalism encapsulates excitation, relaxation, and off-resonance with accessible matrix representation for a straightforward implementation with high computational efficiency.
Theory
The SBE is derived using spherical tensor operators to encapsulate the spin-3/2 dynamics in biological systems in a single system matrix, a formalism akin to the well-known Bloch Equations (BE).
Methods
Using the proposed SBE, simulations of three classical 23Na pulse sequences were performed to demonstrate the versatility and applicability of the model, returning the evolution of the 23Na spin system during these experiments: soft rectangular and adiabatic inversion recovery (IR) and triple-quantum filtering. IR simulations were compared with two existing spin-3/2 simulators and the adaptive BE as a first-order approximation.
Results
The proposed SBE is straightforward to implement and facilitates accurate and fast simulations of the underlying higher order coherence in sodium experiments of biological tissues. SBE simulations and comparison spin-3/2 simulators outperform the BE simulations as expected, with the SBE offering superior computational efficiency achieved by the single system matrix formalism.
Conclusion
The proposed SBE enables comprehensive and accurate simulations for spin-3/2 systems in biological tissue. With a one-line call to an ordinary differential equation solver, it offers a computationally efficient and accessible method for use in 23Na pulse sequence design.