Abstract
Magnetic resonance spectroscopy and imaging experiments in which spatial dynamics (diffusion and flow) closely coexists with chemical and quantum dynamics (spin-spin couplings, exchange, cross-relaxation, etc.) have historically been very hard to simulate - Bloch-Torrey equations do not support complicated spin Hamiltonians, and the Liouville-von Neumann formalism does not support explicit spatial dynamics. In this paper, we formulate and implement a more advanced simulation framework based on the Fokker-Planck equation. The proposed methods can simulate, without significant approximations, any spatio-temporal magnetic resonance experiment, even in situations when spatial motion co-exists intimately with quantum spin dynamics, relaxation and chemical kinetics.
Highlights
Spatial coordinates are more convenient in magnetic resonance than temporal ones – unlike the indelible past, any voxel in the R3 can be accessed repeatedly
We present the corresponding algebraic and numerical framework, and illustrate its performance by simulating a singlet state diffusion and flow imaging experiment, and single-scan diffusionordered spectroscopy (DOSY) experiments that rely on spatial encoding of the diffusion dimension
M(r,t) = rTÁv(r,t) + vT(r,t)Ár + rTÁD(r,t)Ár where r = [q/qx q/qy q/qz]T is the gradient operator, v(r,t) is the flow velocity and D(r,t) is the translational diffusion tensor. The latter rarely depends on time; the flow is often stationary, meaning that the spatial dynamics generator in eqn (3) is time-independent and may be consigned to the background evolution operator – this is the primary advantage of the Fokker–Planck formalism over other simulation methods for the problem in question.[33]
Summary
Spatial coordinates are more convenient in magnetic resonance than temporal ones – unlike the indelible past, any voxel in the R3 can be accessed repeatedly. Very complicated systems of precisely this kind are emerging in all areas of magnetic resonance: hyperpolarised pyruvate imaging[15] requires simultaneous modelling of quantum spin dynamics, diffusion, hydrodynamics, chemical kinetics and spin relaxation theory; the same applies to hyperpolarised singlet state imaging.[16] Pure-shift NMR,[8] ultrafast NMR,[17] metaboliteselective imaging,[18] PARASHIFT contrast agents[19] and other similar recent developments are all united by their theory and simulation infrastructure requirements that are a level above anything that is currently available. The only formalism that simultaneously supports diffusion, hydrodynamics, chemical kinetics and Liouville-space spin dynamics in a general and computationally affordable way is the Fokker–Planck equation.[29,30,31] In this paper, we present the corresponding algebraic and numerical framework, and illustrate its performance by simulating (on the MRI side) a singlet state diffusion and flow imaging experiment, and (on the ultrafast NMR side) single-scan diffusionordered spectroscopy (DOSY) experiments that rely on spatial encoding of the diffusion dimension
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