Quantum mechanical simulation of magnetic resonance imaging

Abstract

Many recent magnetic resonance imaging experiments involve spin states other than longitudinal or transverse magnetization. For example, singlet state imaging requires two-spin correlations to be taken into account, hyperpolarised imaging should correctly account for multi-spin dynamics, multiple-quantum imaging must consider the dynamics of the corresponding coherences, etc. Simulation of such experiments requires accurate quantum mechanical treatment of spin processes together with an accurate treatment of classical processes, such as diffusion and convection.

In this thesis I report the theoretical and software infrastructure for magnetic resonance simulations using the Fokker-Planck formalism, which simultaneously accounts for spatial dynamics and quantum mechanical spin processes. Fokker-Planck equation is superior to the usual Liouville - von Neumann equation formalism in that spatial dynamics processes (diffusion, hydrodynamics, magic angle spinning, off-resonance pulses, etc.) are represented by constant matrices that are more convenient from the programming and numerical efficiency point of view than time-dependent Hamiltonians in the Liouville - von Neumann equation formalism.

It is demonstrated below that NMR and MRI experiments with elaborate spatial encoding and complicated spatial dynamics are no longer hard to simulate, even in the presence of spin-spin couplings and exotic relaxation effects, such as singlet state symmetry lockout. Versions 2.3 and later of Spinach library support arbitrary stationary flows and arbitrary distributions of anisotropic diffusion tensors in three dimensions simultaneously with Liouville-space description of spin dynamics, chemical kinetics and relaxation processes. The key simulation design decision that has made this possible is the abandonment of Bloch-Torrey and Liouville - von Neumann formalisms in favour of the Fokker-Planck equation. The primary factors that have facilitated this transition are the dramatic recent improvement in the speed and capacity of digital computers, the emergence of transparent and convenient sparse matrix manipulation methods in numerical linear algebra, and the recent progress in matrix dimension reduction in magnetic resonance simulations.

The principal achievement of this thesis is in programming and software engineering – the reader is encouraged to take a look at the Fokker-Planck module of the Spinach library that the work described in this thesis has made possible, and that was programmed as a part of this work: most of the writing performed within this project was writing code.

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Identifiers

ORCID for Ilya Kuprov: orcid.org/0000-0003-0430-2682

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