We study the influence of diffusion on NMR experiments when the molecules undergo random motion under the influence of a force field, and place special emphasis on parabolic (Hookean) potentials. To this end, the problem is studied using path integral methods. Explicit relationships are derived for commonly employed gradient waveforms involving pulsed and oscillating gradients. The Bloch-Torrey equation, describing the temporal evolution of magnetization, is modified by incorporating potentials. A general solution to this equation is obtained for the case of parabolic potential by adopting the multiple correlation function (MCF) formalism, which has been used in the past to quantify the effects of restricted diffusion. Both analytical and MCF results were found to be in agreement with random walk simulations. A multi-dimensional formulation of the problem is introduced that leads to a new characterization of diffusion anisotropy. Unlike for the case of traditional methods that employ a diffusion tensor, anisotropy originates from the tensorial force constant, and bulk diffusivity is retained in the formulation. Our findings suggest that some features of the NMR signal that have traditionally been attributed to restricted diffusion are accommodated by the Hookean model. Under certain conditions, the formalism can be envisioned to provide a viable approximation to the mathematically more challenging restricted diffusion problems. * Electronic address: evren.ozarslan@boun.edu.tr

Purpose: Definition of a macromolecular MR spectrum based on diffusion properties rather than relaxation time differences and characterization of non-Gaussian diffusion of brain metabolites with strongly diffusion-weighted MR spectroscopy. Methods: Short echo time MRS with strong diffusion-weighting with b-valuesup to 25 ms/μm 2 at two diffusion times was implemented on a Connectom system and applied in combination with simultaneous spectral and diffusion decay modeling. Motion-compensation was performed with a combined method based on the simultaneously acquired water and a macromolecular signal. Results: The motion compensation scheme prevented spurious signal decay reflected in very small apparent diffusion constants for macromolecular signal.Macromolecular background signal patterns were determined using multiple fit strategies. Signal decay corresponding to non-Gaussian metabolite diffusion was represented by biexponential fit models yielding parameter estimates for human gray matter that are in line with published rodent data. The optimal fit strategies used constraints for the signal decay of metabolites with limited signal contributions to the overall spectrum. Conclusion:The determined macromolecular spectrum based on diffusion properties deviates from the conventional one derived from longitudinal relaxation time differences calling for further investigation before use as experimental basis spectrum when fitting clinical MR spectra. The biexponential characterization of metabolite signal decay is the basis for investigations into pathologic alterations of microstructure.

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