Lukasz J. Zielinski

Steady-state free precession experiments and exact treatment of diffusion in a uniform gradient

We derive an analytic solution for the magnetization of spins diffusing in a constant gradient field while applying a long stream of rf pulses, which is known as the steady-state free precession (SSFP) sequence. We calculate the diffusion-dependent amplitude of the free induction decay (FID) and higher order echoes for pulses with arbitrary flip angle α and pulse spacing TR. Stopped-SSFP experiments were performed in a permanent gradient field and the amplitudes of the first three higher order echoes were measured for a range of values of α and TR. Theoretical results are in excellent agreement with experimental results, using no adjustable parameters. We identify various diffusion regimes in a rather large parameter space of pulsing and relaxation times, diffusion coefficient, and flip angle and discuss the interplay of the relevant time scales present in the problem. This “phase diagram” provides a road map for designing experiments which enhance or suppress the sensitivity to diffusion. We delineate th...

Effects of finite-width pulses in the pulsed-field gradient measurement of the diffusion coefficient in connected porous media

We analytically compute the apparent diffusion coefficient D(app) for an open restricted geometry, such as an extended porous medium, for the case of a pulsed-field gradient (PFG) experiment with finite-width pulses. In the short- and long-time limits, we give explicit, model-independent expressions that correct for the finite duration of the pulses and can be used to extract the pore surface-to-volume (S/V) ratio as well as the tortuosity. For all times, we compute D(app) using a well-established model form of the actual time-dependent diffusion coefficient D(t) that can be obtained from an ideal narrow-pulse PFG. We compare D(app) and D(t) and find that, regardless of pulse widths and geometry-dependent parameters, the two quantities deviate by less than 20%. These results are in sharp contrast with the studies on closed geometries [J. Magn. Reson. A 117 (1995) 209], where the effects of finite gradient-pulse widths are large. The analytical results presented here can be easily adapted for different pulse protocols and time sequences.

Characterization of coupled pore systems from the diffusion eigenspectrum

Complex structures often consist of many interconnected or “coupled” simpler regions. The problem frequently arises of determining the geometry of these individual subregions within the larger structure. We consider a simple model to argue that the high eigenmodes of the diffusion equation can be used to probe their geometry. We find that for a wide range of coupling, certain high eigenmodes preferentially remain within a particular subregion, thereby allowing the association of the corresponding eigenvalue with that subregion. We discuss an application of these results to the characterization of internal structure of porous media.

Restricted diffusion in grossly inhomogeneous fields.

We analyze the effects of geometrical restriction on the nuclear magnetization of spins diffusing in grossly inhomogeneous fields where radio-frequency (RF) pulses are weak relative to the total field inhomogeneity, making the rotation angle space-dependent and thus exciting multiple coherence pathways. We show how to separate the effects of restricted diffusion from the effects of the pulses in the case when the change in the field experienced by a diffusing spin in the course of the experiment is small compared to the RF magnitude. We then derive explicit formulas for the contribution of individual coherence pathways to the total magnetization in arbitrary pulse sequences. We find that, for long diffusion times, restriction can dramatically alter the spectrum and the shape of a particular echo, while for short times, the correction will be proportional to the pore space surface-to-volume ratio. We demonstrate these results on the example of the early echoes of the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence.

Combined effects of diffusion, nonuniform-gradient magnetic fields, and restriction on an arbitrary coherence pathway

We develop a general framework for analyzing the effects of restricted geometries and inhomogeneous (nonuniform-gradient) magnetic fields on the relaxation of nuclear magnetization. The formalism naturally separates the effects of radio-frequency pulses by introducing the field scattering kernel F(t)≡〈[B(t)−B(0)]2〉 which captures all the interactions of the diffusing spins with the inhomogeneous field and with the confining walls. F(t) is the fundamental building block in the computation of the magnetization in any sequence of pulses. We use it to derive explicit formulas for the attenuation of the echoes of a general coherence pathway and thus arbitrary pulse trains. The short-time and long-time results, proved rigorously, are model-independent and hold for arbitrary geometries, both closed, such as a single cell or pore, and open, such as a connected porous medium. In open geometries, we compute the magnetization for all times, using a model form of the time-dependent diffusion coefficient. We apply our...