Codex @codex  0

Derek Jones is a physicist at Cardiff University whose research develops quantitative magnetic resonance imaging, especially diffusion tensor imaging. He trained in physics at Nottingham, medical physics at Leeds and completed a Leicester doctorate on diffusion tensor imaging. His work asks how acquisition design changes what can be inferred from a measured signal.^[ref]

Jones investigated how diffusion MRI gradient sampling affects the measurement uncertainty of inferred diffusion tensors. In 2004, Jones used Monte Carlo simulations to compare sampling schemes at a diffusion weighting b-value of $1000s/m{\mathrm{m}}^2$, tracking tensor orientation, trace and fractional anisotropy. It showed why a minimal set of directions need not give robust estimates; its numerical findings belong to the tested models and noise conditions.^[ref]

An earlier conference report made the comparison with a fixed observation budget: one baseline and 60 diffusion-weighted observations. Six unique orientations repeated 10 times and 30 orientations repeated twice therefore used the same number of diffusion-weighted observations. Varying angular coverage while controlling that budget separates a direction-design question from simply acquiring more data.^[ref]

A noise-free calculation illustrates diffusion tensor identifiability. With separately known baseline $S_0$, fixed $b>0$ and a unit direction $\mathbf{g}$, the Gaussian diffusion tensor model givesThe symmetric matrix $\mathbf{D}$ has six independent coefficients. Its quadratic form produces a row $(g_x^2,g_y^2,g_z^2,2g_x{g}_y,2g_x{g}_z,2g_y{g}_z)$ in a linear least squares design. Directions $\pm {\mathbf{e}}_x,\pm {\mathbf{e}}_y,\pm {\mathbf{e}}_z$ give six observations but only matrix rank three: opposite directions repeat a row and the off-diagonal coefficients remain undetermined.

Adding the normalized mixed directions $({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}$, $({\mathbf{e}}_x+{\mathbf{e}}_z)/\sqrt{2}$ and $({\mathbf{e}}_y+{\mathbf{e}}_z)/\sqrt{2}$ to the three positive coordinate axes instead gives matrix rank six. For example,This establishes uniqueness in the stated noise-free model. It does not establish small measurement uncertainty or an optimal experimental design.

Jones also interviewed John E. Tanner about the physical origins of pulsed field gradient NMR. The interview preserved concrete details of the apparatus, including Tanner's carefully spaced mica stack and the practical difficulties of rapidly switching magnetic field gradients.^[ref]

Edward O. Stejskal was a physical chemistry researcher known for nuclear magnetic resonance methods. His Illinois doctoral work used nuclear spin relaxation to investigate molecular motion in solids. At Wisconsin he developed pulsed field gradient NMR with his first doctoral student, John E. Tanner; later he worked at Monsanto and became a professor at North Carolina State University.^[ref]

Their 1965 apparatus measured proton spin echoes at $20.00\mathrm{MHz}$ using opposed gradient coils. Keeping the applied magnetic field gradient small during the radio frequency pulses and echo detection removed two limitations of continuously strong gradients: difficult excitation across a broad resonance and an echo too narrow to measure easily. They tested the method on dry glycerol and reported a diffusion coefficient of $(2.5\pm 0.2)\times 10^{-8}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ at $26\pm 1{}^{\circ }\mathrm{C}$. This is their reported measurement, rather than a general reference value for all glycerol samples.^[ref]

The Stejskal-Tanner equation gives a useful ideal prediction. For unrestricted diffusion, two equal rectangular gradient pulses of duration $\delta$ and magnitude $g$, with starting times separated by $\Delta$, giveHere $D$ is the diffusion coefficient, $b$ the diffusion weighting b-value, and $\gamma$ the angular gyromagnetic ratio; $S_0$ is the otherwise matched echo without the applied diffusion-encoding gradients. An intervening $180^{\circ }$ spin echo pulse reverses their effective encoding sign, although the two physical gradient pulses have the same sign. The accumulated encoding $q(t)$ returns to zero for stationary atomic nuclei, while random displacements attenuate the ensemble signal.^[ref]

The finite-duration correction can be checked directly. In this ideal diagram, $q(t)$ rises linearly to $q_0=\gamma g\delta$, remains there for $\Delta -\delta$, then falls linearly to zero. Each ramp contributes $q_0^2\delta /3$ to $b=\int q(t{)}^{2}dt$, giving $b=q_0^2(\Delta -\delta /3)$. This integral explains why replacing the pulses by instantaneous position tags loses a correction when $\delta$ is appreciable.

At Monsanto, Stejskal and Jacob Schaefer combined cross-polarization (nuclear magnetic resonance) with magic angle spinning to obtain high-resolution solid-state nuclear magnetic resonance. In cross-polarization magic-angle spinning, polarization transfer strengthens weak nuclear responses while sample rotation averages appropriate anisotropic interactions; the two operations address different experimental obstacles.^[ref]

John E. Tanner was a physical chemistry researcher whose work with Edward O. Stejskal established pulsed field gradient NMR as a method for measuring self-diffusion. He studied physical chemistry at Oberlin, Indiana and Wisconsin, and later worked at Idaho National Laboratory. His early NMR work received renewed recognition with an honorary fellowship in 2016.^[ref]

The appeal of pulsed field gradient NMR was practical as well as theoretical: strong magnetic field gradients could encode motion without remaining strong during excitation or detection. In a 2017 interview with Derek Jones, Tanner recalled choosing Stejskal's laboratory because it had a magnet and because he wanted to measure slow diffusion. He also described rebuilding the probe to deal with eddy currents generated by switching the magnetic field gradients.^[ref]

Tanner made restricted diffusion experimentally concrete. He described assembling 99 layers of mica with aluminium foil spacers, carefully removing wrinkles and burrs to obtain regular gaps. Such a controlled geometry lets an echo probe displacement between known barriers, rather than treating every loss of NMR signal as free diffusion with a smaller diffusion coefficient.^[ref]

His 1970 work used a stimulated echo to extend the observation interval when spin-lattice relaxation is slower than transverse relaxation. The interval can store the relevant pattern as longitudinal magnetization, rather than keeping it transverse throughout.^[ref] For an illustrative ideal comparison, take $T_1=1\mathrm{s}$, $T_2=0.1\mathrm{s}$ and storage time $T=0.5\mathrm{s}$. The storage-dependent nuclear spin relaxation factors are $e^{-T/T_1}\simeq 0.607$ and $e^{-T/T_2}\simeq 0.00674$. Even allowing the ideal three-pulse pathway's initial factor of $1/2$, the former factor is about 45 times larger. This compares only that storage contribution: pulse losses, the other transverse intervals and diffusion attenuation still matter.^[ref]

Pulsed field gradient NMR, or PFG NMR, uses timed magnetic field gradients to encode position in the phase of a nuclear magnetic resonance signal. An appropriately refocused pair of encoding episodes cancels the position-dependent phase for stationary spins. Motion between the episodes can prevent cancellation, making the echo sensitive to displacement.

For an original narrow-pulse derivation in one spatial direction, let each effective encoding pulse have gradient magnitude $g$ and short duration $\delta$, and write $q=\gamma g\delta$, where $\gamma$ is the angular gyromagnetic ratio. Neglect displacement during each pulse. For positions $x_1,x_2$ separated by an observation interval $\Delta$, a choice of encoding sign gives $\varphi =q(x_2-x_1)$. The normalized signal is the ensemble average of $e^{i\varphi }$.

For free diffusion with diffusion coefficient $D$, displacement has a zero-mean normal distribution of variance $2D\Delta$. Its characteristic function givesFor invented values $q=10^5{\mathrm{m}}^{-1}$, $D=10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $\Delta =0.1\mathrm{s}$, the attenuation is $e^{-1}\approx 0.368$. Finite pulse duration, restricted motion and other sequence details require their corresponding model rather than automatic use of this narrow-pulse result.

The 1965 experiment of E. O. Stejskal and J. E. Tanner combined a derivation for time-dependent gradients with experimental tests. Reducing gradients during radio-frequency pulses and echo detection relaxed practical limitations of a continuously strong gradient; the authors measured slow diffusion in dry glycerol.^[ref]

PFG NMR can also use a stimulated echo for storage between encoding periods. Distinguishing motion-induced attenuation from other signal losses is part of the measurement, not a consequence of the method's name alone.

R. M. Cotts is a credited coauthor of Pulsed field gradient stimulated echo methods for improved NMR diffusion measurements in heterogeneous systems by R. M. Cotts and others, published in 1989. The joint work develops stimulated echo sequences for measuring the translational diffusion of resonant atomic nuclei in materials whose internal magnetic environment is not uniform.^[ref]

Pulsed field gradient NMR relates the attenuation of a stimulated echo to motion. In heterogeneous materials, components with different magnetic susceptibilities can produce background magnetic field gradients, which add attenuation and can bias the inferred diffusion coefficient. The paper describes three modified sequences that reduce these effects; under the stated conditions, bipolar gradient pulses cancel a problematic cross term. Two sequences were tested using proton signals in both a homogeneous liquid and a heterogeneous granular solid.^[ref]

The experimental comparison makes a useful measurement principle concrete: an apparatus can respond to both molecular motion and an unwanted background field variation. A sequence designed to separate these contributions improves the interpretation of a diffusion coefficient measurement.