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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]

Diffusion MRI gradient sampling is the choice of magnetic field gradient directions, strengths and repetitions used to acquire diffusion magnetic resonance imaging measurements. Under a Gaussian diffusion tensor model, different directions probe different values of a quadratic form. Acquiring many measurements in an uninformative set of directions cannot determine every tensor coefficient.

For a unit gradient direction $\mathbf{g}$, a separately measured matched baseline $S_0$ and fixed positive diffusion weighting b-value $b$, the ideal signal model isThe symmetric matrix $\mathbf{D}$ has six independent coefficients. Opposite directions $\mathbf{g}$ and $-\mathbf{g}$ give the same model prediction. Thus counting orientations without considering their independent information can overstate diffusion tensor identifiability.

A 2003 Monte Carlo simulation study controlled the observation budget: one baseline plus60 diffusion-weighted observations. Six distinct orientations repeated ten times and30 distinct orientations repeated twice each use60 diffusion-weighted observations. The comparison asks how angular coverage and repetition affect estimated orientation, rather than simply comparing a longer acquisition with a shorter one.^[ref]

The subsequent 2004 study compared direction schemes through estimated orientation, trace and fractional anisotropy at $b=1000s/m{\mathrm{m}}^2$. These are distinct outputs of the same tensor fit. An acquisition suited to one readout need not minimize uncertainty in another.^[ref]

An original design comparison separates two questions. Repeating the three coordinate axes may reduce random variation in their measured attenuations, but it adds no equations for the three off-diagonal tensor entries. Adding suitable mixed directions can supply those equations. Full matrix rank establishes uniqueness in the ideal model; practical measurement uncertainty, motion and model mismatch still require assessment.

The diffusion weighting b-value quantifies the encoding applied by a magnetic field gradient waveform in a diffusion measurement. For a refocused experiment in one spatial direction, defineHere $\gamma$ is the angular gyromagnetic ratio, in radians per second per tesla, and $G_{\mathrm{eff}}$ includes the sign reversal caused by the refocusing pulse. The echo time is $T$, with $q(T)=0$ for refocusing of stationary spins. Thus $q$ has units ${\mathrm{m}}^{-1}$ and $b$ has units $\mathrm{s}{\mathrm{m}}^{-2}$. The squared encoding history matters even though its final value is zero.^[ref]

For unrestricted, isotropic Gaussian diffusion with constant diffusion coefficient $D$, the otherwise matched signal ratio is $S/S_0=e^{-bD}$. Here $S_0$ uses the same pulse timing and detection conditions with the applied diffusion-encoding gradients removed. The diffusion weighting b-value describes the experiment's weighting; $D$ describes the assumed sample model.

For an original unit check, $1000\mathrm{s}\mathrm{m}{\mathrm{m}}^{-2}=10^9\mathrm{s}{\mathrm{m}}^{-2}$. With $D=10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the ideal attenuation is $e^{-1}\approx 0.368$. Quoting a numerical $b$ without its units can therefore cause a millionfold scaling error.

The Stejskal-Tanner equation gives $b=(\gamma g\delta {)}^2(\Delta -\delta /3)$ for its equal rectangular pulses. More general waveforms require their own integral. For direction-dependent diffusion, a scalar $b$ alone cannot specify the complete directional encoding and response; a tensor description is needed.

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]

The Stejskal-Tanner equation relates diffusion-sensitive echo attenuation to a rectangular gradient-pair experiment:It assumes free Gaussian diffusion with a constant scalar diffusion coefficient $D$, ideal refocusing and two equal nonoverlapping rectangular gradient pulses of magnitude $g$ and duration $\delta$, whose starting times are separated by $\Delta$. The gyromagnetic ratio $\gamma$ is angular. The baseline $S_0$ is an otherwise matched echo without the applied diffusion-encoding gradients. These definitions are part of the formula, rather than optional choices of notation.^[ref]

The finite-duration term follows directly from the encoding history. Write $q_0=\gamma g\delta$. The effective encoding rises linearly from zero to $q_0$, stays constant for $\Delta -\delta$, then falls linearly to zero. Each ramp contributes $q_0^2\delta /3$ to the diffusion weighting b-value, soThe two physical gradients can have the same polarity: the intervening $180^{\circ }$ pulse reverses the effective phase-encoding sign.

For the illustrated $\delta =2\mathrm{ms}$ and $\Delta =30\mathrm{ms}$, the normalized integral $b/q_0^2$ is $29\frac{1}{3}\mathrm{ms}$, rather than $30\mathrm{ms}$. This isolates the pulse-duration correction without assigning an unmeasured gradient amplitude.

The 1965 work combined theory with apparatus tests, including slow self-diffusion in dry glycerol.^[ref] Restricted diffusion, flow, background gradients and measurement noise require their own interpretation; fitting a single attenuation to this equation does not establish all of its assumptions.