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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.

Diffusion tensor identifiability asks whether the measurements determine a unique diffusion tensor within a stated model. For a constant symmetric matrix $\mathbf{D}$, known matched baseline $S_0$ and known $b_i>0$, defineFor unit directions ${\mathbf{g}}_i$, these equations give a linear system $A\mathbf{d}=\mathbf{y}$ with rowAll six coefficients are uniquely determined in this unrestricted noise-free linear model exactly when $\mathrm{rank}(A)=6$. The condition concerns independent equations, not merely the number of acquired images. Direction design and uncertainty in tensor-derived measurements were examined experimentally through controlled simulations by Derek Jones.^[ref]

For a counterexample, use the six directions $\pm {\mathbf{e}}_x,\pm {\mathbf{e}}_y,\pm {\mathbf{e}}_z$. Each antipodal pair repeats a row, so the matrix rank is three. Only $D_{xx},D_{yy},D_{zz}$ are determined. Even restricting to physically positive tensors does not generally repair this: the identity matrix and a matrix with unit diagonal and $D_{xy}=D_{yx}=\epsilon$ both fit those data for any $|\epsilon |<1$, while having different off-diagonal entries.

Replace the negative axes by $({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}$, $({\mathbf{e}}_x+{\mathbf{e}}_z)/\sqrt{2}$ and $({\mathbf{e}}_y+{\mathbf{e}}_z)/\sqrt{2}$. The axes first give the diagonal entries, thenand similarly for $D_{xz}$ and $D_{yz}$. This directly proves uniqueness for the new design. If $D_{xx}=1.8$, $D_{yy}=0.9$ and $y_{({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}}=1.55$, all in units of $10^{-3}\mathrm{m}{\mathrm{m}}^2/s$, then $D_{xy}=0.20$ in those units.

Full matrix rank is not a guarantee of a stable estimate in noisy data. If $S_0$ must also be fitted, its logarithm is another unknown; the augmented signal model needs a corresponding independent baseline constraint. The six-coefficient argument above explicitly assumes $S_0$ is separately known. Likewise, tractography requires additional interpretation beyond uniqueness of a local tensor.

A reflecting boundary condition prevents normal particle or probability flux through a wall. For drift-free scalar diffusion with $D>0$, the condition isHere $\mathbf{n}$ is the outward unit normal. The last expression is a homogeneous Neumann boundary condition. Integrating the diffusion equation and applying the divergence theorem gives conservation of total probability or material in a closed container.^[ref]

For an original example, choose units in which the interval is $0\le x\le 1$. The densitysatisfies $p_t=D{p}_{xx}$, has zero endpoint derivatives and integrates to one. With $a=0.6$, $D=0.5$ and $t=0.5$, its cosine amplitude is about $0.050883$. A nonuniform density therefore relaxes while remaining inside the interval; reflection does not hold the boundary concentration fixed.

For anisotropic diffusion, impose zero normal diffusive flux using the full diffusion tensor. At a wall with $\mathbf{n}=(1,0)$, letin consistent units. Then ${\mathbf{n}}^T\mathbf{D}\nabla C=0$, although ${\partial }_{n}C=1$. Setting the normal derivative to zero would impose a different condition. Absorption and finite permeability require their own boundary models; a reflecting wall allows continued motion within the available space.