Diffusion MRI gradient sampling

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.