Isotropic and anisotropic diffusion imaging

Early investigations of ADC changes during ischaemia used isotropic diffusion weighting i.e. single axis diffusion sensitisation (25,39,216,221,222,245). If diffusion o f water in the CNS is isotropic delineation of the ischaemic lesion is independent of the applied sensitising gradient and unidirectional PGSE techniques would be adequate. Diffusion is a three dimensional process and molecular mobilities may not be the same in all directions. Anisotropic diffusion was first reported in skeletal muscle (49). Brain diffusional anisotropy was initially identified in white matter (169,172,174). Grey matter anisotropy has been observed in normal rat brain (74) but has received less attention although methodologies to eliminate effects o f grey-white matter contrast to avoid ambiguous interpretations of ischaemic areas have been proposed (107). Anisotropic diffusion in tissues is caused by the restriction of free motion in certain directions by biological membranes. It is thought to be a macroscopic measure of neural fibre tract orientation (14). Anisotropic diffusion imaging is a technique gaining recognition in studying tissue architecture (12). Various pathological states have been studied, mainly in white matter, such as demyelination (219), and stroke (16,248), using anisotropic diffusion imaging techniques.

While the apparent (scalar) diffusivity depends on the direction of the applied magnetic field gradient (47,49,66,138,172) this observation is only indicative of diffusion

anisotropy"^. Furthermore, depending on the relative orientation of the diffusion gradients and the subject, individually measured ADCs can vary by as much as a factor of two to three (170). This orientational dependence prevents meaningful comparisons between and within experiments in addition to impairing the contrast between ischaemic and normal tissue. The diffusion tensor is a geometric mathematical entity that deals with the orientational dependence of the diffusive flux. A full tensor analysis provides the most robust approach in assessing diffusion because the off- diagonal elements of the tensor matrix are included in the calculations. Its three dimensional symmetric properties result in useful measures describing the geometry of the diffusion ellipsoid. Diffusion ellipsoid maps provide a graphical means to display the diffusion tensor field from which the fibre-tract direction field (the eigen vectors), corresponding to the major axis of the ellipsoid, can be inferred (14). Isotropic diffusion would be seen as a sphere, the radius of which is given by the mean-squared displacement of a particle from the centre of the sphere in the diffusion time; while anisotropic diffusion would be seen as a linear or planar ellipsoid. The length of the ellipsoid in any direction in space is given by the diffusion distance covered in that direction (134). Ellipsoids may appear spherical in anisotropic tissue if the diffusion time corresponds to a mean diffusion distance that is too short for the majority of spin- labelled protons to encounter diffusional barriers (13). Analysis of ellipsoids is complicated by the difficulty in handling the diffusion times used with the complex MR sequences requiring multiple gradient pulses. In the reference frame that coincides with the principal or main directions of diffusivity (which is not necessarily the laboratory frame of reference), the tensor is reduced only to its diagonal terms, Dxx,

Dyy, Dzz which represent molecular mobility along the x-, y-, z-axes (correlations between molecular displacements in the same direction which occur in anisotropic media). Since the diagonal and off diagonal terms may not cancel in the diffusion tensor unless the gradient coordinates (laboratory frame) coincide with the principle direction of the material, measurement of the orientationally invariant tensor requires six gradient directions to include three off diagonal elements (correlations in anisotropic media between molecular displacements in orthogonal or perpendicular directions). The relationship between the echo magnitude in each voxel and all the imaging and diffusion gradient sequences is given by:

ln(A(b)/A(0)) = - (bxxDxx + 2bxy Dxy + 2bxzDxz + byyDyy + 2byzDyz +bzzDzz)

(3.34)

where A(b) and A(0) are respectively the echo intensities with and without diffusion weighting; the b-value in single axis diffusion imaging is replaced by a symmetric b- matrix calculated from all three applied gradient sequences (12) and Dxx etc are elements of the diffusion tensor D. Since the b-matrix as calculated for each DW- image contains interactions among diffusion gradient and imaging gradient pulses that may be applied in the same direction or along orthogonal directions the cross-terms are potentially significant (and more complicated than those occurring with single axis diffusion weighting) and may affect the diffusion tensor. To account for coupling between these various gradient pulses, analytical expressions for the b-matrix have been derived for various commonly used MRI sequences (15,154). For an isotropic sample the relationship between the echo magnitude and the gradient sequences is reduced to

ln(A(b)/A(0)) = - (bxx + byy + bzz)D

(3.35)

Characterising the isotropic part of the diffusion tensor has become increasingly important clinically where tissue anisotropy produces experimental artifact that is difficult to distinguish from a pure physiological change (12). Diffusion tensor imaging is not easily applied on most clinical imaging systems due to the modest maximum gradient outputs and the prohibitively long time required for acquisition of a tensor in each voxel. Consequently simpler but reliable anisotropic diffusion imaging techniques are being sought. Associated with each diffusion tensor are scalar quantities known as scalar invariants that are intrinsic to the medium and reflect subtle changes in the microstructure of the medium e.g. mean water mobility (14). These scalar invariants are independent of the laboratory frame of reference and the direction or orientation of structures that reduce the observed diffusion anisotropy. Several scalar invariants have been proposed as MR imaging parameters of which the Trace (D) (13) is becoming an accepted imaging parameter in vivo (248). Trace (D) is proportional to the orientationally or isotropically averaged ADC and is given by the equation (12):

Trace(D) = Dxx + Dyy + Dzz =3(D)

(3.36)

Van Gelderen (248) calculates the trace by summing ADCs measured in three orthogonal directions:

Trace(D) = ADCx +ADCy + ADCz =3(ADC)

(3.37)

where each ADC is estimated from the one-dimensional diffusion imaging equation whereby the b-value is calculated for each pulse sequence and the ADC is derived from

linear regression of the equation used by Tanner (237) to estimate the effective diffusion coefficient in microscopically heterogeneous but macroscopically isotropic media:

ln(A(b)/A(0)) = -b ADC

(3.38)

where A(0) is the signal intensity for b = 0 and in calculating the b-value the contributions of all imaging gradients on the measured echo are ignored while Basser (12) accounts for all cross terms. The two methods would only be equivalent when no localisation gradients are applied or when all cross-terms can be shown to vanish. Cross-terms can be dealt with by some of the methods discussed in section 3.4.2.2. Van Gelderen (248) used an MRI technique based on a stimulated-echo sequence. Spoiler gradients were kept constant and in directions different to the diffusion gradient to avoid cross-terms. Imaging gradients were much lower in amplitude and applied after the diffusion sensitising gradients. A series of diffusion gradient strengths was used on each axis to minimise the influence of background gradients. Using these techniques it was possible to demonstrate an absence of anisotropy and orientation effects in the trace images before and after induction of acute stroke by MCAO in cats. A similar method to van Gelderen’s method of estimating the trace (and from it an average ADCav) of the diffusion tensor is applied in this thesis using a stimulated imaging sequence without localisation gradients or phase-encoding gradients to minimise cross-term effects. Unipolar diffusion sensitising gradients are applied in opposite polarities in sequential scans and the method of Neeman is applied in calculating an ADC free from cross-terms (180). A more robust and time efficient method of acquiring the trace using a single scan method rather than in three separate

scans has been described (170). Various combinations o f bipolar gradient pairs were tested in measuring the trace o f the diffusion tensor in chicken gizzard. Three simultaneous bipolar gradients applied in three directions over four time units i.e. four sets o f bipolar gradients, in the TE period were found to be the most satisfactory method and the most feasible for clinical imaging.

The techniques described so far aim to eliminate contrast due to anisotropy. Before the advent o f diffusion tensor imaging and where it has been desirable to measure the diffusion anisotropy in tissue several scalar indices have been applied. Moseley (172) characterised diffusion anisotropy in a voxel by the ratio o f differences and sums o f DW-images:

D W I z - D W l y

D W I z + D W l y

(3 39)

Douek (66) used the ratio o f two ADCs measured with diffusion sensitising gradients applied in two perpendicular directions:

A D C x

A D C y

The maximum value o f the ratio was assumed to indicate a ratio o f ADCs perpendicular and parallel to the local fibre direction. Beaulieu (20) used a ratio o f two ADCs as above for ADCs measured parallel and perpendicular to the fibre axis o f

isolated nerves to evaluate the effects o f local magnetic-susceptibility-induced gradients on anisotropic diffusion. The advantage o f using an isolated nerve preparation is that alignment at specific angles to a selected gradient axis or the main field is possible.

Van Gelderen (248) proposed a scalar anisotropy index that is proportional to the standard deviation o f three diffusion coefficients measured in three orthogonal directions:

V (Dxx - Dav)^ -f (D y y -D a v )^ -f (D z z-D a v )^

A = y/ \ / 6^--- (3.41)

Dav

^

Cross term effects on ADC measurements have already been discussed and are likely to affect any index based on unidirectional ADC measurement by the original method o f Stejskal and Tanner (231). These indices also depend on tissue orientation with respect to the applied diffusion gradients i.e. they are not rotational ly invariant. Anisotropy indices based on quantities derived from the full diffusion tensor measurement to include the off-diagonal elements i.e. from a b-matrix element, have been proposed. These indices are dimensionless ratios o f the eigenvalues or principal diffusivities and have been shown to be independent o f the sample orientation with respect to the laboratory frame o f reference (12,14,114).