An Optimal Strategy for Precise
Determination of the Diffusion Tensor
D.K. Jones, M.A. Horsfield, A. Simmons*
Division of Medical Physics, University of Leicester, Leicester Royal Infirmary, Leicester LEl 5WW, UK.
*Neuroimaging Research Group, Institute of Psychiatry, De Crespigny Park, Denmark Hill, London, SE5 SAF, UK.
INTRODUCTION
Optimisation of data collection schemes for precise estimation
of the diffusion tensor has not been addressed. Here, we suggest a
strategy for optimising the echo time, and the amplitude and
direction of the set of diffusion weighting gradient vectors for
precise measurement of diffusion in anisotropic systems.
THEORY
Estimation of the diffusion tensor matrix requires a minimum
of 7 signal attenuation measurements with gradients applied along
at least six non-collinear directions’.
In developing our optimisation procedure, we first considered
the gradient vector directions. Since a priori information is
generally unavailable about tissue anisotropy or the orientation of
the principal axes of the tensor, the bias inherent in making
measurements with a fixed set of directions should be minimised
by spreading out measurements in 3-dimensional gradient vector
space. We have therefore developed an algorithm to distribute any
number of gradient vectors uniformly in 3-dimensional space.
Next, the gradient vector amplitudes were considered. It has
been shown’ that measurement of diffusion in isotropic systems is
optimal when just just two weightings (a low and a high b-factor)
are used, with multiple measurements being made with each’bfactor. To illustrate optimisation of gradient amplitudes, consider
the simplest case of 7 measurements - one with a zero b-factor and
six with a high b-factor. The six vector orientations are derived
using our spatial distribution algorithm, and substituted into the
expression for signal attenuation’ to give a set of six simultaneous
equations. Solving these yields the six unique tensor elements
expressed in terms of the gradient b-factors and the unweighted
and weighted signal intensities (S, and &, respectively). The error
in the estimate of each element is subsequently determined from
the propagation of errors3. For example, the variance in the
estimate of D, when a total of Nr measurements are made, is:
shielded gradients (max. amplitude: 22 mT mm’). Images were
acquired with a pulsed gradient spin-echo EPI sequence,
peripherally gated for the volunteer studies. The echo time, bmatrices and the number of measurements made with each
diffusion-weighting were varied according to the medium imaged
and the scheme being evaluated. After correction for eddy-current
induced image distortion, the tensor was estimated for each pixel,
and images of the tensor trace’ and fractional anisotropy created.
Prior to optimisation, we employed the tensor imaging scheme
originally suggested by Basser’ involving measurement along 7
non-collinear directions (x, y, Z, q, XZ, yz and xyz) with Nb bfactors (equally spaced in b) per direction, and a maximum bfactor = 3.3/Tr(D). Experiments were designed to compare this
scheme with the optimal scheme for the same scan time.
RESULTS
Estimates of trace and fractional anisotropy (FA) in the centre
of the water phantom obtained with the conventional (‘Conv’) and
optimal (‘Opti’) schemes are given below. b,,, is the maximum bfactor (s mtxi’), Tr(D) is in mm2 s-l and 6 and TE are in ms.
The standard deviation in Tr(D) is = 40% lower, and the
artefactual anisotropy in the phantom is reduced by = 60% when
using the optimised scheme for the same scan time. A series of
phantom experiments also confirmed that Rapt agreed with the
theoretical value when Tz effects were included.
where NL is the number of measurements made with the lower
(zero) b-factor. The sum of the variances of the six unique tensor
elements is then minimised with respect to the diffusion-weighting
and NL, to derive an optimal weighting (bopt) of 3.27/Tr(D) (for the
chosen set of gradient vectors), where Tr(D) is the tensor trace,
and an optimal ratio (Rapt) of the number of measurements made
with the larger weighting to that with no weighting of 11.3: 1.
With finite gradient strengths, large b-factors require a long
echo time and so the effects of Tz relaxation cannot be ignored.
The duration (6) and separation (A) of the diffusion-encoding
gradients can be expressed in terms of TE such that 6=TE/2-tA and
A=TEl2+t,,
where tA and tB are pulse sequence dependent
constants. 6 and A are then substituted into the Stejskal-Tanner
expression for the b-factor: b = $G2?? (A-S/3) and the resulting
cubic solved for TE in terms of the b-factor. The optimal
parameters
obtained
following
this
modification
were
considerablv different from those when T? effects were ignored:
R,
b cmt
Medium
b,,, L+(D)
T2
Water Phantom
62 ms
2.56
453 s mmm2 8.;;
White Matter
80 ms
2.45
1048 s mmm2 8.63
d
METHODS
Data were acquired from a doped water phantom and a healthy
volunteer using a 1.5 T Signa Echospeed -system with actively
conventional (left) and opthnised (right) schemes, both with N~=28.
CONCLUSION
An optimisation strategy for precise measurement of anisotropic
diffusion has been developed. This results in shorter scan times,
or improved quality of diffusion tensor images.
REFERENCES
1. Basser P.J., Matiello J. et al., J. Muglz. Reson. B., 103,247 1994.
2. Bito Y., Hirata S. et al., “Proceedings of ISMRM, 1995,” p. 913.
3. Bevington P.R., Robinson D.K., Data Reduction and Error Analysis
for the Physical Sciences 2”d Edition, New York, McGraw-Hill, 1992.
4. Basser PI., Pierpaoli C., J. Magn. Reson. B. 111,209, 1996.
This work was supported by the Wellcome Trust (grant
043235/Z/94) to whom the authors express their gratitude.