# Unique and Flexible Isotropic Diffusion Encoding

An important requirement for diffusion tensor imaging (DT-MRI) or advanced approaches (e.g. Q-ball imaging) is a set of many uniformly distributed gradient directions (scheme) for diffusion encoding. One can imagine points on a sphere to represent the direction vectors. However, distributing points uniformly on a sphere is mathematically not trivial (1). The most established schemes fulfilling this criterion are the Jones schemes which are realised by means of a computationally very demanding method (2). Such schemes are unfortunately not accessible instantly for arbitrary numbers, nor are they unique.

Our approach is to mimic the uniformity of the Jones schemes for arbitrary numbers by a crude but effective geometrical simplification. Such an instantly assignable DIrection SCheme Obtained By ALigning points on Lattitudes (DISCOBALL) is shown to perform as well as the corresponding Jones scheme (3). In contrast to the latter, the DISCOBALLs are well defined and provide the possibility to use available MRI acquisition time most efficiently by adapting the number of applied gradient directions.

Fig. 1 Direction schemes and corresponding precision of the fractional anisotropy as a function of diffusion tensor orientation.

Two compared 30 direction schemes are displayed on the left-hand side in Fig. 1 as an example. Black and white dots represent the tip and the base of the direction vector arrows. In contrast to the DISCOBALL, the Jones scheme is the result of a very demanding, randomly initialised optimisation process. The surface plots on the right-hand side (standard deviation of the fractional anisotropy, SFA, vs. diffusion tensor orientation) display results of a Monte Carlo simulation indicating that both schemes perform equally precise for DTI. This is generally not the case for less uniform schemes or fewer directions.

Fig. 2 Weighted standard deviation of the SFA, SMD and CU response surfaces vs. the number of unique DW directions. Simulation parameters: prolate diffusion tensors (FA=0.9), MD=0.7*10-3 mm2/s, 484 different orientations defined by a regular polar grid, SNR(non-weighted)=15. "EffDual" refers to the efficient dual gradient scheme (SNR(non-weighted)=19.9), "Skare" refers to a Jones30 scheme from the literature (5).

Further precision "response surfaces" can be determined with the Monte Carlo simulation, e.g. for the mean diffusivity (SMD) and the principal eigenvector (CU = cone of uncertainty) (4). The weighted standard deviation of such precision response surfaces, plotted in Fig. 2, is a measure for the undesirable precision variance over all possible tensor orientations. In this context, a smaller variance for a certain scheme means a better DT-MRI applicability, e.g. for fibre tracking without a priori knowledge of the actual white matter fibre bundle orientation. This applies essentially for every in vivo DT-MRI experiment. The graphs in Fig. 2 show that Jones schemes and DISCOBALLs are equivalent for N>=10 directions, the latter, however, need not to be precomputed.

## References

﻿Stirnberg R, Stöcker T, Shah NJ. A New and Versatile Gradient Encoding Scheme for DTI: a Direct Comparison with the Jones Scheme. In: Proc. Intl. Soc. Mag. Reson. Med. 17.; 2009.

Jones DK. The Effect of Gradient Sampling Schemes on Measures Derived From Diffusion Tensor MRI: A Monte Carlo Study. Magn Reson Med. 2004;51:807-815.

Skare S, Hedehus M, Moseley ME, Li T-Q. Condition Number as a Measure of Noise Performance of Diffusion Tensor Data Acquisition Schemes with MRI. J Magn Reson. 2000;147:340-352.

Jones DK, Horsfield MA, Simmons A. Optimal Strategies for Measuring Diffusion in Anisotropic Systems by Magnetic Resonance Imaging. Magn Reson Med. 1999;42:515-525.

Saff EB, Kuijlaars AB. Distributing Many Points on a Sphere. The Mathematical Intelligencer. 1997;19(1):5-11.