Lars Mueller^{1}, Tristan Anselm Kuder^{1}, and Frederik Bernd Laun^{1,2}

The possibility to simultaneous compensate for flow, concomitant fields and eddy currents in diffusion weighted MRI were examined by means of numerical simulations. For this purpose, sequences with three to five gradient pulses and one or two refocusing pulses were examined. It is shown that it is possible to effectively minimize all three effects with different sequences. For short to intermediate echo times, it is beneficial to use only one refocusing pulse, while for long echo times two refocusing pulses can yield higher b-values. There is a trade-off between compensation of more effects and the achievable b-value.

Input parameters for the optimization were: gradient amplitude 40 mT/m, RF pulse duration 5 ms, duration of echo planar readout train before spin echo 10 ms. As shorthand notation for the diffusion encoding, a + or – sign is used to denote a gradient with positive/negative amplitude and the symbol | is used for a 180° RF pulse. The considered sequences consisted of three to five rectangular gradient pulses of identical amplitude and one or two refocusing RF pulses. Adjacent gradients needed to switch signs or be separated by a RF pulse (e.g. +–|–+ was considered but ++|– – was not). A sign change of all gradients yields essentially the same sequence, and so they were excluded from consideration. For the optimization, the gradient durations ($$$d_i$$$) and the times between the time between gradients ($$$t_i$$$) were varied. $$$d_i=0$$$ was allowed. To limit the number of gradient profiles further, only the sequences, that can possibly compensate for concomitant fields were taken into account, as this was fairly easy to verify under the assumption of gradient reversal. The gradient reversal was enforced by fixing the first gradient duration such that:

$$\sum_i{s_i^{\prime}\,d_i}=0,$$

where $$$s_i^\prime$$$ stands for the effective sign of the gradient taking the inversions by 180° RF pulses into account.

In the case of two RF pulses, the echo time TE is determined by the separation of these two, which allows one to fix an additional timing parameter, which was chosen to be the time between the gradients ($$$t_3$$$). The actual distance of the first RF pulse to the excitation is in this case irrelevant for TE, which allows one to fix the total duration of the diffusion encoding to the maximal available time (TE - [time for excitation] - [time for EPI echo train until the actual echo]), without loss of generality, and with that an additional timing parameter ($$$d_2$$$). In the cases with only one RF pulse, these parameters cannot be fixed. Timing constraints were used to ensure that the RF pulse is at the specified time and the diffusion encoding is played out in the allowed time.

For flow-compensation, the first gradient moment needs to be nulled, where $$$t_{i,s}$$$ denotes the starting time of the i-th gradient:

$$m_1=\sum_i{s_i^{\prime}\left(\frac{\delta_i^2}{2}+\delta_i\,t_{i,s}\right)}=0.$$

The condition for concomitant field compensation can be written as:

$$\sum_i{s_i^{\prime\prime}\,\delta_i}=0,$$

where

$$s_i^{\prime\prime}=\begin{cases}+1,\;\text{if 0 or 2 RF pulses have been applied before gradient pulse}\\-1,\;\text{if 1 RF pulse has been applied before gradient pulse}\end{cases}.$$

As model
for eddy currents, it was assumed that each change in gradient amplitude
produces eddy currents proportional to it, which then decay exponentially with
the decay time $$$\tau=70$$$ ms^{4,5}. The condition to null eddy
currents is thus:

$$\sum_i{s_i\left(\exp{\left(-\frac{T-t_{i,s}}{\tau}\right)}-\exp{\left(-\frac{T-\delta_i-t_{i,s}}{\tau}\right)}\right)}=0,$$

which has to be fulfilled, where $$$s_i$$$ gives the actual sign of the i-th gradient.

For the
optimization, Matlabs Global Optimization Toolbox (The Mathworks, Naticks, MA)
was used with the GlobalSearch algorithm, which uses stochastic seed points and
a local minimizer to find the global minimum^{6}. The different compensation
conditions were used as constraints in the algorithm.

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