Óscar Peña-Nogales^{1}, Yuxin Zhang^{2,3}, Rodrigo de Luis-Garcia^{1}, Santiago Aja-Fernández^{1}, James H. Holmes^{3}, and Diego Hernando^{4}

Diffusion-Weighted MRI (DW-MRI) often suffers from signal attenuation due to long TE, motion-related artefacts, dephasing due to concomitant gradients (CGs), and image distortions due to eddy currents (ECs). Further, the application of rapidly switching gradients may cause peripheral nerve stimulation (PNS). These challenges hinder the progress, application and interpretability of DW-MRI. Therefore, based on the Optimized Diffusion-weighting Gradient waveforms Design (ODGD) formulation, in this work we design optimal (minimum TE) nth-order moment-nulling diffusion-weighting gradient waveforms with or without CG-nulling able to reduce EC induced distortions and PNS-effects. We assessed the feasibility of these waveforms in simulations and phantom experiments.

Diffusion-Weighted MRI (DW-MRI) has a unique ability to probe tissue microstructure through the application of strong switching diffusion-weighting gradients. However, the application of these gradients results in significant challenges including 1) signal attenuation^{[1]}, 2) motion-related artifacts^{[2,3]}, 3) signal dephasing due to concomitant gradients (CGs)^{[4,5]}, 4) image distortions (shearing and scaling) due to eddy currents (ECs)^{[6,7,8]}, and 5) might cause peripheral nerve stimulation (PNS)^{[9,10]}.

Particularly, ECs and PNS appear when the gradient amplifiers are switched fast to drive large gradient intensities. Firstly, switching gradients result in current inductions that persist after the gradients are switched off^{[6,7]}. Although, ECs are usually tackled through pre-emphasis and post-processing, there are also suboptimal (long TE) diffusion-weighting waveforms (e.g., BIPOLAR^{[11]}) that achieve intravoxel rephasing and EC-nulling. Secondly, they cause an electric depolarization that may lead to PNS^{[9]}. PNS is usually diminished by limiting the maximum slew rate of the waveform resulting in sub-optimal waveforms.

For these reasons, there is an unmet need for optimized diffusion-weighting waveforms able to overcome these challenges. Recently, a nonlinear constrained optimization formulation for the design of high-order moment-nulled and/or CG-nulled waveforms, termed Optimized Diffusion-weighting Gradient waveform Design (ODGD)^{[12]}, has shown great potential in-vivo DW-MRI. Therefore, the purpose of this work is to design optimized waveforms (minimum TE) based on the ODGD formulation that achieve EC- and PNS-nulling, while preserving previous properties.

The ECs impulse response is given by $$$e(t,\tau)=H(t)\sum_\text{n}\alpha_\text{n}\exp{\left(-\frac{t}{\tau_\text{n}}\right)}$$$, where $$$\tau_\text{n}$$$,$$$\alpha_\text{n}$$$, n and H(t) are a characteristic time constant, amplitude, number of induced currents, and the step function, respectively. Thus, we can reduce EC induced distortions (EC-nulling) for a set of time constants ($$$\tau$$$) by adding the linear constraints $$$g(t,\tau)=-\frac{dG(t)}{dt}\circledast~e(t,\tau)\Bigr\rvert_{t=\text{T}_\text{Diff}}=0$$$ to the formulation^{[6,8]}.

A conservative nerve impulse response is given by^{[10,13]} $$$n(t)=\frac{αc}{r(c+t)^2}$$$, where the gradient-coil specific constants $$$\alpha$$$, r, and c are the effective coil length, rheobase, and chronaxie time, respectively. Thus, for waveforms designs with the maximum slew rate supported by the MRI unit we can reduce the nerve stimulation (PNS-nulling) by adding the linear constraints $$$R(t)=\frac{dG(t)}{dt}\circledast~n(t)<Plimit=1$$$, where the PNS limit ($$$\text{P}_\text{Limit}$$$) is established by the International Electrotechnical Commission (IEC 60601-2-33). For the sake of clarity, Table 1 summarizes all constraints of the extended ODGD formulation.

In this work, MONO^{[12]}, BIPOLAR, MOCO^{[12]} and nth-order moment-nulled ODGD waveforms without and with EC- and PNS-nulling were designed for b-values=100-2000s/mm², and T_{EPI} (EPI readout time to the center of the k-space) of 16.4-46.4ms.

A 7.5L cylindrical water phantom doped with NaCl and NiCl_{2}^{[14]} was scanned at a 3T unit (GE Healthcare, Waukesha, WI) using a spin echo DW-EPI sequence to assess EC induced image distortions for several diffusion-weighting waveforms without and with EC-nulling, and to assess the implementation feasibility and SNR of PNS-nulling waveforms. Acquisition parameters were TR=4s, slice thickness=5mm, FOV=24x24cm, in-plane resolution=1.88x1.88mm, full k-space, six diffusion-weighting directions (±A/P,±R/L,±S/I), and b-values(averages)=[0(1),100(2),1000(4)]s/mm^{2}. SNR maps were computed following Ref^{[15]}.

Figure 1 shows examples of various waveforms designed for the experiments. Figure 2 shows the TE dependence on the b-value of MONO, BIPOLAR, MOCO and various ODGD waveforms, and the TE reduction achieved using ODGD-M_{1}-CG compared to ODGD-M_{1}-CG-EC, and ODGD-M_{1}-EC without and with CG-nulling compared to BIPOLAR. ODGD-M_{1}-CG results in TE reductions between 3.8-14.7% compared to ODGD-M_{1}-CG-EC. Relative to BIPOLAR, ODGD-M_{1}-EC and ODGD-M_{1}-CG-EC result in TE reductions between 1.9-27.9%, and -7-11.22%, respectively.

Figure 3 and 4 show the difference between DW images of opposite diffusion-weighting polarity, and their distribution, respectively. MONO is visually severely affected by EC induced distortions showing strong mismatch between opposite DW images. BIPOLAR, and ODGD-M_{1}-EC are slightly more homogeneous than ODGD-M_{1}. This is clearly visible at the phantom’s border where the shearing and scaling effects are apparent when showing the difference between opposite DW images. Further, BIPOLAR, and ODGD-M_{1}-EC have tight confidence intervals with no bias on the phantom’s border.

Slew rate of ODGD-M_{n}-PNS are slightly different to ODGD-M_{n} (Figure 1.e)) resulting in TE reductions between 0-0.94%, 0-0.74%, and 0-0.75% for M_{0}, M_{1}, and M_{2} respectively. Further, there is no SNR difference between ODGD-M_{0} and ODGD-M_{0}-PNS, neither between ODGD-M_{1} and ODGD-M_{1}-PNS (results not shown).

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Table_1. Optimized Diffusion-weighted Gradient waveform Design (ODGD) formulation. G_{Max} and SR_{Max} are the maximum gradient intensity and slew rate respectively, T_{RF-90} and T_{RF-180} the excitation and refocusing radiofrequency pulses, T_{EPI} the time required by the EPI readout to reach the center of the k-space, T_{Diff} the end of the diffusion-weighting gradients, A1 and A2 the time periods before and after T_{RF-180}, and G(t) is the optimal diffusion-weighting gradient waveform. In this work, G_{Max}=70mT/m, and T_{RF-90}=T_{RF-180}=5.5ms. Waveforms without PNS-nulling were computed with SR_{Max}=100T/m/s (limit usually used to avoid PNS-effects) and with PNS-nulling with SR_{Max}=150T/m/s (hardware limit), respectively.

Figure_1.Traditional waveforms (a), first-order moment-nulled (M_{1}) Optimized Diffusion-weighted Gradient waveform Designs (ODGD-M_{1}) without and with CG-nulling^{[12]} (b), ODGD-M_{1} waveforms with EC-nulling for a time constant of τ=10ms without and with CG-nulling (c), ODGD-M_{0} waveforms and their slew rate without and with PNS-nulling (d) and (e), respectively. (a-c) waveform designs were computed with a time resolution of 0.5ms and (d-e) with 0.04ms. All waveforms were designed for b-value=1000s/mm^{2} and T_{EPI} of 24.6ms. ODGD-M_{0} and ODGD-M_{0}-PNS were computed with SR_{Max} of 100T/m/s (limit used to avoid PNS-effects) and 150T/m/s (hardware limit), respectively. Note the different slew rate of ODGD-M_{0}-PNS compared to ODGD-M_{0}.

Figure_2.Minimum TE achieved for a range of b-values 100-200s/mm^{2} and T_{EPI}=26.4ms for ODGD-M_{n}, ODGD-M_{n}-CG, ODGD-M_{n}-EC, ODGD-M_{n}-CG-EC, and the traditional waveforms (MONO, BIPOLAR, and MOCO) for M_{0}, M_{1}, and M_{2} moment constraint, respectively (a). TE reductions (ΔTE) achieved using ODGD-M_{1}-CG compared to ODGD-M_{1}-CG-EC, and ODGD-M_{1}-EC and ODGD-M_{1}-CG-EC compared to BIPOLAR for the same range of b-values and T_{EPI} in the range of 16.4-46.4ms (b). EC-nulling waveforms were designed to null time constants of τ=10ms. ODGD waveforms with EC-nulling achieve higher minimum TE than the corresponding ODGD waveform without EC-nulling. In these cases, ΔTE is larger for higher b-values and lower T_{EPI}.

Figure_3. Difference between diffusion-weighted images acquired with opposite diffusion-weighting polarity (±A/P), b-value=1000s/mm^{2}, and T_{EPI}=11ms with waveforms MONO, ODGD-M_{1}, BIPOLAR, and ODGD-M_{1}-EC of the phantom experiment (a). Masks for ROI analysis (b). MONO is severely affected by eddy currents induced distortions showing strong mismatch between opposite DW images. At the phantom’s border, ODGD-M_{1} is slightly affected by EC induced distortions. ODGD-M_{1}, BIPOLAR, and ODGD-M_{1}-EC show more homogeneous difference between opposite DW images than MONO. Compared to ODGD-M_{1}, BIPOLAR and ODGD-M_{1}-EC show similar variability between opposite DW images despite their longer TE.

Figure_4.Distribution of the difference between diffusion-weighted images acquired with three opposite diffusion-weighting directions (±A/P,±R/L,±S/I) for waveforms of Figure 3. Red and black boxes show the distribution of the phantom and border masks, respectively, shown in Figure 3. Both masks of ODGD-M_{1}, BIPOLAR, and ODGD-M_{1}-EC show less variation between opposite DW images (tighter confidence interval and lower height) proving better EC performance than MONO. Compared to ODGD-M_{1}, BIPOLAR and ODGD-M_{1}-EC show similar variability between opposite DW images (width of the phantom mask) despite their longer TE, and no bias on the border mask (height of the boxplot) showing better EC performance.