Optimized Diffusion-Weighting Gradient Waveform Design (ODGD) formulation for motion compensation and concomitant gradient nulling

Abstract

Purpose: To present a novel Optimized Diffusion-weighting Gradient waveform Design (ODGD) method for the design of minimum echo time (TE), bulk motion-compensated, and concomitant gradient (CG)-nulling waveforms for diffusion MRI.

Methods: ODGD motion-compensated waveforms were designed for various moment-nullings M_n (n = 0, 1, 2), for a range of b-values, and spatial resolutions, both without (ODGD-M_n ) and with CG-nulling (ODGD-M_n -CG). Phantom and in-vivo (brain and liver) experiments were conducted with various ODGD waveforms to compare motion robustness, signal-to-noise ratio (SNR), and apparent diffusion coefficient (ADC) maps with state-of-the-art waveforms.

Results: ODGD-M_n and ODGD-M_n -CG waveforms reduced the TE of state-of-the-art waveforms. This TE reduction resulted in significantly higher SNR (P < 0.05) in both phantom and in-vivo experiments. ODGD-M₁ improved the SNR of BIPOLAR (42.8 ± 5.3 vs. 32.9 ± 3.3) in the brain, and ODGD-M₂ the SNR of motion-compensated (MOCO) and Convex Optimized Diffusion Encoding-M₂ (CODE-M₂ ) (12.3 ± 3.6 vs. 9.7 ± 2.9 and 10.2 ± 3.4, respectively) in the liver. Further, ODGD-M₂ also showed excellent motion robustness in the liver. ODGD-M_n -CG waveforms reduced the CG-related dephasing effects of non CG-nulling waveforms in phantom and in-vivo experiments, resulting in accurate ADC maps.

Conclusions: ODGD waveforms enable motion-robust diffusion MRI with reduced TEs, increased SNR, and reduced ADC bias compared to state-of-the-art waveforms in theoretical results, simulations, phantoms and in-vivo experiments.

Keywords: Concomitant Gradient (CG)-nulling; Diffusion-Weighted Imaging (DWI); diffusion-weighting gradient waveforms; motion compensation; optimization.

Figure 1:

Traditional waveforms (a), Convex Optimized Diffusion Encoding (CODE) gradient waveforms (b), and Optimized Diffusion-weighting Gradient waveform Designs (ODGD) without (c) and with (d) concomitant gradients (CGs) nulling for b = 1000 s/mm² and T_EPI = 26.4 ms. CODE and ODGD waveforms without and with CG-nulling reduced the TE of the traditional waveforms (MONO, BIPOLAR, and MOCO) in all cases. These traditional waveforms have equivalent or symmetric shapes before and after the refocusing pulse, and therefore lead to dead times between the radiofrequency pulses (due to the need for additional time for the EPI readout between the end of the diffusion waveform and the echo time). In contrast, the CODE framework and the ODGD formulation seek to use the available time optimally in order to minimize the TE. Note that the ODGD-based diffusion gradients have the same waveform along the three main gradient axes, although generally with different scaling depending on the diffusion direction. Therefore, only one gradient axis is considered active for the optimization of these waveforms.

Figure 2:

Minimum TE achieved for the range of b-values 100 - 2000 s/mm² in steps of 100 s/mm² and T_EPI (EPI readout time to the center of k-space) of 26.4 ms for ODGD-M_n, CODE-M_n, and the traditional waveforms (MONO, BIPOLAR, and MOCO) for different moment constraints, M₀, M₁, and M₂, respectively (a). TE reduction (ΔTE) achieved using ODGD-M_n compared to CODE-M_n for the same range of b-values and T_EPI in the range of 16.4 - 46.4 ms in steps of 5 ms (b). There is no TE reduction (ΔTE = 0) for zeroth-order moment-nulling (M₀), but there is TE reduction for first- and second-order moment-nulling (M₁ and M₂, respectively). The TE reduction is greater for higher b-values, longer T_EPI, and higher-order moment-nulling. Note that only one gradient axis is considered active for the optimization of these waveforms.

Figure 3:

Minimum TE achieved for a range of b-values 100 - 2000 s/mm² in steps of 100 s/mm² and T_EPI (EPI readout time to the center of k-space) of 26.4 ms for ODGD-M_n-CG and the traditional waveforms (MONO, BIPOLAR, and MOCO) for different moment constraints, M₀, M₁, and M₂, respectively (a). TE reduction (ΔTE) achieved using ODGD-M_n-CG compared to the traditional waveforms for the same range of b-values and T_EPI in the range of 16.4 - 46.4 ms in steps of 5 ms (b). ΔTE is larger for higher b-values and T_EPI, and larger for the M₁ constraint than for M₀, or M₂. Note that only one gradient axis is considered active for the optimization of these waveforms.

Figure 4:

Diffusion-weighted images of the acetone phantom experiments acquired with CODE-M₂ and ODGD-M₂ at b-value 1000 s/mm² in a slice at the isocenter of the magnet (a). SNR of the same images (b). Top row of the set of 10 vials have short T₂ ≈ 39.5 ms (vial numbers 1, 3, 5, 7, 9), and bottom row have long T₂ ≈ 83.5 ms (vial numbers 2, 4, 6, 8, 10). Distribution of the SNR values of each vial grouped by T₂ ≈ 39.5 ms (c), and T₂ ≈ 83.5 ms (d). From left to right, distributions are ordered as vials in (a). Red boxes show the distribution of CODE-M₂ and black boxes the distribution of ODGD-M₂. There is a statistically significant (P < 0.05) SNR increase for every vial of ODGD-M₂ compared to CODE-M₂ except for vials with ADC = 1.9×10⁻³ mm²/s.

Figure 5:

Axial trace diffusion-weighted images (DWI) acquired at 4.5 cm from isocenter of a representative brain are shown acquired with BIPOLAR, ODGD-M₁, and ODGD-M₁-CG with a b-value of 100 s/mm² (a). Corresponding average trace ADC map of the BIPOLAR acquisition (b). ODGD-M₁ and ODGD-M₁-CG average trace ADC maps pixelwise subtracted with the BIPOLAR average trace ADC map (c). Mean ± 95% CI SNR values of the trace DW images (d), and average trace ADC values (e) across ROIs set on white matter of the 10 volunteers. ODGD-M₁ leads to higher statistically significant SNR than BIPOLAR and ODGD-M₁-CG, P < 1 × 10⁻⁶ and P < 0.05, respectively. ODGD-M₁-CG also leads to statistically significant higher SNR than BIPOLAR with P < 0.005. ODGD-M₁ average trace ADC map is positively biased (P < 0.005) with respect to BIPOLAR. There is no statistically significant difference between the average trace ADC maps of ODGD-M₁-CG and BIPOLAR. Note that for this set of waveforms ODGD-M₁ and CODE-M₁ are the same.

Figure 6:

Axial diffusion-weighted images of a representative liver are shown acquired with MONO, CODE-M₂, and ODGD-M₂ with a b-value of 500 s/mm² (a). Corresponding ADC maps (b). MONO ADC maps have heterogeneous positive bias throughout the liver due to intravoxel signal dephasing at b-value of 500 s/mm² produced by bulk motion. ODGD-M₂ and CODE-M₂ waveforms achieve more spatially homogeneous DW images and ADC maps than MONO, showing better motion robustness. ADC values on a ROI on segment II of the liver (blue ROI) of a representative volunteer are 2.46±0.38 × 10⁻³ mm²/s for MONO, 1.85±0.2 × 10⁻³ mm²/s for CODE-M₂, and 1.53±0.21 × 10⁻³ mm²/s for ODGD-M₂.

Figure 7:

Axial diffusion-weighted images of the liver are shown acquired with MOCO, CODE- M₂, ODGD-M₂, and ODGD-M₂-CG with a b-value of 100 s/mm² (a). Signal-to-noise ratio (SNR) maps of these acquisitions (smoothed with an average filter for better representation) (b). Liver SNR measurements in the 10 volunteers, for each of the diffusion waveforms (c). ODGD-M₂ and ODGD-M2-CG lead to statistically significant (P < 0.05) higher SNR than MOCO and CODE-M₂. There is no statistically significant difference between ODGD-M₂ and ODGD-M₂-CG.

Figure 8:

Cropped k-space of the water phantom experiments of slices at isocenter (0 cm) and 4.5 cm from isocenter with MONO, ODGD-M₀, and ODGD-M₀-CG, b-value of 1000 s/mm², and diffusion-weighting direction D_xyz, (a) and (b), respectively. FWHMy indicates the full-width-half-maximum along the phase-encoding direction (y-axes). Shifting indicates the displacement of the k-space from its center. ODGD-M₀ shows broader FWHMy than MONO and ODGD-M₀-CG, and k-space shifting towards the upper left corner at 4.5 cm from isocenter. MONO and ODGD-M₀-CG show little blurring and no k-space shifting. Note that for this set of waveforms ODGD-M₀ and CODE-M₀ are the same.

Figure 9:

Measured ADC maps along each gradient direction combination (D_n) of the water phantom experiment with the waveforms BIPOLAR (a), ODGD-M₁ (b), and ODGD-M₁-CG (d). Acquisitions with BIPOLAR and ODGD-M₁-CG waveforms considerably reduced the bias of the ADC maps introduced by the concomitant gradients. Note that for this set of waveforms ODGD-M₁ and CODE-M₁ are the same.