Mapping the Impact of Non-Linear Gradient Fields on Diffusion MRI Tensor Estimation

Abstract

Non-linear gradients impact diffusion weighted (DW) MRI by corrupting the experimental setup and lead to problems during image encoding including the effects in-plane distortion, in-plane shifts, intensity modulations and phase errors. Recent studies have been shown this may present significant complication in the interpretation of results and conclusion while studying tractography and tissue microstructure in data. To interpret the degree in consequences of gradient non-linearities between the desired and achieved gradients, we introduced empirically derived gradient nonlinear fields at different orientations and different tensor properties. The impact is assessed through diffusion tensor properties including mean diffusivity (MD), fractional anisotropy (FA) and principal eigen vector (PEV). The study shows lower FA are more susceptible to LR fields and LR fields with determinant <1 or >1 corrupt tensor more. The corruption can result in significantly different FA based on true-FA and LR field. Apparent MD decreases for negative determinant, on the other hand positive determinant shows the opposite effect. LR field have a larger impact on PEV when FA value is small. The results are dependent on the underlying orientation, non-linear field corruption can cause both increase and decrease of estimated FA, MD and PEV value. This work provides insight into characterizing the non-linear gradient error and aid in selecting correction techniques to address the inaccuracies in b-values.

1. INTRODUCTION

Diffusion weighted magnetic resonance imaging (DWMRI) allows measurement of tissue diffusivity and is a well-established non-invasive technique for identification of ischemic stroke, differentiation of acute from chronic stroke, multiple sclerosis, schizophrenia or Alzheimer’s disease [1–4]. However, the underlying system inaccuracies in DWMRI cause bias or distortion during acquisition in diffusivity measurement over clinically relevant field of view [5]. One such distortion is due to the presence of the magnetic field inhomogeneities. When not at the center, the magnetic gradient coils induce non-linear gradients [6, 7], producing spatial and geometric distortion in the applied gradient strength. This deviation from the desired gradients can cause errors in the diffusion attenuation factor and b-value miscalculations. Previous research shows various gradient nonuniformities corrections techniques including relating the mismatch between the true and intended gradients with L(r) by Bammer et al. [8], rotation of gradient non-linearity tensor L(r) into the diffusion gradient frame [9] and velocity reconstruction using matrix formalization [8, 10]. Despite extensive methods proposed for correcting the spatial-temporal gradients and the b-fields variations, the underlaying implications of the nonlinearities has not been investigated in depth. To bridge this gap, we present the analysis of effects of gradient coil tensor L(r) on tensor simulations. We map the dependance of diffusion tensor properties, 1) mean diffusivity (MD) 2) fractional anisotropy (FA) and 3) principal eigen vector (PEV) on L(r) field.

2. METHODS

2.1. Tensor simulation

To investigate the effects of gradient nonlinearities, we simulated a range of ‘ground truth’ axially symmetric diffusion tensors with varying FA and orientations, and corrupted the signal using empirically determined gradient nonlinearities realized on a clinical scanner.

The mean diffusivity of white and grey matter is approximately between 0.6 x 10 ⁻³ mm²s⁻¹ and 0.8 x 10 ⁻³ mm²s⁻¹ [11]. We simulate ground truth diffusion tensors, D, with a constant MD = 0.8 x 10 ⁻³ mm²s⁻¹ The mean diffusivity (MD) is calculated as,

$$MD=\frac{Trace(D)}{3}$$ (1)

$$D_n=\left[\begin{array}{ccc}{\lambda }_{\perp }& 0& 0\\ 0& {\lambda }_{\perp }& 0\\ 0& 0& {\lambda }_{\parallel }\end{array}\right]$$ (2)

where n refers to the number of the ground truth tensors, λ _⊥ is radial diffusivity and λ _∥ is axial diffusivity. For axially symmetric D_n as shown in equation 2, radial diffusivity is equal. Given the MD and FA, the radial diffusivity equation can be written as

$${\lambda }_{\perp }=MD\frac{1-FA}{\sqrt{3-2F{A}^2}}$$ (3)

Then, the axial diffusivity can be calculated based on radial diffusivity by [12]

$${\lambda }_{\parallel }=3MD-2{\lambda }_{\perp }$$ (4)

One hundred FA values are varied from 0 to 1 (evenly spaced) in equation (2). Tensors are simulated at different orientations that are uniformly spaced with one hundred θ ∈ [0, π] and hundred φ ∈ [0, 2π] over a spere using dipy that implemented the electrostatic repulsion from Jones1999 [13]. The empirical field maps used in this study are acquired from a 3T MRI scanner. It is a 94 cm bore Philips Intera Achieva MR whole-body system and has a gradient strength of 80 mT/m and, 200 slew-rate. The data was acquired during a previous work [14]. The gradient coil tensor L(r) was generated from the empirical field mapping procedure as described by Rogers et al [15]. The determinant of the expected L(r) fields was rank-ordered and selected by choosing the first through 100^th percentiles. The ground truth tensors are simulated with varying 19406 angles on sphere for orientation, 100 L(r) fields, 100 FA values as shown in Figure 2a, thus n will be 1 x 10⁸.

Figure 2.

Experimental design implementation for tensor simulation, (a) Ground truth tensors are simulated for a range of empirically derived nonlinear fields (ranked by L(r) determinants), a range of orientations, and a range of anisotropy values. (b) These ground truth tensors are corrupted by the nonlinear fields, and the corrupted tensor and derived indices are recomputed in order to determine the impact of nonlinear fields on DTI estimation.

2.2. Tensor corruption

Using conventions described by Rogers et al. [15, 16], the ground truth tensors D_n simulated are corrupted by applying the L(r) tensor that relates the achieved magnetic gradient to the intended one to the signal. The signal S for diffusion gradient value b given by the Stejskal-Tanner equation [17]:

$$S=S_o{e}^{-bD}$$ (5)

where S_o is MR signal at baseline and D is diffusion coefficient. The achieved gradients g′ can be calculated by [14]

$$g\prime =Lg$$ (6)

$$g″=\frac{g\prime }{\left|g\prime \right|}$$ (7)

where g is the intended gradient, g′ is the actual gradient vector and the normalized achieved gradients g″ can be calculated by equation (8). Then, the achieved b-value b′ is the computed by adjusting the intended b-value b by the square of the length change.

$$b\prime =b{\left|g\prime \right|}^2$$ (8)

By rewriting equation 1 with g′ and b′ the signal ${\widehat{S}}_i$ for the i^th diffusion acquisition is:

$$\begin{gathered} {\widehat{S}}_i=S_o{e}^{-bg{\prime }_i^{T}Dg{\prime }_i} \\ =S_o{e}^{-b\prime g{″}_i^{T}Dg{″}_i} \\ =S_o{e}^{-b{g}_i^T{L}^{T}DL{g}_i} \end{gathered}$$ (9)

In consequence, the resulting tensors $\widehat{D_n}$ will contain the effect of the gradient non-linearities field.

$$\widehat{D_n}=\frac{ln\frac{{\widehat{S}}_i}{S_o}}{-b{g}_i^T{L}^{T}DL{g}_i}$$ (10)

This corruption was applied to all the ground truth tensors. We compute the MD, FA and PEV measurements and evaluate it against the ground truth as shown in Figure 2b.

3. RESULTS

3.1. Effects of nonlinear fields on estimated FA, MD, PEV

Figure 4 shows the effects of non-linear field gradients on estimated FA (Figure 4a), MD (Figure 4b), and PEV orientation (Figure 4c), where the total change (or corruption) of each parameter is shown for all simulated FA values, for all gradient nonlinearities colored by the determinant of the L(r) tensor. Several trends are immediately apparent: (1) The L(r) fields with the highest and lowest determinants (dark red and dark blue) result in the largest changes of FA, (2) lower ground truth FA are more affected by nonlinear fields than higher FA, (3) FA is, on average (over all orientations), increased due to nonlinear fields, and (4) the corruption can result in significantly different FA depending on true-FA and field. Corruption to MD also shows intuitive trends, including (1) larger positive detenninant of L(r) resulting in higher diffusivities, (2) changes of diffusivity on the order of 1*10^-4 (about 10% of the simulated diffusivity), and little dependence on FA, although values are averaged over all orientations. Finally, the PEV angular error is highly dependent upon FA, with lower FA values resulting in much larger angular changes after corruption with nonlinear fields.

Figure 4.

Effects of nonlinear fields on estimated FA (a), MD (b), and PEV (c). The change in derived parameters is dependent upon the magnitude of the nonlinear field (color coded based on determinant) and underlying anisotropy. Smaller FA values are more susceptible to nonlinear fields (a), MD corruption is dependent upon the magnitude of the field (b), and angular error depends upon ground truth FA (c).

3.2. Orientation dependency of nonlinear fields on estimated FA, MD, PEV

Figure 5–7 show that nonlinear fields have an orientation-dependency when corrupting measures of FA (Figure 5), MD (Figure 6), and PEV (Figure 7). For these, we show corruption caused by two exemplary LR fields (with a determinant <1 and determinant >1), for 3 different simulated FA values, across all orientations over a sphere (interpolate θ ∈ [−π/2, π/2] degrees). For FA (Figure 5), corruption with nonlinear fields can cause both an increase and decrease in estimated value, with a sinusoidal pattern over a sphere that coincide in an intuitive way with the primary eigenvector (or orientation) of the L-tensor that is corrupting the signal (supporting studies that have shown the errors in FA occur in an intuitive fashion with gradient sampling schemes [18, 19]). Again, lower FA values show greater magnitude of corruption than higher FA, with the same trends over a sphere. For MD (Figure 6), trends are flatter across all orientations, although orientation-dependency is still apparent. Again, a negative determinant yields a decreased apparent MD, whereas a positive L-tensor determinant shows the opposite effect. Here, the primary eigenvector of the L-field seems to result in the greatest change in diffusivity, with similar trends across all FA values. Finally, for angular error (Figure 7), trends are similar to that of FA. Many orientations experience large deviations in PEV (up to and greater than 10 degrees for all FA values), with a general pattern of smaller change in orientation when the true tensor is aligned with the eigenvector of the nonlinearity field.

Figure 5.

Effects of nonlinear fields on FA are dependent upon orientation. For two nonlinear fields (with determinant = 0.86 and 1.12) the change in FA is shown for all ground truth orientations, for FA values of 0.25, 0.5, and 0.75. The primary, secondary, and tertiary eigenvectors of the LR field are designated with dots (note complex eigenvectors may overlap in ‘real’ space). Different nonlinear fields effect the FA in different ways, with both increases and decreases in estimated values, dependent upon orientation and underlying ground truth.

Figure 7.

Effects of nonlinear fields on PEV are dependent upon orientation. For two nonlinear fields (with determinant = 0.86 and 1.12) the change in PEV is shown for all ground truth orientations, FA values of 0.25, 0.5, and 0.75. The primary, secondary, and tertiary eigenvectors of the LR field are designated with dots (note complex eigenvectors may overlap in ‘real’ space). Different nonlinear fields effect the PEV in different ways, with both increases and decreases in estimated values, dependent upon orientation and underlying ground truth.

Figure 6.

Effects of nonlinear fields on MD are dependent upon orientation. For two nonlinear fields (with determinant = 0.86 and 1.12) the change in MD is shown for all ground truth orientations, for FA values of 0.25, 0.5, and 0.75. The primary, secondary, and tertiary eigenvectors of the LR field are designated with dots (note complex eigenvectors may overlap in ‘real’ space). Different nonlinear fields effect the MD in different ways, with both increases and decreases in estimated values, dependent upon orientation and underlying ground truth.

4. DISCUSSION AND CONCLUSIONS

In this work, we have characterized the impact of nonlinear gradient fields on the estimation of diffusion tensors and derived quantitative metrics. We show that nonlinear fields on clinical scanners can results in substantial changes to diffusion tensor estimation, which can lead to incorrect scientific conclusions when using DTI to infer microstructure or connectivity through fiber tractography [20]. Understanding changes due to these fields, where they occur, and in what cases they are most pronounced, is the first step towards correcting the effects of these fields in practice.

We find that nonlinear effects depend on not only the magnitude of the nonlinear field, but also on orientation and anisotropy of the underlying tissue. In particular, voxels with lowest FA showed greatest susceptibility to nonlinear fields, with large positive changes to estimated FA and large angular error, often in excess of 10 degrees. In addition, a nonlinear field with determinant far from one resulted in the greatest changes of all derived measures. Thus, we expect that voxels nearest the periphery of the field of view, and with the lowest FA will be most affected by nonlinear effects, in particular most gray matter voxels of the brain. Finally, effects showed non-intuitive trends with the ground truth orientation of the tissue and showed nonlinear effects can lead to both increases and decreases of all estimated parameters, depending on orientation of the tissue and orientation of the nonlinear field.

One limitation of this study is that b-values are assumed to be from a Gaussian distribution but in the brain the biological tissues due to the presence of cell membranes, and intracellular and extracellular spaces have non-gaussian diffusion behavior [21]. Diffusion Kurtosis Imaging (DKI) describe the degree of non-gaussian water molecular diffusion. Studies have shown the effect of L(r) fields for DKI properties are smaller in comparison to DTI properties [22, 23]. However, DKI properties have tissue specific b-values and require nonuniformities to be acknowledged to avoid errors in classifying changes in tissue microstructure. This study can be extended to investigate inaccuracy due to gradient non-linear fields in diffusions from DKI model.

This study provides a theoretical framework to interpret characteristics of empirically derived non-linear fields, how it alters tensors and impacts the precision and accuracy of fractional anisotropy (FA), mean diffusivity (MD) and principal eigenvector (PEV). The next step is to confirm these findings in empirically acquired data [14], and to investigate these effects in systems that may suffer from more drastic nonlinearities [20]. Finally, we intend to use the insight from these studies to investigate correction methods and their performance in different acquisition conditions and processing implementations [8–10, 24]

Figure 1.

In diffusion MRI, where magnetic field gradients are used to sensitize the signal to the diffusion process, gradient nonlinearities result in spatially varying diffusion sensitization. The intended b-value is constant across the image (top; left), however, when these are realized in practice in clinical scanners, they are irregular depending on the spatial effects of nonlinear gradients (top; middle) which results in b-values varying up to 10% clinically (top; right). These nonlinearities may result in signal changes that affect diffusion tensor estimation (middle row) which can result in improper anisotropy and diffusivity measures, which is shown in empirically acquired data (bottom row), where we can see a change of 10% in diffusivity (bottom right).

Figure 3.

The map of L(r) determinants throughout the brain at isocenter. The brain is mostly between 0.9 and 1 but if we are at the edge of FOV we can expect bigger changes