# Computation of the mid-sagittal plane in diffusion tensor MR brain images

TL;DR: This plane is estimated as the one that best superposes the two hemispheres of the brain by reflection symmetry via the automated minimisation of a correlation-type global criterion over the tensor image via the NEWUOA algorithm in a multiresolution framework.

Abstract: We propose a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images. We estimate this plane as the one that best superposes the two hemispheres of the brain by reflection symmetry. This is done via the automated minimisation of a correlation-type global criterion over the tensor image. The minimisation is performed using the NEWUOA algorithm in a multiresolution framework. We validate our algorithm on synthetic diffusion tensor MR images. We quantitatively compare this computed plane with similar planes obtained from scalar diffusion images (such as FA and ADC maps) and from the B0 image (that is, without diffusion sensitisation). Finally, we show some results on real diffusion tensor MR images.

## Summary (2 min read)

### 1. INTRODUCTION

- The human brain displays an approximately bilateral symmetry with respect to the mid-sagittal plane (MSP).
- After correction of the tilt, the remaining normal or abnormal asymmetries can be assessed qualitatively or quantitatively, which can be relevant for many pathologies.
- A more general approach is to view the mid-sagittal plane as the one that best separates the brain into two similar parts.
- 5–12 Diffusion Tensor Magnetic Resonance Imaging (DT-MRI or DTI for short) complements conventional, anatomical, MR imaging in that it adds specific structural information about the brain.
- Such an algorithm has the same potential applications as the methods developed for anatomical and functional images.

### 2.1. Data acquisition

- A single-shot echo-planar sequence combined with coil sensitivity encoding was used to acquire the diffusion-weighted MR images with a spin-echo Stejskal-Tanner sequence (TE/TR(ms) = 56/7991, flip angle = 90◦).
- For each subject, an image without diffusion sensitisation was acquired (b value = 0 sec/mm2), together with diffusion-weighted MR images (b value = 800 sec/mm2) with diffusion-sensitising gradients applied in 16 different directions.

### 2.2. Pre-processing

- The pre-processing stage includes the following steps: Correction of distortions.
- Diffusion-weighted images acquired with echo-planar sequences are significantly distorted, mainly due to the eddy currents induced by the large diffusion gradients.
- To deal with this artefact, each diffusion-weighted image was registered to the image without diffusion sensitisation.
- The alignment is based on the maximisation of a similarity measure on image intensities (the local correlation) and a transformation model based on the imaging physics.
- 24 Estimation of the diffusion tensor by multiple linear regression on a voxel-by-voxel basis.25 Diagonalisation of the diffusion tensor by classical linear algebra (computation of the eigenvalues and eigenvectors).

### 2.4. Formulation of the problem

- If the brain was perfectly symmetrical, there would exist a symmetry plane P superposing each voxel v of the grid with its anatomical counterpart.
- In real life, the brain is only grossly symmetrical, and such an ideal symmetry plane does not exist.
- This average distance would be null for a perfect symmetry plane.

### 2.5.2. Interpolation scheme

- Thus, an interpolation is needed to estimate the tensor value I(v′) at this non-grid point.
- A natural choice would be to interpolate in log-Euclidean or affine-invariant frameworks, which would preserve the constraint of symmetric positive definiteness.
- 26–29 This unfortunately leads to a high computational cost and a prohibitive computation time for the overall algorithm.
- The authors choice is to simply use the nearest neighbor interpolation instead, which also guarantees that the interpolated value is a SPD matrix.

### 2.5.4. Optimisation

- The authors choose the algorithm NEWUOA (NEW Unconstrained Optimisation Algorithm) to solve the optimisation problem.
- Powell,30 has been shown to be significantly more accurate, more robust and faster than other deterministic, derivative-free algorithms such as Nelder-Mead’s downhill simplex or Powell’s direction set algorithms for intensity-based rigid-body image registration.
- A multiresolution (simple decimation) approach is implemented to increase the robustness, accuracy and speed of the algorithm.
- A subsampled image is built by taking only one voxel out of fx (resp. fy, fz) in the x- (resp. y-, z-) direction in the original image.

### 2.5.5. Realignment of the plane

- Once the MSP is estimated, it is realigned at the centre of the image grid for improved image display.
- Such a realignment is performed by applying the rigid transformation R to both the anatomical and the tensor information.
- The log-Euclidean interpolation28 is used for computation of the realigned DTI.

### 3. VALIDATION AND RESULTS

- The authors use the measures δ and to evaluate the capture range (i.e. the maximal tilt that it is able to correct) and the accuracy (i.e. the closeness between the estimated plane and the ground truth plane) of the algorithm, as shown in Fig.
- To investigate this question, the authors have modified the algorithm to make it suitable for scalar images.
- It appears that the latter is significantly more accurate than the three other ones (p < 5.10−4).
- The computation of the MSP in these three scalar images is much faster and seems as robust as the computation of the MSP in the DTI.

### 4. CONCLUSION

- The authors have proposed a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images.
- Moreover, the authors introduced a new optimisation algorithm to deal with the high computational burden of DTI data.
- The most straightforward modifications could be made in the choice of the interpolation scheme and of the distance function.
- The authors will also evaluate the robustness and accuracy properties of the algorithm in presence of image artefacts (noise, intensity inhomogeneities) and brain normal and pathological asymmetries.

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###### Q2. What are the future works mentioned in the paper "Computation of the mid-sagittal plane in diffusion tensor mr brain images" ?

In this paper, the authors have proposed a method for the automated computation of the mid-sagittal plane of the brain in diffusion tensor MR images. In the future, the authors plan to experiment other implementation choices and compare them with the present method.