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\title{Increased Hindbrain Motion in Chiari I Malformation Patients Measured Through 3D Amplified MRI (3D aMRI)}

\author{Javid Abderezaei,$^{1,*}$ Aymeric Pionteck,$^{1,*}$ Ya-Chen Chuang,$^{1}$ Alejandro Carrasquilla,$^{2}$ Gizem Bilgili,$^{1}$ Tse-An Lu,$^{3}$ Miriam Scadeng,$^{4}$ Patrick Fillingham,$^{5, 6}$ 
Peter Morgenstern,$^{2}$ Michael R. Levitt,$^{1,5,6}$ Richard G. Ellenbogen,$^{5, 6}$ Yang Yang,$^{7,8}$ Samantha J. Holdsworth,$^{4,9}$ Raj Shrivastava,$^{2}$ and Mehmet Kurt$^{1,8}$}
%,}$\footnote[2]{\email{Corresponding author.\\ \indent \indent mkurt@stevens.edu}}}

\address{\it$^{1}$Department of Mechanical Engineering, University of Washington, Seattle, WA, USA\\
$^{2}$Department of Neurosurgery, Icahn School of Medicine at Mount Sinai, New York, NY, USA\\
$^{3}$ Department of Biostatistics, Boston University, Boston, MA, USA\\
$^{4}$Department of Anatomy and Medical Imaging, Faculty of Medical and Health Sciences \& Centre for Brain Research University of Auckland, Auckland, New Zealand\\
$^{5}$Department of Neurological Surgery, University of Washington, Seattle, WA, USA\\
$^{6}$Stroke and Applied Neuroscience Center, University of Washington, Seattle, WA, USA\\
$^{7}$Department of Radiology and Biomedical Imaging, University of California, San Francisco, CA, USA\\
$^{8}$BioMedical Engineering and Imaging Institute, Icahn School of Medicine at Mount Sinai, New York, NY, USA\\
$^{9}$M$\bar{a}$tai Medical Research Institute, Gisborne-Tair$\bar{a}$whiti, New Zealand\\
$^{*}$ These authors contributed equally}


\end{frontmatter}

\section{3D aMRI algorithm}
The 3D aMRI method \cite{abderezaei2021development,terem20213d} is an extension of the 2D aMRI algorithm \cite{terem2018revealing,holdsworth2016amplified}, allowing for the visualization and quantification of sub-voxel level motion in 3 directions. In the first step, a complex 3D steerable pyramid in the spherical domain is applied to the input 3D cine MRI datasets, which decomposes the volumetric data at different angles and scales.
% which in our study, is 3D cine MRI datasets of healthy and CM-I subjects. By using the 3D steerable pyramid the volumetric data is decomposed at different angles and scales.
In the next step, the temporal phase differences of the decomposed data are calculated and a temporal band-pass filter is applied to isolate the motion at the selected frequency bands.
% The band-pass frequency range is selected based on the application of interest, which in the case of cardiac induced brain motion 
To reduce the noise and artifacts, a Gaussian spatial smoothing filter ($\sigma$) can be optionally added to the next step. The processed phase differences at each angle and scale are then amplified using an amplification factor $\alpha$ and then added back to the original reference phase. Finally, the 3D steerable pyramid is collapsed and the decomposed data is reconstructed to produce the 3D amplified data. For more information about the 3D aMRI algorithm refer to \cite{abderezaei2021development, terem20213d}.

\section{Semi-automatic brain segmentation technique}
In order to create masks and measure the displacements of each brain region, a semi-automated brain segmentation method was used (Manuscript Fig.~1). The T1-weighted MRI of each subject was initially processed with FreeSurfer which allowed extracting masks for the cerebellum, brainstem, and medulla \cite{fischl2002whole,dale1999cortical,iglesias2015bayesian}. The cerebellar tonsils of each subject were then manually segmented. The automatically generated masks of each subject were then manually modified in order to improve their accuracy. The general pipeline of the semi-automatic image segmentation can be seen in Fig.~1 in the Manuscript. An automatic intensity based image registration method was next used in MATLAB (MathWorks, Natick, MA, version R2020b) to find the transformation matrix between the T1-weighted MRI of each subject and its corresponding 3D cine MRI. The calculated transformation matrix was then applied to the created masks for the cerebellum, cerebellar tonsils, brainstem, and medulla, to register the T1-weighted MRI based masks to their corresponding 3D cine MRI. Each mask for the 3D cine MRI data was then manually inspected in order to confirm its accuracy.

\section{Morphometric analysis}
We measured various morphological parameters for each of the CM-I patients. These parameters include: cerebellar tonsil herniation, the area of foramen magnum, the ratio of neural brain tissue that occupies the foramen magnum, and the $4^{th}$ ventricular volume.\\
\indent To measure the cerebellar tonsil herniation, the images were first centered with respect to the coronal and axial planes. The mid-sagittal slice was then selected and the foramen magnum was indicated by connecting a line from the basion to the opisthion (McRae’s line). Cerebellar tonsil herniation was measured by calculating the perpendicular distance from the McRae’s line to the most inferior end of the tonsil.\\
\indent The axial plane at the foramen magnum was then selected to measured the cross-sectional areas of the cervicomedullary junction $A_{C}$, the tonsillar cross-sectional area $A_{T}$, and the total cross-sectional area at the foramen magnum ($A_{FM}$). Finally, the ratio of neural tissue in the foramen magnum was calculated using $(A_{T} + A_{C}) / A_{FM}$. These measurements were performed by 3 of the authors using FreeSurfer and Horos (GNU Lesser General Public License, Version 3 (LGPL-3.0)), and the results were averaged. Finally, the $4^{th}$ ventricle segmentation was initially estimated by the FreeSurfer software and was then manually corrected by one of the authors, and the resulting volume size was calculated.

\section{Statistical analysis}
Our statistical analysis includes two parts. In the first part, we compared the healthy brain motion with the CM-I subjects. In the second part, we only considered the CM-I subjects and analyzed the correlations between the morphological variables and ROI brain displacements. We then analyzed the surgical outcomes and the symptomatology of these subjects using the morphometrics and brain motion.\\
\indent To statistically compare the CM-I and healthy brain motion, t-tests were performed between 3 directional displacements (\textit{i.e.} $P_{90\%U_{Ap, Max}}$, $P_{90\%U_{SI, Max}}$,  $P_{90\%U_{ML, Max}}$, $P_{90\%U_{Mag, Max}}$) at the cerebellum, cerebellar tonsils, brainstem, and medulla.\\
\indent In the next step, we only used the CM-I dataset and analyzed the correlations between the brain motion, and morphometrics. First, we calculated the Pearson correlation coefficient ($r$) and linear regression between the morphometrics including the cerebellar tonsil herniation, the ratio of neural tissue at the foramen magnum ($(A_{T} + A_{C}) / A_{FM}$), and the $4^{th}$ ventricular volume and each of the 3 directional brain displacements in the selected ROIs. In the next step, we used the ROI brain displacements and morphometrics and performed a t-test comparison between the subjects with and without the symptoms. Next, similar t-test analyses were performed between the subjects with and without improved surgical outcomes using the morphomgetrics and brain motion. Finally, for the above symptomatology and surgical outcome analysis, logistic regression was used to calculate accuracy scores by using the Area Under the receiver operating characteristics (ROC) Curve (AUC) \cite{james2013introduction, urbizu2017machine, narkhede2018understanding}.\\
\indent For all the above statistical analyses, the differences between the two groups were considered to be statistically different if $p<0.05$. Additionally, since a large number of parameters were compared we also reported the family-wise error corrected of the differences by adjusting the statistical significance threshold \cite{bland1995multiple}. For the comparison between healthy and CM-I brain motion, the threshold was $p<0.0031$ $(0.05/16)$, whereas for the effect of the brain motion and morphology on surgical outcomes and symptomatology the threshold was $p<0.0028$ $(0.05/18)$.
% Additionally, since a large number of parameters were compared in this study (4 parameters in 4 different brain regions), we also reported the family-wise error corrected of the differences by adjusting the statistical significance to $p<0.0031$ $(0.05/16)$ by using the Bonferroni correction \cite{bland1995multiple}.


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