Adam M. J. van Niekerk^{1}, Ernesta M. Meintjes^{1}, and Andre J. W. van der Kouwe^{1,2,3}

^{1}Division of Biomedical Engineering, Human Biology, University of Cape Town, Cape Town, South Africa, ^{2}Athinoula A. Martinos Center, Massachusetts General Hospital, Charlestown, MA, United States, ^{3}Radiology, Harvard Medical School, Boston, MA, United States

### Synopsis

In
this work we aim to address the challenges of fusing motion parameters measured
using different sensing modalities, as temporal
resolutions differ widely between navigator and external motion tracking
techniques. A model is presented in which head motion is characterised as simple rolling motion. The resulting equations describe subject
motion to within 2 mm when applied in an open loop manner. A filter, with feedback, is then implemented where navigator data is used to estimate model parameters. The filtered translation output is smooth without the cost
of increased latency due to the fast orientation estimates.

### Purpose

Since different
patient motion sensing strategies have unique advantages and disadvantages, selection
of an optimal technique is often task specific. Generally, orientation and
translation are estimated with different sensitivities. In marker-based optical
motion tracking^{1}, orientation is encoded using retrograte markers and in active marker-based approaches^{2} through position measurements
of rigidly constrained points. In both
cases orientation is affected by factors (for example, Moire pattern encoding resolution or the physical distance between active
markers) that don’t affect the translation measurements. Navigator approaches
are no different as the MR signal behaves differently for changes in
orientation than translations.
In this work we
present a model that can be used to fuse independently acquired orientation and
translation estimates. Previously^{3} we presented a low cost device
(VectOrient) that uses a vector-based approach to precisely measure orientation
in the MRI scanner independently of the pulse sequence. Here we aim to fuse the
high temporal resolution orientation estimates from (VectOrient) with noisier
translation estimates obtained using high-speed linear navigators.### Methods

Patient set-up is well defined for MRI brain imaging. The subject is
placed head-first-supine into the MRI scanner. The reference frames in Figure 1
are therefore defined for the model description. As neck muscles aren’t effective
at lifting the head in this position, it is assumed that the subject’s head
remains in contact with the scanner bed.
The vector $$$\vec{P}$$$ represents
the position of the origin of the field of view with respect to the point at
which the patient’s head makes contact with the scanner. Unit vector $$$\vec{v}$$$ is the axis of rotation and $$$\vec{n}$$$ is the translation of the field of view after
undergoing the rotation about $$$\vec{v}$$$ by an angle $$$\alpha$$$. Modelling subject head motion as rolling (Figure 2); in much the same way as a ball/sphere rolls over a surface, a
relationship between orientation and translation can be established. To simplify co-ordinate transforms, the
predicted displacement is written in terms of two orthogonal reference frames
aligned to the vertical (Y -axis) and horizontal (parallel to the XZ plane)
components of the unit axis of rotation $$$\vec{v}$$$. The rotation $$$\alpha$$$ about $$$\vec{v}$$$ is therefore broken up into two
rotations, an angle $$$\delta$$$ about $$$\vec{v}_y$$$ and then an
orthogonal rotation of $$$\theta$$$ about $$$\vec{v}_{xz}$$$ as shown in figure 2. The translation of the field of view is:$$\vec{n}=\vec{n}_y+\vec{n}_{xz}$$It follows that,$$\vec{n}_{xz}=R_{\delta}\vec{P}_{xz}-\vec{P}_{xz},$$$$\vec{n}_{xz}=\begin{pmatrix}P_x(\cos\delta-1)+P_z\sin\delta \\ 0 \\ P_z(\cos\delta-1)-P_x\sin\delta \end{pmatrix}$$The translation term on the plane normal to $$$\vec{v}_{xz}$$$:$$\vec{n}_{y}=\big( \vec{l}P_y\tan\theta+\vec{p}_{\perp}\sin\theta\big)+\epsilon (\theta,\vec{p},r),$$where $$$\vec{l} =\frac{\vec{v}_{xz}}{\|\vec{v}_{xz}\|}\times\vec{y}$$$ and $$$\vec{p}_{\perp}=\vec{y}\|\vec{P}_{xz}\|\sin\beta=\vec{P}_{xz}\times\frac{\vec{v}_{xz}}{\|\vec{v}_{xz}\|}$$$ giving,$$\vec{n}_y = \begin{pmatrix} -\frac{v_z}{\|\vec{v}_{xz}\|} P_y \tan \theta \\ \big(v_xP_z-v_z P_x \big) \frac{\sin\theta}{\|\vec{v}_{xz}\|} \\ \frac{v_x}{\|\vec{v}_{xz}\|} P_y \tan\theta \end{pmatrix} + \begin{pmatrix} -\frac{v_z}{\|\vec{v}_{xz}\|} \big( r(\theta-\sin\theta)+p_{\perp}(1-\cos\theta) \big) \\ (r-P_y)(1-\cos\theta) \\ \frac{v_x}{\|\vec{v}_{xz}\|} \big( r(\theta - \sin\theta)+p_{\perp}(1-\cos\theta) \big) \end{pmatrix}$$assuming small angles,$$\vec{n}\approx\begin{pmatrix} P_z\delta-\frac{v_z}{\|\vec{v}_{xz}\|}P_y\theta \\ \frac{\theta}{\|\vec{v}_{xz}\|}\big( v_xP_z-v_zP_x\big ) \\ \frac{v_x}{\|\vec{v}_{xz}\|}P_y\theta-P_x\delta\end{pmatrix}$$
The vertical
displacement is defined with the location of the field of view and is subject independent.
The horizontal displacement $$$\vec{P}_{xz}$$$, is more
difficult to measure although one can assume it remains smaller than $$$P_y$$$ and its effect is scaled by $$$\beta$$$ which is small with symmetrical brain imaging. The model was validated using in vivo scan data where orientation and
translation estimates were available. In an example application a filter was
developed which fuses navigator translation data and orientation measurements, while
observing the vector $$$\vec{P}$$$ (Figure 3). This is achieved by continuously feeding back the navigator and orientation data to rotate and scale $$$\vec{P}$$$ such that the predicted translation follows the navigator. The estimate of $$$\vec{P}$$$ is tracked at a slower rate than the displacement $$$\vec{n}$$$ effectively smoothing noise in the navigator estimates.### Results

As can be seen
from Figure 4, the model explains subject translation even in cases where
complex head motion occurs. There is very little difference between the outputs
showing the assumption of no slipping remains valid.
Figure 5, shows
the ability of the observer to track the contact point using the model when
orientation and translation data is available. The synthesised translation
output is much smoother as the time response for tracking the properties of $$$\vec{P}$$$ using the navigator data can be adjusted with $$$K_p$$$ and $$$K_{\omega}$$$(Figure 3).
It is important to note that filter translation output still has the quick
response of the orientation estimates.### Discussion

The
robustness to subject variance shows that this model could prove useful when noise
in orientation and translation data are complementary in nature. Another application is the fusion of low temporal resolution estimates (vNav^{4}) with faster sequence independent approaches. ### Conclusion

Patient
orientation and translation are very strongly correlated. Further constraints
do exist and methods of fully quantifying subject motion using very few sensors
may be possible. ### Acknowledgements

National Institutes of Health
under grants R01HD071664, R21MH096559, the NRF/DST through the South
African Research Chairs Initiative and the University of Cape Town through
the RCIPS Explorer fund EX15-009.

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