References

Unconstrained Kinematic MRI Tracking of Wrist Carpal Bones

Mohammad Zarenia∗, Volkan Emre Arpinar, Andrew S. Nencka, L. Tugan Muftuler,
and Kevin M. Koch

Department of Radiology, Medical College of Wisconsin, Milwaukee, WI, USA.

∗Address correspondence to:

Mohammad Zarenia, Ph.D.
Medical College of Wisconsin
Department of Radiology
8701 West Watertown Plank Road
Milwaukee, WI 53226
E-mail<EMAIL_ADDRESS>
## Abstract

In this preliminary study technical methodology for kinematic tracking and
profiling of wrist carpal bones during unconstrained movements is explored.
Heavily under-sampled and fat-saturated 3D Cartesian MRI acquisition were used
to capture temporal frames of the unconstrained moving wrist of 5 healthy
subjects. A slab-to-volume point-cloud based registration was then utilized to
register the moving volumes to a high-resolution image volume set collected at
a neutral resting position. Comprehensive error analyses for different
acquisition parameter settings were performed to evaluate the performance
limits of several derived kinematic metrics. Computational results suggested
that sufficient volume coverage for the dynamic acquisitions was reached when
collecting 12 slice-encodes at 2.5mm resolution, which yielded a temporal
resolution of and 2.57 seconds per volumetric frame. These acquisition
parameters resulted in total absolute errors of 1.9${}^{\circ}\pm$1.8∘
(3${}^{\circ}\pm$4.6∘) in derived rotation angles and 0.3mm$\pm$0.47mm
(0.72mm$\pm$0.8mm) in center-of-mass displacement kinematic profiles within
ulnar-radial (flexion-extension) motion. The results of this study have
established the feasibility of kinematic metric tracking of unconstrained
wrist motion using 4D MRI. Temporal metric profiles derived from ulnar-radial
deviation motion demonstrated better performance than those derived from
flexion/extension movements. Future work will continue to explore the use of
these methods in deriving more complex kinematic metrics and their application
to subjects with symptomatic carpal dysfunction.

KEY WORDS: Wrist carpal bones, Registration, 4D MRI, Kinematic

## INTRODUCTION

Imaging of the moving joint has been hypothesized to provide differentiating
diagnostic information in orthopedic assessments and longitudinal management
(1, 2, 3, 4, 5, 6, 7). Connective tissue injuries, structural deformities, and
structural integrity loss can be difficult to characterize in single-frame
static diagnostic images. Improvements in dynamic image acquisition technology
and analytic methods continue to offer potential mechanisms to harness the
diagnostic information contained in dynamic images of moving joints.

Dynamic imaging of the wrist is a topic of particular interest in the
orthopaedic community. Specifically, it is hypothesized that kinematic
analysis of diagnostic images help uncover the role of extrinsic and intrinsic
ligaments in wrist dysfunction(8, 9). This concept has been previously
explored and utilized to qualitatively identify abnormalities in wrist carpal
bone movements and their correlation with wrist instability (10, 4, 11). More
recent work has also performed preliminary kinematic wrist investigations
utilizing x-ray fluoroscopy (3) and CT (12, 13, 11, 14, 15, 16, 17).

The use of MRI for kinematic analysis of the wrist was first explored by
Foster et al, where MRI wrist scans of healthy individuals were acquired at
multiple static positions and the carpal bone displacements were studied
through a principal component analysis (18). Foster et al’s study collected
data on an asymptomatic cohort undergoing quasi-static radial–ulnar deviation
movements. Using this data, several metrics, including the scaphotrapezium
joint and the capitate-to-triquetrum distance were derived from the imaging
data and analysed to explore modal trends within the study cohort. (19).

In the present study, we demonstrate an approach utilizing a 4D MRI
acquisition of the continuously moving and unconstrained wrist to perform
kinematic tracking of individual carpal bones. The acquired dynamic imaging
data is registered to a high-resolution complete static volume using a point
cloud based registration method. Metrics derived from these dynamic
registrations are then utilized to construct kinematic profiles of the moving
wrist.

In a test cohort of 5 subjects performing unconstrained radial-ulnar and
flexion-extension deviations with their dominant hand, kinematic profiles
constructed with the described methodology were analyzed for consistency and
potential viability in ongoing and future studies. The relative accuracy of
the constructed profiles was examined, and the results for the signal to error
analysis of the sample metric profiles were computed.

## METHODS

The MRI acquisition in this study deploys an unconventional application of the
Liver Acquisition with Volume Acceleration (LAVA Flex, GE Healthcare) pulse
sequence to capture high resolution static images and 4D kinematic imaging of
the moving wrist.

LAVA Flex is a 3D, Fast Spoiled Gradient Recalled (FSPGR) sequence that
collects in-phase and out-of phase echoes for two-point Dixon-based fat-water
separation. (20). In the present study, the ”Water” image output of this
technique is utilized for further analysis. To our knowledge, this is the
first application of the 3D LAVA Flex sequence for the study of joint
kinematics using MRI.

Fig. 1: Orthogonal imaging planes through the carpal bones of (a) 3D SPGR and
(b) LAVA Flex acquisitions. LAVA Flex was utilized as a source to manually
segment the bones of interest. (c,d) Sample slices of 3D dynamic LAVA Flex
images utilized to track the static segmentation. (e) Resulting point cloud of
boundary points derived from the high-resolution segmentation. The thick
points show the boundaries of the moving slices which are registered to the
static point clouds.

### MRI acquisition

Imaging data were collected on a GE Heathcare Signa Premier 3T MRI scanner
using a 16 channel large flex coil. Healthy subjects with no prior known wrist
pathology or bone disease were recruited into a locally approved IRB protocol
and provided written consent to participate. Subjects were placed in the MRI
bore in a prone ”superman” position. The dominant arm of each subject was
placed in the center of the receiver array, using sufficient positioning pads
to allow the necessary range of motion for required tasks (i.e. ulnar-radial
and flexion-extension deviations in this feasibility study). A picture of the
wrist positioned in the scanner is shown in Fig. S1 of the Supplementary
Material. No motion-restriction constraints were utilized. Instead, visual
cues were used to pace the motion and the subjects were trained prior to the
exam using same visual cues (see the training videos in the Supplementary
Information).

Static images were acquired using a 3D LAVA Flex sequence with $0.9\times
0.9\times 1$mm voxel size and acquisition matrix size of $224\times 224\times
60$. No acceleration was used for static imaging. Other acquisition parameters
were TE $=$ 1.7ms, TR $=$ 5.3ms, FA $=$ 10∘, NEX $=$ 1, and BW$/$Pixel $=$
417Hz. Forty dynamic sub-volumes with a temporal resolution of 2.57s were
acquired using multi-phase 3D LAVA Flex series with $1.6\times 1.6\times
2.5$mm voxel size with $128\times 128\times 12$ acquisition matrix size. Phase
acceleration of 2 and compressed sensing factor of 1.4 were used for dynamic
imaging. The other acquisition parameters were TE $=$ 1.2, 3.4ms, TR $=$ 4ms,
FA $=$ 10∘, NEX $=$ 1, and BW$/$Pixel $=$ 977Hz.

The visual guidance utilized to direct subject motion indicated 3 cycles of
motion during the 103 second acquisition duration. This rate of motion was
found to cause minimal motion artifacts in the described dynamic 3D
acquisition and was a fast enough rate for subjects to complete without
discomfort.

Figures 1(a-d) provide demonstrative images from one subject. High resolution
static images are shown in 1(a) and 1(b) obtained using 3D SPGR and LAVA Flex
MRI acquisitions, respectively. The LAVA Flex water (i.e. fat-suppressed)
reconstructed image series provided excellent bone/tissue contrast, which is
essential for accurately segmenting the carpal bones. Two sample images from
the 3D dynamic LAVA Flex series during an ulnar-radial deviation are shown in
Figs. 1(c) and 1(d).

### Registration approach

The average number of slices acquired during the dynamic imaging sequence of
ulnar-radial (flexion-extension) motion covered respectively 88$\pm$7.8
(70.1$\pm$9.8), 95$\pm$7.7 (98$\pm$12.9), and 90$\pm$7.2 (83$\pm$9.7) percent
of the scaphoid, capitate, and lunate. These values are obtained by averaging
over 200 time frames of 5 subjects, i.e. 40 acquired frames for each subject.

The mapping of sub-volume images to a given volumetric (3D) reference image is
a challenging image registration problem (21). Here, this problem is addressed
using a point-cloud based registration algorithm on manually segmented carpal
bones. The manual segmentation was done on both the high resolution static
images and the dynamic image volumes using the OHIF viewer (22).

Registration of dynamic to static point clouds derived from the OHIF RTStruct
DICOM segmentation outputs was performed in MATLAB (23) using its iterative-
closest-point registration function, pcregistericp. This function accepts the
moving and static point clouds as inputs and performs a 6-degrees of freedom
rigid-body transformation by minimizing the root-square-mean-square error
(RMSE) of the Euclidean distance between the coordinates of the matched pairs
defined by,

${\rm RMSE}=\left(\frac{\sum_{i=1}^{N_{m}}|{\bf R}_{m}-{\bf
R}_{f}|^{2}}{N_{m}}\right)^{1/2},$ (1)

where, ${\bf R}_{m}$ and ${\bf R}_{f}$ are the coordinates of the nearest
pairs in the moving and static point clouds, respectively. $N_{m}$ is number
of boundary points of the moving point cloud.

The outputs of the registration function are a $4\times 4$ transformation
matrix, the registered point cloud, and the resulting RMSE. The transformation
matrix consists of a $3\times 3$ rotation sub-matrix and 3 spatial
transformation components where the rotation elements are represented in an
angle-axis form. Figure 1e provides an example surface point cloud of the
moving slab (thick points) that has been registered to the surface point cloud
of the static volume.

Fig. 2: Workflow for processing of error analysis in each time frame.

## Error analysis

The error analysis process used to calculate the registration error and the
resulting absolute errors in reported metrics is depicted in a flowchart in
Figure 2.

Fig. 3: Root-Mean-Square-Error (RMSE), Eq. (1), averaged over 10 different
time frames and for all possible slab locations as functions of slab thickness
for 1mm and 2.5mm slice thicknesses of moving slabs of scaphoid, capitate, and
lunate. Error bars indicate the standard deviation.

The high-resolution static volume is first transferred to the moving reference
frame using the inverse of the transfer matrix obtained through the
registration of the segmented moving slab to the static volume. The resulting
moving volume is then resampled to create a moving volume with 2.5mm slice
thicknesses (i.e. the same slice thickness as the actual moving slabs). From
the resampled moving volume, a sub-volume with $N$ slices is chosen ($N$ is
the number of segmented slices in actual moving images). The value of $N$ and
location of sub-volume with respect to the full bone was varied for this
analysis. Using this synthetically generated sub-volume, a new registration
matrix was calculated with respect to the original high-resolution image. The
absolute errors were calculated between the metrics derived from the actual
motion and this synthetic data, such as rotation angles and center-of-mass, as
well as the RMSE between the registered and moving slab.

Fig. 4: Kinematic profiles of scaphoid, capitate, and lunate rotation angles
in relation to their starting positions. The black solid curve represents the
profile of simultaneously registered bones. The absolute errors are
represented by the shaded error bands, calculated using the process shown in
Fig. 2 Fig. 5: Error-to-signal ratio of rotation angle profiles (absolute-
error$/\theta$) that are obtained by averaging over the ratio values at the
extremum points of the profiles of Fig. 4.

## RESULTS

To characterize the performance of the registration as a function of the slice
thickness and number of collected slices (i.e. the collected slab thickness),
Fig. 3 compares the averaged RMSEs over 10 different kinematic position frames
for all possible sub-volume slab locations within the bone at each frame.

As the slab thickness is increased, the registration performance improves.
These findings also demonstrate that slab thickness (bone coverage) has a
greater impact on registration performance than slice resolution. The RMSE
difference between slabs with 1mm and 2.5 slice thicknesses is found to be
less than 0.5mm (i.e. less than the 1mm slice thickness of the static volume)
at slab thicknesses with more than 70$\%$ bone coverage – which is the case
for the data collected and analyzed in this pilot study.

Figure 4 provides derived kinematic metric profiles for the carpal bone
rotation angles with respect to the initial hand position. To obtain the
rotation angle with respect to the initial position, the high-resolution
static volume was first registered to the initial moving slab and the
resulting volume was then utilized to register the other moving slabs. The
shaded bands are the absolute errors evaluated through the process
demonstrated in Fig. 3. During the error analysis, $N$ was fixed to the number
of segmented slices and all possible slab locations with respect to the full
volume were considered at each time frame. In order to simplify the profile
and clarify the modal nature of the motion, a smoothing filter is applied. The
smoothing was performed in MATLAB using rloess method with a span value of 20%
of the total number of points. This method assigns zero weight to data outside
six mean absolute deviations. To compare the carpal bone’s rotation with the
wrist, rotation angles of simultaneously (i.e. bulk) registered scaphoid,
capitate, and lunate (black curves) are superimposed. The maximum rotation
angles for the scaphoid, capitate, and lunate in ulnar-radial (flexion-
extension) motion were found to be $28.6^{\circ}\pm 3.4^{\circ}$
($27^{\circ}\pm 7.8^{\circ}$), $34.6^{\circ}\pm 13.2^{\circ}$
($24.6^{\circ}\pm 7.2^{\circ}$), and $31.4^{\circ}\pm 12^{\circ}$
($17.5^{\circ}\pm 6.8^{\circ}$), represented as means and standard deviations
of 5 subjects. An error-to-signal ratio for the rotation angle profiles, i.e.,
(absolute-error)$/\theta$, is provided in Fig. 5. For the ulnar-radial
(flexion-extension) motion, the average ratios of 5 subjects for scaphoid,
capitate, and lunate are respectively 0.26$\pm$0.08 (0.57$\pm$0.21),
0.06$\pm$0.07 (0.27$\pm$0.12), 0.29$\pm$0.06 (0.57$\pm$0.66).

Fig. 6: Scaphoid, capitate, lunate (red) and their simultaneously registered
(black) angular axes of rotation of 15 time frames corresponding to
10s$<$time$<$62s for subject 3.

Figure 6 provides exemplary angular axes of rotation over 15 time frames
(corresponding to 10s$<$time$<$62s) of scaphoid, capitate, and lunate as well
as the results of their simultaneous registration in subject 3. In each case,
the rotation axes pass through the center-of-mass. As expected, the principal
axes of rotation of simultaneously-registered bones (black lines in Fig. 6)
are directed along the rotation axes of the wrist motion that is perpendicular
to the orthogonal plane during an ulnar-radial deviation and parallel to the
axial plane during flexion-extension.

Fig. 7: (a) Point clouds of scaphoid, capitate, lunate, and articular surface
of radius. (b) Kinematic profiles of the scaphoid, capitate, and lunate
center-of-masses with respect to the radius center-of-mass. The mean value is
subtracted from each profile. The shaded error bands are the absolute errors,
calculated using the process shown in Fig. 2

.

Fig. 8: Error-to-signal ratio of center-of-mass profiles (absolute-
error$/$center-of-mass) that are obtained by averaging over the ratio values
at the extremum points of the profiles of Fig. 7.

Kinematic profiles of the center-of-mass displacement of carpal bones with
respect to radius are shown in Fig. 7. To derive these profiles, the articular
surface of the radius was segmented out of the static and moving images and
registered using the discussed point cloud approach. The center-of-mass
profiles of simultaneously registered scaphoid, capitate, and lunate are
compared to the combined canter-of-mass of individually registered bones in
Fig. S3 of the Supplementary materials. As expected, the results are close to
one other, indicating logically consistent registration performance. Averaging
over 200 time frames of 5 subjects, the range of variations (mean $\pm$
standard deviation) in carpal bones-radius center-of-mass distances during the
ulnar-radial (flexion-extension) motion are respectively obtained as
16.29$\pm$1.05 mm (15.98$\pm$0.63 mm), 24.44$\pm$0.63 mm (23.55$\pm$0.6 mm),
and 11.53$\pm$0.62 mm (11.71$\pm$0.58 mm)) for scaphoid, capitate, and lunate.
Fig. 8 provides the results for the error-to-signal ratio of center-of-mass
profiles (absolute-error$/$center-of-mass) which are calculated at the
extremum points. The average ratios of 5 subjects for the ulnar-radial
(flexion-extension) motion are 0.11$\pm$0.03 (1.54$\pm$1.28), 0.1$\pm$0.075
(0.4$\pm$0.45), 0.13$\pm$0.09 (0.55$\pm$0.36) for the scaphoid, capitate, and
lunate, respectively.

Table 1 provides error analysis results of the rotation angle and center-of-
mass metrics as functions of temporal coverage and number of acquired moving
slices in each time frame. Using the fewer number of acquired slices at
temporal resolutions greater than 2.57s/frame, registration errors were
significantly increased. For poorer temporal resolutions using higher numbers
of slices per volume, the registration accuracy slightly improved (as
expected), however empirical testing identified difficulties for subjects to
move their wrists at such a slow rate. Therefore, the utilized approach (12
2.5mm slices acquired at 2.57s/frame ) provided an optimal balance of subject
comfort, image/data quality, and registration accuracy.

Table 1: Absolute errors of rotation angle and center-of-mass metrics for different acquired slices and temporal resolutions. Due to slice overlapping (to prevent foldover aliasing artifacts), 4 of the initially acquired slices in each volume were removed. The preliminary kinematic profiling of carpal bones in this study was performed using 12 slices (out of 16 initially acquired slices). This case is highlighted in bold text. The error in each case is obtained by averaging errors computed over the 200 time frames of 5 subjects. acquired slices |  temporal resolution (s) |  l— |  l—
---|---|---|---
l— |  l— |  l— |  l—
S | C | L | S | C | L | S | C | L | S | C | L
4 |  2.1 |  30.5$\pm 15.6$ |  17.7$\pm 12.3$ |  50$\pm 15.6$ |  42.8$\pm$19.2 |  19.5$\pm$16.5 |  56.5$\pm$19.2 |  2.28$\pm$1.3 |  2.6$\pm$2.3 |  4.01$\pm$2.6 |  3.4$\pm$2.2 |  2.7$\pm$2.1 |  3.8$\pm$2.3
8 |  2.3 |  6.3$\pm 3.8$ |  1.8$\pm 1$ |  10.1$\pm 8$ |  21.6$\pm$14.9 |  4.05$\pm$2.4 |  16.06$\pm$15.4 |  2.4$\pm$2.1 |  0.38$\pm$0.42 |  1.11$\pm$0.9 |  1.6$\pm$1.1 |  1.07$\pm$0.7 |  1.02$\pm$1.1
12 |  2.57 | 3.6$\pm$2.7 | 0.61$\pm$0.38 | 2.7$\pm$1.9 | 7.17$\pm$ 5.2 | 2.2$\pm$+1.9 | 2.4$\pm$1.7 | 1.04$\pm$0.85 | 0.3$\pm$0.18 | 0.97$\pm$0.8 | 1.41$\pm$1.4 | 0.62$\pm$0.47 | 0.67$\pm$0.36
16 |  3.27 |  2.35$\pm 1.9$ |  0.54$\pm 0.38$ |  2.12$\pm 1$ |  3.1$\pm$2.6 |  2.05$\pm$1.9 |  1.7$\pm$0.98 |  0.77$\pm$0.5 |  0.23$\pm$0.17 |  0.73$\pm$0.6 |  1.0$\pm$0.64 |  0.60$\pm$0.48 |  0.42$\pm$0.34
20 |  3.72 |  2.16$\pm 1.9$ |  0.58$\pm 0.38$ |  2.1$\pm 1$ |  2.7$\pm$2.2 |  2.05$\pm$1.9 |  1.7$\pm$0.97 |  0.77$\pm$0.5 |  0.23$\pm$0.17 |  0.7$\pm$0.6 |  0.97$\pm$0.65 |  0.60$\pm$0.48 |  0.40$\pm$0.28
|  |  |  |  |  |  |  |  |  |  |  |  |

## DISCUSSION

In the present study, the motions of scaphoid, capitate, and lunate carpal
bones within 5 subjects were analyzed during unconstrained ulnar-radial
deviation and flexion-extension movements. Nine basic metrics were analyzed as
proof-of-concept kinematic metric profiles: 6 angular rotation angles and axes
and 3 center-of-mass displacements of carpal bones with respect to the radius.

Individual carpal bone rotation angles followed similar profiles as the bulk
composite of the 3 bones within the ulnar-radial motion (Fig. 4). An
interesting exception was observed in subject 1, whereby the capitate
displayed a different profile compared to the other subjects and a
significantly decreased range of rotation ($0<\theta<13^{\circ}$) was seen.
The registration performance in this bone/subject was inspected and found to
be accurate. After an examination of other high resolution static images of
this subject’s wrist, a structural abnormality was detected in capitate (see
Fig. S2 in the Supplementary Material). Although further examination is needed
to determine the nature and impact of this abnormality, this unexpected
finding is noteworthy.

The center of mass kinematic profiles (Fig. 7) suggest that the scaphoid and
capitate move mainly in-phase, while the lunate moves out of phase (relative
to the proximal radius surface) during the ulnar-radial motion. Again, there
was an exception to this trend for the capitate profile of subject 1.

The kinematic profiles of flexion-extension motion, showed that range of
rotation of the carpal bones were lower in capitate and lunate by about 30%
and 45%, respectively, compared to their range of rotation within ulnar-radial
motion. However, the range of center-of-mass displacement relative to radius
remains comparable (less than 5% difference) during the both ulnar-radial and
flexion-extension motions.

The results of this feasibility study are encouraging. First, the study has
demonstrated an ability to extract repeatable kinematic metric profiles of
carpal bones from 4D MRI of unconstrained wrist motion. Of the 9 studied
metrics, the individual rotation angles of carpal bones within the ulnar-
radial deviation showed the most promise as normative MRI-derived kinematic
wrist profiles. The kinematic metrics derived from flexion-extension motions
demonstrated more variation and lower signal to error signatures in this
preliminary study. However, more complex metric derivations may improve this
performance of the flexion-extension analysis.

Quantitative analysis of unconstrained wrist motion using MRI offers a rich
set of potential tools that could be utilized to study joint degeneration and
dysfunction. The relatively fast kinematic profiling MR acquisitions
demonstrated in this feasibility study could be added to routine orthopedic
MRI exams with relative ease, therefore providing optimal diagnostic imaging
in both static and kinematic contexts in a single visit. Of note, the unique
acquisition approach demonstrated in this study does not use any prototype
pulse-sequences and is commercially available on most GE Healthcare scanner
platforms. Similar sequences and capabilities are also available on other
vendor platforms. As a result, the acquisition methods demonstrated in this
work could easily be implemented within many clinical imaging environments.

In addition to profiling of rotation angles and center-of-mass displacements
of scaphoid, lunate, capitate and their combined metrics, many more metrics
such as gaps between the carpal bones, rotation angles of carpal bones
relative to radius, etc., across several more wrist motions can be tracked in
similar fashion. Future work will continue to deploy these new technologies to
explore more complete kinematic tracking and profiling of unconstrained wrist
motion.

## ACKNOWLEDGEMENTS

Research reported in this publication was supported in part by National
Institute of Health (NIH) grant R21AR075327.

## ETHICAL APPROVAL

All procedures performed in studies involving human participants were in
accordance with the ethical standards of the institutional and/or national
research committee and with the 1964 Helsinki declaration and its later
amendments or comparable ethical standards.
Informed Consent: Informed consent was obtained from all individual
participants included in the study.

## Conflict of Interest

The authors declare that they have no conflicts of interest.

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## FIGURE LEGENDS

Figure 1: Orthogonal imaging planes through the carpal bones of (a) 3D SPGR
and (b) LAVA Flex acquisitions. LAVA Flex was utilized as a source to manually
segment the bones of interest. (c,d) Sample slices of 3D dynamic LAVA Flex
images utilized to track the static segmentation. (e) Resulting point cloud of
boundary points derived from the high-resolution segmentation. The thick
points show the boundaries of the moving slices which are registered to the
static point clouds.

Figure 2: Workflow for processing of error analysis in each time frame.

Figure 3: Root-Mean-Square-Error (RMSE), Eq. (1), averaged over 10 different
time frames and for all possible slab locations as functions of slab thickness
for 1mm and 2.5mm slice thicknesses of moving slabs of scaphoid, capitate, and
lunate. Error bars indicate the standard deviation.

Figure 4: Kinematic profiles of scaphoid, capitate, and lunate rotation angles
in relation to their starting positions. The black solid curve represents the
profile of simultaneously registered bones. The absolute errors are
represented by the shaded error bands, calculated using the process shown in
Fig. 2

Figure 5: Error-to-signal ratio of rotation angle profiles (absolute-
error$/\theta$) that are obtained by averaging over the ratio values at the
extremum points of the profiles of Fig. 4.

Figure 6: Scaphoid, capitate, lunate (red) and their simultaneously registered
(black) angular axes of rotation of 15 time frames corresponding to
10s$<$time$<$62s for subject 3.

Figure 7: (a) Point clouds of scaphoid, capitate, lunate, and articular
surface of radius. (b) Kinematic profiles of the scaphoid, capitate, and
lunate center-of-masses with respect to the radius center-of-mass. The mean
value is subtracted from each profile. The shaded error bands are the absolute
errors, calculated using the process shown in Fig. 2.

Figure 8: Error-to-signal ratio of center-of-mass profiles (absolute-
error$/$center-of-mass) that are obtained by averaging over the ratio values
at the extremum points of the profiles of Fig. 7.

## SUPPLEMENTAL MATERIAL

Videos of ulnar-radial (Video1) and flexion-extension (Video2) motions that
are used to train subjects to follow during the scan:
Video1: link will be provided by Publisher
Video2: link will be provided by Publisher

Fig. S1:  Unconstrained wrist positioning in the flex coil during scanning of
static and moving images. Fig. S2:  Orthogonal planes of (a) 3D-SPGR, (b) ZTE,
and (c) LAVA Flex MRI modalities of wrist carpal bones of subject 1. Red arrow
points the abnormality observed in capitate. Fig. S3:  Kinematic profiles of
center-of-mass displacements of simultaneously registered scaphoid, capitate,
and lunate (blue) and the combined canter-of-mass of individually registered
bones (red).