# A Semi-automatic Cell Tracking Process Towards Completing the 4D Atlas of C.
elegans Development

Andrew Lauziere
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742
<EMAIL_ADDRESS>
&Ryan Christensen
Laboratory of High Resolution Optical Imaging
National Institutes of Health
Bethesda, MD 20892
<EMAIL_ADDRESS>
&Hari Shroff
Laboratory of High Resolution Optical Imaging
National Institutes of Health
Bethesda, MD 20892
<EMAIL_ADDRESS>
  * Abstract. The nematode Caenorhabditis elegans (C. elegans) is used as a model organism to better understand developmental biology and neurobiology. C. elegans features an invariant cell lineage, which has been catalogued and observed using fluorescence microscopy images. However, established methods to track cells in late-stage development fail to generalize once sporadic muscular twitching has begun. We build upon methodology which uses skin cells as fiducial markers to carry out cell tracking despite random twitching. In particular, we present a cell nucleus segmentation and tracking procedure which was integrated into a 3D rendering GUI to improve efficiency in tracking cells across late-stage development. Results on images depicting aforementioned muscle cell nuclei across three test embryos suggest the fiducial markers in conjunction with a classic tracking paradigm overcome sporadic twitching.

##  1 Introduction

While the cell lineage and 4D atlas has been completed for the pre-twitch
embryo, the post-twitch embryo remains a challenge [15]. The onset of muscle
cells in the developing embryo causes sporadic muscular twitching. Current
approaches to cell tracking in the post-twitch embryo require the
identification of seam cells to approximate the coiled posture [4]. In brief,
a set of skin cells, termed seam cells describe anatomical structure in the
coiled embryo, acting as a type of “skeleton” outlining its body. A pair of
neuroblasts appear in the final hours of development, forming an extra pair of
fiducial markers. The seam cells and neuroblasts form in lateral pairs along
the left and right sides of the worm, resulting in eleven pairs upon hatching
[16]. The pairs of cells are named, posterior to anterior: T, V6, V5, Q
(neuroblasts), V4, V3, V2, V1, H2, H1, and H0. Each pair’s left and right cell
is named accordingly; for example, H1L and H1R comprise the H1 pair. Fig 1-A
depicts center points of seam cell nuclei located in an example image volume
as imaged in the eggshell (left) and straightened to reveal the bilateral
symmetry in seam cell locations (right). Fig 1-B shows four sequential images
of an embryo, five minutes between images.

Figure 1: High spatial resolution, low temporal resolution imaging
necessitates posture identification. A: Manually identified and seam cell
nuclei from an imaged C. elegans embryo. The cells form in pairs; they are
labelled posterior to anterior: T, V6, …, H0. The identification of all seam
cells reveals the embryo’s posture. Natural cubic splines through the left and
right-side seam cells estimate the coiled body. The left image depicts
identified nuclei connected to outline the embryonic worm. The fit splines are
used to untwist the worm, generating the remapped straightened points in the
diagram on the right. B: Labelled nuclear coordinates from a sequence of four
images. The embryo repositions in the five minute intervals between images,
causing failure of traditional tracking approaches.

Untwisting the worm describes the computational process of using the
identified seam cell nuclei coordinates to remap coordinates of other imaged
cells to a straightened coordinate system [4]. Subsets of cells can be imaged,
detected, remapped, then tracked. Fig 2-C illustrates the untwisting process
on a magnified portion of the right coordinate set in Fig 2-A. The remapping
uses the left, right, and midpoint splines to establish a change of basis to a
straightened coordinate space defined by the tangent (black), normal (blue),
and binormal (cyan) vectors. Encountered nuclei are remapped according to the
position along the spline and location within the closest inscribed ellipse.
Black lines connect cell nuclei in the left frame (red) to their locations in
the middle frame (blue); muscle cell names arise from the four bands (A-D),
anterior to posterior (Fig 2-D).

Figure 2: Posture identification allows the tracking of other cells during
late-stage embrygenesis. A: Seam cell nuclei coordinates (black) and muscle
nuclei coordinates (red) in a sequence of three sequential volumetric images.
The untwisting process (green arrows) uses the seam cells to remap muscle
coordinates to a common frame of reference. B: The remapped muscle nuclei are
tracked frame-to-frame (blue arrows). C: A higher magnification view from the
right coordinate plot of A. The left, right, and midpoint splines are used to
create a change of basis defined by the tangent (black), normal (blue), and
binormal (cyan) vectors. Ellipses are inscribed along the tangent of the
midpoint spline, approximating the skin of the coiled embryo. D: A portion of
the left (red) and center (blue) remapped muscle coordinates. Black lines
connect the coordinates, frame-to-frame.

Current methods for seam cell identification rely on trained users to manually
annotate the imaged nuclei using a 3D rendering tool [11]. The process takes
several minutes per image volume and must be performed on approximately 100
image volumes per embryo [4]. See [1] for a approaches to automatic seam cell
identification. Fig 3 depicts manually identified seam cells in the first two
successive image volumes of Fig 1-B. Manual identification is performed in
Medical Imaging, Processing, Analysis and Visualization (MIPAV), a 3D
rendering program used for manual annotation [11].

Figure 3: Manual posture identification in two successive image volumes of Fig
1-B using MIPAV. The 20 fluorescently imaged seam cell nuclei rendered in two
successive image volumes. Scale bar: 10 $\mu m$. A & B: Seam cell nuclei
appearing in two successive image volumes visualized in MIPAV. The five minute
interval allows the embryo to reposition between images, yielding entirely
different postures. C & D: Manual seam cell identification by trained users
reveals the posture. The curved lines are cubic splines as described in Fig
2-C.

##  2 Methods

###  2.1 Untwisting the embryo

Consider detected cell nucleus center positions across two sequential frames.
The displacement of the cell nucleus i from frame t-1 to t can be
approximately decomposed:

$\|\mathbf{x}^{(t)}_{i}-\mathbf{x}^{(t-1)}_{i}\|_{2}=\Delta
D^{(t)}_{i}\approx\Delta^{I}D^{(t)}_{i}+\Delta^{O}D^{(t)}_{i}$ (1)

where $\Delta^{I}D^{(t)}_{i}$ measures movement of nucleus i inside the embryo
and $\Delta^{O}D^{(t)}_{i}$ measures nuclear movement attributed to the embryo
repositioning between images. The untwisting process mitigates
$\Delta^{O}D^{(t)}_{i}$, movement attributed to the embryo moving. The
remaining term $\Delta^{I}D^{(t)}_{i}$ can be modeled with MOT methods.
However, the untwisting process is imperfect; the seam cells represent an
approximation of the coiled embryo.

Frame-to-frame displacement of cell i at time t: $\Delta D^{(t)}_{i}$ is
unpredictable due to the embryo repositioning between five minute images. The
seam cells are used to estimate complete cubic splines through the left and
right sides of the embryo [4]. The remapped coordinates (Fig. 2) mitigate the
displacement attributed to embryo movement $\Delta^{O}D^{(t)}_{i}$, leaving
only the inter-worm cell movement $\Delta^{I}D^{(t)}_{i}$. This movement can
be assumed to be Brownian between frames, allowing for the application of MOT
strategies to track “straightened” nuclei. Sample data of the 85 muscle nuclei
(Fig. 2-B) are used to evaluate the efficacy of automated tracking approaches.

###  2.2 Nuclei Detection

Detect-and-track MOT paradigms rely to finding objects in images then
associating unique objects between frames. Nuclei are detected in the sample
images and compared to accompanying annotated coordinate sets. Convolutional
neural networks (CNNs) achieve state of the art performance in image
segmentation tasks. Two approaches are evaluated in this exploration: a 3D
U-Net [14, 21] and Stardist 3D [18, 17]. The 3D U-Net achieves the best
results in image segmentation. Two 3D U-Net models are trained either from a
random initialization or from similar data and of differing sizes to achieve
voxel-level segmentations. The architecture itself remains constant, but the
number of filters in each layer varies according to each model. The $L$ model
has twice as many filters in each layer as $M$, which has twice as many
filters as $S$. 3D U-Net models are trained to maximize the dice coefficient
via the Adam optimization scheme [9]. Stardist 3D combines elements of a FCN
such as the 3D U-Net but with a focus on identifying disjoint objects. Two
configurations of Stardist 3D are evaluated; the first uses a pretrained model
trained on seam cell nuclei while the second is trained from a random
initialization.

###  2.3 Nuclei Tracking

The GNN filter serves as a canonical tool for MOT. The method describes a
linear program (LP) in which measurements are uniquely associated to tracks.
Distances between tracks and measurements are used to find a globally optimal
pairing between points of the two sets. The GNN is expressed as the linear
assignment problem (LAP). The assignment constraints enforce each track
$i=1,2,\dots,n$ being uniquely matched to one detection $j=1,2,\dots,n$. The
data association problem leverages the LAP to perform MOT.

Define
$\mathbf{Z}^{(t)}=[\mathbf{z}^{(t)}_{1},\mathbf{z}^{(t)}_{2},\dots,\mathbf{z}^{(t)}_{n}]$
as the states of each object $i=1,2,\dots,n$ at time $t=1,2,\dots,T$. State
$\mathbf{z}^{(t)}_{i}$ describes the center position of object $i$ at time
$t$. Similarly, define
$\mathbf{O}^{(t)}=[\mathbf{o}^{(t)}_{1},\mathbf{o}^{(t)}_{2},\dots,\mathbf{o}^{(t)}_{m^{(t)}}]$
as the set of measurements at frame $t$, indexed $j=1,2,\dots,m^{(t)}$,
$t=1,2,\dots,T$. The detection step generates sets
$\mathbf{O}^{(t)},t=1,2,\dots,T$, while the association step concerns using
the detections to update tracks $\mathbf{Z}^{(t)},t=1,2,\dots,T$.

The GNN cost matrix $\mathbf{C}\in R^{n\times(m{(t)}+n)}$ (Eq 2) specifies
costs for associating measurements $j=1,2,\dots,m^{(t)}$ to states
$i=1,2,\dots,n$. The matrix comprises two blocks of sizes $n\times m(t)$ and
$n\times n$. The first block measures the euclidean distance between track
$\mathbf{z}^{(t-1)}_{i}$ and detection $\mathbf{o}^{(t)}_{j}$ while entries in
the second block consists of costs of non association known as gates. Each
gate $d^{(t)}_{i}$ allows for track $i$ to receive no measurements at time
$t$. In the context of the GNN filter, the gate specifies a distance radius
about each track $\mathbf{z}_{i}^{(t-1)}$ in which measurements
$\mathbf{o}_{j}^{(t)}$ must reside in order to associate to track $i$.

$\mathbf{C}^{(t)}=\begin{bmatrix}\|\mathbf{z}_{1}^{(t-1)}-\mathbf{o}^{(t)}_{1}\|_{2}&\|\mathbf{z}_{1}^{(t-1)}-\mathbf{o}^{(t)}_{2}\|_{2}&...&\|\mathbf{z}_{1}^{(t-1)}-\mathbf{o}^{(t)}_{m(t)}\|_{2}&d_{1}&\infty&\infty&...&\infty\\\
\|\mathbf{z}_{2}^{(t-1)}-\mathbf{o}^{(t)}_{1}\|_{2}&\|\mathbf{z}_{2}^{(t-1)}-\mathbf{o}^{(t)}_{2}\|_{2}&...&\|\mathbf{z}_{2}^{(t-1)}-\mathbf{o}^{(t)}_{m(t)}\|_{2}&\infty&d_{1}&\infty&...&\infty\\\
\|\mathbf{z}_{3}^{(t-1)}-\mathbf{o}^{(t)}_{1}\|_{2}&\|\mathbf{z}_{3}^{(t-1)}-\mathbf{o}^{(t)}_{2}\|_{2}&...&\|\mathbf{z}_{3}^{(t-1)}-\mathbf{o}^{(t)}_{m(t)}\|_{2}&\infty&\infty&d_{3}&...&\infty\\\
...&...&...&...&\infty&\infty&\infty&...&\infty\\\
\|\mathbf{z}_{n}^{(t-1)}-\mathbf{o}^{(t)}_{1}\|_{2}&\|\mathbf{z}_{n}^{(t-1)}-\mathbf{o}^{(t)}_{2}\|_{2}&...&\|\mathbf{z}_{n}^{(t-1)}-\mathbf{o}^{(t)}_{m(t)}\|_{2}&\infty&\infty&\infty&...&d_{n}\\\
\end{bmatrix}$ (2)

The cost matrix C defines the GNN filter objective, while the LAP constraints
complete the optimization problem. The resulting linear program is then
solvable in polynomial time [10, 8], yielding globally optimal assignments
between tracks and measurements. The columns $m(t)+1,m(t)+2,\dots,m(t)+n$
correspond to a track receiving no detection update are included in the one-
to-one constraints.

min
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{m^{(t)}}\mathbf{C}^{(t)}_{ij}x_{ij}$
(3) s.t. $\displaystyle\sum_{i=1}^{n}x_{ij}=1\hskip 28.45274ptj=1,\dots
n+m^{(t)}$ $\displaystyle\sum_{j=1}^{m^{(t)}}x_{ij}=1\hskip
28.45274pti=1,2,\dots n$ $\displaystyle x_{ij}\in\\{0,1\\}$

The simplest paradigm for MOT is to solve sequential frame-to-frame GNN LPs
(Eq 3). The random motion within the embryo is amenable to the L2 norm cost;
however, the lack of trajectory to cellular motion invalidates a dynamical
model such as the Kalman Filter [6]. However, the coordinate remapping is
often imperfect and may shift nuclei from their true locations due to the
spline curves not perfectly adhering to the embryo body wall. A graph can
specify interactivity among adjacent nuclei, allowing for a stronger
representation of nuclear movement than is possible with the GNN [3, 20].
Graph matching may provide stronger results, but the computation required due
to the high number of muscle nuclei forces the use of heuristic algorithms.
Kernelized Graph Matching (KerGM) [19] is the most recent development in
heuristic graph matching and is applied to track straightened muscle nuclei.
While GNN based solutions have been adapted for object disappearance,
reappearance, merging, and splitting [7, 5], KerGM and such graphical methods
have not been adapted. Indeed, the methods can perform one-to-many or many-to-
one assignment, but the methods cannot handle the common scenario in which a
cell disappears in one image, and a different cell appears in the subsequent
image.

However if an LAP yields strong solutions, then evaluating multiple solutions
via a more complex cost could improve the tracking quality. Murty’s algorithm
allows for returning the $K$ best solutions to an LAP with complexity
$\mathcal{O}(Kn^{3})$ [13, 12]. The $K$ hypothesized solutions to the LAP can
then be evaluated on a quadratic cost. Evaluating a higher order cost is
computationally inexpensive compared to searching the entire space for the
assignment that minimizes the quadratic cost. If the LAP returns high quality
solutions, then sampling and evaluating at the QAP cost could improve
performance by further applying a more complex model to discriminate between
competing hypotheses.

Both KerGM and the evaluation of a graph cost in Murty’s algorithm require a
specified graph for each detection set. The choice in graph greatly influences
the results. A QAP with no edge-wise connections reduces to an LAP. On the
other hand, an overly dense edge set may not generalize well to unseen data.
Thus several types of graphs are evaluated. The first four graphs connect
nuclei in the straightened space that are withing $5\mu m$, $7.5\mu m$, $10\mu
m$, and $12.5\mu m$ respectively. Delaunay Triangulation is used as another
method of generating a graphical representation of the nuclei in the
straightened space at each image [2].

###  2.4 MIPAV Interative Tracking Plugin

Sequential detection and matching allows for accurate frame-to-frame tracking
of the straightened coordinates. However, the simple matching paradigm is
susceptible to large errors from false positives and false negatives. Near
perfect detections are necessary for an effective frame-to-frame matching
paradigm. The Untwisting plugin [4] for MIPAV [11] was augmented to perform
semi-automatic frame-to-frame tracking. The 3D rendering allows for
interactive point identification and the import and export of such data. An
initial detection set is superimposed on the volume in a 3D viewer. The user
is able to perfect the detection set by removing false detections, adding
missed nuclei, and splitting erroneously clumped nuclei. A first pass of the
matching model links detections in the next frame to named nuclei in the
prior. The user can then make corrections if necessary, and then resolve the
linear program problem given user verified matched nuclei being passed as
encoded constraints. This process can be repeated to completion in real time
due to the polynomial complexity of solving the GNN LP [8].

Fig 4 illustrates the semi-automated process. The seed volume detections are
assessed and nuclei are identified (step 1). Then, the sequential volume
detections are similarly assessed to yield an accurate detection set (step 2).
Steps 3A and 3B form a recursion whereby the user generates predicted
identities of the second volume nuclei (step 3A) and then edits the predicted
nuclear identities. Another round of matching with added constraints will
regenerate predictions (step 3B). The process is expedited with accurate
detections.

Figure 4: Iterative tuning helps overcome noisy detections. The semi-automatic
densely labelled nuclei tracking paradigm. Sets of detections in the second
fluorophore are remapped via the seam cell lattice. The initial seed
detections are identified (step 1). Then, the sequential frame detections are
simply edited to account for all nuclei in the image (step 2). The recursive
tracking procedure iteratively corrects the nuclear alignments in a manner
designed to minimize the manual burden.

##  3 Results

A set of $n=233$ images with densely labelled cell nuclei are manually
annotated to train a model to better segment smaller, closer cell nuclei. The
$n$ pairs of fluorescently imaged embryos and binary annotation masks are used
to fit segmentation models as described above.

Images from three embryos bred to illuminate the muscle cell nuclei were
annotated for evaluating the tracking methodology. The strain KP9305 targets
the four bands of muscle cell nuclei within the worm embryo, each band
composing approximately 21 muscle nuclei. The 85 nuclei are to the best
ability of the researchers identified in each image volume throughout
development for each of the three sampled worms. Nuclei centroids with
associated identity are provided for all expertly detected nuclei in each
image across the three worms. Figure 5 depicts XY maximum projections from two
sequential image volumes. Cell nuclei homogeneity and density contribute to
the tracking challenge.

Figure 5: Densely packed muscle cell nuclei are difficult to accurately
segment. XY maximum intensity projections of two sequential image volumes
depicting muscle cell nuclei. Tracking the nuclei is only feasible using seam
cells as fiducial markers to remap coordinates of detected nuclei.

###  3.1 Detection

Voxel-wise evaluation does not adequately measure the performance of nuclei
detection models. Each model and postprocessing combination yields a set of
detections to be compared to annotation volume nuclei sets. 3D U-Net models
[14, 21] are compared to Stardist 3D [18, 17]. Table 1 highlights nuclei
centroid matching results from processed image volumes in the held-out test
set. Stardist 3D is the most precise, but tends to miss more dim nuclei than
the 3D U-Net models. The nuclei that can appear extremely close vary in size
and intensity. Building an effective model for nuclear identification remains
a barrier to automatic cell tracking.

| Precision | Recall | F1
---|---|---|---
Stardist 3D | 0.81 | 0.63 | 0.68
3D U-Net - S | 0.64 | 0.67 | 0.65
3D U-Net - M | 0.69 | 0.77 | 0.72
3D U-Net - L | 0.67 | 0.75 | 0.70
Table 1: Dense nuclei detection is more challenging than seam cell nuclei
detection. The nuclei are smaller and tend to be more clustered upon each
other. Stardist 3D excels in this setting as it is the most precise model. All
models struggle to find all nuclei, evidenced by the low recall measure.

###  3.2 Tracking

Manually identified nuclei positions served as test data to evaluate and
compare methods. Expert annotators missed in this data, and will miss nuclei
in frames due to nuclear dimming or occlusion. Tracking results arise from
assuming expert level detections, i.e. that the detections are of the same
quality of the annotations. Frame-to-frame accuracy is reported by the
proportion of correct matches to total nuclei in the subsequent frame. Longer
term frame-to-frame tracking is prone to errors, and thus tracking is
currently assumed to be done in a semi-automated fashion in which a trained
user edits and verifies tracks in successive frames.

The data as presented features many frames with missing nuclei. The most
challenging of scenarios occurs when two sequential frames have nuclei not
present in each other. The augmented LAP is able to account for these cases by
matching candidates from each set to nothing. However, the methodology is not
designed for highly variant movement within a frame [7]. Nuclear movement
introduced by untwisting error, worm elongation, and general nuclear movement
are exacerbated by inaccurate detections. The GNN LP (Eq 3) was applied at
four varying $\mu m$ gates for nuclear movement: $10\mu m$, $15\mu m$, $20\mu
m$, $100\mu m$. Nuclear movement beyond each interval is a signal that a
nucleus has dimmed in the subsequent frame while a newly emerging detection
outside the cutoff is labelled a false positive.

Table 2 reports median matching accuracies for all gates over the test set.
The gated LAP median accuracies converge to the non-gated LAP accuracy as the
gate values increase. Varying the $\mu m$ cutoffs cause significant changes in
the median accuracy due to highly variant movement between frames. Certain
frame show low nuclear movement while others show larger elongation of the
worm. The elongation stretches coordinates along the body of the worm. Often
nuclei move large distances, but they do so together. A checked detection set
with an high cutoff would then allow the gated GNN LP to match nuclei despite
highly variant bouts of movement.

| Median | IQR
---|---|---
GNN | 0.952 | 0.120
GNN \- 10 $\mu m$ | 0.615 | 0.529
GNN \- 15 $\mu m$ | 0.702 | 0.564
GNN \- 20 $\mu m$ | 0.708 | 0.610
GNN \- 100 $\mu m$ | 0.952 | 0.145
Murty - $K$ 5 - 10 $\mu m$ | 0.609 | 0.505
Murty - $K$ 5 - 15 $\mu m$ | 0.690 | 0.569
Murty - $K$ 5 - 20 $\mu m$ | 0.708 | 0.604
Murty - $K$ 5 - 100 $\mu m$ | 0.952 | 0.134
Murty - $K$ 30 - 10 $\mu m$ | 0.609 | 0.505
Murty - $K$ 30 - 15 $\mu m$ | 0.701 | 0.554
Murty - $K$ 30 - 20 $\mu m$ | 0.704 | 0.597
Murty - $K$ 30 - 100 $\mu m$ | 0.952 | 0.135
Table 2: The augmented LAP allows for flexibility in handling imperfect
detections. However, a cardinal assumption of the method is that nuclear
movement is stationary [7]. This assumption does not hold due to the error
injected from the untwisting process. A semi-automated approach in which a
user corrects detections enables the GNN model effectively match nuclei in the
straightened space.

##  4 Discussion

Tracking in the remapped coordinate space using a simple GNN tracker
accurately tracks the majority of cell nuclei, but requires perfect
annotations to be effective. The sources of nuclear movement are challenging
to model accurately enough for reliable tracking despite the majority of
frame-to-frame displacement being explained by embryo repositioning. The high
variance in frame-to-frame displacement arises from error introduced by the
coordinate remapping process. The body wall approximation via splines
introduces systemic misplacement in some positions. The tracking approach and
interface were designed to be independent of the underlying MOT approach.
Future efforts to build stronger association models could be used in-place of
the GNN.

The burden thus ultimately falls on the detection process. The FCNs discussed
in this work achieved results that correctly identified most of the nuclei,
but are far from reliable in a fully automatic GNN tracker. The results point
to a semi-automated solution. First image volumes are processed to detect
nuclei. A trained user then remove debris, splits touching nuclei, and adds
annotations for missed nuclei. The GNN tracker works in realtime, allowing for
an refining approach to tracking nuclei. An initial pass will tend to
correctly identify all nuclei from the prior frame. However, simply correcting
any errors and rerunning while anchoring the correct predictions iteratively
will produce correct frame-to-frame associations. This process applied
recursively throughout an imaged worm drastically reduces both the manual
effort and the time spent to track all densely labelled nuclei in a strain.

##  5 Acknowledgements

Post-Baccalaureate research fellows Gabi Schwartz and Daniel Del Toro-Pedrosa
were assisted by testing and giving feedback on the tracking interface.
Alexandra Bokinksy was instrumental for the integration of the tracking
backend into the MIPAV Untwisting plugin. This research is supported by the
Intramural Research Program within National Institute of Biomedical Imaging
and Bioengineering at the National Institutes of Health. Andrew Lauziere’s
contribution to this research was supported in part by NSF award DGE-1632976.
This work used the computational resources of the NIH HPC Biowulf cluster.
(http://hpc.nih.gov).

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