Many aspects of animal morphogenesis are thought to be controlled by positional information (Wolpert, 1969)⁠, where cells can sense their position in a developing organ and respond accordingly. This phenomenon may be even more pervasive in plants as cells cannot relocate within organs and must decide their fate based on their location. For example, root morphogenesis appears to be controlled by an organizing center at the root tip that provides founder cells and positional information to the growing structure (Scheres et al., 2002)⁠. Ablation of cortical cell initials in the root meristem causes the neighboring pericycle cells to divide and fill the available space, subsequently adopting the fate associated to their new location (van den Berg et al., 1995)⁠. A similar effect has been demonstrated for a variety of cell types in the Arabidopsis root (Marhava et al., 2019)⁠. In leaves, development is thought to be coordinated by polarity fields oriented from leaf base to tip (Kierzkowski et al., 2019; Kuchen et al., 2012)⁠. Over time organs can initiate new growth axes, such as when serrations or leaflets develop in more complex leaves (Barkoulas et al., 2008; Kierzkowski et al., 2019)⁠, or lateral roots emerge from the primary root (Scheres et al., 2002)⁠. In these cases, information from several organizers must be integrated to direct cell response.

To understand how positional information controls morphogenesis, it is necessary to quantify cell shape, gene expression, and morphogen concentration changes over time, preferably at the cellular level. This information then needs to be related to its position relative to the organizers controlling development within the organ. As computational power and imaging methods improve, new software packages for cell segmentation and lineage tracking are being developed (Sommer et al., 2011; Stegmaier et al., 2016⁠), including many specialized for plants (Barbier de Reuille et al., 2015; Eschweiler et al., 2019; Fernandez et al., 2010; Schmidt et al., 2014; Wolny et al., 2020)⁠. This progress has enabled the segmentation of time-lapse data at increasingly higher resolution and throughput (Hervieux et al., 2016; Kierzkowski et al., 2019; Sapala et al., 2018; Willis et al., 2016)⁠. Although this increase in data volume offers tremendous potential to understand how genes control form, the analysis of geometric data from thousands of cells is nontrivial. Information about a cell’s shape, gene expression, and growth directions is of limited value when the cell’s spatial context within the developing organ is unknown.

MorphoGraphX is a computer software platform that is specialized for image processing on surface layers of cells (Barbier de Reuille et al., 2015)⁠. It has proven especially useful for the analysis of confocal microscopy images from time-lapse data in order to quantify the cellular-level dynamics of growth, cell division, and gene expression (e.g., Bringmann and Bergmann, 2017; Feng et al., 2018; Hervieux et al., 2016; Hong et al., 2016; Kierzkowski et al., 2019; Louveaux et al., 2016; Sapala et al., 2018; Scheuring et al., 2016; Tsugawa et al., 2017; Vlad et al., 2014; Zhang et al., 2020; Zhu et al., 2020; Fridman et al., 2021)⁠. Key to the approach taken in the software is the representation of cell layers as curved, triangulated surface meshes that capture the overall 3D shape of organs, which retains much of the simplicity of 2D segmentation and lineage tracking. These ‘2.5D’ images contain the geometry of the sample at two scales. The global shape of the organ is captured by the mesh’s geometry, while a cellular-scale representation is obtained from the confocal signal projected onto the mesh, which is segmented to extract the shape of individual cells on the surface (Figure 1A–C). When combined with time-lapse data acquisition and cell lineage tracking, MorphoGraphX allows cell growth and its relationship to gene expression to be quantified (Figure 1D and E; Kierzkowski et al., 2019; Sapala et al., 2018; Vlad et al., 2014)⁠. In addition to cell surface analysis, MorphoGraphX also supports the creation and analysis of full 3D meshes with volumetric cells (Figure 1—figure supplement 1; Vijayan et al., 2021)⁠. Here, we describe new methods we have developed in MorphoGraphX to understand these data by additionally annotating cells with positional information. Not unlike the annotation of sequence data, this allows cellular data to be given spatial context, and a frame of reference within the organ relative to its developmental axes and the organizers instructing morphogenesis.

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Most workflows in MorphoGraphX begin by converting 3D image stacks into meshes of 2.5D or 3D cellular segmentations, which are created directly on voxels in the case of 3D segmentation, or on surface meshes in the case of 2.5D data (Figure 1A–C, Figure 1—figure supplement 1A and B). Recent advances in voxel classification with convolutional neural networks (CNNs) for cell boundary prediction can improve input images and the resulting segmentation (Eschweiler et al., 2019; Eschweiler et al., 2021; Wolny et al., 2020)⁠, especially for 3D segmentations. Although a selection of these and other image denoising and preprocessing tools are available directly within MorphoGraphX, it is also possible to preprocess and/or segment 3D images in other software and import them into MorphoGraphX for further analysis. After the initial segmentation, cellular features can be quantified, such as cell area, shape, and gene expression for single time points, or cell proliferation and growth for time-lapse data (Figure 1D and E). These data can then be annotated with positional information to aid the understanding of the development of the organism under study. Even with deep learning techniques, image quality needs to be very high for full 3D segmentation, and this is often not possible with live imaged data. By enabling image processing on 2.5D surface images, MorphoGraphX can be used in many systems on live imaged data where full 3D segmentation is currently not possible (Hervieux et al., 2016; Kierzkowski et al., 2019; Sapala et al., 2018; Silveira et al., 2022; Vlad et al., 2014)⁠. Annotation with coordinate systems can further reduce image quality requirements in cases where growth along a single dimension is required (Liu et al., 2022)⁠.

Defining coordinates within an organ

The simplest method to provide positional information for the cells in a sample is by aligning the sample with a set of 3D coordinate axes (Figure 2A). For example, a developing root meristem can be aligned and positioned such that the organizing quiescent center is at the origin with the Y-axis increasing in the longitudinal direction of the root. Provided the sample is reasonably straight, this allows cellular measures to be compared with their distance from the quiescent center (Figure 2B).

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However, for curved organs significant errors will occur, especially in more distal regions, further from the origin. In this case, the central axis can be defined by a curved line that conforms to the curvature of the organ (Figure 2C; Schmidt et al., 2014; Montenegro-Johnson et al., 2015)⁠. In MorphoGraphX, this line can be represented by a Bezier spline (Bézier, 1968)⁠ with control points positioned using either interactive manipulation or automatically from a selected file of cells. Distance can then be calculated along the line and transferred to cells in the cross section perpendicular to the line (Figure 2C). MorphoGraphX also allows a 2D Bezier surface to be positioned next to or within a sample, enabling two directions to be aligned with the natural curvature of the sample.

Placing the Bezier curve or surface to curved organs with more complex shape is challenging. An alternative method is to select one or more cells at a reference position and calculate the distance relative to the selection (Figure 2D, Video 1). This offers an easy method to create a distance field and greatly increases the variety of organs that can be accommodated. The distance is determined by computing the shortest path along cells through the tissue, causing it to naturally follow the curvature of the organ.

Video 1

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Once cells have been annotated with positional information, it can be used for the analysis of cell-level data, such as growth, cell proliferation, cell shape, and gene expression. Using the distance measure to define the proximal-distal (PD) axis on an Arabidopsis sepal (Figure 2D), geometric measures can be plotted against the local coordinate system. In Figure 2F, cell area extension was plotted against distance from the base of the sepal. On the full 7-day sepal time-lapse shown in Figure 2—figure supplement 1 (Hervieux et al., 2016)⁠, initially growth is predominantly located at the distal parts of the organ, followed by the progressive displacement of the high growth zone towards the base of the sepal (Figure 2—figure supplement 1A). By time point 6, the growth has slowed and become more uniform as the organ differentiates. Proliferation is initially more uniform, but otherwise follows a pattern similar to growth, progressing basally as the organ matures (Figure 2—figure supplement 1B). The data can be indexed by position from the base of the sepal and plotted, showing how growth and proliferation vary along the PD developmental axis as a percentage of the total sepal length (Figure 2—figure supplement 1D and E). It can be seen in the graphs that although the sepal appears to undergo a similar growth arrest starting at the tip as the Arabidopsis leaf, there are subtle differences. The growth in the early stages is more distal in the sepal, along with the proliferation, and the zone of higher growth moves towards the base as the sepal develops. This is in contrast to the Arabidopsis leaf where the growth and proliferation zones remain relatively fixed with respect to the total leaf length (Figure 1K and M; Kierzkowski et al., 2019)⁠. The use of relative coordinates also makes it possible to pool data from multiple samples (Vijayan et al., 2021; Zhang et al., 2020)⁠ and compare data from different genotypes (Kierzkowski et al., 2019; Montenegro-Johnson et al., 2019; Zhang et al., 2020)⁠.

Deriving directions from organ coordinates

In addition to scalar information such as areal growth rate or cell volume, MorphoGraphX can also quantify directional information, such as the principal directions of growth (PDGs) that represent the maximal and minimal directions of growth for each cell (Figure 3A–C). A common problem with the interpretation of such directional information is the tendency for directions to be locally heterogeneous when growth is nearly isotropic. This happens because the maximal and minimal growth amounts are almost the same, and the displayed directions become arbitrary and heavily influenced by noise. This can make the comparison of growth directions between neighboring cells difficult. A more informative approach is to look at growth with respect to the directions of the developmental axes. This can be done by first setting up an axis defining the positional information for the leaf, for example, by using the previously introduced distance field (Figure 2D, Figure 3B). The growth directions are then projected onto this developmental axis and separated into components that are parallel (Figure 3D) and perpendicular (Figure 3E) to the axis. Using this method on the Arabidopsis thaliana leaf primordium different developmental zones with varying growth rates along the PD and medial-lateral (ML) axis can be revealed (Figure 3C–E): while the area extension differs greatly in midrib/petiole (low) and leaf blade cells (high), it can be seen that those differences mainly follow from the ML growth rate, whereas the PD growth map is more similar in these domains. Moreover, an increase in ML growth around the forming serration can be seen, separating it from the surrounding leaf blade cells that show less growth along this direction (Figure 3E). From the original PDG visualization, it is not immediately apparent that the varying ML growth is the main cause of the differences in the domains (Figure 3C).

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Another benefit of looking at PDGs in the context of a local coordinate system is that it can provide a more direct comparison to the outputs of computational simulations. Developmental models of emergent organ shape often use morphogens that are thought to specifically control growth in parallel and perpendicular to a developmental axis (Kierzkowski et al., 2019; Kuchen et al., 2012; Whitewoods et al., 2020)⁠. By projecting the PDGs onto this axis, it is possible to directly compare model growth rates in the different directions to experiments. Since MorphoGraphX can load a wide variety of mesh formats, this allows the direct comparison of similar quantifications made on templates extracted from model simulations from various sources.

Figure 3F–K shows a similar growth quantification for the tomato meristem (Kierzkowski et al., 2012)⁠, where local organ coordinates were created by using cell distance measures around each emerging leaf primordium with directions pointing towards (radial) and around (circumferential) their respective center. In addition to growth, the signal intensity of the auxin reporter DR5 was quantified in the same sample, allowing a direct comparison to auxin signaling levels and cellular growth. For both primordia, we found radial growth to have a high negative correlation with DR5 signal intensity. Figure 3J shows that the radial growth (red) peaks on the abaxial side of the emerging primordium, whereas the DR5 signal (blue) is higher on the adaxial side. Circumferential growth was more or less constant. The DR5 maximum tends to be on the adaxial side of the initiating leaf, whereas growth is much higher on the opposing abaxial side. This supports the idea that auxin acts as a trigger for primordium initiation (Reinhardt et al., 2003; Smith et al., 2006)⁠ rather than via controlling growth rates directly.

Combining directions

In 2D or on 2.5D surfaces, local directions can be fully defined by a single distance measure by taking one direction aligned with the gradient of the distance field or a Bezier curve, and the other perpendicular to the first. This is similar to the methods used to specify directions in developmental modeling in plants (Green et al., 2010; Kennaway and Coen, 2019; Kierzkowski et al., 2019; Kuchen et al., 2012; Whitewoods et al., 2020)⁠, and thus facilitates direct comparison between models and experimentally observed patterns of growth and gene expression.

In 3D, a third direction must be defined (Kennaway and Coen, 2019; Whitewoods et al., 2020)⁠. In MorphoGraphX, this can be done by combining the directions defined by different distance measures. The 3D Cell Atlas add-on (Montenegro-Johnson et al., 2015)⁠ combines several distance measures for radially symmetric structures such as root and hypocotyls. A Bezier curve is placed along the center in the longitudinal direction and combined with a surface mesh to obtain radial directions (Figure 4B). This also puts bounds on the radial direction (and also implicitly on the circumferential direction), which allows relative coordinates to be assigned to cells in addition to absolute values. The relative radial coordinate will follow the layer as the root narrows towards the tip. The relative distance between the central axis and the organ surface can then be used to annotate and classify 3D segmented cells in organs with a layered cellular organization. Figure 4D shows classification of layers with relative coordinates on an Arabidopsis root (Montenegro-Johnson et al., 2015)⁠. Figure 4C shows layer classification using absolute distance from a surface mesh (Montenegro-Johnson et al., 2019)⁠, which can be used as a starting point for layer classification in any organ.

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The mature ovule in Arabidopsis shows a more complicated structure than a root or sepal, with five layers of integument cells encapsulating the nucellus that contains the embryo sac (Schneitz et al., 1995; Vijayan et al., 2021⁠; Figure 1—figure supplement 1, Figure 5). In this example, combining different directions allows the establishment of organ-centric coordinates in the outermost layer of cells in the ovule (the outer integument). After segmentation and 3D mesh extraction in MorphoGraphX, directions normal to the surface (Figure 5A) were combined with those of a Bezier curve computed from a user-selected cell file (Figure 5B and C). Similar to the root data, relative coordinates facilitate the classification of cells into layers.

Figure 5

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The Bezier curve defined the longitudinal direction, whereas the surface directions obtained from the organ surface mesh were used to compute perpendicular width and depth axes and distances. Cell volume and geometry acquired from 3D segmentation and mesh extraction (Figure 1—figure supplement 1) were calculated along the various directions of the organ axes and analyzed (Figure 5D–G).

Moving along the PD axis from 0 to 260 μm, we found variations in cell volume with a clear minimum at around 100 μm and a steady increase towards proximal and distal end (Figure 5G). Measuring the length, width, and depth of cells separately revealed the underlying cause for differences in cell volume between different PD regions of the outermost integument layer. At the proximal end (at 0 μm), the cell shape is relatively isotropic with similar values in cell length, width, and depth. Moving along the PD axis, cell anisotropy slowly increased, first mostly due the decreasing depth and width (until around 100 μm), later mainly due to the increased cell length (from about 150 μm). The increased cell length suggests a highly anisotropic growth along the longitudinal axis in this area. A potential proliferative region could be suspected in the region between 100 and 150 μm, where cell volume and length are the smallest. The quantification using the organ coordinates in this study allows spatial information to be linked to 3D cellular properties such as cell volumes and the associated shape anisotropies along different cell axes.

Using positional information for automatic cell-type classification

Plant organs typically emerge as primordia consisting of undifferentiated tissue. Cells subsequently differentiate, acquiring a unique identity that depends on their location within the organ, via genetic processes that integrate spatial and environmental cues. Although cell differentiation is ultimately controlled by differential gene expression, it is often the case that cell fate can be predicted by geometry, even at very early stages (Yoshida et al., 2014)⁠. It is rare that cells with different cell types have identical morphological features.

MorphoGraphX supports a large variety of measures to quantify different features of cell morphology, including simple geometric measures (area, volume, perimeter, surface area, min and max axis), shape quantifiers (convexity, circularity, lobeyness, largest empty circle, aspect ratio), neighborhood measures (number of neighbors, variability), gene expression (average, total, near boundary), and cell network measures (betweenness centrality, betweenness current flow). Most measures can be used on time-lapse data to quantify changes over time (growth rates, gene expression changes, cell proliferation). For a complete list of the measures implemented in MorphoGraphX, see Table 1 and Table 2. The modular architecture of the software also allows custom measures tailored to specific problems to be easily added through its plug-in interface. More sophisticated calculations, for example, the averaging of data over multiple samples, can be calculated externally in packages such as R and imported back into MorphoGraphX for visualization on segmented meshes. Here, the development of more complex data-flows is enabled by the use of a standardized attribute system to store and visualize cell data for both scalar values and tensor (directional) information.

Table 1

Measure	Unit	Description
Geometry
Area	µm2	Area of the cell (sum of its triangle area)
Aspect ratio	-	Ratio of length major axis and length minor axis (see below)
Average radius	µm	Average distance from the center of gravity of a cell to its border
Junction distance	µm	Max or min distance between neighboring junctions of a cell
Length major axis	µm	Length of the major axis of the 2D shape analysis (computes a PCA on the triangle positions weighted by their area)
Length minor axis	µm	Length of the minor axis of the 2D shape analysis (computes a PCA on the triangle positions weighted by their area)
Maximum radius	µm	Maximum distance from the cell center to its border
Minimum radius	µm	Minimum distance from the cell center to its border
Perimeter	µm	Sum of the length of the border segments of a cell
Lobeyness
Circularity	-	Perimeter^2/(4*PI*Area)
Lobeyness	-	Ratio of cell perimeter to convex hull perimeter (1 for convex shapes)
Rectangularity	-	Ratio of cell area to the area of the smallest rectangle that can contain the cell (1 for rectangular shapes, lower values for irregular shapes)
Solidarity	-	Ratio of the convex hull area to the cell area (1 for convex shapes, higher values for complicated shapes)
Visibility pavement	-	1-(visibility stomata) (see below)
Visibility stomata	-	Estimate of visibility in the cell 1 for convex shapes, decreases with the complexity of the contour
Location
Bezier coord	µm	Associated Bezier coordinate of a cell Requires a Bezier grid
Bezier line coord	µm	Associated Bezier coordinate of a cell Requires a Bezier curve
Cell coordinate	µm	Cartesian coordinate of a cell
Cell distance	µm/cells	Distance to the nearest selected cell (finds the shortest path to a selected cell through the cell connectivity graph, edge weights: Euclidean, cell number or 1/wall area)
Distance to Bezier	µm	Euclidean distance to the Bezier curve or grid
Distance to mesh	µm	Euclidean distance to the nearest vertex in the other mesh
Major axis theta	°	Angle between the long axis of the cell and a reference direction
Polar coord	°/µm	Polar coordinate around a specified Cartesian axis
Network
Neighbors	Count	Number of neighbors of a cell
Betweenness centrality	-	Computes the betweenness centrality of the cell connectivity graph Edges can be weighted by the length of the shared wall between neighboring cells
Betweenness current flow	-	Computes the betweenness current flow of the cell connectivity graph Edges can be weighted by the length of the shared wall between neighboring cells
Signal
Signal border		Average amount of border signal in a cell
Signal interior		Average amount of interior signal in a cell
Signal parameters		Advanced and general process that allows the setting of parameters to compute different kinds of signal quantifications
Signal total		Average amount of total (=border + interior) signal in a cell
Other measure processes
Mesh/lineage tracking/heat map proliferation	Cells	Proliferation
Mesh/cell axis/custom/custom direction angle	°	Angle between a cell axis and a custom axis
Mesh/division analysis/compute division plane angles	°	Angle between division planes and/or cell axes

1. PCA: principal component analysis.

Table 2

Measure	Unit	Description
Cell atlas
Cell length (circumferential, radial, longitudinal)	µm	Cell length as determined by 3D Cell Atlas root (shoot rays from the cell center to the side walls to measure cell size along organ-centric directions)
Coord (circumferential, radial, longitudinal)		3D organ coordinates as determined by 3D Cell Atlas root
Geometry
Cell length (custom, X, Y, Z)	µm	Cell length along the specified direction (cell size is measured as in 3D Cell Atlas root [see above])
Cell wall area	µm2	Total area of the cell wall
Cell volume	µm3	Volume of the cell
Outside wall area	µm2	Cell wall area that is not shared with another neighboring cell
Outside wall area ratio	%	Proportion of cell wall area that is not shared with a neighbor cell
Location
Bezier coord	µm	Associated Bezier coordinate of a cell Requires a Bezier curve or grid
Cell coordinate	µm	Cartesian coordinate of a cell
Cell distance	µm/cells	Distance to the nearest selected cell (finds the shortest path to a selected cell through the cell connectivity graph, edge weights: Euclidean, cell number or 1/wall area)
Distance to Bezier	µm	Euclidean distance to the Bezier curve or grid
Mesh distance	µm	Euclidean distance to the nearest vertex in the other mesh
Network
Neighbors	Count	Number of neighbors of a cell
Betweenness centrality	-	Computes the betweenness centrality of the cell connectivity graph Edges can be weighted by the length of the shared wall between neighboring cells
Betweenness current flow	-	Computes the betweenness current flow of the cell connectivity graph Edges can be weighted by the length of the shared wall between neighboring cells

During the segmentation process, MorphoGraphX assigns a unique label to each cell. When tracking cell lineages over multiple time points, a secondary label called the ‘parent label’ is provided. Other secondary labelings are also possible, for example, for cell type, cell layer, or zones within an organ. MorphoGraphX has several methods to assign these labels to classify cell types and layers. These labels can be assigned manually by interactively selecting cells or by employing a number of processes that use heat map measure data to assign secondary labels (Figure 4E–H). Positional information from the distance measures can be combined with measures of cell morphology and gene expression, where a secondary labeling can be used to provide additional context.

Since cells of a common biological cell type have similarities in one or more geometrical, positional, or gene expression attributes, the values of these attributes will often form a cluster, facilitating their automatic classification. An example can be seen in the 3D Cell Atlas add-on for MorphoGraphX (Figure 4—figure supplement 1A–C, Montenegro-Johnson et al., 2015)⁠ that clusters cells by relative radial distance and cell size to classify the cell layers of the root, hypocotyl, mature embryo, or other radially symmetric plant organs. To aid in optimizing cell clustering, MorphoGraphX offers a 2D interactive heat map, where information from two independent measures can be visualized, and clusters selected (Figure 4G, Figure 4—figure supplement 1B). These methods can be used repeatedly on subsets of cells to enable a classification of cells that differ across more than two features.

Multifeature classifications tasks can be solved automatically by machine learning approaches (Cortes and Vapnik, 1995)⁠ when provided with sufficient training data. Of particular relevance are support vector machines (SVMs) that have been used to classify cell types based on geometrical features of plant cells (de Reuille and Ragni, 2017; Sankar et al., 2014)⁠. MorphoGraphX provides a simple interface to the libSVM SVM library (Chang and Lin, 2011)⁠. Cells can be selected and classified into different cell types for use as training data (Figure 4H, Video 2). Any cell attribute or measure that can be quantified in MorphoGraphX can be used by the classifier. These include all the morphological and gene expression measures, time-lapse measures, positional information created via distance maps or other coordinate systems, and even custom measures created via plug-ins or calculated externally with R or MATLAB. Once trained on a small group of cells with the desired measures, the classifier can be used to classify all the cells in a sample (Figure 4H). After manual curation, the classification can then be used as additional training data, improving the power of the classifier. Figure 4G and H and Figure 4—figure supplement 1F and G show the cell types of the Arabidopsis gynoecium, which consists of the lateral valve tissues that are fused to the replum. Within the valve, stomata are homogeneously distributed, consistent with the uniform growth and differentiation of this tissue. In contrast, no stomata cells can be found within the replum, which is made up of smaller, more homogenously sized cells. Cell typing this organ can be useful to identify the region of the valve margin, where the fully matured fruit will dehisce to release the seeds (Eldridge et al., 2016; Ripoll et al., 2019)⁠.

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Mapping positional information through time

In the analysis of morphogenesis, many key quantifications such as growth depend on the ability to track samples through time. In MorphoGraphX, this can be done following cell segmentation by manually assigning parent labels to the second time point, a process that has been highly streamlined in the user interface for 2.5D surfaces. However for full 3D samples, or large 2.5D samples with multiple time points, this method can be cumbersome. One method to address this problem is to find a nonlinear coordinate transformation or deformation that maps all the points from one time point onto the next. Parent labeling can then be determined by mapping cell centers of the later time point to an earlier one, allowing the cell they came from to be identified. This can be used to directly assign lineage or to seed algorithms that use more involved methods, such as the minimization of the total distances between the mapped cells (Fernandez et al., 2010)⁠. In MorphoGraphX, to define a mapping between the meshes of two successive time points (Figure 6A), we have implemented a 3D deformation function based on scattered data point interpolation (using cubic radial-basis functions; Duchon, 1977; Turk and O’Brien, 1999)⁠. An initial transformation is computed based on a few preassigned landmarks by matching several cells with their parents in the previous time point (Figure 6C). A deformation field is then calculated that provides a mapping for all points in 3D. This is then used to assign parent labels in the second time point based on their closest match in the previous time point. Close to the landmark points, this mapping will be very accurate, with accuracy decreasing with distance from the landmarks. The decrease in accuracy away from landmarks is larger if the deformation between the time points is highly nonuniform. After assigning all the cells to their closest parent, the mapping is then verified by checking the correspondence of neighborhoods between each cell and its parent. The labeling for cells that do not match is cleared, and the process is repeated. This causes correctly labeled regions to ‘grow’ out from the initially placed landmarks until the entire sample is correctly labeled (Figure 6C, Video 3). At each step, only correctly labeled cells remain. Sometimes the iterative cell labeling process can get stuck in highly proliferative areas where cells have divided repeatedly between time points. In this case, a few additional landmarks can be manually added at trouble spots. One significant advantage of the method is that incorrect cells remain unlabeled, making manual curation straightforward. Once all of the parents are assigned and have passed the neighborhood correspondence check, one can be assured that both the lineage and the underlying segmentations are correct.

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Deformation functions can also be used to create animations of organ development from 2.5D or 3D time-lapse data. This requires two or more time points of a segmented mesh with corresponding cell lineages. The cell centers and/or junctions are used as the landmarks defining the deformation function that maps one mesh onto the other. Interpolating mesh vertices between stages creates a smooth animation with as many intermediary steps as desired (Figure 6—figure supplement 1, Video 4). MorphoGraphX has a user-friendly pipeline to record animations directly from the GUI with options to adjust the camera angle and visualize cell lineages, heat maps, and cell outlines during the animation. Temporal smoothing of morphing animations created from more than two time points is achieved using Catmull–Rom splines to interpolate the position of mesh vertices over time (Catmull and Rom, 1974)⁠. Heat and signal values in the mesh, such as cell area, growth rates, or gene expression, can also be interpolated along with vertex positions.

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In large cells, growth can vary significantly within the same cell (Armour et al., 2015; Elsner et al., 2018)⁠. As the deformation function provides a smooth mapping between time points, its gradient can be used to create a continuous growth map at any point on a mesh. This enables the approximation of areal expansion and PDGs at a subcellular level, where the quality of the approximation is limited by the number and placement of landmarks (junctions). It is also possible to apply the process to subcellular landmarks, such as those obtained by tracking microbeads, as done previously for 2D images (Armour et al., 2015; Elsner et al., 2018)⁠. Our 3D implementation of this method has been used to compute growth directions on curved surface meshes (Figure 6E) and volumetric meshes (Figure 7I).

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A comparison of the areal growth and PDGs calculated with deformation functions vs. the cell-based method is shown in Figure 6D and E. The deformation function captures differences in growth within single cells, as is often apparent in larger cells that straddle areas of fast and slow growth. Figure 7A–I shows a 3D time lapse of the Arabidopsis root where the deformation functions have been used to perform lineage tracking in 3D. It can be seen in Figure 7D that the four tissue types – epidermis, cortex, endodermis, and pericyle – all show a similar pattern of cell volume increase, with slow growth in the meristem and faster growth near the transition zone, which can also be seen when displayed as a function of the distance from the root tip (Figure 7E). Most of the volume growth occurs along the longitudinal direction and is reflected by the increase in cell length, which is highly synchronized due to the physical connections of the cell layers (Figure 7F). Growth along width and depth directions is less synchronized, but much smaller than the length increase, reflecting the strong anisotropic growth of this system (Fridman et al., 2021)⁠.

Advanced geometric analysis

In addition to tools for creating organ coordinates and deformation functions, we have implemented a range of additional new processes in MorphoGraphX 2.0 for advanced image analysis.

3D lineage tracking and growth analysis

While MorphoGraphX was created to work with 2.5D surface projections, it now supports a complete set of tools for full 3D image processing, and in many samples advantages can be gained from combining both techniques. One example is automated cell lineage tracking, originally implemented for surfaces, which has now been extended to facilitate growth analysis in full 3D. Lineage tracking in 3D is a much harder task than on 2.5D surface images as 3D cellular meshes lack the cellular junctions that serve as material points for surfaces. However, in many cases, entire organs are well defined by their surfaces meshes, allowing landmarks on the surface to be used to construct a 3D deformation function to aid lineage tracking in full 3D. Surface landmarks can also be combined with 3D cell centers and/or face centers as material points to improve the internal resolution of the deformation functions for full 3D samples. These techniques allow the methods used to calculate growth rates and PDGs in 2.5D to be extended to full 3D (Figure 7; Fridman et al., 2021)⁠. Cell proliferation and most of the other measures can also be quantified from 3D time-lapse data (Figure 7—figure supplement 1). In addition to the automated tools, improved manual 3D parent labeling and the ability to relabel cells so that adjacent cells are always a different color aid in the manual curation of 3D lineage maps.

Cell division analysis

One of the more advanced quantifications from time-lapse data is the analysis of cell divisions. As plant cells cannot move, cell division and growth are the main determinants of morphogenesis. MorphoGraphX has processes to identify dividing cells from time-lapse data and quantify the orientation of the division wall in both 2.5D and 3D (Figure 8A–I, Figure 8—figure supplement 1). In 2.5D, the best-fit line to the division wall is calculated (Figure 8A and D), whereas in 3D the best-fit plane is used (Figure 8B, C and G). There are also measures to determine the asymmetry of the daughter cells. The use of positional information to give organ context is especially important for directional information, such as the orientation of cell division. Quantifying the orientation of the division plane is of little use without knowing how it relates to the developmental axes. Orientations can be computed with respect to the axes of a local coordinate system defined for the organ, along with its associated positional information (Figure 8E and I, Figure 8—figure supplement 1A). It is also possible to quantify how close cell divisions are to common division rules proposed in the literature, such as the shortest wall through the center of the cell including local minima (Figure 8H; Besson and Dumais, 2011; Yoshida et al., 2014; Vaddepalli et al., 2021) along the PDG (Hejnowicz, 2014)⁠, or rules based on network measures (Jackson et al., 2019)⁠.

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Cell connectivity networks

The organization of cells in organs may be analyzed through the extraction of cell connectivity networks from 2.5D or 3D segmented data. The physical associations between cells (cell-cell wall areas) can be extracted and converted into networks where they are analyzed using network measures (Figure 8J–K). Local measures such as the number of immediate neighbors (degree) can be calculated, along with more global measures, such as betweenness centrality based on the number of shortest paths cells lie upon, or random walk centrality (Figure 8K). These global measures are central to understanding how information flows within tissues (Jackson et al., 2017; Jackson et al., 2019)⁠. The use of these measures uncovered the presence of a global property in cellular organization within the Arabidopsis shoot apical meristem (SAM) (Jackson et al., 2019)⁠. Namely, the length of paths between cells is maximized, whereby cells that lie upon more shortest paths have a great propensity to divide, and the orientation of this division tends to leave the two daughter cells on the fewest number of shortest paths. Using this approach, the local geometric properties of cells can be related to the emergent global organization of cellular arrangements. Perturbation of cell shape in the katanin1 mutant led to alterations in path length in the SAM, which correlated with defects in phyllotaxis (Jackson et al., 2019)⁠.

Cell polarity

Cellular signals, such as proteins tagged with fluorescent reporters, can also be quantified in MorphoGraphX. After segmenting a surface into cells by using a cell wall stain or marker line, a signal collected in a second channel can be projected onto the surface mesh, and the abundance, orientation, and polarity of signals can be computed. Examples are the PIN-FORMED (PIN) auxin transporter report line (Benková et al., 2003)⁠ or the GFP:MBD (Van Bruaene et al., 2004)⁠ line that tags microtubules (MTs) (Figure 8L–N, Figure 8—figure supplement 2). For the quantification of cell polarity, MorphoGraphX implements a process where the projected signal along the cell border is binned based on its position in relation to the cell center to obtain its predominant direction and its intensity. Figure 8L shows an example of the PIN1 polarity quantification at the cell wall of surface segmented cells in the SAM. A similar quantification can be performed for 3D meshes as shown in Figure 8N and Figure 8—figure supplement 2A and B, where we computed the PIN2 polarity in epidermis and cortex cells of an Arabidopsis root. Again, this directional information can be combined with the organ coordinates to compute the angle between cell polarity and the organ axis (Figure 8—figure supplement 2C). Another example is the quantification of MT alignment using our implementation of FibrilTool (Boudaoud et al., 2014)⁠ that has been adapted for processing on surfaces. After projecting the MT signal onto the surface, the alignment direction and strength of the signal can be quantified at the subcellular level (Figure 8M) or for entire cells (Figure 8—figure supplement 2E). Again, this information can be interpreted using organ coordinates as we demonstrate on cells of a SAM that tend to have their MTs aligned circumferentially from the meristem center (Figure 8—figure supplement 2E,F).

3D visualization and interactive tools

MorphoGraphX has a flexible rendering engine that can handle meshes containing millions of triangles. It supports the independent rotation and translation of different stacks and meshes in the same world space and the ability to render both voxel and geometric data together with blending and transparency. It has adjustable clipping plane pairs and a bendable Bezier cutting surface that can be used to look inside 3D samples, and an interface to support the creation of animations (Video 5). However, visualizing and interacting with 3D data on a 2D computer screen still remains a challenge. A particular problem is the validation and correction of 3D segmentations of organs as internal cells are obscured by outer layers. 3D segmentation curation and correction is a bottleneck when developing training sets for deep learning tools, and MorphoGraphX has become a useful tool for this purpose (Wolny et al., 2020)⁠. In addition to clipping planes, exploded views are a commonly used method to visualize the internal structure of multicomponent 3D objects (Li et al., 2008)⁠, which can be used on mesh data in MorphoGraphX. These are particularly useful for visualizing the internal structure of entire organs with small cell numbers such as embryos (Figure 8—figure supplement 1C) and can be used in combination with multiple clipping planes for larger samples. To add biological meaning to the exploded visualization, cells can be bundled by their parent or cell-type label to visualize key aspects of biological data sets such as cell divisions or to separate organs into cell layers or by cell type (Figure 7D, Figure 8—figure supplement 1C). Furthermore, cells sharing the same cell-type label can be easily manually selected, moved, or deleted for improved visualization of specific tissues and groups of cells. These processes can also aid user interactions, making cell selection and annotation more straightforward when users are curating 3D cellular segmentations and lineage maps.

Video 5

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In addition to mesh editing tools, MorphoGraphX has several tools to edit voxel data. The simplest is an eraser tool that can be used to remove portions of the stack that would otherwise interfere with processing. An example is the digital deletion of the peripodial membrane overlying the Drosophila wing disc, which needs to be removed to allow for the extraction of the organ’s surface (Aegerter-Wilmsen et al., 2012)⁠. A typical workflow for 3D segmentation starts with a 3D image of a cell boundary marker. This is then preprocessed with operations such as blurring to reduce noise or background removal filters, before segmentation with algorithms such as the Insight Toolkit’s (ITK) Morphological Watershed filter (https://www.itk.org). More recently, deep learning methods with CNNs have been developed to predict cell boundaries, such as the 3D U-Net model (Çiçek et al., 2016; Ourselin et al., 2016)⁠, that can improve the stacks for downstream segmentation. The modular structure of MorphoGraphX has allowed us to interface an implementation of the 3D U-Net model developed by Eschweiler et al., 2019⁠. This enables the interactive use of the CNN boundary prediction tool from within MorphoGraphX (Video 6) and simplifies experimentation with different networks or downstream segmentation strategies. Several networks are currently implemented (Eschweiler et al., 2019; Eschweiler et al., 2021; Wolny et al., 2020)⁠, although the add-on should work with any libtorch traced model. It also avoids the requirement to set up a full Python environment and comes stand-alone as a package built for the most common nVidia GPU architecture.

Video 6

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Once data is segmented, it often requires some manual correction before it is ready for final analysis in a chosen coordinate system. MorphoGraphX has interactive tools that operate on voxel data both to combine and split labels (cells), although typically it is easier to oversegment and combine, rather than undersegment and split (Video 6). This can be used to correct segmentations, which can then be used to help train deep learning networks to further improve automatic segmentation (Vijayan et al., 2021)⁠. In this context, MorphoGraphX has been used to segment and curate Arabidopsis ovule data to create ground truth for confocal prediction networks (Wolny et al., 2020)⁠.

Software design

The internal architecture of MorphoGraphX has been designed to make it easily extendable, while retaining the speed of the fully compiled, statically typed language C++. The relatively small visualization and data management core is augmented with processes that are loaded dynamically at startup and provide almost all of the software’s functionality. MorphoGraphX has grown to provide a wealth of custom processes for 2.5D and 3D image processing and coordinate system creation. Additionally, it has become a platform to integrate published tools and methods that have no visual interface to the data of their own, which increases their accessibility and ease of exploration for biologists. Examples include the previously mentioned deep learning tools and the processes that interface to the Insight Toolkit (ITK) C++ image processing libraries. This greatly reduces the learning curve required to use ITK tools, such as the popular Morphological Watershed filter, which can be run from MorphoGraphX with a click of the mouse. For more advanced ITK pipelines, we have recently integrated an open-source tool that provides an XML Pipeline Wrapper for the Insight Toolkit (XPIWIT) that allows the development of ITK image processing pipelines interactively without any C++ programming required (Bartschat et al., 2016). These pipelines can be packaged into processes and called directly from the MorphoGraphX GUI, allowing the easy exploration of complex ITK image processing pipelines. An example is the ‘Threshold of weighted intensity and seed-normal gradient dot product image’ (TWANG) pipeline (Stegmaier et al., 2014)⁠, a fast parallel algorithm for nuclear segmentation. A selection of pipelines along with the XPIWIT software is bundled as an add-on for MorphoGraphX and requires no additional installation.

Another example of software integration with popular open-source tools is the processes we have developed to interface with R (Video 7) that provide plots for basic statistical analysis on attributes created in MorphoGraphX, including positional information provided by organ-centric coordinate systems. This simplifies the creation of the most commonly used plots without the need for export files and the mastery of ggplot. In addition to directly linking to C++ libraries, MorphoGraphX can be scripted with Python, allowing repetitive functions to be performed in batch, and the possibility to integrate Python-based tools. Operations interactively performed in MorphoGraphX are written to a Python log file for reproducibility and logging, and to allow easy cut-paste script creation.

Video 7

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Since its inception, a major focus of MorphoGraphX has been in the creation, manipulation, and visualization of geometry in the form of surface meshes for 2.5D and full 3D cellular meshes. This is in addition to the 3D voxel source data from microscopy images. There is currently very little software available that can handle both and enable the interaction between the two. Most image processing software work only with voxels, whereas most computer graphics, animation, and engineering software deal only with meshes. If the goal is simply to segment 2D or 3D cells, there are many options available, for example, Fiji (Schindelin et al., 2012)⁠ or ilastik (Berg et al., 2019)⁠. If the processing of surface projections is required, advanced geometrical quantification like cell division analysis, or annotation with custom designed coordinate systems, then MorphoGraphX should be considered.

All new data collected for this article, along with the specific version of MorphoGraphX (https://www.MorphoGraphX.org) used for analysis, and the tutorial videos are available online, and can be downloaded from Dryad at: https://doi.org/10.5061/dryad.m905qfv1r.

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Author details

1. Sören Strauss

   Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

   Contribution

   Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7141-1750

2. Adam Runions

   Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

   Present address

   Department of Computer Science, University of Calgary, Calgary, Canada

   Contribution

   Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7758-7423

3. Brendan Lane

   1. Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany
   2. John Innes Centre, Norwich Research Park, Norwich, United Kingdom

   Contribution

   Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – review and editing

   Competing interests

   No competing interests declared

4. Dennis Eschweiler

   Institute of Imaging and Computer Vision, RWTH Aachen University, Aachen, Germany

   Contribution

   Software, Writing – review and editing

   Competing interests

   No competing interests declared

5. Namrata Bajpai

   Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

   Contribution

   Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – review and editing

   Competing interests

   No competing interests declared

6. Nicola Trozzi

   1. Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany
   2. John Innes Centre, Norwich Research Park, Norwich, United Kingdom

   Contribution

   Data curation, Investigation, Resources, Validation, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3951-6533

7. Anne-Lise Routier-Kierzkowska

   IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

   Contribution

   Investigation, Methodology, Software, Validation, Visualization, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-0383-0811

8. Saiko Yoshida

   Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

   Contribution

   Data curation, Formal analysis, Investigation, Writing – review and editing

   Competing interests

   No competing interests declared

9. Sylvia Rodrigues da Silveira

   IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

   Contribution

   Investigation, Writing – review and editing

   Competing interests

   No competing interests declared

10. Athul Vijayan

    Plant Developmental Biology, TUM School of Life Sciences, Technical University of Munich, Freising, Germany

    Contribution

    Data curation, Formal analysis, Investigation, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-1837-6359

11. Rachele Tofanelli

    Plant Developmental Biology, TUM School of Life Sciences, Technical University of Munich, Freising, Germany

    Contribution

    Investigation, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0002-5196-1122

12. Mateusz Majda

    1. Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany
    2. John Innes Centre, Norwich Research Park, Norwich, United Kingdom

    Contribution

    Data curation, Investigation, Validation, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-3405-2901

13. Emillie Echevin

    IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

    Contribution

    Investigation, Writing – review and editing

    Competing interests

    No competing interests declared

14. Constance Le Gloanec

    IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

    Contribution

    Investigation, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0002-7959-6307

15. Hana Bertrand-Rakusova

    IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

    Contribution

    Investigation, Writing – review and editing

    Competing interests

    No competing interests declared

16. Milad Adibi

    Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

    Contribution

    Data curation, Validation, Writing – review and editing

    Competing interests

    No competing interests declared

17. Kay Schneitz

    Plant Developmental Biology, TUM School of Life Sciences, Technical University of Munich, Freising, Germany

    Contribution

    Funding acquisition, Supervision, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0001-6688-0539

18. George W Bassel

    School of Life Sciences, University of Warwick, Coventry, United Kingdom

    Contribution

    Formal analysis, Investigation, Supervision, Validation, Writing – review and editing

    Competing interests

    No competing interests declared

19. Daniel Kierzkowski

    IRBV, Department of Biological Sciences, University of Montreal, Montreal, Canada

    Contribution

    Investigation, Supervision, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0002-1947-8691

20. Johannes Stegmaier

    Institute of Imaging and Computer Vision, RWTH Aachen University, Aachen, Germany

    Contribution

    Software, Writing – review and editing

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-4072-3759

21. Miltos Tsiantis

    Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany

    Contribution

    Conceptualization, Funding acquisition, Resources, Supervision, Writing – review and editing

    Competing interests

    No competing interests declared

22. Richard S Smith

    1. Max Planck Institute for Plant Breeding Research, Department of Comparative Development and Genetics, Cologne, Germany
    2. John Innes Centre, Norwich Research Park, Norwich, United Kingdom

    Contribution

    Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing – review and editing

    For correspondence

    Richard.Smith@jic.ac.uk

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0001-9220-0787

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