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Computational reconstruction of cell and tissue surfaces for

modeling and data analysis

Frederick Klauschen1,*, Hai Qi2, Jackson G. Egen2, Ronald N. Germain1,2, and Martin Meier-

Schellersheim1,*

1Program in Systems Immunology and Infectious Disease Modeling, National Institute of Allergy

and Infectious Diseases, National Institutes of Health, 9000 Rockville Pike, Bethesda MD 20892,

USA

2Lymphocyte Biology Section, Laboratory of Immunology, National Institute of Allergy and

Infectious Diseases, National Institutes of Health, 9000 Rockville Pike, Bethesda MD 20892, USA

Abstract

We present a method for computational reconstruction of the 3-D morphology of biological

objects, such as cells, cell conjugates or 3-D arrangements of tissue structures, using data from

high-resolution microscopy modalities. The method is based on the iterative optimization of

Voronoi representations of the spatial structures. The reconstructions of biological surfaces

automatically adapt to morphological features of varying complexity with flexible degrees of

resolution. We show how 3-D confocal images of single cells can be used to generate numerical

representations of cellular membranes that may serve as the basis for realistic, spatially resolved

computational models of membrane processes or intracellular signaling. Another example shows

how the protocol can be used to reconstruct tissue boundaries from segmented two-photon image

data that facilitate the quantitative analysis of lymphocyte migration behavior in relation to

microanatomical structures. Processing time is of the order of minutes depending on data features

and reconstruction parameters.

INTRODUCTION

The increased availability and use of advanced 3-D imaging techniques such as fluorescence

confocal and multi-photon or electron tomographic microscopy has enabled researchers to

investigate biological structures and processes with a high degree of spatial resolution. The

applications range from high-resolution mapping of (sub-)cellular morphology and dynamic

tracking of fluorescently labeled proteins in sub-cellular compartments to in situ imaging of

cell population behavior1–3. Because many biological processes are closely related to and

influenced by the spatial context in which they occur – information made accessible by 3-D

microscopy – it is essential to include these spatial properties in the analysis of such data.

Moreover, because data sets acquired in these experiments are usually large and the relevant

biological objects are numerous and/or the spatial properties complex, manual analyses are

laborious and frequently involve subjective choices that render them problematic for

quantitative data analysis. Here, we describe the application of an approach, implemented in

a user-friendly software tool, that allows for the automated three-dimensional reconstruction

of the surfaces of biological objects ranging from (sub-) cellular membranes to tissue/organ

*Correspondence to Frederick Klauschen (fklauschen@niaid.nih.gov) or Martin Meier-Schellersheim (mms@niaid.nih.gov).

COMPETING FINANCIAL INTERESTS STATEMENT

The authors declare that they have no competing financial interests.

NIH Public Access

Author Manuscript

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Published in final edited form as:

Nat Protoc. 2009 ; 4(7): 1006–1012. doi:10.1038/nprot.2009.94.

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boundaries and the subsequent integration of these reconstructions with automated tissue-

contextual cell migration data analysis and modeling.

Introduction to Voronoi diagrams

Many different strategies for computational surface reconstruction have been developed,

frequently based either on higher-order polygonal 4, 5 or triangular 6, 7 surface meshes (for a

review, see e.g. 8). Most approaches were designed mainly for visualization purposes in

software used to process microscopy data, as for instance in Imaris® (Bitplane). The

difference between those approaches and the technique introduced here are that our

approach uses adaptive resolution of surface features, can reduce artefacts resulting from

lower out-of-plane resolution, and is capable of high-quality mesh generation required for

computational modeling. The price that has to be paid for the combination of these

advantages is that the iterative optimization procedure may cause longer processing times

compared to conventional approaches if high mesh resolutions are desired. Our method,

used to obtain the results published in ref. 9 is based on the geometric concept of ‘Voronoi

diagrams10’ that combines the concepts of polygonal and triangular meshes and offers

specific advantages for numerical simulations 11. In two dimensions, a Voronoi diagram

(also called Dirichlet tesselation12) of a set of points, here called ‘vertices’, is constructed by

subdividing the area containing the vertices into geometric mesh elements in such a way that

the Voronoi element of each vertex comprises the region surrounding the vertex that is

closer to this than to any other vertex13 (Fig. 1). Because Voronoi diagrams can be

computed for arbitrary vertex distributions, their shapes can also be highly variable (Fig.

1A). There exist, however, vertex distributions for which the Voronoi diagrams have

properties that are particularly desirable for computational analyses: the variation of the

distances between neighboring vertices is minimized (equally spaced mesh) and the ratio of

the element circumference and element area are minimal. In 2-D, these optimized Voronoi

diagrams, or meshes, are hexagonal lattices (Fig. 1D). They occur naturally, for instance, in

bee honeycombs, minimizing the material needed for building robust planar structures.

Voronoi-like shapes generated during isotropic growth or diffusion processes starting from

initial seed points, as in turtle carapaces or giraffe fur, show similar but sometimes less

optimized hexagonal structures. Perfectly hexagonal structures can be viewed as limiting

cases toward which ‘real’ meshes, i. e. 2-D meshes constrained by boundaries or embedded

in 3-D, may evolve if appropriate algorithms are used. Such meshes are called ‘optimal’ (or

also ‘high-quality’ meshes).

An important characteristic of optimal Voronoi meshes is that the center points (also called

forming points) have the same coordinates as the centroids of the elements. While vertex

distributions resulting in optimal Voronoi meshes can be easily generated in a two

dimensional plane without boundaries or with rectangular borders, this task is nontrivial if

curved boundaries are present or for curved surfaces embedded in 3-D, which is the case for

computational reconstructions of cell/tissue surfaces. The computational method we

describe here uses an iterative optimization algorithm that produces optimal meshes for

arbitrary shapes and boundaries from arbitrary initial vertex distributions. Initial “seed”

vertices are distributed randomly within the confines of the surface layer defined by a 3-

dimensional binary mask created from microscopy image series (Figure 2). Then the

Voronoi surface elements are computed for the initial distribution and the vertices are

subsequently moved to the centroids of these elements. In the next step the Voronoi

elements are recomputed for the centroids, which starts an iterative process (computation of

Voronoi diagram → computation of Voronoi centroid → replacement of Voronoi center

vertex with centroid ↲) that converges to an optimal Voronoi mesh with hexagonal element

geometry (Figures 1E,F and 3).

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Our approach permits to automatically and adaptively produce optimized meshes and

resolve morphological details of varying complexity with an optimal number of elements for

a given desired accuracy. More precisely, our method offers two reconstruction alternatives

that may also be combined. The first option requires the definition of the total number of

elements/vertices used in the reconstruction. The method then iteratively optimizes the

distribution of the vertices so that a uniform (high-quality) mesh is generated (Figure 4 and

Box 1 ‘Description of the reconstruction algorithm’), which approximates the surface with

an accuracy not lower than the average distance between neighboring vertices. The second

(and default) option allows for the automatic adaptation of the reconstruction to local

differences in spatial detail in the image data and ensures that a certain accuracy of the

surface reconstruction is reached with a minimal number of elements (Figure 5). This

approach is based on the computation of local quality measures (for instance, angles

between surface elements or Euclidean vs. geodesic vertex distances) and uses local mesh

refinement if quality criteria are not met (see Box 2 ‘Local adaptivity’ for details). These

numerical Voronoi reconstructions are useful whenever quantitatively precise

representations of spatial features have to be extracted from 3D microscopy images. Here

we focus on two example applications at distinct scales of biological resolution.

BOX 1

Description of the reconstruction algorithm

The following steps are performed automatically without any user intervention and are

listed here only to present details of the algorithm.

1)

Extraction/labeling of the surface voxels.

2.1)

Initial random distribution of a number (depending on the intended

resolution) of seed vertices within the surface layer defined in step 1 and

2.2)

Definition of a minimum reconstruction accuracy (optional, see below for

details).

3)

Computation of the Voronoi diagram for the current vertex locations

embedded in the voxel surface data. Algorithm: Assign each surface voxel

the label of the closest seed vertex.

4)

Computation of the centroid for each Voronoi element. Algorithm: Compute

the arithmetic means of the coordinates of all surface voxels of each Voronoi

element and assign the centroid to the surface voxel closest to the mean

coordinates.

5)

Replacement of the current Voronoi vertices with the centroids of the

Voronoi element.

6)

Iteration of steps 3–6 for a pre-defined number of loops or until the average

distance between the centroid and the Voronoi center vertex drops below a

(default or user-definable) threshold.

BOX 2

Local adaptivity

The reconstruction algorithm without local adaptivity results in a homogeneous mesh,

i.e., the mesh vertices are equally distributed over the surface resulting in an optimal

mesh approximating the surface with a accuracy given by the mesh constant (average

distance between neighboring vertices) (Supplementary Figure 1). With locally adaptive

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reconstruction, the surface reconstruction adaptively resolves morphological details until

a certain, definable accuracy is reached. This is achieved by computing the angles

between every pair of adjacent triangles. If this angle is smaller than a certain threshold,

the mesh is locally refined by inserting an additional vertex at the center of the common

side of the two adjacent triangles. To maintain a locally optimized grid structure, one

‘regular’ iteration step is performed following the insertion of additional vertices. This

adaptivity generates surface parameterization of higher accuracy with a locally higher

density of vertices in regions of higher spatial complexity, while limiting the total

number of elements (Figure 5). The relevant options and parameters can be set in the

MorphologyModeler Params menu.

Example applications

The first example describes surface reconstruction based on microscopy images of cellular

morphology. Computational studies in cell biology increasingly take into account that many

aspects of cellular function cannot be understood or modeled without proper treatment of

cellular morphology14–17. Analyzing or simulating intra-cellular biochemical dynamics,

such as non-isotropic reaction-diffusion processes in the cytoplasm or on the cell membrane

or intra-cellular transport mechanisms, requires computational representations (spatial

discretizations) of cellular morphology. The elements of high quality Voronoi meshes are

particularly suitable for such discretizations because they ensure locally homogeneous mesh

spacings even for complex cell morphologies and the lines connecting the Voronoi centroids

are perpendicular to the element boundaries, properties that greatly facilitate mathematical

analyses and simulation of diffusion and fluxes18. Because our reconstruction method can

adapt its spatial resolution to the morphological properties of the cells being analyzed, we

obtain high quality representations of cellular structure with a minimal number of volume

elements, thereby minimizing computational costs of numerical analyses and simulations

(Fig. 5).

On a slightly higher spatial scale, analyzing the dynamic organization of physically

interacting cell populations in specialized tissues requires computational representations of

micro-anatomical structures and cellular motion tracks. Recent advances in in-situ two-

photon microscopy have facilitated the investigation of cell behavior dynamics within

complex tissue environments. These technological developments have been particularly

beneficial for immunological research, as the analysis of migration and interaction of

leucocytes, lymphocytes, and antigen presenting cells in secondary lymphoid organs and at

sites of chronic inflammation is critical for a deeper understanding of the immune

response19. Because the microanatomy of secondary lymphoid organs reflects the

compartmentalization of T and B cells and antigen presenting cells and provides guidance

for entry, internal movement, and exit of cells, cell migration analysis must take into account

the micro-anatomical context. In the tissue surface reconstruction example presented here

we specifically look at germinal centers (GCs), which are organized lymphoid tissue

structures where high affinity antibody responses are generated. Lymphocytes that occupy

GCs mainly include antigen-specific B and T cells, while naïve B cells are largely excluded

from the GC proper. Intravital 2-photon microscopy has been increasingly utilized to

understand how the GC tissue structure is dynamically organized and maintained and how

the trafficking of its cellular components is controlled20–22. To analyze imaging data from

such studies, it is often necessary to define GC-related tissue domains. Typically, this is

accomplished by manual annotation of tissue boundaries on optical sections according to

relevant fluorescently-labeled tissue landmarks. Ideally, these boundaries defined on series

of axially adjacent 2-D imaging planes are further digitally combined to allow visualization

and mathematic description of the reconstructed tissue structure in full 3-D space (Figures

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7). The tissue reconstructions generated using this protocol have been used in a recent

research publication9 to quantitatively analyze the cellular migration behavior of

lymphocytes in the context of germinal center tissue structure. In that study computing and

comparing the speed and location/distance of a SLAM-associated-protein and wild-type

lymphocytes relative to the virtual germinal center surface over each track length permitted

the automatic quantification and classification of the cell behaviour. The analysis revealed

that in contrast to wild-type, SAP KO T lymphocytes are incapable of crossing the border of

and moving into germinal centers (see 9 for details).

In addition to these types of applications, our method may improve the computational

efficiency of standard surface reconstruction applications, such as morphological

measurements or surface visualization, because it minimizes the number of surface elements

for a given reconstruction accuracy. Conventional approaches, on the other hand, use

triangular meshes with high homogeneous resolution and cannot locally adapt the mesh to

surface features.

MATERIALS

Reagents

•

Image data, see Reagent Setup for details of preparation/segmentation. The

Morphology Modeler requires (binary) segmented image data as input, which may

be either generated manually using a standard image manipulation software such as

Photoshop® (http://www.adobe.com/products/photoshop/) or GIMP (http://

gimp.org) or automatically from fluorescence microscopy data using image

segmentation software (e. g., 2PISA (Klauschen et al., in preparation)).

Equipment

•

Reconstruction software: The software package MoMo (Morphology Modeler) we

developed and use here is available for Windows XP/Vista, MacOS X and Linux

32-/64-bit and may be obtained free of charge for non-commercial use. It can be

downloaded at http://www3.niaid.nih.gov/labs/aboutlabs/psiim/

computationalBiology. The surface reconstruction and cell track analysis method

we present here may also be implemented using any modern programming

language and the explanations of the method we present would be sufficient for that

purpose.)

•

Computer hardware: The computations can be performed on most computers with

current standard specifications (OpenGL capabilities are required for surface

visualization), limited only by the memory required to handle 3-D microscopy data.

The reconstructions shown here were performed on an Apple MacPro®

Workstation with 16GB main memory (http://www.apple.com) running 64bit SuSE

Linux (http://www.opensuse.org).

Reagent set up

Image data preparation—The surface reconstruction protocol described in the procedure

may be used with any kind of segmented image data whose format represents an object

whose surface is to be reconstructed with intensity values equal to or larger than 1 while the

background pixels are zero. The images have to be provided in individual z -stack files and

may have the following formats: JPEG, TIFF, or PNG. Files have to be named using a prefix

followed by numbers indicating the z slice (the index format is detected automatically) and

the corresponding image format suffix.

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To generate the binary segmentations of the 3-D confocal microscopy images of the T cell

used in our first example application, we manually defined an intensity threshold to

differentiate between cell and background pixels using standard image processing software

(e. g. Photoshop or GIMP) and applied that threshold to the whole z-stack (Figure 2). In case

of image data of lower quality, for instance due to z-stack intensity inhomogeneities, we

recommend using automatic segmentation software (see Reagents). To define binary masks

of the germinal center area on individual imaging planes, which serve as the input for the

tessellation program, we took advantage of the fact that naive B cells as a population are

substantially excluded from GCs (Figure 7A). Therefore, fluorescent signals associated with

naïve B cells were time-averaged for each imaging time series, and the follicular area and

approximation of the GC–mantle border on individual optical sections were manually

marked using the Photoshop mask function. Such marked images were then transformed into

binary images in Photoshop, with pixel size and dimension maintained. The resulting z-stack

of segmented germinal center masks essentially represents the average configuration of the

visualized GC during the imaging period (Figure 7B,C).

PROCEDURE

Surface Reconstruction (using the Morphology Modeler MoMo)

1

Start the Morphology Modeler software MoMo.

2

In the SurfParam menu select item Load Microscopy Data.

3

Select the first slice of the data set to load the complete image series and click

‘open’.

4

Choose between homogeneous vertex distribution or locally adaptive surface

reconstruction. For homogeneous reconstruction, set the number of vertices/

mesh elements. For locally adaptive reconstruction, set the refinement threshold

angle parameter (between −1.0 for minimum and +1.0 for maximum

refinement).

?Troubleshooting.

5

Enter the z stack factor, which is the ratio of the out-of-plane and in-plane

resolution.

6

Select if top or bottom elements should be labeled in case the z stack does not

fully contain the microanatomical object.

7

The software will automatically check if input data contains more than one class

label. If data contains only background and a single object/class label the

program automatically continues to step 8 and no user action is required. If data

contains more than one class label choose a class label for reconstruction or

enter ‘0’ to merge all labels.

8

In the SurfParam menu select item Save Surface Tesselation to save the result.

9

Rotate the mouse wheel to visualize the surface and hold the left mouse button

clicked and move the mouse to rotate the surface rendering.

Load and visualize surface

10

In the SurfParam menu select item Load Surface Tesselation to load surface.

11

Scroll mouse wheel to visualize surface.

12

In the GridVis menu select toggle Transparency, toggle Lighting, or ClipPlane

to modify rendering options.

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Transform coordinates to true spatial distances

13

In the SurfParam menu select item Transform to real space.

14

Enter the true/real spatial length of a voxel unit in meters.

15

In the SurfParam menu select item Save Surface Tesselation to save the result.

Cell track analysis

16

If you wish to use the protocol for surface reconstruction with subsequent cell

track analysis, follow the extended protocol (Supplementary Manual 1).

TROUBLESHOOTING

Since the Voronoi mesh computation utilizes discrete voxel coordinates given by the image

data, the precision of the Voronoi centroid computation decreases when a Voronoi element

contains only relatively few voxels, i.e., the Voronoi resolution comes close to the image

resolution. However, this is not a limitation of the reconstruction method, but related to the

image quality and if necessary has to be overcome by an increase in image resolution. Low

image resolutions mainly pose a problem in form of discrepancies between a higher in-plane

and a lower out-of-plane (z-direction) resolution and visible ‘steps’ or ‘kinks’ may occur

along the z direction. One intrinsic feature of the method presented here is that such z-

direction ‘steps’ can be automatically interpolated. However, this only works as long as the

z-steps are smaller than the reconstruction mesh element size (see Supp. Fig. 2, 3). If the

Voronoi mesh elements have a higher resolution than the z-resolution of the image data, the

z-steps are ‘precisely’ reconstructed (see Supp. Fig. 3 and 4), an unwanted phenomenon that

is present in surface visualizations of microscopy data performed with most conventional

software packages (e.g. Imaris®) that are widely used in biological microscopy that do not

offer adaptive reconstructions. Thus, there is an optimum number of surface vertices that

may correct limitations of microscopy data and more accurately represent the actual, real

biological surface than higher-resolution surface meshes. In the method described here, this

problem is automatically avoided when the number of vertices is adaptively adjusted (option

B), but it has to be taken into account if the number of surface vertices used for the

reconstruction is defined by the user (more is not necessarily better!). If adaptive

reconstruction (option B) is used it has to be kept in mind that setting the quality parameter

to high values (approaching the maximum 1.0) may in certain cases lead to very long

processing times due to a large number of iterations. We therefore recommend starting with

medium quality values.

TIMING

The computational time required to perform the task outlined above depends on the size of

the microscopy data set, the intended reconstruction accuracy and the performance of the

computer hardware. Typical times using recent computers for surface reconstruction are on

the order of minutes.

ANTICIPATED RESULTS

Application of this protocol to segmented image data produces surface tesselations that

consist of mesh points (‘vertices’) and a mesh topology (vertex connections, ‘edges’). The

general features of the results that can be anticipated from the surface reconstruction

protocol in general depend on the features of the true object, the image resolution with

which the object is represented and the choice of the reconstruction parameters and their

combination: First, it can be expected that with increasing number of Voronoi mesh

elements the image data is more accurately reconstructed. We call this type of accuracy

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image-related accuracy and define it as the distance of the centroids of the triangular mesh

that is formed by connected the Voronoi center points (the ‘dual’ mesh) from the voxel

image surface. The second type of reconstruction quality is the accuracy with which the true

object is represented by the surface tesselation, called ‘object-related accuracy’. While an

increasing mesh resolution always leads to an increasing image-related accuracy, an increase

in the number of mesh elements does not always produce better object-related accuracy and

in such cases, a lower number of mesh elements might lead to a higher-object accuracy due

to smoothing effects (see also ‘Troubleshooting’ and Supp. Fig. 3 and 4).

The iterative optimization process (both the homogenous and locally adaptive options)

converges to optimal (high-quality) meshes (see Fig. 1), if the resolution of the voxel data is

sufficiently fine in relation to the mesh element size (see also ‘Troubleshooting’). The effect

of locally adaptive surface reconstruction on the results depends on the spatial properties of

the image data. A spherical object will produce a result similar to that of homogeneous

reconstruction, because the iterative refinement process depends on local membrane

curvature, which is constant on a sphere. However, when objects have special spatial

features (which are resolved in the image data), the local adaptivity produces

inhomogeneously resolved surface meshes. An example for the latter case is the T cell

surface reconstruction in which the membrane protrusion (shown in the lower right part of

the cell in Fig. 5C and middle part of Fig. 5D) is automatically resolved by local adaptive

mesh refinement.

The results of the surface reconstruction of the T cell alone, the T-B-cell conjugate and the

germinal center border reconstruction are presented in Figures 4–6. Figure 4 (+ Supp. Video

1) and Fig. 5 (+ Supp. Video 2) offer a comparison between adaptive and homogeneous

reconstruction and illustrate the optimization process. See also Supplementary Figures for

additional examples. Supplementary Figure 5 shows example results obtained when two

separate surfaces present in the same data set are reconstructed simultaneously. If regions

exist where the distance between the surfaces is equal to or smaller than the reconstruction

mesh spacing the separate surfaces might merge/attach to each other. This problem can be

solved by increasing the mesh resolution or processing the surfaces separately. Exemplary

results of the cell tracking analysis can be found in the supplementary material section ‘cell

track analysis’, Figure 7 Supplementary Figure 6.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.

Acknowledgments

This research was supported by the Intramural Research Program of NIAID, NIH.

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Figure 1.

Iterative Voronoi mesh optimization illustrated in 2-D plane. Voronoi cells are depicted by

center point (blue) and cell border (magenta). A: Initial random distribution of vertices. B,

C, D: optimized mesh after 1, 10 and 500 iterations. E: Movement of Voronoi cell centroids

during optimization process indicated by grey trajectory. F: Average movement of centroids

during the optimization process. The linear plot shows the relatively fast convergence after

~30 iterations. The logarithmic plot shows local maxima prominent between 200 and 400

iterations, which indicate that the final neighborhood configuration has not been reached. At

more than 400 iterations the optimum hexagonal structure is accomplished and the average

centroid movement falls strictly monotonously.

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Figure 2.

Binary image generation from fluorescence microscopy data. A: Original confocal image

(example z-slice of 3-D stack data) of a T cell; B: binary segmentation result used as input

for surface reconstruction (image size: 500×500 pixel).

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Figure 3.

Illustration of Voronoi surface elements. Reconstruction result after optimization (20

iterations) with adaptive vertex insertion (using T cell data shown in Figure 2). Because

surfaces that are not perfect spheres cannot be approximated by 2-D planar polygonal

Voronoi elements that have continuous transitions between neighboring cells. To achieve

continuity at the element boundaries Voronoi elements are used that correspond to

projections of the 2-D planar elements onto the voxel surface. These elements still fulfill the

Voronoi criteria and are bended along the connections of neighboring Voronoi center points

that form the edges of the dual surface triangulation. The Voronoi center points are located

directly on the surface given by the voxel image data.

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Figure 4.

Optimizing surface reconstruction for a given number of vertices/elements (n=500) for the T

cell shown in Figure 2. Blue: Lines connecting the Voronoi cell centers. A, initial arbitrary

vertex distribution. B, C and D after 2, 10 and 100 optimization iterations.

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Figure 5.

Automatic adaptive surface reconstruction of the T cell microscopy data shown in Figure 1,

minimizing the required number of vertices/elements for a given reconstruction accuracy

(ratio out-of-plane/in-plane resolution 9:1, average cell radius ~4.5μm, compare scale in Fig.

1). Blue lines: Voronoi mesh; Grey lines: Corresponding ‘dual’ triangular mesh connecting

the Voronoi cell centers. A: Initial surface approximation with a small number of vertices

(n=20). B, C: Optimization results after 11 (B) and 26 (C) iterations based on acute angle

refinement criterion: Adaptive vertex insertion if angles between adjacent triangles are

smaller than 135 degrees allows for accurately resolving morphological features while

minimizing the number of surface vertices (n(final)=289) (C, D: rotation and magnification

of C).

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Figure 6.

Surface reconstruction of cell conjugates. A: Confocal microscopy data of a cognate T cell

(green) forming a synapse with an antigen-presenting cell (blue). B: Surface reconstruction

result (ratio out-of-plane/in-plane resolution 9:1).

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