A Dynamic Non-Manifold Mesh Data Structure to RepresentBiological Materials

Endre SomogyiDept. of Intelligent Systems Engineering, Indiana University Bloomington, IN 47405

{somogyie} @ indiana.edu

ABSTRACTComputational models of biological materials enable researchers to gain insight and make testable predictionsof quantitative dynamic responses to stimuli. These models are particularly challenging to develop because bi-ological materials are (1) highly heterogeneous containing both biological cells and complex substances such asextra-cellular medium, (2) undergo structural rearrangement (3) couple biological cells with their environment viachemical and mechanical processes. Existing numerical approaches excel at either describing biological cells orsolids and fluids, but have difficulty integrating them into a single simulation approach. We present a novel dy-namic non-manifold mesh data structure that naturally represents biological materials with coupled chemical andmechanical processes and structural rearrangement in a unified way.

KeywordsPhysically Based Modeling, Biological Simulation, Dynamic Meshing, Finite Element Simulation.

1 INTRODUCTIONResearchers increasingly build computational modelsof biological materials to gain insight and make testablepredictions about responses to stimuli. Mechanisticmodels of biological tissues are particularly challeng-ing to develop because biological materials are highlyheterogeneous across a broad range of scales. Biolog-ical cells exist in a dynamic, spatial fluid environmentand create and respond to a range of physical and chem-ical stimuli with complex behaviors, including move-ment, changes in morphology and mechanics, prolif-eration, death, differentiation and modification of thelocal environment. Biological materials combine ac-tive agents such as biological cells with highly hetero-geneous visco-elastic substances such as extra-cellularmedium (ECM), fluids and solids. We use the termphysical agent to refer to parcels of biological mate-rial. Physical agents may be active or passive, may ormay not have sub-structures, and may or may not havenatural boundaries.

As a key enabling component of a tool for modeling bi-ological materials, we have generalized existing man-ifold mesh data structures into a novel dynamic non-manifold mesh data structure that can consistently rep-resent smooth deformations and dynamic topological

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rearrangement. This mesh forms the basis of our Me-chanica [13] environment for simulation of cells andtissues. Our mesh represents physical agents as poly-topes in contact through shared surfaces, where bothsides of the surface maintain their own identity. We per-form smooth mesh deformations via the explicit finiteelement method; we use three mesh update operations,radial edge split, radial edge collapse, and vertex splitto perform discrete topological changes.

Biological material models are diverse, so tools to buildthese models must be extremely flexible and able to eas-ily accommodate new constructs. Computational bio-material model development tools thus need to be ableto conveniently represent the interactions and dynamicsof a wide range of different agents with varied materialproperties. Most numerical methods naturally repre-sent only a limited and fixed subset of agent behaviors.In engineered applications designed for modularity andtestability, this specialization does not usually pose aproblem. For example, simulations of mechanical partssuch as an aircraft wing, usually need to account pri-marily for mechanical and thermal properties; tradi-tional finite-element simulators suffice for such sim-ulations. Even complex engineered systems, such ascombustion dynamics tend to involve a limited num-ber of pre-specified physical processes such as reaction-advection-diffusion, for which finite-volume simulationis appropriate. Biological materials often lack strictmodularity, and tend to be highly interdependent withlarge numbers of coupled chemical and mechanical pro-cesses. Fig.1, illustrates cell and tissue rearrangementand dynamics that occur during body elongation (epi-boly) in early embryonic development. Epiboly’s com-

plex dynamics requires coordinated material transferand rearrangement, formation and loss of intercompo-nent boundaries and intra- and extra-cellular regulationof coupled chemical and mechanical processes.

While building computational models of the complexbehaviors and interactions found in nature will in-evitably be cognitively challenging, the lack of toolsthat describe naturally the mechanics of materialscomposed of continually rearranging agents createsan additional and unnecessary computational burdenon model developers. Many computational-mechanics

(a): 0.75 hours (b): 1 hours (c): 1.2 hoursFigure 1: Cell rearrangement and identity changein embryonic development. Time series of images(0.8 mm×0.8 mm) of epiboly in a developing zebrafishembryo. During epiboly, ectodermal cells proliferateand flow from the dorsal to the ventral side of the em-bryo, as cell layers fold inwards and the yolk is incor-porated into the embryo. Epiboly illustrate the complexinterplay between biochemistry and mechanics duringembryogenesis and the changes in agent number, shape,properties and relative positions which a tissue model-ing platform must describe (from [4]).

simulation methodologies require the creation ofmeshes to approximate the shape and structure ofphysical agents. A manifold mesh is locally smoothand flat (homeomorphic to a disc), so a tessellatedmanifold surface mesh contains only edges of degreetwo; that is, exactly two faces share each edge. Anon-manifold mesh is not restricted to locally-smoothsurfaces, so a tessellated non-manifold surface cancontains edges of degree one, two, three, or higher.

Existing dynamic manifold mesh data structures [10]often suffice to represent single cells in isolation, aslong as their movement dynamics are not too com-plex. However biological tissues consist of large num-bers of cells in contact, and these contacts continuallyform and disappear as cells rearrange, change shapesand adjacency, divide, merge and disappear. In mes-enchyme (three dimensional connective tissue), both thetopology and shape of cell-cell contacts can be com-plex. In epithelia (sheet-like tissues), cells contact eachother via numerous locally flat surfaces, whose inter-sections often define geometric edges and vertices. Inboth cases, manifold mesh data structures are a poormatch to the underlying physical reality.

When a biological cell contacts its neighboring cellsand its environment Fig.1, its volume and surface

both maintain their identities. With few exceptions, amembrane (and possible additional cell wall structures)separate cells from other cells and the surroundingenvironment. This membrane is thin relative to thesize of the enclosed cell cytoplasm, with a typicalsize ratio of ∼ 2,000 : 1. In many cases, when weare modeling cell and tissue-scale phenomena, we canapproximate the membrane as a quasi-two-dimensionalmanifold surface embedded in three-dimensionalspace. The properties of living materials depend onthe interplay of quasi-one-dimensional fibers (e.g., inthe extracellular matrix and intracellular cytoskeleton),quasi-two-dimensional sheets and membranes andfully three-dimensional structures (e.g., biologicalcells, organelles, micelles or fluid droplets). Treatingone- two- and three- dimensional agents consistently inthe same mathematical and computational frameworkis challenging and defines the aspect ratio problem.This paper will focus on how our data structure rep-resents the interaction of three-dimensional volumeswith quasi-two-dimensional surfaces. We will discussthe use of the data structure to represent fibers in asubsequent paper.

In Fig.2, we categorize key identity changes that occurwhen physical agents consisting of a three-dimensionalvolume with a two-dimensional surface change adja-cency. When two such initially separate agents con-tact each other, either: (1) the agents behave like waterdroplets: the contact surface between the agents dis-appears, while single volume and surface replace theoriginal agents’ volumes and remaining surfaces, (2)the agents behave like soap bubbles: the contact sur-face between the agents persists and the agents’ vol-umes maintain their identity, but a single surface re-places the original agents’ surfaces, or (3) the agents be-have like biological cells: the contact surface betweenthe agents persists and both agents maintain their vol-umes and surfaces. Traditional data structures usuallynaturally support only one of these adjacency-changeprocesses (either 1) or 3)) and require awkward manipu-lations to implement the others. A single agent can alsosplit partially or completely through the inverse of anyof these processes. Our data structure enables us to ef-ficiently represent all three types of adjacency-change.

2 RELATED WORKMany numerical approaches address some of the com-putational challenges of representing biological materi-als, however no single existing approach can representthe variety of these materials in a self-consistent way.

The Finite Element Method (FEM) [14] is convenientfor modeling solids under small deformations. TheFEM discretizes materials into “elements” represent-ing finite regions of space. An FEM solver generates

(1) (2) (3)

Figure 2: Topological and identity changes on adja-cency change. When two physical agents come intocontact: (1) They may merge both volumes and surfacesand lose their respective identities, like water droplets.(2) They may merge surfaces but maintain independentvolumes, like soap bubbles. (3) They may maintain in-dependent volumes and surfaces, like biological cells.

a set of nodes connecting adjacent elements, and a setof algebraic equations that define the time evolution ofthe nodal positions, velocities and properties at eachnode. Standard FEM discretizations can describe com-plex geometries and constitutive relations. However,most FEM methods can only naturally describe ma-terials with a fixed dimensionality (one-, two or threedimensions). E.g., to describe a composite agent withmembrane and volume, standard FEM would representthe membrane using a very large number of small tetra-hedra, which is computationally costly. Most FEMpackages do not naturally handle large-scale materialrearrangement or changing neighbor relationships.

The Finite Volume Method (FVM) [5] partitions spaceinto a set of finite, connected volumes, and integratesgoverning equations for fluxes between volumes to cal-culate time evolution. FVM can represent fluid trans-port and reactions in complex geometries. However,most FVM approaches cannot naturally describe dy-namic geometries or changing neighbor relationships.

Agent-based simulations of biological tissues typicallyemploy lattice-, particle-, vertex-, or more recentlyFEM-based approaches to represent cells. Some ofthese approaches explicitly represent cell membranes,but many use implicit representations. Here we usethe term implicit to mean data that is derived fromexplicit quantities. Because these are derived values,they do not have state variables, hence they can notdefine their own time evolution. Implicit surfaces havedifficulty representing important biological processessuch as surface chemistry, advection-transport, surfaceand edge contraction and adhesion, local signaling, etc.Explicit representation do not ipso facto mean that thatthey support these biological processes, but rather thatthey can support them.

Lattice based approaches have explicit volumes and canrepresent both implicit and explicit surfaces, but edgesand vertices are implicit. The Cellular Potts Model(CPM) usually defines an implicit representation of

membranes, edges and vertices between voxels of dif-ferent cells. Volume advection can be challenging inCPM as well.

Particle based approaches include center models [5]and sub-cellular element models [11]. Center modelstreat cells as single point particles that interact via non-bonded forces, and define the boundary of a cell im-plicitly. Subcellular element models represent cells ascollections of point particles and permit both explicitand implicit surface representations.

Vertex models [6] represent biological cells as con-nected, relatively simple (∼ 6− 15) faceted convexpolyhedra, with implicit surfaces, but explicit verticesand volumes. Modern FEM approaches [1] have ex-plicit vertices, edges and surfaces, with volumes ex-plicit or implicit depending on the representation. Withfew exceptions [9], most vertex or FEM type biologi-cal cell simulations are hard-coded to solve specific bi-ological problems and are not available as simulationenvironments.

Surface Evolver [3] is a program which determines theminimal energy configurations of surfaces such as soapfilms. Surface Evolver represents surfaces using a dy-namic non-manifold mesh. However it can only imple-ment case (2) in Fig. 2 and cannot handle explicit mem-branes in contact. It also does not support descriptionsof the complex biochemistry of biological cells.

While existing dynamic manifold mesh data struc-tures [10] can conveniently represent individualphysical agents and sets of agents with a limitednumber of contacts, they typically do not directlysupport the variety of identity changes which occurwhen agents change adjacency. Most do not explicitlytrack cell neighbor relationships and contact areas,but calculate them as needed, which can be slow. Wepreviously developed a numerical simulation engineusing the deformable manifold mesh from the BulletPhysics library. We found that this data structure wasable to calculate deformations for at most 20 cells incontact, before performance dropped to unacceptablelevels. Benchmarking showed that because all cells arein physical contact with each other, collision detectionwas computationally expensive. Furthermore, deter-mining cell neighbor relationships and contact areasbetween cells (which Bullet Physics required us to ateach time step) was computationally costly. Computetime using our non-manifold boundary representationmesh scales linearly with the number of vertices plusthe number of triangles; essentially, performance isproportional to the total surface area rather than thetotal volume.

3 APPROACHTo represent and simulate physical agents, we mustconsider two related questions: (1) How do we rep-

resent the structure of these agents when the physicalproperties of the agent’s constitutive elements may alldiffer? and (2) How do we represent dynamics of thesephysical agents including deformation and structural re-arrangement.

3.1 Physical StructureTraditional numerical approaches do not adequately ad-dress agents with changing adjacency. Tissues containmany cells (composite agents) which frequently changetheir adjacency, so we need to efficiently represent: (1)multiple agents in contact, (2) individual cell surfacechemical process occurring on each cell’s surface, and(3) chemical processes occurring between neighboringcells. Physical agents (fluid droplets, soap bubbles,biological cells and tissues) can have a well-definedboundary that has intrinsic material and chemical prop-erties distinct from the agents they envelop. We rep-resent biological materials with an explicit boundary,dynamic non-manifold mesh data structure, inspired byHun and Lee’s partial entity [8] and Weiler’s radial edge[16] structures. This mesh data structure enables usto faithfully represent changing neighbor relationshipsand chemical and mechanical processes in a unifiedway.

Consider two biological cells are in contact asin Fig.2.3. When we take a section of this contactingregion, we can make the abstraction that there arebasically three kinds of physical materials here: (1)the membrane that belongs to the first cell, (2) theinterstitial material between the cells, and (3) themembrane of the second cell. Each of these regions isthin relative to the size of the cells, but still has finitethickness. We represent this stack of physical materialswith the Triangle data structure. The triangle itself is acomposite type that can be thought of as a “sandwich”formed from a left Partial Triangle, the triangle itself,and a right partial triangle. The triangle representsa complete section of this contact region, where thepartial triangles represent one side of a boundary, i.e.the biological cell’s membrane.

Our mesh data structure consists of four key data types:vertices, partial triangles, triangles and cells. Verticesrepresent a position in space, maintain a list of incidenttriangles and cells, and they presently do not store anyother state variables. We measure mass at each vertexas the barycentric area (1/3 the area) weighted sum ofeach incident triangle and partial triangle’s mass.

The partial triangle Fig.3 represents one side of a phys-ical boundary, i.e., a biological cell membrane. Eachpartial triangle belongs to the boundary of a specificcell, and the cell’s boundary is defined as a set of con-nected partial triangles that form a closed manifold sur-face. Each partial triangle has pointers to its base tri-angle, the cell that it belongs to, its opposing partial

triangle, and to its three neighboring partial trianglesthat are also part of the same cell’s boundary. A par-tial triangle has an explicit mass, and can contain otherstate variables such as chemical amounts. The trian-gle data structure represents a section of shared bound-ary between physical objects, and is itself a compositestructure of two partial triangles. Each triangle containspointers to three vertices. Like the partial triangle, thetriangle has an explicit mass, and can contain a vectorof attached chemical amounts. We call the edge wheretwo or more triangles intersect a radial edge.

A cell represents a closed physical region of space de-marcated by an explicit boundary composed of partialtriangles. A cell type can represent a physical agent,such as fluid droplets, fluid volumes, soap bubbles, orbiological cells. A cell contains pointers to the partialtriangles that comprise the cell’s boundary. Presently,we approximate physical agents as homogeneous de-formable solids, but we plan to add more complex inter-nal structures in future versions. The cell has an explicitmass and can contain a vector of chemical amounts.Two cells in contact form a manifold surface, three ormore cells can intersect at a radial edge as in Fig.3.bforming a non-manifold intersection between three ormore surfaces. The partial triangles in a non-manifoldintersection are adjacent on only one side of the trian-gles incident to the radial edge.

(a) (b)

Figure 3: The partial triangle data structure enables usto represent both (a) manifold and (b) non-manifoldmeshes. (a) shows a manifold surface between a yellowand a red cell, (b) shows a non-manifold edge betweena red, green and yellow cells.

3.2 DynamicsThe physical agents that we represent are not static– both their positions and their states evolve in time.They move around and past each other, and regu-larly come into and out of contact with each other(i.e., change neighbor relationships). We representcell motion, membrane deformation, and chemicalprocesses such as cell-signaling, membrane transportas continuous, ordinary differential equation definedprocesses. We implement large-scale structural rear-rangement and topological change, such as large shapechanges and cell attachment/detachment as discreterule-based events. Vertices move according to the lawsof classical mechanics, and we calculate their time

evolution using the Propagator data type. Chemicalreaction-transport processes are well-studied, and were-use our existing solver [12] to implement them.The Rules Engine performs large scale rearrangementand topological transformations with rule triggeredmesh reconnect operations. Mesh reconnect operationsalter the underlying mesh, they define how elementssplit and merge - that is, they define local topologicalrearrangement. Mesh operations can create and deletemesh objects (triangles, and ultimately cells) and altertheir neighbor pointers and vertex positions. Exactlywhen to apply a mesh update operation is highlymaterial and simulation specific. Users need the abilityto define custom triggering rules for mesh updates inorder to model a range of different physical materials.Thus, we have developed a general rule-based meshupdate system that separates the mesh update operationfrom the triggering rule, and enables the user to readilyadd new mesh update operations and to write custom,material-specific trigger rules.

The Propagator calculates the continuous-time evolu-tion of the material vertices, as well as the fluid materi-als and state variables located on the triangle, partial tri-angle and cell elements. In other words, the propagatorcalculates the subsequent simulation state based on thecurrent state. The propagator queries the model for thenet force on each vertex and the rate of change of eachfluid or other state variables associated with the cellsand surfaces. The propagator presently uses a Runge-Kutta integrator, however we plan on adding more so-phisticated integrators in future versions.

In order to enforce constraints, the propagator imple-ments a Position Based Dynamics (PBD) constraintsolver [2]. The PBD solver enforce constraints by ad-justing the positions of the mesh vertices. The solverapplies a correction displacement to the vertex posi-tions. The basic idea of the PBD constraint solver isto first project all of the mesh vertices forward in un-constrained motion, according to the net force actingon each vertex. The unconstrained motion most likelyviolates some constraints. Each constraint object con-tains two functions: a) a constraint function of the cur-rent model state that yields a scalar constraint value,and b) a function that calculates the rate of change ofthe constraint function relative to the mesh vertices, i.e.the Jacobian of the constraint function. The constraintis satisfied when the constraint function evaluates tozero. The propagator sequentially adjusts the vertexpositions in a Gauss-Siedel fashion, according the con-straint Jacobian until the constraint function reaches atolerance. We have found that volume constraints aresatisfied with only 2-3 iterations.

The Rules Engine is responsible for discontinuous timestate changes in the mesh and other state variables. Therules engine maintains a list of rules that associate a

trigger condition and energy function with a discretemesh update operation. The Rules Engine monitorsthe mesh for positional and topological changes andlooks for mesh configuration patterns that match a trig-ger condition. When a configuration matches a trigger,the rules engine applies the corresponding mesh opera-tion to the mesh, thus modifying the mesh. We presentthree mesh update operations: radial edge split, radialedge collapse, and vertex split. Each mesh update rulealso defines an energy function. We associate an en-ergy with each mesh topological configuration, analo-gous to the way a potential energy is associated withdifferent configurations in physics. For example, usersmay wish to define a rule that penalizes excessivelylong edges, where the energy function could be pro-portional to the square of the edge length, say kx2, andthen associate the edge split operation with this energyfunction. Here, the rules engine might find an edge, andperform a trial edge split. The initial energy of this edgewould be kx2, and the energy of the split edge wouldbe 2∗ k(x/2)2 = (1/2)kx2, thus the change in energy is−k/2x2. The rules engine determines that this is an en-ergetically favorable operation and applies it. The rulesengine stores all triggered rules in a priority queue, or-dered energy value, such that the most energetically fa-vorable rules are evaluated first. This approach is simi-lar to [6], where they perform triangle to edge and edgeto triangle operations, and store all pending operationsin a priority queue ordered on edge length. Because therules engine successively applies rules sorted by energylevel, each time a rule is evaluated, the rules enginelooks at which mesh objects were altered by the ruleand removes any pending operation that also dependson these objects. If that pending operation is still valid,it will be triggered and queued in the next time step.These rules enable us to to represent objects separat-ing and rearranging, and we intend to add more rules asneeded.

3.2.1 Radial Edge Split

The triangles in an evolving mesh can become too largeto adequately represent an object’s surface which canresult in increased numerical error. In order to accu-rately sample and represent spatially variant physicalquantities, we must refine large triangles. Triangle re-finement replaces a single large triangle with two ormore smaller triangles. Numerous approaches subdi-vide a single triangle into three triangles, but the morecommon approach is split a single triangle into two tri-angles. In a manifold mesh, when a single triangle issplit in two, the neighboring triangle incident to the splitedge must also be split, hence the name, “edge split”.The radial edge split operation is a generalization ofthe commonly used manifold edge split, with the man-ifold edge split being a special case of the radial edgesplit. The radial edge split creates a new vertex at the

midpoint of edge, and identifies all incident trianglesaround this edge, and splits each one of them into twotriangles. The new radial triangles share the space asthe original triangles did. Because the radial edge splitdoes not move any existing vertices and only inserts anew vertex, the operation does not need to check forany topological or geometric violations - the radial edgesplit operation is always topologically valid. As such, itis one of the simplest mesh operation to implement.

For each triangle in a radial edge, the radial edge splitoperation identifies the outer-most vertex, and createstwo new triangles and removes the original triangle.Each of the two new triangles share an edge formed bythe new center vertex and the outer vertex. Radial edgesplit then reconnects these new triangle neighbor point-ers to the triangles adjacent to the removed triangles.

3.2.2 Radial Edge Collapse

As a simulation evolves in time, a triangles can becomeexcessively small. This over-refinement leads to wastedcompute resources and performance degradation. Theradial edge collapse operation removes small trianglesand re-connects these triangles’ neighbors. The radialedge collapse is a generalization of the conventionalmanifold edge collapse operation as studied by [15, 7],where the manifold edge collapse is a special case ofthe radial edge collapse. In a manifold edge collapse,an edge can be incident to exactly two triangles. A ra-dial edge however can have any number of triangles ar-ranged radially around an edge as in Fig.3. The ideahowever is the same: we want to remove the edge andre-connect all of the neighboring triangles. A varietyof different trigger conditions may be appropriate forinitiating a radial edge collapse. Say one would liketo coarsen a mesh in areas of low curvature, or, sayone wants to remove all triangles below a certain edgelength threshold, as is the case in [6]

A radial edge collapse traverses every triangle in a ra-dial edge and collapses the triangle along the edge side,and re-connects the neighboring triangles on the col-lapsed triangle’s two remaining edges. The edge col-lapse removes a region of mesh which consists of theedge itself and its two incident triangles and enlargesthe neighboring triangles to fill the void. The edge col-lapse operation reduces the dimensionality of a mesh.As such, it is only applicable under specific conditions.For an edge collapse (or any other operation) to bevalid, it must not invert any triangles, i.e., it must notchange any triangle’s normal by more than 90 degrees.Vieira et al. [15] identified that a manifold edge collapseis valid if it does not violate the link condition. The linkcondition sates that an edge e = {u,v} can be collapsedif and only if link(u)∩ link(v) = link(e). In a triangularmesh, the star of a vertex v is the set of triangles andedges that are incident to v. The link of a vertex is the

frontier of the star. The frontier of the star is the set ofvertices that are incident to every edge and triangle inthe star, not including v. The links of the edge and eachvertex for non-manifold meshes are slightly more com-plex to calculate than the manifold case. We first buildthe edge link by iterating over every triangle in the ra-dial edge and inserting the outer vertices into the edgelink set. Because we only need to test that the intersec-tion of the u and v link sets are equal to the edge linkset, we do not need to build the entire link set for eachvertex. Rather, we only need to build the link(u) set outof the vertices adjacent to u, that are not in the link(e)set. If we find a vertex adjacent to the other edge vertexv that is not in link(e) and not in the link(u), the radialedge violates the link condition.

In a radial edge collapse, the link condition alone doesnot guarantee that a vertex move will not invert a tri-angle. We must also explicitly test triangle incident toeach of the radial edge endpoints. We test each of thesetriangles by test-moving the edge endpoint vertex intoa trial position and testing for a change in the trianglenormal direction. Presently, we also test to ensure thata radial edge collapse will not collapse a tetrahedrondown to a triangle. We check this by looking at the par-tial triangles on each face of a radial edge triangle. Ifthat partial triangle’s two outer neighbors are adjacentto each other, then this partial triangle and its two neigh-bors define three faces of a tetrahedron, with an openbase. The collapse of this triangle will result its twoneighboring triangles collapsing down to each other -that is, collapsing a tetrahedron to a triangle.

Figure 4: The radial edge collapse operation collapsesa set of radial triangles (green) down to a single vertex,pulls in the remaining triangles (pink) to fill the newlycreated hole, and connects each remaining triangle tomaintain mesh connectivity.

3.2.3 Vertex Split

The cell valence count of a vertex is the number of cellsthat share that vertex. The previously described radialedge collapse removes a vertex from a mesh and at-taches all of this vertex’s cells to another vertex, thuspotentially increasing the cell valence count of the re-maining vertex. Many mesh generation packages, tendto produce meshes that are predominantly pentahedron

and hexahedron, thus contain a large number of ver-tices with eight cell valences. Eight-valence vertices areunusual in nature, and high or unbalanced vertex cellvalences also tend increase computational cost and nu-merical error. The vertex split operation here is a non-manifold generalization of Hoppe’s [7] manifold vertexsplit. The vertex split operation here splits a single ver-tex into two vertices and detaches a cell from a vertexin order to reduce vertex cell valence count. We cansee in in Fig. 5 an example of a vertex with a high cellvalence count, (five in this case), and relaxed configu-ration after five vertex splits. The vertex split operationenables cells to reconnect to each other. In a manifoldmesh, a vertex split is the exact inverse of the edge col-lapse operation. In our non-manifold mesh, the vertexsplit operation shares a similar relationship with the ra-dial edge collapse, though they are not exact inverses ofeach other.

A

BC

DE

uA

BC

DE

uv

A

BC

DE

uv

A

BC

DE

uv

1 2 3 4

Figure 5: A sequence of vertex split operations in 2D.The central vertex initially has a cell valence count of5, and the C cell (in red) is selected for ejection. Thevertex split locates the shared face between C and itsneighboring cells, creates new vertex towards the cen-ter of C, pulls the neighboring face of C to this new ver-tex, and creates a set of triangles to fill this void. Theoriginal vertex now has a cell valence count of 4. Thevertex split then selects the A cell for ejection, createsa new vertex, moves the shared face, and fills the voidagain. The final images show the mesh after two subse-quent vertex split operations. Note, the vertex split is alocalized operation, the frontier elements of these cellsare not modified.

A vertex split begins with a candidate vertex u andidentifies a target cell C to disconnect and pull awayfrom u. Vertex split will then split u into two ver-tices, u and v, where v is attached to the target cell,C, which is then no longer incident to u. As C is dis-connected from u, u’s cell valence count is reducedby one. The cell valence count of v is then the num-ber of cells that that are both incident to the originalvertex u, and adjacent to C, i.e. valence_count(v) =|incident_cells(u)∩ad jacent_cells(C)|+ 1. Thus, inorder to reduce net valence counts, it is important forvertex split to choose C such that valence_count(v) <valence_count(u). Otherwise, vertex split will indeedreduce the valence count of u, but it will also create an-other vertex v that has the same valence count that uoriginally had.

The most challenging task of the vertex split operationis filling the hole created by splitting and moving thevertex. This task is illustrated in Fig. 6. The vertexmove creates a void between the cell being moved (C)and the set of cells adjacent to C, and incident to theoriginal vertex u. In order to fill this void, the vertexsplit iterates around the triangle fan centered at v, com-posed of triangles that face C. Every triangle in this fan,by definition has exactly one face in the C cell surface.Consider a sequential pair of triangles in this fan. Thenon-C facing partial triangles either belong to a samecell, or they point to different cells. If both partial tri-angles point to the same cell on both sides, then thereis nothing to do, these triangles simply shift slightly, astheir center vertex gets attached to a new vertex v. If,however, there is a cell change, (i.e., the first triangle’snon-C facing partial triangle points to a different cellthan the second triangle’s non-C facing partial triangle)then the vertex split inserts a new triangle at this edge.We say that triangle t1 is incident to cells A and C, andtriangle t2 is incident to cells B and C. Vertex split thencreates a new triangle, tn with vertices v, u, and v f , thevertex on the frontier of the fan that is incident to botht1 and t2. The tn triangle faces cell A on one side, andcell B on the other. The vertex split will continue it-erating around this fan until it encounters the startingtriangle. Vertex split here creates a set of new triangles,one for each pair of triangles that have a cell change.All of these new triangles share the new {u,v} edge,and when vertex split completes creating these new tri-angles, it then connects their partial triangles togetherappropriately. We can see that vertex split is essentiallythe inverse of the radial edge collapse, in that a radialedge collapse removes a set of triangles around a radialedge, and vertex split creates a set of triangles around aradial edge.

A

B B B

CCC

A A

(1) (2) (3)

t1t1t1

t2 t2 t2

tnvf vf vf

vvu

u u

Figure 6: Three stages in a vertex split operation. Thisoperation first identifies a target cell, C in yellow, andpulls this cell away, then splits the vertex u into u andv, and finally fills in the resulting holes with new trian-gles. Cell C is adjacent to cells A and B. Vertex splitensures that the partial triangles on the new trianglesare connected to the correct cell boundaries.

3.3 Implementation DetailsBiological cells and membranes participate in chemi-cal processes that can often occur at significantly fastertimescales than the motion of the objects themselves. In

biological material simulations, the most computation-ally expensive task is evaluating these continuous timechemical reactions local to cells, surface elements andthe extracellular milieu, as well as the chemical fluxesbetween neighboring spatial objects. The topologicalrearrangement operations previously discussed performa significant number of mesh traversals when evaluatingmesh operation rule triggers, and when performing themesh update operations themselves. In order to max-imize the performance of the chemical process solverand mesh traversals, we choose to explicitly store allneighbor relationships in the mesh at the expense of in-creased memory usage and software complexity.

We provide two basic mesh traversal operations: ra-dial edge and triangle fan traversals. A radial edgeis an edge connecting two or more incident triangles.To enumerate these incident triangles, the radial edgetraversal (1) starts with a partial triangle and a pair ofvertices that define an edge, (2) finds the next partialtriangle that is both a neighbor of the current partial tri-angle, and incident to the edge, (3) identifies the op-posite partial triangle (4) continues the iteration until itcome across the starting partial triangle. A triangle fanis the set of triangles that are incident to both the samevertex and the same cell. (i.e., a triangle fan is a set oftriangles, all of which face the boundary of a cell andsurround a specified vertex.) The triangle fan traversal(1) starts with a triangle, vertex and cell, (2) followsthe partial triangle neighbor pointers until the originaltriangle is encountered again.

4 RESULTSWe have created a set of simple models that demon-strate the mesh reconnect rules. These models read aninitial configuration from a gmesh file. These modelsall implement a surface tension term (a force that actsin a direction tangent to the surface), and a volume con-straint (acts perpendicular to the surface).

We use Zheng’s method [17] to calculate the surfacetension force. This method calculates the surface ten-sion contribution of a triangle’s three barycentric re-gions to each corresponding vertex. Each barycentricregion pulls its’ incident vertex in towards the triangle’sbarycenter. The magnitude of each Voronoi region’ssurface tension is proportional to the 1/2 the length ofopposite edge. For details, see page 39 in [17].

We define the volume constraint on a per cell basis asthe difference between a target volume and the presentvolume, C(X) = λ (vt − v), where vt is the target vol-ume, v is the current volume, and λ represents the stiff-ness of the constraint. Cells with a lower λ are “softer”,as the constraint solver adjusts the cell positions morerapidly in cells with a larger λ . We approximate thevolume constraint Jacobian, as in [2] as area weightedsurface normal to each vertex. We use the barycentric

area of each triangle to estimate the surface area of eachvertex, and we compute the total volume of each cellusing the divergence theorem.

To illustrate the effect of the edge collapse and vertexsplit operations, we performed an experiment with fourcells and applied a harmonic bond force to the center ofmass of two cells in order to bring the cells in contactdepicted in Fig. 7. The initial condition contained thetwo red cells topologically connected with each other -they share triangles, and the two blue cells are discon-nected.

We then apply a harmonic force to the center of massof each blue cell which caused the blue cells to comecloser, and start to push the red cells apart. We can seethe edge length shrinking between the blue cells. Whenthe edge length drops below a threshold, the radial edgecollapse rule culls these short edges. This operationhowever increases the cell valence count on the centralvertexes to four. The increased valence count causesthe rules engine to invoke the vertex split operation. Wespecified a rule that causes the cell with the highest cur-vature to be ejected from the vertex. The vertex splitoperation splits these vertices and creates the new trian-gles in yellow. We can then see that as these two bluecells continue moving towards each other, these cen-tral edges continue to fall below the threshold, and con-tinue to be culled, thus invoking further vertex splits.In the last frame, we can see that all of the shared con-tacts between the red cells has been eliminated. At thispoint, we remove the force between the two blue cells,and allow system to equilibrate. We can see that in thefinal frame, the material is topologically very distinctfrom the initial configuration. In the final frame, theblue cells are not connected and the red cells are dis-joint. The mesh operations provide one possible wayto maintain a memory on materials, as these are atomicoperations that alter the mesh structure and topology.

To illustrate the effects of differential surface tensionand demonstrate the mesh update operations when cellsseparate, we created a model that mimics a simple bi-ological model of cellular mitosis. Here, we representa biological cell that is about to undergo mitosis as apair of Mechanica cells in contact, separated by a mem-brane. We first create two cubic cells in contact andset the surface tension of the membrane that separatesthese two cells to zero and set the surface-tension ofthe membrane not in contact to a positive value andallow the simulation to relax. As expected, the twocells in contact in a relaxed configuration form a nearlyperfect sphere as in Fig. 8. Because the shared mem-brane is initially not under tension, it does not exert anyforce on the outside membranes. Once the system isrelaxed (no net motion), we increase the surface ten-sion on the shared membrane to a positive value. Wecan see that this shared membrane begins to pull in-

Figure 7: A series of frames from a simulation thatstarts with an initial configuration with the two red cellsconnected. We then apply a force between the two bluecells, which causes the blue cells to come towards eachother and push the red cells out of the way. We can seethe surface shrinking between the blue cells, as theseedges shrink. The rules engine invokes the edge col-lapse operation to cull the short edges, which results ina number of vertices with a large cell valence count.The rules engine then invokes the vertex split operationto peel the red cells away. We then remove the harmonicbond between the blue cells, the system then relaxes onits own. The yellow regions indicate triangles that weregenerated by vertex split operations.

wards on the outer membrane, and the two Mechan-ica cells in contact begin to take the classic shape ofa biological cell undergoing mitosis. We then increasethe shared membrane surface tension to a value largerthan the outer membrane surface tension. At this point,the shared membrane will continue pulling the outermembranes inwards, and shared contact area betweenthe cells shrinks. As this shared area shrinks, the edgelength of triangles within this shared area drop belowthe edge length cutoff, and the rules engine applies theradial edge collapse rule to cull these offending trian-gles. The shared surface area continues to shrink andpull inwards on the outer membranes until the sharedarea eventually disappears. The shared area disappearswhen the rules engine culls the last remaining trianglein that area. At this point, the two cells are no longerin contact, and their respective surface areas relax toapproximate a sphere.

5 CONCLUSIONSResearchers who simulate biological systems must takeinto account both mechanical and chemical processes.They must also represent objects that continually moveand change neighbor relationships. Most existing nu-merical simulation approaches such as the finite ele-ment or the finite volume methods excel at represent-ing certain aspects of physical objects. Finite elementis ideal for simulating mechanical properties of mate-rials with limited structural rearrangement. Finite vol-ume methods excel at simulating complex fluids in rel-atively stationary geometries. Few, if any simulation

Figure 8: A simple cellular mitosis model. Two cellsare initially in contact with a shared membrane, andeach cell’s individual membrane has a surface tensionof σ , and the the shared membrane’s net surface ten-sion is zero. As the shared membrane is completelyrelaxed, the two cells in contact form a sphere. We thenincrease the surface tension of the membrane to a pos-itive value which causes the membrane to shrink. Asthe membrane shrinks, the mesh update operations au-tomatically cull the triangles with short edges and splitlarge triangles. The mesh update operations enable theshared membrane to shrink and eventually disappear, atwhich point the cells separate.

platforms that can do both at once using the same simu-lation approach. The novel mesh data structure we havedeveloped meets this challenge and enables us to repre-sent a wide range of physical and biological materialsin a unified way. As an example of the applicabilityof our platform, consider again the epiboly process de-picted in Fig. 1. This biological process is challeng-ing to represent using current simulation approachesbecause it involves a variety of different chemical andmechanical processes. Presently, researchers must re-sort to integrating a range of different numerical sim-ulation methodologies and programs. Using our newnumerical simulation engine, the representation of theepiboly process is much simpler, is done entirely selfconsistently, and requires no integration with externalsolvers. We are presently developing a cell-divisionmesh update operation that will split a single cell intotwo cells. When this new operation is ready, we willbe able to directly represent this key biological process.Take, for example, Fig. 1. This image depicts a clus-ter of biological cells on top of a yolk sack. The cellsstart out as a blob on top of the yolk sack. This blobthen gradually spreads out across the yolk, ultimatelyenveloping most of the yolk. We could represent eachembryo cell as specific Mechanica cell type and rep-resent the yolk as another cell type. We know thatthere are a number of chemical reaction processes thatoccur inside each embryonic cell. We presently sup-port implementing chemical reaction networks insideour cell types. We also know that these embryo cellscommunicate with one another via intricate signalingpathways, and we implement these as chemical fluxesbetween cells. We also know that the embryo cells areinitially blob-like, and later move to envelop the yolk.We can represent this transition with a surface tension

like force between the cells and the yolk. Initially wecould have a high surface tension between the embryocells and yolk, and lower surface tension between em-bryo cells of the same type. Later, possibly based onsome chemical state variable in each embryo cell, wecould have the embryo-yolk surface tension reduce, andthe embryo-embryo surface tension increase. Here, wewould expect the embryo cells to then decrease theiraffinity towards each other, and their affinity towardsthe yolk to increase and spread out over the yolk. Theembryo development process exemplifies how biologi-cal processes couple chemical and mechanical interac-tions - a feature that the novel numerical simulation en-gine we have developed uniquely captures, enabling re-searchers to model these kinds of processes. The meshdata structure presented here lays the groundwork forthis numerical simulation engine.Our simulation code is currently under active devel-opment, all source code and binaries will be madefreely available on the Mechanica website, (http://www.mechanica.org) under a open source (GPL)license.

6 ACKNOWLEDGMENTSWe acknowledge generous financial support the Na-tional Institutes of Health, National Institute of Gen-eral Medical Sciences, grant R01 GM122424, and Na-tional Science Foundation grant 1720625. We thank Dr.David Umulis, Dr. Marie Gingras, Dr. Amit Hagar andDr. James P. Sluka for their discussion and insights.

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