Computers & Graphics 34 (2010) 294–303

Contents lists available at ScienceDirect

Computers & Graphics

0097-84

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journal homepage: www.elsevier.com/locate/cag

Technical Section

Developmental modelling with SDS

Benjamin Porter �, Jon McCormack

Centre for Electronic Media Art, Monash University, Clayton, 3800 Victoria, Australia

a r t i c l e i n f o

Keywords:

Developmental model

Morphogenesis

Physical simulation

Tetrahedral mesh

93/$ - see front matter & 2010 Elsevier Ltd. A

016/j.cag.2010.05.008

esponding author.

ail address: benjamin.porter@gmail.com (B. P

e term ‘model’ has a dual, context dependen

mental context it means a formal mathematic

stem; in a computer graphics context it mean

e generated for image synthesis and anim

ent geometric mesh). We anticipate the

g based on context.

a b s t r a c t

This paper describes modelling methods based on biological development for use in computer graphics

applications, specifically the automated growth and development of complex organic shapes that are

difficult to model directly. We examine previous approaches, including grammar-based methods,

embedded systems and cellular models. Each system can be classified as endogenous (internally

determined) or exogenous (externally determined), with some models exhibiting features of both. We

then introduce a new model, the Simplicial Developmental System (SDS), which simulates individual

cells embedded in a physical environment, with cell division, movement and growth controlled by

morphogenetic chemical simulation. SDS uses a tetrahedral mesh as its base representation for

geometric modelling and physical simulation. Cell growth, movement and division are determined by

simulating chemical morphogens that are diffused between cells according to a set of user defined

rules. We discuss the advantages and disadvantages of this model in terms of the competing goals of

user control, developmental complexity and open-ended development (the ability to generate new

component structures without explicit specification). Examples highlighting the strengths of the model

are illustrated.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

One of the greatest challenges in biology is understanding themechanisms that enable a single, fertilised cell to develop into acomplex, multi-cellular organism. The developmental processesthat lead to interacting functional components and self-organis-ing, heterogeneous structures are complex and numerous,remaining the subject of on-going research. In this paper wefocus on systems, inspired by biological processes of develop-ment, that begin with concise specifications for initial conditionsand developmental rules, then proceed to grow and developautonomously in simulation. Our application is in computergraphics, so we concentrate on the process of growing three-dimensional form rather than a complete simulation of biologicalfunction. Our goal is to define systems that allow the specificationof complex organic shape and form, removing the tedium oftrying to assemble such forms manually.

The process of simulating development in this more generalcontext is referred to as developmental modelling. As a modellingmethodology, developmental models1 offer a vastly different

ll rights reserved.

orter).

t meaning in this paper: in a

al or procedural specification

s a spatiotemporal geometric

ation purposes (e.g. a time-

reader will distinguish the

modelling paradigm over commonly used methods in computergraphics. ‘Traditional’ modelling focuses on processes organisedaround the design of artefacts: the artist conceptualises a form,and then proceeds towards the goal of building the formexplicitly. This is normally achieved through the direct instantia-tion and manipulation of geometry and texture to achieve thefinished result. Procedural modelling uses the specification ofprocedures that automate the geometry and texture buildingprocess. Generative models exploit the generative properties ofcomputation to achieve database amplification [2], where complexstructures are built from relatively simpler specification (oftenby several degrees of magnitude). In this sense, developmentalmodels are a subset of generative models.

There have been two complementary approaches to modellingdevelopment in computer graphics. In endogenous systems,development originates from, and is driven by, internal, rule-based mechanisms. Typically, the rules are specified by the user ofthe system. Endogenous models work well for the proceduralspecification of complex, static or time-varying geometry; how-ever, there are limitations in how models generated by the systemcan be affected by, and interact with, their environment. For asystem to develop in this way—an exogenous system—the formand space in which it develops must be coupled.

In this paper we will first review some of the literature for bothendogenous and exogenous methods, with a brief discussion onthe advantages and limitations of each approach (Section 2). InSection 3 we introduce the Simplicial Developmental System (SDS),

Fig. 1. (left) An abstract organic form generated using the system introduced in

this paper. (right) A related form manufactured in stainless steel (by Shapeways

[1]).

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303 295

a tetrahedral mesh based system that supports exogenous growththrough the physical simulation of a developing 3D form (e.g.Fig. 1). Finally, Section 4 discusses implications and furtherdevelopment of the SDS.

2. Modelling development

There are many established systems in computer graphics thatincorporate aspects of biological development for modellingshape. The fields of theoretical biology and artificial embryologyalso contain models that are useful when considering shapeformation from a developmental perspective. Surveys that coverthese models in the literature are numerous [3–11] and henceonly a brief overview is given here, with discussion pertinent tothe model introduced later in the paper.

2.1. Grammar-based approaches

L-systems, introduced by Lindenmayer [12] to model thedevelopment of multicellular organisms, have been extensivelyused to simulate the development of trees, herbaceous plants, andmany other biological structures. Early L-system models wereendogenous, operating at a symbolic level, with component partsand their development specified using a set of distinct symbols,and the development simulated by parallel re-writing of thosesymbols. As development proceeds the set of active symbolschanges in discrete time steps. Symbols representing componentparts are converted to geometry as a unidirectional, post-development operation, i.e., no information from the environmentor built geometry flows back to the symbols to affect theirdevelopment.

Researchers recognised these limitations, over subsequentyears devising enhancements and additional mechanisms tocircumvent them. The primary goal in computer graphicsapplications has been the synthesis of visually realistic models,with an emphasis on the final, rather than developing, geometricform. L-system extensions include: continuous mechanisms[6,13], the modelling of physical and mechanical effects [14,15],specifying explicit hierarchy [16] and coupling the developmentalprocess more closely to the environment [17]. These extensionsaddress specific issues, a number incorporating exogenousdevelopment, but they diminish the simplicity and elegance ofthe original formalism.

In general, L-systems simulate growth and development at anabstracted macro-level (e.g. florets, meristems, leaves, branchingsegments), due to the complexity of defining interactions at themicro-level (cells, molecules). The appropriate choice of macro-level abstraction relies on a structural understanding of the entity

being modelled, along with a means of inferring developmentalrelationships at the level chosen. In general, this is an ill-definedand ad hoc process, making it difficult for computer graphicsusers without significant experience to devise grammars thatgenerate the desired form.

L-systems abstract the processes behind development to that ofreplacement of parts by other parts. In this respect they are related toother grammar-based approaches such as shape grammars [18],graph grammars and graph-based representations [19]. They havealso been generalised to grammars that operate on polygonalsurface representations [20–22]. These methods address thetopologically focused bias of tree L-systems and provide generativemethods for more complex geometric surfaces. Developments haveenabled the application of grammar-based methods to a broaderclass of shape and form, such as architectural design [23] andlegged animals [24].

2.2. Embedded models

Many systems directly embed development in a spatialenvironment, an idea that dates back many years in graphics[25,26]. A structure that is being generated within a space can beaffected by its own parts, other static and dynamic objects, andenvironmental factors such as light and gravity. These interac-tions provide a mechanism for directing the growth of specificforms, and also result in an emergent complexity difficult toobtain using non-embedded systems. While embedded systemsappear to be a natural and biologically realistic approach (after all,all real biological growth is embedded in a physical environment),taking full advantage of this embedded approach presents anumber of challenges [27], particularly if the open-endedcomplexity of real biology is sought.

Cellular automata models demonstrate that exogenous growthfactors, including environmental effects and spatial limitations,can contribute greatly to the complexity of a developing form[26,28–33]. Even if the mechanism behind the development issimple, environmental interaction can result in complex creationsthat far exceed the simple specification from which they emerge[34,35].

A developmental model of accretive growth that illustrates thecombination of a geometric surface-based developmental modelwith a physical model of nutrients and hydrodynamics ispresented by Kaandorp and Kubler [36,37]. These experimentsreinforce the notion that a simple growth logic combined with aphysical model can result in complex organic forms. This isfurther exemplified by Combaz and Neyret, who demonstrated asystem that generates rich and abstract organic form throughphysical simulation and growth [38]. The user paints growthchemicals onto a surface, which causes that part of the surface toexpand and grow, resulting in naturally wrinkled surfaces due togrowth-induced deformation. Similar systems in computationalbiology can also be found [39,40].

2.3. Cellular models

Cellular models utilise the idea of building complex structurefrom a single underlying primitive—the cell—which typicallydivides, moves or changes based on an abstraction of chemicalsignalling or protein synthesis. Some of these models areembedded in restricted spatial arrangements (cartesian grids,isospatial sites), others operate at a more symbolic level. Incomputer graphics applications they have been used to modelthree-dimensional form and two-dimensional textures.

Fleischer and Barr [41] introduced a cellular programmingmodel that could grow cellular texture elements over pre-defined

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303296

surfaces [42]. Each cell consisted of a cellular program (a time-varying first order differential equation) that could control itsplacement over the surface based on, for example, simulation ofchemical reaction–diffusion over the surface. The method offeredcapabilities such as cell movement, adhesion and changes in sizedue to cell–cell interaction.

Kumar and Bentley used a method of ‘oriented cell division’controlled by a simple genetic regulatory system as a basis for anevolutionary form design system [9]. Each cell was represented bya sphere with isospatial sites for cell division. They experimentedwith several cell division methods, but were only able to evolvesimple shapes such as a line of cells and compound sphere,partially due to the difficulty of defining fitness measures forevolving development into more complex shapes.

Miller describes an embedded, lattice-based cellular systemthat uses genetically evolvable feed-forward Boolean networks foreach cell’s program [43]. Binary state information is used as amodel of ‘chemical signalling’ and the networks determinechanges in cell state and growth. Miller’s goal was to overcomethe difficulty of defining the cell rules that lead to a specificpattern, inherent in systems such as that of Fleischer and Barr.Using an evolutionary algorithm, he was able to evolve a 2D cellpattern that resembled the French flag from a single zygote cell.

Some developmental models incorporate hierarchies as anexplicit feature, for example, P-systems [44], Vaario’s Multi-LevelInteraction Simulation (MLIS) language [16] and McCormack’sCellular Developmental Model (CDM) [45]. To date, the predominantapplications for P-systems and MLIS have not been in the creativedomain (P-systems have been used primarily for studying computa-tion and MLIS was designed to evolve artificial neural networks).CDM, however, was designed to generate complex time-varying3D form.

CDM uses a hierarchical specification of developing cells, eachwith a set of predicate-action rules. In addition to these rules, cellscontain a continuous state vector that is updated as the celldevelops and rules are applied. Rules control changes in cell state,and if certain conditions are met, cell division, replacement ordeath. Rather than representing cells as simple geometricprimitives, geometry in CDM is built using generalised cylinders,a fundamental design element in the functional morphology ofmany species [46]. Sequences of cells are interpreted as instruc-tions to a state machine that builds the geometry. The hierarchicalspecification of the CDM allows low-level details of modelconstruction to be encapsulated, parameterised, and reused byhigher-level cells that specify body parts and other arrangements,overcoming the problems with single macro-level specificationdiscussed in Section 2.1.

Developmental complexity is reflected not only in the numberof individual parts that make up a model, but also in temporaldevelopmental interactions. While L-systems, for example, arecapable of increasing the number of developing symbols dis-cretely over time, continuous state systems, such as CDM exploitthe non-discrete, temporal nature of the development, allowinganimation effects in models such as continuous growth orsimulating the gaits of legged figures.

While CDM is a flexible and powerful method for develop-mental modelling of time-variant natural forms, it still shareswith its predecessors some of the limitations of endogenoussystems. That is, development is at best only partially embeddedin the environment of final representation. Cells develop instructures with limited spatial relations, hence physical relation-ships may influence development. However, a cell may representcomplex geometric development that is realised ‘post-develop-ment’, making physical interaction with the environment thataffects development difficult. While physical parameters can befed back into cell development (e.g. tropisms, chemical gradients),

physical interaction between the development process and thegeometry it generates is limited.

On the other hand, fully embedded cellular systems (such asMiller’s Boolean network cells) trade developmental and visualcomplexity for spatial interaction. The emergent nature of spatialinteraction makes it difficult to intuitively design underlying rulesthat will grow specific forms, hence the use of evolutionarysearches to try and find the rules necessary to grow specific shapes.

3. The simplicial developmental system

A new system, the Simplicial Developmental System (SDS),was designed to address environmental and physical effects on agrowing structure, with the goal of overcoming the limitationsdiscussed in the previous section. SDS models a developingorganism as a simplicial complex within a spatial environment.More specifically an organism in k dimensions is a collection ofconnected non-overlapping k-simplexes joined together by(k�1)-simplexes. In 2D this is a collection of triangles connectedby their edges and in 3D it is a collection of tetrahedra connectedby their faces. Driven by an internal program, the cells of theorganism grow, divide and move—transforming the simplicialcomplex. Through a morphogen-based cell communicationmodel, cells can coordinate their activity and develop coherentmodules within the larger organism.

A mass–spring model defines energy minimising forces thatact upon the simplicial complex, resulting in a soft-body elasticappearance. Additionally a non-overlap constraint results in asurface that interacts in space. Other spatial and physicalelements such as static geometries, tropism sources, directionalgravity, or even other developing organisms can be included inthe simulation environment. This results in a type of exogenous

complexity that is difficult to achieve with previous endogenousmethods, such as CDM and L-systems, but still permits a relativelyflexible degree of control that was often lacking in priorexogenous systems.

SDS was developed with three dimensional form generation inmind, however it can be instantiated in either two or threedimensions. The discussion here is limited to the three dimen-sional case: SDS3. Details on the two dimensional case, SDS2, canbe found in [47]. The basic concepts in SDS are the organism andthe cell. An organism is composed of cells that are spatiallysituated and topologically related through the geometry whichtransforms over time (Section 3.1). A physical model (Section 3.2)uses the geometry and a mass–spring model to give the organisma dynamic elastic behaviour. The final component of the system isthe process model (Section 3.3), which defines the cells asindividual agents, communicating with each other and perform-ing various actions that drive the geometric transformation of theorganism. Each of these aspects are now considered in turn,followed by some examples (Section 3.4).

3.1. Geometry

A shape in SDS3 is represented as a set of connected non-overlapping tetrahedra joined to each other by their faces. Anorganism consists of a set of cells, with each cell corresponding toa unique vertex of the organism’s shape. Fig. 2 illustrates anexample organism and its shape. The edges of the shape define atopology amongst the cells—if an edge connects two cells thenthose cells are topological neighbours. This shape representationhas many benefits including conceptual simplicity, the ability torepresent detail over many scales, and the ability to modelarbitrary forms.

Fig. 2. An example SDS organism. It has (a) vertices, edges and (b) tetrahedra. It

also has (c) spherical cells. These are all views on the same structure. There is a

one-to-one mapping between cells and vertices, for example, vertex v corresponds

to cell c.

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303 297

In SDS, the starting geometry of an organism is usually verysimple. This initial state is analogous to an axiom in L-Systems orthe root system of CDM. Through a set of local transformationsthe geometry gains complexity. These transformations are celldivision, cell movement and cell growth.

3.1.1. Cell division

Cell division is the primary transformation; it adds new cells,edges and tetrahedra to the geometry, providing the accumula-tion of complexity over time. A cell may elect to divide in aspecific direction, at which point it is removed and replaced withtwo or more cells. The tetrahedral complex is modified toaccommodate them. The new cells occupy different positionsand have an equal distribution of the mass of the original cell.Topologically, the new cells are neighbours and have the localneighbourhood of the original cell distributed evenly amongstthem. An internal cell can divide in any direction, whereas asurface cell can divide tangential to the surface or perpendicularlyoutwards or inwards. An ideal transformation would minimisesudden jumps in the physical model and localise changes to thetopology of the system as much as possible.

Internal cell division was originally modelled in SDS3 bysubdividing a tetrahedron adjacent to the dividing cell; however,this resulted in a large variation of vertex degree. This isundesirable as it decreases the stability of the physical model.The division algorithm was then modified to achieve better vertexdistribution through optimal tetrahedralisation. This algorithm isas follows:

1.

Let c be the dividing cell, and d be the desired direction ofdivision.

2.

Let Tc be the set of all tetrahedra adjacent to c.S 3. Let F be the hull surrounding Tc.S 4. Remove Tc and c. 5. Add two cells a and b, with ax ¼ cxþEd and bx ¼ cx�Ed, where

ax, bx, cx are the positions of the cells and E is such that a and b

are contained within F.

6.

Case 1 Case 2

Fig. 3. Two different situations that cause a tetrahedron to become flat. Case 1. A

vertex enters the opposite face of the tetrahedron. Case 2. An edge intersects the

opposite edge. These cases can be distinguished by observing that if any vertex of

the tetrahedron lies within the face opposite then Case 1 has occurred, and if not

then Case 2 has occurred.

Tetrahedralise the structure F [ fa,bg.

This last step can be performed in a number of ways. We choseto generate the Delaunay tetrahedralisation using an externallibrary, Tetgen [48].

Following the tetrahedralisation phase, the mass of the mothercell cm is distributed evenly amongst the daughter cells, am¼

bm ¼12 cm. The radii of the new cells are then computed from these

masses assuming a uniform density (e.g., am ¼43pa3

r ). All newtetrahedra and edges then have their rest sizes calculated (seeSection 3.2). This allows the local volume to be preserved andexisting stress the occurs in that area will be transferred into thenew configuration.

Cells on the surface can be similarly divided, with theimportant observation that c is now on the boundary of Tc.Hence we must remove those faces of F that are adjacent to c,resulting in an open hull. The resulting structure F [ fa,bg can be

tetrahedralised using a variety of different methods, in our caseTetgen was used for this.

3.1.2. Cell movement

Given a set of cells, the simplicial complex defines the topologyamongst them. As cells move through space the simplexes(tetrahedra) change shape and occasionally become flat (Fig. 3).When this occurs the topology changes via local adjustments tothe structure. At every time step, we detect whether the signedvolume of each tetrahedron has become negative. The signedvolume of a tetrahedron, t, is:

volðtÞ ¼ 16ðv1�v0Þ � ððv2�v0Þ � ðv3�v0ÞÞ, ð1Þ

where the vi are the positions of the four vertices of thetetrahedron. An inverted tetrahedron is one where volðtÞr0, andthe time of inversion occurs at the point when vol(t)¼0.

Multiple tetrahedra may invert during one single time step. Todeal with this we rewind the simulation back to the firstinversion, perform a movement transformation, and then restartthe simulation from that point. Given that we are at time step u2

and the last time step was u1, for each inverted tetrahedron t, wecompute the time u that the inversion occurs by assuming linearmotion of the vertices and solving vol(t)¼0. This equates tosolving the following cubic (discarding the roots that fall outsidethe interval [u1,u2]):

ðv1ðuÞ�v0ðuÞÞ � ððv2ðuÞ�v0ðuÞÞ � ðv3ðuÞ�v0ðuÞÞÞ ¼ 0,

where

viðuÞ ¼ viðu1Þþu�u1

u2�u1ðviðu2Þ�viðu1ÞÞ:

Restarting the simulation from the first inversion is notefficient if there are many inversions occurring in one time step.This problem has also been identified in the collision literatureand some systems solve it by handling multiple collisions at atime [49]. In SDS, however, handling multiple inversions simulta-neously is non-trivial because of the topological modificationsinvolved. More specifically, our movement transformations givenbelow use the assumption that there are no other invertedtetrahedra besides the target one in order to enumerate thetopological configurations that arise. We now discuss thetransformations of the two inversion cases illustrated in Fig. 3.

Case 1: This occurs when a vertex in a tetrahedron attempts topass through the opposite face, and results in the transformationdescribed in Fig. 4. The figure illustrates the case where neither v

nor f lie on the surface. If v lies on the surface then thetransformation is exactly the same. If f is on the surface then t0

does not exist and hence new tetrahedra are not generated. If v

and f are both on the surface then the transformation cannot beapplied (as this would result in an infinitely thin section). Thissituation has yet to arise in the simulations performed; however,if it does then another transformation could be designed.

Fig. 5. Case 2. (a) Consider the tetrahedron in the diagram. Call its upper edge eu

and its lower edge el. (b) Create the sets U and L by considering all tetrahedra that

share edge eu and el, respectively. (c) Consider the hull around the union of those

sets, U [ L. When the tetrahedron’s volume becomes zero the task is to

tetrahedralise the hull of U [ L.

Fig. 4. Case 1. (a) Consider a tetrahedron t and a vertex v. Let f be the opposite

face. (b) Let tu be the tetrahedron that is joined to face f, and vu be the vertex in tu

that is opposite to f. (c) v is assumed to have just intersected face f. (d) We remove

f and t but keep the faces adjacent to v. We then add a new edge connecting v and

vu thus implicitly tetrahedralising the hull of t [ tu.

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303298

Case 2: This case occurs when opposite edges in a tetrahedronintersect. Let these edges be named eu and el, respectively.Consider the case where eu and el are not on the surface, resultingin a closed hull of all tetrahedra attached to either edge (seeFig. 5). The mesh is transformed at the time of inversion byremoving all tetrahedra within the hull of U [ L and thentetrahedralising the empty space formed. Tetgen was used togenerate the delaunay tetrahedralisation of the empty hull. Ingeneral any tetrahedralisation approach will work, with a tradeoffexisting between running speed and quality of tetrahedra. If eu orel is on the surface then the hull is not closed, and additionaltetrahedra must be added to cover each edge before applying themethod above.

3.1.3. Cell growth

Cells have a radius and mass. These are used by the physicalmodel to determine the dynamics of the organism. By changingsize, cells can affect the lengths of edges and the shape oftetrahedra, allowing the development of different sized regions.Hence one part of an organism can be coarsely modelled withlarge tetrahedra, while simultaneously another part can havemany small tetrahedra modelling smaller features. This is aprimary benefit of the tetrahedral complex representation, asopposed to, for example, a voxel-based representation. Therelationship between cell size and the shape of an organism isdiscussed next.

3.2. Physical simulation

SDS incorporates a physical model in order to generate organicand natural-looking forms. The approach can be classified as amass–spring model [50], a popular technique that has been usedpreviously to model cellular complexes [22,51,52].

Each cell c has a radius cr and a mass cm, where cmpc3r . Every

edge is modelled as a spring. While under compression the spring

simulates internal cell pressure and under tension simulates celladhesion. Following the approach of Teschner et al. [53]tetrahedra also resist compression and tension by modellingthem as generalised springs. The energy within such a spring s canbe defined by generalising Hooke’s Law using the followingpotential energy equation:

EðsÞ ¼sk

2

VðsÞ�RðsÞ

RðsÞ

� �2

,

where sk is the spring stiffness, V(s) denotes the current size, andR(s) denotes the rest size. The difference between the two sizes isnormalised to make the spring stiffnesses scale-free as in [53], thisallows us to use the same stiffnesses throughout the entirestructure. For an edge, e, connecting cells a and b, VðeÞ ¼ jax�bxj

and R(e)¼ar+br. For a tetrahedron, t, we have V(t)¼vol(t) (Eq. (1))and R(t) as shown in Appendix A. At each time step the force fromeach spring, s, acting on each cell, c, is computed as:

FsðcÞ ¼ �@EðsÞ

@cx

Fs(c) can be understood as the direction and magnitude that c hasto be pushed in one particular instant in order to minimise theenergy of the spring. For each cell, all the forces from all thesprings adjacent to it are summed, and then integrated to obtainan updated velocity and position.

An adaptive Verlet integration technique [54] was used as ithas been shown to be efficient and sufficiently stable for this typeof model [53]. An adaptive step size is necessary because thetopological cell movement transformation may require the timestep to be modified (Section 3.1.2). Damping is incorporated intothe system as both damping on the tetrahedral springs andviscous damping. The spring stiffness and damping coefficientscould easily be specified per spring, but for the experimentspresented here, uniform spring stiffness and damping coefficientsare used for the springs on the surface, and another set for theinternal springs. The actual coefficients vary from experiment toexperiment and generally have to be fine-tuned to achieve thedesired results.

So far we have described a model that moves cells aroundspace based on mass–spring equations. This correctly simulatesthe interactions between topologically adjacent cells, but fails tocapture the interactions between cells that are spatially adjacent.If a surface cell collides with another part of the surface we wouldexpect an interaction. The simulator that has been built uses thecollision detection and response scheme of Teschner et al. [55,56]to prevent surface cells from penetrating another part of thesurface. The same system is used to confine the organism within aphysically bounded region, and to model collisions with otherobjects in the world.

3.2.1. Initial conditions and spring multipliers

The SDS simulator takes as input a tetrahedral complexgenerated by Tetgen from a surface mesh. To convert this to anorganism we compute the radii (and hence mass) of all cells. Thisis done based on the edge lengths, computing the radius of a cell,cr, as:

cr ¼1

jNðcÞj

XcuANðcÞ

1

2jcx�cuxj,

where N(c) is the set of all neighbours of c. From the cell radii wethen compute the rest sizes, R(s), for all edges and tetrahedra.Initially this procedure seemed adequate, however, importing atetrahedral complex that has non-regular tetrahedra (a commonoccurrence) resulted in a deformation of the initial shape. This isundesirable from a control point of view, especially as we oftenbegin with a sphere and wish to preserve its smooth surface. To

Fig. 6. A starfish-like form grown with the SDS2 system. Here we are visualising

the cells as polygons, where polygons are adjacent only if there is an edge between

the cells in the corresponding (two-dimensional) simplicial complex.

Fig. 7. A visualisation of how the edges (left) or the tetrahedra (right) of a

developing organism change over time. (top) An organism at the start of a

simulation has a cell chosen to be a growing tip (circled). (from top to bottom) As

morphogen diffuses, cells begin to grow and divide near the growing tip, forcing

the tip to the right and creating a limb-like structure.

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303 299

solve this issue we could set R(s) to be equal to V(s) whenimporting the complex; however, this solution is inadequate ascells often change size and R(s) is hence recalculated. To handlethis, we introduced a spring multiplier constant, sm, that isinitialised as sm¼V(s)/R(s). Then when querying the rest size forthe energy calculations, we use a modified rest size: RuðsÞ ¼ smRðsÞ.We now discuss our model of cell behaviour.

3.3. Process model

From the process perspective cells are viewed as autonomousindividuals executing cell programs within the simulation, similarto CDM. Existing methods of modelling cell processes includeprogramming languages [57], GRN-based network models[58–60], and rule-based techniques [61]. The effectiveness ofthe latter inspired the model used here. A simple morphogen andrule-based approach to modelling cell state and behaviour is used.Given a set of morphogens F¼ ff1, . . . ,fng, each cell, c, contains aquantity of each, cfi

, bounded above by the cell volume.Morphogens can be created and destroyed within cells, andisotropically diffuse between adjacent cells. Diffusion and decayof morphogens are modelled using the standard particle diffusionequation:

@cf@t¼Dr2cf�Ccf,

where cf is the amount of morphogen f in cell c, D is the diffusionrate and C is the decay rate. Each morphogen has its own diffusionand decay rates. This equation is approximated using the discreteLaplacian:

r2cf ¼X

nANðcÞ

minðnf,volðcÞÞ�minðcf,volðnÞÞ

jcx�nxj,

where N(c) is the set of cells adjacent to c and volðcÞ ¼ 43pc3

r . Forsimplicity, the morphogen equations are solved in step with thephysical simulator using the explicit Euler method. At the end of asimulation time step the values of the morphogens may activate aset of rules. For example, the following rule:

cf40:5-divideðrfÞ,

specifies that a cell, upon becoming half full of morphogen f,should divide in the direction where f is greatest. The specifica-tion of the rules and parameters are the user’s interface to thesystem.

Having now discussed all the major components of SDS, wenext provide some examples of how the system can be used togenerate organic forms.

3.4. Examples

In a previous paper [47] a model of primitive limb growth forSDS2 was presented (see Fig. 6). This limb growth model has beenadapted and simplified for the 3D system and the examplespresented here use a derivative of this model. For a discussion ofthe biological inspiration for the model we refer the reader to theoriginal paper.

There is only one morphogen: f. Let ct be a cell state variable,equal to 0 or 1, that indicates whether a cell is a growing tip ornot. The rules are:

r1 : ct ¼ 1-cf ¼ volðcÞ1,

r2 : ct ¼ 0 and cf4K-dcr

dt¼D,

r3 : ct ¼ 0 and cr 4R-divideðrfÞ:

Rule r1 specifies that a cell of type 1 constantly creates f. Rule r2

specifies that a cell of type 0 starts growing at a linear rate, D, when athreshold, K, of the morphogen has been reached. Rule r3 then statesthat once a cell has reached a given size, R, then it will divide towardsthe source of the morphogen. This basic process results in a localisedgrowth towards a growing tip, if the tip is on the surface then we seelimb-like structures forming. This entire process is possible due to thecell rules, the physical model, and the geometric transformations.

Fig. 7 demonstrates the basic sequence of one growing limb.The parameters of the model can affect the rate of growth and sizeof the limb. The model is quite sensitive to the parameters, andsome parameter ranges can result in either no limb growth oruncontrolled tumour-like growth. Finding the appropriateparameter ranges can be done experimentally, but an idealgrowth model would have a minimal set of parameters, eachrelating to some visual aspect of the model. In the case of limbgrowth, an ideal set of parameters would be limb size, growth

Fig. 8. A time series of an organism with six limbs. The limbs eventually collide with the bounding box. They continue to grow in the constrained space and the physical

model causes them to curve and bend, eventually filling the space as shown in Fig. 1.

Fig. 9. This simulation was initialised with a radially symmetric geometry with six growing tips placed upon a rock. Various snapshots of the simulation are shown that

demonstrate the limbs growing outwards and over the rock. The growth model includes a rule to allow the body to grow slowly over time ðr4 : ct ¼ 0 and

cfoK-dcr=dt¼DuÞ. The stripes are generated by a morphogen timer that slowly decays within the growing tip. Once depleted the growing tip signals all nearby cells to

become stripe cells and then the timer is reset. The rate of decay of the timer directly influences how apart the stripes are.

Fig. 10. This simulation incorporated an attraction point (shown as a black dot), just above the initial organism. Starting in the same configuration as in Fig. 8, the growing

tips had a small attraction force applied to them, which resulted in the developmental sequence shown.

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303300

speed, and limb length. Future work will look at building thesekinds of models.

As shown in Fig. 8, multiple limbs can be grown in the sameorganism, simply by specifying multiple growing tips. The re-useof components like this is an important feature of a proceduralgeometry system, as it allows us to abstract away from the level ofcells to the level of limbs and bodies. However, because the limbmodel operates on the level of geometry we get additionalbenefits such as a complex boundary forming between the limbmodule and the body it grew from. This complexity is difficult toachieve using module-based geometric methods [62] andnormally has to be achieved as a post-process.

SDS forms can be influenced by spatial and environmentalfactors, providing another level of control over a growinggeometry. For example, we can restrict the growth of a structurewithin a confined space (Fig. 8), incorporate other objects in thespace (Fig. 9), or include an attraction point which affects thegrowth direction of the limbs (Fig. 10).

Many environmental phenomenon such as phototropism orchemotaxis could be included within the simulation. Therequirements of the user dictate which aspects are importantfor each specific case. For example a form can be made to stick to ageometric object as it grows, allowing vines to be grown on a wall.This embedded physical interaction provides another level ofcontrol over the system and the variety of situations it can model.

Fig. 11 illustrates a variation derived from the basic limb model. Inthis example a timer (non-diffusing morphogen that slowly decays)was added to the growing tip, causing it to stop releasing the limb

growth morphogen after a certain delay. It then activates anothermorphogen which causes nearby cells to grow slowly until they reacha specific size, resulting in buds that form at the end of the limbs.

3.5. Efficiency

SDS can be computationally intensive, particularly for complexmodels with many cell divisions. As an example, the experimentshown in Fig. 9 is analysed here. This model took approximately5.7 min to generate to the stage shown in the figure. The simulationmachine was a standard desktop PC with a dual core 2 GHz processorand 2 Gb RAM. Fig. 12 shows plots of the virtual simulation time onthe x-axis versus the real time to compute (blue) and the number ofcells currently in the simulation (red). In this experiment the numberof cells increases linearly over time. Cell division events for each limboccur more or less simultaneously, due to the symmetric initialconditions. The computation time plot shows the number ofmilliseconds the simulator takes to compute each step. The baseline is punctuated by large peaks that occur when the cells divide,illustrating that the division algorithms take much longer to processthat the base algorithm for physical simulation.

3.6. Current limitations of the limb bud model

The limb growth experiments presented here generate limbsthat are, internally, only one cell thick (see Fig. 7). Conceptuallythe growth model is capable of generating limbs of arbitrary

Fig. 11. This budded form was created using the limb bud model with an extra

rule added that caused the tentacle tips to start expanding after a short time.

0 2 4 6 8 10 12 14

0

500

1000

1500

2000

2500

simulation time s

time

perf

ram

e m

s

0

100

200

300

400

num

ber

of c

ells

Fig. 12. Timing information for the model shown in Fig. 9: simulation time on the

horizontal axis versus CPU time per time step (blue) and number of cells (red). The

spikes in CPU time correspond to division of cells in the growing model. (For

interpretation of the references to colour in this figure legend, the reader is

referred to the web version of this article.)

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303 301

thickness (as demonstrated in [47]) simply by increasing the rateof diffusion; however, generating different limb widths has notyet been explored in the 3D model. Generating these thin limbsresults has some consequences which we briefly outline.

As discussed above, the limb structure develops because cellsare dividing toward the growing tip, pushing it outwards. Whengenerating limbs that are internally one cell thick, we need toensure that the initial configuration has a cell directly below thegrowing tip, for without it there would be no internal pressure forthe tip to move outwards. To guarantee this constraint, weestablish the initial conditions manually by adding a vertex belowall the growing tips. An important goal for future work is toeliminate this manual intervention.

4. Discussion and future work

This paper presents a new approach in developmentalmodelling for computer graphics. The Simplicial DevelopmentalSystem, or SDS, addresses some of the disadvantages of previoussystems, such as L-systems or the CDM. It takes a fully embeddedand physically simulated approach, effectively making it capableof a more accurate and diverse set of simulation possibilities. SDSuses only a single geometric primitive (tetrahedral meshes),simplifying physical and developmental simulation, but conver-sely requiring far greater processing power than previous systemsto simulate the development of complex structures. SDS is a newapproach to generating forms and is still the subject of activeresearch. As such the examples do not encompass all aspects ofwhat is possible within the general framework. Combining

physical and biological simulation of growth processes appearsto be a very powerful methodology, opening many possibilities forfurther research, a number of which will be now be discussed.

Growing other sub-structures: The examples presented here areonly the initial steps towards what could be a highly expressiveand powerful method for synthesising complex organic form. Thelimb bud model was exploited in the examples to generate limb-and tentacle-like sub-forms, and adding additional processes toexpand the palette of interesting sub-forms is currently underinvestigation. We envisage a wide range of different processeswithin SDS that provide the artist with a toolbox of different sub-form and pattern generating tools. It would be highly beneficial tofind a set of axiomatic processes forming the core of this toolbox,and upon which other processes could be built. One of these, forexample, might be local diffusion and directional proliferation,which forms the basis of the limb bud model. Sub-forms such astubular growths, folds, rings and bone-like structures are amongstthe kind of things we could expect to see in a toolbox, along withmethods for laying them out using morphogen patterns andcomposing them into more complex structures.

Patterning: Morphogen creation, destruction, diffusion anddecay plays the role of patterning within SDS. Distributions ofmorphogens can direct specific parts of a mesh to grow, forexample, in the regions where limb development occurs. Morecomplex patterns could be achieved by multiplying two differentmorphogen gradients together. In general, functionally composingmorphogen gradients can lead to complex patterns [63]. Anotherapproach is to specify the equivalent of reaction–diffusionequations [64], which could be achieved through the existingrule-based framework. This latter approach, while difficult tocontrol, would allow the specification of complex texturalpatterns, such as spots and stripes.

User friendliness: The procedural specification of an SDS forminvolves the creation of cell rules that act on, and are activated by,temporal patterns. In addition a number of different parameters,such as spring stiffness, tropism strength, viscosity, and morpho-gen decay rate may be adjusted. Many of these parameters areabstract or irrelevant to the end user, and the specification of rule-sets becomes more difficult for increasingly complex forms. Asdiscussed in Section 2.3, exogenous systems are often difficult tocontrol from a form design perspective, which is why pastapproaches have used, for example, evolutionary search techni-ques to find appropriate parameters and rules. In contrast, SDSallows the possibility of presenting higher-level concepts to theuser, such as limb, pattern and attractor. These building blockscould then be connected together and coupled with a geometricdescription of the scene built using standard 3D modelling tools. Aprocess of translation would then take the high-level descriptionand generate the appropriate cell programs.

Simulation method: The choice of integration schemes for thesoft-body and morphogen simulations were adequate to achievethe initial results presented in this paper. However, thesemethods are crude and fail to correctly conserve physicalproperties, such as the energy within the soft-body or theconcentration of morphogen in the cells. They are sometimesunstable and without the correct choice of time step the physicalsimulation can ‘blow up’. This is an especially significant problemwhen finer geometric details are required. The use of a betterintegration technique, specifically for the soft body simulation,would improve the reliability and robustness of the simulation.Further work is required to compare performance of differentschemes against different requirements, for example, speedversus geometric detail.

Heterogenous material: The simulations presented in this paperhave all been performed using a homogeneous material (all thespring stiffnesses are the same). One exciting avenue of research

B. Porter, J. McCormack / Computers & Graphics 34 (2010) 294–303302

would be to explore the creative possibilities of allowing differentmaterial properties within the same organism. This would allowus to have, for example, rigid bones, softer muscle mass, and ajelly-like material all within the same structure. Aside from theuser-control issues (i.e., how would a user specify the formationof different materials), a major impediment to achieving this isthe physical simulation of such a material. Incorporating highlystiff springs requires a large amount of computation time due tothe increased instability: a commonly observed problem withmass–spring simulation [65]. Research into real-time simulationof deformable bodies for surgical simulation may providesolutions, for example the hybrid approach of Lin et al. [66]combines rigid and non-rigid materials in a conceptually simpleand fast model. Another simulation issue is the refinement ofheterogeneous material. For example, when a cell divides in aheterogeneous region, what is an appropriate procedure fordeciding the physical properties of the new edges and tetrahedra.Recent work into numerical coarsening [67] may offer insight intoa possible solution.

Transformations: Cell death is an obvious exclusion from thetransformations presented here. There are numerous ways that atransform analogous to cell death could be implemented—forexample performing cell division in ‘‘reverse’’. Aside from meshsimplification it is not clear the role cell death would play in thissystem. Other interesting operators that could be researched areones that allow the surface to fuse together and split apart,permitting changes in the topological quality of the surface (andwould allow biological phenomenon like gastrulation and neur-ulation to be modelled.) In addressing some these factors, thesimulation-based approach of SDS will ideally become more user-friendly and expressive.

In conclusion, computational simulation of biological morpho-genesis and development holds great promise as a generalpurpose tool for computer graphics modelling. The idea of beingable to grow a myriad of complex forms without the need todirectly and painstakingly model them by hand has obviousappeal. The next stage in modelling development is to combinethe geometric complexity possible with models like CDM with thedirect embedding and full physical simulation of SDS. This raisesthe issues of computational power (complex physical simulationis computationally expensive) and useful user specification (thetradeoff between specification abstractions, control and complex-ity). Addressing these issues is an active area of on-going research.

Acknowledgements

The authors would like to thank the reviewers for their helpfulcomments and suggestions, in particular for highlighting thenumerical issues that can occur with the simulation techniquesused. This research was supported by an Australian ResearchCouncil Discovery Project Grant DP0772667.

Appendix A. Calculating the rest size of a tetrahedron

The rest size of a tetrahedron t with cells a, b, c and d can beeasily derived from Heron’s formula, where the lengths of thetetrahedron’s edges are the sum of the adjacent cell’s radii. For thesake of completeness it is included here:

RðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiABCDp

3ðarþdrÞðbrþdrÞðcrþdrÞ

with

A¼wþxþy�z,

B¼wþx�yþz,

C ¼w�xþyþz,

D¼�wþxþyþz,

w¼ arbrcr ,

x¼ dr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibrcrðarþbrþdrÞðarþcrþdrÞ

p,

y¼ dr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiarcrðarþbrþdrÞðbrþcrþdrÞ

p,

z¼ dr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiarbrðarþcrþdrÞðbrþcrþdrÞ

p:

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