Lattice-gas cellular automaton models forbiology: from fluids to cells

Bastien Choparda, Rafik Ouareda, Andreas Deutschb,Haralambos Hatzikiroub, Dieter Wolf-Gladrowc

a University of Geneva, Switzerlandb Center for Information Services and High Performance Computing,Technische Universität Dresden, Germanyc Alfred Wegener Institute for Polar and Marine Research, Bremerhaven,Germany

February 28, 2009

1 Introduction

Cellular automata (CA) are discrete, local dynamical systems. They have beenintroduced by J. v. Neumann and S. Ulam in the 1950s as collective model of in-dividual reproduction. During the last years it has become clear that CA havea much broader potential as models for physical, chemical and biological sys-tems. In particular, CA models have now been proposed for a large number ofbiological applications in which one is interested in the emergence of collectivemacroscopic behaviours arising from the microscopic interaction of individualcomponents - being it molecules, cells or organisms. However, there are draw-backs: currently, there exists a huge jungle of different rules for often the sameor similar processes. Therefore, there is need for a specification and classifica-tion of CA rules. Wolfram made a promising start with the classification of 1Dautomata. Furthermore, analysis of CA models is still rather limited and oftenrestricted to visual superficial inspection of simulation outcomes.

In the same spirit as CA, lattice-gas cellular automaton (LGCA) and latticeBoltzmann (LB) models are promising models for studying transport and in-teraction processes in biological systems [5, 31, 9]. In these models the updaterule is split into two parts which are called collision (interaction) and prop-agation, respectively. The collision rule of LGCA can be compared with theupdate rule for CA in that it assigns new states to each cell based on the statesof the sites in a local neighborhood. After the collision step the state of eachnode is propagated to a neighboring node. This split of the update rule guar-antees propagation of quantities while keeping the rules simple. The desiredbehaviour, e.g. spatio-temporal pattern formation of a LGCA shows up in themacroscopic limit which can be derived from a theory of statistical mechanicson a lattice. Instead of discrete particles, LB models deal with continuous dis-tribution functions which interact locally and which propagate after “collision”to the next neighbour node.

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In this chapter, we will emphasise the use of LGCA/LB models and thederivation and analysis of LGCA models ranging from the classical exampledynamics of fluid flow to clotting phenomena in cerebra and the invasion oftumour cells.

2 LGCA for fluid flows

LGCA were introduced by Hardy, Pomeau, and de Pazzis ([15]; HPP) as sim-ple systems of classical particles in order to study approach to equilibrium,time correlation functions, and transport phenomena in fluids. The term ’cel-lular automata’ is never mentioned in HPP. LGCA allowing fluid flow simu-lations were introduced one decade later by Frisch, Hasslacher, and Pomeau([13], FHP) who used the term ’lattice-gas cellular automata’ and by Wolfram[32] who used the term ”cellular automaton fluids“ and considered cellularautomata as discrete analogues to molecular dynamics. The Boolean dynam-ics of LGCA avoids amplification of round-off errors and thus is unconditionalstable. Local interaction and propagation allows very efficient integrations onmassively parallel computers.

The dynamics of fluids in the real world is based on the interaction ofmolecules. Despite the complicated paths of the molecules their interactionrespects local conservation of mass and momentum. On the macroscopiclevel fluid flows obey partial differential equations, namely the continuity andNavier-Stokes equations. The form of these equations can actually be derivedfrom conservation of mass and momentum alone via the Chapman-Enskog ex-pansion. The viscosity depends, however, on details of the interaction. Scalinglength and velocity by characteristic scales of the flow leads to a dimensionlessform of the Navier-Stokes equation that is characterized by the dimensionlessReynolds number Re = L U

ν , where L is the characteristic length scale, U isthe characteristic velocity scale, and ν is the kinematic viscosity of the fluid.The law of dynamic similarity states that two flows behave similar when thescaling leads to the same geometry and Reynolds number. Thus, the transi-tion from laminar to turbulent flow in similar geometry occurs at the sameReynolds number for fluids as different as air and water.

The basic idea of LGCA for simulation of fluid flow is to construct a simpleartificial micro-world of particles which interact while conserving mass andmomentum. The most simple dynamical systems are provided by cellular au-tomata (CA) which are discrete in space, time, and state. CA develop in time byupdate rules identical and synchronous for each cell. The state st+1

i of cell i attime t + 1 is a function of the states of cell i and of n neighboring cells 1, 2, ..., nat time t, i.e. st+1

i = f(st

i , st1, st

2, ..., stn).

In order to use CA to simulate flows one has to define mass and momentumin the artificial micro-world. For convenience let us consider identical cells thatcompletely fill an unbounded checkerboard. Each cell is in one of two possi-ble states: empty or occupied. The definition of mass of a single cell is quiteobvious: empty cells possess mass zero, occupied cells possess unit mass. Thedefinition of momentum requires the introduction of vectors. A possible choiceare the links between the cell centre and the centres of its nearest neighbors. Inthis way each cell would possess several vectors and it it not obvious how tosingle out a momentum vector. HPP [15] added an explicit lattice structure to

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the simple array of CA to solve this problem. They considered a square latticeand set one cell at each link of a lattice node to its nearest neighbor (Fig. 1).Thus each cell possesses a single vector that connects two lattice nodes. Thesevectors are called ’lattice velocities’ , cj (Fig. 1, upper right). Momentum cannow be defined for each cell simply as mass times lattice velocity.

Update rules for for LGCA have to conserve mass and thus cannot be de-fined for single cells because any change between empty and occupied violatesmass conservation. One has to choose a small bunch of cells and update themtogether (which is different from synchronous updating)

(st+1

1 , st+12 , ..., st+1

n

)=

F(st

1, st2, ..., st

n)

. The joint update of all cells at a node is an obvious choice. Forthe sake of simplicity the uptake rules depend on this bunch of cells only.

An obvious question is: Do nontrivial update rules exist, i.e. can one findtransitions between different joint states that conserve mass and momentum?For the HPP model there exists just one nontrivial transition which can be in-terpreted as the collision (or scattering) of two particles that leads to a rotationby 90o (head-on collision, Fig. 1, middle right).

The collision step changes the states of cells at a single node but has no im-pact on the state of cells at other nodes. The propagation of mass and momen-tum to neighboring nodes has to be implemented explicitly: after the collisionstep each particle (= occupied cell) is transfered along the links to the nearestneighbor node (Fig. 1), i.e. mass and momentum propagates like in a gas be-tween collisions. The complete update rule consists of a collision step followedby propagation.

Macroscopic mass and momentum density are calculated by coarse-graining (averaging over larger domains including order of 1000 lattice sites).It came at a surprise that the HPP recipe does not lead to Navier-Stokes inthe macroscopic limit: the non-linear advection term is non-isotropic. It tookone decade to identify the cause of this problem: insufficient lattice symmetry.Frisch, Haslacher, and Pomeau [13] could show that a sixfold symmetry in twodimensions is large enough to yield an isotropic advection term.

There is no single-speed lattice with sufficient symmetry in three dimen-sions. Frisch, Haslacher, and Pomeau [13] and Wolfram [32] proposed to usethe face-centered hyper-cube in 4D and to project onto 3D. The collision rulesfor this LGCA with 24 lattice velocities per node and thus 224 = 16777216 pos-sible states per node are quite complicated ([18, 25, 26]). An alternative for 3Dsimulations is the pair interaction model by [22] that uses a simple cubic latticefor the prize of separate cells for mass and momentum components.

LGCA for the simulation of fluid flows have been applied very success-fully. In order to avoid the noise in LGCA simulations that stems from theBoolean dynamics, lattice Boltzmann (LB) models have been developed firstas an extension of LGCA. Further developments of LB methods have led tomodels that are much more flexible than LGCA and that allow, for example,to simulate turbulent (high Reynolds number) flows ([31, 27], and literaturecited therein). Although in fluid dynamics LGCA have been largely replacedby LB methods, the splitting of CA update rules in interaction and propagationis very useful in other fields, for example if one is interested in the dynamics ofinteracting cells (section 4).

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Figure 1: HPP: collision and propagation. Filled circles denote occupied cellsand open circles empty cells. The lattice velocities cj are shown in the upperright. Head-on collision leads to rotation of particles by 90o (middle right).a) Part of the lattice before collision (e.g. after propagation); there is only onecollision configuration (two particles in opposite cells at the same node; onthe left). b) After collision (e.g. before propagation): the configuration of thecells at one node has changed. c) After propagation: all particles have movedalong the links to their nearest neighbors (the lattice outside the part shownwas assumed to be empty, e.g. no propagation of particles from ‘outside’).

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Figure 2: Left: an aneurysm from an angiogram. Right: illustration of a stent.

3 The lattice Boltzmann approach to simulate clot-ting in cerebral aneurysms

Aneurysms are abnormal bubble-like local deformations of a blood vessel (seefig. 2). Under some circumstances which are still not well understood, theymay rupture and cause patient death. Among the causes that impact thegrowth and evolution of aneurysms, it is well accepted that haemodynamicsplays a central role [19].

A natural mechanism of self-repair is the occurrence of thrombosis insidethe aneurysm which eventually leads to the cavity occlusion. Since recently,stent (see fig. 2) endovascular procedures are considered as a promising non-invasive technique to canalize the blood flow in the parent vessel, thus reduc-ing flow in the aneurysm and promoting thrombosis [20] therein.

A lattice Boltzmann based numerical model for blood flow and blood clot-ting in a vessel on which an aneurysm has grown has been proposed firstin [24, 1] while more mathematical insight has been given recently in [7]. Inthis approach, we are interested in capturing the large time and space scaleshence ignoring a molecular level description. We will consider a mesoscopicmodel only including the features that are relevant to the aneurysm occlusion.Such a simulation tool will be of central importance to predict the evolutionof thrombogenesis within an aneurysm, be it driven by anatomy or/and bystents.

Here, blood flow is described by a lattice Boltzmann (LB) fluid model. TheLB method is an extension of the lattice-gas cellular automata discussed previ-ously (see section (2)), where the state of the cells is described by real-valuedvariables instead of Boolean ones. Fig. 3 illustrates one time iteration δt of a 2DLB-fluid model, at a given spatial location r. It can be shown that running theprocess described in fig. 3 is equivalent to model a fluid flow provided that thecollision step is properly chosen. More information on the LB method can befound in text books and reviews [4, 27, 6, 31].

On top of a LB-fluid model, the advection-diffusion of passive scalars canbe added to include the transport of platelets and red blood cells. From themodeling point of view we describe these particles with a multiparticle cellularautomaton (CA) (see [6]). At our description scale, clotting is simply the resultof the aggregation of ”blood particles”, under suitable conditions.

Our central hypothesis is that clotting is initiated in low shear rate regions

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Collision Propagation

Figure 3: Illustration of the collision and propagation phases in a LB modeldefined on a 2D square lattice (the so-called D2Q9 model). The size of thearrows is proportional to the density of particles entering the lattice site alongthe corresponding direction. As suggested by the figure, mass and momentumare conserved in the collision process.

velocity for geom2 at low shear

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low shear-stress cutoffmedium shear-stresscutoff high shear-stress cutoff

Figure 4: Simulations of low-shear rate induced clotting in aneurysms, for var-ious valued of the threshold Sstart = Sstop.

[1]. There is clear evidence for endothelium dysfunction when the bloodflowdoes not produce a high enough shear rate at a vessel wall [8]. As a result,platelets can adhere on the sub-endothelium. Some biological studies [21, 2, 30]clearly show that platelets adhere and aggregate when the shear rate decreases.Platelet aggregation is then followed by the formation of a thrombus. Thegrowth of this thrombus is maintained as long as the blood shear rate at theinterface region remains low enough.

The above mechanisms can be translated rather easily in a LB-CA modelingframework [24, 1, 7]. The clotting model is based on two shear rate thresholds:Sstart below which clotting can starts at a wall and Sstop ≥ Sstart above whichthe thrombus stops growing). These cutoff values are model parameters thateventually have to be obtained from biological studies.

Fig. 4 shows the typical scenarios that can be expected from this model(assuming Sstart = Sstop). When the threshold is low, clotting is almost nonex-istent. Such an aneurysm will not clot spontaneously and inserting a stent asa flow diverter is a solution to decrease the shear rate in the aneurysm belowthe clotting threshold. For higher values of the threshold, we observe partialor full clotting. If the threshold is too large, the clot can even invade the par-ent vessel. All the scenarios proposed in fig. 4 are also observed in clinicalinvestigations [10, 29]. In addition, the simulation shows that the clot forms insuccessive layers, a feature well observed in giant aneurysms that clot sponta-neously.

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Figure 5: Left: clotting fraction as a function of the aneurysm aspect ratio (clin-ical observations by O. Brina, Hospital of Geneva). Right: shear rate at theaneurysm wall as a function of the aspect ratio (LB simulations).

Our LB simulations also indicate that the value of the shear rate at theaneurysm wall (be it endothelium or forming clot membrane) is related tothe geometry of the cavity. A key parameter is the aspect ratio (ratio of theaneurysm diameter to its neck size). For small aspect ratios, the wall shear rateis higher than for large aspect ratios. That explains why giant aneurysms arelikely to clot spontaneously [10, 29] unlikewise for smaller aneurysms.

Fig. 5 compares clinical observations with simulations and clearly suggestsa critical value of the aspect ratio around 1.8. Aneurysms with aspect ratiobelow this value do not clot and are characterized by high shear rate value. Onthe other hand, aneurysms with high aspect ratios are subject to low wall shearrate and exhibit partial to full clotting. This observation provides a promisingqualitative validation of our LB-CA model of clotting.

4 Lattice-gas cellular automata models for the anal-ysis of tumor invasion

Cancer describes a group of genetic and epigenetic diseases, characterized byuncontrolled growth of cells, leading to a variety of pathological consequencesand frequently death. Cancer has long been recognized as an evolutionarydisease [23]. In particular, cancer progression can be depicted as a sequence oftraits or phenotypes that cells have to acquire if a neoplasm (benign tumor) is tobecome an invasive and malignant cancer. Thereby, a phenotype refers to anykind of observed morphology, function or behavior of a living cell. Hanahanand Weinberg [14] have identified six cancer phenotypes: unlimited prolifera-tive potential, environmental independence for growth, evasion of apoptosis,angiogenesis, invasion and metastasis.

Here, we concentrate on the invasive phenotype. The progression of a be-nign tumor with limited growth to a tumor that is invasive and potentiallymetastatic is the major cause of poor clinical outcome in cancer patients, interms of therapy and prognosis. Understanding tumor invasion could poten-tially lead to the design of novel therapeutical strategies. Biomedically, inva-sion involves the following tumor cell processes:

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• tumor cell proliferation,

• tumor cell migration, which is a result of down-regulation of cadherinsand corresponding loss of cell-cell adhesion, and

• tumor cell-extracellular matrix (ECM) interactions, such as cell-ECM ad-hesion, and ECM degradation/remodeling, by means of proteolysis.These processes allow for the penetration of the migrating tumor cellsinto host tissue barriers, such as basement and interstitial stroma [12].

Our aim is to formulate a mathematical model that allows the prediction oftumor invasion behavior under controlled conditions. Our approach relies ona suitable abstraction of the tumor invasion problem. In particular, we restrictthe tumor invasion problem to the interplay of the two main tumor cell pro-cesses, i.e. proliferation and migration. Our strategy is to describe the relevantphenomena that influence tumor cell migration and proliferation (e.g. ECMcues) at the cellular scale and to predict the emergent invasive tumor behaviorat the macroscopic scale. Motivated by the success of the lattice-gas cellularautomaton method to model interacting particles in a fluid in a very abstractform, which however facilitates prediction of macroscopic flow properties (seesection (2) we have extended the approach to a model of interacting cells in aheterogeneous environment [17]. In the following, we summarize our resultswith respect to the tumor invasion problem.

Firstly, we investigate the influence of tumor cell proliferation and migra-tion on the invasive behavior in a homogeneous environment (i.e. we neglecteffects of the ECM). The main question is how fast the tumor expands or, inother words, what is the invasion speed of a tumor. Therefore, we developa LGCA model of tumor cell proliferation, necrosis and tumor cell migration(see fig. 6, [16] for details). Our analysis aims at predicting the velocity of thetraveling invasion front, which strongly depends upon fluctuations that arisefrom the motion of the discrete cells at the front. We find an excellent quanti-tative agreement between the velocities measured in simulations of the LGCAand an analytical estimate derived in the cut-off mean-field approximation viathe discrete lattice Boltzmann (LB) equation and its linearization. In particu-lar, we predict the front velocity to scale with the square root of the product ofprobabilities for mitosis and the migration coefficient.

In a further model we have analyzed the influence of a heterogeneousmicroenvironment. The tumor microenvironment is a highly heterogeneousmedium including the extracellular matrix composed of fibrillar structures,collagen matrices, diffusible chemical signals as well as other mobile and im-mobile cells [12]. A LGCA model of cell migration together with a mathe-matical characterization of different biological environments can contribute tounderstand the interplay of moving cells with their heterogeneous environ-ment [17]. In particular, we have shown how the mathematical analysis of theLGCA model can yield an estimate for the cell dispersion speed within a givenenvironment.

The ultimate goal is to use mathematical models and corresponding simula-tions to predict invasion behavior in real patients. We have moved initial stepsinto this direction by focusing our attention to a very malignant and invasivebrain tumor, namely glioblastoma multiforme tumor (or just gliomas). Gliomacells tend to spread faster along fiber tracks. Diffusion Tensor Imaging (DTI)

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Figure 6: Left: Typical spatio-temporal pattern formation of in vitro tumors(reprinted with permission from Folkmann et al. [11]). One observes the pres-ence of a necrotic core and an outer rim of proliferative tumor cells. Right:LGCA simulations exhibit a similar structure. In the simulation, tumor cellsare depicted in grey, necrotic entities in white, and empty nodes in black.

is a Magnetic Resonance Imaging (MRI)-based method that provides the localanisotropy information in terms of diffusion tensors. High anisotropy pointsindicate the brain’s white matter consisting of fiber tracks. A preprocessing ofthe diffusion tensor field can lead to the principle eigenvectors’ extraction ofthe diffusion tensors, that provides us with the local principle axis of motion.By considering a proliferative cell population, like in [16], and using the result-ing eigenvector field we can model and simulate glioma cell invasion. In fig.7, we simulate an example of glioma growth and show the effect of the fibertracks in tumor growth using the DTI information.

Finally, in order to predict the spatio-temporal in vivo tumor invasion alsoquantitatively, we need to design parameter optimzation algorithms. There-fore, we developed an evolutionary algorithm [28] that estimates the parame-ters of a tumor growth LGCA model based on time-series of patient medicaldata (in particular MR and DT Imaging data). In the future, these parametersmay allow to reproduce clinically relevant tumor growth scenarios for a spe-cific patient, providing a prediction of the tumor growth at a later time stage.

5 Discussion

In this article, we have shown how lattice-gas cellular automaton and latticeBoltzmann (LB) models that originated in fluid dynamical applications can beextended to analyze important medical problems. In fluid dynamics LGCAhas been replaced to a large extent by LB methods. However, mean-field meth-ods such as LB methods do not provide a good description when spontaneousfluctuations and many-particle correlations play an essential role (compare[9], p.51). Spontaneous fluctuations and many-particle correlations might beimportant in various biological systems (e.g. in “small cellular systems” thenumber of cells is much smaller than the number of molecules in fluid flow)and thus LGCA models are advantageous compared to LB methods. Corre-

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Figure 7: Brain’s fiber track effect on glioma growth. We use a LGCA of a pro-liferating cancer cell population moving in a tensor field provided by clinicalDTI data, representing the brain’s fiber tracks. Left: Initial tumor mapped (red)on the DTI data of a real brain. Right: Final state of tumor growth along thefiber tracks.

sponding approximation methods are especially interesting if one is interestedin predicting macroscopic behavior from a well-known mesoscopic behavior.

The need for discrete models, especially cellular automata, goes beyond theexamples presented here. A discrete cell-oriented approach is required if thedynamic system behaviour depends on fluctuations at the individual cell level.This is, for example, the case at the front of invading tumours and crucial forthe formation of metastases. Lately, experimental findings of Bru et al. [3] in-dicate that many tumors share the same surface dynamics. This finding moti-vated the analysis of the tumor interface by means of a fractal scaling analysis.Obviously, corresponding cancer models have also to be of a discrete natureand CA models are promising candidates to identify growth mechanisms thatlead to a particular scaling.

Based on the variability in the local dynamics, we demonstrated that the“interaction-modul oriented” cellular automaton modeling provides an intu-itive and powerful approach to capture essential aspects of complex phenom-ena on various scales. In conclusion, there are both challenging future per-spectives with regards to interesting biological applications of the lattice-gascellular automaton idea and possible refinements of analytical tools for theinvestigation of lattice-gas cellular automata. Surely, the potential of cellularautomata for modeling essential aspects of biological systems will be furtherexploited in the future.

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