pam: particle automata in modeling of multiscale...

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PAM: Particle Automata in Modeling of Multiscale Biological Systems

WITOLD DZWINEL and RAFAL WCISŁO, AGH University of Science and Technology, PolandDAVID A. YUEN, China University of Geosciences, Wuhan, China; Minnesota SupercomputingInstitute, Minneapolis, MNSHEA MILLER, Ottawa Research and Development Centre, Agriculture and Agri-Food Canada, Ottawa,Canada

Serious problems with bridging multiple scales in the scope of a single numerical model make computersimulations too demanding computationally and highly unreliable. We present a new concept of modelingframework that integrates the particle method with graph dynamical systems, called the particle automatamodel (PAM). We assume that the mechanical response of a macroscopic system on internal or externalstimuli can be simulated by the spatiotemporal dynamics of a graph of interacting particles representingfine-grained components of biological tissue, such as cells, cell clusters, or microtissue fragments. Meanwhile,the dynamics of microscopic processes can be represented by evolution of internal particle states representedby vectors of finite-state automata. To demonstrate the broad scope of application of PAM, we present threemodels of very different biological phenomena: blood clotting, tumor proliferation, and fungal wheat infection.We conclude that the generic and flexible modeling framework provided by PAM may contribute to moreintuitive and faster development of computational models of complex multiscale biological processes.

CCS Concepts: � Computing methodologies → Model development and analysis; � Appliedcomputing → Computational biology;

Additional Key Words and Phrases: Particle automata model, modeling using particles, graph dynamicalsystems, blood flow, tumor growth, F. graminearum proliferation

ACM Reference Format:Witold Dzwinel, Rafal Wcisło, David A. Yuen, and Shea Miller. 2016. PAM: Particle automata in modelingof multiscale biological systems. ACM Trans. Model. Comput. Simul. 26, 3, Article 20 (January 2016), 21pages.DOI: http://dx.doi.org/10.1145/2827696

1. INTRODUCTION

The majority of biological processes have a very complex multiscale character. More-over, for the most of them, the macroscopic scales (e.g., the tissue scale) are tightlycoupled with microscopic processes (e.g., occurring at a molecular or cellular level).The nonlinear interactions across spatiotemporal scales make their modeling and nu-merical simulation both very demanding computationally and unreliable in the scope ofclassical modeling paradigms (e.g., Dzwinel [2012] and Weinan [2011]). To address thisissue, we propose a cross-scale modeling method—the particle automata model (PAM).It combines diverse spatiotemporal scales in the scope of a computational frameworkof a generalized graph dynamical system (GDS) (e.g., Aledo et al. [2015a, 2015b]). In

This research was financed by the Polish National Center of Science (NCN) project DEC 2013/10/M/ST6/00531 and partially by AGH grant 11.11.230.124.Authors’ addresses: W. Dzwinel; email: [email protected]; R. Wcisło; email: [email protected]; D. A. Yuen;email: [email protected]; S. Miller; email: [email protected] to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrights forcomponents of this work owned by others than ACM must be honored. Abstracting with credit is permitted.To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of thiswork in other works requires prior specific permission and/or a fee. Permissions may be requested fromPublications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or [email protected]© 2016 ACM 1049-3301/2016/01-ART20 $15.00DOI: http://dx.doi.org/10.1145/2827696

ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.

http://dx.doi.org/10.1145/2827696

http://dx.doi.org/10.1145/2827696

20:2 W. Dzwinel et al.

comparison to other GDS approaches, in PAM the graph architecture undergoes spa-tiotemporal evolution stimulated by particle (vertexes) interactions and Newtoniandynamics.

In general, the GDS (also referred to as the generalized cellular automata (GCA)when automata are synchronously updated) are computation models that capture thegraph (network) of interacting objects where each object behaves like a finite-stateautomaton. Marr and Hutt [2012] demonstrate the application of cellular automata ongraphs for exploring the relationship between network architecture and dynamics fromthe perspective of pattern formation. This facilitates an evolutionary interpretation ofreal biological networks in the light of dynamical function. The integrated modelingframeworks for GDS were proposed in Kulman et al. [2011] and Rosenkrantz et al.[2015], who provide a powerful formalism to model and analyze agent-based systems(ABS). In the recent publications of Arendt and Blaha [2015] and Liu and Wei [2014],the GDS were employed in community detection in social networks and modeling somesocial behaviors. Many papers use GDS in studying nonlinear dynamics of biologicalsystems, such as disease outbreaks [Eubank et al. 2004] and regulatory networks [Marrand Hutt 2012]. The GDS methodology is tightly connected with dynamics on complexnetworks [Strogatz 2001, 2014; Watts and Strogatz 1998].

In the context of biological systems, PAM is a kind of GDS in which the objects (graphvertexes) correspond to interacting and moving particles representing tissue fragmentssuch as cells or clusters of cells. The particles are endowed with some inherent at-tributes defined by the vector of particle states. In particular, the particle network canbe the nearest neighbor graph. The graph edges define interactions between particlesand their neighbors, which directly stimulates the particle spatiotemporal dynamics.Simultaneously with particle dynamics, the particle states evolve in time accordingto a set of rules that correspond to microscopic processes occurring “inside” a particleand/or in its neighborhood. These microscopic rules directly affect the behavior of theentire system.

In this article, we present the main concept and assumptions staying behind PAM.We show that PAM reflects a specific coarse-graining procedure in which the micro-scopic degrees of freedom are encapsulated inside a particle and manifest in changesof its state vector, consequently influencing local and global dynamics of the entiresystem. This way, the particle automata ensemble can simulate multiscale systems byusing only one macroscopic spatiotemporal scale controlled by particle dynamics and amicroscopic rule-based operator.

We demonstrate the applicability of PAM in the modeling of complex multiscalebiological systems. As a proof of concept, we present three examples. First, we describethe particle model (PM), which we used earlier in modeling of the blood-clotting processin capillaries [Boryczko et al. 2004]. We show that the microscopic process of clotformation can be mimicked by a simple finite-state automata rule integrated withblood flow modeled by using fluid particle dynamics. Then we describe two models: themodel of cancer proliferation and the model of wheat colonization by a fungal pathogen,which present more advanced aspects of the PAM framework. We close the article withour conclusions.

2. PARTICLE AUTOMATA MODEL

2.1. Motivation

The model of interacting particles is one of the most popular and intuitive modelingparadigms (e.g., see Dzwinel [2012], Espanol [1998], Haile [1992], Spohn [2012], andWeinan [2011]). However, due to many difficulties with realistic representation of manymultiscale phenomena in the scope of this paradigm, and surrealistic computational


PAM: Particle Automata in Modeling of Multiscale Biological Systems 20:3

Fig. 1. The diagrams demonstrating computationally irreducible systems described by microscopic models:the molecular dynamics (MD) and the nonequilibrium molecular dynamics (NEMD) (a); the system withclearly separable scales: MD, the fast multipole method (FMM), DPD and thermodynamically consistentDPD (TC-DPD), smoothed particle hydrodynamics (SPH), and partial differential equations (PDEs) (b); andthe systems with strong cross-scales interactions (c). The spheres represent consecutive spatiotemporalscales, and arrows denote the couplings between them. In (c), the arrows are drawn arbitrarily and showpossible cross-scale interactions.

load when a macroscopic system is simulated directly from its atomistic or molecularscale, the question about its computational reducibility should be decided.

As shown in several papers (e.g., Dzwinel et al. [2006], Espanol [1998], and Grootand Warren [1997]), the dynamics of an ensemble of atoms can be coarse grained byinteracting clusters of atoms and, going up to the scale, by interacting droplets orpieces of matter. As demonstrated in Figure 1, such coarse graining can decouple theconsecutive spatiotemporal scales. Similarly to discrete wavelet transform (DWT) (e.g.,Goswami and Chan [2011]), coarse graining splits the space time into layers (tiles)in which coarse-grained models of various resolution are defined. Unlike DWT, theboundaries between the layers are fuzzy, and a simple mathematical formalism of alayer-to-layer transformation does not exist. However, by using the DWT analogy, theaveraged properties from the finer scales can be passed up to the coarser ones. Theycan manifest there as emergent global properties of the system or local features ofits components, such as potential fields (as it is in the particle in cell (PIC) method[Birdsall 1991; Hockney and Eastwood 2010]) or collision operators (e.g., as it is indissipative particle dynamics (DPD) [Groot and Warren 1997; Li et al. 2014; Pastorino



and Goicochea 2015]), respectively. However, the complexity of the particle collisionoperator in the coarser spatiotemporal scales (clusters of molecules, fluid particles,cells, lumps of fluid, droplets, tissue components, etc.) increases exponentially with theprecision of reconstruction of the corresponding fine-grained system evolution [Dzwinelet al. 2006; Li et al. 2014; Serrano and Espanol 2001; Yaghoubi et al. 2015]. Conse-quently, the parameter space of the model inflates what results in overfitting. Anyway,as shown in several papers (e.g., Dzwinel et al. [2006], Dzwinel [2012], Israeli andGoldenfeld [2006], Magiera and Dzwinel [2014], Dzwinel and Magiera [2015], andWeinan [2011]), coarse graining is a very effective procedure for bridging spatiotem-poral scales of many real-world systems provided that the scales can be separable.It allows for simulating real-world phenomena in the scale of interest defined by acomputational setup attributed by microscopic features.

Biological systems can be located between systems of both fully separable and in-separable scales. On the one hand, the strong dependence of macroscopic behavior onmolecular processes makes them chaotic and unpredictable. On the other hand, manyof microscopic biological processes are local and well separated, especially those tiedto particular biological structures such as DNA chains, molecular films, or those en-capsulated inside closed subsystems such as organelles, cells, and microvessels. Thus,the various phases of those physicochemical processes can be represented by internalstates of corresponding discrete biological agents, which define their properties. For ex-ample, internal molecular processes occurring in a cell can define its state as newborn,mitotic, apoptotic, or hypoxic, for example. Moreover, the state vector of a particulardiscrete object (particle) may depend on the state vectors of other objects, such as thoselocated in its nearest neighborhood in the Euclidean or abstract spaces. For example,signaling cells may modify both their own states and properties of the cells in theirvicinity. Analogously, the financial condition of a company may change depending onboth its internal status and the statuses of all cooperating partners. Therefore, the mi-croscopic processes can be reflected in the coarser scales by both individual evolution ofparticle attributes (states) caused by an internal process (e.g., cell life cycle) and similarto the GDS [Aledo et al. 2015a, 2015b], local rules. Consequently, the time evolutionof the vector state of discrete particles can influence overall system dynamics. On theother hand, the spatiotemporal evolution of the particle system may affect the evolu-tion of their individual state vectors. It is worth mentioning that the set of rule-basedmicroscopic models may correspond to various (sometimes remote) temporal scales. Byusing the DWT analogy, they represent “details” from various resolution levels. ThePAM approach allows for coupling all of them within the GDS framework of spatialdynamics of particles on the coarsest approximation level.

Moreover, the particle dynamics and its vector state evolution may occur in electrical,temperature, pressure, or concentration continuous fields. These fields are usually theresult of some physical processes (e.g., diffusion of substances secreted by cells, such assignaling proteins) modified by the system environment and boundary conditions. Theevolution of the internal states of biological objects, their spatiotemporal dynamics,and the dynamics of continuous fields are tightly coupled, creating an integrated andgeneralized framework for modeling of multiscale systems.

2.2. Particle Automata as an Evolving Graph Dynamical System

We regard a particle system as an undirected dynamic graph Gt = (Vt, Et), whereVt, Et are the sets of particles and particle-particle interactions eij ∈ Et, respectively.We assume that Vt = {v1, v2, . . . , vN}t, where Nt = #Vt is the number of particles intime t and undirected edge eij = (vi, v j) ∈ Et. For any vertex (particle) vi ∈ V, theneighborhood of vi, Ng(vi), is the set {vi} ∪ {v j ∈ V|(vi, v j) ∈ Et} that contains vi and



Fig. 2. The snapshots from 2D simulations of dynamics of particle systems represented by evolving graphs:the nearest neighbor graph that corresponds to the ensemble of Nt = 5 × 104 particles interacting with theirclosest neighbors (a) and the social network where vertexes (Nt = 1.1 × 106 particles) interact with theirneighbors connected by the outgoing edges (b).

all neighbors of vi. The particle network evolves in Euclidean space. Let us define theparticle automata as a generalized GDS.

Definition 2.1 (Particle Automata). Particle automata is a GDS represented by atuple Tt = (Gt, R, A,�,�, V0) evolving in discretized time t, where

—Gt = (Vt, Et) is an undirected graph, whose nodes vt = (rt, at)∈Vt are called particles.—V0 = {v0 = (r0, a0)} is the graph initial configuration.—rt ∈ R are coordinates of a particle vt in a geographical space R (here, R = R3 is the

Euclidean space).—∀vt ∈ Vt,�rt : R → R is the local collision/translation operator, which defines the

particle spatial evolution in discretized time with timestep �t and

rt+�t = �rt. (1)

—at ∈ A is the particle state vector, where A is a set of subsets of states representingstate vector coordinates.

—∀at ∈ Vt,�at : ANg(vt) → A is the local transition operator, which defines temporalevolution of the particle state vector in discretized time with timestep �τ and

at+�τ = �at. (2)

The respective attributes from vector state ai may evolve in various temporal scales,and they can correspond to multiple scales of many microscopic cellular and molec-ular processes. For the sake of simplicity, we assume that the ratios of the greatesttimestep to the smaller ones are the integers.

By using the PAM approach, one can model not only the dynamics of systems withshort-range interactions but also the networks with long-range connections, such associal networks. For example, in Figure 2, we show the snapshots from 2D simulationsof particle systems represented by these two types of evolving graphs.

This way, the particle automata ensemble can simulate multiscale systems withonly one “distinguished” spatiotemporal scale controlled by the � operator and manymicroscopic processes simulated by the rule-based � operator. We understand the “rule”as a simple if_then_else relation or more complicated function (e.g., of cellular automatatype) that modifies the vector state of vertexes in the subsequent timesteps. It may



depend on the current vector state, the vector states of directly connected neighbors,and/or the values of density fields in the vicinity of the graph vertexes. The otherassumptions are as follows:

(1) The force component F = FC + FD + FB of the operator �, such as in DPD [Grootand Warren 1997; Hoogerbrugge and Koelman 1992] or in the fluid particle model(FPM) [Espanol 1998], consists of the conservative FC , dissipative FD, and randomFB (e.g., the Brownian) components. The conservative factor corresponds to thecumulative impact from two-body central interactions between the particles (ver-texes) connected by the edge eij �= 0. The dissipative and fluctuation components of� are responsible for controlling the amount of kinetic energy in the system—thatis, FD alone freezes the system and FB melts it down. According to the fluctuation-dissipation theorem [Groot and Warren 1997; Weber 1956], these two componentsof the collision operator are responsible for the “temperature” of the entire system.The temperature can reflect stochastic instability of the particle system caused bythe coarse-grained degrees of freedom.

(2) The particle attribute operator � acts only on these components of state vector at,which evolve with time t according to rule-based principles involving the particle’sneighborhood or its internal scenario (e.g., cell mitosis, apoptosis). It means thatthe following state at+1 of a particle vi depends on its current state and/or the statesof the neighboring particles v j , provided that eij �= 0.

(3) Additionally, the � operator follows the time evolution of some “concentration fields”in R3, such as concentration of diffusive substances, and assigns the values of thesefields as particle’s state vector attributes. In this case, the fields can be obtainedsolving some reaction-diffusion equations.

(4) The particles Vt representing Gt nodes and its edges Et can proliferate or die.(5) The particle system can be bounded or unbounded—that is, various periodic bound-

ary conditions can be applied (e.g., Dzwinel et al. [1991] and Hockney and Eastwood[2010]).

Summing up, the entire PAM of a complex system consists of the following tightlycoupled components:

—The ensemble of interacting particles, which represents the coarsest spatiotemporalscale.

—The GDS of automata, which mimics the evolution of hidden microscopic scales ofthe system.

—The varying concentration fields resulting from continuous processes, such as diffu-sion, advection, and fluid flow, which go with the system evolution.

In the following sections, as a proof of concept of PAM, we present its applications inthe modeling of selected biological processes.

3. BIOLOGICAL APPLICATION OF THE PARTICLE AUTOMATA MODEL

In this section, we briefly describe three computational models of complex biologicalphenomena developed by using the particle automata paradigm, which combines par-ticle dynamics with rule-based microscopic models. The models differ in both the scopein which the GDS automata rules are exploited and the context in which they wereused. We discuss the models of

—blood clotting in capillary vessels,—cancer dynamics, and—wheat colonization by the Fusarium graminearum fungal pathogen.



First, we briefly describe the model of blood clotting in microcapillaries, which wasdeveloped a relatively long time ago [Boryczko et al. 2003, 2004; Dzwinel et al. 2003]but still continues to be developed (e.g., Guy et al. [2007], Soares et al. [2013], andStorti et al. [2014]). This model is described to demonstrate the main assumptions ofthe FPM, which we use as a computational framework of PAM. Moreover, in this blooddynamics setup, we implemented a simple automata rule (Algorithm 1), which mimicsthe microscopic model of fibrinogen aggregation. The second example (based on Wcisłoet al. [2009, 2010]) presents a much more complicated PAM of cancer proliferation inthe presence of the process of angiogenesis. The last model is a metaphor of fungalwheat infection. This is a new application of PAM that is developed for a differentsetup than the previous two examples.

3.1. Particle Model of Blood Clotting

Hemorrhage over a vast tissue volume caused by blood clotting in capillary vessels isvery feasible under multiple g jet accelerations or violent deceleration during a caraccident. In Boryczko et al. [2003], we presented a microscopic model of blood, validover short time scales, which allows for simulating such shocking events.

We assumed that the system, which consists of a capillary, blood plasma, fibrinogen,and red blood cells (RBCs), is made of particles vi = (ri, ai) of various types. The at-tributes of a discrete particle are as follows: particle type, mass, moment of inertia,and translational and angular momenta. The capillary, filled with plasma and RBCs,is made of motionless particles. Its diameter is of the size of a single RBC (i.e., ap-proximately 10μm). Each fluid (plasma) particle represents coarse-grained clusters ofmolecules, whereas RBCs are the structures made of particles interacting via elasticforces [Boryczko et al. 2003]. Two particles, vi and v j, interact with each other by acollision operator, Fi j , defined as a sum of constituent forces: central and noncentralconservative FC , dissipative FD, and Brownian FB. The collision operator is defined asfollows [Espanol 1998]:

Fi j = FC +FD +FB,

Fi j = −F(rij)ei j −γ [A(rij1 + B(rij)ei jei j] ◦[vi j + 1

2ri j × ( �ωi + �ω j)

]+FB(rij),

(3)

where F(rij) is the module of central conservative force, such as in DPD [Hoogerbruggeand Koelman 1992]; γ is a scaling factor of dissipative component FD; ωi is the angularvelocity of particle vi; ei j is the unit vector; and rij = ||ri −r j || is the separation distancebetween particles vi and v j . A(rij) and B(rij) are the weighting functions, whereas FBis the Brownian component. The value of Fi j is equal to 0 if the separation distance rijbetween two particles i and j exceeds a cut-off radius Rcut.

The total force Fi acting on a particle i is the sum of forces between i and otherparticles located within the sphere O(ri, Rcut). The dynamics of the particle systemobeys the Newtonian laws of motion:

Pi =∑

j;rij<rcut

Fi j(ri, vi, ωi),

ri = vi,

ωi = 1Ii

∑j;rij<rcut

Ni j(ri, vi, ωi),

Ni j = −12

ri j × Fi j,

(4)

where Pi and Ni are translational and angular momenta, respectively. To retainthe shape and elastic properties of both the endothelial wall and the RBCs, we use



two-body harmonic force FC(rij) instead of the repulsive conservative force employedfor fluid particles. The parameters of the interactions were adapted by employing avail-able physical and mechanical properties of the blood system [Fung 1993].

This part of the blood flow model represents the classical method of interactingparticles. However, to simulate blood clotting, the microscopic process of fibrin poly-merization should be considered. During the clotting process, the large number offibrins starts to aggregate, forming long threads, which cross over each other and forma weblike structure that entraps the RBCs. Because the fibrinogen molecule is oneorder of magnitude smaller than the fluid particle, modeling of fibrinogen and fluid asseparate particle types is very demanding computationally [Boryczko et al. 2004]. Tosimulate this process, one can assume that the fibrin fragments are dissolved withinthe particles of plasma fluid. Therefore, the formation of a hydrated network of fibrinshas to be described by a different, nonformalized, rule-based model, which representsthe coarse-grained microscopic process of fibrin formation. Next, we present its shortdescription. More details are given in Boryczko et al. [2004].

The fibrin network—using PAM terminology—is represented by a graph Gt = (Vt, Et)of fluid particles. Its nodes are the fluid particles (droplets), and the edges correspondto the bonds between them. The bond eij > 0 appears when the nodes fulfill someprescribed conditions. The pseudocode that described the rules responsible for thenetwork creation is shown in Algorithm 1.

We assumed additionally that

—Two bonded i and j particles interact via a conservative elastic force F f (.):

F f (rij) = χ · (|rij | − aij) · ei j, (5)

where χ defines the elasticity and aij is the bonding distance.—The particles, which create a fibrin, interact with other fluid particles, RBCs, and

the capillary walls through the force F (see Equation (3)).

Because there are only about 10 fibrin monomers per fluid particle of the length 1/7of the average distance between fluid particles [Bark et al. 1999], it is not feasible thatthe bond between particles, which are too far one from another, will be created. We haveassumed that it appears when the distance between particles rij < RAttach. Moreover,we assumed arbitrarily that one fluid particle (with fibrin monomers) can have only twoother particles attached. There is a small feasibility for creating more than two bondsfor such a low concentration of rod-shaped fibrins. Thus, the state ai of fluid particlei can be equal to 0 (detached), 1 (one particle attached), or 2 (two particles attached).We have assumed that the particles forming fibrins and other fluid particles interactwith a repulsive force FC . The interaction between RBC particle and fluid particle withfibrins is attractive at longer distances. It agrees with the laboratory observations ofJesty and Nemerson [1995]. The snapshots from simulation showing the process of clotformation are shown in Figure 3.

Other biological systems have been simulated using a similar approach. For example,in Klann et al. [2013], the intracellular processes (signaling transduction) are mod-eled, assuming that the molecules diffuse inside the cell and the colliding complexesstochastically interact according to a rule-based scheme. Another model of this type ispresented in Bittig et al. [2014]. It is used for simulation of intracellular processes ofcells at the interface to an implant surface triggered by the extracellular surroundings.In these two examples, the spatial motion of molecules and/or cell components aremodeled by employing simplistic Brownian dynamics.

All of these models, including the blood clotting model, do not fully exploit PAMadvantages. They are biased by the particle motion, whereas the rule-based opera-tor is of a very secondary character. In the following examples, we demonstrate the



Fig. 3. Snapshots from simulation of clot formation. The clot formed from a fibrin web with RBCs entrappedis shown in (a), whereas in (b) we can observe formation of fibrins between stacked RBCs and the capillarywall. (Courtesy of Dr. Krzysztof Boryczko, AGH Department of Computer Science.)

ALGORITHM 1: The pseudocode describing the rules of hydrated bond creation.for all pairs of particles vi , v j , {i, j = 1 . . . N} do

rij ← ||ri − r j || ; // distance between vi, v jif �∈ list of neighbors( j) then // if they are not linked

if rij < RAttach and number of bounds(i) < 2 thenr ← rand(0, 1);p ← calculate bound probability(rij); // probability of creating a boundif r < p then

number of bounds(i) ← number of bounds(i) + 1; // create a boundnumber of bounds( j) ← number of bounds( j) + 1;actualize list of neighbors(i);actualize list of neighbors( j);a(i) ← a( j) ← 1; // change a state vector a

endend

elser ← rand(0, 1);q ← calculate bound break probability(rij); // decide about bound breakingif r < q and rij > Rbrake then

number of bounds(i) ← number of bounds(i) − 1;number of bounds( j) ← number of bounds( j) − 1;actualize list of neighbors(i);actualize list of neighbors( j);a(i) ← a( j) ← 0; // change a state vector a

endend

end

model of tumor proliferation and wheat infection, in which the tumor and fungus cellsdemonstrate considerably richer individual behavior.

3.2. Tumor Proliferation

Tumor proliferation is a very nonlinear and heterogeneous process occurring in mul-tiple spatiotemporal scales. It develops in five main phases: oncogenesis, avascular



growth, angiogenesis, vascular growth, and metastasis. Oncogenesis represents a cas-cade of biological processes occurring at molecular level stimulated by serious mutation-based disruptions in DNA repair system. In a larger spatiotemporal scale, an avasculartumor develops in the absence of blood supply. For tumors of size of a few millimetersin diameter, hypoxic cells from the avascular tumor mass produce and release manysignaling proteins called tumor angiogenic factors (TAFs). They diffuse throughout thesurrounding tissue and, by hitting vasculature, trigger a cascade of events leadingto vascularization of the tumor. In the vascular stage, the tumor has access to vir-tually unlimited resources, so it can grow beyond any limits. Moreover, it acquires ameans of transport for cancerous cells that penetrate into the vasculature and can formmetastases in any part of the host organism.

A plethora of computer models of both the separate subprocesses of cancer prolif-eration and more general multiscale approaches exist. The main modeling techniquesare presented in many books and overviews [Adam and Bellomo 2012; Barillot et al.2012; Cristini and Lowengrub 2010; Deisboeck and Stamatakos 2011]. However, mostmethods used for simulating tumor growth assume the coarse-graining scenario shownin Figure 1(b). Meanwhile, just the cross-scale influence of microscopic processes occur-ring inside a single cell and its nearest neighborhood on the overall tumor dynamicscan be crucial for its growth dynamics.

For example, some of internal cell processes, such as those influencing the cell lifecycle, depend on its local environmental conditions stimulated by cancerogenesis. Thediversity of possible situations will lead to the development of a heterogeneous tumorthat promotes the cancer cells having the highest survival abilities (e.g., see Smith[2013]). These malignant cells decide about the fatal prognoses of tumor proliferation.

Therefore, the tissue cells—especially tumor cells—have to be treated as indepen-dent, partly autonomous objects that reveal very sophisticated behavior. Thus, in com-parison to the blood clotting model presented earlier, the complexity of the model ofcancer dynamics must be considerably higher. We present such the model in Wcisłoet al. [2009]. Next, we briefly describe it in terms of the PAM framework.

Let us assume that a small piece of tissue is made of an ensemble of Nt particlesvi = (ri, ai, t). Each particle represents a single tissue cell with a fragment of extracel-lular matrix (ECM). As shown in Figure 4, for the sake of simplicity, we assume thatthe blood vessel is constructed of endothelial tubelike “particles”—EC tubes—made oftwo particles connected by a rigid spring. Consequently, three types of interactions—particle-particle, particle-tube, and tube-tube—have to be considered. The forces be-tween particles mimic mechanical repulsion from squashed cells, attraction due to celladhesiveness, and depletion interactions between the ECM and the cell. The collisionoperator has a general form similar to Equation (3). The vessel and cell dynamics aresimulated by using the Newtonian equations of motion (see Equation (4)). The entiresystem is confined in a cubical box under constant external pressure.

The vector of states and attributes ai = (a1, a2, . . . , ak) for every cell consists of theparticle type {tumor cell (TC), normal cell (NC), EC-tube}; cell life cycle state {newlyformed, mature, in hypoxia, after hypoxia, apoptosis, necrosis}; cell size; cell age lifetimein hypoxia; and values of continuous fields, such as concentrations of TAF, oxygen, andtotal pressure exerted on particle i from its nearest neighbors. The cell states change intime according to their own clocks (cell life cycle states, cell size, time spent in hypoxiastate) and the dynamics of continuum concentration fields. The latter are computedby solving continuous reaction-diffusion equations (e.g., see Cristini and Lowengrub[2010]) and employing the virial theorem for calculating local pressure [Hockney andEastwood 2010]. The blood pressure in vessels is calculated by using the Kirchhoff law[Wcisło et al. 2009].



Fig. 4. Tissue particles and tubelike EC-tube particles made of two spherical “vessel particles.”

Fig. 5. The snapshots from PAM simulation of two phases of tumor growth. (a) In the avascular phase, onecan see the tumor blob consisting of tumor cells in various stage of hypoxia (oxygen deficiency). The cellscloser to the blood vessels are better oxygenated. The normal tissue cells are invisible in this figure. (b) Inthe vascular phase, the new sprouts and newly formed vessels are created. One can see the tumor cells invarious stages of hypoxia and normal cells (white particles). In both simulations, the dynamics of about 105

particles were simulated.

After the initialization phase, in subsequent timesteps we calculate: forces acting onparticles, new particle positions, the diffusion fields of active substances (i.e., nutri-ents, TAFs, pericytes), and the pressure in blood vessels. Simultaneously, we changethe states of individual cells, whose evolution is controlled by the factors mentionedpreviously. In Algorithm 2, we present selected automata rules used in our PAM.

As shown in Figure 5, our model of tumor proliferation allows for investigating thetumor dynamics in both avascular and angiogenic stages of growth. The life cycle ofindividual cells—from mitosis to its apoptosis (or hypoxia)—changes the number of



ALGORITHM 2: The pseudocode describing some rules of the cells state vector.if state = Alive and type = Normal and nei tumor cnt > nei normal cnt then

// normal, alive cell will change state to apoptosis if majority of itsneighbors is tumor

state ← Apoptosis;endif state = Hypoxia then

// cell secretes TAF in hypoxiaproduce TAF();

endif state �= Necrosis then

// cell consumes oxygen if not in necrotic stateconsume O2();

endif state = Alive and age > time to apoptosis then

// old cell changes its state to apoptoticstate ← Apoptosis;

endif state = Alive and O2 concentration < O2 hypoxia then

// cell changes its state to hypoxic if oxygen concentration drops below definedthreshold

state ← Hypoxia;endif (state = Apoptosis or state = Hypoxia) and state age > time to necrosis then

// cell changes its state to necrotic after defined time in hypoxia or apoptosisstate ← Necrosis;

endif state = Necrosis and state age > time in necrosis then

// cell is removed from simulation after defined time in necrotic stateremove cell();

end

particles in the system. Consequently, the increase of tumor size in a restricted volumecan influence global properties of the system, such as internal pressure, which triggersa cascade of both microscopic and macroscopic events. For example, purely mechanicalprocesses, such as vessel remodeling, result in tumor heterogeneity and developmentof most malignant tumor cells.

In the PAM, the local particle interactions and local automata rules, together withNewton laws of motion, decide about the system dynamics. However, in colonies oforganisms such as the coral reef, bacterial biofilm, and human civilization, the longdistance interactions interfere with those of short-range character. Similar situationscan be observed for the following PAM.

3.3. Fusarium Graminearum Infection

The fungal pathogen F. graminearum is the cause of many devastating cereal diseases,such as Gibberella ear rot of maize, Fusarium head blight (FHB), and scab of wheatand barley. This kind of infections cause significant crop and food quality losses.

The micrographs from Figure 6 show the F. graminearum colony in two differentsetups. It develops weblike structures of long threads of connected Fusarium cells. Asshown in the right panel of Figure 6, the plant colonization process is more globallyand locally constrained than the previous modeling examples. This is mainly due to theparticular structure of the plant and the sparse localization of nutrient sources, as well



Fig. 6. Micrographs demonstrating the F. graminearum network expanding in a Petri dish (left panel) andthe cross section (right panel) of rachis of Chinese Spring wheat cultivar, showing in green the fungus colonygrowing in the vascular bundles (arrows).

as the elongated shape of a single fungi cell and its directional reproduction mechanism[Boswell et al. 2007; Miller et al. 2004; Meskauskas et al. 2004]. As shown in Figures 6and 7, all of these factors favor a directional type of growth. The Fusarium colonyspreads mainly through vascular bundles, also penetrating the closest neighborhood(see Figure 7(c)). When it finds the part of plant that is rich in nutrients, the growthtype changes from a linear to an extensive (undirected) one. It is very similar to thatfrom the micrograph shown in the left panel of Figure 6. This way, F. graminearum isable to completely exploit and destroy invaded plant fragments.

The widespread colony had to develop other (faster) mechanisms of nutrients distri-bution than diffusion. Otherwise, the motionless colony section placed in the distantand exploited plant area would die from starvation. The transduction mechanism usesthe topological properties of the colony and its long web threads. It allows for transport-ing the nutrients throughout the colony body from nutrient-rich to nutrient-poor plantsections. Most fungal and plant cells, except those close to the penetrating hyphal tipcells, are motionless. Therefore, the PAM of F. graminearum differs considerably fromthat of tumor proliferation and blood clotting. The evolution of the particle attributesprevails that of cell dynamics.

According to the definition of PAM, the simulated system (i.e., the plant cells andFusarium colony) is made of a set of particles vi = (ri, ai, t). The vector of attributes aiis defined by the particle type {plant cell, wall cell, Fusarium}, current cell state {tip,active, inactive, spore, dead}, size, and concentration of nutrients. According to Boswellet al. [2007], the tip and active cells are involved in nutrient uptake, threads branch-ing, and nutrient translocation. Particularly, the tip cells are responsible for colonygrowth and, together with active cells, secrete the enzymes and toxins. Moreover, theypenetrate mechanical barriers (e.g., capillary walls) and disarm the plant’s immuno-logical system. The inactive cells are the cells that are no longer directly involved intranslocation, branching, or nutrient uptake. The spores are reproductive entities thatare adapted for dispersal and survival for extended periods of time in unfavorableconditions.

The state of each cell changes with concentration of diffusive substances and totalpressure exerted by its closest neighbors. The plant cell is represented by a single par-ticle (spherical cell), whereas the Fusarium cell (due to its elongated rodlike shape) ismade of two particles separated by a distance li, similar to EC-tubes from the cancer



Fig. 7. (a) Fluorescence micrograph showing green fungus in an infected wheat floret. The fungus bodyis denoted by arrows. (b) Snapshots from PAM simulation of F. graminearum growth in rachis. The arrowshows the fungus breaking throughout the plant wall. The dynamics of about 105 particles were simulated.

model. The interaction F(rij) between these two particles is harmonic and very stiffin order to maintain the constant length li of Fusarium cells. To reduce the compu-tational load, we assume that all plant cells are motionless and interact only with F.graminearum tubelike cells. The repulsive forces acting on the plant cells are ignoredand assumed to be dissipated in the plant body. Thus, we can define only sphere-tube(plant cell–fungal cell) and tube-tube (fungal cell–fungal cell) interactions. The plantcell walls can be degraded by DON toxins and acid secreted by Fusarium cells, and,finally, their interior can be consumed.

The sphere-tube interactions are represented by the force acting between a plantparticle and the two particles, which build the Fusarium cell body. The particle-particleinteractions represented by the force F(rij) mimic both the mechanical repulsion fromneighboring cells and attraction due to cell adhesiveness and depletion forces. Themechanical repulsion is approximated by Hooke’s law. We assume that the attractivetail of the interaction force has similar character but is less rigid than its repulsive part.

The tube-tube (fungi-fungi) interactions are of diverse characters. Typically, theFusarium cells create both chain and anastomosing web structures. We assume thatthe cell located at the tip of the growing web attracts another Fusarium cell strongerthan other cells. Similarly, the particles that form the nodes of the Fusarium thread orbranching sites are firmly glued. The heuristic formulas for various types of fungi-fungiinteractions are enumerated in Boswell et al. [2007]. Unlike in the tumor model, weassume a much simpler time evolution of a particle ensemble:

midVi

dt= −a · ∇�(dij) − λvi,

dri

dt= vi, rij = (ri − r j) · (ri − r j)T , (6)



where λ is a friction coefficient. This set of equations of motion is solved numericallyby using the direct leap-frog numerical scheme (e.g., see Haile [1992] and Hockneyand Eastwood [2010]). The Fusarium cells form the threadlike structures shown inFigures 6 and 7, which can be represented by a disconnected graph Gt = (Vt, Et).

The attributes of particle i are updated according to the state of cells in its neighbor-hood and prescribed finite-state automata rules. For example, new Fusarium cells canappear, cell function may change (tip → active → inactive), and their mass, stiffness,and size can evolve due to cell growth or degradation. Next, we collect the most impor-tant biological mechanisms simulated by using simple automata rules and thresholdingrelations.

3.3.1. Anastomosis. The tip is attracted to the active cell and can create a loop.

3.3.2. Branching. Active Fg cells can create a new colony branch in the node k (a newgraph edge and node) with the following probability:

p2(k) = c2sf (k, t) · �t. (7)

Branching can occur only once in the node.

3.3.3. Fusarium DON Toxins. DON is an inhibitor of protein synthesis and thus stopsdefense mechanisms [Miller et al. 2004]. The Fg cell secretes don f (k, t) toxins. Theamount of secreted toxins is proportional to DON activator substances and nutrientconcentration sf (k, t) in neighboring plant cell i. Both wall-degrading substances andDON spread due to diffusion.

3.3.4. Maintenance. An amount of food is needed per unit of time and length of the Fgcell. In every timestep, the concentration of nutrients in the Fg cell will decrease as

sf (k, t) − c1�t�x. (8)

If sf (k, t) < w f , the Fusarium cell becomes inactive or produces a spore. The sporeundergoes slow linear degradation, and finally it is removed from the system.

3.3.5. Nutrition.Uptake. The amount of food the Fg cell k drains out from attacked plant cell i in time

on the unit length is as follows:

�uptake(k) = c3sf (k, t) ·∑Sk

se(i, t)sf (k, t)∑Si

s f (k, t)�t. (9)

The summation goes through all attacked plant cells in the Fg cell neighborhood Sk.The attacked cell redistributes nutrients proportionally to each of Fusarium cells in itsnearest neighborhood Si.

Depletion. The decrease in nutrient concentration in the Fusarium nearest neigh-borhood is

�depletion(k) = −c4sf (k, t) ·∑Sk

se(i, t)si(k, t)∑Si

si(k, t)�t c4 > c3. (10)

If se(i, t) < we (we is a threshold), the Fusarium cell becomes inactive. After somedegradation time, it is removed from the system.

3.3.6. Secretion of Substances Degrading the Cellular Membranes. The Fg cell secretesaf (k, t) cell wall degradation substances (enzymes). The amount of substances is pro-portional to the nutrient concentration sf (k, t). When the amount of nutrients in plantcells attacked by Fg cells drops below a threshold, the cell dries out and dies.



By comparing tumor cells in the PAM of cancer dynamics to Fusarium cells, thelatter can “communicate” not only with their nearest neighbors. The nutrients can betranslocated directionally from active to the tip cells or the colony body. We recognizetwo mechanisms of nutrients dispersion:

—Active. This occurs only in the tip cell k1 (sprout) direction. An additional amount ofnutrients is transported from the neighboring Fusarium active cell k2 (it is not a tipcell). This amount is proportional to the nutrient concentration in k2, sf (k, t)—thatis,

�active(k2 → k1) = c5sf (k2, t) · �t. (11)The same amount is deducted from k2.

—Passive. This is caused by diffusion in the hyphal network. Diffusion between Fusar-ium cells is modeled via nutrient exchange processes between neighboring active(and inactive) cells:

�Diff(k2 → k1) = c6(sf (k1, t) − sf (k2, t)) · (�t/�x2). (12)

In this article, considering that we concentrate on the definition of a PAM methodin the context of F. graminearum growth, the parameters c1 through c6 were matchedcoarsely using published data (mainly Boswell et al. [2002, 2007] and Boenisch andSchafer [2011]) and observations of micrographs from laboratory experiments.

As shown in Figure 7(b) and (c), due to the degradation influence of Fusarium tipcells on capillary walls and other host tissue, the model is able to simulate realis-tic penetration properties of the colonization process, reproducing both vertical andlateral Fusarium invasion scenarios. As shown in Figure 7, the comparison of sim-ulation results with the fluorescence micrographs from laboratory experiments showencouraging qualitative agreement between the two.

4. DISCUSSION AND CONCLUSIONS

In this work, we define a new modeling approach—PAM—which combines the inter-acting PM with rule-based GDS. PAM is more generic version of the complex automatamodel (CxA), which we presented in Dzwinel [2012]. In that work, we focused on sys-tems with clearly separable scales (see Figure 1(b)) and various approaches to coarsegraining of particle-based and cellular automata models [Magiera and Dzwinel 2014;Dzwinel and Magiera 2015]. In this work, we concentrate on generalization of dynami-cal systems represented by graphs such as GDS. Unlike other GDS approaches, in PAMthe graph architecture undergoes spatiotemporal evolution stimulated by Newtoniandynamics of interacting particles.

We emphasize on PAM applicability in simulating biological systems with not onlylocal and separable spatiotemporal scales but also long-range and strong cross-scaleinteractions. In this context, we present a novel PAM of fungal infection and its spreadin wheat. We compare it to other rule-based models that we developed earlier, such asblood-clotting and cancer proliferation models.

We demonstrate that by representing a biological system as a generalized GDSin which microscopic processes are simulated by automata rules on graph vertexes,while its overall (macroscopic) architecture is driven by the interacting PM, we cancreate a robust modeling framework for simulating multiscale processes. In general,the automata rules may correspond to microscopic processes occurring in multipletemporal scales. Thus, they can be treated in a similar way as the “details” in DWT.Consequently, the reality can be approximated by the sum of “details” from the rule-based part of the model and the particle-based coarse-grained system dynamics.

The great advantage of PAM is its reasonable computational complexity in spatiotem-poral scales of processes occurring in microscopic tissue (approximately 10−9 m3), which



Fig. 8. Diagram presenting the speedup with the number of cells simulated for the PAM tumor modelimplemented in a CUDA environment on a GPU (Nvidia GeForce GTX 295, 240 threads) compared to a serialcode run on an Intel Xeon [email protected]. The upper plot shows the speedup for interaction calculations(PM), whereas the lower plot shows the overall speedup of the code.

cannot be described by continuum models. Moreover, as shown in Figure 8 and in Wcisłoet al. [2010, 2013] and Worecki and Wcisło [2012], PAM simulations can be consider-ably accelerated because the method can be efficiently parallelized both on multicoreCPU and GPU processors.

The simplicity and the clear scheme of multiple scales bridging are also advantagesof PAM over other modeling approaches. The PAM is scalable in the context of boththe system size and modeling fidelity. The latter means that the PAM can be easilyextended, including additional fine-grained processes, provided that they can be ex-pressed in terms of particle states and/or complexity of the particle collision operator.The computational complexity scales roughly proportionally with the number of spa-tiotemporal scales (states) simulated. Thus, the model can be computationally efficientfor a reasonable number of particles and particle states. For example, tumors up to1mm in diameter can be simulated on a laptop computer equipped with a standardGPU card (up to 1 million particles can be simulated with a few particle states) andtheoretically two orders of magnitude greater on medium-ranged MPI/GPU servers[Grinberg et al. 2011; Tang and Karniadakis 2013].

The PM, which defines the system spatial resolution and stays on top of the hierarchyof spatiotemporal scales, is responsible for the system dynamics and its mechanicalproperties. They are modified by microscopic processes represented by evolving particlestates and simple automata rules. The ability to mimic mechanical interactions ofactive biological systems such as blood cells, tumor, and fungal colonies with the restof tissue shows that PAM can reproduce realistic 3D dynamics of complex biologicalprocesses. This way, the PAM approach merges the advantages of both PM and cellularautomata paradigms. In the models presented here, only basic principles of cancer andfungal growth were taken into account. However, the addition of more sophisticated



processes to the framework of PAM, such as heterogeneous tumor cells, more angiogenicfactors, immunological mechanisms, and toxin effects, appears to be straightforward.Our modeling approach also has some important drawbacks. Next, we discuss only themost serious ones.

The PM represents the coarsest scale of the PAM. If a particle mimics a cell, thesize of tissue modeled is limited by the number of particles N that currently can besimulated. The linear size of 3D object increases with N1/3. Thus, PAM applicability tosimulate large systems, such as a tumor of diameter greater than a few millimeters,is limited by at least O(N) computational and memory complexities [Dzwinel 2012].Therefore, to simulate the tissue scale larger than 1cm, the continuum level of thedescription should be considered or a new coarse-graining procedure developed (e.g.,in which a particle can represent a tissue fragment instead of a single cell).

Since this work focuses on PAM formulation, less attention was paid to its calibration.The parameters of models presented here were matched coarsely using data publishedearlier, rule of thumb, and visual comparisons. This is a consequence of serious diffi-culties with data assimilation to all models based on particle interactions caused bythe lack of a viable formal procedure, which is able to represent mechanical propertiesof the tissue in terms of forces and their parameters. Because they are chosen mainlyusing rule of thumb, the PAM in its current form can rather be applied as a usefulqualitative metaphor of multiscale growth than a quantitative and predictive tool.

Further generalization of the PAM, more rigorous integration of interacting PMwith rule-based GDS and continuous models, and above all, releasing it from variousconstraints imposed by physics, would allow for discovering even more areas of PAMapplication. The interactions between a pair of objects cannot only be physical andcannot concern only the nearest neighbors. For example, people and robots can com-municate remotely, and the result of such interactions cannot be described by simplelaws of motion. Moreover, the particles can be used in function optimization or as theuniversal solver for finding a global minimum of multimodal functions [Dzwinel 1997].In machine learning, the particles can correspond to feature vectors in low-dimensionalembeddings, whereas their state vectors may consist of the class labels, the class mem-bership of their k-nearest neighbors, or text annotations. By using particle-based mul-tidimensional scaling, it is possible to develop interactive visual classifiers [Dzwineland Wcisło 2015].

Summing up, although the PAM approach is not yet a “silver bullet” in the modelingof complex multiscale phenomena, it is an interesting concept and an alternative forexisting continuous-discrete modeling strategies.

ACKNOWLEDGMENTS

We are very grateful to Dr. Arkadiusz Dudek, Professor of Medicine at the Division of Hematology/Oncology,University of Illinois, for his collaboration in development of the cancer model. We also thank Dr. KrzysztofBoryczko, Professor of Computer Science at the AGH Department of Computer Science for providing us withthe figure of blood clotting simulation.

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Received March 2014; revised June 2015; accepted September 2015


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