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Towards a Two-Scale Cellular Automata Model of Tumour-Induced Angiogenesis Pawel Topa AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krak´ ow, Poland [email protected] Abstract. This paper presents a new framework for modelling tumourinduced angiogenesis. Classical Cellular Automata approach is employed to model cellular and intracellular processes that occur in cancer tissue and neighbourhood. Vascular system is modelled by using Graph of Cellular Automata, which combines graph theory with Cellular Automata paradigm. A new model is proposed as a starting point for further investigations on multiscale model covering wide range of spatio-temporal scales including blood flow processes. The basis of the model with the algorithms are presented. Preliminary results with short discussion are also included.

1

Introduction

Angiogenesis is the process of formation of blood vessels. It occurs in embryogenesis after the vasculogenesis stage. During vasculogenesis, primary, chaotic network of capillaries are formed from the endothelial precursors [1]. Next, during the angiogenesis stage, the network is rebuilt into a fully functional network of arteries, capillary vessels and veins. In adulthood angiogenesis is rigorously controlled by wide range of stimulators and inhibitors [2]. Their very precise balance makes this process quiescent except tissue healing, placenta forming during pregnancy and in the cycling ovary. The angiogenic process can be activated by metabolic stress e.g. low O2 (hypoxia), low pH or hypoglycemia. Other conditions such as mechanical stress (pressure generated by proliferating cells), immune response and genetic mutations [2] can activate angiogenesis too. Oxygen and nutrients penetrate the tissue only in a certain distance from the vessel. Distant cells, influenced by metabolic stress, synthesise angiogenesis stimulators such as VEGF (Vascular Endothelial Growth Factor) and bFGF (Basic Fibroblast Growth Factor) [2], [3]. Stimulators migrate towards the nearest blood vessels. When they reach the vessel, the endothelial cells (ECs) that lines the wall of this vessel are activated. They start to proliferate and migrate towards the tumour cell attracted by VEGF and other stimulators. The wall of the parent blood vessel becomes degraded and it opens to a new capillary. Migrating and proliferating ECs form a hollow tube-like cavity (the lumen), which are stabilised later by smooth muscle cells and perycites. Finally a new capillary vessel becomes fully functional. S. El Yacoubi, B. Chopard, and S. Bandini (Eds.): ACRI 2006, LNCS 4173, pp. 347–356, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Uncontrolled proliferation of tumour cells makes that existing blood vessels cannot supply them with oxygen and nutrients. In consequence ”starving” tumour cells produce VEGF, bFGF and other stimulators of angiogenesis described in this case as Tumour Angiogenesis Factors (TAFs). Neighbouring vessels, activated by TAFs, start sprouting, and develop toward tumour tissue. Due to imbalance of angiogenic factors new vessels form a highly chaotic and disorganised network [6]. Moreover, their walls have a pathological form, i.e. they are thin and permeable, their diameter changes abruptly etc. Inhibition of tumour-induced angiogenesis is the most promising strategy in anti–cancer therapy [2], [1], [5]. Most of the currently tested therapies targeted endothelial cells. The inhibitors not only suppress ECs proliferation but also initiate their death what follows to vessels regression. Anti-VEGF treatment also normalises chaotic structure and abnormal architecture of tumour induced vessels what improves drug delivery to tumour tissue. However, clinical tests show that none of the tested inhibitors did success in broad range types of cancers [1]. Monotherapies fail because angiogenesis is controlled by very complex balance of stimulators and inhibitors. Therefore, further investigations have to concentrate on researches including wider range of angiogenic factors. 1.1

Models of Angiogenesis

Angiogenesis is modelled by using continuous and discrete approaches [3]. Continuous models employ Partial Differential Equations in order to reflect distributions of endothelial cells and angiogenic factors. Stochastic movement of endothelial cells are represented by diffusion equation. Other factors are included as additional terms to the original diffusion equation. Anderson and Chaplain [7] simulate diffusion of ECs governed by angiogenic stimulators and fibronectin influence. Plank et al. [8] meets wider range of angiogenic stimulators including angiopoetins to their model. Discrete approach assumes that modelled molecule as endothelial cells or angiogenic factors are treated individually. Anderson and Chaplain proposed one of the most often cited discrete model of angiogenesis [7]. They assume that growth of the single vessel is governed by move of the endothelial cell located at the sprout tip. This cell moves across regular, rectangular network according to defined rules. At each step of simulation the cell moves in one of the four directions or stays with a certain probability. The probabilities are calculated by using continuous approach, i.e. diffusion equation supplied with terms reflecting VEGF and fibronectin influence. Additional rules which model vessels branching and anastomosing are also defined. Stokes and Lauffenberg presented a bit different approach [9]. They also model sprouting vessels as separate structures. Each sprout is described by the position and velocity of its tip at a given time step of simulation. The velocity is calculated by using stochastic differential equation that combines viscous damping term, random motion term and chemotactic term (models TAFs influence).

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Recent researches focus on multiscale model that are able to cover all phenomena contributing to cancer development. Alacorn et al. [10], [11] presented model that couples processes occurring on different spatio-temporal scales: – vascular scale that includes vascular network adaptation and blood flow, – cellular scale: cell-cell interaction (e.g. tumour-normal competition) and cell spatial distribution, – intracellular scale: cell division, TAFs secretion and apoptosis. Alacorn et al. [10] focuses on modelling whole cancer rather than on angiogenic processes, thus their vascular network has a form of simple hexagonal mesh. In this paper a new framework for modelling tumour–induced angiogenesis is proposed. The presented model is regarded as a test for this framework, therefore, some problems connected with angiogenesis were substituted by their simplifications. The investigation is targeted on building a modelling environment that besides wide spectrum of angiogenesis factors, is able to consider blood flow processes and their influence on vascular system development.

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The Model of Tumour-Induced Angiogenesis

The model is founded upon the concept of transportation network and consuming (or producing) environment [12], [13]. The network delivers certain resources to the system, where they are absorbed and changed into progress of the environment. Changes in the environment influence network structure. The system leads to the state of dynamic balance, when the whole environment is equally supplied. An anastomosing river [14] as well as a vascular system are good examples of such the phenomenon. We can also consider the system in which the resources are transported in the opposite direction i.e. the resources are collected from the environment and transported outside. Branching network formed by a river and its tributaries fits this scheme. This approach postulates partial separation of the two time scales represented by formation of the channel and the environmental factor, respectively. Instead of modelling local relations between the endothelial cells, blood, nutrients and oxygen, we consider now the global interactions between the blood vessels network and tissue. The network edges (vessels) can be added or removed accordingly to the local distribution of TAFs. Conversely, the nutrients distribution is formed by the entire river network. The feedback between environmental changes and evolution of the network should be faster allowing for modelling vessels over larger spatial scales. The tissue is represented by a mesh of cellular automata. The distribution of oxygen, nutrients and TAFs is modelled by using Cellular Automata rules of local interaction [15]. The transportation network is represented by the Graph of Cellular Automata (GCA) which is built over the CA mesh [12], [13]. The graph is constructing by choosing some cells from the regular mesh, and connecting them with edges that represent the sections of the vessels. The angiogenesis is an extremely complex process which is still not fully understood. Therefore, the following assumptions had to be made to this model:

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– – – –

tumour cells do not migrate nor proliferate, ”hungry” tumour cells produce TAFs at constant rate, TAFs migrating through the tissue establish gradient of TAFs concentration, TAFs concentration exceeding certain threshold activate endothelial cells in existing vessels, – only ”mature” vessels are able to create sprouts, – sprouts grow attracted by TAFs concentration, – new vessel have to ”mature” before it can be able to fulfil they function — it corresponds to the process of covering endothelial cells by smooth muscle cells an perycites. Most of these assumptions are based on real observations, however some of them are only hypothesis or they were included for the sake of clarity of the algorithms. The model can be defined in a more formal way as follows: CAAN G =< Z 2 , GCA , XK , S, δ >,

where :

– Z 2 — a collection of cells ordered as a square mesh of Z × Z cells, – GCA — a planar and acyclic graph defined as (VG , EG ), where VG ⊂ Z 2 and EG ⊂ Z 2 ×Z 2 are a finite set of vertices and a finite set of edges, respectively, – XK (i, j) — neighbourhood for the (i, j) cell in regular mesh of automata, – S — is the set of state vectors corresponding to each cell: S = Sm × Sg , • Sm — represents states corresponding to all cells in the mesh: ∗ tij — state of a single tumour cell, two possible values: ”full” and ”hungry”, ∗ fij — TAFs concentration, ∗ nij — nutrient (oxygen) concentration, • Sg — represents states corresponding to the cells that belong to the Graph of Cellular Automata: ∗ aij — ”age”, maturation level, ∗ pij — indicate ”tip” cell (boolean), The cells which form the graph are the sources of nutrients (e.g. oxygen). Nutrients are distributed to the surrounding cells, providing certain gradient of concentration. Tumour cells with nutrients concentration below the certain value turn their state into ”hungry” and start producing TAFs. TAFs distribute through the mesh of automata and establish certain gradient of concentration, in a similar way as in case of nutrients. When the TAFs concentration in cell that belongs to the graph exceed certain threshold, a new branch is initiated. The vessel grows attracted by higher TAFs concentration. Similarly to the discrete Chaplain and Anderson model [7], growth of the single vessel is governed by the move of its tip. The consecutive tip cells are calculated based on local TAFs gradient. Initially a new vessel is not mature enough to be able to supply nutrients. The maturation level of each cell in the graph is increased at each step of simulation until it reach the state ”mature”. The ”mature” cells become the source of nutrients.

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Vessel forks when the local condition (TAFs concentration exceed certain threshold and cell is mature enough) are fulfilled. When growing vessel meet other vessel, it joins them creating anastomosis. The set of the parameters that tune the model is as follows: – – – – – – 2.1

ρT AF — gradient of TAFs distribution, ρN — gradient of nutrient(oxygen) distribution, Tb — TAFs threshold that triggers branch forming, Nt — nutrients threshold that triggers production of TAFs in tumour cells, Ms — maturation speed, Pb — branch probability. The Algorithm

Fig. 1 presents an outline of the algorithm. At each step of simulation, procedures that implement the defined rules are applied to cellular automata and graph of cellular automata.

Fig. 1. An outline of the main algorithm

Procedure updateTumourCells() tests the nutrients concentration in the cells that represent tumour and triggers the production of TAFs if necessary.

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Procedure 1 An outline of the updateTumourCells() procedure for all cell in the mesh do if nij < Nt then tij ⇐ ”hungry”; else tij ⇐ ”full”;

Procedures updateNutrients() and updateTAFdistribution() distribute nutrients and TAFs, respectively, and they have very similar form. New values of nutrients and TAFs concentrations are calculated based on maximum concentration within their neighbourhoods. Cij denotes a cell identified by using indexes i, j.

Procedure 2 Procedures updateTAFdistribution() and updateNutrients() TAFs distribution

Nutrients distribution

for all cell in the mesh do if tij = ”hungry” then fij ⇐ 1.0; else fij ⇐ ρT AF max(fXK (ij) );

for all cell in the mesh do if Cij ∈ VG and aij = ”mature” then nij ⇐ 1.0; else nij ⇐ ρO max(nXK (ij) );

Procedure updateVessels() is responsible for vessels growth and maturation (see Procedure 3). If ”tip cell” marker (pij ) is set, procedure addNextTip() calculates which cell will be added to the branch next. Growing vessels are attracted by TAFs thus the cell with the largest increase of the tij value will be added. If there is no such a cell a simple random-walk procedure is applied. An new cell becomes a ”tip cell”. If selected cell already belongs to other branch, the ”tip” marker pij is unset, and the branches join creating anastomosis. At each step of simulation only one cell is added to each sprout. Procedure 3 An outline of updateVessels() procedure for all cell Cij ∈ VG do if pij then addNextTip(Cij ); if aij < ”mature” then aij ⇐ aij + Ms ;

Procedure branchVessels() tests whether the conditions for branching occur. If necessary it initiates a new sprout with probability Pb , by setting ”tip cell” marker — pij . New sprout starts to grow on the next run of the updateVessels() procedure.

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Procedure 4 An outline of the branchVessels() procedure for all cell Cij ∈ VG do if fij > Tb and aij = ”mature” and random() < Pb then pij ⇐ 1;

3

Results

We present preliminary results obtained by using described model. The model was implemented in C++ language as a sequential program and run on workstation under Linux operation system. Results were postprocessed and visualized by using Amira program (www.tgs.com). Fig. 2 presents snapshot taken after 200 steps of simulation performed on mesh 100 × 100 cell. Apart the vascular network, the TAFs distribution is visualized. Single sprouts grow attracted by higher TAFs concentrations. The nodes of Graph of Cellular Automata were intentionally emphasised.

Fig. 2. Simulation results: vessels network and TAFs distribution

Fig. 3 presents snapshots taken after 100, 120, 220 and 1200 steps of simulation. Simulations were performed on 100 × 100 mesh. The vascular network is presented together with the TAFs distribution. The algorithm of branch growth neglects random motion and considers only the TAFs influence. It results in the

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Fig. 3. Snapshots from simulation performed on 100 × 100 cells after 100, 120, 220 and 1200 steps. Primary vessel is located on the right edge of the mesh.

clearly visible direction of growth. The assumption that only ”mature” vessels can branch makes that we observe anastomosing vessels without branching at all. Fig. 4 presents the network of blood vessels together with the nutrients and the TAFs distribution. Simulation was performed on mesh 100 × 100 cells and snapshot was taken after 1200 steps. Compared with Fig. 3 simulations were performed with higher maturation speed Ms . As a result most of the sprouts are already mature enough to supply nutrients and the network has much more complex structure. Single sprouts have already reached tumour cells, however they are still not mature enough to supply oxygen and nutrients to starving cells.

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Fig. 4. Network of blood vessels growing towards tumour cells. Primary vessel is located on the bottom edge of the mesh. Green area represents tumour cells producing TAFs. Yellow-to-red area represents nutrients.

4

Conclusions

The general framework combining Cellular Automata and Graph of Cellular Automata seems to be suitable for modelling tumour-induced angiogenesis. The preliminary results are promising, however at this stage it is pure phenomenological model, and it has numerous oversimplifications. Future investigations have to carefully revise the rules, defined for this model and include the new ones that will be able to consider other factors and subprocesses that contributes to angiogenesis. One of the major simplification is the assumption that nutrients can be supplied by any mature vessel. In fact it is not true if the vessel is not a part of closed circuit. Thus, the further work on this model will be targeted on considering blood flow processes in capillary vessels and their influence on cancer development [16]. Moreover, the investigations on blood flow in tumour induced vessels are another very promising area in anti-cancer researches e.g. improved drug delivery increases chemotherapy efficiency. Another issue, which is going to be investigated deeply is quantitative comparison of real and simulated vessels networks. Graph representation of vessels networks facilitates calculating the descriptors for the simulation results. Network descriptors for real vascular networks will be obtained from pictures with tumour tissue through the pattern recognition process.

Acknowledgements The author thanks Prof. A. Dudek (University of Minnesota Cancer Center), Prof. D.A. Yuen (Minnesota Supercomputing Institute), Prof. W. Dzwinel,

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T. Arod´z (AGH University of Science and Technology) and Dr. M. Paszkowski (Polish Academy of Science) for valuable comments and discussions on the model assumptions. Dr. J. Tyszka and Dr. W. Alda kindly revised this paper. This research is partially supported by Polish Ministry of Education and Science (grant no. 3 T11F 010 30).

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