Dynamically Reorganising Vascular Networks Modelled Using ...

Dynamically Reorganising Vascular Networks Modelled Using Cellular Automata Approach Pawel Topa AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krak´ ow, Poland [email protected]

Abstract. We present the model of tumour induced angiogenesis that apart from predominating phenomena, as influence of VEGF, includes newly discovered factors such as Dll4/Notch signalling and remodelling processes. The Cellular Automata approach is employed to model cellular and intracellular processes that occur in cancer tissue and surroundings. Vascular system is modelled by using the Graph of Cellular Automata, which combines graph theory with the Cellular Automata paradigm. Additionally, an outline of model verification method which uses graph descriptors is presented and exemplified.

1

Introduction

Angiogenesis is the process of blood vessels formation [1]. The cells influenced by the stresses (e.g., low O2 , low pH), synthesise angiogenic stimulators (most of all VEGF — Vascular Endothelial Growth Factor) [1]. Stimulators diffuse towards the nearest blood vessels and activate the endothelial cells (ECs) that lines the vessel walls. In the response the endothelial cells start to proliferate and migrate being attracted by the gradient of VEGF. The wall of parent blood vessel becomes degraded and a lumen of a new capillary is formed. The process of angiogenesis has a crucial role in solid tumour growth. Clusters of growing tumour cells are short of oxygen and nutrients. The ”starving” tumour cells produce VEGF and other angiogenic stimulators (Tumour Angiogenesis Factors — TAFs) that activate neighbouring vessels. Tumour induced angiogenesis is a very promising target in anti-cancer therapy [3]. Inhibition of angiogenesis or regression of existing vasculature may suppress tumour development. Treatment targeted on improving vasculature around tumour can be helpful for drug delivery during chemotherapy. The computer models of angiogenesis employ continuous or discrete approaches [2]. The continuous models base on differential equations and as a result we obtain distributions of endothelial cells in the tissue. The main disadvantage of the continuous approach is the lack of information about the structure of vascular network. In contrast, the discrete models are able to produce vascular network of a given topology [2]. These models mostly base on the Cellular Automata (CA) paradigm and assume that modelled species (endothelial cells, angiogenic factors H. Umeo et al. (Eds): ACRI 2008, LNCS 5191, pp. 494–499, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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etc.) are treated individually. In [6], Anderson and Chaplain assumed that growth of a single sprout is governed by move of the endothelial cell located at its tip. It moves across regular, rectangular network of CA according to predefined rules. At each step of simulation the tip cell moves in one out of four directions or waits with a certain probability. The probabilities are calculated by using diffusion equation with terms reflecting VEGF and fibronectin influence [6]. Additional rules that model vessels branching and anastomosing were also defined. In [8] we proposed a new framework for modelling multiscale phenomena in tumour induced angiogenesis. Our approach combines the Cellular Automata paradigm with a graph structure into a new modelling tool which we call the Graph of Cellular Automata [7]. We assumed that the tissue is represented by a regular mesh of cellular automata and some of the accompanying diffusion processes are modelled by using simple rules of local interaction. The vascular network is represented by the graph structure constructed over the regular mesh. The graph is made of some CA cells, for which we define additional relations of neighbourhood with other cells. The role of Dll4/Notch in angiogenesis has been investigated experimentally quite recently [9]. This ligand regulates the processes of sprouting and vessels maturation. Inhibition on Dll4/Notch causes creation of dense but dysfunctional (immature) vascular network, while its stimulation results in creation of sparse though functional network. Both the inhibition and stimulation can be considered as anti-cancer therapeutic strategy, The influence of Dll4/Notch on the dynamics of the process of angiogenesis has not been modelled so far. Unlike in the normal angiogenesis (e.g. during embryogenesis), tumour induced angiogenesis forms mostly pathological vascularisation [4]. Defected vessels and vessels, which do not transport blood decline and are removed from the tissue. Conversely, the vessels with circulating blood survive [5]. Models of any physical processes require verification. In the case of angiogenesis this verification is not trivial. We use the feature vectors of statistical descriptors, hoping that these descriptors, as an ensemble, will have sufficient generalisation power to classify various network structures. The analysis of the networks, i.e., investigation of their similarities and dissimilarities can be made then by using pattern recognition tools, such as classifiers, clustering and feature extraction techniques. In this paper we present extended version of our previous Cellular Automata model [8]. First, we shortly introduce its foundation and our recent extensions. Then we discuss exemplary results. We propose also the method of quantitative analysis of vascular networks, which employs the statistical graph descriptors [11]. Finally, we discuss the conclusions.

2

Tumour-Induced Angiogenesis Model with Vascular Network Reorganisation

The formal definition of the model and the outline of the essential algorithms were presented in [8]. Shortly, the model works as follows:

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1. The cells, which form the graph (“vessel” cells) are the sources of nutrients (e.g. O2 ). Nutrients are distributed to surrounding cells producing a certain gradient of concentration. 2. Tumour cells with nutrients concentration below a certain threshold change their states into ”hungry” and start producing TAFs. TAFs are distributed over the mesh of CA with certain gradient of concentration. 3. The growth of a single vessel is governed by the move of its tip. The consecutive tip cell positions follow local TAFs gradient. 4. Initially, a newly formed vessel is not mature enough to be able to supply nutrients. The maturation level of each cell in the graph is incremented at every step of simulation until it reaches the ”mature” state. 5. When TAFs concentration in “vessel” cell exceeds a certain threshold and the cell is “mature” enough, a new sprout is generated. The vessels, which glue together produce anastomoses.

Fig. 1. A: The outline of the model of tumour-induced angiogenesis [8]. B: Initial geometry for model of tumour induced angiogenesis enforces the direction of blood flow.

In Fig. 2 we present an outline of the model. At each step of simulation, procedures, which implement the defined CA rules are applied both to the Cellular Automata and the Graph of Cellular Automata. The model is very flexible and new modules can be easily included. In this paper we investigate the influence pf Dll4/Notch ligand on sprouting and maturation processes. The process of removing of unused and dysfunctional vessels (the vessels without circulating blood) is also taken into account. The new rules are as follows: 1. The “mature“ vessels supply oxygen only if they create the closed circuit with blood flowing due to pressure gradient. 2. Dysfunctional vessels, i.e., the vessels which do not transport blood are removed gradually. 3. The ”vessel“ cells have additional parameters corresponding to Dll4/Notch signalling. It describes sensitivity of the cells to the TAFs and the rate of

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maturation. The value of Dll4/Notch, which exceeds a given threshold stimulates sprouting but delays the process of maturation. Conversely, low value of this parameter keeps low level of sprouting but speeds up maturation. We define the following set of the control parameters: 1. 2. 3. 4.

tg — TAFs(VEGF) gradient coefficient. og — Oxygen gradient coefficient. dll4 — Dll4/Notch signalling level in ”vessel“ cells. tt — TAFs threshold that activates ”vessel“ cells.

We assume also that some previously defined parameters [8] are functions of TAFs and Dll4/Notch concentration, i.e., the probability of branching is directly proportional, while maturation speed is inversely proportional to this value. Calculating blood flow in such a complex, chaotic and dynamically changing network structure is extremely difficult. We assume, that blood flows only in the vessels that create closed circuits. We detect such the circuits, calculating ”pressure“ at each node of the network and ”blood flow” in every edge. The edges with nonzero pressure gradient between corresponding nodes are treated as “efficient” ones. To simplify the model, we construct initial conditions of simulations with two vessels: artery and vein (see Fig. 2B). We assume that only the artery has the ability of sprouting. The sprouts attracted by TAF gradient grow towards the vein. This way the blood flow direction from artery to vein is biased (see Fig. 2B). Closed circuits and blood flow in the edges are updated at each step of simulation. At each step of simulation the cells in vessels network, which do not transport blood, are marked as inactive. After a certain amount of time-steps, these cells are removed from the vascular network.

3

Results

The simulations were performed on the mesh of 100 × 100 cells. Figure 2A displays vascular network generated without removal of unused vessels while in Fig. 2B we present the results of simulation with this option turned on. As shown in Fig. 2C the “brush border effect” (sudden increase of density of sprouts) appears as a result of existence of TAF threshold, which activates endothelial cells. The networks displayed in Fig.2A and Fig.2B were generated with lower value of dll4 parameter (Dll4/Notch signalling) than those shown in Fig.2C. As a result we obtain considerably sparser vascular network with a better organised structure. 3.1

Quantitative Verification

To verify the model, the networks obtained from simulations have to be compared to the networks extracted from images of real vascular systems. We require a parameter space (feature space) that will be able to differentiate their structural properties. To make such the structural analysis possible we use the feature vectors composed of statistical descriptors of complex networks [11]. This method

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Fig. 2. Sample simulation results (see comments in the text). The vessels with blood flowing are depicted in black while those without blood circulation are marked in grey.

Fig. 3. Three sets of simulated network topologies (0, 1, 2) representing 3 different values of parameters. The result of visualisation of descriptor feature space with MDS method is displayed. Two different views of reduced 3-D feature space are shown.

will allow for investigating model parameters, which influence topology of the simulated vascular networks. These parameters can be tuned reflecting various types of drug treatment (e.g. inhibiting of VEGF). Thus our model can serve as a kind of virtual laboratory for testing of various therapy strategies. We applied a selected set of graph descriptors [11] to calculate feature vectors and compare them using pattern recognition tools such as the multidimensional scaling [10]. Each network is described by 11 component feature vector. MDS method translates 11-D feature space into its 3-D representations. In Fig.3 we show the sample results of simulation generated for three different combinations of parameters (sets marked as (0, 1, 2)). The feature vectors have been calculated for each set. We visualise the results using multidimensional scaling procedure. We can see clusters in the feature space corresponding to each class of generated networks (0, 1 and 2). This way we can estimate the influence of various biological

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factors on the network topology. There are only preliminary tests but we show that the method can effectively separates network samples generated for various model parameters.

4

Conclusions

We show that the general framework of the Graph of Cellular Automata used for modelling tumour-induced angiogenesis [8] can be easily extended by new factors and processes. We have defined additional parameter that reflects influence of Dll4/Notch signalling on sprouting and maturation processes. It controls both the reaction of endothelial cells in the vessels, VEGF concentration and the rate of vessels maturation. We show that by changing the parameter corresponding to the Dll4/Notch activity we can obtain instead of rich but chaotic and dysfunctional vascular network more sparse structure but potentially efficient. The model has been extended with algorithms that detect vessels with blood flowing. The process of remodelling of vascular network in case of tumour induced angiogenesis is very important. Treatments targeted against tumour induced angiogenesis aim at vessels regression. Also therapies that support vessels network normalisation require rules for removal of unused and defective sprouts. In this way the network system tries to build up more and more efficient structure. The application of graph descriptors for comparison of the vessel networks, their classification and model validation is a very promising method. In the future we plan to select from tens of known descriptors [11], those which are the most relevant in the detection and generalisation of dominant networks topologies. Acknowledgements. The author thanks A. Dudek (University of Minnesota Medical School), W. Dziwnel and T. Arod´z (AGH University of Science and Technology) for valuable discussions. This research is partially supported by the Polish Ministry of Education and Science (grant no. 3 T11F 010 30).

References 1. 2. 3. 4. 5. 6. 7. 8.

Carmeliet, P., Jain, R.K.: Nature, 407 (2000) Mantzaris, N.V., Webb, S., Othmer, H.G.: J. Math. Biol. (2004) Ferrara, N., Kerbel, R.S.: Nature, 438 (2005) Tonini, T., Rossi, F., Claudio, P.P.: 22 (2003) Gee, M.S., et al.: American J. of Patbiology, 162 (2003) Anderson, A.R.A., Chaplain, M.A.J.: Bull. of Math. Biol. 60 (1998) Topa, P., Dzwinel, W., Yuen, D.: Int. J. Mod. Phys. C. 17 (2006) Topa, P.: In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173. Springer, Heidelberg (2006) 9. Hellstrom, M., et al.: Nature, 445 (2007) 10. Dzwinel, W., Blasiak, J.: Future Generation Computer Systems, 15 (1999) 11. Goryczka, S., Arod´z, T.: Bio-algorithms and med-systems 2 (2006)