Particle Model of Tumor Growth and Its Parallel ... - Springer Link

Particle Model of Tumor Growth and Its Parallel Implementation Rafal Wcislo and Witold Dzwinel AGH University of Science and Technology, Institute of Computer Science, Cracow, Poland {wcislo,dzwinel}@agh.edu.pl

Abstract. We present a concept of a parallel implementation of a novel 3-D model of tumor growth. The model is based on particle dynamics, which are building blocks of normal, cancerous and vascular tissues. The dynamics of the system is driven also by the processes in microscopic scales (e.g. cell life-cycle), diffusive substances – nutrients and TAF (tumor angiogenic factors) – and blood flow. We show that the cell life-cycle (particle production and annihilation), the existence of elongated particles, the influence of continuum fields and blood flow in capillaries, makes the model very tough for parallelization in comparison to standard MD codes. We present preliminary timings of our parallel implementation and we discuss the perspectives of our approach.

1

Introduction

Typically, tumor growth is divided into three stages: avascular growth, angiogenesis, and vascular growth [1]. In the earliest stage, the tumor develops in the absence of blood supply and grows up to a maximum size. Its growth is limited by the amount of nutrients the tumor can obtain through its surface. In angiogenic phase [1], some of the cells of avascular tumor mass produce and release substances called tumor angiogenic factors (TAFs). TAF diffuses through the surrounding tissue, and, upon arrival to the vasculature, it triggers a cascade of events which stimulates the growth of vasculature towards the tumor. In vascular stage the tumor has access to virtually unlimited resources, so it can grow beyond its limited avascular size. Moreover, it can use the blood network for spreading cancerogenic material to form metastases in any part of the host organism. Thus, whereas in the avascular phase tumors are basically harmless, once they become vascular they are potentially fatal. A better understanding of the dynamics of tumor formation and growth can be expected from mathematical modeling and computer simulation. A rich bibliography about mathematical models of tumor growth driven by the process of angiogenesis is overviewed in [2,3,4,5]. The models fall into four categories: (a) continuum models that treat the endothelial cell (EC) and chemical species densities as continuous variables that evolve according to a reactiondiffusion system, (b) mechano-chemical models that incorporate some of the mechanical effects of EC-ECM (extracellular matrix) interactions, (c) discrete, R. Wyrzykowski et al. (Eds.): PPAM 2009, Part I, LNCS 6067, pp. 322–331, 2010. c Springer-Verlag Berlin Heidelberg 2010

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cellular automata or agent based models in which cells are treated as units which grow and divide according to prescribed rules, (d) hybrid multiscale models involving processes from micro-to-macroscale. Multiscale and multiphysic models represent the most advanced simulation methodologies. Disregarding all microscopic and mesoscopic biological and biophysical processes, in macroscopic scales, tumor growth looks like a purely mechanical phenomenon. It involves dissipative interactions between the main actors: normal and cancerous tissues and vascular network. Any of existing computational paradigms is not able to reproduce this basic process, which influences the most tumor and vasculature remodeling, tumor shape and its directional progression. In [6,7] we present the concept of tumor growth model which is driven by particle dynamics [8]. In fine-grained models a particle mimics a single cell or components of ECM. Nevertheless, this assumption becomes very computationally demanding for modeling tumors of realistic sizes. The largest MD (molecular dynamics) simulations involve 2×1010 particles [9] for rather not so impressive (only 50-100) number of timesteps. The computational restrictions impose current limits on spatiotemporal scales simulated by particle model of tumor growth. Tumor of 1mm in diameter consists at least of a million cells [2]. To simulate in 3-D the tumor of this size and its closest environment, one needs at least 1.6×107 particles simulated in 105 timesteps. This requires approximately the same computational resources as the largest MD simulations. However, particle method can also be used in its coarse-grained form. Unlike in purely fine-grained particle model, where all cells and tissue ingredients are represented by particles, in our model a particle can mimics a cluster of cells in ECM envelope. The particle system representing growing tumor is very different than standard particle codes such as molecular dynamics (MD) (e.g. in [10,11,12]). The number of cells increases and/or fluctuates because they can replicate or disappear. The particles have attributes which evolves according to the rules of cellular automata. Moreover, their values depend on concentration fields which are coupled with particle dynamics. Thus, the process of parallelization is expected to be more complicated than for standard particle codes. Below, we give a brief description of our model, and we discuss its parallel implementation. We present also examples of trial simulations and preliminary computational timings. Finally, we discuss the conclusions.

2

Model Description

In this section we give a brief description of particle model of tumor growth. More detailed discussion can be found in [7]. We assume that the fragment of tissue is made of a set ΛN = {Oi : O(r i , V i , ai ), i = 1, . . . , N } of particles where: i - the particle index; N - the number of particles, r i , V i , ai - particle position, velocity and attributes, respectively. The attributes are defined by particle type {tumor cell (TC), normal cell (NC), endothelial cell (EC)}, cell life cycle state {newly formed, mature, in hypoxia, after hypoxia, apoptosis, necrosis}, cell size, cell

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age, hypoxia time, concentrations of TAF, O2 and other factors, total pressure exerted on particle i from its closest neighbors and the walls of the computational box. The particles are confined in the cubical computational box of volume V. For the sake of simplicity we assume here that the vessel is constructed of tubelike “particles” – EC-tubes – each made of many endothelial cells. Summarizing, we define three types of interactions: sphere-sphere, sphere-tube, and tube-tube. The forces between cells should mimic both mechanical repulsion from squashed cells and attraction due to cell adhesiveness and depletion interactions. The particle dynamics is governed by the Newtonian laws. During the life-cycle, the normal and tumor cells change their states from new to apoptotic (or necrotic) according to their individual clock (and/or oxygen concentration). The cells of certain age and size, being well oxygenated, undergo the process of mitosis. They form two daughter cells. Finally, after given time threshold, the particles die due to apoptosis, i.e., programmed cell death. Dead cells are removed from the system. For oxygen concentration smaller than a given threshold, the living cell changes its state to hypoxia. Such the cells become the source of TAF. The cells, which are in hypoxia for a period of time longer than a given threshold, die and become necrotic. We assume that in the beginning, the diameter of necrotic cell decreases twice and, after some time, the cell is removed. This is contrary to apoptotic cells, which are rapidly digested by their neighbors or by macrophages. The duration of phases of the cell cycle for normal and tumor cells differs considerably. Moreover, the proliferation rate of tumor cells, which were in hypoxia state, can change [7]. The cell cycle for EC-tubes is different than for spherical cells. In fact, ECtubes are clusters of endothelial cells. The tubes grow both in length and in diameter. They can collapse due to a combination of reduced blood flow, the lack of VEGF, dilation, perfusion and solid stress exerted by the tumor. Because the EC-tube is a cluster of EC cells, its division onto two adjoined tubes does not represent the process of mitosis but is a computational metaphor of vessel growth. Unlike normal and tumor cells, the tubes can appear as tips of newly created capillaries sprouting from existing vessels. The new sprout is formed when the TAF concentration exceeds a given threshold and its growth is directed to the local gradient of TAF concentration. The transport of bloodborn oxygen and TAF into the tissue is modeled by means of reaction-diffusion equations. The distribution of hematocrit is the source of oxygen, while the distribution of tumor cells in hypoxia is the source of TAF. We assume that the cells of any type consume oxygen and the rate of oxygen consumption depends on both cell type and its current state. We assume additionally that only EC-tubes absorb TAF. TAF is washed out from the system due to blood flow. Diffusion of oxygen and TAF is many orders of magnitude faster than the process of tumor growth. On the other hand, the blood circulation is slower than diffusion but still faster than mitosis cycle. Therefore, we can assume that both the concentrations and hydrodynamic quantities are in steady state in the time scale defined by the timestep of numerical integration of equations of motion.


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This allows for employing fast approximation procedures for both calculation of blood flow rates in capillaries and solving reaction-diffusion equation (see [7]).

3

Data Structure and Algorithm

The simulation of tumor growth using particles involves data structures allowing for efficient performance of multiple, multiscale concurrent processes such as cell cycle, lumen formation, development and maturation of vascular network, blood flow and others. As was shown before, our system consists of two types of particles: spherical and “tube-like” particles, representing tissue and vasculature, respectively (see Fig. 1). The state of each particle (cell, fragment of capillary) is defined by several attributes and tens of minor parameters connected with their physical and biochemical properties. These data are stored in a shared array P of records. Each record contains information corresponding to a single particle. As shown in Fig. 1, during simulation, the number of spherical and “tube-like” cells can both increase due to mitosis and decrease as the result of apoptosis and necrosis. Information on newly formed cells is added at the end of particle array. In case of lack of free slots, the array is enlarged dynamically. The information on dead cell is removed from P and the empty space is replaced by the record from the end of array. During simulation particle indices in the array P become mixed. Consequently, positions of the particles in the array do not reflect their position in space. This fact has advert effect on the efficiency of the parallel implementation. The main loop of simulation representing time evolution of the particle system is shown in Fig. 2. array of cells

new cell

unused records

dead cell

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array of cells

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Fig. 1. The mapping of particles onto P array records containing current state of cells during the changes cased by cell cycle

After initialization phase, in subsequent timesteps we calculate new position of particles, the diffusion of active substances (nutrients, TAFs, pericytes), the intensity of blood flow in the vessels and the states of individual cells triggered by previous three factors and constrained by individual cell time clocks. These modifications of cell state can result in cell proliferation, its death and changes in its functions (hypoxic cells), size and properties.

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From the point of view of efficient computations, the linear computational complexity O(N ) of the algorithm is the most important requirement. However, both the calculation of forces and approximate procedure used for solving diffusion equation, involve the pairs of particles. Fortunately, both the particle interactions and approximation kernel [7] are short-ranged. By using well known procedures such as Hockney boxes (as in [10,11,12]) the approximate linear computational complexity is guaranteed. The computational box is divided onto cubic sub-boxes with edges equal to the interaction (kernel) range. As demonstrated in Fig. 3, the particle located in a given sub-box interacts with other particles located in this sub-box and in adjacent sub-boxes.

initialization for each time step for every pair of interacting particles compute forces

over 90% of simulation time (in sequential version)

compute gradients of densities for every particle calculate new velocitiy, size and position calculate new densities change state { may lead to creation of new cell or vessel or death (removal) of cell }

for every vessel particle calculate blood flow and pressure

Fig. 2. The main loop of the simulation code

Despite its apparent efficiency this classic algorithm has an annoying deficiency. As was shown in [11,12], the execution time for sequential fluid particle codes increases substantially during simulation. This happens when the particles cross the sub-box boundaries and the attributes of particles residing in the same sub-box are located in distant memory addresses. Therefore, to obtain the efficient code, the neighboring particles should be closer one another in the computer memory. This effect becomes more evident for multithread implementations on shared memory computers [12]. To avoid frequent cache misses and delays caused by programs retaining cache coherency, the particles should be renumbered every some period of time and the processors with large cache memories are preferred. Consequently, the particles in the same cell are classified with consecutive numbers. However, the gap between particle numbers still exists for the particles from neighboring cells used for computing the forces. This is due to the incoherence between the linear addressing of memory and indexing of 3D particle system. This cache effect is even stronger in our model, where incoherency between particle indices and their spatial position are larger than for the systems with conserved number of particles. This problem becomes serious even on the single sub-box level, where the particle addresses residing in the single sub-box can be


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distant in P array. To avoid this problem, we use another way of data storing in sub-boxes. The P array is divided onto blocks of similar size. The size is equal to the maximum number of particles which can be located in the subblock. The blocks correspond to individual sub-boxes. Every block in a certain moment of time has filled as many records as many particles (cells) are found in the corresponding sub-box (see Fig. 4). The rest of records are empty. The newly formed cells are located directly in corresponding blocks so not at the end of the P array.

array of boxes

many-to-many

array of cells

r_cut

r_cut unused records

Fig. 3. The array of sub-boxes

As we demonstrate in the following section, the preliminary timings made for improved code are encouraging. On the other hand we realize that this procedure allows for avoiding the cache-space incoherency on the level of individual subboxes, though the problem with neighboring sub-boxes still exists. As we show in the previous section the blood vessels are simulated using “tubelike” particles. Moreover these tubes can grow. Just the tubes are very serious complication disabling direct application of Hockney algorithm. The tubes can cover even 5 sub-boxes. Thus, in forces calculation procedure, one has to use additional tokens preventing from multiple calculations of insignificant interactions where at least one out of two particles is the “tube-like.”

4

Parallelization of the Simulation Program

In order to speed up the computations, the sequential code has been rewritten as a parallel program which can be executed on SMP machines with a shared memory access. We mainly focused on the efficient parallelization of computations of particle-particle interactions – as we annotated in Fig. 2, those computations were the most time consuming part of a simulation. In the parallel version of the program computations threads are created for every available processor core. During the simulation, every thread performs the

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R. Wcislo and W. Dzwinel array of cells

box 1

box 2

box 3

box 4 unused records

box n

Fig. 4. The sparse array of cells

computations for the assigned subset of sub-boxes only. The threads, however, cannot run independently: the computations involving the boundary sub-boxes require access (both read and write) to data assigned to neighboring threads. The proper order of sub-box computations allows to avoid the locking problems. Currently, we are porting the simulation code to nVidia GPU cards to test the possibility of achieving better speed-ups than shown in Fig. 7.

5

Results and Timings

In Figs. 5, 6 we display the effects of mechanical interaction between vessels and growing tumor simulated by using our particle based model. The figures clearly show how important role this process plays in tumor progression. In Fig. 5A, B two vessels are pushed away by growing tumor opening the gap filled with cells in hypoxia (pointed by white arrow in Fig. 5C). The cells, being in hypoxia during a period of time longer than a given threshold, die and are removed from the system. This process produces necrotic region in the middle of growing tumor which has principal consequences on tumor dynamics in its both avascular and angiogenic phases. In avascular phase the existence of the necrotic region decreases the internal tumor pressure slowing down its growth. As shown in Fig. 5C, D, where the tumor progression goes with vascular network evolution (the angiogenic phase), the evolving network is being continuously remodeled by mechanical forces exerted by growing tumor. On the other hand, the vessels reshape the tumor body and influence its compartmentalization. The separate regions of low oxygen concentration become the sources of TAF. Due to chemotaxis, these regions attract the growing vessels, which can be the sources of oxygen. Supply of oxygen down-regulates TAF production and can rebuild tumor tissue in the necrotic regions. This process occurs in various spatial scales resulting in the transformation of the regular vasculature in normal tissue into a highly inhomogeneous vasculature of the tumor.


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Fig. 5. The process of tumor growth with angiogenesis process switched on. Due to better oxygenation of the cells, the tumor grows faster. The necrotic region (C) is reborn (D) because of oxygen supply by the newly formed vessels.

Fig. 6. The snapshots from 3-D simulation of angiogenesis and tumor progression. We display only the evolution of the vascular network. The angiogenesis starts from a single vessel (A). The sprouts of blood vessels, stimulated by TAF secreted by tumor cells, emerge (B). They are represented by black threads. Only those threads, which can conduct blood – i.e., the closed loops (anstomoses) – can survive. They are marked in red. The unproductive vessels are removed from the system after some time.

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interactions overall

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5.2 5

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seconds

140 120 100

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new array organization

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Fig. 7. Timings and speed-ups obtained as the result of reorganization of particle ordering (left) and influence of a greater number of threads (right)

The progression of tumor vasculature is shown in Fig. 6. The source, initially straight vessel, is remodeled due to tumor growth dynamics. Because of increasing TAF concentration we can observe newborn capillaries sprouting from the source vessel (black threads). The sprouts can bifurcate and merge creating

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anastomoses. The blood flow is stimulated by pressure difference in anastomosing vessels. Only productive vessels – marked in red – have a chance to survive if the TAF concentration is sufficiently high. Unproductive vessels undergo regression and disappear after some time. The preliminary timings of the model computer implementation are presented in Fig. 7. For testing simulations, employing 2×105 particles in the 3-D cubic box divided onto 52×52×52 cells, the procedure calculating interactions between the particles consumes about 90% of the total computational time. As shown in Fig. 7 (left), the application of proposed data structure (see Fig. 4) allows for 10% decreasing of the total computational time. More meaningful acceleration of the code was obtained by increasing the number of threads on SMP server equipped with two dual core AMD Opteron processors. The speed ups are shown in the right panel of Fig. 7.

6

Conclusions

The particle model developed in this study can be used as a robust modeling framework for developing more advanced tumor growth models. Consequently, these models can be used to evaluate effects of individual factors on the process of tumor growth. The particle based framework is easy both to develop principal mechanisms of tumor dynamics such as cell proliferation, vascularization, vessels remodeling and incorporate sub-models of other important processes like: microscopic intracellular phenomena, blood circulation and extracellular matrix models. The particle model can be supplemented easily by more precise models of reproduction mechanism, cell-life cycle, lumen growth dynamics, haptotaxis mechanism and others. Some models of intracellular phenomena can be copied directly from the existing bibliography (e.g from [2,3,4,5,6]). The quantitative verification of our model with experiment requires performing large simulations involving millions of particles to obtain realistic tumor sizes. In this paper, we present some ideas for better parallelization of the code and we demonstrate preliminary timings. In the nearest future we are aimed at obtaining efficient parallel codes on the newest multicore and GPU processors, to provide flexible simulation tool for oncologists, which can be used on small but strong stand alone workstations. Acknowledgments. Thanks are due to Professor Dr A. Dudek from Hematology, Oncology and Transplantation (HOT) Division of the University of Minnesota Medical School for his contribution to this paper and priceless expertise. This research is financed by the Polish Ministry of Education and Science, Project No. 3 T11F 010 30 and internal AGH Institute of Computer Science grant No. 11.11.120.865. The exemplar movies from simulations can be downloaded from http://www.icsr.agh.edu.pl/∼wcislo/Angiogeneza/index.html


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