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Vol. 16 no. 2 2000 Pages 96–100 BIOINFORMATICS Self-organization and competition in the immune response to cancer invasion: a phase-orientated computational model of oncogenesis Sorinel Adrian Oprişan 1, Aurel Ardelean 2 and Petre T. Frangopol 2 1 ‘Al.I.Cuza’ University, Department of Theoretical Physics, Blvd. Carol I, no. 11, 6600 Iaşi, Romania and 2 Western University ‘Vasile Goldis’, Department of Biophysics, Blvd. Revolutiei no. 81, 2900 Arad, Romania Received on July 6, 1999; revised on September 15, 1999; accepted on September 17, 1999 Abstract Motivation: Recent studies indicate that fractal dimensions can uncover aspects of cellular dynamics prior to pathological manifestation. In this respect we are interested in building a computational model of oncogenesis able to generate patterns with the same fractal dimension spectrum as the in vivo tumor. Results: A new theoretical model incorporating a systemic view of oncogenesis in a computational model was proposed. The tumor growth is viewed as competition for resources between the two self-organizing subsystems: the neoplastic and the immune. Numerical simulations revealed that tumor escape can be uncovered in some earlier stage of the immune-system–tumor interaction using multifractal measures. The described computational model is able to simulate also the case of immune, surgical, chemical and radiotherapeutical treatment, as well as their effects. Availability: The software used is available on request from the authors. Contact: Sorinel Oprisan, University of New Orleans, Department of Psychology, New Orleans, LA 70148, USA. [email protected] Introduction A theoretical approach of tumor invasion based on the development of mathematical models appears interesting from at least two points of view. First, one can expect that using mathematical models will help to fill the gap between the quantitative and qualitative information provided at the level of population by the development of medical imaging techniques and the description and comprehension of oncogenesis at the cellular level, with obvious implications for the optimization of therapeutical strategies. Second, the morphological differences and transitions between well and smooth defined benign tumor 96 and tentacular malignant tumor suggest a theoretical analysis of tumor invasion based on the development of mathematical models exhibiting bifurcations of spatial patterns in tumor cells density and fractal geometry (Tracqui, 1995). A large number of studies have been devoted to the modeling of tumor growth. The computer modeling of highly complex biological systems requires a breakdown of the problem and appropriate simplifications. In fact a computational model is a ‘model of the model’ (Duchting, 1996). Deterministic models usually consider tumor growth as a wave propagation phenomenon controlled by various factors like mitotic inhibitors and nutrient depletion, immune response or competition with normal cells, and alteration of mitotic rates. Alternatively, stochastic and cellular automata approaches have been proposed to describe the spread of tumor cells into the normal tissue (Keen and Spain, 1992; Meinhard, 1982; Qi et al., 1993; Smolle and Grimstad, 1992; Smolle and Stettner, 1993) with eventual consideration of cell status with regard to the cell cycle (Duchting, 1990; Duchting and Vogelsaenger, 1985). Some of these works have combined computer simulations and image analysis of histological tumor sections to characterize the tumor invasion patterns along a spectrum ranging from broad front invasion to tentacular invasion (Smolle and Grimstad, 1992). By comparing experimentally observed patterns with patterns generated by computer simulations, it was possible to establish a relationship between tumor cell motility and the resulting morphological tumor patterns controlled by the probability of cell division, death and migration (Smolle and Stettner, 1993; Hogeweg, 1987). One of the challenges in cancer research is to understand how cell division, migration, death and effector cells (EC) attack interplay in the modulation and control of the spatio-temporal dynamics of tumor growth. c Oxford University Press 2000 A phase-orientated computational model of oncogenesis At the cellular level it must be stressed that the natural immune EC population is highly heterogeneous. Among other sub-populations with cytotoxic action CTL and natural killer NK lymphocytes are the most studied. Even a simple group, like NK cells, is morphologically and phenotypically heterogeneous. Our present study does not take into account such diversity and consider the EC mostly a homogeneous class. System and methods Our present work is placed at the cellular level and is focused on constructing feedback control models that describe the cell division of normal and tumoral cells. There is general consensus that cell proliferation plays an important role in the carcinogenic process (Cohen and Ellwein, 1990; Moolgavkar and Knudson, 1981; Moolgavkar and Luebeck, 1995). In our computational model, malignant cells are clonal, i.e. they arise from a single progenitor cell that is malignantly transformed. A malign focus consists of the descendents of a single cell which acquired the altered phenotype, either spontaneous or in response to an initiating agent. Altered cells undergo cell division, migration and cell death. The model includes apoptosis, or programmed cell death, as an important feature along the division and migration. Algorithm The computational model stands for most representative and clinically relevant features of oncogenesis viewed as a fight between two distinct subsystems: the immune system of the host and the neoplastic system (Oprişan et al., 1998). For the neoplastic subsystem (S I) the details of simulation strategy are as follows: S I.1. (‘active’) if the distance between the tumor cell and the nutrient medium is less than three cell layers, the tumor cell is in active [mitosis (M)] stage, with two different distant-dependent dynamical behaviors S I.1a. a tumor cell placed on the tumor boundary undergoes cell division phase A major simplification in our model refers to cell phase duration that is supposed to take place during a single time step. We are aware that, for example, cell division is going on along a series of subphases [G1 in which proteins and ribonucleic acid are synthesized; S in which DNA is duplicated; G2 in which the cell synthesizes RNA and protein molecules required for transit through this phase; M leading to cell division (Duchting, 1996)]. Our model combines all the above stages, called M phase, offering the possibility to the physician or biophysicist of setting a different duration of the cell division phase. It is important, from a clinical point of view, to develop a phase orientated mathematical model of tumor growth since the sensitivity of tumor cells differs from one cell phase to another. This is a decisive fact for tumor treatment, because radiation therapy and chemotherapeutic drugs are killing tumor cells during distinct cell-cycle phase only . The second subsystem considered (S II) consists of cytotoxic active cells EC with a very broad phenotype ranging from NK cells to CTL cells, macrophages, etc. These kind of cells are endowed with the following properties: S II.1. they can perform a random walk through the extracellular matrix (ECM) S II.2. they suffer an aging effect depending on their interaction history S II.2a. a free EC increases its age with one unit every time step S II.2b. an EC interacting with a tumor cell increases its age with differing amount, depending on tumor cell phase S II.3. an EC will die after a given age. Removal of an EC can be deterministic or stochastic. In the deterministic case, all EC older than a given limit age with disappear. Instead, in a stochastic removal, an EC has a surveillance probability depending on its age: Psurv = 1 , 1 + exp−γ (t−t0 ) S I.1b. a tumor cell placed on the second layer will force boundary cells diffusion. where t0 is the limit age of the immune cells, γ is the characteristic aging factor of the immune cells. Some authors (Duchting, 1989) also consider the possibility of tumor cell division even if there is no room left around the mother cell; Detailed balance requirements and the Metropolis Monte Carlo numerical algorithm inspired the above relationship (Eisen, 1979; Martin et al., 1990). The described complex model was transformed into a computer program (Borland/Turbo Pascal and MATHEMATICA language). The computer experiment simulating in vitro growth of a tumor, starting with a single tumor cell, in mitotic phase, placed in the center of a nutrient medium of a two-dimensional cell space. The model is based on the cell cycle regulation mechanisms described above. S I.2. (‘dormant’) if the distance between the tumor cell and the nutrient medium is more than three cell layers, the tumor cell will enter the resting (dormant) phase S I.3. (‘necrosis’) a tumor cell resting in the dormant state for more than a critical number of time steps enters the necrosis phase (cell death). 97 S.A.Oprişan et al. Discussion and conclusion Based on the mathematical model described above it is possible to evaluate the dynamics of tumor cell growth. The simulation runs start with a single tumor cell that resides in the state of mitosis at t = 0. The status of cells is updated in units of one cell cycle time. Figure 1 indicates the actual cell cycle phase in which every tumor cell is residing according to computer tomography and magnetic resonance imaging findings. Our major concern here addresses not only the qualitative resemblance of the tumor structures but also the quantitative fit. In this respect, the scaling exponents of the cross section of tumor and of the total perimeter of the cross section were evaluated. Our findings indicate that the cross section of tumor spheroid scales with tumor cells number as follows: ∝ nα , and the perimeter L ∝ nβ . These scaling lows resemble quite well the definition of capacity dimension of the fractal objects and we are confident that a natural approach of the tumoral pattern trace form fractal theory. The scaling exponents α and β include global information on cell cycle phase and they can be measured from computer generated images using the algorithm described above. In the absence of immune response the tumor pattern is done only by tumor cell division rate and tumor cell intrinsic diffusion rate. The cross section shown in Figure 1 has a more regulate structure and than the contour on Figure 3 (with active immune response). Using box counting algorithm we found that the corresponding capacity dimension for the compact structure shown in Figure 1 is 1.95 ± 0.05. For the case shown in Figure 3 the capacity dimension is 1.15 ± 0.05. Figure 2 shows the population cell dynamics for three cell types: A, mitotic phase cells; B, dormant phase; C, 98 Cells number Fig. 1. The stroboscopic evolution of the tumor cross section followed with a time sampling rate of 100 (in Monte Carlo time steps). The simulations were performed with the suppressed immune response. The central necrotic compact domain and a general regulate shape of the tumor characterized by a fractal capacity dimension 1.95 ± 0.05 is shown. 240 220 200 180 160 140 120 100 80 60 40 20 0 –20 –200 A B C 0 200 400 600 800 1000 1200 1400 1600 Time Fig. 2. The time evolution of the cell number in different cell phase (A, mitotic; B, dormant; C, necrotic phase) without immune response. It can be observed the saturation of the growth rate for the cells in the mitotic phase due to the limited carrying capacity of the surrounding tissue. The initial slope of this curve (prior to saturation effect) gives the exponent β. The long term behavior indicated a constant, high, growth rate for the dormant cells. The slope of this curve gives the exponent α. The fact that α > β results in the tumor invading the whole tissue. necrotic phase. As the time elapses the rate of active cell generation on the tumor boundary decreases and, at the same time, the rate of necrotic process remains constant and higher than the mitotic process. Therefore, the final stage of the system can be easily predicted—necrotic cells will invade the whole tissue after enough time has elapsed. Our cross section and perimeter measurements indicate the smoothness of the topology (regular pattern). The growth curve of the active cells in Figure 2 can be described mathematically by the so-called Gompartz function (Duchting, 1996) and the fractal capacity dimension is high, near the saturation threshold which is given by the topological dimension of the cross section. In the presence of the immune response it can be qualitatively observed (see Figure 3 and quantitatively estab- A phase-orientated computational model of oncogenesis Fig. 3. The stroboscopic evolution of the tumor cross section followed with a time sampling rate of 100 (in Monte Carlo time steps). The simulations were performed in the presence of immune response but with a finite, constant, flow of CTL cells and considering the apoptosis effect. The shape of the tumor is less regular and its fractal capacity dimension is 1.15 ± 0.05. (a) Cell number Active cells Times Cell number (b) lished that the tumor developed pattern is more ‘fractured’. In that case the fractal exponents indicate the existence of a fractal structure. The possible explanation for the observed difference can be that the immune cells attack and destroy the tumor cells on the boundary. Our assumption is supported by numerical findings, which provide data on smoothing of the boundary after suppression of immune response due to complete destruction of immune cells. The conclusiveness of our statements must be related to extrapolation from in vitro to in vivo experiments. Therefore, a major concern at play is the systematic reduction of the simplified assumptions of modeling. Based on the successful correlation between the reliability of model predictions and experimental observations, it seems possible to use computer simulations to assist the evaluation and optimization of clinical treatment schemes. According to our computer modeling tumor escape can be uncovered in some earlier stages of the immune system— tumor interaction by image processing techniques. In this respect the use of multifractal measures seems to be an adequate tool. References Times Fig. 4. The time evolution of the cell number in different cell phase can be traced. (a) Cells in the mitotic phase EC and EC-tumoral complex. It can be observed that the growth rate for the cells in the mitotic phase in the presence of the immune response (a) is higher than in the absence of immune response (Figure 2). There is a very strong mechanism of tumor activation via EC interaction. Due to the constant inflow of EC and the effect of apoptosis the total number of EC will decrease and finally the dynamics of the tumor will be identical with the previous one. The balance between the inflow of EC and rate of apoptosis controls the time evolution of the tumor. Cohen,S.M. and Ellwein,L.B. 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