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Modeling the growth of in-vitro tumor spheroidsDirk Drasdo, Stefan Höhme Interdisciplinary Centre for Bioinformatics University of Leipzig 
Figure 1: Typical growth scenario of monoclonal multicellular spheroid growth.
The color codes in the upper image denote the nutrient concentration (left half) and the proliferative activity (right half),
while the lower image illustrates the volume distribution for the shown cross-section a tumor spheroid. As observed experimentally
(Ref. 6), the largest cells are close to the border One commonly used experimental setting is monolayer cultures where cells grow on a Petri dish coated with specific proteins as extracellular matrix components, and with liquid media of specific compositions of e.g. nutrients and growth factors. Some cell lines are able to grow in suspension, without anchorage to a surface. Above a certain size they form three-dimensional multicellular aggregates with a characteristic layer-like structure: a necrotic core is surrounded by viable rim, that consists of an inner layer of quiescent cells and an outer layer of proliferating cells. A step closer towards the in-vivo situation are growing cell populations in environmental media of varying biomechanical properties, for example agarose gel, or co-cultures in which a monoclone of one cell type grows in an environment of non-proliferating cells of another cell type. In these settings the mechanical properties of the embedding medium have been observed to trigger a growth saturation at different population sizes. State of the art: Biomechanical influences are increasingly identified to play an important role in the growth control of tissues and tumors. There are a number of models to explain the spatial structure and growth kinetics of tumor spheroids and of monolayers in liquid medium as well as under mechanical pressure. These include continuum and cellular automata models and phenomenologically motivated growth laws such as the Gompertz-law. Problem: We have shown that this law is misleading since it suggests saturation of growth in situations where a careful data analysis indicates power-law growth of the cell population. Furthermore, most models neither provide a direct comparison of experiments and of computer simulations, nor do they consider the discrete cellular structure of tissue. Idea: We developed a biophysical model (1-3) to study the spatio-temporal growth dynamics of two-dimensional tumor monolayers, three-dimensional tumor spheroids and co-culture spheroids (and monolayers) as a complementary tool to in-vitro experiments. Within our model each cell is represented as an individual object and parameterized by cell-biophysical and cell kinetic parameters that can all be experimentally determined. Hence our modeling strategy allows to study which mechanisms on the microscopic level of individual cells may affect the macroscopic properties of a growing tumor. Results: Mono-cultures of multicellular spheroids and monolayers: Quantitative comparisons of computer simulations with our model to published experimental observations on monolayer cultures suggest a biomechanically-mediated form of growth inhibition during the experimentally observed transition from exponential to sub-exponential growth at sufficiently large tumor sizes. Our simulations show that this transition during the growth of avascular tumor spheroids of EMT6/Ro cells can be explained by mainly the same mechanism; nutrient (glucose) depletion has only a minor influence on the expansion and seems to mainly trigger the size of the necrotic core (Fig.1). We explore the consequences of the biomechanical form of contact inhibition in co-cultures if the mechanical and kinetic properties of the embedding cells are modified. First simulations suggest that the physical properties of the environmental cells may determine the kinetics and morphometry of the expanding cell population in both, 3D cultures (Fig.2) and in 2D-monolayers. 
Figure 2: Multicellular spheroids growing in a co- culture of non-dividing cells: spheroids are almost
smooth (left) while for environmental cells with a very low motility the surface is rough (right).
Benefit and outlook: Mono-cultures: Based on our simulations we propose a characteristic phenomenological growth law in early expansion phases in which specific biological small-scale processes are subsumed in a small number of effective parameters. The simplicity of the growth law permits a test by experimentalists and in case it applies to a specific experimental situation, an easy classification and quantification of the experimental results. This can be demonstrated for NIH3T3-tumors growing in nu-/nu- mice for which we aim at analysing the underlying growth mechanisms. Co-cultures: The developed models eventually permit to predict the influence of the embedding medium (including other cells) on an expanding cell clone as a further step towards a detailed model of tumor spheroids with which also simulations of therapies of tumors in vitro and in vivo becomes feasible. We are currently exploring the expansion of monolayers and multicellular spheroids in media of different biomechanical composition (with emphasis on co-cultures) systematically and test competing hypotheses on the growth control in extensive computer simulations in comparison with experimental data.
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