@article{d1d96e00463d49f2a30a0d2e9b89a8e3,
title = "Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion",
abstract = "Cancer invasion is the process of cells detaching from a primary tumor and infiltrating the healthy tissue. Cancer invasion has been recognized as a complex system, since a tumor's invasive behavior emerges from the combined effect of tumor cell proliferation, tumor cell migration and cell-microenvironment interactions. Cellular automata (CA) provide simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting {"}simple{"} components. Here, we introduce a lattice-gas cellular automaton (LGCA) model of tumor cell proliferation, necrosis and tumor cell migration. The impact of the tumor environment on tumor cells has been investigated in a previous study. Our analysis aims at predicting the velocity of the traveling invasion front, which depends upon fluctuations that arise from the motion of the discrete cells at the front. We find an excellent agreement between the velocities measured in simulations of the LGCA and an analytical estimate derived in the cut-off mean-field approximation via the discrete Lattice Boltzmann equation and its linearization. In particular, we predict the front velocity to scale with the square root of the product of probabilities for mitosis and the migration coefficient. Finally, we calculate the width of the traveling front which is found to be proportional to the front velocity.",
keywords = "Cut-off mean-field approximation, Lattice Boltzmann equation, Lattice-gas cellular automata, Traveling fronts, Tumor invasion",
author = "H. Hatzikirou and L. Brusch and C. Schaller and M. Simon and A. Deutsch",
note = "Funding Information: We are grateful to A. Chauviere, F. Peruani and M. Tektonidis for fruitful discussions. Special thanks go to Elan Gin for proof-reading the English of our manuscript. We acknowledge support by the systems biology network HepatoSys of the German Ministry for Education and Research through grant 0313082J, and support by the Gottlieb Daimler- and Karl Benz-Foundation through their research program “From bio-inspired logistics to logistics-inspired bio-nano-engineering”. Andreas Deutsch is a member of the DFG Research Center for Regenerative Therapies Dresden–Cluster of Excellence–and gratefully acknowledges the support by the Center. The research was supported in part by funds from the EU Marie Curie Network “Modeling, Mathematical Methods and Computer Simulation of Tumor Growth and Therapy” (EU-RTD IST-2001-38923). Appendix In this Appendix , we present the details of the micro-dynamical Eqs. (5) and (6) . In the following for simplicity reasons and without any loss of generality, we drop the spatial and the temporal arguments of the functions. The Heaviside functions Θ ( θ M − n C ) and Θ ( n C − θ N ) can be alternatively written in terms of random variables: (40) Θ ( θ M − n C ) = ∑ l = 1 θ M δ ( n C = l ) = { 1 , if n C ≤ θ M 0 , else (41) Θ ( n C − θ N ) = ∑ l = θ N b{\~ } δ ( n C = l ) = { 1 , if n C ≥ θ N 0 , else where the δ ( n C ) functions represent the possible node configurations that account for n C number of cells, defined in the general form: (42) δ ( n = N ) = ∑ l = 1 ( b{\~ } N ) ∏ i ∈ M l n η i ( r , k ) ∏ j ∈ M / M l n ( 1 − η j ( r , k ) ) = { 1 , if n = N 0 , else where N ∈ { 0 , … , b{\~ } } and the index set M = 1 , … , b{\~ } and M l n denotes the n th subset of M with l elements. Using Eqs. (40)–(42) in combination with Eq. (5) , (6) , we can write the micro-dynamical equations in terms of occupation numbers. Now let us evaluate the expected collision operators C{\~ } σ , i from Eqs. (10) and (11) . We assume that θ M = 4 and θ N = 6 and that the system is in the steady state ( f{\= } C , f{\= } N ) . Moreover, we fix the node capacity as b{\~ } = 4 . Therefore, Eqs. (10) and (11) yield: (43) C{\~ } C , i = 1 8 [ r M ( f{\= } C ( 1 − f{\= } C ) 7 + 28 f{\= } C 2 ( 1 − f{\= } C ) 6 + 56 f{\= } C 3 ( 1 − f{\= } C ) 5 + 70 f{\= } C 4 ( 1 − f{\= } C ) 4 ) − r N ( 28 f{\= } C 6 ( 1 − f{\= } C ) 2 + 8 f{\= } C 7 ( 1 − f{\= } C ) 1 + f{\= } C 8 ) ] , (44) C{\~ } N , i = 1 8 [ r N ( 1 − f{\= } N ) ( 28 f{\= } C 6 ( 1 − f{\= } C ) 2 + 8 f{\= } C 7 ( 1 − f{\= } C ) 1 + f{\= } C 8 ) ] . Setting C{\~ } σ , i = 0 , we can calculate the exact values of the steady states ( f{\= } C , f{\= } N ) in (12) . ",
year = "2010",
month = apr,
doi = "10.1016/j.camwa.2009.08.041",
language = "British English",
volume = "59",
pages = "2326--2339",
journal = "Computers and Mathematics with Applications",
issn = "0898-1221",
publisher = "Elsevier Ltd",
number = "7",
}
