A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

A cellular automaton model for collective neural dynamics

A stochastic epidemic model for the collective behaviour of a large set of Boolean automata placed upon the sites of a complete graph is revisited. In this paper we study the generalisation of the model to take into account inhibitory neurons. The resulting stochastic cellular automata are completely defined by five parameters: the number of excitatory neurons, N, the number of inhibitory neurons, M, the probabilities of excitation, α, and inhibition, γ, among neurons and the spontaneous transition rate from the firing to the quiescent state, β.We propose that the background of the electroencephalographic signals could be mimicked by the fluctuations in the total number of firing neurons in the excitatory subnetwork. These fluctuations are Gaussian and the mean-square displacement from an initial state displays a strongly subdiffusive behaviour approximately given by , where NA=β/(β+Mγ), τ=2(Nα−β). Comparison with real EEG records exhibits good agreement with these predictions.     

The social impact in a high-risk community: A cellular automata model

This research examines the spread of criminal behavior and hard drug consumption using a mathematical approach called cellular automata (CA). This CA model is based on two behavioral concepts. Firstly, peer association impacts criminal involvement. Secondly, addiction can heighten criminal activity. The model incorporates four types of actors who interact in a high-risk social community and one intervention method. The actors exert a social influence on each other by encouraging or discouraging drug use and criminal behavior. The intervention method called Incapacitation has a probabilistic impact on the individuals in the model. The results identify the threshold where positive influences on a population reduce the number of high-rate offenders in the community. These results are discussed to further the knowledge about the social influences in a high-risk community and how these influences can effect decisions on offender management.     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Computational methods for Conway's Game of Life cellular automaton

The cellular automaton model of computation has drawn the interest of researchers from different disciplines including computer science, biology, mathematics, economy, biochemistry and philosophy. Although a cellular automaton is based on a set of simple rules, over time complex patterns may evolve. We present computational methods for implementing and optimizing a well known two-state cellular automaton, Conway's Game of Life, on a 16-core Intel Xeon. The evaluation is based on three multicore algorithms. The first algorithm is coherent and utilizes shared memory and barrier synchronization. The remaining two algorithms are distributed and utilize private memories and explicit core-to-core message passing. We provide a link to our open source simulation software.     

On irreversibility of von Neumann additive cellular automata on grids

The von Neumann cellular automaton appears in many different settings in Operations Research varying from applications in Formal Languages to Biology. One of the major questions related to it is to find a general condition for irreversibility of a class of two-dimensional cellular automata on square grids (σ⁺-automata). This question is partially answered here with the proposal of a sufficient condition for the irreversibility of σ⁺-automata.     

The dynamics of expansive invertible onesided cellular automata

Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.

     

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ_c. Moreover the dynamical system (cellular automaton F, invariant measure μ_c) has still the μ_c-almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ_c. We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple (μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.     

A new cellular automaton model for traffic flow

Establishment of effective traffic models to reveal fundamental traffic characteristics is an essential requirement in the design, planning and operation of transportation systems. In 1992 Nagel and Schreckenberg presented a cellular automaton model describing traffic flow of N cars on a single lane and applied it in the famous project TRANSIMS on transportation simulation. In this paper, the author proposes a new model for the same problem and gives a comparison of simulation results with the former ones. The comparison shows that the new model works better under the condition of high traffic density.     

Reversibility of linear cellular automata

In this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled. Specifically, a criterion for the reversibility in terms of the number of cells is stated and, in these cases, the inverse cellular automata are explicitly computed.     

Modelling nonlinear sequence generators in terms of linear cellular automata

In this work, a wide family of LFSR-based sequence generators, the so-called clock-controlled shrinking generators (CCSGs), has been analyzed and identified with a subset of linear cellular automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.     

Modelling forest fire spread using hexagonal cellular automata

In this paper a new mathematical model for predicting the spread of a fire front in homogeneous and inhomogeneous environments is presented. It is based on a bidimensional cellular automata model, whose cells stand for regular hexagonal areas of the forest. The results obtained are in agreement with the fire spreading in real forests.     

Synthesis of cryptographic interleaved sequences by means of linear cellular automata

This work shows that a class of pseudorandom binary sequences, the so-called interleaved sequences, can be generated by means of linear multiplicative polynomial cellular automata. In fact, these linear automata generate all the solutions of a type of linear difference equations with binary coefficients. Interleaved sequences are just particular solutions of such equations. In this way, popular nonlinear sequence generators with cryptographic application can be linearized in terms of simple cellular automata.     

A single-copy minimal-time simulation of a torus of automata by a ring of automata

We consider cellular automata on Cayley graphs and we simulate the behavior of a torus of n×m automata (nodes) by a ring of n·m automata (cells). Our simulation technique requires the neighborhood of the nodes to be preserved. We achieve this constraint by copying the contents of nodes on the cells. We consider the problem of minimizing the number of the copies. We prove that it is possible to simulate the behavior of a torus on a ring with a single copy on each cell if and only if n and m satisfy a given condition. In that case we propose a time-optimal algorithm. We thus improve a previous work done by Martin where two copies were requested. When the condition on n and m is not fulfilled one can use the previous algorithm.     

A CA-based epidemic model for HIV/AIDS transmission with heterogeneity

The complex dynamics of HIV transmission and subsequent progression to AIDS make the mathematical analysis untraceable and problematic. In this paper, we develop an extended CA simulation model to study the dynamical behaviors of HIV/AIDS transmission. The model incorporates heterogeneity into agents’ behaviors. Agents have various attributes such as infectivity and susceptibility, varying degrees of influence on their neighbors and different mobilities. Additional, we divide the post-infection process of AIDS disease into several sub-stages in order to facilitate the study of the dynamics in different development stages of epidemics. These features make the dynamics more complicated. We find that the epidemic in our model can generally end up in one of the two states: extinction and persistence, which is consistent with other researchers’ work. Higher population density, higher mobility, higher number of infection source, and greater neighborhood are more likely to result in high levels of infections and in persistence. Finally, we show in four-class agent scenario, variation in susceptibility (or infectivity) and various fractions of four classes also complicates the dynamics, and some of the results are contradictory and needed for further research.     

A secret sharing scheme based on cellular automata

A new secret sharing scheme based on a particular type of discrete delay dynamical systems: memory cellular automata, is proposed. Specifically, such scheme consists of a (k, n)-threshold scheme where the text to be shared is considered as one of the k initial conditions of the memory cellular automata and the n shares to be distributed are n consecutive configurations of the evolution of such cellular automata. It is also proved to be perfect and ideal.     

Genetic algorithms and cellular automata in aquifer management

A groundwater management problem is presented involving pumping cost minimization with both well discharges and well locations as decision variables. A grid of candidate well locations is set up and optimal arrangements of wells are sought within this discrete space. A genetic algorithm approach is presented with the following particular features: (a) A suitable scaling is applied to the objective function in order to alleviate its regionally flat behavior. (b) No penalty functions are involved in constraint handling. Instead, the feasible region is transformed into a rectangular domain. The transformation introduced is proved to be bijective. (c) A binary representation of well configurations is presented and compared to a combinatorial one. The binary representation necessitates the introduction of specially designed genetic operators. Besides purely genetic algorithms, the concept of cellular automaton is introduced as the basis of an alternative formulation of the optimization problem. The lattice of the cellular automaton provides the discrete set of candidate well positions. The well configuration is represented by a group of agents occupying an equal number of lattice sites. The agents change positions as dictated by the structure of the automaton and, also, by an associated genetic algorithm, which directs the evolution of the whole scheme toward an optimal configuration. An improved performance of this approach is noted and discussed in comparison to the purely genetic algorithm schemes of the present work. A simulated annealing approach is also applied to the same problem for comparison purposes. Finally, a new and more efficient hybrid annealing–genetic approach is introduced and discussed.     

Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics

Cellular automata (CA) and ordinary differential equation (ODE) models compete for dominance in microscopic pedestrian dynamics. There are two major differences: movement in a CA is restricted to a grid and navigation is achieved by moving directly in the desired direction. Force based ODE models operate in continuous space and navigation is computed indirectly through the acceleration vector. We present the Optimal Steps Model and the Gradient Navigation Model, which produce trajectories similar to each other. Both are grid-free and free of oscillations, leading to the conclusion that the two major differences are also the two major weaknesses of the older models.