Multiscale Biological Tissue Models and Flux-Limited Chemotaxis for Multicellular Growing Systems

Bellomo, Nicola; Bellouquid, Abdelghani; Nieto, Juan J.; Soler, Juan

doi:10.1142/s0218202510004568

Cited by 166 publications

(117 citation statements)

References 47 publications

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“…This approach has been initiated in biology by Othmer, Dunbar and Alt, 156 where the main idea consists in perturbing the transport equation by a velocity jump process, which appears appropriate to model the velocity dynamics of cells modeled as living particles. This method, which leads to hyperbolic transport equation, has been subsequently developed by various authors, among others, 2,13,14,22,46,48,68,76,97,118,157 who contributed to improve the approach and developed various applications in biology. The book by Banasiak and Lachowicz 8 is an important source of additional bibliography.…”

Section: Asymptotic Methods From the Micro-scale To The Macro-scalementioning

confidence: 99%

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Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

927

818

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This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as 1663

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Section: Asymptotic Methods From the Micro-scale To The Macro-scalementioning

confidence: 99%

“…The contents refer to an already vast literature (see Refs. 2,9,11,[13][14][15]22,51,[46][47][48]68,69,76,97,118,157,173) from which three different techniques are presented/have been chosen. The first one, given in Sec.…”

Section: Aims and Plan Of The Papermentioning

confidence: 99%

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

927

818

View full text Add to dashboard Cite

show abstract

“…Some hybrid models that combine cellular automata with partial differential equations were also used to describe interactions between a tumor and the immune system of the host organism [15] or morphologies of an avascular tumor [16]. One should also mention that related mathematical and computational approaches have been used to model of some properties of tissues or genomes [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Coexistence and critical behavior in a lattice model of competing species

Wendykier¹,

Lipowski²,

Ferreira³

2011

Phys. Rev. E

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In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d = 1,2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor cells. With such an interpretation, examples of tumor remission, recurrence, and different morphologies are presented. In the d = 1 and d = 2 models, we analyze the nature of phase transitions: they are either discontinuous or belong to the directed-percolation universality class, and in some cases they have an active subcritical phase. In the d = 2 case, one of the transitions seems to be characterized by critical exponents that differ from directed-percolation ones, but this transition could be also weakly discontinuous. In the d = 3 version, Monte Carlo simulations are in a good agreement with the solution of the mean-field approximation. This approximation predicts that oscillatory behavior occurs in the present model but only for d 2. For d 2, a steady state depends on the initial configuration in some cases.

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“…Generalized kinetic models transfer the methodology developed for systems of a great number of interacting particles (such as Boltzmann and Vlasov equations, with direct interactions among the particles or mean field terms and external forces [17]) to various other fields of research, such as traffic dynamics (see, for instance, the review paper [11]), cellular dynamics (see, among others, [18,9,10,13,11]), social and population dynamics [27,14,[32][33][34]11,24] and biological systems in general [8,11]. Various models of the classical mathematical kinetic theory have been investigated and generalized in several contexts, [22,2].…”

Section: Comparison With the Kinetic Modelmentioning

confidence: 99%

An artificial neural network approach for modeling the ward atmosphere in a medical unit

Schiavo

Prinari

Gronski

et al. 2015

Mathematics and Computers in Simulation

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Artificial neural networks (ANNs) have been developed, implemented and tested on the basis of a four-year-long experimental data set, with the aim of analyzing the performance and clinical outcome of an existing medical ward, and predicting the effects that possible readjustments and/or interventions on the structure may produce on it. Advantages of the ANN technique over more traditional mathematical models are twofold: on one hand, this approach deals quite naturally with a large number of parameters/variables, and also allows to identify those variables which do not play a crucial role in the system dynamics; on the other hand, the implemented ANN can be more easily used by a staff of non-mathematicians in the unit, as an on-site predictive tool. As such, the ANN model is particularly suitable for the case study. The predictions from the ANN technique are then compared and contrasted with those obtained from a generalized kinetic approach previously proposed and tested by the authors. The comparison on the two case periods shows the ANN predictions to be somewhat closer to the experimental values. However, the mean deviations and the analysis of the statistical coefficients over a span of multiple years suggest the kinetic model to be more reliable in the long run, i.e., its predictions can be considered as acceptable even on periods that are quite far away from the two case periods over which the many parameters of the model had been optimized. The approach under study, referring to paradigms and methods of physical and mathematical models integrated with psychosocial sciences, has good chances of gaining the attention of the scientific community in both areas, and hence of eventually obtaining wider diffusion and generalization.

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Multiscale Biological Tissue Models and Flux-Limited Chemotaxis for Multicellular Growing Systems

Cited by 166 publications

References 47 publications

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Coexistence and critical behavior in a lattice model of competing species

An artificial neural network approach for modeling the ward atmosphere in a medical unit

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