Mathematical modelling of cancer invasion : phenotypic transitioning provides insight into multifocal foci formation

Funding: Z. Szymańska acknowledge the support from the National Science Centre, Poland – grant No. 2017/26/M/ST1/00783. N. Sfakianaki’s scientific visit to the University of Warsaw was partially supported by the Excellence Initiative Research University Programme at the University of Warsaw. M. Lachowicz is happy to acknowledge the support from the New Ideas Grant - ”Równania kinetyczne w opisie zjawisk samoorganizacji” funded by the Excellence Initiative Research University Programme at the University of Warsaw.The transition from the epithelial to mesenchymal phenotype and its reverse (from mesenchymal to epithelial) are crucial processes necessary for the progression and spread of cancer. In this paper, we investigate how phenotypic switching at the cancer cell level impacts on behaviour at the tissue level, specifically on the emergence of isolated foci of the invading solid tumour mass leading to a multifocal tumour. To this end, we propose a new mathematical model of cancer invasion that includes the influence of cancer cell phenotype on the rate of invasion and metastasis. The implications of model are explored through numerical simulations revealing that the plasticity of tumour cell phenotypes appears to be crucial for disease progression and local invasive spread. The computational simulations show the progression of the invasive spread of a primary cancer reminiscent of in vivo multifocal breast carcinomas, where multiple, synchronous, ipsilateral neoplastic foci are frequently observed and are associated with a poorer patient prognosis.Publisher PDFPeer reviewe

Intracellular protein dynamics as a mathematical problem

International audienceIn this paper we undertake a mathematical analysis of a model of intracellular protein dynamics , i.e. protein and mRNA transport inside a cell, proposed by Szymanska at al. in 2014. The model takes into account diffusive transport in the nucleus and cytoplasm, as well as active transport of protein molecules along microtubules in the cytoplasm. The model reproduces, at least in numerical simulations, the oscillatory changes in protein concentration observed in the experimental data. To our knowledge this is the first paper that, in the multidimensional case, deals with a rigorous mathematical analysis of a model of intracellular dynamics with active transport on microtubules. In particular, in the present paper we prove well-posedness of the model in any space dimension. The model is a complex system of nonlinear PDEs with specific boundary conditions. It may be adapted to other signaling pathways

Review: Nonlinear Dynamical Systems and Chaos Edited by H. W. Broer, S. A. van Gils, I. Hoveijn and F. Takens

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Mathematical models in biology - an introduction

Artykuł nie zawiera streszczeniaThe article contains no abstrac

Mathematical models in biology - an introduction

Artykuł nie zawiera streszczeniaThe article contains no abstrac

Review: Nonlinear Dynamical Systems and Chaos Edited by H. W. Broer, S. A. van Gils, I. Hoveijn and F. Takens

Artykuł nie zawiera streszczeniaThe article contains no abstrac

Some remarks on applied mathematics – written by the applied mathematician

Terminy ,,zastosowania matematyki'' i ,,matematyka stosowana'' są często używane wymiennie.  Niektórzy jednak nadają im różne znaczenia. Ja wolę ten drugi termin, bo tak określa się tę dziedzinę w większości języków, w nazwach wydziałów, nazwach znanych czasopism, etc. O ile termin  ,,zastosowania matematyki'' sugeruje wykorzystanie wcześniej opracowanych metod, to termin ,,matematyka stosowana'' wskazuje na nową jakość odróżniającą ją od matematyki teoretycznej, nazywanej również niezręczną kalką z angielskiego ,,matematyką czystą'' - Pure and Applied Mathematics, n.p. z nazwy czasopisma Communications on Pure and Applied Mathematics, podobnie Journal des Mathematiques Pures et Appliquees, lub Annali di Matematica Pura ed Applicata, żeby pozostać tylko przy nazwach znanych (i o dużej tradycji) czasopism.Terms ,,the application of mathematics '' and ,,applied mathematics'' are often used interchangeably. Some people, however, give them different meanings. I prefer the latter term, because it is the name of  this field in most languages, the names of departments, names of well-known magazines, etc. While the term "the application of mathematics" suggests the use of previously developed methods, the term "applied mathematics" indicates a new quality that distinguishes it from theoretical mathematics, also known as a carbon copy of the English awkward "pure mathematics" - Pure and Applied mathematics, e.g. the name of the journal Communications on Pure and Applied Mathematics, like the Journal des Mathèmatiques Pures et Appliquèes, Annali di Matematica Pura ed Applicata, to remain only the names of famous (and with high tradition) magazines

A nonlocal coagulation-fragmentation model

A new nonlocal discrete model of cluster coagulation and fragmentation is proposed. In the model the spatial structure of the processes is taken into account: the clusters may coalesce at a distance between their centers and may diffuse in the physical space Ω. The model is expressed in terms of an infinite system of integro-differential bilinear equations. We prove that some results known in the spatially homogeneous case can be extended to the nonlocal model. In contrast to the corresponding local models the analysis can be carried out in the $L_1(Ω)$ setting. Our purpose is to study global (in time) existence, mass conservation and well-posedness of the model

Diauxic Growth at the Mesoscopic Scale

In the present paper, we study a diauxic growth that can be generated by a class of model at the mesoscopic scale. Although the diauxic growth can be related to the macroscopic scale, similarly to the logistic scale, one may ask whether models on mesoscopic or microscopic scales may lead to such a behavior. The present paper is the first step towards the developing of the mesoscopic models that lead to a diauxic growth at the macroscopic scale. We propose various nonlinear mesoscopic models conservative or not that lead directly to some diauxic growths