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Mathematics
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Mathematical Models for Cell Migration and Spread
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Mathematical Models for Cell Migration and Spread

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E3: Mathematical Biology".

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 10961

Special Issue Information

Dear Colleagues,

Real world problems from Biology and Medicine arise challenging mathematical issues and often require a broad knowledge of methods from several mathematical disciplines, along with the ability to connect them. One of the main directions in Mathematical Biology addresses continuous time problems described by way of ordinary/stochastic/random/partial differential equations, often coupled in systems comprising one or several such types.

Cell motility is a hallmark of many biological processes related, among others, to development of cancer and building of metastases, wound healing, tissue regeneration, embryonic development, etc. This special issue focuses on problems related to cell migration and spread in a heterogeneous environment containing soluble and insoluble signals (chemoattractants and  -repellents,  tissue components, various types of cells, interstitial fluid, etc.) and modelled by systems like those mentioned above. Thus, it welcomes manuscripts addressing questions related to modeling, analysis and/or numerical simulations of such systems. 

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical models related to cell motility
  • systems of ordinary, stochastic, random, and/or partial differential equations
  • mathematical analysis (well-posedness, qualitative properties of solutions, long-time behavior, traveling waves, etc.)
  • numerical simulations

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Published Papers (4 papers)

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Research

17 pages, 859 KiB  
Article
Space and Genotype-Dependent Virus Distribution during Infection Progression
by Nicholas Bessonov, Gennady Bocharov and Vitaly Volpert
Mathematics 2022, 10(1), 96; https://doi.org/10.3390/math10010096 - 28 Dec 2021
Cited by 4 | Viewed by 1740
Abstract
The paper is devoted to a nonlocal reaction-diffusion equation describing the development of viral infection in tissue, taking into account virus distribution in the space of genotypes, the antiviral immune response, and natural genotype-dependent virus death. It is shown that infection propagates as [...] Read more.
The paper is devoted to a nonlocal reaction-diffusion equation describing the development of viral infection in tissue, taking into account virus distribution in the space of genotypes, the antiviral immune response, and natural genotype-dependent virus death. It is shown that infection propagates as a reaction-diffusion wave. In some particular cases, the 2D problem can be reduced to a 1D problem by separation of variables, allowing for proof of wave existence and stability. In general, this reduction provides an approximation of the 2D problem by a 1D problem. The analysis of the reduced problem allows us to determine how viral load and virulence depend on genotype distribution, the strength of the immune response, and the level of immunity. Full article
(This article belongs to the Special Issue Mathematical Models for Cell Migration and Spread)
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21 pages, 4346 KiB  
Article
Mathematical Modelling of Glioblastomas Invasion within the Brain: A 3D Multi-Scale Moving-Boundary Approach
by Szabolcs Suveges, Kismet Hossain-Ibrahim, J. Douglas Steele, Raluca Eftimie and Dumitru Trucu
Mathematics 2021, 9(18), 2214; https://doi.org/10.3390/math9182214 - 9 Sep 2021
Cited by 11 | Viewed by 3329
Abstract
Brain-related experiments are limited by nature, and so biological insights are often limited or absent. This is particularly problematic in the context of brain cancers, which have very poor survival rates. To generate and test new biological hypotheses, researchers have started using mathematical [...] Read more.
Brain-related experiments are limited by nature, and so biological insights are often limited or absent. This is particularly problematic in the context of brain cancers, which have very poor survival rates. To generate and test new biological hypotheses, researchers have started using mathematical models that can simulate tumour evolution. However, most of these models focus on single-scale 2D cell dynamics, and cannot capture the complex multi-scale tumour invasion patterns in 3D brains. A particular role in these invasion patterns is likely played by the distribution of micro-fibres. To investigate the explicit role of brain micro-fibres in 3D invading tumours, in this study, we extended a previously introduced 2D multi-scale moving-boundary framework to take into account 3D multi-scale tumour dynamics. T1 weighted and DTI scans are used as initial conditions for our model, and to parametrise the diffusion tensor. Numerical results show that including an anisotropic diffusion term may lead in some cases (for specific micro-fibre distributions) to significant changes in tumour morphology, while in other cases, it has no effect. This may be caused by the underlying brain structure and its microscopic fibre representation, which seems to influence cancer-invasion patterns through the underlying cell-adhesion process that overshadows the diffusion process. Full article
(This article belongs to the Special Issue Mathematical Models for Cell Migration and Spread)
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22 pages, 5419 KiB  
Article
Anisotropic Network Patterns in Kinetic and Diffusive Chemotaxis Models
by Ryan Thiessen and Thomas Hillen
Mathematics 2021, 9(13), 1561; https://doi.org/10.3390/math9131561 - 2 Jul 2021
Cited by 2 | Viewed by 2169
Abstract
For this paper, we are interested in network formation of endothelial cells. Randomly distributed endothelial cells converge together to create a vascular system. To develop a mathematical model, we make assumptions on individual cell movement, leading to a velocity jump model with chemotaxis. [...] Read more.
For this paper, we are interested in network formation of endothelial cells. Randomly distributed endothelial cells converge together to create a vascular system. To develop a mathematical model, we make assumptions on individual cell movement, leading to a velocity jump model with chemotaxis. We use scaling arguments to derive an anisotropic chemotaxis model on the population level. For this macroscopic model, we develop a new numerical solver and investigate network-type pattern formation. Our model is able to reproduce experiments on network formation by Serini et al. Moreover, to our surprise, we found new spatial criss-cross patterns due to competing cues, one direction given by tissue anisotropy versus a different direction due to chemotaxis. A full analysis of these new patterns is left for future work. Full article
(This article belongs to the Special Issue Mathematical Models for Cell Migration and Spread)
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19 pages, 754 KiB  
Article
Modeling Interactions among Migration, Growth and Pressure in Tumor Dynamics
by Beatriz Blanco, Juan Campos, Juan Melchor and Juan Soler
Mathematics 2021, 9(12), 1376; https://doi.org/10.3390/math9121376 - 14 Jun 2021
Cited by 4 | Viewed by 2532
Abstract
What are the biomechanical implications in the dynamics and evolution of a growing solid tumor? Although the analysis of some of the biochemical aspects related to the signaling pathways involved in the spread of tumors has advanced notably in recent times, their feedback [...] Read more.
What are the biomechanical implications in the dynamics and evolution of a growing solid tumor? Although the analysis of some of the biochemical aspects related to the signaling pathways involved in the spread of tumors has advanced notably in recent times, their feedback with the mechanical aspects is a crucial challenge for a global understanding of the problem. The aim of this paper is to try to illustrate the role and the interaction between some evolutionary processes (growth, pressure, homeostasis, elasticity, or dispersion by flux-saturated and porous media) that lead to collective cell dynamics and defines a propagation front that is in agreement with the experimental data. The treatment of these topics is approached mainly from the point of view of the modeling and the numerical approach of the resulting system of partial differential equations, which can be placed in the context of the Hele-Shaw-type models. This study proves that local growth terms related to homeostatic pressure give rise to retrograde diffusion phenomena, which compete against migration through flux-saturated dispersion terms. Full article
(This article belongs to the Special Issue Mathematical Models for Cell Migration and Spread)
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