Traveling Waves of a Go-or-Grow Model of Glioma Growth

Stepien, Tracy L.; Rutter, Erica M.; Kuang, Yang

doi:10.1137/17m1146257

Cited by 21 publications

(29 citation statements)

References 36 publications

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“…Nevertheless, substantial interest remains and numerous mathematical models have incorporated its central tenet (e.g. [39,40,41,42]), typically through supposing two cell-state variables and incorporating switching between states. On a similar note, a recent study has investigated how malignant invasion varies in nutrient-depleted environments, via two cell-state models where cells can have distinct chemotactic sensitivity and/or nutrientdependent growth [43].…”

Section: Discussion and Research Perspectivesmentioning

confidence: 99%

“…Moreover, if assumptions ( 5)-( 7) hold then R(Y, 0, •) = 0. Hence, the relation (36), the monotonicity property (40) along with the asymptotic condition (39), the property ( 51) and the monotonicity result (52) allow one to conclude that the position of the leading edge of a travelling-wave solution ȳ(z) that satisfies the differential equation ( 37) subject to the asymptotic condition (41) coincides with the unique point ∈ R such that ȳ( ) = Y , i.e.…”

Section: Travelling-wave Analysismentioning

confidence: 94%

“…assumptions ( 4) or ( 6)), differentiating the relation (36) with respect to z gives the following differential relation We start by noting that when cell proliferation is independent from the concentration of the attractant, i.e. when assumptions (4) hold, the asymptotic condition (41) along with the relation (36) gives…”

Section: Travelling-wave Analysismentioning

confidence: 99%

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Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves

Lorenzi,

Painter

2021

Preprint

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Chemotaxis-driven invasions have been proposed across a broad spectrum of biological processes, from cancer to ecology. The influential system of equations introduced by Keller and Segel has proven a popular choice in the modelling of such phenomena, but in its original form restricts to a homogeneous population. To account for the possibility of phenotypic heterogeneity, we extend to the case of a population continuously structured across space, time and phenotype, where the latter determines variation in chemotactic responsiveness, proliferation rate, and the level of chemical environment modulation. The extended model considered here comprises a nonlocal partial differential equation for the local phenotype distribution of cells which is coupled, through an integral term, with a differential equation for the concentration of an attractant, which is sensed and degraded by the cells. In the framework of this model, we concentrate on a chemotaxis/proliferation trade-off scenario, where the cell phenotypes span a spectrum of states from highly-chemotactic but minimally-proliferative to minimally-chemotactic but highlyproliferative. Using a combination of numerical simulation and formal asymptotic analysis, we explore the properties of travelling-wave solutions. The results of our study demonstrate how incorporating phenotypic heterogeneity may lead to a highly-structured wave profile, where cells in different phenotypic states dominate different spatial positions across the invading wave, and clarify how the phenotypic structuring of the wave can be shaped by trade-offs between chemotaxis and proliferation.

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Section: Discussion and Research Perspectivesmentioning

confidence: 99%

Section: Travelling-wave Analysismentioning

confidence: 94%

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Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves

Lorenzi,

Painter

2021

Preprint

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“…Our starting point is a nonlinear reaction-diffusion system of partial differential equations that governs the evolution of GBM cells and the concentration of oxygen in a microfluidic device [ 28 ]. Although some works support the use of two [ 34 – 36 ] or even three [ 27 ] phenotypes for describing the cell population, we use a previously validated model offering some flexibility for modeling the switch between proliferative and migratory activity. In particular, the equations of the fields evolution are: …”

Section: Methodsmentioning

confidence: 99%

Understanding glioblastoma invasion using physically-guided neural networks with internal variables

et al. 2022

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Microfluidic capacities for both recreating and monitoring cell cultures have opened the door to the use of Data Science and Machine Learning tools for understanding and simulating tumor evolution under controlled conditions. In this work, we show how these techniques could be applied to study Glioblastoma, the deadliest and most frequent primary brain tumor. In particular, we study Glioblastoma invasion using the recent concept of Physically-Guided Neural Networks with Internal Variables (PGNNIV), able to combine data obtained from microfluidic devices and some physical knowledge governing the tumor evolution. The physics is introduced by means of nonlinear advection-diffusion-reaction partial differential equation that models the Glioblastoma evolution for defining the network structure. On the other hand, multilayer perceptrons combined with a nodal deconvolution technique are used for learning the go or grow metabolic behavior which characterises the Glioblastoma invasion. The PGNNIV is here trained using synthetic data obtained from in silico tests created under different oxygenation conditions, using a previously validated model. The unravelling capacity of PGNNIV enables discovering complex metabolic processes in a non-parametric way, thus giving explanatory capacity to the networks, and, as a consequence, surpassing the predictive power of any parametric approach and for any kind of stimulus. Besides, the possibility of working, for a particular tumor, with different boundary and initial conditions, permits the use of PGNNIV for defining virtual therapies and for drug design, thus making the first steps towards in silico personalised medicine.

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“…In [30] it was assumed that hypoxia (lack of oxygen) triggers the switch in the long term dynamics of the system, by selection of the migrating phenotype, but in a global manner (oxygen supply was accounted for via the constant carrying capacity, as one parameter of the cellular automaton). Later contributions considered PDE models with density-dependent switch (see [63], as opposed to [25] where the switching rate is not modulated, and also the experimental design of density-dependent motility in bacteria [44]).…”

Section: Scenario 2: Cell Leakage Compensated By Growthmentioning

confidence: 99%

Mathematical Modeling of Cell Collective Motion Triggered by Self-Generated Gradients

Cálvez

Demircigil

Sublet

2021

Modeling and Simulation in Science, Engineering and Technology

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Self-generated gradients have atttracted a lot of attention in the recent biological literature. It is considered as a robust strategy for a group of cells to find its way during a long journey. This note is intended to discuss various scenarios for modeling traveling waves of cells that constantly deplete a chemical cue, and so create their own signaling gradient all along the way. We begin with one famous model by Keller and Segel for bacterial chemotaxis. We present the model and the construction of the traveling wave solutions. We also discuss the limitation of this approach, and review some subsequent work addressing stability issues. Next, we review two relevant extensions, which are supported by biological experiments. They both admit traveling wave solutions with an explicit value for the wave speed. We conclude by discussing some open problems and perspectives, and particularly a striking mechanism of speed determinacy occurring at the back of the wave. All the results presented in this note are illustrated by numerical simulations.

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Traveling Waves of a Go-or-Grow Model of Glioma Growth

Cited by 21 publications

References 36 publications

Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves

Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves

Understanding glioblastoma invasion using physically-guided neural networks with internal variables

Mathematical Modeling of Cell Collective Motion Triggered by Self-Generated Gradients

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