Classification and stability of global inhomogeneous solutions of a macroscopic model of cell motion

In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions. doi:10.1017/S1446181119000130

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A MULTIPHASE MULTISCALE MODEL FOR NUTRIENT-LIMITED TISSUE GROWTH, PART II: A SIMPLIFIED DESCRIPTION

The ANZIAM Journal ◽

10.1017/s1446181119000130 ◽

2019 ◽

Vol 61 (4) ◽

pp. 368-381

Author(s):

E. C. HOLDEN ◽

S. J. CHAPMAN ◽

B. S. BROOK ◽

R. D. O’DEA

Keyword(s):

Multiple Scale ◽

Multiscale Model ◽

Tissue Growth ◽

Macroscopic Model ◽

Multiphase Model ◽

Multiphase Mixture ◽

Computational Implementation ◽

Scaffold Structure ◽

Interstitial Growth ◽

Cell Motion

In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.

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Chemotactic Responses of Jurkat Cells in Microfluidic Flow-Free Gradient Chambers

Micromachines ◽

10.3390/mi11040384 ◽

2020 ◽

Vol 11 (4) ◽

pp. 384 ◽

Cited By ~ 1

Author(s):

Utku M. Sonmez ◽

Adam Wood ◽

Kyle Justus ◽

Weijian Jiang ◽

Fatima Syed-Picard ◽

...

Keyword(s):

Immune Response ◽

Cancer Progression ◽

Soft Lithography ◽

Cell Movement ◽

Population Level ◽

Jurkat Cells ◽

Dependent Manner ◽

Diverse Range ◽

Microfluidic Flow ◽

Cell Motion

Gradients of soluble molecules coordinate cellular communication in a diverse range of multicellular systems. Chemokine-driven chemotaxis is a key orchestrator of cell movement during organ development, immune response and cancer progression. Chemotaxis assays capable of examining cell responses to different chemokines in the context of various extracellular matrices will be crucial to characterize directed cell motion in conditions which mimic whole tissue conditions. Here, a microfluidic device which can generate different chemokine patterns in flow-free gradient chambers while controlling surface extracellular matrix (ECM) to study chemotaxis either at the population level or at the single cell level with high resolution imaging is presented. The device is produced by combining additive manufacturing (AM) and soft lithography. Generation of concentration gradients in the device were simulated and experimentally validated. Then, stable gradients were applied to modulate chemotaxis and chemokinetic response of Jurkat cells as a model for T lymphocyte motility. Live imaging of the gradient chambers allowed to track and quantify Jurkat cell migration patterns. Using this system, it has been found that the strength of the chemotactic response of Jurkat cells to CXCL12 gradient was reduced by increasing surface fibronectin in a dose-dependent manner. The chemotaxis of the Jurkat cells was also found to be governed not only by the CXCL12 gradient but also by the average CXCL12 concentration. Distinct migratory behaviors in response to chemokine gradients in different contexts may be physiologically relevant for shaping the host immune response and may serve to optimize the targeting and accumulation of immune cells to the inflammation site. Our approach demonstrates the feasibility of using a flow-free gradient chamber for evaluating cross-regulation of cell motility by multiple factors in different biologic processes.

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MICRO AND MESO SCALES OF DESCRIPTION CORRESPONDING TO A MODEL OF TISSUE INVASION BY SOLID TUMOURS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202505000935 ◽

2005 ◽

Vol 15 (11) ◽

pp. 1667-1683 ◽

Cited By ~ 53

Author(s):

MIROSŁAW LACHOWICZ

Keyword(s):

Mathematical Models ◽

Type Equation ◽

Reaction Diffusion ◽

Solid Tumours ◽

Macroscopic Model ◽

Scale Level ◽

Micro Scale ◽

Tissue Invasion ◽

Mathematical Relationships ◽

Meso Scale

In this paper two new mathematical models are proposed that correspond to a macroscopic model of tissue invasion of solid tumours, in terms of a system of reaction-diffusion-chemotaxis equations. The first model is defined at the micro-scale level of a large number of interacting individual entities, and is in terms of a linear (Markov) equation. The second model refers to the meso-scale level of description of test-entities and is given in terms of a bilinear Boltzmann-type equation. Mathematical relationships among these three possible descriptions are formulated. Explicit error estimates are given.

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Deterministic convection-diffusion approach for modeling cell motion and spatial organization: Experimentation on avascular tumor growth

2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) ◽

10.1109/bibm.2017.8217709 ◽

2017 ◽

Author(s):

Cheikhou Oumar Ka ◽

Jean-Marie Dembele ◽

Christophe Cambier ◽

Serge Stinckwich ◽

Moussa Lo ◽

...

Keyword(s):

Tumor Growth ◽

Spatial Organization ◽

Convection Diffusion ◽

Cell Motion ◽

Avascular Tumor

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Numerical Investigation of the Coulter Principle in a Microfluidic Device

Volume 2, Fora: Cavitation and Multiphase Flow; Fluid Measurements and Instrumentation; Microfluidics; Multiphase Flows: Work in Progress ◽

10.1115/fedsm2013-16011 ◽

2013 ◽

Author(s):

Muheng Zhang ◽

Yongsheng Lian

Keyword(s):

Electric Field ◽

Stokes Equations ◽

Equations Of Motion ◽

Navier Stokes ◽

Working Mechanism ◽

Navier Stokes Equations ◽

Sheath Flow ◽

Coulter Counters ◽

Pass Through ◽

Cell Motion

Coulter counters are analytical microfluidic instrument used to measure the size and concentration of biological cells or colloid particles suspended in electrolyte. The underlying working mechanism of Coulter counters is the Coulter principle which relies on the fact that when low-conductive cells pass through an electric field these cells cause disturbances in the measurement (current or voltage). Useful information about these cells can be obtained by analyzing these disturbances if an accurate correlation between the measured disturbances and cell characteristics. In this paper we use computational fluid dynamics method to investigate this correlation. The flow field is described by solving the Navier-Stokes equations, the electric field is represented by a Laplace’s equation in which the conductivity is calculated from the Navier-Stokes equations, and the cell motion is calculated by solving the equations of motion. The accuracy of the code is validated by comparing with analytical solutions. The study is based on a coplanar Coulter counter with three inlets that consist of two sheath flow inlet and one conductive flow inlet. The effects of diffusivity, cell size, sheath flow rate, and cell geometry are discussed in details. The impacts of electrode size, gap between electrodes and electrode location on the measured distribution are also studied.

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MIXED CONFORMING ELEMENTS FOR THE LARGE-BODY LIMIT IN MICROMAGNETICS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202513500486 ◽

2013 ◽

Vol 24 (01) ◽

pp. 113-144 ◽

Cited By ~ 1

Author(s):

MARKUS AURADA ◽

JENS M. MELENK ◽

DIRK PRAETORIUS

Keyword(s):

Mixed Finite Element ◽

Large Body ◽

Mixed Finite Element Method ◽

Adaptive Strategy ◽

Posteriori Error ◽

Higher Order ◽

Three Dimensions ◽

Macroscopic Model ◽

Error Estimator ◽

Discrete Problem

We introduce a stabilized conforming mixed finite element method for a macroscopic model in micromagnetics. We show well-posedness of the discrete problem for higher order elements in two and three dimensions, develop a full a priori analysis for lowest order elements, and discuss the extension of the method to higher order elements. We introduce a residual-based a posteriori error estimator and present an adaptive strategy. Numerical examples illustrate the performance of the method.

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Cancer Cell Invasion of Brain Tissue: Guided by a Prepattern?

Journal of Theoretical Medicine ◽

10.1080/1027366042000334144 ◽

2005 ◽

Vol 6 (1) ◽

pp. 21-31 ◽

Cited By ~ 13

Author(s):

Michael Wurzel ◽

Carlo Schaller ◽

Matthias Simon ◽

Andreas Deutsch

Keyword(s):

Lattice Gas ◽

Individual Cell ◽

Physical Structure ◽

Tumour Cells ◽

Malignant Cells ◽

Malignant Brain Tumour ◽

Tumour Bulk ◽

Cell Motion ◽

The Brain ◽

Global Growth

The malignant brain tumourGlioblastoma multiforme(GBM) displays a highly invasive behaviour. Spreading of the malignant cells appears to be guided by the white matter fibre tracts within the brain. In order to understand the global growth process we introduce a lattice-gas cellular automaton model which describes the local interaction between individual malignant cells and their neighbourhood. We consider interactions between cells (brain cells and tumour cells) and between malignant cells and the fibre tracts in the brain, which are considered as a prepattern. The prepattern implies persistent individual cell motion along the fibre structure. Simulations with the model show that only the inclusion of the prepattern results in invading tumour and growing tumour islets in front of the expanding tumour bulk (i.e. the growth pattern observed in clinical practice). Our results imply that the infiltrative growth of GBMs is, in part, determined by the physical structure of the surrounding brain rather than by intrinsic properties of the tumour cells.

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