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Cancer cell migration between different body parts is the driving force behind cancer metastasis, which is the main cause of mortality of patients. Migration of cancer cells often proceeds by penetration through narrow cavities in locally stiff, yet flexible tissues. In our previous work, we developed a model for cell geometry evolution during invasion, which we extend here to investigate whether leader and follower (cancer) cells that only interact mechanically can benefit from sequential transmigration through narrow micro-channels and cavities. We consider two cases of cells sequentially migrating through a flexible channel: leader and follower cells being closely adjacent or distant. Using Wilcoxon's signed-rank test on the data collected from Monte Carlo simulations, we conclude that the modelled transmigration speed for the follower cell is significantly larger than for the leader cell when cells are distant, i.e. follower cells transmigrate after the leader has completed the crossing. Furthermore, it appears that there exists an optimum with respect to the width of the channel such that cell moves fastest. On the other hand, in the case of closely adjacent cells, effectively performing collective migration, the leader cell moves $12\%$ faster since the follower cell pushes it. This work shows that mechanical interactions between cells can increase the net transmigration speed of cancer cells, resulting in increased invasiveness. In other words, interaction between cancer cells can accelerate metastatic invasion.

The phenomenological model for cell shape deformation and cell migration (Chen et.al. 2018; Vermolen and Gefen 2012) is extended with the incorporation of cell traction forces and the evolution of cell equilibrium shapes as a result of cell differentiation. Plastic deformations of the extracellular matrix are modelled using morphoelasticity theory. The resulting partial differential differential equations are solved by the use of the finite element method. The paper treats various biological scenarios that entail cell migration and cell shape evolution. The experimental observations in Mak et.al. (2013), where transmigration of cancer cells through narrow apertures is studied, are reproduced using a Monte Carlo framework.

More than eighty percent of pancreatic cancer involves ductal adenocarcinoma with an abundant desmoplastic extracellular matrix surrounding the solid tumor entity. This aberrant tumor microenvironment facilitates a strong resistance of pancreatic cancer to medication. Although various therapeutic strategies have been reported to be effective in mice with pancreatic cancer, they still need to be tested quantitatively in wider animal-based experiments before being applied as therapies. To aid the design of experiments, we develop a cell-based mathematical model to describe cancer progression under therapy with a specific application to pancreatic cancer. The displacement of cells is simulated by solving a large system of stochastic differential equations with the Euler–Maruyama method. We consider treatment with the PEGylated drug PEGPH20 that breaks down hyaluronan in desmoplastic stroma followed by administration of the chemotherapy drug gemcitabine to inhibit the proliferation of cancer cells. Modeling the effects of PEGPH20 + gemcitabine concentrations is based on Green’s fundamental solutions of the reaction–diffusion equation. Moreover, Monte Carlo simulations are performed to quantitatively investigate uncertainties in the input parameters as well as predictions for the likelihood of success of cancer therapy. Our simplified model is able to simulate cancer progression and evaluate treatments to inhibit the progression of cancer., Numerical Analysis

More than eighty percent of pancreatic cancer involves ductal adenocarcinoma with an abundant desmoplastic extracellular matrix surrounding the solid tumor entity. This aberrant tumor microenvironment facilitates a strong resistance of pancreatic cancer to medication. Although various therapeutic strategies have been reported to be effective in mice with pancreatic cancer, they still need to be tested quantitatively in wider animal-based experiments before being applied as therapies. To aid the design of experiments, we develop a cell-based mathematical model to describe cancer progression under therapy with a specific application to pancreatic cancer. The displacement of cells is simulated by solving a large system of stochastic differential equations with the Euler–Maruyama method. We consider treatment with the PEGylated drug PEGPH20 that breaks down hyaluronan in desmoplastic stroma followed by administration of the chemotherapy drug gemcitabine to inhibit the proliferation of cancer cells. Modeling the effects of PEGPH20 + gemcitabine concentrations is based on Green’s fundamental solutions of the reaction–diffusion equation. Moreover, Monte Carlo simulations are performed to quantitatively investigate uncertainties in the input parameters as well as predictions for the likelihood of success of cancer therapy. Our simplified model is able to simulate cancer progression and evaluate treatments to inhibit the progression of cancer., Numerical Analysis

During metastasis of cancer, cell migration plays a crucial role, which is normally accompanied by morphological evolution. To simulate cell deformation, we develop a phenomenological, computational model involving deformation of a cell as well as its nucleus. The migration of a single cell is orchestrated by a generic signal (e.g. a chemokine or a stiffness stimulus), the microvascular flow and stochastic processes, which are dealt with by using Green’s Fundamental solutions, Poisseuille flow and a vector Wiener process, respectively. Moreover, due to the uncertainties in the input variables, Monte Carlo simulations are carried out to evaluate the correlations between various parameters and quantitatively predict the likelihood of vessel transmigration of one cell during cancer metastasis., Numerical Analysis

Cell migration plays an essential role in cancer metastasis. In cancer invasion through confined spaces, cells must undergo extensive deformation, which is a capability related to their metastatic potentials. Here, we simulate the deformation of the cell and nucleus during invasion through a dense, physiological microenvironment by developing a phenomenological computational model. In our work, cells are attracted by a generic emitting source (e.g., a chemokine or stiffness signal), which is treated by using Green’s Fundamental solutions. We use an IMEX integration method where the linear parts and the nonlinear parts are treated by using an Euler backward scheme and an Euler forward method, respectively. We develop the numerical model for an obstacle-induced deformation in 2D or/and 3D. Considering the uncertainty in cell mobility, stochastic processes are incorporated and uncertainties in the input variables are evaluated using Monte Carlo simulations. This quantitative study aims at estimating the likelihood for invasion and the length of the time interval in which the cell invades the tissue through an obstacle. Subsequently, the two-dimensional cell deformation model is applied to simplified cancer metastasis processes to serve as a model for in vivo or in vitro biomedical experiments., Numerical Analysis

During metastasis of cancer, cell migration plays a crucial role, which is normally accompanied by morphological evolution. To simulate cell deformation, we develop a phenomenological, computational model involving deformation of a cell as well as its nucleus. The migration of a single cell is orchestrated by a generic signal (e.g. a chemokine or a stiffness stimulus), the microvascular flow and stochastic processes, which are dealt with by using Green’s Fundamental solutions, Poisseuille flow and a vector Wiener process, respectively. Moreover, due to the uncertainties in the input variables, Monte Carlo simulations are carried out to evaluate the correlations between various parameters and quantitatively predict the likelihood of vessel transmigration of one cell during cancer metastasis., Numerical Analysis

Cell migration plays an essential role in cancer metastasis. In cancer invasion through confined spaces, cells must undergo extensive deformation, which is a capability related to their metastatic potentials. Here, we simulate the deformation of the cell and nucleus during invasion through a dense, physiological microenvironment by developing a phenomenological computational model. In our work, cells are attracted by a generic emitting source (e.g., a chemokine or stiffness signal), which is treated by using Green’s Fundamental solutions. We use an IMEX integration method where the linear parts and the nonlinear parts are treated by using an Euler backward scheme and an Euler forward method, respectively. We develop the numerical model for an obstacle-induced deformation in 2D or/and 3D. Considering the uncertainty in cell mobility, stochastic processes are incorporated and uncertainties in the input variables are evaluated using Monte Carlo simulations. This quantitative study aims at estimating the likelihood for invasion and the length of the time interval in which the cell invades the tissue through an obstacle. Subsequently, the two-dimensional cell deformation model is applied to simplified cancer metastasis processes to serve as a model for in vivo or in vitro biomedical experiments., Numerical Analysis

Cell migration, known as an orchestrated movement of cells, is crucially important for wound healing, tumor growth, immune response as well as other biomedical processes. This paper presents a cell-based model to describe cell migration in non-isotropic fibrin networks around pancreatic tumor islets. This migration is determined by the mechanical strain energy density as well as cytokines-driven chemotaxis. Cell displacement is modeled by solving a large system of ordinary stochastic differential equations where the stochastic parts result from random walk. The stochastic differential equations are solved by the use of the classical Euler–Maruyama method. In this paper, the influence of anisotropic stromal extracellular matrix in pancreatic tumor islets on T-lymphocytes migration in different immune systems is investigated. As a result, tumor peripheral stromal extracellular matrix impedes the immune response of T-lymphocytes through changing direction of their migration., Numerical Analysis

Cell migration, known as an orchestrated movement of cells, is crucially important for wound healing, tumor growth, immune response as well as other biomedical processes. This paper presents a cell-based model to describe cell migration in non-isotropic fibrin networks around pancreatic tumor islets. This migration is determined by the mechanical strain energy density as well as cytokines-driven chemotaxis. Cell displacement is modeled by solving a large system of ordinary stochastic differential equations where the stochastic parts result from random walk. The stochastic differential equations are solved by the use of the classical Euler–Maruyama method. In this paper, the influence of anisotropic stromal extracellular matrix in pancreatic tumor islets on T-lymphocytes migration in different immune systems is investigated. As a result, tumor peripheral stromal extracellular matrix impedes the immune response of T-lymphocytes through changing direction of their migration., Numerical Analysis

The aqueous solutions of the anionic hybrid fluorocarbon/hydrocarbon surfactant sodium 1-oxo-1[4-(tridecafluorohexyl)phenyl]-2-hexanesulfate (FC6HC4) shows peculiar rheological behavior. At 25 °C the viscosity vs concentration curve goes successively through a maximum and a minimum, while the viscosity vs temperature curve of the 10 wt% aqueous FC6HC4 solution goes through a marked maximum at 36 °C [Tobita et al., Langmuir 13 (1997) 5054]. In an attempt to explain these properties the microstructure of aqueous solutions of FC6HC4 has been investigated by means of digital light microscopy, transmission electron microscopy at cryogenic temperature (cryo-TEM), rheology, and self-diffusion NMR. At 20 °C, the increase of the FC6HC4 concentration was found to result in a progressive change of structure of the surfactant assemblies from mainly spherical micelles at 0.5 wt% to mainly cylindrical micelles at 10 wt%. At intermediate concentrations small disklike micelles and small complete and incomplete vesicles coexisting with cylindrical micelles were visualized. The occurrence of stretched cylindrical micelles is responsible for the effect of the surfactant concentration on the solution viscosity. Cryo-TEM, rheology, and self-diffusion NMR all suggest that an increase of the temperature brings about a growth of the assemblies present in the 10 wt% solution of FC6HC4. The structure of the assemblies present at the temperature where the viscosity is a maximum could not be elucidated by cryo-TEM because of the probable occurrence of an on-the-grid phase transformation, the result of blotting during specimen preparation. Nevertheless, the results show that the observed large assemblies break up at higher temperature to give rise to a more labile bicontinuous structure that consists of multiconnected disordered lamellae, with many folds and creases, and that may well be the L3 phase.