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6 Examples of contact inhibition of growth reported in the experiments by Tsukatani et al. (Tzukatani et al., 1997) using human breast epithelial cells and by Orford et al. (Orford et al., 1999) using canine kidney-derived nontransformed epithelial cells.
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This chapter provides the basics to deduce multiphase models for the essential constituents present in tumours (cells, extracellular matrix, extracellular liquid, and possibly blood and limphatic vasculature). All the steps of the modelling procedure are explained in detail, special attention being paid to the meaning of all the different terms inv...
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... is integrated in Fig. 1.16. If the compression applies for a long enough time, then Ψ will tend towards that value Ψ ∞ > ψ 0 corresponding to the vanishing of the square parenthesis in (1.118), ...
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... This would allow us to perform in silico experiments combining these two mechanisms and make predictions on the possible way cells sort or combine the two processes. Existing models regarding cell migration on ECM with a particular focus on the role of confinement and the influence of the ECM porosity/density on the cell speed include individualbased models [44,45,46,47], kinetic models [24], and mechanical models [20,35]. On the other hand, contact guidance has also been successfully described at the mesoscopic level through kinetic equations in [9,18], where the authors propose models that allow to take into account the variation of the microscopic velocities in response to a given ECM fiber network. ...
We propose a non-local model for contact guidance and steric hindrance depending on a single external cue, namely the extracellular matrix, that affects in a twofold way the polarization and speed of motion of the cells. We start from a microscopic description of the stochastic processes underlying the cell re-orientation mechanism related to the change of cell speed and direction. Then, we formally derive the corresponding kinetic model that implements exactly the prescribed microscopic dynamics and, from it, it is possible to deduce the macroscopic limit in the appropriate regime. Moreover, we test our model in several scenarios. In particular, we numerically investigate the minimal microscopic mechanisms that are necessary to reproduce cell dynamics by comparing the outcomes of our model with some experimental results related to breast cancer cell migration. This allows us to validate the proposed modeling approach and, also, to highlight its capability of predicting the qualitative cell behaviors in diverse heterogeneous microenvironments.
... In addition, at low densities, χ is close to zero, being negative and strongly increases for growing density above D b . A standard choice is the Cahn-Hilliard potential,(see Appendix E) but here, by reason of numerical convergence the potential introduced by Byrne and Preziosi 41,42 has been preferred. Its derivative reads: Heterogeneity under tumor growth. ...
... With numerical simulations, we understood how modifications of the environment, in which the tumor expands, can abruptly change its long time behavior. In the future, a clear understanding of the mechanical and physical properties of CSCs can improve the model, especially when considering adhesion energy, elastic deformation and cell-cell interaction 41,46 . In this spirit, a more complicated multiphase model can be constructed, taking into account immune cells, the host cells and their www.nature.com/scientificreports ...
... Due to the number of independent parameters, it is more convenient to fix the 3 parameters: α, B, d 0 which partially define the initial fixed point 2 up to the value of D 2 , so all the curves C 0 (k) will start at the same value and will differ because of D 2 . A representation of a spinodal decomposition can be found in the Section (Spatio-Temporal), see Fig. (3c,d)) with the Byrne-Preziosi potential 41,42 . ...
We present a multiphase model for solid tumor initiation and progression focusing on the properties of cancer stem cells (CSC). CSCs are a small and singular cell sub-population having outstanding capacities: high proliferation rate, self-renewal and extreme therapy resistance. Our model takes all these factors into account under a recent perspective: the possibility of phenotype switching of differentiated cancer cells (DC) to the stem cell state, mediated by chemical activators. This plasticity of cancerous cells complicates the complete eradication of CSCs and the tumor suppression. The model in itself requires a sophisticated treatment of population dynamics driven by chemical factors. We analytically demonstrate that the rather important number of parameters, inherent to any biological complexity, is reduced to three pivotal quantities.Three fixed points guide the dynamics, and two of them may lead to an optimistic issue, predicting either a control of the cancerous cell population or a complete eradication. The space environment, critical for the tumor outcome, is introduced via a density formalism. Disordered patterns are obtained inside a stable growing contour driven by the CSC. Somewhat surprisingly, despite the patterning instability, the contour maintains its circular shape but ceases to grow for a typical size independently of segregation patterns or obstacles located inside.
... In the same spirit, in the absence of fluid flow, macroscopic models have been developed for cell adhesion force (Preziosi and Vitale 2011), where bonds are described as a distribution function. Its dynamics follows a maturation-rupture equation (also called renewal equation). ...
... The bond forces are described by linear elastic forces. In the case where binding and dissociation rates are constant, averaging leads to a deterministic linear Volterra integro-differential equation similar to the one considered in Preziosi and Vitale (2011), Milišić and Oelz (2011), that provides information on the cell location. Linear continuous models are not satisfactory as they can not describe the strong dependency of cells arrest on shear flow. ...
Cell dynamics in the vicinity of the vascular wall involves several factors of mechanical or biochemical origins. It is driven by the competition between the drag force of the blood flow and the resistive force generated by the bonds created between the circulating cell and the endothelial wall. Here, we propose a minimal mathematical model for the adhesive interaction between a circulating cell and the blood vessel wall in shear flow when the cell shape is neglected. The bond dynamics in cell adhesion is modeled as a nonlinear Markovian Jump process that takes into account the growth of adhesion complexes. Performing scaling limits in the spirit of Joffe and Metivier (Adv Appl Probab 18(1):20, 1986), Ethier and Kurtz (Markov processes: characterization and convergence, Wiley, New York, 2009), we obtain deterministic and stochastic continuous models, whose analysis allow to identify a threshold shear velocity associated with the transition from cell rolling and firm adhesion. We also give an estimation of the mean stopping time of the cell resulting from this dynamics. We believe these results can have strong implications for the understanding of major biological phenomena such as cell immunity and metastatic development.
... This approach allowed to explain the dispersion of rolling velocity data acquired under different experimental conditions. In the same spirit, in the absence of blood flow, macroscopic models have been developed for cell adhesion force (Preziosi and Vitale, 2011), where bonds are described as a distribution function. Its dynamics follows a maturation-rupture equation, also called renewal equation. ...
... The bond forces are described by linear elastic forces. For constant binding and dissociation rates, the averaged problem writes as a deterministic linear Volterra integro-differential equation similar to the one considered by Preziosi and Vitale (2011); Milišić and Oelz (2011), and provides some information on the cell location. Linear continuous models are not satisfactory as they can not describe the strong dependency of the cell arrest on the shear flow. ...
This paper deals with the adhesive interaction arising between a cell circulating in the blood flow and the vascular wall. The purpose of this work is to investigate the effect of the blood flow velocity on the cell dynamics, and in particular on its possible adhesion to the vascular wall. We formulate a model that takes into account the stochastic variability of the formation of bonds, and the influence of the cell velocity on the binding dynamics: the faster the cell goes, the more likely existing bonds are to disassemble. The model is based on a nonlinear birth-and-death-like dynamics, in the spirit of Joffe and Metivier (1986); Ethier and Kurtz (2009). We prove that, under different scaling regimes, the cell velocity follows either an ordinary differential equation or a stochastic differential equation, that we both analyse. We obtain both the identification of a shear-velocity threshold associated with the transition from cell sliding and its firm adhesion, and the expression of the cell mean stopping time as a function of its adhesive dynamics.
The aim of this contribution is to put together in a systematic way several approaches operating at different scales that were recently developed to describe the phenomenon of physical limit of migration, that occurs when the environment surrounding cells results restrictive, and to apply it to tumour growth and invasion. In particular, we will present: (i) a mechanical model of the behaviour of a cell within a microchannel that gives a blockage criterium for its penetration; (ii) a cellular Potts model to describe the dependence of the speed of a malignant cell from the mechanical characteristics both of its compartments (i.e., nucleus and cytosol) and of its environment; (iii) a multiphase model embodying such effects; (iv) the proper interface conditions to implement to deal with tumour invasion across matrix membranes and cell linings.
Gliomas are primary brain tumours arising from the glial cells of the nervous system. The diffuse nature of spread, coupled with proximity to critical brain structures, makes treatment a challenge. Pathological analysis confirms that the extent of glioma spread exceeds the extent of the grossly visible mass, seen on conventional magnetic resonance imaging (MRI) scans. Gliomas show faster spread along white matter tracts than in grey matter, leading to irregular patterns of spread. We propose a mathematical model based on Diffusion Tensor Imaging, a new MRI imaging technique that offers a methodology to delineate the major white matter tracts in the brain. We apply the anisotropic diffusion model of Painter and Hillen (J Thoer Biol 323:25-39, 2013) to data from 10 patients with gliomas. Moreover, we compare the anisotropic model to the state-of-the-art Proliferation-Infiltration (PI) model of Swanson et al. (Cell Prolif 33:317-329, 2000). We find that the anisotropic model offers a slight improvement over the standard PI model. For tumours with low anisotropy, the predictions of the two models are virtually identical, but for patients whose tumours show higher anisotropy, the results differ. We also suggest using the data from the contralateral hemisphere to further improve the model fit. Finally, we discuss the potential use of this model in clinical treatment planning.
The aim of this article is to propose a simple way of describing a tumour as a linear elastic material from a reference configuration that is continuously evolving in time due to growth and remodelling. The main assumption allowing this simplification is that the tumour mass is a very ductile material, so that it can only sustain moderate stresses while the deformation induced by growth, that can actually be quite big, mainly induces a plastic reorganisation of malignant cells. In mathematical terms this means that the deformation gradient can be split into a volumetric growth term, a term describing the reorganisation of cells, and a term that can be approximated by means of the linear strain tensor. A dimensional analysis of the importance of the different terms also allows to introduce a second simplification consisting in the decoupling of the equations describing the growth of the tumour mass from those describing the flow of the interstitial fluid.
In this paper we study the mechanical behavior of multicellular aggregates under a cycle of compressive loads and releases. Some analytical properties of the solution are discussed and numerical results are presented for a compressive test under constant force imposed on a cylindrical specimen and for a cycle of compressions and releases. We show that a steady loaded configuration is achieved. The analytical determination of the steady state value allows to obtain mechanical parameters of the cellular structure that cannot be obtained on the basis of creep tests at constant stress.
Several mathematical formulations have analyzed the time-dependent behavior of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the thermodynamically constrained averaging theory. A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TCs), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HCs); and an interstitial fluid for the transport of nutrients. The equations are solved by a finite element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTSs) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behavior: initially, the rapidly growing TCs tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable TCs whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case—mostly due to the relative adhesion of the TCs and HCs to the ECM, and the less favorable transport of nutrients. In particular, for HCs adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas TC infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a three-dimensional geometry. It is shown that TCs tend to migrate among adjacent vessels seeking new oxygen and nutrients. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on TC proliferation.