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Mechanical aspects of tumour growth: Multiphase modelling, adhesion, and evolving natural configurations
... The mass-effect on tumor growth has been mainly modeled using two different perspectives: i) analyzing the stress, strain, and deformations produced by a solid tumor with nonlinear viscoelastic behavior [41][42][43][44] and, ii) analyzing the physical interactions among a multicomponent tumor constituted by deformable sphere-like structures residing in biological fluids [45][46][47][48]. The second scheme's analysis of the brain tumor has been described at different levels of complexity, ranging from mixtures from two to multiple species [45][46][47][48]. ...
... The mass-effect on tumor growth has been mainly modeled using two different perspectives: i) analyzing the stress, strain, and deformations produced by a solid tumor with nonlinear viscoelastic behavior [41][42][43][44] and, ii) analyzing the physical interactions among a multicomponent tumor constituted by deformable sphere-like structures residing in biological fluids [45][46][47][48]. The second scheme's analysis of the brain tumor has been described at different levels of complexity, ranging from mixtures from two to multiple species [45][46][47][48]. These models are based on the principle that all the components co-exist simultaneously, and evolve according to balance laws and conservation principles. ...
... The model reproduces some phenomenological states associated with the volumetric growth, such as the progressive compression within the tumor itself, and the compression caused to the surrounding tissue. A more sophisticated model is proposed by Preziosi et al. [45], who represent the brain-tumor system as a mixture of solid components embedded in a liquid media interacting between them. The analysis of the mass-effect is performed through the mechanical interactions among a set of multiphase spheroids representing the different kind of cells, all embedded in a fluid media. ...
The growth of a tumor within a finite domain (skull) generates mechanical forces that alter the physical interactions among cells. The relationship between these forces and the tumor architecture remains an open problem subjected to extensive research. Recently, it has been determined that those regions of high mechanical compression can accelerate and intensify the invasive capacity of the malignant cells, forming an irregular tumor whose full extent and edges are difficult to identify.
In the present paper, we propose a one-dimensional mathematical model that describes the process of proliferation and diffusion of glioma cells taking into account the mechanical compression generated during its expansion. Supported on the mixture theory, we model the brain-tumor system as a multiphase mixture of cancer cells, healthy cells, biological fluids and extracellular matrix whose densities determine the mechanical loads generated during the volumetric growth. Our model provides a detailed understanding of the pressure distribution on the interface boundary between healthy and cancer cells. It validates the hypothesis that the conferred ability of cancer cells to proliferate depends strongly on the mechanical pressure sensed. Through the analysis of the mechanical pressure, we determine that the anisotropic loads promote cancer cells to grow preferentially in the directions of low mechanical compression.
... In Korn and Schwarz (2008), the Brownian motion of the sphere is taken into account in order to model the spatial receptor-ligand encountering in more details. In the absence of fluid flow, macroscopic models have been developed for cell adhesion force (see Preziosi and Vitale (2011)). In our approach, bonds are not described individually but as a distribution function. ...
... This bonds distribution follows a maturation-rupture equation (also called renewal equation) as in Oelz and Schmeiser (2010). In the limit of large lig-ands binding turnover, a friction coefficient can be computed, Preziosi and Vitale (2011); Milisic and Oelz (2011). ...
... Therefore, in a second step we are interested in averaging the model based on Poisson point process. Averaging this stochastic model leads to a deterministic Volterra integro-differential equation similar to the one considered in Preziosi and Vitale (2011);Milisic and Oelz (2011). The deterministic one-dimensional model provides the particle location. ...
Cell adhesion on the vascular wall is a highly coupled process where blood flow and adhesion dynamics are closely linked. Cell dynamics in the vicinity of the vascular wall is driven mechanically by the competition between the drag force of the blood flow and the force exerted by the bonds created between the cell and the wall. Bonds exert a friction force. Here, we propose a mathematical model of such a competitive system, namely leukocytes whose capacity to create bonds with the vascular wall and transmigratory ability are coupled by integrins and chemokines. The model predicts that this coupling gives rise to a dichotomic cell dynamic, whereby cells switch from sliding to firm arrest, through non linear effects. Cells can then transmigrate through the wall. These predicted dynamic regimes are compared to in-vitro trajectories of leukocytes. We expect that competition between friction and drag force in particle dynamics (such as shear stress-controlled nanoparticle capture) can lead to similar dichotomic mode.
... Courbe représentant le rapport de volume d'un constituant, φ i en fonction de la taille de l'échantillon pris pour faire la moyenne τ . Figure extraite de[132]. ...
... Figure extraitede[132]. . . . . . . . . . . . . . . . . . . . . . . . .. 40 2.5 Représentation de la surpression cellulaire f (φ c ). ...
Le mélanome est un cancer dont la mortalité augmente rapidement avec le temps. Afin d'assurer une détection précoce, des campagnes de sensibilisation ont été menées donnant des critères morphologiques pour le distinguer des grains de beauté. Mais, l'origine des différences d'aspects entre lésions bénignes et malignes reste inconnue. L'objectif est ici de relier les effets des modifications génétiques à l'aspect des tumeurs, en utilisant des outils venus de la physique macroscopique. Les mélanomes ont l'avantage d'être facilement observables et fins, ce qui en font un système idéal. Ce travail commence par rappeler les aspects physiologiques des cancers de la peau. On explique le fonctionnement de la peau saine, puis nous décrivons les différents types de lésions cutanées, et enfin nous donnons un bref aperçu des différents chemins génétiques connus menant au mélanome. Ensuite, nous faisons un rappel des différents modèles mathématiques du cancer. Nous nous attardons sur l'utilisation de la théorie des mélanges comme base théorique de mise en équation des tumeurs. Nous l'appliquons ensuite dans un modèle simplifié à deux phases en deux dimensions. Puis, nous analysons ces équations. Une étude des composantes spatiales montre la possibilité d'un processus de séparation de phases : la décomposition spinodale. L'étude temporelle permet de montrer que ces équations contiennent les ingrédients nécessaires à décrire plusieurs types de mélanomes observés in vivo. Nous terminons par l'étude des effets de la troisième dimension jusqu'alors mis de côté dans le modèle. Nous mettons en équation des mélanomes évoluant sur un épiderme ondulé, au niveau des mains et des pieds.
... However, they decided to ignore viscoelastic effects and modelled both brain and tumour tissue as elastic materials; this is likely to be an unrealistic approach. More recently, Preziosi and Vitale carried out a study on the mechanical aspects of tumour growth, pointing out that tumour tissue can be regarded as stiffer than healthy tissue; while they do not refer specifically to neural tissue, it may be safe to assume that this is true for brain tumours as well [30]. ...
... Because of the scarcity of detailed experimental data regarding the biomechanical characteristics of cancerous tissue, it was assumed to behave similarly to healthy brain tissue [9,29]. However, tumours are generally stiffer than normal tissue [30]. For this reason, and to allow visible distinction between the two in the FE simulations, the bulk modulus and the density of the tumour were set to be slightly higher than those of brain matter. ...
This thesis proposes a novel data-based approach to model the deformation of a brain tumour for augmented reality (AR)-based applications in neurosurgery. AR systems have the potential to assist surgeons in the most delicate operations, such as tumour removal, by providing intuitive 3-dimensional visualisations of anatomical structures and targets. One of the major challenges to tackle in order to enhance the clinical impact of this technology is computing and displaying the deformation of the virtual overlay accurately and in real time. This is fundamental to maintain depth perception and thus avoid the degradation of surgical performance. Current soft tissue deform- ation models are characterised by a trade-off between speed and accuracy, or are not apt for patient-specific implementations. The method developed in this thesis employs statistical techniques and machine learning algorithms to derive a model computable in real time. The model is trained on a large set of data (7 × 106 data points) ob- tained through force-driven finite element simulations. Data analysis is carried out to identify the relationship between predictors and output, and to perform feature selection. Three different machine learning algorithms — Artificial Neural Networks, Support Vector Regression, and Gaussian Process Regression — are tested and their performance is evaluated, resulting in real-time predictions of the tumour’s deform- ation with mean errors below 0.4 mm, a value that is considerably lower than the threshold of surgical accuracy and AR’s position uncertainty. The results compare well with the existing literature, providing smaller errors, instantaneous computation, and high patient-specificity. The proposed approach addresses the current needs of image-guided surgical systems, and has the potential to be employed to model the deformation of any type of soft tissue.
... Because of the scarcity of detailed experimental data regarding the biomechanical characteristics of cancerous tissue, it was assumed to behave similarly to healthy brain tissue [9,48]. However, tumours are generally sti↵er than normal tissue [49]. For this reason, and to allow a noticeable distinction between the two in the FE simulations, the density and bulk modulus of the tumour were set at twice the value than those of brain matter: 2160 kg m 3 and 2000 kPa respectively for the tumour, compared to 1080 kg m 3 and 1000 kPa for the brain. ...
Objectives:
Accurate reconstruction and visualisation of soft tissue deformation in real time is crucial in image-guided surgery, particularly in augmented reality (AR) applications. Current deformation models are characterised by a trade-off between accuracy and computational speed. We propose an approach to derive a patient-specific deformation model for brain pathologies by combining the results of pre-computed finite element method (FEM) simulations with machine learning algorithms. The models can be computed instantaneously and offer an accuracy comparable to FEM models.
Method:
A brain tumour is used as the subject of the deformation model. Load-driven FEM simulations are performed on a tetrahedral brain mesh afflicted by a tumour. Forces of varying magnitudes, positions, and inclination angles are applied onto the brain's surface. Two machine learning algorithms-artificial neural networks (ANNs) and support vector regression (SVR)-are employed to derive a model that can predict the resulting deformation for each node in the tumour's mesh.
Results:
The tumour deformation can be predicted in real time given relevant information about the geometry of the anatomy and the load, all of which can be measured instantly during a surgical operation. The models can predict the position of the nodes with errors below 0.3mm, beyond the general threshold of surgical accuracy and suitable for high fidelity AR systems. The SVR models perform better than the ANN's, with positional errors for SVR models reaching under 0.2mm.
Conclusions:
The results represent an improvement over existing deformation models for real time applications, providing smaller errors and high patient-specificity. The proposed approach addresses the current needs of image-guided surgical systems and has the potential to be employed to model the deformation of any type of soft tissue.
... In the same spirit, in the absence of fluid flow, macroscopic models have been developed for cell adhesion force [36]. Bonds are not described individually but as a distribution function. ...
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.
... Such models are capable not only of describing the variation of mass density within the tumour and the host tissue, but also of evaluating the evolution of stresses and interstitial pressure, linking the mechanics of tumours to their growth and selected interactions with the outer environment. For more details the reader is referred to the following reviews [1][2][3][4][5][6]. Most of the models describe the tumour mass as a fluid, which is of course a strong simplification. ...
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.
... A biphasic mixture consisting of a solid and a fluid phase is perhaps the most essential model of multicellular aggregates [1,2,5,11,15]. Cells and the network of fibers form the solid elastic skeleton of the mixture, whereas the fluid phase stands for the interstitial fluid, that completely saturates the pores of the solid and may move throughout it. ...
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.
... The permeability, k(φ ECM ), which is a scalar only if the ECM is isotropic (with respect to the flow), is generally a function of porosity (e.g. Kozeny-Carman or Holmes-Mow [25]), but it is often assumed to be a constant [35,39,53]. Sometimes the orientation of ECM fibres is considered [5,6,36]. ...
The mechanical properties of cell nuclei have been demonstrated to play a fundamental role in cell movement across extracellular networks and micro-channels. In this work, we focus on a mathematical description of a cell entering a cylindrical channel composed of extracellular matrix. An energetic approach is derived in order to obtain a necessary condition for which cells enter cylindrical structures. The nucleus of the cell is treated either (i) as an elastic membrane surrounding a liquid droplet or (ii) as an incompressible elastic material with Neo-Hookean constitutive equation. The results obtained highlight the importance of the interplay between mechanical deformability of the nucleus and the capability of the cell to establish adhesive bonds and generate active forces in the cytoskeleton due to myosin action.
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.
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.
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