A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features

Behavior of cell aggregates under force-controlled compression

In this paper we study the mechanical behavior of multicellular aggregates under compressive loads and subsequent 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. The case of a cycle of compressions at constant force and releases is also considered. We show that a steady state configuration able to bear the load is achieved. The analytical determination of the steady state value allows to obtain mechanical parameters of the cellular structure that are not estimable from creep tests at constant stres

Statistical mechanics model of angiogenic tumor growth

We examine a lattice model of tumor growth where survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the supply is correlated with distribution of tumor cells, which as we suggest might mimick the angiogenic growth, the extinction shows different, and most likely novel critical behaviour. Such a correlation affects also the morphology of the growing tumors and drastically raise tumor survival probability.Comment: 4 page

Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications

Resorting to a multiphase modelling framework, tumours are described here as a mixture of tumour and host cells within a porous structure constituted by a remodelling extracellular matrix (ECM), which is wet by a physiological extracellular fluid. The model presented in this article focuses mainly on the description of mechanical interactions of the growing tumour with the host tissue, their influence on tumour growth, and the attachment/detachment mechanisms between cells and ECM. Starting from some recent experimental evidences, we propose to describe the interaction forces involving the extracellular matrix via some concepts coming from viscoplasticity. We then apply the model to the description of the growth of tumour cords and the formation of fibrosis

Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation

Experiments of in vitro formation of blood vessels show that cells randomly spread on a gel matrix autonomously organize to form a connected vascular network. We propose a simple model which reproduces many features of the biological system. We show that both the model and the real system exhibit a fractal behavior at small scales, due to the process of migration and dynamical aggregation, followed at large scale by a random percolation behavior due to the coalescence of aggregates. The results are in good agreement with the analysis performed on the experimental data.Comment: 4 pages, 11 eps figure

Relationship between regional fat distribution and hypertrophic cardiomyopathy phenotype

Hypertrophic cardiomyopathy (HCM), the most common genetic heart disease, is characterized by heterogeneous phenotypic expression. Body mass index has been associated with LV mass and heart failure symptoms in HCM. The aim of our study was to investigate whether regional (trunk, appendicular, epicardial) fat distribution and extent could be related to hypertrophy severity and pattern in HCM

Modelling of Tirapazamine effects on solid tumour morphology

Bioreductive drugs are in clinical practice to exploit the resistance from tumour microenvironments especially in the hypoxic region of tumour. We pre-sented a tumour treatment model to capture the pharmacology of one of the most prominent bioreductive drugs, Tirapazamine (TPZ) which is in clinical trials I and II. We calculated solid tumour mass in our previous work and then integrated that model with TPZ infusion. We calculated TPZ cytotoxicity, concentration, penetra-tion with increasing distance from blood vessel and offered resistance from micro-environments for drug penetration inside the tumour while considering each cell as an individual entity. The impact of these factors on tumour morphology is also showed to see the drug behaviour inside animals/humans tumours. We maintained the heterogeneity factors in presented model as observed in real tumour mass es-pecially in terms of cells proliferation, cell movement, extracellular matrix (ECM) interaction, and the gradients of partial oxygen pressure (pO2) inside tumour cells during the whole growth and treatment activity. The results suggest that TPZ high concentration in combination with chemotherapy should be given to get maximum abnormal cell killing. This model can be a good choice for oncologists and re-searchers to explore more about TPZ action inside solid tumour

Modelling physical limits of migration by a kinetic model with non-local sensing

Migrating cells choose their preferential direction of motion in response to different signals and stimuli sensed by spanning their external environment. However, the presence of dense fibrous regions, lack of proper substrate, and cell overcrowding may hamper cells from moving in certain directions or even from sensing beyond regions that practically act like physical barriers. We extend the non-local kinetic model proposed by Loy and Preziosi (J Math Biol, 80, 373–421, 2020) to include situations in which the sensing radius is not constant, but depends on position, sensing direction and time as the behaviour of the cell might be determined on the basis of information collected before reaching physically limiting configurations. We analyse how the actual possible sensing of the environment influences the dynamics by recovering the appropriate macroscopic limits and by integrating numerically the kinetic transport equation

A Statistical Mechanics Approach to Describe Cell Reorientation Under Stretch

Experiments show that when a monolayer of cells cultured on an elastic substratum is subject to a cyclic stretch, cells tend to reorient either perpendicularly or at an oblique angle with respect to the main stretching direction. Due to stochastic effects, however, the distribution of angles achieved by the cells is broader and, experimentally, histograms over the interval [0(?), 90(?)] are usually reported. Here we will determine the evolution and the stationary state of probability density functions describing the statistical distribution of the orientations of the cells using Fokker-Planck equations derived from microscopic rules for describing the reorientation process of the cell. As a first attempt, we shall use a stochastic differential equation related to a very general elastic energy that the cell tries to minimize and, we will show that the results of the time integration and of the stationary state of the related forward Fokker-Planck equation compare very well with experimental results obtained by different researchers. Then, in order to model more accurately the microscopic process of cell reorientation and to shed light on the mechanisms performed by cells that are subject to cyclic stretch, we consider discrete in time random processes that allow to recover Fokker-Planck equations through classical tools of kinetic theory. In particular, we shall introduce a model of reorientation as a function of the rotation angle as a result of an optimal control problem. Also in this latter case the results match very well with experiments

Stability of a non-local kinetic model for cell migration with density-dependent speed

The aim of this article is to study the stability of a non-local kinetic model proposed by Loy & Preziosi (2020a) in which the cell speed is affected by the cell population density non-locally measured and weighted according to a sensing kernel in the direction of polarization and motion. We perform the analysis in a d-dimensional setting. We study the dispersion relation in the one-dimensional case and we show that the stability depends on two dimensionless parameters: the first one represents the stiffness of the system related to the cell turning rate, to the mean speed at equilibrium and to the sensing radius, while the second one relates to the derivative of the mean speed with respect to the density evaluated at the equilibrium. It is proved that for Dirac delta sensing kernels centered at a finite distance, corresponding to sensing limited to a given distance from the cell center, the homogeneous configuration is linearly unstable to short waves. On the other hand, for a uniform sensing kernel, corresponding to uniformly weighting the information collected up to a given distance, the most unstable wavelength is identified and consistently matches the numerical solution of the kinetic equation