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In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient $\sigma (r,t)$ through blood vessels, and the tumor attracts blood vessels at a rate proportional to $\alpha (t)$. So, the boundary value condition

In this paper, we derive the mathematical model that allows chemotaxis in avascular tumor growth in a two-phase medium. The two phases are the viscous cell phase and the inviscid fluid phase. The conservation of mass–momentum is incorporated in each phase, and appropriate constitutive laws are applied to formulate the governing equations. Further, these equations are simplified into three main variables: cell volume fraction, cell velocity, and nutrient concentration. These variables generate a coupled system of nonlinear partial differential equations. A numerical scheme based on the finite volume method is applied to approximate the solution of cell volume fraction. The finite element method is applied to approximate the solutions of cell velocity and nutrient concentration. In this research paper, we investigate tumor growth when tumor cells move along a fluid containing a diffusible nutrient to which the cells are drawn. We perform some numerical simulations to show the effect of the parameters. The findings of this literature are compatible with the existing literature.

In this paper, the diffusion equation is used to model the spatio-temporal dynamics of a tumor, taking into account the heterogeneity of the medium. This approach allows us to take into account the complex geometric shape of the tumor when modeling. The main purpose of the work is to demonstrate the applicability of this approach by comparing the results obtained with the data from clinical observations. We use an algorithm based on an explicit finite-difference approximation of differential operators to solve the diffusion equation. The ranges of possible values that can take the input parameters of the model to match the results of clinical observations are obtained. On the basis of the data of clinical observations, the relative error of the results of computational experiments was determined, which lies in the range from 1.8% to 14.6%. It is concluded that the heterogeneity of the physical parameters of the model, in particular the diffusion coefficient, has a significant effect on the shape of the tumor.

Solanum nigrum L., an edible plant and local dish, has been assigned anticancer activities. However, the anticancer mechanisms of S. nigrum are poorly understood. Here, we investigated whether the water or polyphenol extracts of S. nigrum (SNWE or SNPE) could inhibit angiogenesis-mediated tumor growth. In nude mice bearing tumor xenografts, SNWE or SNPE significantly reduced the volume and weight of the tumors, and decreased the expression of CD31, a marker for angiogenesis. SNWE or SNPE was found to inhibit the VEGF-induced capillary structure formation of endothelial cells. The chicken egg chorioallantoic membrane (CAM) and matrigel plug assays showed further that SNWE or SNPE inhibited tumor angiogenesis. In human umbilical vascular endothelial cells (HUVECs), SNWE or SNPE suppressed the VEGF-induced activation of AKT and mTOR. Moreover, SNWE or SNPE inhibited the viability of human hepatoma HepG2 cells, and these effects were correlated with the extent of inhibition of the AKT/mTOR pathway. Taken together, our data imply that SNWE or SNPE downregulated the AKT/mTOR pathway in HUVECs and HepG2 cells, which lead to reduced tumor growth and angiogenesis.

The dynamical behavior of tumor growth model driven by Lévy noise terms is investigated. For α = 2 and β = 0, the process driven by white Lévy noise approach to the standard Gaussian white noise can be viewed in the analysis of the steady-state probability distribution and the mean first-passage time. When β → 0, the index α would increase the mean first-passage time as scale σ < 0 and shorten the mean first-passage time as scale σ > 0. A nonzero β parameter induces α to decrease the mean first-passage time. Thus analyzing the initial situation of tumor is very important to obtain more therapy time.

This paper presents some analytical criteria for local activity principle in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever–Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua's oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or nearby the edge of chaos.

This work extends a previous model that described the evolution of a tumor cord (a cylindrical arrangement of tumor cells, generally surrounded by necrosis, growing around a blood vessel of the tumor) under the activity of cell killing agents. In the present model we include the most relevant aspects of the dynamics of extracellular fluid, by computing the longitudinal average of the radial fluid velocity and of the pressure field. We still assume that the volume fraction occupied by the cells always keeps the same constant value everywhere in the cord. The necrotic region is treated as a "fluid reservoir". To improve the modelling of therapeutic treatment, we have subdivided the viable cell population into a proliferating and a quiescent subpopulation. The transitions between the two states are both permitted, and are regulated by rates depending on the local oxygen concentration. For simplicity, the rates of death induced by treatment are assumed to be known functions of the radial distance and time. Existence and uniqueness of the stationary state in the absence of treatment has been shown, as well as the existence and uniqueness of the evolution that arises following a cell killing treatment.

This work is concerned with a model which has been proposed to describe the growth of solid tumors. More precisely, the model under consideration provides a procedure to extract information about the growth dynamics from the analysis of the geometrical properties of the interface tumor-host tissue. In particular, it is suggested that the tumor boundary should evolve according to some stochastic evolution equation. This is herein compared with other dynamic equations related to the growth of rough surfaces, and a number of questions concerning the asymptotics of the corresponding solutions (and its relation to that of their deterministic counterparts) are discussed.

In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body.

Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena.

The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.

Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell–cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.

While a large and growing literature exists on mathematical and computational models of tumor growth, to date tumor growth models are largely qualitative in nature, and fall far short of being able to provide predictive results important in life-and-death decisions. This is largely due to the enormous complexity of evolving biological and chemical processes in living tissue and the complex interactions of many cellular and vascular constituents in living organisms. Several new technologies have emerged, however, which could lead to significant progress in this important area: (i) the development of so-called phase-field, or diffuse-interface models, which can be developed using continuum mixture theory, and which provide a general framework for modeling the action of multiple interacting constituents. These are based on generalizations of the Cahn–Hilliard models for spinodal decomposition, and have been used recently in certain tumor growth theories; (ii) the emergence of predictive computational methods based on the use of statistical methods for calibration, model validation, and uncertainty quantification; (iii) advances in imaging, experimental cell biology, and other medical observational methodologies; and (iv) the advent of petascale computing that makes possible the resolution of features at scales and at speeds that were unattainable only a short time in the past.

Here we develop a general phenomenological thermomechanical theory of mixtures that employs phase-field or diffuse interface models of surface energies and reactions and which provides a framework for generalizing existing theories of the types that are in use in tumor growth modeling. In principle, the framework provides for the effects of M solid constituents, which may undergo large deformations, and for the effect of N - M fluid constituents, which could include highly nonlinear, non-Newtonian fluids. We then describe several special cases which have the potential of providing acceptable models of tumor growth. We then describe the beginning steps of the development of Bayesian methods for statistical calibration, model validation, and uncertainty quantification, which, with further work, could produce a truly predictive tool for studying tumor growth. In particular, we outline the processes of statistical calibration and validation that can be employed to determine if tumor growth models, drawn from the broad class of models developed here, are valid for prediction of key quantities of interest critical to making decisions on various medical protocols. We also describe how uncertainties in such key quantities of interest can be quantified in ways that can be used to establish confidence in predicted outcomes.

The simulation of the creation and evolution of biological forms requires the development of computational methods that are capable of resolving their hierarchical, spatial and temporal complexity. Computations based on interacting particles, provide a unique computational tool for discrete and continuous descriptions of morphogenesis of systems ranging from the molecular to the organismal level. The capabilities of particle methods hinge on the simplicity of their formulation which enables the formulation of a unifying computational framework encompassing deterministic and stochastic models. In this paper, we discuss recent advances in particle methods for the simulation of biological systems at the mesoscopic and the macroscale level. We present results from applications of particle methods including reaction–diffusion on deforming surfaces, deterministic and stochastic descriptions of tumor growth and angiogenesis and discuss successes and challenges of this approach.

The mechanical tumor-growth model of Jackson and Byrne is analyzed. The model consists of nonlinear parabolic cross-diffusion equations in one space dimension for the volume fractions of the tumor cells and the extracellular matrix (ECM). It describes tumor encapsulation influenced by a cell-induced pressure coefficient. The global-in-time existence of bounded weak solutions to the initial-boundary-value problem is proved when the cell-induced pressure coefficient is smaller than a certain explicit critical value. Moreover, when the production rates vanish, the volume fractions converge exponentially fast to the homogeneous steady state. The proofs are based on the existence of entropy variables, which allows for a proof of the non-negativity and boundedness of the volume fractions, and of an entropy functional, which yields gradient estimates and provides a new thermodynamic structure. Numerical experiments using the entropy formulation of the model indicate that the solutions exist globally in time also for cell-induced pressure coefficients larger than the critical value. For such coefficients, a peak in the ECM volume fraction forms and the entropy production density can be locally negative.

In this paper we propose a model for the evolution of a tumor spheroid assuming a structure in which the central necrotic region contains an inner liquid core surrounded by dead cells that keep some mechanical integrity. This partition is a consequence of assuming that a finite delay is required for the degradation of dead cells into liquid. The phenomenological assumption of constant local volume fraction of cells is also made. The above structure is coupled with a mechanical two-phase model that views the cell component as a Bingham-like fluid and the extracellular liquid as an inviscid fluid. By imposing the continuity of the normal stress throughout the whole spheroid, we can describe the spheroid evolution and characterize the possible steady state. Depending on the values of mechanical parameters, the model predicts either an evolution toward the steady state or an unbounded growth. An existence and uniqueness result has been proved under suitable assumptions, along with some qualitative properties of the solution.

The development of predictive computational models of tumor initiation, growth, and decline is faced with many formidable challenges. Phenomenological models which attempt to capture the complex interactions of multiple tissue and cellular species must cope with moving interfaces of heterogeneous media and the sprouting vascular structures due to angiogenesis and their evolution. They must be able to deliver predictions consistent with events that take place at cellular scales, and they must faithfully depict biological mechanisms and events that are known to be associated with various forms of cancer. In the present work, a ten-species vascular model for the tumor growth is presented which falls within the framework of phase-field (or diffuse-interface) models suggested by continuum mixture theory. This framework provides for the simultaneous treatment of interactions of multiple evolving species, such as tumor cells, necrotic cell cores, nutrients, and other cellular and tissue types that exist and interact in living tissue. We develop a hybrid model that couples the tumor growth with sprouting through angiogenesis. The model is able to represent the branching of new vessels through coupling a discrete model for which the angiogenesis is started upon pre-defined conditions on the nutrient deprivation in the continuum model. Such conditions are represented by hypoxic cells that release tumor growth factors that ultimately trigger vascular growth. We discuss the numerical approximation of the model using mixed finite elements. The results of numerical experiments are also presented and discussed.

Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions.

We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases.

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matrix (ECM), together with chemotaxis, haptotaxis, apoptosis, nutrient distribution, and cell-to-matrix adhesion. We provide a rigorous proof of the existence of solutions of the coupled system with gradient-based and adhesion-based haptotaxis effects. In addition, we discuss finite element discretizations of the model, and we present the results of numerical experiments designed to show the relative importance and roles of various effects, including cell mobility, proliferation, necrosis, hypoxia, and nutrient concentration on the generation of MDEs and the degradation of the ECM.

In this paper, we introduce the problem of parameter identification for a coupled nonlocal Cahn–Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to be identified play the role of controls, and a suitable cost functional of tracking-type is introduced to account for the discrepancy between some a priori knowledge of the parameters and the controls themselves. The analysis is carried out for several classes of models, each one depending on a specific relaxation (of parabolic or viscous type) performed on the original system. First-order necessary optimality conditions are obtained on the fully relaxed system, in both the two- and three-dimensional cases. Then, the optimal control problem on the non-relaxed models is tackled by means of asymptotic arguments, by showing convergence of the respective adjoint systems and the minimization problems as each one of the relaxing coefficients vanishes. This allows obtaining the desired necessary optimality conditions, hence to solve the parameter identification problem, for the original PDE system in case of physically relevant double-well potentials.

We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions $d=2$ by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. $\mathrm{\Delta }t\le C{h}^2$, is required.

Moreover, in arbitrary dimensions $d\in \{2,3\}$, we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions $d\in \{2,3\}$ and in two dimensions $d=2$, where a CFL condition is required. Then, in two dimensions $d=2$, we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions $d=2$.