Model Of Avascular Tumor Growth Research Articles

Mathematical Model of Avascular Tumor Growth: Necrosis and Apoptosis

The first section of this paper reviews the research on the development of avascular tumours, which are tumours without a blood artery network to supply their cells with nutrients. The process of avascular tumours is then illustrated utilising classical PDE model, which is composed of an outer layer of actively dividing cells and an inner core of dead cells (referred to as necrotic cells) (called proliferating cells). The model takes into account both necrosis (a type of cell death brought on by changes in the cell’s microenvironment) and apoptosis (a sort of programmed cell death that takes place when a cell reaches the end of its normal lifespan). We offer a modified, simplified model and study how the relative importance of these mechanisms shifts as the tumour enlarges using a mix of numerical and analytical methods.

Two Numerical Solutions for Solving a Mathematical Model of the Avascular Tumor Growth

Objectives: Cancer which is one of the most challenging health problems overall the world is composed of various processes: tumorigenesis, angiogenesis, and metastasis. Attempting to understand the truth behind this complicated disease is one of the common objectives of many experts and researchers from different fields. To provide deeper insights any prognostic and/or diagnostic scientific contribution to this topic is so crucial. In this study, the avascular tumor growth model which is the earliest stage of tumor growth is taken into account from a mathematical point of view. The main aim is to solve the mathematical model of avascular tumor growth numerically. 
 Methods: This study has focused on the numerical solution of the continuum mathematical model of the avascular tumor growth described by Sharrett and Chaplin. Unlike the existing recent literature, the study has focused on the methods for the temporal domain. To obtain the numerical schemes the central difference method has been used in the spatial coordinates. This discretization technique has reduced the main partial differential equation into an ordinary differential equation which will be solved successively by two alternative techniques: the 4th order Runge-Kutta method (RK4) and the three-stage strongly-stability preserving Runge-Kutta method (SSP-RK3). 
 Results: The model has been solved by the proposed methods. The numerical results are discussed in both mathematical and biological angles. The biological compatibility of the methods is depicted in various figures. Besides biological outputs, the accuracies of the methods have been listed from a mathematical point of view. Furthermore, the rate of convergence of the proposed methods has also been discussed computationally.
 Conclusion: All recorded results are evidence that the proposed schemes are applicable for solving such models. Moreover, all exhibited figures have proved the biological compatibility of the methods. It is observed that the quiescent cells which are one of the most mysterious cells in clinics tend to become proliferative for the selected parameters.

A multi-scale agent-based model for avascular tumour growth

In this paper, we have developed a multi-scale, lattice-free, agent based model of avascular tumour growth in epithelial tissue. The model integrates different events to identify the underlying diversity within intracellular, cellular, and extracellular layer dynamics. The model considers every cell as an agent. A cellular agent may proliferate, spawns two identical daughter agents, or it may be transformed into other phenotypes during its life time depending on its internal proteins′ activity as well as its external microenvironment. In this context, a simplified age-structured cell cycle model is adopted from the existing literature. The model considers that the intracellular events are regulated by p27 gene expression. In this model, p27 protein controls the overall tumour growth dynamics. Moreover, p27 is controlled by the external oxygen and nutrients that are modelled with the reaction-diffusion equations. The model also considers several biophysical forces which directly effect on the tumour growth dynamics.This modelling framework offers biologically realistic outcomes and also covers important criteria of the hallmarks of cancer which include oxygen and nutrient consumptions, micro-environmental heterogeneity, tumour cell proliferation by avoiding growth suppressor signals, replication of tumour cells at an abnormally faster rate, and resistance of apoptosis. The avascular tumour growth model is validated with immunohistochemistry and histopathology data. The outcome of the proposed model is very close to the range of the patient data, which concludes that the model is capable enough to mimic these complex biophysical phenomena.

A numerical method based on the moving mesh for the solving of a mathematical model of the avascular tumor growth

Using adaptive mesh methods is one of the strategies to improve numerical solutions in time dependent partial differential equations. The moving mesh method is an adaptive mesh method, which, firstly does not need an increase in the number of mesh points, secondly reduces the concentration of points in the steady areas of the solutions that do not need a high degree of accuracy, and finally places the points in the areas, where a high degree of accuracy is needed. In this paper, we improved the numerical solutions for a three-phase model of avascular tumor growth by using the moving mesh method. The physical formulation of this model uses reaction-diffusion dynamics with the mass conservation law and appears in the format of the nonlinear system of partial differential equations based on the continuous density of three proliferating, quiescent, and necrotic cell categorizations. Our numerical results show more accurate numerical solutions, as compared to the corresponding fixed mesh method. Moreover, this method leads to the higher order of numerical convergence.

Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate function

Tumor growth models have proved as an important tool to produce an engineering background for cancer therapy either by designing therapeutic procedures combined with control engineering or by using the models for simulations and evaluation of treatment procedures. Mathematical modeling has been critical for the description of tumor growth, which is a highly complex process, as a finely chiseled tumor growth model always outlines the measurements and the physiological processes of the tumors. Therefore, a mathematical model involving partial differential equations for the growth of tumor have been studied, modified and developed in this paper. It is based on the deterministic model of an avascular tumor growth which is framed in a system of nonlinear coupled PDEs, describing the proliferating, quiescent, necrotic, and surrounding cells densities, accompanied by a supply of the nutrients. These equations are explained numerically by the Finite Difference Method. Furthermore, the simulations are carried out by introducing the Gompertz growth rate function for mitosis rate function g(c) of proliferating cells in the model. The discretization forms a system of coupled nonlinear difference equations which are solved at each iteration. The algorithm is implemented sequentially in MATLAB 2018a and numerical results suggest that the employed model is an authentic tool for analyzing the dynamics of tumor.

A New Mathematical Model for Controlling Tumor Growth Based on Microenvironment Acidity and Oxygen Concentration.

Hypoxia and the pH level of the tumor microenvironment have a great impact on the treatment of tumors. Here, the tumor growth is controlled by regulating the oxygen concentration and the acidity of the tumor microenvironment by introducing a two-dimensional multiscale cellular automata model of avascular tumor growth. The spatiotemporal evolution of tumor growth and metabolic variations is modeled based on biological assumptions, physical structure, states of cells, and transition rules. Each cell is allocated to one of the following states: proliferating cancer, nonproliferating cancer, necrotic, and normal cells. According to the response of the microenvironmental conditions, each cell consumes/produces metabolic factors and updates its state based on some stochastic rules. The input parameters are compatible with cancer biology using experimental data. The effect of neighborhoods during mitosis and simulating spatial heterogeneity is studied by considering multicellular layer structure of tumor. A simple Darwinist mutation is considered by introducing a critical parameter (Nmm) that affects division probability of the proliferative tumor cells based on the microenvironmental conditions and cancer hallmarks. The results show that Nmm regulation has a significant influence on the dynamics of tumor growth, the growth fraction, necrotic fraction, and the concentration levels of the metabolic factors. The model not only is able to simulate the in vivo tumor growth quantitatively and qualitatively but also can simulate the concentration of metabolic factors, oxygen, and acidity graphically. The results show the spatial heterogeneity effects on the proliferation of cancer cells and the rest of the system. By increasing Nmm, tumor shrinkage and significant increasing in the oxygen concentration and the pH value of the tumor microenvironment are observed. The results demonstrate the model's ability, providing an essential tool for simulating different tumor evolution scenarios of a patient and reliable prediction of spatiotemporal progression of tumors for utilizing in personalized therapy.

Global asymptotic stability of positive steady states of a solid avascular tumor growth model with time delays

In this work, global stability of a free boundary problem modeling solid avascular tumor growth is studied. The model is considered with time delays during the proliferation process. We prove that the unique positive constant steady state is globally asymptotically stable under some assumptions. The proof uses the comparison principle and the iteration method.

Self-influenced growth through evolving material inhomogeneities

We reformulate a model of avascular tumour growth in which the tumour tissue is studied as a biphasic medium featuring an interstitial fluid and a solid phase. The description of growth relies on two fundamental features: one of those is given by the mass transfer among the constituents of the phases, which is taken into account through source and sink terms; the other one is the multiplicative decomposition of the deformation gradient tensor of the solid phase, with the introduction of a growth tensor, which represents the growth-induced structural changes of the tumour. In general, such tensor is non-integrable, and it may allow to define a Levi-Civita connection with non-trivial curvature. Moreover, its evolution is related to the source and sink of mass of the solid phase through an evolution equation. Our goal is to study how growth can be influenced by the inhomogeneity of the growth tensor. To this end, we study the evolution of the latter, as predicted by two different models. In the first one, the dependence of the growth tensor on the tumour’s material points is not explicitly considered in the evolution equation. In the second model, instead, the inhomogeneity of the growth tensor is resolved explicitly by introducing the curvature associated with it into the evolution equation. We perform numerical simulations to compare the results produced by these two models, and evaluate a possible role of the material inhomogeneities on growth.

A new method for shrinking tumor based on microenvironmental factors: Introducing a stochastic agent-based model of avascular tumor growth

In this paper, by using cellular automata formalism, a two-dimensional stochastic agent-based model for avascular tumor growth is presented. The model is based on biological assumptions, physical structure, agents and their states, and transition rules. The interaction between immune and cancer cells is probabilistic. The immune cell recruitment, which usually occurs after the detection of tumor cells, is considered. The insights gain in the proposed work to model the immune-tumor interaction is of great importance to control the tumor growth via micro-environment factors. The parameters used in the present model are in compatible with cancer biology using in vivo experimental data. The results show that the proposed model not only is able to simulate the tumor growth graphically, but also the in vivo tumor growth quantitatively and qualitatively. In this paper, by introducing a new concept; critical point, a new idea is proposed to shrink the tumor or slow down its growth rate. The tumor will grow slowly if the division probability of the proliferative tumor cells depends on the microenvironmental conditions. The proposed idea has been validated using an in silico simulation.

A DCE-MRI Driven 3-D Reaction-Diffusion Model of Solid Tumor Growth.

Predicting tumor growth and its response to therapy remains a major challenge in cancer research and strongly relies on tumor growth models. In this paper, we introduce, calibrate, and verify a novel image-driven reaction-diffusion model of avascular tumor growth. The model allows for proliferation, death and spread of tumor cells, and accounts for nutrient distribution and hypoxia. It is constrained by longitudinal time series of dynamic contrast-enhancement-MRI images. Tumor specific parameters are estimated from two early time points and used to predict the spatio-temporal evolution of the tumor volume and cell densities at later time points. We first test our parameter estimation approach on synthetic data from 15 generated tumors. Our in silico study resulted in small volume errors (<5%) and high Dice overlaps (>97%), showing that model parameters can be successfully recovered and used to accurately predict the tumor growth. Encouraged by these results, we apply our model to seven pre-clinical cases of breast carcinoma. We are able to show promising preliminary results, especially for the estimation for early time points. Processes like angiogenesis and apoptosis should be included to further improve predictions for later time points.

Finite Element Modeling of Avascular Tumor Growth Using a Stress-Driven Model

Tumor growth being a multistage process has been investigated from different aspects. In the present study, an attempt is made to represent a constitutive-structure-based model of avascular tumor growth in which the effects of tensile stresses caused by collagen fibers are considered. Collagen fibers as a source of anisotropy in the structure of tissue are taken into account using a continuous fiber distribution formulation. To this end, a finite element modeling is implemented in which a neo-Hookean hyperelastic material is assigned to the tumor and its surrounding host. The tumor is supplied with a growth term. The growth term includes the effect of parameters such as nutrient concentration on the tumor growth and the tumor's solid phase content in the formulation. Results of the study revealed that decrease of solid phase is indicative of decrease in growth rate and the final steady-state value of tumor's radius. Moreover, fiber distribution affects the final shape of the tumor, and it could be used to control the shape and geometry of the tumor in complex morphologies. Finally, the findings demonstrated that the exerted stresses on the tumor increase as time passes. Compression of tumor cells leads to the reduction of tumor growth rate until it gradually reaches an equilibrium radius. This finding is in accordance with experimental data. Hence, this formulation can be deployed to evaluate both the residual stresses induced by growth and the mechanical interactions with the host tissue.

An avascular tumor growth model based on porous media mechanics and evolving natural states

Mechanical factors play a major role in tumor development and response to treatment. This is more evident for tumors grown in vivo, where cancer cells interact with the different components of the host tissue. Mathematical models are able to characterize the mechanical response of the tumor and can provide a better understanding of these interactions. In this work, we present a biphasic model for tumor growth based on the mechanics of fluid-saturated porous media. In our model, the porous medium is identified with the tumor cells and the extracellular matrix, and represents the system’s solid phase, whereas the interstitial fluid constitutes the liquid phase. A nutrient is transported by the fluid phase, thereby supporting the growth of the tumor. The internal reorganization of the tissue in response to mechanical and chemical stimuli is described by enforcing the multiplicative decomposition of the deformation gradient tensor associated with the solid phase motion. In this way, we are able to distinguish the contributions of growth, rearrangement of cellular bonds, and elastic distortions which occur during tumor evolution. Results are shown for three cases of biological interest, addressing (i) the growth of a tumor spheroid in the culture medium, and (ii) the evolution of an avascular tumor growing first in a soft host tissue and then (iii) in a three-dimensional heterogeneous region. We analyze the dependence of tumor development on the mechanical environment, with particular focus on cell reorganization and its role in stress relaxation.

An agent-based model of avascular tumor growth: Immune response tendency to prevent cancer development

Mathematical and computational models are of great help to study and predict phenomena associated with cancer growth and development. These models may lead to introduce new therapies or improve current treatments by discovering facts that may not be easily discovered in clinical experiments. Here, a new two-dimensional (2D) stochastic agent-based model is presented for the spatiotemporal study of avascular tumor growth based on the effect of the immune system. The simple decision-making rules of updating the states of each agent depend not only on its intrinsic properties but also on its environment. Tumor cells can interact with both normal and immune cells in their Moore neighborhood. The effect of hypoxia has been checked off by considering non-mutant proliferative tumor cells beside mutant ones. The recruitment of immune cells after facing a mass of tumor is also considered. Results of the simulations are presented before and after the appearance of immune cells in the studied tissue. The growth fraction and necrotic fraction are used as output parameters along with a 2D graphical growth presentation. Finally, the effect of input parameters on the output parameters generated by the model is discussed. The model is then validated by an in vivo study published in medical articles. The results show a multi-spherical tumor growth before the immune system strongly involved in competition with tumor cells. Besides, considering the immune system in the model shows more compatibility with biological facts. The effect of the microenvironment on the proliferation of cancer and immune cells is also studied.

A chemical energy approach of avascular tumor growth: multiscale modeling and qualitative results.

In the present manuscript we propose a lattice free multiscale model for avascular tumor growth that takes into account the biochemical environment, mitosis, necrosis, cellular signaling and cellular mechanics. This model extends analogous approaches by assuming a function that incorporates the biochemical energy level of the tumor cells and a mechanism that simulates the behavior of cancer stem cells. Numerical simulations of the model are used to investigate the morphology of the tumor at the avascular phase. The obtained results show similar characteristics with those observed in clinical data in the case of the Ductal Carcinoma In Situ (DCIS) of the breast.

Two Dimensional Mathematical Model of Tumor Angiogenesis: Coupling of Avascular Growth and Vascularization

Introduction As a tumor grows, the demand for oxygen and nutrients increases and it grows further if acquires the ability to induce angiogenesis. In this study, we aimed to present a two-dimensional continuous mathematical model for avascular tumor growth, coupled with a discrete model of angiogenesis. Materials and Methods In the avascular growth model, tumor is considered as a single mass, which uptakes oxygen through diffusion and invades the extracellular matrix (ECM). After the tumor reaches its maximum size in the avascular growth phase, tumor cells may be in three different states (proliferative, quiescent and apoptotic), depending on oxygen availability. Quiescent cells are assumed to secrete tumor angiogenic factors, which diffuse into the surrounding tissue until reaching endothelial cells. The mathematical model for tumor angiogenesis is consisted of a five-point finite difference scheme to simulate the progression of endothelial cells in ECM and their penetration into the tumor. Results The morphology of produced networks was investigated, based on various ECM degradation patterns. The generated capillary networks involved the rules of microvascular branching and anastomosis. Model predictions were in qualitative agreement with experimental observations and might have implications as a supplementary model to facilitate mathematical analyses for anti-cancer therapies. Conclusion Our numerical simulations could facilitate the qualitative comparison between three layers of tumor cells, their TAF-producing abilities and subsequent penetration of micro-vessels in order to determine the dynamics of microvascular branching and anastomosis in ECM and three different parts of the tumor.

Mathematical modeling of anisotropic avascular tumor growth

Cancer represents one the most challenging problems in medicine and biology nowadays, and is being actively addressed by many researchers from different areas of knowledge. The increasing development of sophisticated mathematical models and computer-based procedures has had a positive impact on our understanding of cancer-related mechanisms and the design of anticancer treatment strategies. However, further investigation and experimentation are still required to completely elucidate the tumor-associated mechanical responses, as well as the effect of mechanical forces on the net tumor growth. In this work we develop a theoretical framework in the context of continuum mechanics to investigate the anisotropic growth of avascular tumor spheroids. To that end, a specific anisotropic growth deformation tensor is considered, which also describes an isotropic growth law as a particular case. Mixtures theory and the notion of multiple natural configurations are then used to formulate a mathematical model of avascular tumor growth. More precisely, mass, momentum balance and nutrients diffusion equations are derived, where solid tumors are assumed as hyperelastic and compressible materials. Moreover, mechanical interactions with a rigid extracellular matrix (ECM) are considered, and the mechanical modulation of growing tumors in a rigid surrounding tissue is investigated by means of numerical simulations. Finally, the model results are compared with experimental data previously reported in the literature.

A 3D COUPLED MATHEMATICAL MODEL FOR THE GROWTH OF AVASCULAR SOLID TUMOR

We develop a coupled mathematical model of avascular tumor growth based on porous media mechanics. This comprises of the migration of tumor cells (TCs), the degradation of extracellular matrix (ECM), the transport of matrix-degrading enzymes (MDEs), the seepage of tissue fluid and the supplement and consumption of oxygen. The simulation that a solid tumor grows in the micro-environment composed of the pre-existing capillaries and the surrounding tissues, and the specific property of varying porosity with the growth of TCs in a tumor micro-environment are taken into account. We propose functional coefficients for fluid seepage and oxygen diffusion, and incorporate the convection–diffusion of oxygen and the convection of MDEs. From this modified model the main findings include: first, a solid tumor originating in the inlet region undergoes necrosis in the outlet region because of a low supply of oxygen, while a solid tumor originating in the outlet region undergoes necrosis at the primary site because of overconsumption of oxygen; second, tumors further from capillaries grow faster than tumors close to adjacent capillaries; third, the pre-existing capillaries cause some impact to the transport of those chemical factors involved in tumor growth, and further affect tumor migration and necrosis.

A System of Coupled Nonlinear Partial Differential Equations Describing Avascular Tumour Growth Are Solved Numerically Using Parallel Programming to Assess Computational Speedup

The challenging issues of cancer prevention and cure lie in the need for a more detailed knowledge of the internal processes and mechanisms of tumour growth. We present a mathematical model of avascular tumour growth formulated in a system of coupled nonlinear PDEs. The interaction between the surrounding tissue and cell motility of the developing tumour are also included to more realistic replicate an in-vivo environment. The mathematical model is solved using finite difference methods and implemented in the C programming language. The CUDA programming framework is then introduced to allow a parallelisation of the sequential C implementation. Results show a dramatic Speedup of around 26x that of conventional implementations in C. Such increased computational efficiency clearly highlights the possibility of improvements in the numerical simulation of more complex mathematical models of 2D and 3D tumour growth, such as angiogenesis and vascularisation. Parallelisation of such models can greatly facilitate researchers, clinicians and oncologists by performing time-saving in-silico experiments that have the potential to highlight new cancer treatments and therapies without the need for the use of valuable resources associated with excessive pre-clinical trials.

Patterns in melanocytic lesions: impact of the geometry on growth and transport inside the epidermis.

In glabrous skin, nevi and melanomas exhibit pigmented stripes during clinical dermoscopic examination. They find their origin in the basal layer geometry which periodically exhibits ridges, alternatively large (limiting ridges) and thin (intermediate ridges). However, nevus and melanoma lesions differ by the localization of the pigmented stripes along furrows or ridges of the epidermis surface. Here, we propose a biomechanical model of avascular tumour growth which takes into account this specific geometry in the epidermis where both kinds of lesions first appear. Simulations show a periodic distribution of tumour cells inside the lesion, with a global contour stretched out along the ridges. In order to be as close as possible to clinical observations, we also consider the melanin transport by the keratinocytes. Our simulations show that reasonable assumptions on melanocytic cell repartition in the ridges favour the limiting ridges of the basal compared with the intermediate ones in agreement with nevus observations but not really with melanomas. It raises the question of cell aggregation and repartition of melanocytic cells in acral melanomas and requires further biological studies of these cells in situ.

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) [J. Math. Biol. 45(2) (2002) 125–152]. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.