Numerical simulation of solid tumor angiogenesis with Endostatin treatment： a combined analysis of inhibiting effect of anti-angiogenic factor and micro mechanical environment of extracellular matrix

To investigate the influence of anti-angiogenesis drug Endostatin on solid tumor angiogenesis, a mathematical model of tumor angiogenesis was developed with combined influences of local extra-cellular matrix mechanical environment, and the inhibiting effects of Angiostatin and Endostatin. Simulation results show that Angiostatin and Endostatin can effectively inhibit the process of tumor angiogenesis, and decrease the number of blood vessels in the tumor. The present model could be used as a valid theoretical method in the investigation of anti-angiogenic therapy of tumors.     

Coupled modeling of tumour angiogenesis,tumour growth,and blood perfusion

This paper proposes a more realistic mathematical simulation method to investigate the dynamic process of tumour angio-genesis by fully coupling the vessel growth, tumour growth and associated blood perfusion. The tumour growth and angiogenesis are coupled by the chemical microenvironment and the cell-matrix interaction. The haemodynamic calculation is carried out on the new vasculature, and an estimation of vessel collapse is made according to the wall shear stress criterion. The results are consistent with physiological observations, and further confirm the application of the coupled model feedback mechanism. The model is available to examine the interactions between angiogenesis and tumour growth, to study the change in the dynamic process of chemical environment and the vessel remodeling.     

3D numerical simulation of avascular tumour growth: effect of hypoxic micro-environment in host tissue

A three-dimensional （3D） mathematical model of tumour growth at the avascular phase and vessel remodelling in host tissues is proposed with emphasis on the study of the interactions of tumour growth and hypoxic micro-environment in host tissues. The hybrid based model includes the continuum part, such as the distributions of oxygen and vascular endothelial growth factors （VEGFs）, and the discrete part of tumour cells （TCs） and blood vessel networks. The simulation shows the dynamic process of avascular tumour growth from a fewinitial cells to an equilibrium state with varied vessel networks. After a phase of rapidly increasing numbers of the TCs, more and more host vessels collapse due to the stress caused by the growing tumour. In addition, the consumption of oxygen expands with the enlarged tumour region. The study also discusses the effects of certain factors on tumour growth, including the density and configuration of pre- existing vessel networks and the blood oxygen content. The model enables us to examine the relationship between early tumour growth and hypoxic micro-environment in host tissues, which can be useful for further applications, such as tumour metastasis and the initialization of tumour angiogenesis.     

内皮抑素抑制肿瘤血管生成的数值模拟

数值模拟抗血管生成药物内皮抑素对肿瘤血管生成的抑制效应. 建立内皮抑素作用下肿瘤内外血管生成的二维、三维离散数学模型,模型中考虑内皮抑素的抑制作用、内皮细胞自身的增殖、促血管生成因子TAF和Fibronectin对内皮细胞产生的趋化性和趋触性以及内皮细胞自身扩散引起的随机性运动,数值模拟肿瘤内外微血管网的生成过程. 模拟结果表明,抗血管生成药物内皮抑素对肿瘤内外血管生成的速度、成熟度以及血管分支数量均有明显的抑制作用,从而有效地抑制肿瘤新生血管的形成. 该模型能够较好地模拟内皮抑素对肿瘤血管内皮细胞迁移与增殖的抑制效应,为临床抗血管生成治疗肿瘤提供有益的信息.      

利用血管生成模型计算实体肿瘤内的血液动力学

肿瘤血管生成(Tumor-induced Angiogenesis)是指在实体肿瘤细胞诱导下毛细血管的生长以及肿瘤中血液微循环的建立。肿瘤内血液、组织液等流体流动在肿瘤药物输运过程中扮演着重要作用,而这些流动受到肿瘤内微血管网络结构的直接影响。目前要获得精确的肿瘤内外的毛细血管拓扑结构存在一定困难,因此给肿瘤内的血液动力学研究带来困难。本文根据肿瘤内外的复杂生理特性,建立肿瘤内外血管生成的二维离散模型,在获得相对真实的毛细血管网络拓扑结构基础上对肿瘤内的血液动力学进行初步计算,数值计算的结果加深了对肿瘤的复杂生理特性的理解,同时也给肿瘤内的药物输运给予一定的提示。     

Symmetry Analysis and Numerical Modelling of Invasion by Malignant Tumour Tissue

We develop a model for the early stages of malignant tumour invasion dueto random motility, cellular proliferation, proteolysis and haptotaxis.At early times in the absence of tumour cell diffusion, a compressedtumour layer is evident. Transient protease production-decay dynamicsand diffusion, must be present in order for the tumour concentrationpeak to be smoothed to realistic levels.We demonstrate that invasion profiles asymptotically evolve totravelling wave solutions and that kink-like profiles, previouslythought to be due to contact inhibition and haptotaxis, can equally beexplained by cellular diffusion with a decreasing nonlinear diffusivity.As well as generalising the model and examining its robustness, afull Lie Symmetry classification is carried out.     

实体肿瘤血管生成和肿瘤生长的耦合研究

研究宏观尺度上肿瘤向外浸润式生长的动态变化对微观尺度上肿瘤内部微血管网生成过程产生的影响。建立肿瘤组织生长动力学模型与促血管生成的二维离散数学模型,模型考虑促血管生成因子诱导下血管内皮细胞的随机性、趋化性和趋触性运动以及血管内皮细胞与胞外基质的相互作用,并且肿瘤组织的生长满足经典的Compertz函数形式,通过耦合研究模拟了肿瘤组织动态生长过程中微脉管系统生成的时空演化,数值生成肿瘤动态生长下内外异构的微血管网和肿瘤内部分层的网络结构。该模型可以产生相对真实的具有接近肿瘤病理生理特性的血管网,为临床研究提供有益的信息。     

Two-dimensional discrete mathematical model of tumor-induced angiogenesis

A 2D discrete mathematical model of a nine-point finite difference scheme is built to simulate tumor-induced angiogenesis. Nine motion directions of an individual endothelial cell and two parent vessels are extended in the present model. The process of tumor-induced angiogenesis is performed by coupling random motility, chemotaxis, and haptotaxis of endothelial cell in different mechanical environments inside and outside the tumor. The results show that nearly realistic tumor microvascular networks with neoplastic pathophysiological characteristics can be generated from the present model. Moreover, the theoretical capillary networks generated in numerical simulations of the discrete model may provide useful information for further clinical research.     

A model for cell migration in tumour growth

Tumour growth results, in particular, from cell–cell interaction and tumour and healthy cell proliferation. The complexity of the cellular microenvironment may then be framed within the theory of mixtures by looking at cell populations as the constituents of a mixture. In this paper the balance equations are reviewed to account for directionality onto a collective migration of the tumour cell population, via an attractive force of the chemotactic type, in addition to the customary pressure term. The density of tumour cells turns out to be governed by a hyperbolic differential equation. By neglecting, as usual, the inertia term it follows that the density satisfies a backward, or forward, diffusion equation according as the attraction, or pressure effect, prevails. Uniqueness of the solution to the backward equation is investigated and a family of solutions is described. An estimate is given for the growth rate of a tumour profile.     

A multiphase model of tumour segregation in situ by a heterogeneous extracellular matrix

Normal and tumour cells live in a fibrous environment that is often very heterogeneous, even characterized by the presence of basal membranes and regions with high density of collagen fibres that physiologically comparmentalize cells in well defined regions, as for in situ tumours. In case of metastatic tumours these porous structures are instead invaded by cancer cells. The aim of this paper is to propose a multiphase model that is able to describe cell segregation by thick porous structures and to relate the transition rule that determines whether cells will pass or not to microscopic characteristics of the cells, such as the stiffness of their nucleus, their adhesive and traction abilities, the relative dimension of their nucleus with respect to the dimension of the pores of the extra-cellular matrix.     

3D numerical study of tumor blood perfusion and oxygen transport during vascular normalization

The changes of blood perfusion and oxygen transport in tumors during tumor vascular normalization are studied with 3-dimensional mathematical modeling and numerical simulation. The models of tumor angiogenesis and vascular-disrupting are used to simulate "un-normalized" and "normalized" vasculatures. A new model combining tumor hemodynamics and oxygen transport is developed. In this model, the intravasculartransvascular-interstitial flow with red blood cell(RBC) delivery is tightly coupled, and the oxygen resource is produced by heterogeneous distribution of hematocrit from the flow simulation. The results show that both tumor blood perfusion and hematocrit in the vessels increase, and the hypoxia microenvironment in the tumor center is greatly improved during vascular normalization. The total oxygen content inside the tumor tissue increases by about 67%, 51%, and 95% for the three approaches of vascular normalization,respectively. The elevation of oxygen concentration in tumors can improve its metabolic environment, and consequently reduce malignancy of tumor cells. It can also enhance radiation and chemotherapeutics to tumors.     

Numerical simulation of tumor-induced angiogenesis influenced by the extra-cellular matrix mechanical environment

To investigate tumor-induced angiogenesis under the influence of the mechanical environments inside and outside the tumor, mathematical model of tumor angiogenesis was developed. In the model, extra-cellular matrix （ECM） was treated as a thin plane. The displacement of ECM is obtained from the force balance equation consisted of the ECs traction, the ECM visco-elastic forces and the exter- nal forces. Simulation results show that a layered capillary network is obtained with a well vascularized region at the periphery of the tumor. The present model can be used as a valid theoretical method in the basic researches in tumorinduced angiogenesis.     

Numerical simulation of avascular tumor growth based on p27 gene regulation

A multi-scale continuous-discrete model based on the effects of the p27 gene control is built to simulate the avascular tumor growth. At the tissue level, the continuous Eulerian model is adopted to determine the distribution of the concentration of oxygen, the extracellular matrix (ECM), and the matrix-degradative enzyme (MDE). At the cellular level, the discrete Lagrangien model is adopted to determine the movement, the proliferation, and the death of single tumor cells (TCs). At the genetic level, whether a cell is committed to mitosis is determined by solving a set of equations modeling the effects of the p27 gene control. The avascular morphological evolution of the solid tumor growth is simulated, including the radius of the solid tumor, the number of the TCs, the oxygen distribution over time, and the inhibiting effect of the up-regulating p27 gene expression.     

Simulation of tumor microvasculature and microenvironment response to anti-angiogenic treatment by angiostatin and endostatin

The effects of anti-angiogenesis treatment by angiostatin and endostatin on normalization of tumor microvasculature and microenvironment are investigated, based on mathematical modeling and numerical simulation of tumor anti-angiogenesis and tumor haemodynamics. The results show that after anti-angiogenesis treatment: (i) the proliferation, growth, and branching of neo-vessels are effectively inhibited, and the extent of vascularization in tumors is accordingly reduced. (ii) the overall blood perfusion inside of tumor is declined, the plateau of tumor interstitial fluid pressure (IFP) is relieved, the interstitial fluid oozing out from the tumor periphery into the surrounding normal tissue is reduced, the reduction of overall extravasation across vasculature to tumor interstium is much less than the decreased overall blood perfusion, due to the decline of IFP, the intravasations is remarkablely effected by the change, in some cases there are no intravasation flow appear.     

Numerical simulation of blood flow and interstitial fluid pressure in solid tumor microcirculation based on tumor-induced angiogenesis

A coupled intravascular–transvascular–interstitial fluid flow model is developed to study the distributions of blood flow and interstitial fluid pressure in solid tumor microcirculation based on a tumor-induced microvascular network. This is generated from a 2D nine-point discrete mathematical model of tumor angiogenesis and contains two parent vessels. Blood flow through the microvascular network and interstitial fluid flow in tumor tissues are performed by the extended Poiseuille’s law and Darcy’s law, respectively, transvascular flow is described by Starling’s law; effects of the vascular permeability and the interstitial hydraulic conductivity are also considered. The simulation results predict the heterogeneous blood supply, interstitial hypertension and low convection on the inside of the tumor, which are consistent with physiological observed facts. These results may provide beneficial information for anti-angiogenesis treatment of tumor and further clinical research. The project supported by the National Natural Science Foundation of China (10372026).     

Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations

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 of decoupling the equations describing the growth of the tumour mass from those describing the flow of the interstitial fluid.     

Machine learning of synaptic structure with neurons to promote tumor growth

In this paper, we use machine learning techniques to form a cancer cell model that displays the growth and promotion of synaptic and electrical signals. Here, such a technique can be applied directly to the spiking neural network of cancer cell synapses. The results show that machine learning techniques for the spiked network of cancer cell synapses have the powerful function of neuron models and potential supervisors for different implementations. The changes in the neural activity of tumor microenvironment caused by synaptic and electrical signals are described. It can be used to cancer cells and tumor training processes of neural networks to reproduce complex spatiotemporal dynamics and to mechanize the association of excitatory synaptic structures which are between tumors and neurons in the brain with complex human health behaviors.     

基于自然增长的细胞群粘附数值模拟

细胞群通过表面的粘附分子不断聚集的过程称为细胞群粘附, 是生物学许多研究领域(早期的胚胎发育、组织的新陈代谢以及肿瘤生长等)的基础. 考虑细胞群的自由运动和细胞群互相之间的粘附力作用, 并引入Logistic模型来描述细胞群自身的自然增长, 从宏观的角度研究细胞群粘附现象, 构建出一个细胞群粘附模型; 其数值模拟结果与实验结果较吻合, 能定性地对不同种类细胞群粘附的最终形态做出预判.      

On a Nonlinear Model for Tumor Growth: Global in Time Weak Solutions

We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum Ω with boundary ?Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation.     

Hybrid discrete-continuum model of tumor growth considering capillary points

A hybrid discrete-continuum model of tumor growth in the avascular phase considering capillary points is established. The influence of the position of capillary points on tumor growth is also studied by simulation. The results of the dynamic tumor growth and the distribution of oxygen, matrix-degrading enzymes, and extracellular matrixconcentration in the microenvironment with respect to time are shown by graphs. The relationships between different oxygenated environments and the numbers of surviving, dead, proliferative, and quiescent tumor cells are also investigated.