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.     

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.     

Biochemomechanical poroelastic theory of avascular tumor growth

Tumor growth is a complex process involving genetic mutations, biochemical regulations, and mechanical deformations. In this paper, a thermodynamics-based nonlinear poroelastic theory is established to model the coupling among the mechanical, chemical, and biological mechanisms governing avascular tumor growth. A volumetric growth law accounting for mechano-chemo-biological coupled effects is proposed to describe the development of solid tumors. The regulating roles of stresses and nutrient transport in the tumor growth are revealed under different environmental constraints. We show that the mechano-chemo-biological coupling triggers anisotropic and heterogeneous growth, leading to the formation of layered structures in a growing tumor. There exists a steady state in which tumor growth is balanced by resorption. The influence of external confinements on tumor growth is also examined. A phase diagram is constructed to illustrate how the elastic modulus and thickness of the confinements jointly dictate the steady state of tumor volume. Qualitative and quantitative agreements with experimental observations indicate the developed model is capable of capturing the essential features of avascular tumor growth in various environments.     

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.     

The interplay between stress and growth in solid tumors

A number of biological phenomena are interlaced with classical mechanics. In this review we discuss the role of mechanics in tumor growth, namely the avascular phase of solid tumors. While a growing mass produces a traction of the surrounding tissues, a feedback mechanism controls the proliferation of the malignant cells depending on the tensional state. The formalism of continuum mechanics, possibly accompanied by numerical simulations, is able to shed light on biological controversial subjects. The converse is also true: non-standard mechanical problems suggest new challenging theoretical questions.     

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。     

Multi-level DEM study on silo discharge behaviors of non-spherical particles

The silo discharge of non-spherical particles has been widely practiced in engineering processes, yet the understanding of multi-level mechanisms during solid transportation is still lacking. In this study, a high-fidelity super-ellipsoid Discrete Element Method (DEM) model is established to investigate the discharge behaviors of non-spherical particles with different size distributions. After the comprehensive model validations, we investigated the effects of particle shape (aspect ratio and particle sharpness) on the particle level discharge behaviors. The discharge rates of the ellipsoid particles used in the current work are larger than the spherical particles due to the larger solid fraction. The discharge rates of the cuboid-like particles are determined by the combined effect of the solid fraction and the contact force. Parcel level data show that the translational movements of the ellipsoid particles are more ordered, which is supported by the global level data. Strong correlations exist between the particle level and parcel level data, especially the ellipsoid particles and the large particles in the polydispersed cases.     

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.     

Superquadric DEM-SPH coupling method for interaction between non-spherical granular materials and fluids

The research on the coupling method of non-spherical granular materials and fluids aims to predict the particle–fluid interaction in this study. A coupling method based on superquadric elements is developed to describe the interaction between non-spherical solid particles and fluids. The discrete element method (DEM) and the smoothed particle hydrodynamics (SPH) are adopted to simulate granular materials and fluids. The repulsive force model is adopted to calculate the coupling force and then a contact detection method is established for the interaction between the superquadric element and the fluid particle. The contact detection method captures the shape of superquadric element and calculates the distance from the fluid particle to the surface of superquadric element. Simulation cases focusing on the coupling force model, energy transfer, and large-scale calculations have been implemented to verify the validity of the proposed coupling method. The coupling force model accurately represents the water entry process of a spherical solid particle, and reasonably reflects the difference of solid particles with different shapes. In the water entry process of multiple solid particles, the total energy of the water entry process of multiple solid particles tends to be stable. The collapse process of the partially submerged granular column is simulated and analyzed under different parameters. Therefore, this coupling method is suitable to simulate fluid–particle systems containing solid particles with multiple shapes.     

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.     

Three dimensional microstructural effects on plane strain ductile crack growth

Ductile crack growth under mode I, plane strain, small scale yielding conditions is analyzed. Overall plane strain loading is prescribed, but a full 3D analysis is carried out to model three dimensional microstructural effects. An elastic-viscoplastic constitutive relation for a porous plastic solid is used to model the material. Two populations of second-phase particles are represented, large inclusions with low strength, which result in large voids near the crack tip at an early stage, and small second-phase particles, which require large strains before cavities nucleate. The larger inclusions are represented discretely and the effects of different three dimensional distributions on the crack path and on the overall crack growth rate are analyzed. For comparison purposes, a two dimensional distribution of cylindrical inclusions is analyzed. Crack growth occurs off the initial crack plane in all 3D computations, whereas straight ahead crack growth occurs with the two dimensional cylindrical inclusions. As a consequence, the three dimensional distributions of spherical inclusions exhibit an increased crack growth resistance as compared to the two dimensional distribution of cylindrical inclusions.     

力场-化学场耦合作用对含裂纹固体电解质力学行为的研究

为了研究力场-化学场耦合作用下的含裂纹电解质的断裂问题,本文构造了耦合情况下力场和浓度场的本构关系,并由这些本构关系建立了力场-化学场耦合问题的有限元方程。通过具体的算例,进一步探讨了裂纹尖端应力场和氧空位浓度分布的耦合作用对GDC(氧化钆掺杂的氧化铈)力学行为的影响,发现在耦合作用下,裂尖应力场对氧空位的分布有明显的诱导作用。     

基于人工神经网络的非均匀固相应力模型

最小多尺度理论EMMS已经被引入多相质点网格法MP-PIC中,建立了非均匀EMMS固相应力模型.但现有的非均匀固相应力模型计算中,中间步骤繁琐且花费时间长.采用人工拟合的方式能获得非均匀固相应力表达式,但需要人为确定拟合变量和拟合函数,且针对于非均匀固相应力这种高度非线性函数所得到的拟合精度不高、用于MP-PIC模拟的结果相比原EMMS固相应力模型结果存在偏差.针对上述问题,本文提出通过机器学习的方法,规避对固相体积分数的局部分布情况的表征,并提出和建立能考虑颗粒浓度详细分布的人工神经网络ANN固相应力模型.首先,基于局部颗粒浓度和颗粒非均匀分布指数建立了双变量的ANN固相应力模型;进一步将当前网格及其周边网格颗粒浓度组成的序列来详细表征颗粒浓度分布,并建立相应的ANN固相应力模型.然后,将两种模型与EMMS固相应力模型进行了对比并测试了网格分辨率和粗化率对模型的影响.研究表明:基于ANN固相应力模型的模拟结果对EMMS固相应力模型结果有较高的还原度,同时具有一定的网格分辨率无关性和粗化率无关性.     

圣地亚结构动力学问题模型确认与评估

考虑模型的不确定性与建模误差等因素, 是现代模型确认的研究特点. 为了 研究现代模型确认过程中的建模、模型确认准则等核心问题, 以圣地亚提出的结构动力 学挑战问题为研究实例, 采用正态分布和非参数核密度估计研究了参数不确定性建模, 并且 通过所采用的两条确认准则对所建立的数学模型进行了确认研究, 并最终得到了较好的数学 模型. 通过该实例的具体开展, 发现联合使用多条确认准则可对所建原始模型进行更科学 的评估确认.     

Enhancement of electronic cooling by insertion of foam materials

 A numerical investigation of electronic cooling enhancement is carried out in this study in order to determine how the operating temperature can be maintained under the allowable level. A new technique based on use of porous or foam material inserted between the components on a horizontal board is studied. One energy equation model has been adopted to analyse the thermal field. The control volume method based on finite differences with appropriate averaging for diffusion coefficients is used to solve the coupling between solid, fluid and porous regions. The effect of parameters such as Reynolds number, Darcy number and thermal conductivity ratio are considered in order to look for the most appropriate properties of the foam or porous substrate that allow optimal cooling. The results show that for high thermal conductivity of the porous substrate, substantial enhancement is obtained compared to the fluid case even if the permeability is low. In the mixed convection case, the insertion of the foam between the blocks leads to a temperature reduction of 50%. Received on 14 December 1999     

On plastic localization and failure in plane strain and round void containing tensile bars

Gurson's constitutive model describing effects of void nucleation and growth in ductile metals is adopted for numerical analysis of plastic localization and failure modes in axisymmetric and plane strain tensile tests. A simple numerically oriented treatment of void coalescence is utilized. The analysis is performed in the framework of the bifurcation approach and no initial imperfections are needed to trigger localizations. The unsymmetric eigenvalue problem is solved by using a perturbation scheme which essentially reduces the problem to a symmetric eigenvalue formulation for a recently proposed “symmetrized comparison solid” of nonassociated plasticity plus the easily computable perturbation correction. The results obtained are shown to qualitatively reproduce the essential features observed experimentally in uniaxial tests.     

Modeling dynamic brittle behavior of materials with circular flaws or pores

Compressive failure of brittle materials is driven primarily by crack growth from pre-existing flaws in the material. These flaws, such as grain boundaries, pores, preexisting cracks, inclusions and missing grains, are randomly spaced and have a range of possible shapes and sizes. The current work proposes a micromechanics-based model for compressive dynamic failure of brittle materials with circular pore flaws, which incorporates both the number density and the size distribution of flaws. Results show that the distribution of flaw sizes is very important, particularly at moderate strain rate, since analyses based solely on the mean flaw size overpredict strength. Therefore, in order to increase dynamic strength at low to moderate strain rates, it is most effective to control the presence of large flaws. At very high strain rates, however, crack growth is activated even in small flaws and therefore controlling the total number density rather than the size of the flaws is effective for increasing dynamic strength. Finally, the model shows that neglecting very small flaws in the pore population may not have significant effects on the results in many cases, suggesting that the model is a useful tool for identifying a minimum resolution required for experimental characterization of microstructure.     

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).     

M-integral analysis for a two-dimensional metal/ceramic bimaterial solid with extending subinterface microcracks

Summary  The problem of the extension of subinterface microcracks in an infinite metal/ceramic bimaterial solid is studied. For the microcrack growth, the values of the M-integral are calculated under the assumption of a self-similar growth. First, the role that the M-integral plays in a metal/ceramic bimaterial solid with growing subinterface cracks is analyzed. It is concluded that an inherent relation exists between the value of the M-integral and the decrease of the effective elastic moduli for a bimaterial solid with growing subinterface microcracks. Second, it is concluded that mutual amplification and shielding effects exist during the microcrack extension, while they are substantially dependent on the increment of the microcrack length as well as the geometry of the microcrack arrangement under given loads. This strong mutual shielding effect of interacting microcracks makes the microcrack extension become increasingly difficult, and may stop the growth of the microcracks even under constant loads. Also, it is concluded that for a certain microcrack growth, the value of the M-integral in metal/ceramic bimaterial solid is always larger than that in homogeneous brittle solid for the same crack configuration. This means that the same microcrack growth in the former case shows lower stability than that in the latter one, due to the existence of a ductile phase. Received 3 May 2001; accepted for publication 27 June 2002 This work was supported by the Chinese National Nature Science Foundation (Grant 19472053) and supported by the Doctorate Foundation of Xi'an Jiaotong University (Grant DFXJU2000-15).