On the modelling of finite growth considering the mechanics of cell division

Mitotic cells grow in volume and divide themselves into two identical cells producing at macroscopic scale a volume expansion in living bodies. Due to inhomogeneous distributions of the growth factors, growth occurs at different rates and directions. Focusing into the direction of growth, some living bodies alter their growing behaviour influenced by mechanical loads. If loads appear during the growth process, cell division is reorientated following the main direction of the elastic deformations. Therefore, new cells will be created in this direction while relaxing the stress state of the body at the same time. In this work, we present a modelling approach for growing bodies which change their growth direction depending on mechanical loads. The model is implemented into a finite element framework to be an useful tool for predicting morphological changes in growing bodies. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growing between barriers

Growth in living materials is the result of the changes in volume and mass during their development. If volume expansion occurs in a constrained case, some living materials change its growth behaviour. For example, when growth takes place in an environment with restrictions of volume, living materials will stop their volume expansion under compression due to the high amount of water that makes these bodies nearly incompressible. In case boundary conditions limit the growth of the body, the growth direction changes and gives the body another shape as expected. We present a modelling approach that takes volume and shape restrictions during growth into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the mechanics of fluid-like growth in solid materials

Living materials can change in mass and volume depending on the availability of nutrients and on its mechanical environment. During growth phenomena, even with mechanical properties as solids, some living materials present fluid-like characteristics. Bodies with such behaviour can be deformed as elastic bodies under the application of loads and constrains. However, their growth can be described as the flow of the new material contribution as these loads and constrains do not let the body grow in an isotropic way. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuum model for free growth in living materials

We focus in this work on isotropically growing materials. An adaptive algorithm is used in order to maintain a stress-free state during growth if no external loads are applied, but keeping the volume growth defined by a former kinetic. The proposed model is based on a modified multiplicative split of the deformation gradient into a growth part and an elastic part. The growth part will be isotropic if the elastic deformations are favourable, otherwise the growth will find a more comfortable direction. Three-dimensional examples based on different kinetics are presented and discussed using the numerical model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of the volume effect on a simple batch fermentation process

In this paper two models, one structured and the other unstructured, for the simple batch fermentation process, based on the kinetic scheme

, are compared quantitatively and qualitatively. One model assumes that the volume of the cells in the fermentation process is approximately constant while the other model incorporates the changing cell volume. It is shown that the specific growth rates for each model differ by approximately a factor of 2. Qualitatively, the shape of the trajectories for each model in the specific growth rate–substrate phase plane are very similar. One model is the traditional fixed volume model where the intra- and extracellular substrate levels are equal, while the “new” model incorporates the intracellular substrate (without cellular transport) with changing cell volumes.     

Primary and secondary instabilities in soft bilayered systems

Wrinkling phenomena emerging from mechanical instabilities in inhomogeneously compressed soft bilayered systems can evoke a wide variety of surface morphologies. Applications range from undesired instabilities in engineering structures such as sandwich panels, via fabricating surfaces with controlled buckling patterns of unique properties, to wrinkling phenomena in living matter such as lungs, mucosas, and brain convolutions. While moderate compression evokes periodic sinusoidal wrinkles, higher compression induces secondary instabilities - the surface bifurcates into increasingly complex morphologies. Periodic wrinkling has already been extensively studied, but the rich pattern formation in the highly nonlinear post-buckling regime remains poorly understood. Here, we establish a computational model of differential growth to explore the evolving buckling pattern of a growing layer bonded to a non-growing substrate. Our model provides a mechanistic understanding of growth-induced primary and secondary instabilities. We show that amongst all possible secondary bifurcations, the mode of period-doubling is energetically favorable. We experimentally validate our numerical results by examining buckling of a compressed polymer film on a soft foundation. Our computational studies have broad applications in the microfabrication of distinct surface patterns as well as in the morphogenesis of living systems, where growth is progressive and the formation of structural instabilities is critical to biological function. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth and intersection of cracks using boundary-element analysis

Existing procedures which are available for the determinationof the path traversed by a crack growing in a two-dimensionallinear elastic body are limited to simple geometric configurationswhere the growth of all the crack tips is the same. In thispaper, an important advance has been made; a crack may intersectother cracks, while a crack growing from a boundary may intersectother boundaries. Simulations of crack growth and intersectionsare performed for a number of configurations and computationalprocedures are given which control the rate of crack growth.In order to assess the robustness of these new extensions, cracksat or near circular holes and multiple cracks growing into anedge crack are considered, and detailed results given.     

A novel coupled system of non-local integro-differential equations modelling Young’s modulus evolution,nutrients’ supply and consumption during bone fracture healing

During fracture healing, a series of complex coupled biological and mechanical phenomena occurs. They include: (i) growth and remodelling of bone, whose Young’s modulus varies in space and time; (ii) nutrients’ diffusion and consumption by living cells. In this paper, we newly propose to model these evolution phenomena. The considered features include: (i) a new constitutive equation for growth simulation involving the number of sensor cells; (ii) an improved equation for nutrient concentration accounting for the switch between Michaelis–Menten kinetics and linear consumption regime; (iii) a new constitutive equation for Young’s modulus evolution accounting for its dependence on nutrient concentration and variable number of active cells. The effectiveness of the model and its predictive capability are qualitatively verified by numerical simulations (using COMSOL) describing the healing of bone in the presence of damaged tissue between fractured parts.     

Identification of the material symmetries with anisotropic damage evolution due to a growing mixed-mode crack

A numerical study of a growing mixed-mode internal crack in a unit cell was undertaken with the help of a finite element simulation. The model enables us to measure the components of the elastic compliance tensor modified by damage as the crack grows, showing the evolution of the anisotropic damage and the evolution of the type of material symmetries. The evolution of the elasticity tensor shows that the damage associated with a growing elliptical crack changes the virgin isotropic properties into orthotropic ones and by crack growth the axes of orthotropic symmetry, initially aligned with the local coordinates of the crack, rotate towards the principle loading axes. Crack propagation is simulated using the stepwise method, which consists of the succession of straight segments and crack growth is governed by the principle of maximum driving force which is a direct consequence of the variational principle of a cracked body in equilibrium and considers the effect of all three stress intensity factors. Without any ad hoc assumption, the crack growth rate is calculated using its thermodynamic duality with the local maximum driving force. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

A model of proliferating cell populations with correlation of mother-daughter mitotic times

Summary A kinetic model of proliferating cell populations is studied. The model features the correlation of mother-daughter mitotic times. The model is analysed by means of the theory of strongly continuous semigroups of linear operators. A connection is made between the asymptotic behavior of solutions and the spectral properties of the infinitesimal generator. It is proved that the solutions of the model have the property of asynchronous exponential growth.     

Three-dimensional cellular automaton simulation of tumour growth in inhomogeneous oxygen environment

Cellular automaton theory has previously been used to study cell growth. In this study, we present a three-dimensional cellular automaton model performing the growth simulation of normal and cancerous cells. The necessary nutrient supply is provided by an artificial arterial tree which is generated by constrained constructive optimization. Spatial oxygen diffusion is approximated again by a cellular automaton model. All results could be illustrated dynamically by three-dimensional volume visualization. Because of the chosen modelling approach, an extension of the model to simulate angiogenic processes is possible.     

A Mathematical Model of Prostate Tumor Growth Under Hormone Therapy with Mutation Inhibitor

This paper extends Jackson’s model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical mutation inhibitors. The new model not only considers the mutation by which androgen-dependent (AD) tumor cells mutate into androgen-independent (AI) ones but also introduces inhibition which is assumed to change the mutation rate. The tumor consists of two types of cells (AD and AI) whose proliferation and apoptosis rates are functions of androgen concentration. The mathematical model represents a free-boundary problem for a nonlinear system of parabolic equations, which describe the evolution of the populations of the above two types of tumor cells. The tumor surface is a free boundary, whose velocity is equal to the cell’s velocity there. Global existence and uniqueness of solutions of this model is proved. Furthermore, explicit formulae of tumor volume at any time t are found in androgen-deprived environment under the assumption of radial symmetry, and therefore the dynamics of tumor growth under androgen-deprived therapy could be predicted by these formulae. Qualitative analysis and numerical simulation show that controlling the mutation may improve the effect of hormone therapy or delay a tumor relapse.     

An alternative perspective on the concept of stress in classical continuum mechanics

Within a variational formulation of continuum mechanics, as proposed for instance by Germain [1], the internal virtual work contribution of a continuum is postulated as a smooth density integrated over the deformed configuration of the body. In this smooth density the stress field appears as dual quantity to the gradient of the virtual displacement field. Since the mathematical definition of the volume integral naturally provides a macro-micro relation between infinitesimal volume elements and the continuous body, we propose in this paper an alternative definition of stress on the micro level of the infinitesimal volume elements. In particular, the stress is defined as the internal forces of the body that model the mutual force interaction between neighboring volume elements. The existence of the stress tensor on the macro level is then obtained from the summation of all virtual work contributions within the body, followed by a limit process in which the volume elements are sent to zero. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Dynamic Programming Model for Pig Production

The feeding policy of a pig production unit affects both the cost of production and the weight and carcase composition of the pigs produced. Since the market value of the pigs produced is determined by the weight and composition of the carcase, feeding policy has a major influence on the economic performance of the unit. In order to evaluate possible feeding policies, the effect of feed intake on both the weight and the body composition of the growing pig must be known, and since an optimal policy will involve using least cost rations, it must be possible to determine the least cost rations to produce liveweight gains of specified body composition. A dynamic programming model to determine the optimal feeding policy to produce pigs of specified weight and carcase composition is developed using a published pig growth model which allows the formulation of the required least cost rations, and the use of this dynamic programming model is illustrated.     

Modelling and remodelling of biological tissue in the framework of continuum biomechanics

A biological tissue in general is formed by cells, extracellular matrix (ECM) and fluids. Consequently, its overall material behaviour results from its components and their interaction among each other. Furthermore, in case of living tissues, the material properties do not remain constant but naturally change due to adaptation processes or diseases. In the context of the Theory of Porous Media (TPM), a continuum-mechanical model is introduced to describe the complex fluid-structure interaction in biological tissue on a macroscopic scale. The tissue is treated as an aggregate of two immiscible constituents, where the cells and the ECM are summarised to a solid phase, whereas the fluid phase represents the extracellular and interstitial liquids as well as necrotic debris and cell or matrix precursors in solution. The growth and remodelling processes are described by a distinct mass exchange between the fluid and solid phase, which also results in a change of the constituent material behaviour. To furthermore guarantee the compliance with the entropy principle, the growth energy is introduced as an additional quantity. It measures the average of chemical energy available for cell metabolism, and thus, controls the growth and remodelling processes. To set an example, the presented model is applied for the simulation of the early stages of avascular tumour growth in the framework of the finite element method (FEM). (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The mathematical theory of growing bodies. Finite deformations

The fundamentals of the mathematical theory of accreting bodies for finite deformations are explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes. One of the possible types of accretion, as due to the continuous addition of stressed material surfaces to a three-dimensional body, is considered. The complete system of equations of the mechanics of accreting bodies is presented. Unlike in problems for bodies of constant composition, the tensor field of the incompatible distortion, which can be found from the equilibrium condition for the boundary of growth, that is, a material surface in contact with a deformable three-dimensional body, enters into these equations. Generally speaking, a growing body does not have a stress-free configuration in three-dimensional Euclidean space. However, there is such a configuration on a certain three-dimensional manifold with a non-Euclidean affine connectedness caused by a non-zero torsion tensor that is a measure of the incompatibility of the deformation of the growing body. Mathematical models of the stress-strain state of a growing body are therefore found to be equivalent to the models of bodies with a continuous distribution of the dislocations.     

增长网络的形成机理和度分布计算

关于增长网络的形成机理,着重介绍由线性增长与择优连接组成的BA模型, 以及加速增长模型.此外,我们提出了一个含反择优概率删除旧连线的模型,这个模型能自组织演化成scale-free(SF)网络.关于计算SF网络的度分布,简要介绍文献上常用的基于连续性理论的动力学方法(包括平均场和率方程)和基于概率理论的主方程方法.另外,我们基于马尔可夫链理论还首次尝试了数值计算方法.这一方法避免了复杂方程的求解困难,所以较有普适性,因此可用于研究更为复杂的网络模型.我们用这种数值计算方法研究了一个具有对数增长的加速增长模型,这个模型也能自组织演化成SF网络.     

存在人口增长极限下的改进Ramsey模型

在存在人口增长极限的假设下,对标准的Ramsey模型进行了修正,结果表明:首先,与常数人口增长率的Ramsey模型不同,模型的人均消费和人均资本并不一定随时间的推移而递增;其次,人均消费和人均资本不存在平衡增长路径均衡,但是存在渐进平衡增长路径均衡;再次,在外生参数满足一定条件的前提下,人均消费和人均资本渐近于均衡的过程是单调的.     

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. The time dependence of the metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and “shape”. We show that the time dependency of the material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange–d’Alembert principle with Rayleigh’s dissipation functions to derive the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of the deformation gradient, F=F _e F _g, into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.