A simple mechanistic model of sprout spacing in tumour-associated angiogenesis

This paper develops a simple mathematical model of the siting of capillary sprouts on an existing blood vessel during the initiation of tumour-induced angiogenesis. The model represents an inceptive attempt to address the question of how unchecked sprouting of the parent vessel is avoided at the initiation of angiogenesis, based on the idea that feedback regulation processes play the dominant role. No chemical interaction between the proangiogenic and antiangiogenic factors is assumed. The model is based on corneal pocket experiments, and provides a mathematical analysis of the initial spacing of angiogenic sprouts

Approximate modelling of coal pyrolysis

A new approach to the modelling of the evolution of different chemical species from coal during its thermal decomposition under variable heating rate regimes was developed. The approach, which can be based on experimental data or the multiple-reaction model (MRM) with distributed activation energies, uses an Nth order reaction model where the pre-exponential factor and the apparent activation energy are functions of the rate of heating. A comparison of the predictions of the new model with the MRM was carried out. The new approach significantly cuts down the computation time with almost no loss of accuracy. The functional group pyrolysis model was examined and the parameters for the new Nth order approximation obtained considerably simplifying pyrolysis modelling. In addition, the article shows how to obtain estimates for certain parameters, which characterize chemical processes modelled by the MRM

Optimization of coal pyrolysis modeling

Coal pyrolysis is complex, involving a large number of chemical reactions. The most accurate and up-to-date approach to modeling coal pyrolysis is to adopt the Distributed Activation Energy Model (DAEM), also called the Multiple Reaction Model (MRM). This can be very computationally expensive, since it involves a complicated multiple integration. A novel method of optimizing the mathematical modeling based on the DAEM has been developed. It has been shown that for a given accuracy, the method involves significantly less computational time than standard methods. Another advantage of the new method is that it allows errors to be estimated. Copyright (C) 2000 The Combustion Institute

Condensed phase combustion travelling waves with sequential exothermic or endothermic reactions

The one-dimensional propagation of a combustion wave through a premixed solid fuel for two-stage kinetics is studied. We re-examine the analysis of a single reaction travelling-wave and extend it to the case of two-stage reactions. We derive an expression for the travelling wave speed in the limit of large activation energy for both reactions. The analysis shows that when both reactions are exothermic, the wave structure is similar to the single reaction case. However, when the second reaction is endothermic, the wave structure can be significantly different from single reaction case. In particular, as might be expected, a travelling wave does not necessarily exist in this case. We establish conditions in the limiting large activation energy limit for the non-existence, and for monotonicity of the temperature profile in the travelling wave

Avascular tumour dynamics and necrosis

We consider the dynamic growth of a tumour, concentrating on the possible development of a necrotic region and examine some simple tumour geometries in detail. The growth and death rates of the cells in the viable rim of the tumour are taken to be determined by the local oxygen concentration. Crucially the cell motion is determined by the forces generated by cell affinity, by cell interaction and by the need to get the waste products of cell death, primarily water, out of the tumour and products for cell growth, again primarily water, into the tumour. A consolidation type model with surface tension on the cells, slow viscous flow of the cells and porous media flow of the extracellular water is derived. The dynamic behaviour of this model is examined. Considering the very simple case where resistance to extracellular water flow dominates the problem, the model accounts naturally for the formation of a necrotic region. In regions where the extracellular water pressure gets too large, the cells are assumed to be ripped from the extracellular matrix and die. This model contrasts significantly from previous models which typically assume a necrotic region exists and that its behaviour is primarily governed directly by oxygen concentration. Here, the stress determines the necrotic region behaviour and this is affected by the oxygen only indirectly through the cell growth and death rates. The predicted time-dependent growth of one-dimensional, and spherical tumours are illustrated by numerical calculations

Approximations to the DAEM for Coal Pyrolysis

The Distributed Activation Energy Model (DAEM), used for the pyrolysis of a range of materials (including coal, biomass, residual oils and kerogen), assumes that the thermal decomposition of numerous components is described by a distribution of activation energies. Existing theories are reviewed with particular focus on methods used to evaluate solutions quickly and efficiently. This paper demonstrates that previous approaches taken to simplify the methods of solution can usually be identified as belonging to one of two distinct and physically relevant regimes. A careful analysis in these two regimes is given based upon asymptotic expansions, leading to systematic methods for rapidly finding accurate approximations. The new theory results in simple expressions for the devolatilisation rate of a given distribution of reactants. The method thereby provides a rapid and highly effective method for estimating kinetic parameters and the distribution of activation energies. Comparison of the simplified results with existing theories and with calculations of the full model are given. The methods provide a useful basis for calculations of coupled models of volatilisation and combustion, and for models with spatially varying temperature

Primary alkaline battery cathodes: a three scale model

A mathematical model for the galvanostatic discharge and recovery of porous, electrolytic manganese dioxide cathodes, similar to those found within primary alkaline batteries is presented. The phenomena associated with discharge are modeled over three distinct size scales, a cathodic (or macroscopic) scale, a porous manganese oxide particle (or microscopic) scale, and a manganese oxide crystal (or submicroscopic) scale. The physical and chemical coupling between these size scales is included in the model. In addition, the model explicitly accounts for the graphite phase within the cathode. The effects that manganese oxide particle size and proton diffusion have on cathodic discharge and the effects of intraparticle voids and microporous electrode structure are predicted using the model

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel onedimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted