Mathematical models of calcium and tight junctions in normal and reconstructed epidermis

This project investigated the calcium distributions of the skin, and the growth patterns of skin substitutes grown in the laboratory, using mathematical models. The research found that the calcium distribution in the upper layer of the skin is controlled by three different mechanisms, not one as previously thought. The research also suggests that tight junctions, which are adhesions between neighbouring skin cells, cannot be solely responsible for the differences in the growth patterns of skin substitutes and normal skin.

Simulating urban growth in a developing nation's region using a cellular automata-based model

Urbanization is one of the most evident global changes. Research in the field of urban growth modelling has generated models that explore for drivers and components of the urban growth dynamics. Cellular automata (CA) modeling is one of the recent advances, and a number of CA-based models of urban growth have produced satisfactory simulations of spatial urban expansion over time. Most application and test of CA-based models of urban growth which provide likely and reliable simulations has been developed in urban regions of developed nations; urban regions in the United States, in particular. This is because most of the models were developed in universities and research centers of developed nations, and these regions have the required data, which is extensive. Most of the population growth in the world, however, occurs in the developing world. While some European countries show signs of stabilization of their population, in less developed countries, such as India, population still grows exponentially. And this growth is normally uncoordinated, which results in serious environmental and social problems in urban areas. Therefore, the use of existing dynamic–spatial models of urban growth in regions of developing nations could be a means to assist planners and decision makers of these regions to understand and simulate the process of urban growth and test the results of different development strategies. The pattern of growth of urban regions of developing nations, however, seems to be different of the pattern of developed countries. The former use to be more dense and centralized, normally expanding outwards from consolidated urban areas; while the second is normally more fragmented and sparse. The present paper aims to investigate to how extent existing CA-based urban growth models tested in developed nations can also be applied to a developing country urban area. The urban growth model was applied to Porto Alegre City, Brazil. An expected contiguous expansion from existing urban areas has been obtained as following the historical trends of growth of the region. Moreover, the model was sensitive and able to portray different pattern of growth in the study area by changing the value of its parameters.

The MIRELA framework: modeling and analyzing mixed reality applications using timed automata

Mixed Reality (MR) aims to link virtual entities with the real world and has many applications such as military and medical domains [JBL+00, NFB07]. In many MR systems and more precisely in augmented scenes, one needs the application to render the virtual part accurately at the right time. To achieve this, such systems acquire data related to the real world from a set of sensors before rendering virtual entities. A suitable system architecture should minimize the delays to keep the overall system delay (also called end-to-end latency) within the requirements for real-time performance. In this context, we propose a compositional modeling framework for MR software architectures in order to specify, simulate and validate formally the time constraints of such systems. Our approach is first based on a functional decomposition of such systems into generic components. The obtained elements as well as their typical interactions give rise to generic representations in terms of timed automata. A whole system is then obtained as a composition of such defined components. To write specifications, a textual language named MIRELA (MIxed REality LAnguage) is proposed along with the corresponding compilation tools. The generated output contains timed automata in UPPAAL format for simulation and verification of time constraints. These automata may also be used to generate source code skeletons for an implementation on a MR platform. The approach is illustrated first on a small example. A realistic case study is also developed. It is modeled by several timed automata synchronizing through channels and including a large number of time constraints. Both systems have been simulated in UPPAAL and checked against the required behavioral properties.

Mathematical models of calcium and tight junctions in normal and reconstructed epidermis

This project investigated the calcium distributions of the skin, and the growth patterns of skin substitutes grown in the laboratory, using mathematical models. The research found that the calcium distribution in the upper layer of the skin is controlled by three different mechanisms, not one as previously thought. The research also suggests that tight junctions, which are adhesions between neighbouring skin cells, cannot be solely responsible for the differences in the growth patterns of skin substitutes and normal skin.

Simulation of dynamic recrystallization using random grid cellular automata

Computer simulation is a powerful tool to predict microstructure and its evolution in dynamic and post-dynamic recrystallization. CAFE proposed as an appropriate approach by combining finite element (FE) method and cellular automata (CA) for recrystallization simulation. In the current study, a random grid cellular automaton (CA), as micro-scale model, based on finite element (FE), as macro-scale method, has been used to study initial and evolving microstructural features; including nuclei densities, dislocation densities, grain size and grain boundary movement during dynamic recrystallization in a C-Mn steel. An optimized relation has been established between mechanical variables and evolving microstructure features during recrystallization and grain growth. In this model, the microstructure is defined as cells located within grains and grain boundaries while dislocations are randomly dispersed throughout microstructure. Changes of dislocation density during deformation are described considering hardening, recovery and recrystallization. Recrystallization is assumed to initiate near grain boundaries and nucleation rate was considered constant (site-saturated condition). The model produced a mathematical formulation which captured the initial and evolving microstructural entities and linked their effects to measurable macroscopic variables (e.g. stress).

LATTICE BOLTZMANN METHOD AND CELLULAR AUTOMATA SIMULATION OF PARTICLE MOTION AND DEPOSITION IN 2-D CASE

This technical report discusses the application of the Lattice Boltzmann Method (LBM) and Cellular Automata (CA) simulation in fluid flow and particle deposition. The current work focuses on incompressible flow simulation passing cylinders, in which we incorporate the LBM D2Q9 and CA techniques to simulate the fluid flow and particle loading respectively. For the LBM part, the theories of boundary conditions are studied and verified using the Poiseuille flow test. For the CA part, several models regarding simulation of particles are explained. And a new Digital Differential Analyzer (DDA) algorithm is introduced to simulate particle motion in the Boolean model. The numerical results are compared with a previous probability velocity model by Masselot [Masselot 2000], which shows a satisfactory result.

Exploring the parameter space of a rabbit ventricular action potential model to investigate the effect of variation on action potential and calcium transients

Computational models for cardiomyocyte action potentials (AP) often make use of a large parameter set. This parameter set can contain some elements that are fitted to experimental data independently of any other element, some elements that are derived concurrently with other elements to match experimental data, and some elements that are derived purely from phenomenological fitting to produce the desired AP output. Furthermore, models can make use of several different data sets, not always derived for the same conditions or even the same species. It is consequently uncertain whether the parameter set for a given model is physiologically accurate. Furthermore, it is only recently that the possibility of degeneracy in parameter values in producing a given simulation output has started to be addressed. In this study, we examine the effects of varying two parameters (the L-type calcium current (I(CaL)) and the delayed rectifier potassium current (I(Ks))) in a computational model of a rabbit ventricular cardiomyocyte AP on both the membrane potential (V(m)) and calcium (Ca(2+)) transient. It will subsequently be determined if there is degeneracy in this model to these parameter values, which will have important implications on the stability of these models to cell-to-cell parameter variation, and also whether the current methodology for generating parameter values is flawed. The accuracy of AP duration (APD) as an indicator of AP shape will also be assessed.

Active regulation of the epidermal calcium profile

A distinct calcium profile is strongly implicated in regulating the multi-layered structure of the epidermis. However, the mechanisms that govern the regulation of this calcium profile are currently unclear. It clearly depends on the relatively impermeable barrier of the stratum corneum (passive regulation) but may also depend on calcium exchanges between keratinocytes and extracellular fluid (active regulation). Using a mathematical model that treats the viable sublayers of unwounded human and murine epidermis as porous media and assumes that their calcium profiles are passively regulated, we demonstrate that these profiles are also actively regulated. To obtain this result, we found that diffusion governs extracellular calcium motion in the viable epidermis and hence intracellular calcium is the main source of the epidermal calcium profile. Then, by comparison with experimental calcium profiles and combination with a hypothesised cell velocity distribution in the viable epidermis, we found that the net influx of calcium ions into keratinocytes from extracellular fluid may be constant and positive throughout the stratum basale and stratum spinosum, and that there is a net outflux of these ions in the stratum granulosum. Hence the calcium exchange between keratinocytes and extracellular fluid differs distinctly between the stratum granulosum and the underlying sublayers, and these differences actively regulate the epidermal calcium profile. Our results also indicate that plasma membrane dysfunction may be an early event during keratinocyte disintegration in the stratum granulosum.

Molecular dynamics simulation of mechanical behavior of osteopontin-hydroxyapatite interfaces

Bone is characterized with an optimized combination of high stiffness and toughness. The understanding of bone nanomechanics is critical to the development of new artificial biological materials with unique properties. In this work, the mechanical characteristics of the interfaces between osteopontin (OPN, a noncollagenous protein in extrafibrillar protein matrix) and hydroxyapatite (HA, a mineral nanoplatelet in mineralized collagen fibrils) were investigated using molecular dynamics method. We found that the interfacial mechanical behaviour is governed by the electrostatic attraction between acidic amino acid residues in OPN and calcium in HA. Higher energy dissipation is associated with the OPN peptides with a higher number of acidic amino acid residues. When loading in the interface direction, new bonds between some acidic residues and HA surface are formed, resulting in a stick-slip type motion of OPN peptide on the HA surface and high interfacial energy dissipation. The formation of new bonds during loading is considered to be a key mechanism responsible for high fracture resistance observed in bone and other biological materials.

STI-GMaS : an open-source environment for simulation of sexually-transmitted infections

Background Sexually-transmitted pathogens often have severe reproductive health implications if treatment is delayed or absent, especially in females. The complex processes of disease progression, namely replication and ascension of the infection through the genital tract, span both extracellular and intracellular physiological scales, and in females can vary over the distinct phases of the menstrual cycle. The complexity of these processes, coupled with the common impossibility of obtaining comprehensive and sequential clinical data from individual human patients, makes mathematical and computational modelling valuable tools in developing our understanding of the infection, with a view to identifying new interventions. While many within-host models of sexually-transmitted infections (STIs) are available in existing literature, these models are difficult to deploy in clinical/experimental settings since simulations often require complex computational approaches. Results We present STI-GMaS (Sexually-Transmitted Infections – Graphical Modelling and Simulation), an environment for simulation of STI models, with a view to stimulating the uptake of these models within the laboratory or clinic. The software currently focuses upon the representative case-study of Chlamydia trachomatis, the most common sexually-transmitted bacterial pathogen of humans. Here, we demonstrate the use of a hybrid PDE–cellular automata model for simulation of a hypothetical Chlamydia vaccination, demonstrating the effect of a vaccine-induced antibody in preventing the infection from ascending to above the cervix. This example illustrates the ease with which existing models can be adapted to describe new studies, and its careful parameterisation within STI-GMaS facilitates future tuning to experimental data as they arise. Conclusions STI-GMaS represents the first software designed explicitly for in-silico simulation of STI models by non-theoreticians, thus presenting a novel route to bridging the gap between computational and clinical/experimental disciplines. With the propensity for model reuse and extension, there is much scope within STI-GMaS to allow clinical and experimental studies to inform model inputs and drive future model development. Many of the modelling paradigms and software design principles deployed to date transfer readily to other STIs, both bacterial and viral; forthcoming releases of STI-GMaS will extend the software to incorporate a more diverse range of infections.

On the effect of topology on cellular automata rule spaces

This thesis presents an empirical study of the effects of topology on cellular automata rule spaces. The classical definition of a cellular automaton is restricted to that of a regular lattice, often with periodic boundary conditions. This definition is extended to allow for arbitrary topologies. The dynamics of cellular automata within the triangular tessellation were analysed when transformed to 2-manifolds of topological genus 0, genus 1 and genus 2. Cellular automata dynamics were analysed from a statistical mechanics perspective. The sample sizes required to obtain accurate entropy calculations were determined by an entropy error analysis which observed the error in the computed entropy against increasing sample sizes. Each cellular automata rule space was sampled repeatedly and the selected cellular automata were simulated over many thousands of trials for each topology. This resulted in an entropy distribution for each rule space. The computed entropy distributions are indicative of the cellular automata dynamical class distribution. Through the comparison of these dynamical class distributions using the E-statistic, it was identified that such topological changes cause these distributions to alter. This is a significant result which implies that both global structure and local dynamics play a important role in defining long term behaviour of cellular automata.

Towards a quantitative theory of epidermal calcium profile formation in unwounded skin

We propose and mathematically examine a theory of calcium profile formation in unwounded mammalian epidermis based on: changes in keratinocyte proliferation, fluid and calcium exchange with the extracellular fluid during these cells' passage through the epidermal sublayers, and the barrier functions of both the stratum corneum and tight junctions localised in the stratum granulosum. Using this theory, we develop a mathematical model that predicts epidermal sublayer transit times, partitioning of the epidermal calcium gradient between intracellular and extracellular domains, and the permeability of the tight junction barrier to calcium ions. Comparison of our model's predictions of epidermal transit times with experimental data indicates that keratinocytes lose at least 87% of their volume during their disintegration to become corneocytes. Intracellular calcium is suggested as the main contributor to the epidermal calcium gradient, with its distribution actively regulated by a phenotypic switch in calcium exchange between keratinocytes and extracellular fluid present at the boundary between the stratum spinosum and the stratum granulosum. Formation of the extracellular calcium distribution, which rises in concentration through the stratum granulosum towards the skin surface, is attributed to a tight junction barrier in this sublayer possessing permeability to calcium ions that is less than 15 nm/s in human epidermis and less than 37 nm/s in murine epidermis. Future experimental work may refine the presented theory and reduce the mathematical uncertainty present in the model predictions.