[show abstract][hide abstract] ABSTRACT: Porous architecture has a dramatic effect on tissue formation in porous biomaterials used in regenerative medicine. However, the wide variety of 3D structures used indicates there is a clear need for the optimal design of pore architecture to maximize tissue formation and ingrowth. Thus, the aim of this study was to characterize initial tissue growth solely as a function of pore geometry. We used an in vitro system with well-defined open pore slots of varying width, providing a 3D environment for neo-tissue formation while minimizing nutrient limitations. Results demonstrated that initial tissue formation was strongly influenced by pore geometry. Both velocity of tissue invasion and area of tissue formed increased as pores became narrower. This is associated with distinct patterns of actin organisation and alignment depending on pore width, indicating the role of active cell generated forces. A mathematical model based on curvature driven growth successfully predicted both shape of invasion front and constant rate of growth, which increased for narrower pores as seen in experiments. Our results provide further evidence for a front based, curvature driven growth mechanism depending on pore geometry and tissue organisation, which could provide important clues for 3D scaffold design.
Annals of biomedical engineering 02/2013; · 2.41 Impact Factor
[show abstract][hide abstract] ABSTRACT: The paper models the first stage of the process of pillar extraction in a coal mine. The problem of understanding how a coal mine roof collapses after secondary cutting of the supporting pillars to create small supporting snooks is considered. The fracture of the roof is considered when a set of snooks have failed and the roof must support itself between two pillars. Models that account for the relative importance of the overburden weight on the roof and the compressive stresses in the roof are examined using a simple strut and beam theory.
International Journal of Rock Mechanics and Mining Sciences 01/2013; 64:132–138. · 1.20 Impact Factor
[show abstract][hide abstract] ABSTRACT: Amino acid transfer to the fetus is dependent on several different factors. While these factors can be understood in isolation, it is still not possible to predict the function of the system as a whole. In order to do this an integrated approach is required which incorporates the interactions between the different determinants of amino acid transfer. Computational modelling of amino acid transfer in the term human placenta provides a mechanism by which this integrated approach can be delivered. Such a model would be invaluable for understanding amino acid transfer in both normal and pathological pregnancies. In order to develop a computational model it is necessary to determine all the biological factors which are important contributors to net amino acid transfer and the ways in which they interact. For instance, how different classes of amino acid transporter must interact to transfer amino acids across the placenta. Mathematically, the kinetics of each type of transporter can be represented by separate equations that describe their transfer rate as a non-linear function of amino acid concentrations. These equations can then be combined in the model to predict the overall system behaviour. Testing these predictions experimentally will demonstrate the strengths and weaknesses of the model, which can then be refined with increasing complexity and retested in an iterative fashion. In this way we hope to develop a functional computational model which will allow exploration of the factors that determine amino acid transfer across the placenta. This model may also allow the development of strategies to optimise placental transfer in pathologies associated with impaired amino acid transfer such as fetal growth restriction.
[show abstract][hide abstract] ABSTRACT: Organic diodes and solar cells are constructed by placing together two organic semiconducting materials with dissimilar electron affinities and ionization potentials. The electrical behavior of such devices has been successfully modeled numerically using conventional drift diffusion together with recombination (which is usually assumed to be bimolecular) and thermal generation. Here a particular model is considered and the dark current-voltage curve and the spatial structure of the solution across the device is extracted analytically using asymptotic methods. We concentrate on the case of Shockley—Read—Hall recombination but note the extension to other recombination mechanisms. We find that there are three regimes of behavior, dependent on the total current. For small currents—i.e., at reverse bias or moderate forward bias—the structure of the solution is independent of the total current. For large currents—i.e., at strong forward bias—the current varies linearly with the voltage and is primarily controlled by drift of charges in the organic layers. There is then a narrow range of currents where the behavior undergoes a transition between the two regimes. The magnitude of the parameter that quantifies the interfacial recombination rate is critical in determining where the transition occurs. The extension of the theory to organic solar cells generating current under illumination is discussed as is the analogous current-voltage curves derived where the photo current is small. Finally, by comparing the analytic results to real experimental data, we show how the model parameters can be extracted from the shape of current-voltage curves measured in the dark.
[show abstract][hide abstract] ABSTRACT: In this study, a previously derived two-dimensional model is used to describe the slow spreading of viscous films on the surface of a quiescent deep viscous pool due to gravity. It is assumed that the densities and viscosities of the fluids in the films and pool are comparable, but may be different. It is also assumed that surface tension effects are negligible. The fluid in the films and in the pool are both modelled using the Stokes flow equations. By exploiting the slenderness of the spreading films, asymptotic techniques are used to analyse the flow. It is shown that the dominant forces controlling the spreading are gravity and the tangential stress induced in the films by the underlying pool. As a consequence, the rate of spreading of the films is independent of their viscosity. For the case special of a symmetric configuration of films on the surface of the pool, the flow is studied by assuming that the solution becomes self-similar and hence the problem is recast in a self-similar coordinate system. Stokeslet analysis is then used to derive a singular integral equation for the stresses on the interfaces between the films and the pool. The form of this integral equation depends on the configuration of spreading films that are to be considered. A number of different cases are then studied, namely, a single film, two films, and an infinite periodic array of films. Finally, some results are derived that apply to a general symmetric configuration of films. It is shown that the profile of a spreading film close to its front (where the film thickness becomes zero) is proportional to x1/4. It is also shown that fronts move, and hence, the distance between adjacent fronts increases proportional to t1/3.
Physics of Fluids 06/2012; 24(6). · 1.94 Impact Factor
[show abstract][hide abstract] ABSTRACT: In the field of cartilage tissue engineering, filter cultures are a frequently used three-dimensional differentiation model. However, understanding of the governing processes of in vitro growth and development of tissue in these models is limited. Therefore, this study aimed to further characterise these processes by means of an approach combining both experimental and applied mathematical methods. A mathematical model was constructed, consisting of partial differential equations predicting the distribution of cells and glycosaminoglycans (GAGs), as well as the overall thickness of the tissue. Experimental data was collected to allow comparison with the predictions of the simulation and refinement of the initial models. Healthy mature equine chondrocytes were expanded and subsequently seeded on collagen-coated filters and cultured for up to 7 weeks. Resulting samples were characterised biochemically, as well as histologically. The simulations showed a good representation of the experimentally obtained cell and matrix distribution within the cultures. The mathematical results indicate that the experimental GAG and cell distribution is critically dependent on the rate at which the cell differentiation process takes place, which has important implications for interpreting experimental results. This study demonstrates that large regions of the tissue are inactive in terms of proliferation and growth of the layer. In particular, this would imply that higher seeding densities will not significantly affect the growth rate. A simple mathematical model was developed to predict the observed experimental data and enable interpretation of the principal underlying mechanisms controlling growth-related changes in tissue composition.
European cells & materials 01/2011; 22:377-92. · 4.56 Impact Factor
[show abstract][hide abstract] ABSTRACT: A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of
lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler–Volmer
reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in
agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate
to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the
coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e.g. particle
size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic
methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler
expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current,
regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform
use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of
intercalated lithium concentration, such as Li
C6, but not for materials with a flat discharge curve, such as LiFePO4.
KeywordsButler–Volmer equation–Electrolyte–Homogenisation–Matched asymptotic expansions
Journal of Engineering Mathematics 01/2011; 72(1):41-72. · 1.08 Impact Factor
[show abstract][hide abstract] ABSTRACT: Amino acid transfer from mother to fetus via the placenta plays a critical role in normal development, and restricted transfer is associated with fetal growth restriction. Placental amino acid transfer involves the interaction of 15 or more transporters and 20 amino acids. This complexity means that knowing which transporters are present is not sufficient to predict how they operate together as a system. Therefore, in order to investigate how placental amino acid transfer occurs as a system, an integrated mathematical/computational modelling framework was developed to represent the simultaneous transport of multiple amino acids. The approach was based on a compartmental model, in which separate maternal, syncytiotrophoblast and fetal volumes were distinguished, and transporters were modelled on the maternal- and fetal-facing membranes of the syncytiotrophoblast using Michaelis-Menten-type kinetics. The model was tested in comparison with placental perfusion experiments studying serine-alanine exchange and found to correspond well. The results demonstrated how the different transporters can work together as an integrated system and allowed their relative importance to be assessed. Placental-fetal serine exchange was found to be most sensitive to basal membrane transporter characteristics, but a range of secondary, less intuitive effects were also revealed. While this work only addressed a relatively simple three amino acid system, it demonstrates the feasibility of the approach and could be extended to incorporate additional experimental parameters. Ultimately, this approach will allow physiological simulations of amino acid transfer. This will enhance our understanding of these complex systems and placental function in health and disease.
[show abstract][hide abstract] ABSTRACT: Cartilage tissue repair procedures currently under development aim to create a construct in which patient-derived cells are seeded and expanded ex vivo before implantation back into the body. The key challenge is producing physiologically realistic constructs that mimic real tissue structure and function. One option with vast potential is to print strands of material in a 3D structure called a scaffold that imitates the real tissue structure; the strands are composed of gel seeded with cells and so provide a template for cartilaginous tissue growth. The scaffold is placed in the construct and pumped with nutrient-rich culture medium to supply nutrients to the cells and remove waste products, thus promoting tissue growth. In this paper we use asymptotic homogenization to determine the effective flow and transport properties of such a printed scaffold system. These properties are used to predict the distribution of nutrient/waste products through the construct, and to specify design criteria for the scaffold that will optimize the growth of functional tissue.
Journal of Theoretical Biology 05/2009; 259(3):489-502. · 2.35 Impact Factor
[show abstract][hide abstract] ABSTRACT: This paper presents an elastohydrodynamic model of the human eyelid wiper. Standard lubrication theory is applied to the fluid layer between the eyelid wiper and ocular surface. The role of the lubrication film is to reduce the shear stresses by preventing solid to solid contact between the eyelid wiper and ocular surface. For the lubrication film to be effective, it is required that the orientation of the eyelid wiper changes between the opening and closing phases of a blink. In order to model this, the hydrodynamic model is coupled with an elastic mattress model for the soft tissue of the eyelid wiper and ocular surface. This leads to a one-dimensional non-linear partial differential equation governing the fluid pressure in the lubrication film. In order to solve the differential equation, a loading condition or constraint equation must be specified. The resulting system is then solved numerically. The model allows predictions of the tear film flux from under the upper eyelid, as well as normal and shear stresses acting on the ocular surface. These factors are important in relation to dry eye syndrome, deformation of the cornea and contact lens design. It is found that the pressure and shear stress under the eyelid act across a length of approximately 0.1 mm which is consistent with clinical observations. It order to achieve a flow of tears from under the upper eyelid during a blink, the model requires that the normal force the eyelid applies to the ocular surface during the closing phase of the blink is significantly higher than during the opening phase of the blink.
Bulletin of Mathematical Biology 03/2008; 70(2):323-43. · 2.02 Impact Factor
[show abstract][hide abstract] ABSTRACT: The mechanisms underlying the formation of necrotic regions within avascular tumours are not well understood. In this paper, we examine the relative roles of nutrient deprivation and of cell death, from both the proliferating phase of the cell cycle via apoptosis and from the quiescent phase via necrosis, in changing the structure within multicellular tumour spheroids and particularly the accumulation of dead cell material in the centre. A mathematical model is presented and studied that accounts for nutrient diffusion, changes in cell cycling rates, the two different routes to cell death as well as active motion of cells and passive motion of the dead cell material. In studying the accumulation of dead cell matter we do not distinguish between the route by which each was formed. The resulting mathematical model is examined for a number of scenarios. Results show that in many cases the size of the necrotic core is closely correlated with low levels in nutrient concentration. However, in certain cases, particularly where the rate of necrosis is large, the resulting necrotic core can lead to regions of non-negligible nutrient concentration-dependent upon the mode of cell death.
[show abstract][hide abstract] ABSTRACT: The generation of induced pluripotent stem cells from adult somatic cells by ectopic expression of key transcription factors holds significant medical promise. However, current techniques for inducing pluripotency rely on viral infection and are therefore not, at present, viable within a clinical setting. Thus, there is now a need to better understand the molecular basis of stem cell pluripotency and lineage specification in order to investigate alternative methods to induce pluripotency for clinical application. However, the complexity of the underlying molecular circuitry makes this a conceptually difficult task. In order to address these issues, we considered a computational model of transcriptional control of cell fate specification. The model comprises two mutually interacting sub-circuits: a central pluripotency circuit consisting of interactions between stem-cell specific transcription factors OCT4, SOX2 and NANOG coupled to a differentiation circuit consisting of interactions between lineage-specifying master genes.The molecular switches which arise from feedback loops within these circuits give rise to a well-defined sequence of successive gene restrictions corresponding to a controlled differentiation cascade in response to environmental stimuli. Furthermore, we found that this differentiation cascade is strongly unidirectional: once silenced, core transcription factors cannot easily be reactivated. In the context of induced pluripotency, this indicates that differentiated cells are robustly resistant to reprogramming to a more primitive state. However, our model suggests that under certain circumstances, amplification of low-level fluctuations in transcriptional status (transcriptional "noise") may be sufficient to trigger reactivation of the core pluripotency switch and reprogramming to a pluripotent state. This interpretation offers an explanation of a number of experimental observations concerning the molecular mechanisms of cellular reprogramming by defined factors and suggests a role for stochasticity in reprogramming of somatic cells to pluripotency.
PLoS ONE 02/2008; 3(8):e3086. · 3.73 Impact Factor
[show abstract][hide abstract] ABSTRACT: A mathematical model for the interfacial charge transfer within dye-sensitised solar cells (DSC) is presented for the
semiconductor–dye–electrolyte interface. The model explicitly accounts for each reaction at the interface involving dye molecules,
electrolyte species and adsorbed electrons associated with the conduction band surface states of the semiconductor. Additionally, the
model accounts for photoelectron injection via singlet and triplet excited dye states. The governing equations can be used to describe the
total current produced by the DSC under illuminated and non-illuminated conditions, at steady state. Regular perturbation methods are
applied to the model equations to obtain closed form analytic approximations, resulting in approximate solutions that negate the need
for numerical solution of the model system. All parameter values associated with the model are obtained from the literature and from
experimental data. The presented numerical results and analytic approximations compare favourably to experimental data, capturing the
interfacial characteristics of current versus voltage curves of the DSC.
[show abstract][hide abstract] ABSTRACT: Limited cell ingrowth is a major problem for tissue engineering and the clinical application of porous biomaterials as bone substitutes. As a first step, migration and proliferation of an interacting cell population can be studied in two-dimensional culture. Mathematical modelling is essential to generalize the results of these experiments and to derive the intrinsic parameters that can be used for predictions. However, a more thorough evaluation of theoretical models is hampered by limited experimental observations. In this study, experiments and image analysis methods were developed to provide a detailed spatial and temporal picture of how cell distributions evolve. These methods were used to quantify the migration and proliferation of skeletal cell types including MG63 and human bone marrow stromal cells (HBMSCs). The high level of detail with which the cell distributions were mapped enabled a precise assessment of the correspondence between experimental results and theoretical model predictions. This analysis revealed that the standard Fisher equation is appropriate for describing the migration behaviour of the HBMSC population, while for the MG63 cells a sharp front model is more appropriate. In combination with experiments, this type of mathematical model will prove useful in predicting cell ingrowth and improving strategies and control of skeletal tissue regeneration.
Journal of The Royal Society Interface 01/2008; 4(17):1107-17. · 4.91 Impact Factor
[show abstract][hide abstract] ABSTRACT: This paper analyses a recent mathematical model of avascular tumour spheroid growth which accounts for both cell cycle dynamics and chemotactic driven cell movement. The model considers cells to exist in one of two compartments: proliferating and quiescent, as well as accounting for necrosis and apoptosis. One particular focus of this paper is the behaviour created when proliferating and quiescent cells have different chemotactic responses to an extracellular nutrient supply. Two very different steady-state behaviours are identified corresponding to those cases where proliferating cells move either more quickly or more slowly than quiescent cells in response to a gradient in the extracellular nutrient supply. The case where proliferating cells move more rapidly leads to the commonly accepted spheroid structure of a thin layer of proliferating cells surrounding an inner quiescent core. In the case where proliferating cells move more slowly than quiescent cells the model predicts an interesting structure of a thin layer of quiescent cells surrounding an inner core of proliferating and quiescent cells. The sensitivity of this tumour structure to the cell cycle model parameters is also discussed. In particular variations in the steady-state size of the tumour and the types of transient behaviour are explored. The model reveals interesting transient behaviour with sharply delineated regions of proliferating and quiescent cells.
Bulletin of Mathematical Biology 06/2007; 69(4):1147-65. · 2.02 Impact Factor
[show abstract][hide abstract] ABSTRACT: Improved biological and mechanical functionality of musculoskeletal tissue-engineered constructs is required for clinical application, which can only be achieved by comprehensive multidisciplinary research. This review focuses on the contribution of computational modelling as a framework for obtaining an integrated understanding of key processes, which include: nutrient transport and utilization, matrix formation, cell population dynamics, cell attachment and migration, and local cell-cell interactions. Such an integrated perspective of these key aspects will be critical to open up new directions in tissue engineering research, as significant progress can be made by combining existing computational and experimental methods. Furthermore, theoretical modelling has enormous potential in applications ranging from the interpretation of experimental results and the identification of the main governing processes, to the optimization of practical tissue engineering protocols with implications therein for an increasing ageing population.