Multiscale modelling and homogenisation of fibre-reinforced hydrogels for tissue engineering.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic-poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

Optimal control of diffuser shapes for non-uniform flow.

A simplified model is used to identify the diffuser shape that maximises pressure recovery for several classes of non-uniform inflow. We find that optimal diffuser shapes strike a balance between not widening too soon, as this accentuates the non-uniform flow, and not staying narrow for too long, which is detrimental for wall drag. Three classes of non-uniform inflow are considered, with the axial velocity varying across the width of the diffuser entrance. The first case has inner and outer streams of different speeds, with a velocity jump between them that evolves into a shear layer downstream. The second case is a limiting case when these streams are of similar speed. The third case is a pure shear profile with linear velocity variation between the centre and outer edge of the diffuser. We describe the evolution of the time-averaged flow profile using a reduced mathematical model that has been previously tested against experiments and computational fluid dynamics models. The model consists of integrated mass and momentum equations, where wall drag is treated with a friction factor parameterisation. The governing equations of this model form the dynamics of an optimal control problem where the control is the diffuser channel shape. A numerical optimisation approach is used to solve the optimal control problem and Pontryagin's maximum principle is used to find analytical solutions in the second and third cases. We show that some of the optimal diffuser shapes can be well approximated by piecewise linear sections. This suggests a low-dimensional parameterisation of the shapes, providing a structure in which more detailed and computationally expensive turbulence models can be used to find optimal shapes for more realistic flow behaviour.

A mathematical model for cell infiltration and proliferation in a chondral defect.

We develop a mathematical model to describe the regeneration of a hydrogel inserted into an ex vivo osteochondral explant. Specifically we use partial differential equations to describe the evolution of two populations of cells that migrate from the tissue surrounding the defect, proliferate, and compete for space and resources within the hydrogel. The two cell populations are chondrocytes and cells that infiltrate from the subchondral bone. Model simulations are used to investigate how different seeding strategies and growth factor placement within the hydrogel affect the spatial distribution of both cell types. Since chondrocyte migration is extremely slow, we conclude that the hydrogel should be seeded with chondrocytes prior to culture in order to obtain zonal chondrocyte distributions typical of those associated with healthy cartilage.

Stochastic modelling of membrane filtration.

Membrane fouling during particle filtration occurs through a variety of mechanisms, including internal pore clogging by contaminants, coverage of pore entrances and deposition on the membrane surface. In this paper, we present an efficient method for modelling the behaviour of a filter, which accounts for different retention mechanisms, particle sizes and membrane geometries. The membrane is assumed to be composed of a series of, possibly interconnected, pores. The central feature is a conductivity function, which describes the blockage of each individual pore as particles arrive, which is coupled with a mechanism to account for the stochastic nature of the arrival times of particles at the pore. The result is a system of ordinary differential equations based on the pore-level interactions. We demonstrate how our model can accurately describe a wide range of filtration scenarios. Specifically, we consider a case where blocking via multiple mechanisms can occur simultaneously, which have previously required the study through individual models; the filtration of a combination of small and large particles by a track-etched membrane and particle separation using interconnected pore networks. The model is significantly faster than comparable stochastic simulations for small networks, enabling its use as a tool for efficient future simulations.

Computational modelling of placental amino acid transfer as an integrated system.

Placental amino acid transfer is essential for fetal development and its impairment is associated with poor fetal growth. Amino acid transfer is mediated by a broad array of specific plasma membrane transporters with overlapping substrate specificity. However, it is not fully understood how these different transporters work together to mediate net flux across the placenta. Therefore the aim of this study was to develop a new computational model to describe how human placental amino acid transfer functions as an integrated system. Amino acid transfer from mother to fetus requires transport across the two plasma membranes of the placental syncytiotrophoblast, each of which contains a distinct complement of transporter proteins. A compartmental modelling approach was combined with a carrier based modelling framework to represent the kinetics of the individual accumulative, exchange and facilitative classes of transporters on each plasma membrane. The model successfully captured the principal features of transplacental transfer. Modelling results clearly demonstrate how modulating transporter activity and conditions such as phenylketonuria, can increase the transfer of certain groups of amino acids, but that this comes at the cost of decreasing the transfer of others, which has implications for developing clinical treatment options in the placenta and other transporting epithelia.

Computational modelling of amino acid exchange and facilitated transport in placental membrane vesicles.

Placental amino acid transport is required for fetal development and impaired transport has been associated with poor fetal growth. It is well known that placental amino acid transport is mediated by a broad array of specific membrane transporters with overlapping substrate specificity. However, it is not fully understood how these transporters function, both individually and as an integrated system. We propose that mathematical modelling could help in further elucidating the underlying mechanisms of how these transporters mediate placental amino acid transport. The aim of this work is to model the sodium independent transport of serine, which has been assumed to follow an obligatory exchange mechanism. However, previous amino acid uptake experiments in human placental microvillous plasma membrane vesicles have persistently produced results that are seemingly incompatible with such a mechanism; i.e. transport has been observed under zero-trans conditions, in the absence of internal substrates inside the vesicles to drive exchange. This observation raises two alternative hypotheses; (i) either exchange is not fully obligatory, or (ii) exchange is indeed obligatory, but an unforeseen initial concentration of amino acid substrate is present within the vesicle which could drive exchange. To investigate these possibilities, a mathematical model for tracer uptake was developed based on carrier mediated transport, which can represent either facilitated diffusion or obligatory exchange (also referred to as uniport and antiport mechanisms, respectively). In vitro measurements of serine uptake by placental microvillous membrane vesicles were carried out and the model applied to interpret the results based on the measured apparent Michaelis-Menten parameters Km and Vmax. In addition, based on model predictions, a new time series experiment was implemented to distinguish the hypothesised transporter mechanisms. Analysis of the results indicated the presence of a facilitated transport component, while based on the model no evidence for substantial levels of endogenous amino acids within the vesicle was found.

Pore geometry regulates early stage human bone marrow cell tissue formation and organisation.

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.

A fluid mechanical explanation of the spontaneous reattachment of a previously detached Descemet membrane.

Descemet membrane detachment (DMD) is a rare but potentially serious surgical complication that arises most often during cataract surgery. A recent study (Couch, S. M. & Baratz, K. H. (2009) Cornea, 28, 1160-1163) cited the case of a patient with DMDs in both eyes, noting that though one detachment was surgically repaired, the other spontaneously reattached and needed no further treatment. A fluid mechanical model of buoyancy-driven aqueous humour flow in the anterior chamber around a DMD is developed to explain this phenomenon. The analytical model is based on the lubrication theory limit of the Navier-Stokes equations. The flow is determined for a fixed geometry and the possible motion of the DMD is then analysed. Numerical calculations are also carried out (using COMSOL© Multiphysics) to confirm the lubrication theory results. The analytical and numerical results both show that, under the correct conditions, either spontaneous reattachment or worsening of the tear may occur. We conclude that it is possible that clinical outcomes for DMDs may be controlled by adjusting the temperature difference across the eye and/or the orientation of the patient.

Review: Modelling placental amino acid transfer--from transporters to placental function.

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.

Mathematical modelling of tissue formation in chondrocyte filter cultures.

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.

Computational modelling of amino acid transfer interactions in the placenta.

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.

Design criteria for a printed tissue engineering construct: a mathematical homogenization approach.

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.

Experimental-computational evaluation of human bone marrow stromal cell spreading on trabecular bone structures.

The clinical application of macro-porous scaffolds for bone regeneration is significantly affected by the problem of insufficient cell colonization. Given the wide variety of different scaffold structures used for tissue engineering it is essential to derive relationships for cell colonization independent of scaffold architecture. To study cell population spreading on 3D structures decoupled from nutrient limitations, an in vitro culture system was developed consisting of thin slices of human trabecular bone seeded with Human Bone Marrow Stromal Cells, combined with dedicated microCT imaging and computational modeling of cell population spreading. Only the first phase of in vitro scaffold colonization was addressed, in which cells migrate and proliferate up to the stage when the surface of the bone is covered as a monolayer, a critical prerequisite for further tissue formation. The results confirm the model's ability to represent experimentally observed cell population spreading. The key advantage of the computational model was that by incorporating complex 3D structure, cell behavior can be characterized quantitatively in terms of intrinsic migration parameters, which could potentially be used for predictions on different macro-porous scaffolds subject to additional experimental validation. This type of modeling will prove useful in predicting cell colonization and improving strategies for skeletal tissue engineering.

Modelling the formation of necrotic regions in avascular tumours.

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.

Elastohydrodynamics of the eyelid wiper.

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.

Modelling the cell cycle and cell movement in multicellular tumour spheroids.

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.

Mechanisms to explain the reverse perivascular transport of solutes out of the brain.

Experimental studies and observations in the human brain indicate that interstitial fluid and solutes, such as amyloid-beta (Abeta), are eliminated from grey matter of the brain along pericapillary and periarterial pathways. It is unclear, however, what constitutes the motive force for such transport within blood vessel walls, which is in the opposite direction to blood flow. In this paper the potential for global pressure differences to achieve such transport are considered. A mathematical model is constructed in order to test the hypothesis that perivascular drainage of interstitial fluid and solutes out of brain tissue is driven by pulsations of the blood vessel walls. Here it is assumed that drainage occurs through a thin layer between astrocytes and endothelial cells or between smooth muscle cells. The model suggests that, during each pulse cycle, there are periods when fluid and solutes are driven along perivascular spaces in the reverse direction to the flow of blood. It is shown that successful drainage may depend upon some attachment of solutes to the lining of the perivascular space, in order to produce a valve-like effect, although an alternative without this requirement is also postulated. Reduction in pulse amplitude, as in ageing cerebral vessels, would prolong the attachment time, encourage precipitation of Abeta peptides in vessel walls, and impair elimination of Abeta from the brain. These factors may play a role in the pathogenesis of cerebral amyloid angiopathy and in the accumulation of Abeta in the brain in Alzheimer's disease.

Dynamics of tear film deposition and draining.

This paper investigates the deposition of the tear film on the cornea of the human eye. The tear film is laid down by the motion of the upper eyelid and then subsequently flows and thins. Of particular interest is the stability of the tear layer and the development of dry patches on the cornea. While there has been significant research on the behaviour of tear films between blinks, this paper focuses on understanding the mechanisms which control the shape and thickness of the deposited film and how this affects the subsequent film behaviour. Numerical and analytical methods are applied to a lubrication model which includes the effects of surface tension, viscosity, gravity and evaporation. The model reveals the importance of the eyelid velocity, motion of the surface lipid layer and the storage of tear film between blinks.

Mathematical modelling of skeletal repair.

Tissue engineering offers significant promise as a viable alternative to current clinical strategies for replacement of damaged tissue as a consequence of disease or trauma. Since mathematical modelling is a valuable tool in the analysis of complex systems, appropriate use of mathematical models has tremendous potential for advancing the understanding of the physical processes involved in such tissue reconstruction. In this review, the potential benefits, and limitations, of theoretical modelling in tissue engineering applications are examined with specific emphasis on tissue engineering of bone. A central tissue engineering approach is the in vivo implantation of a biomimetic scaffold seeded with an appropriate population of stem or progenitor cells. This review will therefore consider the theory behind a number of key factors affecting the success of such a strategy including: stem cell or progenitor population expansion and differentiation ex vivo; cell adhesion and migration, and the effective design of scaffolds; and delivery of nutrient to avascular structures. The focus will be on current work in this area, as well as on highlighting limitations and suggesting possible directions for future work to advance health-care for all.

Tumour dynamics and necrosis: surface tension and stability.

A model is developed for the motion of cells within a multicell spherical tumour. The model allows either for the intercellular forces to be in compression and cells to be compacted to a fixed number density, or for the cell number density to fall and cells to become isolated from each other. The model develops necrotic regions naturally due to force balances rather than being directly attributable to a critical oxygen concentration. These necrotic regions may result in a gradual reduction in local cell density rather than jump to a completely dead region. Numerical and analytical analysis of the spherically symmetric model shows that the long time behaviour of the spheroid depends on any surface tension effects created by cells on the outer surface. For small surface tension the spheroid grows linearly in time developing a large necrotic region, while for larger surface tension the growth can be halted. The linear stability to spherically symmetric perturbations of all the possible resulting steady states is revealed.