A mathematical model for bone tissue regeneration inside a specific type of scaffold

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.     

Individual-based modelling of angiogenesis inside three-dimensional porous biomaterials

This paper presents a simulation modelling framework to study the growth of blood vessels and cells through a porous tissue engineering scaffold. The model simulates the migration of capillaries and the formation of a vascular network through a single pore of a tissue engineering scaffold when it is embedded in living tissue. The model also describes how the flow of blood through the network changes as growth proceeds. Results are given for how the different strategies of seeding the pore with cells affects the extent of vascularisation. Also simulations are made to compare results where the values of different model parameters are varied such as the pore dimensions, the density of endothelial cells seeded into the pore, and the release rate of growth factor from the scaffold into the pore. The modelling framework described in this paper is useful for exploring experimental strategies for producing well-vascularised tissue engineered constructs, and is therefore potentially important to the field of regenerative medicine.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.     

Morphological characterization of a novel scaffold for anterior cruciate ligament tissue engineering

Tissue engineering offers an interesting alternative to current anterior cruciate ligament (ACL) surgeries. Indeed, a tissue-engineered solution could ideally overcome the long-term complications due to actual ACL reconstruction by being gradually replaced by biological tissue. Key requirements concerning the ideal scaffold for ligament tissue engineering are numerous and concern its mechanical properties, biochemical nature, and morphology. This study is aimed at predicting the morphology of a novel scaffold for ligament tissue engineering, based on multilayer braided biodegradable copoly(lactic acid-co-(e-caprolactone)) (PLCL) fibers The process used to create the scaffold is briefly presented, and the degradations of the material before and after the scaffold processing are compared. The process offers varying parameters, such as the number of layers in the scaffold, the pitch length of the braid, and the fibers' diameter. The prediction of the morphology in terms of pore size distribution and pores interconnectivity as a function of these parameters is performed numerically using an original method based on a virtual scaffold. The virtual scaffold geometry and the prediction of pore size distribution are evaluated by comparison with experimental results. The presented process permits creation of a tailorable scaffold for ligament tissue engineering using basic equipment and from minimum amounts of raw material. The virtual scaffold geometry closely mimics the geometry of real scaffolds, and the prediction of the pore size distribution is found to be in good accordance with measurements on real scaffolds. The scaffold offers an interconnected network of pores the sizes of which are adjustable by playing on the process parameters and are able to match the ideal pore size reported for tissue ingrowth. The adjustability of the presented scaffold could permit its application in both classical ACL reconstructions and anatomical double-bundle reconstructions. The precise knowledge of the scaffold morphology using the virtual scaffold will be useful to interpret the activity of cells once it will be seeded into the scaffold. An interesting perspective of the present work is to perform a similar study aiming at predicting the mechanical response of the scaffold according to the same process parameters, by implanting the virtual scaffold into a finite element algorithm.     

Thermally induced proliferation of pores in a model fluid membrane.

The growth of thermally induced pores in a two-dimensional model fluid membrane is investigated by Monte Carlo simulation. Holes appear in the membrane via an activated process, and their subsequent growth is controlled by an edge energy per unit length or line tension. The barrier height and line tension, together with a lateral tension, are the independent parameters of the model. In the resulting phase diagram, a rupture transition separates an intact membrane from a disintegrated state. The approach to the ruptured state shows distinct regimes. Reducing the barrier height at large line tension produces multiple, quasi-independent, small holes whose behavior is dominated by their edge energy, whereas at lower line tensions shape fluctuations of the holes facilitate their coalescence into a single large hole. At a small value of line tension and large barrier height, a single hole spontaneously permeabilizes the membrane in an entropically driven phase transition. Entropy dominates pore growth for line tensions not far below those measured for artificial vesicles. Permeabilization of lipid bilayers by certain peptides involves perturbing lipid-lipid cohesive energies, and our simulations show that at small line tensions the entropy of hole shape fluctuations destroys the model membrane's stability.     

The osmotic flow of an electrolyte through a charged porous membrane

A pore model in which the pore wall has a continuous distribution of electrical charge is used to investigate the osmotic flow through a charged permeable membrane separating electrolyte solutions of unequal concentrations. The pore is treated as a long, circular, cylindrical duct. The analysis is based on a continuum formulation in which a dilute electrolyte solution is described by the coupled Nernst-Planck/Poisson creeping flow equations. Account is taken of the significant size of the electrolyte ions (assumed to be rigid spheres) when compared with the diameter of the membrane pores. Analytical solutions for the ion concentrations, hydrostatic pressure and electrostatic potential in the electrolyte solutions are given and an intra-pore flow solution is derived. A mathematical expression for the osmotic reflection coefficient as a function of the solute ion: pore diameter ratio λ and the solute fluxes is obtained. Approximate solutions are quoted which relate the solute fluxes and the solution electrostatic potentials at the membrane surfaces to the bulk solution concentrations, the membrane pore charge and pore geometry. The osmotic reflection coefficient is thus determined as a function of these parameters.     

The exocytotic fusion pore interface: a model of the site of neurotransmitter release

infrastructurel techniques have shown that an early event in the exocytotic fusion of a secretory vesicle is the formation of a narrow, water-filled pore spanning both the vesicle and plasma membranes and connecting the lumen of the secretory vesicle to the extracellular environment. Smaller precursors of the exocytotic fusion pore have been detected using electrophysio-logical techniques, which reveal a dynamic fusion pore that quickly expands to the size of the pores seen with electron microscopy. While it is clear that in the latter stages of expansion, when the size of the fusion pore is several orders of magnitude bigger than any known macromolecule, the fusion pore must be mainly made of lipids, the structure of the smaller precursors is unknown. Patch-clamp measurements of the activity of individual fusion pores in mast cells have shown that the fusion pore has some unusual and unexpected properties, namely that there is a large flux of lipid through the pore and the rate of pore closure has a discontinuous temperature dependency, suggesting a purely lipidic fusion pore. Moreover, comparisons of experimental data with theoretical fusion pores and with breakdown pores support the view that the fusion pore is initially a pore through a single bilayer, as would be expected for membrane fusion proceeding through a hemifusion mechanism. Based on these observations we present a model where the fusion pore is initially a pore through a single bilayer. Fusion pore formation is regulated by a macromolecular scaffold of proteins that is responsible for bringing the plasma membrane into a highly curved dimple very close to a tense secretory granule membrane, creating the architecture where the strongly attractive hydrophobic force causes the membranes to form a ‘hemifusion’ intermediate. Membrane fusion is completed by the formation of an aqueous pore after rupture of the shared bilayer. We also propose that the microenvironment of the interface when the pore first opens, dominated by the charged groups on the secretory vesicle matrix and phospholipids, will greatly influence the release of secretory products.     

Theoretical analysis of insulin-dependent glucose uptake heterogeneity in 3D bioreactor cell culture

Three-dimensional (3D) cell cultures in bioreactors are becoming relevant as models for biological and physiological in vitro studies. In such systems, mathematical models can assist the experiment design that links the macroscopic properties to single-cell responses. We investigated the relationship between biochemical stimuli and cell response within a 3D cell culture in scaffold with heterogeneous porosity. Specifically, we studied the effect of insulin on the local glucose metabolism as a function of 3D pore size distribution. The multiscale mathematical model combines the mass transport within a 3D scaffold and a signaling pathways model. It considers the scaffold heterogeneity, and it describes spatiotemporal concentration of metabolites, biochemical stimuli, and cell density. The signaling model was integrated into this model, linking the local insulin concentration at cell membrane to the glucose uptake rate through glucose transporter type 4 (GLUT4) translocation from the cytosol to the cell membrane. The integrated model determines the cell response heterogeneities in a single channel, hence the biological response distribution in a 3D system. It also provides macroscopic outcomes to evaluate the feasibility of an experimental measurement of the system response. From our analysis, it became apparent that the flow rate is the most important operative variable, and that an optimum value ensures a fast and detectable cell response. This model on insulin-dependent glucose consumption rate offers insight into the cell metabolism physiology, which is a fundamental requirement for the study metabolic disorder such as Type 2 diabetes mellitus, in which the physiological insulin-dependent glucose metabolism is impaired.     

A growth-controlled model of the shape of a sunflower head

A mathematical model is presented which predicts the shape of a sunflower receptacle (or the compact receptacle of various other taxa) and the pattern of its floral parts (florets) from the time of their initiation to maturity. The model assumes that the expansion and curving of the receptacle surface is just sufficient to accommodate the development of the florets, thus minimizing the quantity of plant tissue involved. The model assumes a fixed angular separation (divergence) between successive florets, an S-shaped (sigmoidal) growth function followed by each florets, and a fixed time delay (period) between the initiation of successive florets. It is further assumes that the shape and relative position occupied by the florets on the receptacle surface are invariant in time. By this theory, the shape of the receptacle surface is fully determined once the mathematical form of the growth function is specified. Using the logistic growth function, the theory is tested against the measured shapes of plant receptacles from different taxa at various points in their development. The least-squares adjusted fits to the theory are, in most cases, very good indeed.     

Flow characteristics of red cell containing fluids through pores. The effect of filter plugging, a mathematical model

A mathematical model was developed that describes the effects of filter plugging on flow through 3 micron pore polycarbonate filters as a function of time, pressure, and cell concentration, both under stirring and nonstirring conditions. The mathematical constants for the model were derived from experimental data generated with a filtration apparatus, and were tested by using various concentrations of cells that are able to plug filter pores. A computer simulation program was written to test the model over a wide range of nonfilterable cell concentrations.     

Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds--a modeling approach

Bone tissue engineering is a promising strategy to repair local defects by implanting biodegradable scaffolds which undergo remodeling and are replaced completely by autologous bone tissue. Here, we consider a Keller-Segel model to describe the chemotaxis of bone marrow-derived mesenchymal stem cells (BMSCs) into a mineralized collagen scaffold. Following recent experimental results in bone healing, demonstrating that a sub-population of BMSCs can be guided into 3D scaffolds by gradients of signaling molecules such as SDF-1α, we consider a population of BMSCs on the surface of the pore structure of the scaffold and the chemoattractant SDF-1α within the pores. The resulting model is a coupled bulk/surface model which we reformulate following a diffuse-interface approach in which the geometry is implicitly described using a phase-field function. We explain how to obtain such an implicit representation and present numerical results on μCT-data for real scaffolds, assuming a diffusion of SDF-1α being coupled to diffusion and chemotaxis of the cells towards SDF-1α. We observe a slowing-down of BMSC ingrowth after the scaffold becomes saturated with SDF-1α, suggesting that a slow release of SDF-1α avoiding an early saturation is required to enable a complete colonization of the scaffold. The validation of our results is possible via SDF-1α release from injectable carrier materials, and an adaptation of our model to similar coupled bulk/surface problems such as remodeling processes seems attractive.     

A model of lipid rearrangements during pore formation in the DPPC lipid bilayer

Context: The molecular bases of pore formation in the lipid bilayer remain unclear, as do the exact characteristics of their sizes and distributions. To understand this process, numerous studies have been performed on model lipid membranes including cell-sized giant unilamellar vesicles (GUV). The effect of an electric field on DPPC GUV depends on the lipid membrane state: in the liquid crystalline phase the created pores have a cylinder-like shape, whereas in the gel phase a crack has been observed.

Objective: The aim of the study was to investigate the geometry of pores created in a lipid bilayer in gel and liquid crystalline phases in reference to literature experimental data.

Methods: A mathematical model of the pore in a DPPC lipid bilayer developed based on the law of conservation of mass and the assumption of constant volume of lipid molecules, independent of their conformation, allows for analysis of pore shape and accompanying molecular rearrangements.

Results: The membrane area occupied by the pore of a cylinder-like shape is greater than the membrane area occupied by lipid molecules creating the pore structure (before pore appearance). Creation of such pores requires more space, which can be achieved by conformational changes of lipid chains toward a more compact state. This process is impossible for a membrane in the most compact, gel phase.

Discussion and conclusions: We show that the geometry of the pores formed in the lipid bilayer in the gel phase must be different from the cylinder shape formed in the lipid bilayer in a liquid crystalline state, confirming experimental studies. Furthermore, we characterize the occurrence of the ‘buffer’ zone surrounding pores in the liquid crystalline phase as a mechanism of separation of neighbouring pores.     

Mathematical model and numerical simulation of the cell growth in scaffolds

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l ₂ error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.     

Model of creation and evolution of stable electropores for DNA delivery

Electroporation, in which electric pulses create transient pores in the cell membrane, is becoming an important technique for gene therapy. To enable entry of supercoiled DNA into cells, the pores should have sufficiently large radii (>10 nm), remain open long enough for the DNA chain to enter the cell (milliseconds), and should not cause membrane rupture. This study presents a model that can predict such macropores. The distinctive features of this model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. The model is used to explore the process of pore creation and evolution and to determine the number and size of pores as a function of the pulse magnitude and duration. Next, our electroporation model is combined with a heuristic model of DNA uptake and used to predict the dependence of DNA uptake on pulsing parameters. Finally, the model is used to examine the mechanism of a two-pulse protocol, which was proposed specifically for gene delivery. The comparison between experimental results and the model suggests that this model is well-suited for the investigation of electroporation-mediated DNA delivery.     

In silico multi‐scale model of transport and dynamic seeding in a bone tissue engineering perfusion bioreactor

Computer simulations can potentially be used to design, predict, and inform properties for tissue engineering perfusion bioreactors. In this work, we investigate the flow properties that result from a particular poly‐L ‐lactide porous scaffold and a particular choice of perfusion bioreactor vessel design used in bone tissue engineering. We also propose a model to investigate the dynamic seeding properties such as the homogeneity (or lack of) of the cellular distribution within the scaffold of the perfusion bioreactor: a pre‐requisite for the subsequent successful uniform growth of a viable bone tissue engineered construct. Flows inside geometrically complex scaffolds have been investigated previously and results shown at these pore scales. Here, it is our aim to show accurately that through the use of modern high performance computers that the bioreactor device scale that encloses a scaffold can affect the flows and stresses within the pores throughout the scaffold which has implications for bioreactor design, control, and use. Central to this work is that the boundary conditions are derived from micro computed tomography scans of both a device chamber and scaffold in order to avoid generalizations and uncertainties. Dynamic seeding methods have also been shown to provide certain advantages over static seeding methods. We propose here a novel coupled model for dynamic seeding accounting for flow, species mass transport and cell advection‐diffusion‐attachment tuned for bone tissue engineering. The model highlights the timescale differences between different species suggesting that traditional homogeneous porous flow models of transport must be applied with caution to perfusion bioreactors. Our in silico data illustrate the extent to which these experiments have the potential to contribute to future design and development of large‐scale bioreactors. Biotechnol. Bioeng. 2013; 110: 1221–1230. © 2012 Wiley Periodicals, Inc.     

Cascades of transient pores in giant vesicles: line tension and transport

Under ordinary circumstances, the membrane tension of a giant unilamellar vesicle is essentially nil. Using visible light, we stretch the vesicles, increasing the membrane tension until the membrane responds by the sudden opening of a large pore (several micrometers in size). Only a single pore is observed at a time in a given vesicle. However, a cascade of transient pores appear, up to 30-40 in succession, in the same vesicle. These pores are transient: they reseal within a few seconds as the inner liquid leaks out. The membrane tension, which is the driving force for pore opening, is relaxed with the opening of a pore and the leakage of the inner liquid; the line tension of the pore's edge is then able to drive the closure of a pore. We use fluorescent membrane probes and real-time videomicroscopy to study the dynamics of the pores. These can be visualized only if the vesicles are prepared in a viscous solution to slow down the leakout of the internal liquid. From measurements of the closure velocity of the pores, we are able to infer the line tension,. We have studied the effect of the shape of inclusion molecules on. Cholesterol, which can be modeled as an inverted cone-shaped molecule, increases the line tension when incorporated into the bilayers. Conversely, addition of cone-shaped detergents reduces. The effect of some detergents can be dramatic, reducing by two orders of magnitude, and increasing pore lifetimes up to several minutes. We give some examples of transport through transient pores and present a rough measurement of the leakout velocity of the inner liquid through a pore. We discuss how our results can be extended to less viscous aqueous solutions which are more relevant for biological systems and biotechnological applications.     

Quantification of fluid shear stress in bone tissue engineering scaffolds with spherical and cubical pore architectures

Recent studies have shown that mechanical stimulation, in the form of fluid perfusion and mechanical compression, can enhance osteogenic differentiation of mesenchymal stem cells and bone cells within tissue engineering scaffolds in vitro. The precise nature of mechanical stimulation within tissue engineering scaffolds is not only dictated by the exogenously applied loading regime, but also depends on the geometric features of the scaffold, in particular architecture, pore size and porosity. However, the precise contribution of each geometric feature towards the resulting mechanical stimulation within a scaffold is difficult to characterise due to the wide range of interacting parameters. In this study, we have applied a fluid–structure interaction model to investigate the role of scaffold geometry (architecture, pore size and porosity) on pore wall shear stress (WSS) under a range of different loading scenarios: fluid perfusion, mechanical compression and a combination of perfusion and compression. It is found that scaffold geometry (spherical and cubical pores), in particular the pore size, has a significant influence on the stimulation within scaffolds. Furthermore, we observed an amplified WSS within scaffolds under a combination of fluid perfusion and mechanical compression, which exceeded that caused by individual fluid perfusion or mechanical compression approximately threefold. By conducting this comprehensive parametric variation study, an expression was generated to allow the design and optimisation of 3D TE scaffolds and inform experimental loading regimes so that a desired level of mechanical stimulation, in terms of WSS is generated within the scaffold.     

PhoE protein pore of the outer membrane of Escherichia coli K12 is a particularly efficient channel for organic and inorganic phosphate

This study was undertaken to investigate the proposed in vivo pore function of PhoE protein, an Escherichia coli K12 outer membrane protein induced by growth under phosphate limitation and to compare it with those of the constitutive pore proteins OmpF and OmpC. Appropriate mutant strains were constructed containing only one of the proteins PhoE, OmpF or OmpC, or none of these proteins at all. By measuring rates of nutrient uptake at low solute concentrations, the proposed pore function of PhoE protein was confirmed as the presence of the protein facilitates the diffusion of Pi through the outer membrane, such as a pore protein deficient strain behaves as a Km mutant. Comparison of the rates of permeation of Pi, glycerol 3-phosphate and glucose 6-phosphate through pores formed by PhoE, OmpF and OmpC proteins shows that PhoE protein is the most effective pore in facilitating the diffusion of Pi and phosphorus-containing compounds. The three types of pores were about equally effective in facilitating the permeation of glucose and arsenate. Possible reasons for the preference for Pi and Pi-containing solutes are discussed.     

Fabrication of individual scaffolds based on a patient-specific alveolar bone defect model

Fabricating individualized tissue engineering scaffolds based on the three-dimensional shape of patient bone defects is required for the successful clinical application of bone tissue engineering. However, there are currently no reported studies of individualized bone tissue engineering scaffolds that truly reproduce a patient-specific bone defect. We fabricated individualized tissue engineering scaffolds based on alveolar bone defects. The individualized poly(lactide-co-glycolide) and tricalcium phosphate composite scaffolds were custom-made by acquiring the three-dimensional model through computed tomography, which was input into the computer-aided low-temperature deposition manufacturing system. The three-dimensional shape of the fabricated scaffold was identical to the patient-specific alveolar bone defects, with an average macropore diameter of 380 μm, micropore diameters ranging from 3 to 5 μm, and an average porosity of 87.4%. The mechanical properties of the scaffold were similar to adult cancellous bone. Scaffold biocompatibility was confirmed by attachment and proliferation of human bone marrow mesenchymal stem cells. Successful realization of individualized scaffold fabrication will enable clinical application of tissue-engineered bone at an early date.