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.     

Cell Proliferation and Oxygen Diffusion in a Vascularising Scaffold

The supply of oxygen to proliferating cells within a scaffold is a key factor for the successful building of new tissue in soft tissue engineering applications. A recent in vivo model, where an arteriovenous loop is placed in a scaffold, allows a vascularising network to form within a scaffold, establishing an oxygen source within, rather than external, to the scaffold. A one-dimensional model of oxygen concentration, cell proliferation and cell migration inside such a vascularising scaffold is developed and investigated. In addition, a vascularisation model is presented, which supports a vascularisation front which moves at a constant speed. The effects of vascular growth, homogenous and heterogenous seeding, diffusion of cells and critical hypoxic oxygen concentration are considered. For homogenous seeding, a relationship between the speed of the vascular front and a parameter defining the rate of oxygen diffusion relative to the rate of oxygen consumption determines whether a hypoxic region exists at some time. In particular, an estimate of the length of time that a fixed point in the scaffold will remain under hypoxic conditions is determined. For heterogenous seeding, a Fisher-like travelling wave of cells is established behind the vascular front. These findings provide a fundamental understanding of the important interplay between the parameters and allows for a theoretical assessment of a seeding strategy in a vascularising scaffold.     

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.     

Effect of cell seeding and mechanical loading on vascularization and tissue formation inside a scaffold: A mechano-biological model using a lattice approach to simulate cell activity

Achieving successful vascularization remains one of the main problems in bone tissue engineering. After scaffold implantation, the growth of capillaries into the porous construct may be too slow to provide adequate nutrients to the cells in the scaffold interior and this inhibits tissue formation in the scaffold core. Often, prior to implantation, a controlled cell culture environment is used to stimulate cell proliferation and, once in place, the mechanical environment acting on the tissue construct is determined by the loading conditions at the implantation site. To what extent do cell seeding conditions and the construct loading environment have an effect on scaffold vascularization and tissue growth? In this study, a mechano-biological model for tissue differentiation and blood vessel growth was used to determine the influence of cell seeding on vascular network development and tissue growth inside a regular-structured bone scaffold under different loading conditions. It is predicted that increasing the number of cells seeded homogeneously reduces the rate of vascularization and the maximum penetration of the vascular network, which in turn reduces bone tissue formation. The seeding of cells in the periphery of the scaffold was predicted to be beneficial for vascularization and therefore for bone growth; however, tissue formation occurred more slowly during the first weeks after implantation compared to homogeneous seeding. Low levels of mechanical loading stimulated bone formation while high levels of loading inhibited bone formation and capillary growth. This study demonstrates the feasibility of computational design approaches for bone tissue engineering.     

Curvature- and fluid-stress-driven tissue growth in a tissue-engineering scaffold pore

Cell proliferation within a fluid-filled porous tissue-engineering scaffold depends on a sensitive choice of pore geometry and flow rates: regions of high curvature encourage cell proliferation, while a critical flow rate is required to promote growth for certain cell types. When the flow rate is too slow, the nutrient supply is limited; when it is too fast, cells may be damaged by the high fluid shear stress. As a result, determining appropriate tissue-engineering-construct geometries and operating regimes poses a significant challenge that cannot be addressed by experimentation alone. In this paper, we present a mathematical theory for the fluid flow within a pore of a tissue-engineering scaffold, which is coupled to the growth of cells on the pore walls. We exploit the slenderness of a pore that is typical in such a scenario, to derive a reduced model that enables a comprehensive analysis of the system to be performed. We derive analytical solutions in a particular case of a nearly piecewise constant growth law and compare these with numerical solutions of the reduced model. Qualitative comparisons of tissue morphologies predicted by our model, with those observed experimentally, are also made. We demonstrate how the simplified system may be used to make predictions on the design of a tissue-engineering scaffold and the appropriate operating regime that ensures a desired level of tissue growth.

     

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.     

A numerical model of heterogeneous surface strains in polymer scaffolds

In vitro bone tissue growth inside porous scaffolds can be enhanced by macroscopic cyclic compression of the construct, but the heterogeneous strain generated inside the construct must be investigated to determine appropriate levels of compression. For this purpose a linear micro-finite element (muFE) technique based on micro-computed tomography (muCT) was verified for the calculation of local displacements inside polymer scaffolds, from which local strains may be estimated. Local displacements in the axial direction at the surface of microstructures inside the scaffold in 60 locations were calculated with the muFE model, based on compression simulation of a muCT reconstruction of the scaffold. These displacements were compared with accurately measured displacements in the axial direction in the same polymer scaffold at the same 60 locations, using a micro-compression chamber and muCT reconstructions of the scaffold under two fixed levels of compression (5% and 0%). The correlation between the calculated and the measured displacements, after correction for the dependence of the axial displacement on the axial position, was r=0.786 (r2=0.617). From this we conclude that the linear muFE model is suitable to estimate local surface strains inside polymer scaffolds for tissue engineering applications. This technique can not only be used to determine appropriate parameters such as the level of macroscopic compression in experimental design, but also to investigate the cellular response to local surface strains generated inside three-dimensional scaffolds.     

The interplay between tissue growth and scaffold degradation in engineered tissue constructs

In vitro tissue engineering is emerging as a potential tool to meet the high demand for replacement tissue, caused by the increased incidence of tissue degeneration and damage. A key challenge in this field is ensuring that the mechanical properties of the engineered tissue are appropriate for the in vivo environment. Achieving this goal will require detailed understanding of the interplay between cell proliferation, extracellular matrix (ECM) deposition and scaffold degradation. In this paper, we use a mathematical model (based upon a multiphase continuum framework) to investigate the interplay between tissue growth and scaffold degradation during tissue construct evolution in vitro. Our model accommodates a cell population and culture medium, modelled as viscous fluids, together with a porous scaffold and ECM deposited by the cells, represented as rigid porous materials. We focus on tissue growth within a perfusion bioreactor system, and investigate how the predicted tissue composition is altered under the influence of (1) differential interactions between cells and the supporting scaffold and their associated ECM, (2) scaffold degradation, and (3) mechanotransduction-regulated cell proliferation and ECM deposition. Numerical simulation of the model equations reveals that scaffold heterogeneity typical of that obtained from $\mu $ CT scans of tissue engineering scaffolds can lead to significant variation in the flow-induced mechanical stimuli experienced by cells seeded in the scaffold. This leads to strong heterogeneity in the deposition of ECM. Furthermore, preferential adherence of cells to the ECM in favour of the artificial scaffold appears to have no significant influence on the eventual construct composition; adherence of cells to these supporting structures does, however, lead to cell and ECM distributions which mimic and exaggerate the heterogeneity of the underlying scaffold. Such phenomena have important ramifications for the mechanical integrity of engineered tissue constructs and their suitability for implantation in vivo.     

Determination of mechanical properties of soft tissue scaffolds by atomic force microscopy nanoindentation

While the determination of mechanical properties of a hard scaffold is relatively straightforward, the mechanical testing of a soft tissue scaffold poses significant challenges due in part to its fragility. Here, we report a new approach for characterizing the stiffness and elastic modulus of a soft scaffold through atomic force microscopy (AFM) nanoindentation. Using collagen-chitosan hydrogel scaffolds as model soft tissue scaffolds, we demonstrated the feasibility of using AFM nanoindentation to determine a force curve of a soft tissue scaffold. A mathematical model was developed to ascertain the stiffness and elastic modulus of a scaffold from its force curve obtained under different conditions. The elastic modulus of a collagen-chitosan (80%/20%, v/v) scaffold is found to be 3.69 kPa. The scaffold becomes stiffer if it contains more chitosan. The elastic modulus of a scaffold composed of 70% collagen and 30% chitosan is about 11.6 kPa. Furthermore, the stiffness of the scaffold is found to be altered significantly by extracellular matrix deposited from cells that are grown inside the scaffold. The elastic modulus of collagen-chitosan scaffolds increased from 10.5 kPa on day 3 to 63.4 kPa on day 10 when human foreskin fibroblast cells grew inside the scaffolds. Data acquired from these measurements will offer new insights into understanding cell fate regulation induced by physiochemical cues of tissue scaffolds.     

Mathematical modelling of human mesenchymal stem cell proliferation and differentiation inside artificial porous scaffolds

We present a mathematical model for the proliferation and differentiation of human mesenchymal stem cells grown inside artificial porous scaffolds under different oxygen concentrations. The values of parameters in the model are determined by comparison of the model solutions to published experimental data, complemented with a sensitivity analysis of the fitted parameters. It is shown that a simple hypothesis whereby the secretion of extra-cellular matrix (ECM) is oxygen dependent and that ECM itself stimulates cell proliferation is sufficient to explain the experimental data, which under conditions of low oxygen reveals increased total cell proliferation, upregulation of the numbers of undifferentiated cells, and extended lag phase. These results may help further to understand how cells proliferate inside artificial materials and are of importance to the field of tissue engineering.     

The role of cell-cell interactions in a two-phase model for avascular tumour growth

 A two-phase model is presented to describe avascular tumour growth. Conservation of mass equations, including oxygen-dependent cell growth and death terms, are coupled with equations of momentum conservation. The cellular phase behaves as a viscous liquid, while the viscosity of the extracellular water manifests itself as an interphase drag. It is assumed that the cells become mechanically stressed if they are too densely packed and that the tumour will try to increase its volume in order to relieve such stress. By contrast, the overlapping filopodia of sparsely populated cells create short-range attractive effects. Finally, oxygen is consumed by the cells as it diffuses through the tumour. The resulting system of equations are reduced to three, which describe the evolution of the tumour cell volume fraction, the cell speed and the oxygen tension. Numerical simulations indicate that the tumour either evolves to a travelling wave profile, in which it expands at a constant rate, or it settles to a steady state, in which the net rates of cell proliferation and death balance. The impact of varying key model parameters such as cellular viscosity, interphase drag, and cellular tension are discussed. For example, tumours consisting of well-differentiated (i.e. viscous) cells are shown to grow more slowly than those consisting of poorly-differentiated (i.e. less viscous) cells. Analytical results for the case of oxygen-independent growth are also presented, and the effects of varying the key parameters determined (the results are in line with the numerical simulations of the full problem). The key results and their biological implications are then summarised and future model refinements discussed. Received: 3 May 2001 / Revised version: 7 January 2002 / Published online: 17 July 2002     

Simulation of bone tissue formation within a porous scaffold under dynamic compression

A computational model of mechanoregulation is proposed to predict bone tissue formation stimulated mechanically by overall dynamical compression within a porous polymeric scaffold rendered by micro-CT. Dynamic compressions of 0.5–5% at 0.0025–0.025 s⁻¹ were simulated. A force-controlled dynamic compression was also performed by imposing a ramp of force from 1 to 70 N. The model predicts homogeneous mature bone tissue formation under strain levels of 0.5–1% at strain rates of 0.0025–0.005 s⁻¹. Under higher levels of strain and strain rates, the scaffold shows heterogeneous mechanical behaviour which leads to the formation of a heterogeneous tissue with a mixture of mature bone and fibrous tissue. A fibrous tissue layer was also predicted under the force-controlled dynamic compression, although the same force magnitude was found promoting only mature bone during a strain-controlled compression. The model shows that the mechanical stimulation of bone tissue formation within a porous scaffold closely depends on the loading history and on the mechanical behaviour of the scaffold at local and global scales.     

The impact of phenotypic switching on glioblastoma growth and invasion

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.     

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.     

Analysis of cell growth and diffusion in a scaffold for cartilage tissue engineering

Developments in tissue engineering over the past decade have offered promising future for the repair and reconstruction of damaged tissues. To regenerate three dimensional and weight-bearing implants, advances in biomaterials and manufacturing technologies prompted cell cultivations with natural or artificial scaffolds, in which cells are allowed to proliferate, migrate, and differentiate in vitro. In this article, we develop a mathematical model for cell growth in a porous scaffold. By treating the cell-scaffold construct as a porous medium, a continuum model is set up based on basic principles of mass conservation. In addition to cell growth kinetics, we incorporate cell diffusion in the model to describe the effects of cell random walks. Computational results are compared to experimental data found in the literature. With this model, we are able to investigate cell motility, heterogeneous cell distributions, and non-uniform seeding for tissue engineering applications. Results show that random walks tend to enhance uniform cell spreads in space, which in turn increases the probabilities for cells to acquire nutrients; therefore random walks are likely to be a positive contribution to the overall cell growth on scaffolds.     

Three‐dimensional nanocharacterization of porous hydrogel with ion and electron beams

Porous hydrogels provide an excellent environment for cell growth and tissue regeneration, with high permeability for oxygen, nutrients, and other water‐soluble metabolites through their high water‐content matrix. The ability to image three‐dimensional (3D) cell growth is crucial for understanding and studying various cellular activities in 3D context, particularly for designing new tissue engineering scaffold, but it is still challenging to study cell‐biomaterial interfaces with high resolution imaging. We demonstrate using focused ion beam (FIB) milling, electron imaging, and associated microanalysis techniques that novel 3D characterizations can be performed effectively on cells growing inside 3D hydrogel scaffold. With FIB‐tomography, the porous microstructures were revealed at nanometer resolution, and the cells grown inside. The results provide a unique 3D measurement of hydrogel porosity, as compared with those from porosimetry, and offer crucial insights into material factors affecting cell proliferation at specific regions within the scaffold. We also proved that high throughput correlative imaging of cell growth is viable through a silicon membrane based environment. The proposed approaches, together with the protocols developed, provide a unique platform for analysis of the microstructures of novel biomaterials, and for exploration of their interactions with the cells as well. Biotechnol. Bioeng. 2013; 110: 318–326. © 2012 Wiley Periodicals, Inc.     

Fluid transport and mechanical properties of articular cartilage: a review

This review is aimed at unifying our understanding of cartilage viscoelastic properties in compression, in particular the role of compression-dependent permeability in controlling interstitial fluid flow and its contribution to the observed viscoelastic effects. During the previous decade, it was shown that compression causes the permeability of cartilage to drop in a functional manner described by k = ko exp (epsilon M) where ko and M were defined as intrinsic permeability parameters and epsilon is the dilatation of the solid matrix (epsilon = tr delta u). Since permeability is inversely related to the diffusive drag coefficient of relative fluid motion with respect to the porous solid matrix, the measured load-deformation response of the tissue must therefore also depend on the non-linearly permeable nature of the tissue. We have summarized in this review our understanding of this non-linear phenomenon. This understanding of these flow-dependent viscoelastic effects are put into the historical perspective of a comprehensive literature review of earlier attempts to model the compressive viscoelastic properties of articular cartilage.     

Supercritical fluid-assisted controllable fabrication of open and highly interconnected porous scaffolds for bone tissue engineering

Recently tremendous progress has been evidenced by the advancements in developing innovative three-dimensional(3 D)scaffolds using various techniques for addressing the autogenous grafting of bone. In this work, we demonstrated the fabrication of porous polycaprolactone(PCL) scaffolds for osteogenic differentiation based on supercritical fluid-assisted hybrid processes of phase inversion and foaming. This eco-friendly process resulted in the highly porous biomimetic scaffolds with open and interconnected architectures. Initially, a 2~3 factorial experiment was designed for investigating the relative significance of various processing parameters and achieving better control over the porosity as well as the compressive mechanical properties of the scaffold. Then, single factor experiment was carried out to understand the effects of various processing parameters on the morphology of scaffolds. On the other hand, we encapsulated a growth factor, i.e., bone morphogenic protein-2(BMP-2), as a model protein in these porous scaffolds for evaluating their osteogenic differentiation. In vitro investigations of growth factor loaded PCL scaffolds using bone marrow stromal cells(BMSCs) have shown that these growth factor-encumbered scaffolds were capable of differentiating the cells over the control experiments. Furthermore, the osteogenic differentiation was confirmed by measuring the cell proliferation, and alkaline phosphatase(ALP) activity, which were significantly higher demonstrating the active bone growth. Together, these results have suggested that the fabrication of growth factor-loaded porous scaffolds prepared by the eco-friendly hybrid processing efficiently promoted the osteogenic differentiation and may have a significant potential in bone tissue engineering.     

Predicting the stress distribution within scaffolds with ordered architecture

Current tissue engineering technologies involve the seeding of cells on porous scaffolds, within which the cells can proliferate and differentiate, when cultured in bioreactors. The flow of culture media through the scaffolds generates stresses that are important for both cell differentiation and cell growth. A recent study [Appl. Phys. Lett. 97 (2010), 024101] showed that flow-induced stresses inside highly porous and randomly structured scaffolds follow a three-point gamma probability density function (p.d.f.). The goal of the present study is to further investigate whether the same p.d.f. can also describe the distribution of stresses in structured porous scaffolds, what is the range of scaffold porosity for which the distribution is valid, and what is the physical reason for such behavior. To do that, the p.d.f. of flow-induced stresses in different scaffold geometries were calculated via flow dynamics simulations. It was found that the direction of flow relative to the internal architecture of the scaffolds is important for stress distributions. The stress distributions follow a common distribution within statistically acceptable accuracy, when the flow direction does not coincide with the direction of internal structural elements of the scaffold.