Identifying chondrogenesis strategies for tissue engineering of articular cartilage.

A key step in the tissue engineering of articular cartilage is the chondrogenic differentiation of mesenchymal stem cells (MSCs) into chondrocytes (native cartilage cells). Chondrogenesis is regulated by transforming growth factor-ß (TGF-ß), a short-lived cytokine whose effect is prolonged by storage in the extracellular matrix. Tissue engineering applications aim to maximise the yield of differentiated MSCs. Recent experiments involve seeding a hydrogel construct with a layer of MSCs lying below a layer of chondrocytes, stimulating the seeded cells in the construct from above with exogenous TGF-ß and then culturing it in vitro. To investigate the efficacy of this strategy, we develop a mathematical model to describe the interactions between MSCs, chondrocytes and TGF-ß. Using this model, we investigate the effect of varying the initial concentration of TGF-ß, the initial densities of the MSCs and chondrocytes, and the relative depths of the two layers on the long-time composition of the tissue construct.

Inducing chondrogenesis in MSC/chondrocyte co-cultures using exogenous TGF-ß: a mathematical model.

The differentiation of mesenchymal stem cells (MSCs) into chondrocytes (native cartilage cells), or chondrogenesis, is a key step in the tissue engineering of articular cartilage, where the motility and high proliferation rate of MSCs used as seed cells are exploited. Chondrogenesis is regulated by transforming growth factor-beta (TGF-ß), a short-lived cytokine whose effect is prolonged by storage in the extracellular matrix. Tissue engineering applications require the complete differentiation of an initial population of MSCs, and two common strategies used to achieve this in vitro are (1) co-culture the MSCs with chondrocytes, which constitutively produce TGF-ß; or (2) add exogenous TGF-ß. To investigate these strategies we develop an ordinary differential equation model of the interactions between TGF-ß, MSCs and chondrocyte. Here the dynamics of TGF-ß are much faster than those of the cell processes; this difference in time-scales is exploited to simplify subsequent model analysis. Using our model we demonstrate that under strategy 1 complete chondrogenesis will be induced if the initial proportion of chondrocytes exceeds a critical value. Similarly, under strategy 2 we find that there is a critical concentration of exogenous TGF-ß above which all MSCs will ultimately differentiate. Finally, we use the model to demonstrate the potential advantages of adopting a hybrid strategy where exogenous TGF-ß is added to a co-culture of MSCs and chondrocytes, as compared to using either strategy 1 or 2 in isolation.

Stem Cell Differentiation as a Non-Markov Stochastic Process.

Pluripotent stem cells can self-renew in culture and differentiate along all somatic lineages in vivo. While much is known about the molecular basis of pluripotency, the mechanisms of differentiation remain unclear. Here, we profile individual mouse embryonic stem cells as they progress along the neuronal lineage. We observe that cells pass from the pluripotent state to the neuronal state via an intermediate epiblast-like state. However, analysis of the rate at which cells enter and exit these observed cell states using a hidden Markov model indicates the presence of a chain of unobserved molecular states that each cell transits through stochastically in sequence. This chain of hidden states allows individual cells to record their position on the differentiation trajectory, thereby encoding a simple form of cellular memory. We suggest a statistical mechanics interpretation of these results that distinguishes between functionally distinct cellular "macrostates" and functionally similar molecular "microstates" and propose a model of stem cell differentiation as a non-Markov stochastic process.

Extracellular Matrix Deposition in Engineered Micromass Cartilage Pellet Cultures: Measurements and Modelling.

This article explores possible mechanisms governing extracellular matrix deposition in engineered cartilaginous cell pellets. A theoretical investigation is carried out alongside an experimental study measuring proteoglycan and collagen volume fractions within murine chondrogenic (ATDC-5) cell pellets. The simple mathematical model, which adopts a nutrient-dependent proteoglycan production rate, successfully reproduces the periphery-dominated proteoglycan deposition, characteristic of the growth pattern observed experimentally within pellets after 21 days of culture. The results suggest that this inhomogeneous proteoglycan production is due to nutrient deficiencies at the pellet centre. Our model analysis further indicates that a spatially uniform distribution of proteoglycan matrix could be maintained by initiating the culture process with a smaller-sized pellet. Finally, possible extensions are put forward with an aim to improve the model predictions for the early behaviour, where different mechanisms appear to dominate the matrix production within the pellets.

Integration of computational modeling with membrane transport studies reveals new insights into amino acid exchange transport mechanisms.

Uptake of system L amino acid substrates into isolated placental plasma membrane vesicles in the absence of opposing side amino acid (zero-trans uptake) is incompatible with the concept of obligatory exchange, where influx of amino acid is coupled to efflux. We therefore hypothesized that system L amino acid exchange transporters are not fully obligatory and/or that amino acids are initially present inside the vesicles. To address this, we combined computational modeling with vesicle transport assays and transporter localization studies to investigate the mechanisms mediating [(14)C]L-serine (a system L substrate) transport into human placental microvillous plasma membrane (MVM) vesicles. The carrier model provided a quantitative framework to test the 2 hypotheses that l-serine transport occurs by either obligate exchange or nonobligate exchange coupled with facilitated transport (mixed transport model). The computational model could only account for experimental [(14)C]L-serine uptake data when the transporter was not exclusively in exchange mode, best described by the mixed transport model. MVM vesicle isolates contained endogenous amino acids allowing for potential contribution to zero-trans uptake. Both L-type amino acid transporter (LAT)1 and LAT2 subtypes of system L were distributed to MVM, with L-serine transport attributed to LAT2. These findings suggest that exchange transporters do not function exclusively as obligate exchangers.

The design of microfluidic affinity chromatography systems for the separation of bioanalytes.

The analytical (numerical) design of planar microfluidic affinity chromatography devices, which consist of multiple separation lanes and multiple, different surface-immobilised receptor patterns in each lane, is described. The model is based on the analytical solution of the transport-reaction equations in microfluidic systems of low Gratz number and for injection of small analyte plugs. The results reveal a simple approach for the design of microfluidic affinity chromatography devices tailored to the separation of bioanalytes, where receptors with high binding affinity are available. These devices have been designed for bioanalytical applications in mind, most notably for the proteomics field; the results are illustrated with an example using ß-Amyloid binding peptides.

A continuum model for the development of tissue-engineered cartilage around a chondrocyte.

The limited ability of cartilage to repair when damaged has led to the investigation of tissue engineering as a method for reconstructing cartilage. We propose a continuum multispecies model for the development of cartilage around a single chondrocyte. As in healthy cartilage, the model predicts a balance between synthesis, transport, binding and decay of matrix components. Two mechanisms are investigated for the transport of soluble matrix components: diffusion and advection, caused by displacement of the scaffold medium. Numerical results indicate that a parameter defined by the ratio of the flux of soluble components out of the chondrocyte and its diffusive flux determines which of these mechanisms is dominant. We investigate the diffusion-dominated and advection-dominated limiting cases using perturbation analysis. Using parameter values from the literature, our modelling results suggest that both diffusion and advection are significant mechanisms in developing cartilage. Moreover, in this parameter regime, results are particularly sensitive to parameter values. These two observations could explain differences observed experimentally between various scaffold media. Modelling results are also used to predict the minimum chondrocyte seeding density required to produce functional cartilage.

Stochasticity and the molecular mechanisms of induced pluripotency.

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.

Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration.

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.

Computational modelling of cell spreading and tissue regeneration in porous scaffolds.

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.

A non-invasive method for in situ quantification of subpopulation behaviour in mixed cell culture.

Ongoing advances in quantitative molecular- and cellular-biology highlight the need for correspondingly quantitative methods in tissue-biology, in which the presence and activity of specific cell-subpopulations can be assessed in situ. However, many experimental techniques disturb the natural tissue balance, making it difficult to draw realistic conclusions concerning in situ cell behaviour. In this study, we present a widely applicable and minimally invasive method which combines fluorescence cell labelling, retrospective image analysis and mathematical data processing to detect the presence and activity of cell subpopulations, using adhesion patterns in STRO-1 immunoselected human mesenchymal populations and the homogeneous osteoblast-like MG63 continuous cell line as an illustration. Adhesion is considered on tissue culture plastic and fibronectin surfaces, using cell area as a readily obtainable and individual cell specific measure of spreading. The underlying statistical distributions of cell areas are investigated and mappings between distributions are examined using a combination of graphical and non-parametric statistical methods. We show that activity can be quantified in subpopulations as small as 1% by cell number, and outline behaviour of significant subpopulations in both STRO-1+/- fractions. This method has considerable potential to understand in situ cell behaviour and thus has wide applicability, for example in developmental biology and tissue engineering.

Heterogeneous proliferation within engineered cartilaginous tissue: the role of oxygen tension.

This article investigates heterogeneous proliferation within a seeded three-dimensional scaffold structure with the purpose of improving protocols for engineered tissue growth. A simple mathematical model is developed to examine the very strong interaction between evolving oxygen profiles and cell distributions within cartilaginous constructs. A comparison between predictions based on the model and experimental evidence is given for both spatial and temporal evolution of the oxygen tension and cell number density, showing that behaviour for the first 14 days can be explained well by the mathematical model. The dependency of the cellular proliferation rate on the oxygen tension is examined and shown to be similar in size to previous work but linear in form. The results show that cell-scaffold constructs that rely solely on diffusion for their supply of nutrients will inevitably produce proliferation-dominated regions near the outer edge of the scaffold in situations when the cell number density and oxygen consumption rate exceed a critical level. Possible strategies for reducing such non-uniform proliferation, including the conventional methods of enhancing oxygen transport, are outlined based on the model predictions.

A mathematical model of dynamic glioma-host interactions: receptor-mediated invasion and local proteolysis.

We present a mathematical model of glioma spread based on cellular movement by receptor-mediated haptotaxis, local proteolysis of healthy tissue components by glioma-derived proteinases, malignant proliferative enhancement and host up-regulation of specific key extracellular matrix (ECM) components in response to the invading glioma. We subsequently consider the nature of glioma-host interactions as predicted by our model in order to test the hypothesis given in (Knott et al. (1998) that production of adhesive ECM components by the brain in response to the invading glioma may have the counter-intuitive effect of enhancing glioma invasion by assisting haptotactic migration. We suggest that host production of certain adhesive ECM chemicals can have a profound effect on both glioma invasion speed and the character of the glioma-host interface. In particular, we conclude that up-regulation of host ECM production in the vicinity of the glioma may produce a less diffuse glioma, providing clearer demarcation between glioma and healthy tissue, and thus improving the possibility of surgical resection within reasonable bounds.

Residual stress generation and necrosis formation in multi-cell tumour spheroids.

We consider how cell proliferation and death generate residual stresses within a multi-cell tumour spheroid (MCTS). Previous work by Jones and co-workers [8] has shown that isotropic growth in a purely elastic MCTS produces growth induced stresses which eventually become unbounded, and hence are physically unrealistic. Since viscoelastic materials show stress relaxation under a fixed deformation we consider the effect of the addition of a small amount of viscosity to the elastic system by examining formation of equilibrium stress profiles within a Maxwell type viscoelastic MCTS. A model of necrosis formation based upon that proposed by Please and co-workers (see [16] [17] [18]) is then presented in which necrosis forms under conditions of adverse mechanical stress rather than in regions of extreme chemical stress as is usually assumed. The influence of rheology on necrosis formation is then investigated, and it is shown that the excessive stress generated in the purely elastic tumour can be relieved either by the addition of some viscosity to the system or by accounting for an inner necrotic interface with an appropriate stress boundary condition.