A mathematical model of signalling molecule-mediated processes during regeneration of osteochondral defects after chondrocyte implantation.

Treating bone-cartilage defects is a fundamental clinical problem. The ability of damaged cartilage to self-repair is limited due to its avascularity. Left untreated, these defects can lead to osteoarthritis. Details of osteochondral defect repair are elusive, but animal models indicate healing occurs via an endochondral ossification-like process, similar to that in the growth plate. In the growth plate, the signalling molecules parathyroid hormone-related protein (PTHrP) and Indian Hedgehog (Ihh) form a feedback loop regulating chondrocyte hypertrophy, with Ihh inducing and PTHrP suppressing hypertrophy. To better understand this repair process and to explore the regulatory role of signalling molecules on the regeneration process, we formulate a reaction-diffusion mathematical model of osteochondral defect regeneration after chondrocyte implantation. The drivers of healing are assumed to be chondrocytes and osteoblasts, and their interaction via signalling molecules. We model cell proliferation, migration and chondrocyte hypertrophy, and matrix production and conversion, spatially and temporally. We further model nutrient and signalling molecule diffusion and their interaction with the cells. We consider the PTHrP-Ihh feedback loop as the backbone mechanisms but the model is flexible to incorporate extra signalling mechanisms if needed. Our mathematical model is able to represent repair of osteochondral defects, starting with cartilage formation throughout the defect. This is followed by chondrocyte hypertrophy, matrix calcification and bone formation deep inside the defect, while cartilage at the surface is maintained and eventually separated from the deeper bone by a thin layer of calcified cartilage. The complete process requires around 48 months. A key highlight of the model demonstrates that the PTHrP-Ihh loop alone is insufficient and an extra mechanism is required to initiate chondrocyte hypertrophy, represented by a critical cartilage density. A parameter sensitivity study reveals that the timing of the repair process crucially depends on parameters, such as the critical cartilage density, and those describing the actions of PTHrP to suppress hypertrophy, such as its diffusion coefficient, threshold concentration and degradation rate.

Modeling and design of optimal flow perfusion bioreactors for tissue engineering applications.

Perfusion bioreactors have been used in different tissue engineering applications because of their consistent distribution of nutrients and flow-induced shear stress within the tissue-engineering scaffold. A widely used configuration uses a scaffold with a circular cross-section enclosed within a cylindrical chamber and inlet and outlet pipes which are connected to the chamber on either side through which media is continuously circulated. However, fluid-flow experiments and simulations have shown that the majority of the flow perfuses through the center. This pattern creates stagnant zones in the peripheral regions as well as in those of high flow rate near the inlet and outlet. This non-uniformity of flow and shear stress, owing to a circular design, results in limited cell proliferation and differentiation in these areas. The focus of this communication is to design an optimized perfusion system using computational fluid dynamics as a mathematical tool to overcome the time-consuming trial and error experimental method. We compared the flow within a circular and a rectangular bioreactor system. Flow simulations within the rectangular bioreactor are shown to overcome the limitations in the circular design. This communication challenges the circular cross-section bioreactor configuration paradigm and provides proof of the advantages of the new design over the existing one.

A mathematical model of cartilage regeneration after cell therapy.

Autologous Chondrocyte Implantation (ACI) is a cell-based therapy used mainly for the treatment of chondral defects in the knee. It involves surgically inserting isolated chondrocytes or mesenchymal stem cells (MSCs), previously expanded in culture, into the defect region. These chondrocytes then proliferate and migrate in the process forming extracellular matrix (ECM) and new cartilage. In the case of MSCs, the process of forming new cartilage is initiated only after differentiation of the stem cells into chondrocytes. Many details of the repair process following insertion in humans are unknown. To enable better understanding of the repair process, we present a mathematical model of cartilage regeneration after cell therapy. The key mechanisms involved in the regeneration process are simulated by modelling cell migration, proliferation and differentiation, nutrient diffusion and depletion, and ECM synthesis and degradation at the defect site, both spatially and temporally. The model successfully simulates the progression of cartilage regeneration. The model predicts a time frame of about 18months for the defect to reach full maturation which corresponds with results from clinical studies and demonstrates that cartilage regeneration is a slow process. Moreover, the model also suggests that regeneration using stem cells alone is no better than that using chondrocytes. The stem cells need to first differentiate into chondrocytes before forming ECM and new cartilage, a process that is initiated only after the stem cell density exceeds a threshold value. Furthermore, with chondrocytes alone, the matrix seems to develop from the subchondral bone interface as compared to the normal cartilage interface, in the case of stem cells alone. The influence of initial conditions and parameters, such as the initial cell seeding densities and cell proliferation rates, is shown to not significantly influence the general evolution characteristics other than accelerating the initial growth process. The model presented here is a first approach towards better understanding of cartilage regeneration after cell therapy techniques.

Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor.

We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue-engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy's law, is used to examine the nutrient and shear-stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier-Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution.

A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation - I: the effects of growth factors.

Autologous chondrocyte implantation is a cell-based therapy for treating chondral defects. The procedure begins by inserting chondrocytes into the defect region. The chondrocytes initiate healing by proliferating and depositing extracellular matrix, which allows them to migrate into the defect until it is completely filled with new cartilage. Mesenchymal stem cells can be used instead of chondrocytes with similar long-term results. The main differences are at early times since mesenchymal stem cells must first differentiate into chondrocytes before cartilage is formed. To better understand this repair process, we present a mathematical model of cartilage regeneration after cell therapy. We extend our previous work to include the cell-cell interaction between mesenchymal stem cells and chondrocytes via growth factors. Our results show that matrix formation is enhanced at early times in the presence of growth factors. This study reinforces the importance of mesenchymal stem cell and chondrocyte interaction in the cartilage healing process as hypothesised in experimental studies.

A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation - II: the effects of co-implantation.

We present a mathematical model of cartilage regeneration after cell therapy, to show how co-implantation of stem cells (mesenchymal stem cells) and chondrocytes into a cartilage defect can impact chondral healing. The key mechanisms involved in the regeneration process are simulated by modelling cell proliferation, migration and differentiation, nutrient diffusion and Extracellular Matrix (ECM) synthesis at the defect site, both spatially and temporally. In addition, we model the interaction between mesenchymal stem cells and chondrocytes by including growth factors. In Part I of this work, we have shown that matrix formation was enhanced at early times when mesenchymal stem cell-to-chondrocyte interactions due to the effects of growth factors were considered. In this article, we show that the additional effect of co-implanting mesenchymal stem cells and chondrocytes further enhances matrix production within the first year in comparison to implanting only chondrocytes or only mesenchymal stem cells. This could potentially reduce healing time allowing the patient to become mobile sooner after surgery.

In vitro experiments of cerebral blood flow during aspiration thrombectomy: potential effects on cerebral perfusion pressure and collateral flow.

BACKGROUND: Mechanical thrombectomy with stent retriever devices is associated with significantly better outcomes than thrombolysis alone in the treatment of acute ischemic stroke. Thrombus aspiration achieves high patency rates, but clinical outcomes are variable. The aim of this study was to examine the effect of different suction conditions on perfusate flow during aspiration thrombectomy. METHODS: A computational fluid dynamics model of an aspiration device within a patent and occluded blood vessel was used to simulate flow characteristics using fluid flow solver software. A physical particulate flow model of a patent vessel and a vessel occluded by thrombus was then used to visualize flow direction and measure flow rates with the aspiration catheter placed 1-10 mm proximal of the thrombus, and recorded on video. RESULTS: The mathematical model predicted that, in a patent vessel, perfusate is drawn from upstream of the catheter tip while, in an occluded system, perfusate is drawn from the vessel proximal to the device tip with no traction on the occlusion distal of the tip. The in vitro experiments confirmed the predictions of this model. In the occluded vessel aspiration had no effect on the thrombus unless the tip of the catheter was in direct contact with the thrombus. CONCLUSIONS: These experiments suggest that aspiration is only effective if the catheter tip is in direct contact with the thrombus. If the catheter tip is not in contact with the thrombus, aspirate is drawn from the vessels proximal of the occlusion. This could affect collateral flow in vivo.

Epithelial cell deformation during surfactant-mediated airway reopening: a theoretical model.

A theoretical model is presented describing the reopening by an advancing air bubble of an initially liquid-filled collapsed airway lined with deformable epithelial cells. The model integrates descriptions of flow-structure interaction (accounting for nonlinear deformation of the airway wall and viscous resistance of the airway liquid flow), surfactant transport around the bubble tip (incorporating physicochemical parameters appropriate for Infasurf), and cell deformation (due to stretching of the airway wall and airway liquid flows). It is shown how the pressure required to drive a bubble into a flooded airway, peeling apart the wet airway walls, can be reduced substantially by surfactant, although the effectiveness of Infasurf is limited by slow adsorption at high concentrations. The model demonstrates how the addition of surfactant can lead to the spontaneous reopening of a collapsed airway, depending on the degree of initial airway collapse. The effective elastic modulus of the epithelial layer is shown to be a key determinant of the relative magnitude of strains generated by flow-induced shear stresses and by airway wall stretch. The model also shows how epithelial-layer compressibility can mediate strains arising from flow-induced normal stresses and stress gradients.

Hydrodynamics of a bounded vertical film with nonlinear surface properties.

The drainage of a thin liquid film with an insoluble monolayer down a vertical wall is studied. Lubrication theory is used to develop a model where the film is pinned at the top with a given thickness and the film drains into a bath at the bottom. A nonlinear equation of state is used for the surface tension and the surface viscosity is a nonlinear function of the surfactant concentration; these are appropriate for some aqueous systems. The three partial differential equations are solved via discretization in space and then the resulting differential algebraic system is solved. Results are described for a wide range of parameters, and the conditions under which the free surface is immobilized are discussed.

Dynamics of voluntary cough maneuvers: a theoretical model.

Voluntary cough maneuvers are characterized by transient peak expiratory flows (PEFs) exceeding the maximum expiratory flow-volume (MEFV) curve. In some cases, these flows can be well in excess of the MEFV, generally referred to as supramaximal flows. Understanding the flow-structure interaction involved in these maneuvers is the main goal of this study. We present a simple theoretical model for investigating the dynamics of voluntary cough and forced expiratory maneuvers. The core modeling idea is based on a 1D model of high Reynolds number flow through flexible-walled tubes. The model incorporates key ingredients involved in these maneuvers: the expiratory effort generated by the abdominal and expiratory muscles, the glottis, and the flexibility and compliance of the lung airways. Variations in these allow investigation of the expiratory flows generated by a variety of single cough maneuvers. The model successfully reproduces the transient PEFs, reported in cough studies. The amplitude of the PEFs is shown to depend on the cough generation protocol, the glottis reopening time, and the compliance of the airways. The particular highlight is in simulating supramaximal PEFs for very compliant tubes. The flow-structure interaction mechanisms behind these are discussed. The wave-speed theory of flow limitation is used to characterize the PEFs. Existing hypotheses of the origin of PEFs, from cough and forced expiration experiments, are also tested using this model. This modeling framework could be a first step toward more sophisticated cough models as well as in developing ideas for new bench-top experiments.