Shailesh Naire

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 Darcys 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.

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.

Unsteady bubble propagation in a flexible channel: Predictions of a viscous stick-slip instability

We investigate the unsteady motion of a long bubble advancing under either prescribed pressure p(b) or prescribed volume flux q(b) into a fluid-filled flexible-walled channel at zero Reynolds number, an idealized model for the reopening of a liquid-lined lung airway. The channel walls are held under longitudinal tension and are supported by external springs; the bubble moves with speed U. Provided p(b) exceeds a critical pressure p(crit) the system exhibits two types of steady motion. At low speeds, the bubble acts like a piston, slowly pushing a column of fluid ahead of itself, and U decreases with increasing p(b). At high speeds, the bubble rapidly peels the channel walls apart and U increases with increasing p(b.) Using two independent time-dependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed p(b) > p(crit), peeling motion is stable and the steady pushing solution is unstable; for p(b) > p(crit) the system ultimately exhibits unsteady pushing behaviour for which U continually diminishes with time. When q(b) is prescribed, peeling motion (with large q(b)) is again stable, but pushing motion (with small q(b)) loses stability at long times to a novel instability leading to spontaneous relaxation oscillations of increasing amplitude and period, for which the bubble switches abruptly between slow unsteady pushing and rapid quasi-steady peeling. This stick-slip motion is characterized using a third-order lumped-parameter model which in turn is reduced to a nonlinear map. Implications for the inflation of occluded lung airways are discussed.

LIMITING CASES OF GRAVITATIONAL DRAINAGE OF A VERTICAL FREE FILM FOR EVALUATING SURFACTANTS

The evolution of the deforming free surface of a vertically oriented thin film draining under gravity is examined for the case when there is an insoluble surfactant monolayer on a viscous, incompressible, and free liquid film with finite surface viscosity. Three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity, and surfactant transport are obtained. These equations are derived at leading order and do not have inertial effects. We examine the case where the film is nearly flat so that mean surface tension is negligible; this will be in good agreement with experimental data with respect to long-time behavior of film thickness. This will be shown both analytically and computationally.We will show that in the limit of large surface viscosity, the evolution of the free surface is that obtained for the tangentially immobile case. It is verified that increasing surface viscosity slows down film drainage, thereby enhancing film stability. The Marangoni effec...

An insoluble surfactant model for a vertical draining free film with variable surface viscosity

A mathematical model is constructed to study the evolution of a vertically oriented, thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite variable surface viscosity, on its free surface. Lubrication theory for this free film results in three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity and the surfactant transport, at leading order. In the limit of large surface viscosity and the Marangoni effect, the evolution of the free surface is that of a rigid film. For mobile films with small surface viscosity, transition from a mobile to an essentially immobile film is observed for large Marangoni effects. It is also verified that stable aqueous films can be formed in the regime of high surfactant concentrations. The theoretical results are compared with experiment; the purpose of both is to act as a model problem to evaluate the effectiveness of surfactants for potential use in foam-fabrication processes.

Liquid film dynamics in horizontal and tilted tubes: Dry spots and sliding drops

Using a model derived from lubrication theory, we consider the evolution of a thin viscous film coating the interior or exterior of a cylindrical tube. The flow is driven by surface tension and gravity and the liquid is assumed to wet the cylinder perfectly. When the tube is horizontal, we use large-time simulations to describe the bifurcation structure of the capillary equilibria appearing at low Bond number. We identify a new film configuration in which an isolated dry patch appears at the top of the tube and demonstrate hysteresis in the transition between rivulets and annular collars as the tube length is varied. For a tube tilted to the vertical, we show how a long initially uniform rivulet can break up first into isolated drops and then annular collars, which subsequently merge. We also show that the speed at which a localized drop moves down the base of a tilted tube is nonmonotonic in tilt angle.

Models for gravitationally-driven free-film drainage

The drainage of the thin fluids layers, or lamellae, in a foam may be modeled by a vertical draining thin liquid film. A sequence of mathematical models is described that attempts to explain some aspects of the drainage of the film. Lubrication theory is used to derive the nonlinear partial differential equations (PDE) that describe the film; all models assume an insoluble surfactant in this paper. The models include effects from gravity, viscosity, surface tension and its dependence on surface concentration (the Marangoni effect), and surface viscosity; they may also include nonlinear equations of state. The models are able to predict very well the fast and slow limits of the drainage observed experimentally; a limited range of intermediate drainage rates has been described by these models to date. The limitations of the models and possible extensions will be discussed.

An Asymptotic Model of Unsteady Airway Reopening

We consider a simple physical model for the reopening of a collapsed lung airway involving the unsteady propagation of a long bubble of air, driven at a prescribed flow-rate, into a liquid-filled channel formed by two flexible membranes that are held under large longitudinal tension and are confined between two parallel rigid plates. This system is described theoretically using an asymptotic approximation, valid for uniformly small membrane slopes, which reduces to a fourth-order nonlinear evolution equation for the channel width ahead of the bubble tip, from which the time-evolution of the bubble pressure pb* and bubble speed may be determined. The model shows that there can be a substantial delay between the time at which the bubble starts to grow in volume and the time at which its tip starts to move. Under certain conditions, the start of the bubbles motion is accompanied by a transient overshoot in pb*, as seen previously in experiment; the model predicts that the overshoot is greatest in narrow channels when the bubble is driven with a large volume flux. It is also shown how the threshold pressure for steady bubble propagation in wide channels has distinct contributions from the capillary pressure drop across the bubble tip and viscous dissipation in the channel ahead of the bubble.

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.

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.