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ABSTRACT: Subject collections (253 articles) applied mathematics (211 articles) biomedical engineering (87 articles) mathematical modelling Articles on similar topics can be found in the following collections We present a theoretical description of flow-induced self-excited oscillations in the Starling resistor—a pre-stretched thin-walled elastic tube that is mounted on two rigid tubes and enclosed in a pressure chamber. Assuming that the flow through the elastic tube is driven by imposing the flow rate at the downstream end, we study the development of small-amplitude long-wavelength high-frequency oscillations, combining the results of two previous studies in which we analysed the fluid and solid mechanics of the problem in isolation. We derive a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillations, and determine the slow growth or decay of the normal modes by considering the system's energy budget. We compare the theoretical predictions for the mode shapes, frequencies and growth rates with the results of direct numerical simulations, based on the solution of the three-dimensional Navier–Stokes equations, coupled to the equations of shell theory, and find good agreement between the results. Our results provide the first asymptotic predictions for the onset of self-excited oscillations in three-dimensional collapsible tube flows.
Proc. R. Soc. A Proc. R. Soc. A. ; 466:3635-3657.
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ABSTRACT: A model for fluid and mass transport in a single module of a tissue engineering hollow fibre bioreactor (HFB) is developed. Cells are seeded in alginate throughout the extra-capillary space (ECS), and fluid is pumped through a central lumen to feed the cells and remove waste products. Fluid transport is described using Navier-Stokes or Darcy equations as appropriate; this is overlaid with models of mass transport in the form of advection-diffusion-reaction equations that describe the distribution and uptake/production of nutrients/waste products. The small aspect ratio of a module is exploited and the option of opening an ECS port is explored. By proceeding analytically, operating equations are determined that enable a tissue engineer to prescribe the geometry and operation of the HFB by ensuring the nutrient and waste product concentrations are consistent with a functional cell population. Finally, results for chondrocyte and cardiomyocyte cell populations are presented, typifying two extremes of oxygen uptake rates.
Mathematical Medicine and Biology 11/2011; · 1.82 Impact Factor
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ABSTRACT: Tissue engineering aims to regenerate, repair or replace organs or defective tissues. This tissue regeneration often occurs in a bioreactor. Important challenges in tissue engineering include ensuring adequate nutrient supply, maintaining the desired cell distribution and achieving sufficiently high cell yield. To put laboratory experiments into a theoretical framework, mathematical modelling of the physical and biochemical processes involved in tissue growth is a useful tool. In this work, we derive and solve a model for a cell-seeded porous scaffold placed in a perfusion bioreactor in which fluid delivers nutrients to the cells. The model describes the key features, including fluid flow, nutrient delivery, cell proliferation and consequent variation of scaffold porosity. Fluid flow through the porous scaffold is modelled by Darcy's law, and nutrient delivery is described by a reaction-advection-diffusion equation. A reaction-diffusion equation describes the evolution of cell density, in which cell proliferation is modelled via logistic growth and cell spreading via non-linear diffusion, which depends on cell density. The effect of shear stress on nutrient consumption and cell proliferation is also included in the model. COMSOL (a commercial finite element solver) is used to solve the model numerically. The results reveal the dependence of the cell distribution and total cell yield on the initial cell density and scaffold porosity. We suggest various seeding strategies and scaffold designs to improve the cell distribution and total cell yield in the engineered tissue construct.
Mathematical Medicine and Biology 10/2011; · 1.82 Impact Factor
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ABSTRACT: Motivated by the problem of self-excited oscillations in fluid-filled collapsible tubes, we examine the flow structure and energy budget of flow through an elastic-walled tube. Specifically, we consider the case in which a background axial flow is perturbed by prescribed small-amplitude high-frequency long-wavelength oscillations of the tube wall, with a slowly growing or decaying amplitude. We use a multiple-scale analysis to show that, at leading order, we recover the constant-amplitude equations derived by Whittaker et al. (Whittaker et al. 2010 J. Fluid Mech. 648, 83-121. (doi:10.1017/S0022112009992904)) with the effects of growth or decay entering only at first order. We also quantify the effects on the flow structure and energy budget. Finally, we discuss how our results are needed to understand and predict an instability that can lead to self-excited oscillations in collapsible-tube systems.
Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 07/2011; 369(1947):2989-3006. · 2.77 Impact Factor
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ABSTRACT: This paper presents a mathematical model to describe the growth of tissue into a rapid-prototyped porous scaffold when it is implanted onto the chorioallantoic membrane (CAM). The scaffold was designed to study the effects of the size and shape of pores on tissue growth into conventional tissue engineering scaffolds, and consists of an array of pores each having a pre-specified shape. The experimental observations revealed that the CAM grows through each pore as an intact layer of tissue, provided the width of the pore exceeds a threshold value. Based on these results a mathematical model is described to simulate the growth of the membrane, assuming that the growth is a function of the local isotropic membrane tension. The model predictions are compared against measurements of the extent of membrane growth through the pores as a function of time for pores with different dimensions.
Biomechanics and Modeling in Mechanobiology 10/2010; 10(4):539-58. · 3.19 Impact Factor
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Sarah L Waters,
Jordi Alastruey,
Daniel A Beard,
Peter H M Bovendeerd,
Peter F Davies,
Girija Jayaraman,
Oliver E Jensen,
Jack Lee,
Kim H Parker,
Aleksander S Popel,
Timothy W Secomb,
Maria Siebes,
Spencer J Sherwin,
Rebecca J Shipley,
Nicolas P Smith,
Frans N van de Vosse
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ABSTRACT: A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.
Progress in Biophysics and Molecular Biology 10/2010; 104(1-3):49-76. · 3.20 Impact Factor
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ABSTRACT: We examine the linear stability of two-dimensional Poiseuille flow in a long channel confined by a rigid wall and a massless damped-tensioned membrane. We seek solu-tions that are periodic in the streamwise spatial direction and time, solving the homoge-neous eigenvalue problem using a Chebyshev spectral method and asymptotic analysis. Several modes of instability are identified, including Tollmien–Schlichting (TS) waves and travelling-wave flutter (TWF). The eigenmode for neutrally stable downstream-propagating TWF in the absence of wall damping is shown to have a novel asymptotic structure at high Reynolds numbers, not reported in symmetric flexible-walled chan-nels, involving a weak but destabilising critical layer at the channel centreline where the wavespeed is marginally greater than the maximum Poiseuille flow speed. We also show that TS instabilities along the lower branch of the neutral curve are modified remarkably little by wall compliance, but can be either stabilised or destabilised by wall damping. We discuss the energy budget underlying TWF and briefly describe the structure of other flow-induced surface instabilities.
02/2010;
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Journal of Fluid Mechanics - J FLUID MECH. 01/2010; 648.
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Journal of Fluid Mechanics - J FLUID MECH. 01/2010; 648.
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ABSTRACT: Summary We consider small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross section. The tube is subject to an axial pre-stress, and the deformations result from an applied transmural pressure. An approximate tube law (linking the transmural pressure, the cross-sectional area, and its axial derivatives) is derived from shell theory in the distinguished asymptotic limit in which the tube's behaviour is dominated by the restoring forces from the axial pre-stress and azimuthal bending. This is possible because the deformations of the tube induced by both the transmural pressure and the axial forces can be described, to very good approximation, by a single azimuthal mode of deformation of axially varying amplitude. The resulting tube law is compared with numerical solutions of the full shell equations and good agreement is found (provided the tube is sufficiently long and the wall not too thin so that in-plane shearing is negligible). We discuss the applications of our results to the modelling of flow in collapsible tubes.
Quarterly Journal of Mechanics and Applied Mathematics - QUART J MECH APPL MATH. 01/2010; 63(3).
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ABSTRACT: We investigate the localised spatial growth of two-dimensional disturbances governed by the long-wavelength Orr–Sommerfeld
operator in a rigid channel at sub-transition Reynolds numbers. We discuss the link between this growth and vorticity waves,
a striking feature of the self-excited oscillations that arise when flow is driven through a finite-length channel, one wall
of which contains a segment of flexible membrane.
12/2009: pages 397-402;
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ABSTRACT: 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.
Mathematical biosciences 08/2009; 221(2):101-20. · 1.30 Impact Factor
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ABSTRACT: Angiogenic sprouts at the leading edge of an expanding vascular plexus are recognised as major regulators of the structure of the developing network. Early in sprout development, a vascular lumen is often evident which communicates with the parent vessel while the distal tip is blind-ended. Here we describe the temporal evolution of blind-ended vessels (BEVs) in a small wound made in the panniculus carnosus muscle of a mouse viewed in a dorsal skin-fold window-chamber model with intra-vital microscopy during the most active period of angiogenesis (days 5-8 after injury). Although these structures have been mentioned anecdotally in previous studies, we observed BEVs to be frequent, albeit transient, features of plexus formation. Plasma leakage into the surrounding extracellular matrix occurring from these immature conduits could play an important role in preparing hypoxic tissue for vascular invasion. Although sprout growth is likely to be regulated by its flow environment, the parameters regulating flow into and through BEVs have not been characterised in situ. Longitudinal data from individual animals show that the number of BEVs filled with plasma alone peaks at day 7, when they can exceed 150 microm in length. Additionally, BEVs greater than 40 microm in length are more likely to be filled with stationary erythrocytes than with plasma alone. Using a mathematical model, we show how the flux of 150 kD fluorinated (FITC-) dextran through an individual plasma-filled BEV is related to its geometry being determined primarily by its surface area; by fitting theoretical intensity values to experimental data we assess the permeability of the vessel to FITC-dextran. Plasma skimming provides a mechanistic explanation for the observation that BEVs with larger surface area are more likely to recruit erythrocytes.
Microvascular Research 08/2008; 76(3):161-8. · 2.83 Impact Factor
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ABSTRACT: We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, Recrit, at which the wall begins to extract energy from the flow. We conjecture that Recrit corresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between Recrit and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.
Journal of Fluid Mechanics 04/2008; 601:199 - 227. · 2.46 Impact Factor
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ABSTRACT: 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.
Journal of Theoretical Biology 01/2008; 249(3):543-53. · 2.21 Impact Factor
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ABSTRACT: We consider laminar high-Reynolds-number flow through a finite-length planar channel,
where a portion of one wall is replaced by a thin massless elastic membrane that is held
under longitudinal tension T and subject to an external pressure distribution. The flow
is driven by a fixed pressure drop along the full length of the channel. We investigate
the global stability of two-dimensional Poiseuille flow using a method of matched local
eigenfunction expansions, which is compared to direct numerical simulations. We trace
the neutral stability curve of the primary oscillatory instability of the system, illustrating
a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’
motion at low T . Small-amplitude sloshing at high T can be captured using a low-order
eigenmode truncation involving four surface-based modes in the compliant segment of the
channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that
hydrodynamic modes contribute increasingly to the global instability and we demonstrate
a change in the mechanism of energy transfer from the mean flow, with viscous effects
being destabilising. Simulations of finite-amplitude oscillations at low T reveal a generic
slamming motion, in which the the flexible membrane is drawn close to the opposite rigid
wall before rapidly recovering. A simple model is used to demonstrate how fluid inertia
in the downstream rigid channel segment, coupled to membrane curvature downstream
of the moving constriction, together control slamming dynamics.
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ABSTRACT: We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.
European Journal of Mechanics - B/Fluids.
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ABSTRACT: We analyse the flows in fluid-conveying tubes whose elastic walls perform small-amplitude high-frequency oscillations. We show that the velocity perturbations induced by the wall motion are dominated by their transverse components and use numerical simulations to analyse the two-dimensional flows that develop in the tube's cross-sections. Asymptotic methods are then employed to derive explicit predictions for the flow fields and for the total viscous dissipation, whose magnitude plays an important role in the development of self-excited oscillations.
We show that in cases with fluid–structure interaction, the coupled oscillations are controlled by the ratio of the fluid and wall densities, and by a material parameter that is equivalent to the Womersley number, and indicates the importance of fluid inertia and wall elasticity relative to the fluid's viscosity. We present numerical simulations of the coupled oscillations and use asymptotic techniques to derive explicit predictions for their period and decay rate. Finally, we discuss the implications of our results for the development of self-excited oscillations in three-dimensional collapsible tubes.
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