Linda Cummings

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.

Flow dynamics in a stented ureter.

Vesicorenal reflux is a major side effect associated with ureteric stent placement. In a stented upper urinary tract when the bladder pressure rises, such as during bladder spasms (due to irritation caused by the stent) or voiding of the bladder, it drives urine reflux up the ureter, which, in turn, may be a contributory factor for infections in the renal pelvis. We develop a mathematical model to examine urine flow in a stented ureter, assuming that it remains axisymmetric and treating the wall as a non-linear elastic membrane. The stent is modelled as a rigid, permeable, hollow, circular cylinder lying coaxially inside the ureter. The renal pelvis is treated as an elastic bag, whose volume increases in response to an increased internal pressure. Fluid enters the renal pelvis from the kidney with a prescribed flux. The stent, ureter and renal pelvis are filled with urine, and the bladder pressure is prescribed. We use the model to calculate the total volume of reflux generated during rises in bladder pressure and investigate how it is affected by the stent and ureter properties.

Note on the hydrodynamic description of thin nematic films: Strong anchoring model

We discuss the long-wave hydrodynamic model for a thin film of nematic liquid crystal in the limit of strong anchoring at the free surface and at the substrate. We rigorously clarify how the elastic energy enters the evolution equation for the film thickness in order to provide a solid basis for further investigation: several conflicting models exist in the literature that predict qualitatively different behaviour. We consolidate the various approaches and show that the long-wave model derived through an asymptotic expansion of the full nemato-hydrodynamic equations with consistent boundary conditions agrees with the model one obtains by employing a thermodynamically motivated gradient dynamics formulation based on an underlying free energy functional. As a result, we find that in the case of strong anchoring the elastic distortion energy is always stabilising. To support the discussion in the main part of the paper, an appendix gives the full derivation of the evolution equation for the film thickness via asymptotic expansion.

Tracking large solid constructs suspended in a rotating bioreactor: A combined experimental and theoretical study

We present a combined experimental and theoretical study of the trajectory of a large solid cylindrical disc suspended within a fluid‐filled rotating cylindrical vessel. The experimental set‐up is relevant to tissue‐engineering applications where a disc‐shaped porous scaffold is seeded with cells to be cultured, placed within a bioreactor filled with nutrient‐rich culture medium, which is then rotated in a vertical plane to keep the growing tissue construct suspended in a state of “free fall.” The experimental results are compared with theoretical predictions based on the model of Cummings and Waters (2007), who showed that the suspended disc executes a periodic motion. For anticlockwise vessel rotation three regimes were identified: (i) disc remains suspended at a fixed position on the right‐hand side of the bioreactor; (ii) disc executes a periodic oscillatory motion on the right‐hand side of the bioreactor; and (iii) disc orbits the bioreactor. All three regimes are captured experimentally, and good agreement between theory and experiment is obtained. For the tissue engineering application, computation of the fluid dynamics allows the nutrient concentration field surrounding a tissue construct (a property that cannot be measured experimentally) to be determined (Cummings and Waters, 2007). The implications for experimental cell‐culture protocols are discussed. Biotechnol. Bioeng. 2009; 104: 1224–1234.

Modelling spreading dynamics of nematic liquid crystals in three spatial dimensions

We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth-order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three-dimensional droplets spreading in the presence of anchoring variations that model simple defects in the nematic orientation at the substrate.

Modeling and simulations of the spreading and destabilization of nematic droplets

A series of experiments [C. Poulard and A. M. Cazabat, “Spontaneous spreading of nematic liquid crystals,” Langmuir 21, 6270 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveals a surprisingly rich variety of behaviors. Small droplets can either be arrested in their spreading, spread stably, destabilize without spreading (corrugated surface), or spread with a fingering instability and corrugated free surface. In this work, we discuss the problem of NLC drops spreading in a simplified two-dimensional (2D) geometry. The model that we present is based on a long-wavelength approach for NLCs by Ben Amar and Cummings [“Fingering instabilities in driven thin nematic films,” Phys. Fluids 13, 1160 (2001); L. J. Cummings, “Evolution of a thin film of nematic liquid crystal with anisotropic surface energy,” Eur. J. Appl. Math. 15, 651 (2004)]. The improvements in the model here permit fully nonlinear time-dependent simulations. These simulations, for the appropriate choice of parameter values, exhib...

Mathematical model for determining the binding constants between immunoglobulins, bivalent ligands, and monovalent ligands

AbstractThis paper analyzes the equilibria between immunoglobulins (R2), homo-bifunctional ligands (L2), monovalent ligands (I), and their complexes. We present a mathematical model that can be used to estimate the concentration of each species present in a mixture of R2, L2, and I, given the initial conditions defining the total concentration of R2, L2, I, and four dissociation constants (

Interfacial deformation and jetting of a magnetic fluid

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