Dielectrowetting of a thin nematic liquid crystal layer.

We consider a mathematical model that describes the flow of a nematic liquid crystal (NLC) film placed on a flat substrate, across which a spatially varying electric potential is applied. Due to their polar nature, NLC molecules interact with the (nonuniform) electric field generated, leading to instability of a flat film. Implementation of the long wave scaling leads to a partial differential equation that predicts the subsequent time evolution of the thin film. This equation is coupled to a boundary value problem that describes the interaction between the local molecular orientation of the NLC (the director field) and the electric potential. We investigate numerically the behavior of an initially flat film for a range of film heights and surface anchoring conditions.

Curvature- and fluid-stress-driven tissue growth in a tissue-engineering scaffold pore.

Cell proliferation within a fluid-filled porous tissue-engineering scaffold depends on a sensitive choice of pore geometry and flow rates: regions of high curvature encourage cell proliferation, while a critical flow rate is required to promote growth for certain cell types. When the flow rate is too slow, the nutrient supply is limited; when it is too fast, cells may be damaged by the high fluid shear stress. As a result, determining appropriate tissue-engineering-construct geometries and operating regimes poses a significant challenge that cannot be addressed by experimentation alone. In this paper, we present a mathematical theory for the fluid flow within a pore of a tissue-engineering scaffold, which is coupled to the growth of cells on the pore walls. We exploit the slenderness of a pore that is typical in such a scenario, to derive a reduced model that enables a comprehensive analysis of the system to be performed. We derive analytical solutions in a particular case of a nearly piecewise constant growth law and compare these with numerical solutions of the reduced model. Qualitative comparisons of tissue morphologies predicted by our model, with those observed experimentally, are also made. We demonstrate how the simplified system may be used to make predictions on the design of a tissue-engineering scaffold and the appropriate operating regime that ensures a desired level of tissue growth.

Director gliding in a nematic liquid crystal layer: Quantitative comparison with experiments.

The interaction between nematic liquid crystals and polymer-coated substrates may lead to slow reorientation of the easy axis (so-called "director gliding") when a prolonged external field is applied. We consider the experimental evidence of zenithal gliding observed by Joly et al. [Phys. Rev. E 70, 050701 (2004)PLEEE81539-375510.1103/PhysRevE.70.050701] and Buluy et al. [J. Soc. Inf. Disp. 14, 603 (2006)1071-092210.1889/1.2235686] as well as azimuthal gliding observed by S. Faetti and P. Marianelli [Liq. Cryst. 33, 327 (2006)LICRE60267-829210.1080/02678290500512227], and we present a simple, physically motivated model that captures the slow dynamics of gliding, both in the presence of an electric field and after the electric field is turned off. We make a quantitative comparison of our model results and the experimental data and conclude that our model explains the gliding evolution very well.

Mathematical Modeling of Ischemia-Reperfusion Injury and Postconditioning Therapy.

Reperfusion (restoration of blood flow) after a period of ischemia (interruption of blood flow) can paradoxically place tissues at risk of further injury: so-called ischemia-reperfusion injury or IR injury. Recent studies have shown that postconditioning (intermittent periods of further ischemia applied during reperfusion) can reduce IR injury. We develop a mathematical model to describe the reperfusion and postconditioning process following an ischemic insult, treating the blood vessel as a two-dimensional channel, lined with a monolayer of endothelial cells that interact (respiration and mechanotransduction) with the blood flow. We investigate how postconditioning affects the total cell density within the endothelial layer, by varying the frequency of the pulsatile flow and the oxygen concentration at the inflow boundary. We find that, in the scenarios we consider, the pulsatile flow should be of high frequency to minimize cellular damage, while oxygen concentration at the inflow boundary should be held constant, or subject to only low-frequency variations, to maximize cell proliferation.

Effects of flexoelectricity and weak anchoring on a Freedericksz transition cell.

We consider a mathematical model that consists of a nematic liquid crystal layer sandwiched between two parallel bounding plates, across which an external field is applied. We investigate how the number and type of solutions for the director orientation within the layer change as the field strength, anchoring conditions, and material properties of the nematic liquid crystal layer vary. In particular, we focus on how the inclusion of flexoelectric effects alters the Freedericksz and saturation thresholds.

Substrate-induced gliding in a nematic liquid crystal layer.

We consider the interaction between nematic liquid crystals (NLCs) and polymer substrates. Such substrates can interact with NLCs, exhibiting a phenomenon known as director gliding: the preferred orientation of the NLC molecules at the interface changes on time scales that are slow relative to the elastic relaxation time scale of the NLC. We present two models for gliding, inspired by experiments that investigate the interaction between the NLC and a polymer substrate. These models, though simple, lead to nontrivial results, including loss of bistability under gliding. Perhaps surprisingly, we find that externally imposed switching between the steady states of a bistable system may reverse the effect of gliding, preventing loss of bistability if switching is sufficiently frequent. Our findings may be of relevance to a variety of technological applications involving liquid crystal devices, and particularly to a new generation of flexible liquid crystal displays that implement polymeric substrates.

Electric-field variations within a nematic-liquid-crystal layer.

A thin layer of nematic liquid crystal (NLC) across which an electric field is applied is a setup of great industrial importance in liquid crystal display devices. There is thus a large literature modeling this situation and related scenarios. A commonly used assumption is that an electric field generated by electrodes at the two bounding surfaces of the layer will produce a field that is uniform: that is, the presence of NLC does not affect the electric field. In this paper, we use calculus of variations to derive the equations coupling the electric potential to the orientation of the NLC's director field, and use a simple one-dimensional model to investigate the limitations of the uniform field assumption in the case of a steady applied field. The extension of the model to the unsteady case is also briefly discussed.

Cyclic loading of growing tissue in a bioreactor: mathematical model and asymptotic analysis.

A simplified 2D mathematical model for tissue growth within a cyclically-loaded tissue engineering scaffold is presented and analyzed. Such cyclic loading has the potential to improve yield and functionality of tissue such as bone and cartilage when grown on a scaffold within a perfusion bioreactor. The cyclic compression affects the flow of the perfused nutrient, leading to flow properties that are inherently unsteady, though periodic, on a timescale short compared with that of tissue proliferation. A two-timescale analysis based on these well-separated timescales is exploited to derive a closed model for the tissue growth on the long timescale of proliferation. Some sample numerical results are given for the final model, and discussed.

Bifurcation properties of nematic liquid crystals exposed to an electric field: switchability, bistability, and multistability.

Bistable liquid crystal displays (LCDs) offer the potential for considerable power savings compared with conventional (monostable) LCDs. The existence of two (or more) stable field-free states that are optically distinct means that contrast can be maintained in a display without an externally applied electric field. An applied field is required only to switch the device from one state to the other, as needed. In this paper we examine the basic physical principles involved in generating multiple stable states and the switching between these states. We consider a two-dimensional geometry in which variable surface anchoring conditions are used to control the steady-state solutions and explore how different anchoring conditions can influence the number and type of solutions and whether or not switching is possible between the states. We find a wide range of possible behaviors, including bistability, tristability, and tetrastability, and investigate how the solution landscape changes as the boundary conditions are tuned.

Mathematical model of growth factor driven haptotaxis and proliferation in a tissue engineering scaffold.

Motivated by experimental work (Miller et al. in Biomaterials 27(10):2213-2221, 2006, 32(11):2775-2785, 2011) we investigate the effect of growth factor driven haptotaxis and proliferation in a perfusion tissue engineering bioreactor, in which nutrient-rich culture medium is perfused through a 2D porous scaffold impregnated with growth factor and seeded with cells. We model these processes on the timescale of cell proliferation, which typically is of the order of days. While a quantitative representation of these phenomena requires more experimental data than is yet available, qualitative agreement with preliminary experimental studies (Miller et al. in Biomaterials 27(10):2213-2221, 2006) is obtained, and appears promising. The ultimate goal of such modeling is to ascertain initial conditions (growth factor distribution, initial cell seeding, etc.) that will lead to a final desired outcome.

Defect modeling in spreading nematic droplets.

Experiments by Poulard and Cazabat [Langmuir 21, 6270 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveal a surprisingly rich variety of behavior, including at least two different emerging length scales resulting from a contact line instability. In earlier work [Cummings, Lin, and Kondic, Phys. Fluids 23, 043102 (2011)] we modified a lubrication model for NLCs due to Ben Amar and Cummings [Phys. Fluids 13, 1160 (2001)] and showed that, in a qualitative sense, it can account for two-dimensional (2D) versions of the observed behavior. In the present work we propose a different approach that allows us to explore the effect of anchoring variations on the substrate, again in a 2D geometry. This in turn gives a simple way to model the presence of defects, which are nearly always present in such flows. The present model leads to additional terms in the governing equation. We explore the influence of these additional terms for some simple flow scenarios to gain insight into their influence.

Analysis of Biochemical Equilibria Relevant to the Immune Response: Finding the Dissociation Constants.

This paper analyzes the biochemical equilibria between bivalent receptors, homo-bifunctional ligands, monovalent inhibitors, and their complexes. Such reaction schemes arise in the immune response, where immunoglobulins (bivalent receptors) bind to pathogens or allergens. The equilibria may be described by an infinite system of algebraic equations, which accounts for complexes of arbitrary size n (n being the number of receptors present in the complex). The system can be reduced to just 3 algebraic equations for the concentrations of free (unbound) receptor, free ligand and free inhibitor. Concentrations of all other complexes can be written explicitly in terms of these variables. We analyze how concentrations of key (experimentally-measurable) quantities vary with system parameters. Such measured quantities can furnish important information about dissociation constants in the system, which are difficult to obtain by other means. We provide analytical expressions and suggest specific experiments that could be used to determine the dissociation constants.

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.

Modelling crystal aggregation and deposition in the catheterised lower urinary tract.

Urethral catheters often become encrusted with crystals of magnesium struvite and calcium phosphate. The encrustation can block the catheter, which can cause urine retention in the bladder and reflux into the kidneys. We develop a mathematical model to investigate crystal deposition on the catheter surface, modelling the bladder as a reservoir of fluid and the urethral catheter as a rigid channel. At a constant rate, fluid containing crystal particles of unit size enters the reservoir, and flows from the reservoir through the channel and out of the system. The crystal particles aggregate, which we model using Becker-Döring coagulation theory, and are advected through the channel, where they continue to aggregate and are deposited on the channel's walls. Inhibitor particles also enter the reservoir, and can bind to the crystals, preventing further aggregation and deposition. The crystal concentrations are spatially homogeneous in the reservoir, whereas the channel concentrations vary spatially as a result of advection, diffusion and deposition. We investigate the effect of inhibitor particles on the amount of deposition. For all parameter values, we find that crystals deposit along the full length of the channel, with maximum deposition close to the channel's entrance.

Ureteric stents: investigating flow and encrustation.

Blockages of the ureter, e.g. due to calculi (kidney stones), can result in an increase in renal pelvic pressure. This may be relieved by inserting a stent (essentially a permeable hollow tube). However, a number of complications are associated with stent use. Stents can result in reflux (backflow of urine along the ureter), which will promote recurrent urinary infection and possible renal parenchymal damage. Furthermore, long-term stent use is associated with infection and precipitation of salts from the urine, which can lead to a build-up of crystalline deposits on the stent surface, making stent removal difficult and painful. This paper examines factors governing urine flow in a stented ureter, the implications for reflux, and the processes by which the stent surface encrusts, in particular focusing on the influence of bacterial infection. An interdisciplinary approach is adopted, involving a combination of theoretical investigations and novel experiments.

Toward a mathematical model of the assembly and disassembly of membrane microdomains: comparison with experimental models.

We study a model system in which lipid bilayers are created using variable (precisely known) proportions of phosphatidylcholine and cholesterol. The model membranes exhibit cholesterol-enriched microdomains that are analogous to the so-called "lipid rafts" that form in living cells. After briefly presenting some experimental results, we formulate and solve a novel mathematical model based on the Smoluchowski equations for coagulation and fragmentation. We present a comparison between the distribution of lipid-raft areas observed in experimental lipid bilayers, and that distribution predicted by the theoretical model. Excellent agreement between the experiments and theory is obtained, with minimal parameter fitting.

Tissue growth in a rotating bioreactor. Part II: fluid flow and nutrient transport problems.

Fluid flow and nutrient transport around a growing tissue construct within a cylindrical bioreactor of circular cross-section are considered. The bioreactor is filled with nutrient-rich culture medium, and the growing tissue construct is modelled as a cylindrical obstacle, also of circular cross-section, at a given (moving) position within the nutrient solution. The bioreactor rotates about its cylindrical axis, and its axial length is small relative to its radius (the high-aspect ratio vessel bioreactor). This small-aspect ratio means that a simple idealized model may be considered, in which (leading order) quantities are averaged across the axial direction. The leading-order fluid flow is then of Hele-Shaw type, and may be solved for explicitly. The trajectory of the tissue construct within the rotating bioreactor is determined by analysis of the various forces acting on it. Several different modes of motion are found to be possible, depending on the experimental conditions, and examples of each type of motion are presented. Additionally, we solve the problem for the nutrient transport around the tissue construct in the special case in which the construct remains fixed in the laboratory frame, and (as the cells proliferate in response to the nutrient available locally) deduce growth rates for the construct. Finally, we discuss our results in the light of possible experimental bioreactor set-ups. We note the present model's limitations, and consider how our work could be extended and improved to inform experimental protocols in future.

Tissue growth in a rotating bioreactor. Part I: mechanical stability.

We develop mathematical models to provide insights into the morphology of a tissue construct formed from a single-cell suspension in culture media, within a rotating bioreactor. The bioreactor consists of a cylindrical vessel of circular cross-section rotating about its longitudinal axis with constant angular speed. Experimental studies show that at rotation rates below a critical value, the cells 'self-assemble' to form smooth 'nodules' that are approximately cylindrical with elliptical cross-section; however, at rotation rates above a critical value, an amorphous construct forms with a highly irregular boundary. The construct is denser than the surrounding culture media and histological studies indicate that the interior of the construct, which is a mix of apoptotic cells and culture media, is surrounded by an outer rim of proliferating cells and collagen. The construct is modelled as a viscous fluid drop surrounded by an extensible membrane in a (less dense) immiscible viscous fluid within a rotating bioreactor. We consider both thin-disk and slender-pipe bioreactors for which the aspect ratio, L(*)/a(*) (where L(*) and a(*) are the bioreactor length and radius, respectively), is small and large, respectively, and obtain a series of spatially 2D problems (independent of the axial coordinate). We then examine the hypothesis that the construct morphology is a result of the mechanical forces that it experiences by considering the interfacial stability of an initially circular fluid-fluid interface to small-amplitude, oscillatory perturbations. The instability is driven by the density difference between the two fluids, and we investigate the effect of the rotation rate, the (time-dependent) gravitational field, and the material and geometrical properties of the system on the stability properties.

The effect of ureteric stents on urine flow: reflux.

If the ureter becomes blocked, the resultant increased pressure may be relieved by inserting a double-J stent (a polymer tube, usually punctuated with holes). A major clinical problem associated with stent use is reflux (retrograde flow of urine from the bladder to the kidney), which may result in infections, scarring, and even renal failure. We develop a mathematical model, treating the ureter as an elastic tube and the stent as a permeable rigid tube within it. We investigate how the number of holes in the stent wall affects the total amount of reflux that occurs when bladder pressure rises, by considering the limits of a highly-permeable stent, and an impermeable stent. We find that, in the scenarios we consider, the highly-permeable stent gives rise to less total reflux than the impermeable one.