A continuum model for the elongation and orientation of Von Willebrand factor with applications in arterial flow.

The blood protein Von Willebrand factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates, the protein unfolds and binds to the arterial wall, enabling the rapid deposition of platelets from the blood. We present a novel continuum model for VWF dynamics in flow based on a modified viscoelastic fluid model that incorporates a single constitutive relation to describe the propensity of VWF to unfold as a function of the scalar shear rate. Using experimental data of VWF unfolding in pure shear flow, we fix the parameters for VWF's unfolding propensity and the maximum VWF length, so that the protein is half unfolded at a shear rate of approximately 5000 s - 1 . We then use the theoretical model to predict VWF's behaviour in two complex flows where experimental data are challenging to obtain: pure elongational flow and stenotic arterial flow. In pure elongational flow, our model predicts that VWF is 50% unfolded at approximately 2000 s - 1 , matching the established hypothesis that VWF unfolds at lower shear rates in elongational flow than in shear flow. We demonstrate the sensitivity of this elongational flow prediction to the value of maximum VWF length used in the model, which varies significantly across experimental studies, predicting that VWF can unfold between 2000 and 3200 s - 1 depending on the selected value. Finally, we examine VWF dynamics in a range of idealised arterial stenoses, predicting the relative extension of VWF in elongational flow structures in the centre of the artery compared to high shear regions near the arterial walls.

Role of Kidney Stones in Renal Pelvis Flow.

Ureteroscopy is a commonly performed medical procedure to treat stones in the kidney and ureter using a ureteroscope. Throughout the procedure, saline is irrigated through the scope to aid visibility and wash-out debris from stone fragmentation. The key challenge that this research addresses is to build a fundamental understanding of the interaction between the kidney stones/stone fragments and the flow dynamics in the renal pelvis flow. We examine the time-dependent flow dynamics inside an idealized renal pelvis in the context of a surgical procedure for kidney stone removal. Here, we examine the time-dependent evolution of these vortical flow structures in three dimensions, and incorporate the presence of rigid kidney stones. We perform direct numerical simulations, solving the transient Navier-Stokes equations in a spherical domain. Our numerical predictions for the flow dynamics in the absence of stones are validated with available experimental and numerical data, and the governing parameters and flow regimes are chosen carefully in order to satisfy several clinical constraints. The results shed light on the crucial role of flow circulation in the renal cavity and its effect on the trajectories of rigid stones. We demonstrate that stones can either be washed out of the cavity along with the fluid, or be trapped in the cavity via their interaction with vortical flow structures. Additionally, we study the effect of multiple stones in the flow field within the cavity in terms of the kinetic energy, entrapped fluid volume, and the clearance rate of a passive tracer modeled via an advection-diffusion equation. We demonstrate that the flow in the presence of stones features a higher vorticity production within the cavity compared with the stone-free cases.

Regenerative medicine meets mathematical modelling: developing symbiotic relationships.

Successful progression from bench to bedside for regenerative medicine products is challenging and requires a multidisciplinary approach. What has not yet been fully recognised is the potential for quantitative data analysis and mathematical modelling approaches to support this process. In this review, we highlight the wealth of opportunities for embedding mathematical and computational approaches within all stages of the regenerative medicine pipeline. We explore how exploiting quantitative mathematical and computational approaches, alongside state-of-the-art regenerative medicine research, can lead to therapies that potentially can be more rapidly translated into the clinic.

Experimental and mathematical modelling of magnetically labelled mesenchymal stromal cell delivery.

A key challenge for stem cell therapies is the delivery of therapeutic cells to the repair site. Magnetic targeting has been proposed as a platform for defining clinical sites of delivery more effectively. In this paper, we use a combined in vitro experimental and mathematical modelling approach to explore the magnetic targeting of mesenchymal stromal cells (MSCs) labelled with magnetic nanoparticles using an external magnet. This study aims to (i) demonstrate the potential of magnetic tagging for MSC delivery, (ii) examine the effect of red blood cells (RBCs) on MSC capture efficacy and (iii) highlight how mathematical models can provide both insight into mechanics of therapy and predictions about cell targeting in vivo. In vitro MSCs are cultured with magnetic nanoparticles and circulated with RBCs over an external magnet. Cell capture efficacy is measured for varying magnetic field strengths and RBC percentages. We use a 2D continuum mathematical model to represent the flow of magnetically tagged MSCs with RBCs. Numerical simulations demonstrate qualitative agreement with experimental results showing better capture with stronger magnetic fields and lower levels of RBCs. We additionally exploit the mathematical model to make hypotheses about the role of extravasation and identify future in vitro experiments to quantify this effect.

A temperature model for laser lithotripsy.

OBJECTIVE: To derive and validate a mathematical model to predict laser-induced temperature changes in a kidney during kidney stone treatment. METHODS: A simplified mathematical model to predict temperature change in the kidney for any given renal volume, irrigation flow rate, irrigation fluid temperature, and laser power was derived. We validated our model with matched in vitro experiments. RESULTS: Excellent agreement between the mathematical model predictions and laboratory data was obtained. CONCLUSION: The model obviates the need for repeated experimental validation. The model predicts scenarios where risk of renal tissue damage is high. With real-time knowledge of flow rate, irrigating fluid temperature and laser usage, safety warning levels could be predicted. Meanwhile, clinicians should be aware of the potential risk from thermal injury and take measures to reduce the risk, such as using room temperature irrigation fluid and judicious laser use.

Multiscale modelling and homogenisation of fibre-reinforced hydrogels for tissue engineering.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic-poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

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.

Pattern formation in multiphase models of chemotactic cell aggregation.

We develop a continuum model for the aggregation of cells cultured in a nutrient-rich medium in a culture well. We consider a 2D geometry, representing a vertical slice through the culture well, and assume that the cell layer depth is small compared with the typical lengthscale of the culture well. We adopt a continuum mechanics approach, treating the cells and culture medium as a two-phase mixture. Specifically, the cells and culture medium are treated as fluids. Additionally, the cell phase can generate forces in response to environmental cues, which include the concentration of a chemoattractant that is produced by the cells within the culture medium. The model leads to a system of coupled nonlinear partial differential equations for the volume fraction and velocity of the cell phase, the culture medium pressure and the chemoattractant concentration, which must be solved subject to appropriate boundary and initial conditions. To gain insight into the system, we consider two model reductions, appropriate when the cell layer depth is thin compared to the typical length scale of the culture well: a (simple) 1D and a (more involved) thin-film extensional flow reduction. By investigating the resulting systems of equations analytically and numerically, we identify conditions under which small amplitude perturbations to a homogeneous steady state (corresponding to a spatially uniform cell distribution) can lead to a spatially varying steady state (pattern formation). Our analysis reveals that the simpler 1D reduction has the same qualitative features as the thin-film extensional flow reduction in the linear and weakly nonlinear regimes, motivating the use of the simpler 1D modelling approach when a qualitative understanding of the system is required. However, the thin-film extensional flow reduction may be more appropriate when detailed quantitative agreement between modelling predictions and experimental data is desired. Furthermore, full numerical simulations of the two model reductions in regions of parameter space when the system is not close to marginal stability reveal significant differences in the evolution of the volume fraction and velocity of the cell phase, and chemoattractant concentration.

A mathematical model for cell infiltration and proliferation in a chondral defect.

We develop a mathematical model to describe the regeneration of a hydrogel inserted into an ex vivo osteochondral explant. Specifically we use partial differential equations to describe the evolution of two populations of cells that migrate from the tissue surrounding the defect, proliferate, and compete for space and resources within the hydrogel. The two cell populations are chondrocytes and cells that infiltrate from the subchondral bone. Model simulations are used to investigate how different seeding strategies and growth factor placement within the hydrogel affect the spatial distribution of both cell types. Since chondrocyte migration is extremely slow, we conclude that the hydrogel should be seeded with chondrocytes prior to culture in order to obtain zonal chondrocyte distributions typical of those associated with healthy cartilage.

Mathematical modelling of stretch-induced membrane traffic in bladder umbrella cells.

The bladder is a complex organ that is highly adaptive to its mechanical environment. The umbrella cells in the bladder uroepithelium are of particular interest: these cells actively change their surface area through exo- and endocytosis of cytoplasmic vesicles, and likely form a critical component in the mechanosensing process that communicates the sense of 'fullness' to the nervous system. In this paper we develop a first mechanical model for vesicle trafficking in umbrella cells in response to membrane tension during bladder filling. Recent experiments conducted on a disc of uroepithelial tissue motivate our model development. These experiments subject bladder tissue to fixed pressure differences and exhibit counterintuitive area changes. Through analysis of the mathematical model and comparison with experimental data in this setup, we gain an intuitive understanding of the biophysical processes involved and calibrate the vesicle trafficking rate parameters in our model. We then adapt the model to simulate in vivo bladder filling and investigate the potential effect of abnormalities in the vesicle trafficking machinery on bladder pathologies.

The influence of hydrostatic pressure on tissue engineered bone development.

The hydrostatic pressure stimulation of an appropriately cell-seeded porous scaffold within a bioreactor is a promising method for engineering bone tissue external to the body. We propose a mathematical model, and employ a suite of candidate constitutive laws, to qualitatively describe the effect of applied hydrostatic pressure on the quantity of minerals deposited in such an experimental setup. By comparing data from numerical simulations with experimental observations under a number of stimulation protocols, we suggest that the response of bone cells to an applied pressure requires consideration of two components; (i) a component describing the cell memory of the applied stimulation, and (ii) a recovery component, capturing the time cells require to recover from high rates of mineralisation.

On a poroviscoelastic model for cell crawling.

In this paper a minimal, one-dimensional, two-phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two-phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper-convected Maxwell model and demonstrate that even the simplest of two-phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill-posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling-wave solution in which the crawling velocity has a bell-shaped dependence on adhesion strength, in agreement with biological observation.

A multiscale analysis of nutrient transport and biological tissue growth in vitro.

In this paper, we consider the derivation of macroscopic equations appropriate to describe the growth of biological tissue, employing a multiple-scale homogenization method to accommodate explicitly the influence of the underlying microscale structure of the material, and its evolution, on the macroscale dynamics. Such methods have been widely used to study porous and poroelastic materials; however, a distinguishing feature of biological tissue is its ability to remodel continuously in response to local environmental cues. Here, we present the derivation of a model broadly applicable to tissue engineering applications, characterized by cell proliferation and extracellular matrix deposition in porous scaffolds used within tissue culture systems, which we use to study coupling between fluid flow, nutrient transport, and microscale tissue growth. Attention is restricted to surface accretion within a rigid porous medium saturated with a Newtonian fluid; coupling between the various dynamics is achieved by specifying the rate of microscale growth to be dependent upon the uptake of a generic diffusible nutrient. The resulting macroscale model comprises a Darcy-type equation governing fluid flow, with flow characteristics dictated by the assumed periodic microstructure and surface growth rate of the porous medium, coupled to an advection-reaction equation specifying the nutrient concentration. Illustrative numerical simulations are presented to indicate the influence of microscale growth on macroscale dynamics, and to highlight the importance of including experimentally relevant microstructural information to correctly determine flow dynamics and nutrient delivery in tissue engineering applications.

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.

The interplay between tissue growth and scaffold degradation in engineered tissue constructs.

In vitro tissue engineering is emerging as a potential tool to meet the high demand for replacement tissue, caused by the increased incidence of tissue degeneration and damage. A key challenge in this field is ensuring that the mechanical properties of the engineered tissue are appropriate for the in vivo environment. Achieving this goal will require detailed understanding of the interplay between cell proliferation, extracellular matrix (ECM) deposition and scaffold degradation. In this paper, we use a mathematical model (based upon a multiphase continuum framework) to investigate the interplay between tissue growth and scaffold degradation during tissue construct evolution in vitro. Our model accommodates a cell population and culture medium, modelled as viscous fluids, together with a porous scaffold and ECM deposited by the cells, represented as rigid porous materials. We focus on tissue growth within a perfusion bioreactor system, and investigate how the predicted tissue composition is altered under the influence of (1) differential interactions between cells and the supporting scaffold and their associated ECM, (2) scaffold degradation, and (3) mechanotransduction-regulated cell proliferation and ECM deposition. Numerical simulation of the model equations reveals that scaffold heterogeneity typical of that obtained from [Formula: see text]CT scans of tissue engineering scaffolds can lead to significant variation in the flow-induced mechanical stimuli experienced by cells seeded in the scaffold. This leads to strong heterogeneity in the deposition of ECM. Furthermore, preferential adherence of cells to the ECM in favour of the artificial scaffold appears to have no significant influence on the eventual construct composition; adherence of cells to these supporting structures does, however, lead to cell and ECM distributions which mimic and exaggerate the heterogeneity of the underlying scaffold. Such phenomena have important ramifications for the mechanical integrity of engineered tissue constructs and their suitability for implantation in vivo.

Multiple travelling-wave solutions in a minimal model for cell motility.

Two-phase flow models have been used previously to model cell motility. In order to reduce the complexity inherent with describing the many physical processes, we formulate a minimal model. Here we demonstrate that even the simplest 1D, two-phase, poroviscous, reactive flow model displays various types of behaviour relevant to cell crawling. We present stability analyses that show that an asymmetric perturbation is required to cause a spatially uniform, stationary strip of cytoplasm to move, which is relevant to cell polarization. Our numerical simulations identify qualitatively distinct families of travelling-wave solutions that coexist at certain parameter values. Within each family, the crawling speed of the strip has a bell-shaped dependence on the adhesion strength. The model captures the experimentally observed behaviour that cells crawl quickest at intermediate adhesion strengths, when the substrate is neither too sticky nor too slippy.

On the predictions and limitations of the Becker-Döring model for reaction kinetics in micellar surfactant solutions.

We investigate the breakdown of a system of micellar aggregates in a surfactant solution following an order-one dilution. We derive a mathematical model based on the Becker-Döring system of equations, using realistic expressions for the reaction constants fit to results from Molecular Dynamics simulations. We exploit the largeness of typical aggregation numbers to derive a continuum model, substituting a large system of ordinary differential equations for a partial differential equation in two independent variables: time and aggregate size. Numerical solutions demonstrate that re-equilibration occurs in two distinct stages over well-separated timescales, in agreement with experiment and with previous theories. We conclude by exposing a limitation in the Becker-Döring theory for re-equilibration of surfactant solutions.

A strategy to determine operating parameters in tissue engineering hollow fiber bioreactors.

The development of tissue engineering hollow fiber bioreactors (HFB) requires the optimal design of the geometry and operation parameters of the system. This article provides a strategy for specifying operating conditions for the system based on mathematical models of oxygen delivery to the cell population. Analytical and numerical solutions of these models are developed based on Michaelis-Menten kinetics. Depending on the minimum oxygen concentration required to culture a functional cell population, together with the oxygen uptake kinetics, the strategy dictates the model needed to describe mass transport so that the operating conditions can be defined. If c(min) ≫ K(m) we capture oxygen uptake using zero-order kinetics and proceed analytically. This enables operating equations to be developed that allow the user to choose the medium flow rate, lumen length, and ECS depth to provide a prescribed value of c(min) . When c(min) />>K(m), we use numerical techniques to solve full Michaelis-Menten kinetics and present operating data for the bioreactor. The strategy presented utilizes both analytical and numerical approaches and can be applied to any cell type with known oxygen transport properties and uptake kinetics.

Growth-induced buckling of an epithelial layer.

We use a proof-of-concept experiment and two mathematical models to explore growth-induced tissue buckling, as may occur in colorectal crypt formation. Our experiment reveals how growth of a cultured epithelial monolayer on a thin flexible substrate can cause out-of-plane substrate deflections. We describe this system theoretically using a 'bilayer' model in which a growing cell layer adheres to a thin compressible elastic beam. We compare this with the 'supported-monolayer' model due to Edwards and Chapman (Bull Math Biol 69:1927-1942, 2007) for an incompressible expanding beam (representing crypt epithelium), which incorporates viscoelastic tethering to underlying stroma. We show that the bilayer model can exhibit buckling via parametric growth (in which the system passes through a sequence of equilibrium states, parameterised by the total beam length); in this case, non-uniformities in cell growth and variations in cell-substrate adhesion are predicted to have minimal effect on the shape of resulting buckled states. The supported-monolayer model reveals how competition between lateral supports and stromal adhesion influences the wavelength of buckled states (in parametric growth), and how non-equilibrium relaxation of tethering forces influences post-buckled shapes. This model also predicts that non-uniformities in growth patterns have a much weaker influence on buckled shapes than non-uniformities in material properties. Together, the experiment and models support the concept of patterning by growth-induced buckling and suggest that targeted softening of a growing cell layer provides greater control in shaping tissues than non-uniform growth.

Non-local models for the formation of hepatocyte-stellate cell aggregates.

Liver cell aggregates may be grown in vitro by co-culturing hepatocytes with stellate cells. This method results in more rapid aggregation than hepatocyte-only culture, and appears to enhance cell viability and the expression of markers of liver-specific functions. We consider the early stages of aggregate formation, and develop a new mathematical model to investigate two alternative hypotheses (based on evidence in the experimental literature) for the role of stellate cells in promoting aggregate formation. Under Hypothesis 1, each population produces a chemical signal which affects the other, and enhanced aggregation is due to chemotaxis. Hypothesis 2 asserts that the interaction between the two cell types is by direct physical contact: the stellates extend long cellular processes which pull the hepatocytes into the aggregates. Under both hypotheses, hepatocytes are attracted to a chemical they themselves produce, and the cells can experience repulsive forces due to overcrowding. We formulate non-local (integro-partial differential) equations to describe the densities of cells, which are coupled to reaction-diffusion equations for the chemical concentrations. The behaviour of the model under each hypothesis is studied using a combination of linear stability analysis and numerical simulations. Our results show how the initial rate of aggregation depends upon the cell seeding ratio, and how the distribution of cells within aggregates depends on the relative strengths of attraction and repulsion between the cell types. Guided by our results, we suggest experiments which could be performed to distinguish between the two hypotheses.