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.

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.

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 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.

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.

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.

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.

Definition and validation of operating equations for poly(vinyl alcohol)-poly(lactide-co-glycolide) microfiltration membrane-scaffold bioreactors.

The aim of this work is to provide operating data for biodegradable hollow fiber membrane bioreactors. The physicochemical cell culture environment can be controlled with the permeate flowrate, so this aim necessitates the provision of operating equations that enable end-users to set the pressures and feed flowrates to obtain their desired culture environment. In this paper, theoretical expressions for the pure water retentate and permeate flowrates, derived using lubrication theory, are compared against experimental data for a single fiber poly(vinyl alcohol)-poly(lactide-co-glycolide) crossflow module to give values for the membrane permeability and slip. Analysis of the width of the boundary layer region where slip effects are important, together with the sensitivity of the retentate and permeate equations to the slip parameter, show that slip is insignificant for these membranes, which have a mean pore diameter of 1.1 microm. The experimental data is used to determine a membrane permeability, of k = 1.86 x 10(-16) m(2), and to validate the model. It was concluded that the operating equation that relates the permeate to feed ratio, c, lumen inlet flowrate, Q (l,in), lumen outlet pressure, P (1), and ECS outlet pressure, P (0), is P(1) - P(0) = Q(l),in (Ac + B) where A and B are constants that depend on the membrane permeability and geometry (and are given explicitly). Finally, two worked examples are presented to demonstrate how a tissue engineer can use Equation (1) to specify operating conditions for their bioreactor.

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.

The influence of bioreactor geometry and the mechanical environment on engineered tissues.

A three phase model for the growth of a tissue construct within a perfusion bioreactor is examined. The cell population (and attendant extracellular matrix), culture medium, and porous scaffold are treated as distinct phases. The bioreactor system is represented by a two-dimensional channel containing a cell-seeded rigid porous scaffold (tissue construct), which is perfused with a culture medium. Through the prescription of appropriate functional forms for cell proliferation and extracellular matrix deposition rates, the model is used to compare the influence of cell density-, pressure-, and culture medium shear stress-regulated growth on the composition of the engineered tissue. The governing equations are derived in O'Dea et al. "A Three Phase Model for Tissue Construct Growth in a Perfusion Bioreactor," Math. Med. Biol., in which the long-wavelength limit was exploited to aid analysis; here, finite element methods are used to construct two-dimensional solutions to the governing equations and to investigate thoroughly their behavior. Comparison of the total tissue yield and averaged pressures, velocities, and shear stress demonstrates that quantitative agreement between the two-dimensional and long-wavelength approximation solutions is obtained for channel aspect ratios of order 10(-2) and that much of the qualitative behavior of the model is captured in the long-wavelength limit, even for relatively large channel aspect ratios. However, we demonstrate that in order to capture accurately the effect of mechanotransduction mechanisms on tissue construct growth, spatial effects in at least two dimensions must be included due to the inherent spatial variation of mechanical stimuli relevant to perfusion bioreactors, most notably, fluid shear stress, a feature not captured in the long-wavelength limit.

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.

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.

A mathematical model of liver cell aggregation in vitro.

The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.

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.

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.

In situ monitoring of 3D in vitro cell aggregation using an optical imaging system.

Bioreactor systems that maintain cells and tissues in suspension are increasingly popular for culturing 3D constructs to avoid the loss of in vivo cell function associated with traditional 2D culture methods. There is a need for the online monitoring of such systems to provide better understanding and control of the processes involved and to prevent the disruption of these processes caused by offline sampling and endpoint analysis. We describe a system for the imaging and analysis of cell aggregation, over long periods, within a high aspect rotating vessel (HARV). The system exploits side illumination, using an adjustable beam pattern, to restrict the detected light to that scattered by the cell aggregates, thus eliminating the need for the fluorescent labeling of the cells. The in situ aggregation of mammalian cells (MCF-7 breast carcinoma cells) was monitored over an 8 h period and image sequences showing the growth and motion of the aggregates within the bioreactor were obtained. Detailed size and population data have been derived characterizing the development of the aggregates during this time. We show how the number of resolvable aggregates increases to reach a peak and then declines as these aggregates merge. Once formed, remaining aggregates are found to consolidate to form more tightly packed bodies, typically reducing in cross-sectional area by one third. These results provide the basis for the development of an automated feedback system to control the growth of 3D cell cultures for repeatable, reliable, and quality controlled experimentation.