A Mathematical Model of Liver Cell Aggregation In Vitro

Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth

We employ the multiphase, moving boundary model of Byrne et al. (2003, Appl. Math. Lett., 16, 567-573) that describes the evolution of a motile, viscous tumour cell phase and an inviscid extracellular liquid phase. This model comprises two partial differential equations that govern the cell volume fraction and the cell velocity, together with a moving boundary condition for the tumour edge, and here we characterize and analyse its travelling-wave and pattern-forming behaviour. Numerical simulations of the model indicate that patterned solutions can be obtained, which correspond to multiple regions of high cell density separated by regions of low cell density. In other parameter regimes, solutions of the model can develop into a forward- or backward-moving travelling wave, corresponding to tumour growth or extinction, respectively. A travelling-wave analysis allows us to find the corresponding wave speed, as well as criteria for the growth or extinction of the tumour. Furthermore, a stability analysis of these travelling-wave solutions provides us with criteria for the occurrence of patterned solutions. Finally, we discuss how the initial cell distribution, as well as parameters related to cellular motion and cell-liquid drag, control the qualitative features of patterned solutions.

Thin-film lubrication model for biofilm expansion under strong adhesion

Understanding microbial biofilm growth is important to public health because biofilms are a leading cause of persistent clinical infections. In this paper, we develop a thin-film model for microbial biofilm growth on a solid substratum to which it adheres strongly. We model biofilms as two-phase viscous fluid mixtures of living cells and extracellular fluid. The model explicitly tracks the movement, depletion, and uptake of nutrients and incorporates cell proliferation via a nutrient-dependent source term. Notably, our thin-film reduction is two dimensional and includes the vertical dependence of cell volume fraction. Numerical solutions show that this vertical dependence is weak for biologically feasible parameters, reinforcing results from previous models in which this dependence was neglected. We exploit this weak dependence by writing and solving a simplified one-dimensional model that is computationally more efficient than the full model. We use both the one- and two-dimensional models to predict how model parameters affect expansion speed and biofilm thickness. This analysis reveals that expansion speed depends on cell proliferation, nutrient availability, cell-cell adhesion on the upper surface, and slip on the biofilm-substratum interface. Our numerical solutions provide a means to qualitatively distinguish between the extensional flow and lubrication regimes, and quantitative predictions that can be tested in future experiments.

Endothelial Cells Morphology in Response to Combined WSS and Biaxial CS: Introduction of Effective Strain Ratio

Introduction

Endothelial cells (ECs) morphology strongly depends on the imposed mechanical stimuli. These mechanical stimuli include wall shear stress (WSS) and biaxial cyclic stretches (CS). Under combined loading, the effect of CS is not as simple as pure CS. The present study investigates the morphological response of ECs to the realistic mechanical stimuli.

Methods

The cell population is theoretically studied using our previous validated model. The mechanical stimuli on ECs are described using four parameters; WSS magnitude (0 to 2.0 Pa), WSS angle (- 50° to 50°), and biaxial CS in two perpendicular directions (0 to 10%). The morphology of ECs is reported using four parameters; average shape index (SI) and orientation angle (OA) of the cell population as well as the standard deviation (SD) of SI and OA as measures for scattering of cells' SI and OA from these average values.

Results

A new effective strain ratio (ESR) is defined as the ratio of the undesirable CS to the desirable one. The obtained results of the model, illustrated that the SI and OA of cells increase with absolute value of ESR. In addition, the scattering in the SI of cells decreases with the absolute value of ESR, which means that the cell shapes become more regular. It is shown that the angular irregularity of cells increases at higher ESR values.

Conclusions

The results indicated that, the defined ESR is a stand-alone parameter for describing the realistic mechanical loading on the ECs and their morphological response.

Optimal control strategies for preventing hepatitis B infection and reducing chronic liver cirrhosis incidence

Advanced liver cirrhosis has become life-threatening among non-communicable diseases nowadays. Cirrhosis, the terminal stage of liver diseases in which the liver develops scarring as a result of various long-term continuous damages. Among liver diseases, viral hepatitis is the major risk factor for chronic cirrhosis development. The present paper demonstrates a compartmental model of chronic disease liver cirrhosis describing the transmission dynamics of this disease. Applying the Pontryagin’s maximum principle, the optimal control policies such as vaccination for hepatitis B virus and treatment of other causes of cirrhosis are adopted as control measures. The target of this study is to minimize the number of infected and liver cirrhotic individuals as well as the associated cost of the control. For this purpose, the optimal control strategies are employed according to the underlying causes behind this disease. Our goal is to find the strategy of preventing hepatitis B infection which is considered one of the leading causes of cirrhosis and consequently, reduction of the chronic cirrhosis incidence. Efficiency analysis is also performed to observe the effective control among the two control strategies. The model is investigated both analytically and numerically and the numerical simulations are carried out to illustrate the analytical findings. The analysis reveals that both the vaccination and treatment could be the most fruitful way to reduce the incidence of chronic liver cirrhosis.

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 multiscale approach for determining the morphology of endothelial cells at a coronary artery

The morphology of endothelial cells (ECs) may be an indication for determining atheroprone sites. Until now, there has been no clinical imaging technique to visualize the morphology of ECs in the arteries. The present study introduces a computational technique for determining the morphology of ECs. This technique is a multiscale simulation consisting of the artery scale and the cell scale. The artery scale is a fluid-structure interaction simulation. The input for the artery scale is the geometry of the coronary artery, that is, the dynamic curvature of the artery due to the cardiac motion, blood flow, blood pressure, heart rate, and the mechanical properties of the blood and the arterial wall, the measurements of which can be obtained for a specific patient. The results of the artery scale are wall shear stress (WSS) and cyclic strains as the mechanical stimuli of ECs. The cell scale is an inventive mass-and-spring model that is able to determine the morphological response of ECs to any combination of mechanical stimuli. The results of the multiscale simulation show the morphology of ECs at different locations of the coronary artery. The results indicate that the atheroprone sites have at least 1 of 3 factors: low time-averaged WSS, high angle of WSS, and high longitudinal strain. The most probable sites for atherosclerosis are located at the bifurcation region and lie on the myocardial side of the artery. The results also indicated that a higher dynamic curvature is a negative factor and a higher pulse pressure is a positive factor for protection against atherosclerosis.

A mechanical model for morphological response of endothelial cells under combined wall shear stress and cyclic stretch loadings

The shape and morphology of endothelial cells (ECs) lining the blood vessels are a good indicator for atheroprone and atheroprotected sites. ECs of blood vessels experience both wall shear stress (WSS) and cyclic stretch (CS). These mechanical stimuli influence the shape and morphology of ECs. A few models have been proposed for predicting the morphology of ECs under WSS or CS. In the present study, a mathematical cell population model is developed to simulate the morphology of ECs under combined WSS and CS conditions. The model considers the cytoskeletal filaments, cell-cell interactions, and cell-extracellular matrix interactions. In addition, the reorientation and polymerization of microfilaments are implemented in the model. The simulations are performed for different conditions: without mechanical stimuli, under pure WSS, under pure CS, and under combined WSS and CS. The results are represented as shape and morphology of ECs, shape index, and angle of orientation. The model is validated qualitatively and quantitatively with several experimental studies, and good agreement with experimental studies is achieved. To the best of our knowledge, it is the first model for predicting the morphology of ECs under combined WSS and CS condition. The model can be used to indicate the atheroprone regions of a patient's artery.

Lateral assembly of N-cadherin drives tissue integrity by stabilizing adherens junctions

Cadherin interactions ensure the correct registry and anchorage of cells during tissue formation. Along the plasma membrane, cadherins form inter-junctional lattices via cis- and trans-dimerization. While structural studies have provided models for cadherin interactions, the molecular nature of cadherin binding in vivo remains unexplored. We undertook a multi-disciplinary approach combining live cell imaging of three-dimensional cell assemblies (spheroids) with a computational model to study the dynamics of N-cadherin interactions. Using a loss-of-function strategy, we demonstrate that each N-cadherin interface plays a distinct role in spheroid formation. We found that cis-dimerization is not a prerequisite for trans-interactions, but rather modulates trans-interfaces to ensure tissue stability. Using a model of N-cadherin junction dynamics, we show that the absence of cis-interactions results in low junction stability and loss of tissue integrity. By quantifying the binding and unbinding dynamics of the N-cadherin binding interfaces, we determined that mutating either interface results in a 10-fold increase in the dissociation constant. These findings provide new quantitative information on the steps driving cadherin intercellular adhesion and demonstrate the role of cis-interactions in junction stability.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.