Segregation of co-cultured multicellular systems: review and modeling consideration

Cell segregation caused by collective cell migration (CCM) is crucial for morphogenesis, functional development of tissue parts, and is an important aspect in other diseases such as cancer and its metastasis process. Efficiency of the cell segregation depends on the interplay between: (1) biochemical processes such as cell signaling and gene expression and (2) physical interactions between cells. Despite extensive research devoted to study the segregation of various co-cultured systems, we still do not understand the role of physical interactions in cell segregation. Cumulative effects of these physical interactions appear in the form of physical parameters such as: (1) tissue surface tension, (2) viscoelasticity caused by CCM, and (3) solid stress accumulated in multicellular systems. These parameters primarily depend on the interplay between the state of cell-cell adhesion contacts and cell contractility. The role of these physical parameters on the segregation efficiency is discussed on model systems such as co-cultured breast cell spheroids consisting of two subpopulations that are in contact. This review study aims to: (1) summarize biological aspects related to cell segregation, mechanical properties of cell collectives, effects along the biointerface between cell subpopulations and (2) describe from a biophysical/mathematical perspective the same biological aspects summarized before. So that overall it can illustrate the complexity of the biological systems that translate into very complex biophysical/mathematical equations. Moreover, by presenting in parallel these two seemingly different parts (biology vs. equations), this review aims to emphasize the need for experiments to estimate the variety of parameters entering the resulting complex biophysical/mathematical models.

Simulating photodynamic therapy for the treatment of glioblastoma using Monte Carlo radiative transport

Significance

Glioblastoma (GBM) is a rare but deadly form of brain tumor with a low median survival rate of 14.6 months, due to its resistance to treatment. An independent simulation of the INtraoperative photoDYnamic therapy for GliOblastoma (INDYGO) trial, a clinical trial aiming to treat the GBM resection cavity with photodynamic therapy (PDT) via a laser coupled balloon device, is demonstrated.

Aim

To develop a framework providing increased understanding for the PDT treatment, its parameters, and their impact on the clinical outcome.

Approach

We use Monte Carlo radiative transport techniques within a computational brain model containing a GBM to simulate light path and PDT effects. Treatment parameters (laser power, photosensitizer concentration, and irradiation time) are considered, as well as PDT's impact on brain tissue temperature.

Results

The simulation suggests that 39% of post-resection GBM cells are killed at the end of treatment when using the standard INDYGO trial protocol (light fluence = 200    J / cm 2 at balloon wall) and assuming an initial photosensitizer concentration of 5    μ M . Increases in treatment time and light power (light fluence = 400    J / cm 2 at balloon wall) result in further cell kill but increase brain cell temperature, which potentially affects treatment safety. Increasing the p hotosensitizer concentration produces the most significant increase in cell kill, with 61% of GBM cells killed when doubling concentration to 10    μ M and keeping the treatment time and power the same. According to these simulations, the standard trial protocol is reasonably well optimized with improvements in cell kill difficult to achieve without potentially dangerous increases in temperature. To improve treatment outcome, focus should be placed on improving the photosensitizer.

Conclusions

With further development and optimization, the simulation could have potential clinical benefit and be used to help plan and optimize intraoperative PDT treatment for GBM.

Modelling Keloids Dynamics: A Brief Review and New Mathematical Perspectives

Keloids are fibroproliferative disorders described by excessive growth of fibrotic tissue, which also invades adjacent areas (beyond the original wound borders). Since these disorders are specific to humans (no other animal species naturally develop keloid-like tissue), experimental in vivo/in vitro research has not led to significant advances in this field. One possible approach could be to combine in vitro human models with calibrated in silico mathematical approaches (i.e., models and simulations) to generate new testable biological hypotheses related to biological mechanisms and improved treatments. Because these combined approaches do not really exist for keloid disorders, in this brief review we start by summarising the biology of these disorders, then present various types of mathematical and computational approaches used for related disorders (i.e., wound healing and solid tumours), followed by a discussion of the very few mathematical and computational models published so far to study various inflammatory and mechanical aspects of keloids. We conclude this review by discussing some open problems and mathematical opportunities offered in the context of keloid disorders by such combined in vitro/in silico approaches, and the need for multi-disciplinary research to enable clinical progress.

Mathematical investigation of normal and abnormal wound healing dynamics: local and non-local models

The movement of cells during (normal and abnormal) wound healing is the result of biomechanical interactions that combine cell responses with growth factors as well as cell-cell and cell-matrix interactions (adhesion and remodelling). It is known that cells can communicate and interact locally and non-locally with other cells inside the tissues through mechanical forces that act locally and at a distance, as well as through long non-conventional cell protrusions. In this study, we consider a non-local partial differential equation model for the interactions between fibroblasts, macrophages and the extracellular matrix (ECM) via a growth factor (TGF-$ \beta $) in the context of wound healing. For the non-local interactions, we consider two types of kernels (i.e., a Gaussian kernel and a cone-shaped kernel), two types of cell-ECM adhesion functions (i.e., adhesion only to higher-density ECM vs. adhesion to higher-/lower-density ECM) and two types of cell proliferation terms (i.e., with and without decay due to overcrowding). We investigate numerically the dynamics of this non-local model, as well as the dynamics of the localised versions of this model (i.e., those obtained when the cell perception radius decreases to 0). The results suggest the following: (ⅰ) local models explain normal wound healing and non-local models could also explain abnormal wound healing (although the results are parameter-dependent); (ⅱ) the models can explain two types of wound healing, i.e., by primary intention, when the wound margins come together from the side, and by secondary intention when the wound heals from the bottom up.

Multi-scale nature of the tissue surface tension: Theoretical consideration on tissue model systems

Tissue surface tension is one of the key parameters that govern tissue rearrangement, shaping, and segregation within various compartments during organogenesis, wound healing, and cancer diseases. Deeper insight into the relationship between tissue surface tension and cell residual stress accumulation caused by collective cell migration can help us to understand the multi-scale nature of cell rearrangement with pronounced oscillatory trend. Oscillatory change of cell velocity that caused strain and generated cell residual stress were discussed in the context of mechanical waves. The tissue surface tension also showed oscillatory behaviour. The main goal of this theoretical consideration is to emphasize an inter-relation between various scenarios of cell rearrangement and tissue surface tension by distinguishing liquid-like and solid-like surfaces. This complex phenomenon is discussed in the context of an artificial tissue model system, namely cell aggregate rounding after uni-axial compression between parallel plates. Experimentally obtained oscillatory changes in the cell aggregate shape during the aggregate rounding, which is accompanied by oscillatory decrease in the aggregate surface area, points to oscillatory changes in the tissue surface tension. Besides long-time oscillations, cell surface tension can perform short time relaxation cycles. This behaviour of the tissue surface tension distinguishes living matter from other soft matter systems. This complex phenomenon is discussed based on dilatational viscoelasticity and thermodynamic approach.

The dynamics along the biointerface between the epithelial and cancer mesenchymal cells: Modeling consideration

Epithelial cancer is the one of most lethal cancer type worldwide. Targeting the early stage of disease would allow dramatic improvements in the survival of cancer patients. The early stage of the disease is related to cancer cell spreading across surrounding healthy epithelium. Consequently, deeper insight into cell dynamics along the biointerface between epithelial and cancer (mesenchymal) cells is necessary in order to control the disease as soon as possible. Cell dynamics along this epithelial-cancer biointerface is the result of the interplay between various biological and physical mechanisms. Despite extensive research devoted to study cancer cell spreading across the epithelium, we still do not understand the physical mechanisms which influences the dynamics along the biointerface. These physical mechanisms are related to the interplay between physical parameters such as: (1) interfacial tension between cancer and epithelial subpopulations, (2) established interfacial tension gradients, (3) the bending rigidity of the biointerface and its impact on the interfacial tension, (4) surface tension of the subpopulations, (5) viscoelasticity caused by collective cell migration, and (6) cell residual stress accumulation. The main goal of this study is to review some of these physical parameters in the context of the epithelial/cancer biointerface elaborated on the model system such as the biointerface between breast epithelial MCF-10A cells and cancer MDA-MB-231 cells and then to incorporate these parameters into a new biophysical model that could describe the dynamics of the biointerface. We conclude by discussing three biophysical scenarios for cell dynamics along the biointerface, which can occur depending on the magnitude of the generated shear stress: a smooth biointerface, a slightly-perturbed biointerface and an intensively-perturbed biointerface in the context of the Kelvin-Helmholtz instability. These scenarios are related to the probability of cancer invasion.

The rearrangement of co-cultured cellular model systems via collective cell migration

Cancer invasion through the surrounding epithelium and extracellular matrix (ECM) is the one of the main characteristics of cancer progression. While significant effort has been made to predict cancer cells response under various drug therapies, much less attention has been paid to understand the physical interactions between cancer cells and their microenvironment, which are essential for cancer invasion. Considering these physical interactions on various co-cultured in vitro model systems by emphasizing the role of viscoelasticity, the tissue surface tension, solid stress, and their inter-relations is a prerequisite for establishing the main factors that influence cancer cell spread and develop an efficient strategy to suppress it. This review focuses on the role of viscoelasticity caused by collective cell migration (CCM) in the context of mono-cultured and co-cultured cancer systems, and on the modeling approaches aimed at reproducing and understanding these biological systems. In this context, we do not only review previously-published biophysics models for collective cell migration, but also propose new extensions of those models to include solid stress accumulated within the spheroid core region and cell residual stress accumulation caused by CCM.

Mathematical investigation into the role of macrophage heterogeneity on the temporal and spatio-temporal dynamics of non-small cell lung cancers

Non Small Cell Lung Cancer (NSCLC) is the most common type of lung cancer, and represents the leading cause of cancer-related deaths worldwide. Experimental studies have shown that these solid cancers are heavily infiltrated with macrophages: anti-tumour M1 macrophages, pro-tumour M2 macrophages, and macrophage subtypes sharing both M1 and M2 properties. In this study we aim to investigate qualitatively the role of macrophages with different functional phenotypes (especially those with mixed phenotypes) on cancer dynamics and the success of different immunotherapies for cancer. To this end, we start with two time-evolving mathematical models for cancer-immune interactions that consider: (i) the effect of the two extreme phenotypes, M1 and M2 cells; (ii) the effect of M1 and M2 cells, as well as a macrophage sub-population with a mixed phenotype (throughout this theoretical study we call these cells "M12 cells"). We compare the dynamics of the two models using computational approaches, paying particular attention to the effect of different anti-cancer immunotherapies that focus on macrophages. Since data available for NSCLC and macrophage interactions are incomplete, we perform a global sensitivity analysis to see the influence of input parameters on model outcomes. Finally, we consider extensions of the previous two models to include also the spatial movement of cells, and investigate the role of macrophages with extreme phenotypes and with mixed phenotypes, on the invasion of cancer cells into the surrounding extracellular matrix (ECM). We use numerical simulations to investigate the macrophages phenotypes at the tumour center versus the invasive margin. Again, we examine the impact of immunotherapies for cancer on the spatial dynamics of cancers and immune cells, and observe a shift in the phenotype of macrophages distributed at the tumour center and invasive margin.

A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: application to French data

Many SARS-CoV-2 variants have appeared over the last months, and many more will continue to appear. Understanding the competition between these different variants could help make future predictions on the evolution of epidemics. In this study we use a mathematical model to investigate the impact of three different SARS-CoV-2 variants on the spread of COVID-19 across France, between January-May 2021 (before vaccination was extended to the full population). To this end, we use the data from Geodes (produced by Public Health France) and a particle swarm optimisation algorithm, to estimate the model parameters and further calculate a value for the basic reproduction number R₀. Sensitivity and uncertainty analysis is then used to better understand the impact of estimated parameter values on the number of infections leading to both symptomatic and asymptomatic individuals. The results confirmed that, as expected, the alpha, beta and gamma variants are more transmissible than the original viral strain. In addition, the sensitivity results showed that the beta/gamma variants could have lead to a larger number of infections in France (of both symptomatic and asymptomatic people).

A computational investigation of COVID-19 transmission inside hospital wards and associated costs

The COVID-19 pandemic has placed a particular burden on hospitals: from intra-hospital transmission of the infections to reduced admissions of non-COVID-19 patients. There are also high costs associated with the treatment of hospitalised COVID-19 patients, as well as reductions in revenues due to delayed and cancelled treatments. In this study we investigate computationally the transmission of COVID-19 inside a hospital ward that contains multiple-bed bays (with 4 or 6 beds) and multiple single-bed side rooms (that can accommodate the contacts of COVID-19-positive patients). The aim of this study is to investigate the role of 4-bed bays vs. 6-bed bays on the spread of infections and the hospital costs. We show that 4-bed bays are associated with lower infections only when we reduce the discharge time of some patients from 10 days to 5 days. This also leads to lower costs for the treatment of COVID-19 patients. In contrast, 6-bed bays are associated with reduced hospital waiting lists (especially when there are also multiple side rooms available to accommodate the contacts of COVID-19-positive patients identified inside the 6-bed bays).

Nonlocal multiscale modelling of tumour-oncolytic viruses interactions within a heterogeneous fibrous/non-fibrous extracellular matrix

In this study we investigate computationally tumour-oncolytic virus (OV) interactions that take place within a heterogeneous extracellular matrix (ECM). The ECM is viewed as a mixture of two constitutive phases, namely a fibre phase and a non-fibre phase. The multiscale mathematical model presented here focuses on the nonlocal cell-cell and cell-ECM interactions, and how these interactions might be impacted by the infection of cancer cells with the OV. At macroscale we track the kinetics of cancer cells, virus particles and the ECM. At microscale we track (i) the degradation of ECM by matrix degrading enzymes (MDEs) produced by cancer cells, which further influences the movement of tumour boundary; (ii) the re-arrangement of the microfibres that influences the re-arrangement of macrofibres (i.e., fibres at macroscale). With the help of this new multiscale model, we investigate two questions: (i) whether the infected cancer cell fluxes are the result of local or non-local advection in response to ECM density; and (ii) what is the effect of ECM fibres on the the spatial spread of oncolytic viruses and the outcome of oncolytic virotherapy.

Inverse problem approaches for mutation laws in heterogeneous tumours with local and nonlocal dynamics

Cancer cell mutations occur when cells undergo multiple cell divisions, and these mutations can be spontaneous or environmentally-induced. The mechanisms that promote and sustain these mutations are still not fully understood. This study deals with the identification (or reconstruction) of the usually unknown cancer cell mutation law, which lead to the transformation of a primary tumour cell population into a secondary, more aggressive cell population. We focus on local and nonlocal mathematical models for cell dynamics and movement, and identify these mutation laws from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. In a local cancer invasion model, we first reconstruct the mutation law when we assume that the mutations depend only on the surrounding cancer cells (i.e., the ECM plays no role in mutations). Second, we assume that the mutations depend on the ECM only, and we reconstruct the mutation law in this case. Third, we reconstruct the mutation when we assume that there is no prior knowledge about the mutations. Finally, for the nonlocal cancer invasion model, we reconstruct the mutation law that depends on the cancer cells and on the ECM. For these numerical reconstructions, our approximations are based on the finite difference method combined with the finite elements method. As the inverse problem is ill-posed, we use the Tikhonov regularisation technique in order to regularise the solution. Stability of the solution is examined by adding additive noise into the measurements.

Quantitative predictive approaches for Dupuytren disease: a brief review and future perspectives

In this study we review the current state of the art for Dupuytren's disease (DD), while emphasising the need for a better integration of clinical, experimental and quantitative predictive approaches to understand the evolution of the disease and improve current treatments. We start with a brief review of the biology of this disease and current treatment approaches. Then, since certain aspects in the pathogenesis of this disorder have been compared to various biological aspects of wound healing and malignant processes, next we review some in silico (mathematical modelling and simulations) predictive approaches for complex multi-scale biological interactions occurring in wound healing and cancer. We also review the very few in silico approaches for DD, and emphasise the applicability of these approaches to address more biological questions related to this disease. We conclude by proposing new mathematical modelling and computational approaches for DD, which could be used in the absence of animal models to make qualitative and quantitative predictions about the evolution of this disease that could be further tested in vitro.

Non-local multiscale approach for the impact of go or grow hypothesis on tumour-viruses interactions

We propose and study computationally a novel non-local multiscale moving boundary mathematical model for tumour and oncolytic virus (OV) interactions when we consider the go or grow hypothesis for cancer dynamics. This spatio-temporal model focuses on two cancer cell phenotypes that can be infected with the OV or remain uninfected, and which can either move in response to the extracellular-matrix (ECM) density or proliferate. The interactions between cancer cells, those among cancer cells and ECM, and those among cells and OV occur at the macroscale. At the micro-scale, we focus on the interactions between cells and matrix degrading enzymes (MDEs) that impact the movement of tumour boundary. With the help of this multiscale model we explore the impact on tumour invasion patterns of two different assumptions that we consider in regard to cell-cell and cell-matrix interactions. In particular we investigate model dynamics when we assume that cancer cell fluxes are the result of local advection in response to the density of extracellular matrix (ECM), or of non-local advection in response to cell-ECM adhesion. We also investigate the role of the transition rates between mainly-moving and mainly-growing cancer cell sub-populations, as well as the role of virus infection rate and virus replication rate on the overall tumour dynamics.

Collective Cell Migration in a Fibrous Environment: A Hybrid Multiscale Modelling Approach

The specific structure of the extracellular matrix (ECM), and in particular the density and orientation of collagen fibres, plays an important role in the evolution of solid cancers. While many experimental studies discussed the role of ECM in individual and collective cell migration, there are still unanswered questions about the impact of nonlocal cell sensing of other cells on the overall shape of tumour aggregation and its migration type. There are also unanswered questions about the migration and spread of tumour that arises at the boundary between different tissues with different collagen fibre orientations. To address these questions, in this study we develop a hybrid multi-scale model that considers the cells as individual entities and ECM as a continuous field. The numerical simulations obtained through this model match experimental observations, confirming that tumour aggregations are not moving if the ECM fibres are distributed randomly, and they only move when the ECM fibres are highly aligned. Moreover, the stationary tumour aggregations can have circular shapes or irregular shapes (with finger-like protrusions), while the moving tumour aggregations have elongate shapes (resembling to clusters, strands or files). We also show that the cell sensing radius impacts tumour shape only when there is a low ratio of fibre to non-fibre ECM components. Finally, we investigate the impact of different ECM fibre orientations corresponding to different tissues, on the overall tumour invasion of these neighbouring tissues.

Leadership Through Influence: What Mechanisms Allow Leaders to Steer a Swarm?

Collective migration of cells and animals often relies on a specialised set of “leaders”, whose role is to steer a population of naive followers towards some target. We formulate a continuous model to understand the dynamics and structure of such groups, splitting a population into separate follower and leader types with distinct orientation responses. We incorporate leader influence via three principal mechanisms: a bias in the orientation of leaders towards the destination (orientation-bias), a faster movement of leaders when moving towards the target (speed-bias), and leaders making themselves more clear to followers when moving towards the target (conspicuousness-bias). Analysis and numerical computation are used to assess the extent to which the swarm is successfully shepherded towards the target. We find that successful leadership can occur for each of these three mechanisms across a broad region of parameter space, with conspicuousness-bias emerging as the most robust. However, outside this parameter space we also find various forms of unsuccessful leadership. Forms of excessive influence can result in either swarm-splitting, where the leaders break free and followers are left rudderless, or a loss of swarm cohesion that leads to its eventual dispersal. Forms of low influence, on the other hand, can even generate swarms that move away from the target direction. Leadership must therefore be carefully managed to steer the swarm correctly.

Mathematical investigation of innate immune responses to lung cancer: The role of macrophages with mixed phenotypes

Macrophages' role in the evolution of solid tumours is a well accepted fact, with the M1-like macrophages having an anti-tumour role and the M2-like macrophages having a pro-tumour role. Despite the fact that some clinical studies on lung tumours have emphasised also the presence of macrophages with mixed M1 and M2 phenotypes in addition to macrophages with distinct phenotypes, the majority of studies still use the distinct M1-M2 classification to predict the evolution of tumours and patient survival. In this theoretical study we use a mathematical modelling and computational approach to investigate the role of macrophages with mixed phenotype on growth/control/elimination of lung tumours. We show that tumour control in the presence of M2→M1 re-polarising treatments is mainly the result of macrophages with mixed phenotypes (due to the assumption of short half-life of M1-like macrophages). We also show that the half-life of various macrophage phenotypes (distinct M1 or mixed M1/M2 phenotypes) impacts the outcome of various therapeutic strategies targeting tumour-associated macrophages. All these results suggest the need for a better experimental understanding of the kinetics of macrophages inside solid tumours.

Oncolytic viral therapies and the delicate balance between virus-macrophage-tumour interactions: A mathematical approach

The success of oncolytic virotherapies depends on the tumour microenvironment, which contains a large number of infiltrating immune cells. In this theoretical study, we derive an ODE model to investigate the interactions between breast cancer tumour cells, an oncolytic virus (Vesicular Stomatitis Virus), and tumour-infiltrating macrophages with different phenotypes which can impact the dynamics of oncolytic viruses. The complexity of the model requires a combined analytical-numerical approach to understand the transient and asymptotic dynamics of this model. We use this model to propose new biological hypotheses regarding the impact on tumour elimination/relapse/persistence of: (i) different macrophage polarisation/re-polarisation rates; (ii) different infection rates of macrophages and tumour cells with the oncolytic virus; (iii) different viral burst sizes for macrophages and tumour cells. We show that increasing the rate at which the oncolytic virus infects the tumour cells can delay tumour relapse and even eliminate tumour. Increasing the rate at which the oncolytic virus particles infect the macrophages can trigger transitions between steady-state dynamics and oscillatory dynamics, but it does not lead to tumour elimination unless the tumour infection rate is also very large. Moreover, we confirm numerically that a large tumour-induced M1→M2 polarisation leads to fast tumour growth and fast relapse (if the tumour was reduced before by a strong anti-tumour immune and viral response). The increase in viral-induced M2→M1 re-polarisation reduces temporarily the tumour size, but does not lead to tumour elimination. Finally, we show numerically that the tumour size is more sensitive to the production of viruses by the infected macrophages.

Modelling the effects of environmental heterogeneity within the lung on the tuberculosis life-cycle

•

In silico model of TB in the lung incorporating environmental heterogeneity.

•

Preferential conditions at the apex of lung localise post-primary disease there.

•

Analysis of the key influences driving disease at different regions of the lung.

Progress in shortening the duration of tuberculosis (TB) treatment is hampered by the lack of a predictive model that accurately reflects the diverse environment within the lung. This is important as TB has been shown to produce distinct localisations to different areas of the lung during different disease stages, with the environmental heterogeneity within the lung of factors such as air ventilation, blood perfusion and oxygen tension believed to contribute to the apical localisation witnessed during the post-primary form of the disease.

Building upon our previous model of environmental lung heterogeneity, we present a networked metapopulation model that simulates TB across the whole lung, incorporating these notions of environmental heterogeneity across the whole TB life-cycle to show how different stages of the disease are influenced by different environmental and immunological factors. The alveolar tissue in the lung is divided into distinct patches, with each patch representing a portion of the total tissue and containing environmental attributes that reflect the internal conditions at that location. We include populations of bacteria and immune cells in various states, and events are included which determine how the members of the model interact with each other and the environment. By allowing some of these events to be dependent on environmental attributes, we create a set of heterogeneous dynamics, whereby the location of the tissue within the lung determines the disease pathological events that occur there.

Our results show that the environmental heterogeneity within the lung is a plausible driving force behind the apical localisation during post-primary disease. After initial infection, bacterial levels will grow in the initial infection location at the base of the lung until an adaptive immune response is initiated. During this period, bacteria are able to disseminate and create new lesions throughout the lung. During the latent stage, the lesions that are situated towards the apex are the largest in size, and once a post-primary immune-suppressing event occurs, it is the uppermost lesions that reach the highest levels of bacterial proliferation. Our sensitivity analysis also shows that it is the differential in blood perfusion, causing reduced immune activity towards the apex, which has the biggest influence of disease outputs.

Modelling the transmission of infectious diseases inside hospital bays: implications for COVID-19

Healthcare associated transmission of viral infections is a major problem that has significant economic costs and can lead to loss of life. Infections with the highly contagious SARS-CoV-2 virus have been shown to have a high prevalence in hospitals around the world. The spread of this virus might be impacted by the density of patients inside hospital bays. To investigate this aspect, in this study we consider a mathematical modelling and computational approach to describe the spread of SARS-CoV-2 among hospitalised patients. We focus on 4-bed bays and 6-bed bays, which are commonly used to accommodate various non-COVID-19 patients in many hospitals across the United Kingdom (UK). We investigate the spread of SARS-CoV-2 infections among patients in non-COVID bays, in the context of various scenarios: placing the initially-exposed individual in different beds, varying the recovery and incubation periods, having symptomatic vs. asymptomatic patients, removing infected individuals from these hospital bays once they are known to be infected, and the role of periodic testing of hospitalised patients. Our results show that 4-bed bays reduce the spread of SARS-CoV-2 compared to 6-bed bays. Moreover, we show that the position of a new (not infected) patient in specific beds in a 6-bed bay might also slow the spread of the disease. Finally, we propose that regular SARS-CoV-2 testing of hospitalised patients would allow appropriate placement of infected patients in specific (COVID-only) hospital bays.

Quantitative Predictive Modelling Approaches to Understanding Rheumatoid Arthritis: A Brief Review

Rheumatoid arthritis is a chronic autoimmune disease that is a major public health challenge. The disease is characterised by inflammation of synovial joints and cartilage erosion, which lead to chronic pain, poor life quality and, in some cases, mortality. Understanding the biological mechanisms behind the progression of the disease, as well as developing new methods for quantitative predictions of disease progression in the presence/absence of various therapies is important for the success of therapeutic approaches. The aim of this study is to review various quantitative predictive modelling approaches for understanding rheumatoid arthritis. To this end, we start by briefly discussing the biology of this disease and some current treatment approaches, as well as emphasising some of the open problems in the field. Then, we review various mathematical mechanistic models derived to address some of these open problems. We discuss models that investigate the biological mechanisms behind the progression of the disease, as well as pharmacokinetic and pharmacodynamic models for various drug therapies. Furthermore, we highlight models aimed at optimising the costs of the treatments while taking into consideration the evolution of the disease and potential complications.

Mathematical models of transmission dynamics and vaccine strategies in Hong Kong during the 2017-2018 winter influenza season

Two mathematical models described by simple ordinary differential equations are developed to investigate the Hong Kong influenza epidemic during 2017-2018 winter, based on overall epidemic dynamics and different influenza subtypes. The first model, describing the overall epidemic dynamics, provides the starting data for the second model which different influenza subtypes, and whose dynamics is further investigated. Weekly data from December 2017 to May 2018 are obtained from the data base of the Centre of Health Protection in Hong Kong, and used to parametrise the models. With the help of these models, we investigate the impact of different vaccination strategies and determine the corresponding critical vaccination coverage for different vaccine efficacies. The results suggest that at least 72% of Hong Kong population should have been vaccinated during 2017-2018 winter to prevent the seasonal epidemic by herd immunity (while data showed that only a maximum of 11.6% of the population were vaccinated). Our results also show that the critical vaccination coverage decreases with increasing vaccine efficacy, and the increase in one influenza subtype vaccine efficacy may lead to an increase in infections caused by a different subtype.

Non-local Parabolic and Hyperbolic Models for Cell Polarisation in Heterogeneous Cancer Cell Populations

Tumours consist of heterogeneous populations of cells. The sub-populations can have different features, including cell motility, proliferation and metastatic potential. The interactions between clonal sub-populations are complex, from stable coexistence to dominant behaviours. The cell–cell interactions, i.e. attraction, repulsion and alignment, processes critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this study, we develop a mathematical model describing cancer cell invasion and movement for two polarised cancer cell populations with different levels of mutation. We consider a system of non-local hyperbolic equations that incorporate cell–cell interactions in the speed and the turning behaviour of cancer cells, and take a formal parabolic limit to transform this model into a non-local parabolic model. We then investigate the possibility of aggregations to form, and perform numerical simulations for both hyperbolic and parabolic models, comparing the patterns obtained for these models.

Improving cancer detection through combinations of cancer and immune biomarkers: a modelling approach

Background

Early cancer diagnosis is one of the most important challenges of cancer research, since in many cancers it can lead to cure for patients with early stage diseases. For epithelial ovarian cancer (which is the leading cause of death among gynaecologic malignancies) the classical detection approach is based on measurements of CA-125 biomarker. However, the poor sensitivity and specificity of this biomarker impacts the detection of early-stage cancers.

Methods

Here we use a computational approach to investigate the effect of combining multiple biomarkers for ovarian cancer (e.g., CA-125 and IL-7), to improve early cancer detection.

Results

We show that this combined biomarkers approach could lead indeed to earlier cancer detection. However, the immune response (which influences the level of secreted IL-7 biomarker) plays an important role in improving and/or delaying cancer detection. Moreover, the detection level of IL-7 immune biomarker could be in a range that would not allow to distinguish between a healthy state and a cancerous state. In this case, the construction of solution diagrams in the space generated by the IL-7 and CA-125 biomarkers could allow us predict the long-term evolution of cancer biomarkers, thus allowing us to make predictions on cancer detection times.

Conclusions

Combining cancer and immune biomarkers could improve cancer detection times, and any predictions that could be made (at least through the use of CA-125/IL-7 biomarkers) are patient specific.

Model based analysis of the heterogeneity in the tumour size dynamics differentiates vemurafenib, dabrafenib and trametinib in metastatic melanoma

PURPOSE

Explore the heterogeneity in dynamics of tumour response to vemurafenib, dabrafenib and trametinib using routinely collected clinical trial imaging data.

METHODS

Time-series imaging data from the phase III studies of vemurafenib, dabrafenib and trametinib were collected through a data repository. A mathematical model based on basic mechanisms of tumour growth was placed within a statistical modelling framework to analyse the data.

RESULTS

The analysis revealed: (1) existence of homogeneity in drug response and resistance development within a patient; (2) tumour shrinkage rate does not relate to rate of resistance development; (3) vemurafenib and dabrafenib, two BRAF inhibitors, have different variability in tumour shrinkage rates.

CONCLUSIONS

Overall these results show how analysis of the dynamics of individual lesions can shed light on the within and between patient differences in tumour shrinkage and resistance rates, which could be used to gain a macroscopic understanding of tumour heterogeneity.

Modelling the collective response of heterogeneous cell populations to stationary gradients and chemical signal relay

The directed motion of cell aggregates toward a chemical source occurs in many relevant biological processes. Understanding the mechanisms that control this complex behavior is of great relevance for our understanding of developmental biological processes and many diseases. In this paper, we consider a self-propelled particle model for the movement of heterogeneous subpopulations of chemically interacting cells towards an imposed stable chemical gradient. Our simulations show explicitly how self-organisation of cell populations (which could lead to engulfment or complete cell segregation) can arise from the heterogeneity of chemotactic responses alone. This new result complements current theoretical and experimental studies that emphasise the role of differential cell-cell adhesion on self-organisation and spatial structure of cellular aggregates. We also investigate how the speed of individual cell aggregations increases with the chemotactic sensitivity of the cells, and decreases with the number of cells inside the aggregates.

Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-β pathway in tumour dynamics

The growth and invasion of cancer cells are very complex processes, which can be regulated by the cross-talk between various signalling pathways, or by single signalling pathways that can control multiple aspects of cell behaviour. TGF-β is one of the most investigated signalling pathways in oncology, since it can regulate multiple aspects of cell behaviour: cell proliferation and apoptosis, cell-cell adhesion and epithelial-to-mesenchimal transition via loss of cell adhesion. In this study, we use a mathematical modelling approach to investigate the complex roles of TGF-β signalling pathways on the inhibition and growth of tumours, as well as on the epithelial-to-mesenchimal transition involved in the metastasis of tumour cells. We show that the nonlocal mathematical model derived here to describe repulsive and adhesive cell-cell interactions can explain the formation of new tumour cell aggregations at positions in space that are further away from the main aggregation. Moreover, we show that the increase in cell-cell adhesion leads to fewer but larger aggregations, and the increase in TGF-β molecules - whose late-stage effect is to decrease cell adhesion - leads to many small cellular aggregations. Finally, we perform a sensitivity analysis on some parameters associated with TGF-β dynamics, and use it to investigate the relation between the tumour size and its metastatic spread.

Modelling cell movement, cell differentiation, cell sorting and proportion regulation in Dictyostelium discoideum aggregations

Understanding the mechanisms that control tissue morphogenesis and homeostasis is a central goal not only in developmental biology but also has great relevance for our understanding of various diseases, including cancer. A model organism that is widely used to study the control of tissue morphogenesis and proportioning is the Dictyostelium discoideum. While there are mathematical models describing the role of chemotactic cell motility in the Dictyostelium assembly and morphogenesis of multicellular tissues, as well as models addressing possible mechanisms of proportion regulation, there are no models incorporating both these key aspects of development. In this paper, we introduce a 1D hyperbolic model to investigate the role of two morphogens, DIF and cAMP, on cell movement, cell sorting, cell-type differentiation and proportioning in Dictyostelium discoideum. First, we use the non-spatial version of the model to study cell-type transdifferentiation. We perform a steady-state analysis of it and show that, depending on the shape of the differentiation rate functions, multiple steady-state solutions may occur. Then we incorporate spatial dynamics into the model, and investigate the transdifferentiation and spatial positioning of cells inside the newly formed structures, following the removal of prestalk or prespore regions of a Dictyostelium slug. We show that in isolated prespore fragments, a tipped mound-like aggregate can be formed after a transdifferentiation from prespore to prestalk cells and following the sorting of prestalk cells to the centre of the aggregate. For isolated prestalk fragments, we show the formation of a slug-like structure containing the usual anterior-posterior pattern of prestalk and prespore cells.

Simultaneous use of different communication mechanisms leads to spatial sorting and unexpected collective behaviours in animal groups

Communication among individuals forms the basis of social interactions in every animal population. In general, communication is influenced by the physiological and psychological constraints of each individual, and in large aggregations this means differences in the reception and emission of communication signals. However, studies on the formation and movement of animal aggregations usually assume that all individuals communicate with neighbours in the same manner. Here, we take a new approach on animal aggregations and use a nonlocal mathematical model to investigate theoretically the simultaneous use of two communication mechanisms by different members of a population. We show that the use of multiple communication mechanisms can lead to behaviours that are not necessarily predicted by the behaviour of subpopulations that use only one communication mechanism. In particular, we show that while the use of one communication mechanism by the entire population leads to deterministic movement, the use of multiple communication mechanisms can lead in some cases to chaotic movement. Finally, we show that the use of multiple communication mechanisms leads to the sorting of individuals inside aggregations: individuals that are aware of the location and the movement direction of all their neighbours usually position themselves at the centre of the groups, while individuals that are aware of the location and the movement direction of only some neighbours position themselves at the edges of the groups.

Patterns of spread of influenza A in Canada

Understanding spatial patterns of influenza transmission is important for designing control measures. We investigate spatial patterns of laboratory-confirmed influenza A across Canada from October 1999 to August 2012. A statistical analysis (generalized linear model) of the seasonal epidemics in this time period establishes a clear spatio-temporal pattern, with influenza emerging earlier in western provinces. Early emergence is also correlated with low temperature and low absolute humidity in the autumn. For the richer data from the 2009 pandemic, a mechanistic mathematical analysis, based on a transmission model, shows that both school terms and weather had important effects on pandemic influenza transmission.

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

Interactions between the immune system and cancer: a brief review of non-spatial mathematical models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.

Complex spatial group patterns result from different animal communication mechanisms

We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.

Modeling Group Formation and Activity Patterns in Self-Organizing Collectives of Individuals

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.

The MyoD family of myogenic factors is regulated by electrical activity: isolation and characterization of a mouse Myf-5 cDNA

A full-length cDNA coding for a homolog of the human Myf-5 was isolated from a BC3H-1 mouse library and characterized. The clone codes for a protein of 255 amino acids that is 89%, 88% and 68% identical to the human, bovine and Xenopus myf-5, respectively. The mouse Myf-5 cDNA (mmyf-5), as well as sequences coding for MyoD, myogenin and Mrf-4, were used to probe Northern blots to analyze the effects of innervation on the expression of the MyoD family of myogenic factors. Mouse myf-5, MyoD and myogenin mRNAs levels were found to decline in hind limb muscles of mice between embryonic day 15 (E15) and the first postnatal week, a period that coincides with innervation. In contrast, Mrf-4 transcripts increase during this period and reach steady-state levels by 1-week after birth. To distinguish if the changes in myogenic factor expression are due to a developmental program or to innervation, mRNA levels were analyzed at different times after muscle denervation. Mmyf-5 transcripts begin to accumulate 2 days postdenervation; after 1 week levels are 7-fold higher than in innervated muscle. Mrf-4, MyoD and myogenin transcripts begin to accumulate as soon as 8h after denervation, and attain levels that are 8-, 15- and 40-fold higher than found in innervated skeletal muscle, respectively. The accumulation of these three mRNAs precedes the increase of nicotinic acetylcholine receptor alpha subunit transcripts, a gene that is transcriptionally regulated by MyoD-related factors in vitro. Using extracellular electrodes to directly stimulate in situ the soleus muscle of rats, we found that 'electrical activity' per se, in absence of the nerve, represses the increases of myogenic factor mRNAs associated with denervation.

Myogenin and MyoD join a family of skeletal muscle genes regulated by electrical activity

Myogenin and MyoD are proteins that bind to the regulatory regions of a battery of skeletal muscle genes and can activate their transcription during muscle differentiation. We have recently found that both proteins interact with the enhancer of the nicotinic acetylcholine receptor (nAChR) alpha subunit, a gene that is regulated by innervation. This observation prompted us to study if myogenin and MyoD transcript levels are also regulated by skeletal muscle innervation. Using Northern blot analysis, we found that MyoD and myogenin mRNA levels begin to decline at embryonic day 17 and attain adult levels in muscle of newborn and 3-week-old mice, respectively. In contrast, nAChR mRNAs are highest in newborn and 1-week-old mouse muscle and decline thereafter to reach adult levels in 3-week-old mice. To determine if the down-regulation of myogenin and MyoD mRNA levels during development is due to innervation, we quantitated message levels in adult calf muscles after denervation. We found that in denervated muscle myogenin and MyoD mRNAs reach levels that are approximately 40- and 15-fold higher than those found in innervated muscle. Myogenin mRNAs begin to accumulate rapidly between 8 and 16 hr after denervation, and MyoD transcripts levels begin to increase sharply between 16 hr and 1 day after denervation. The increases in myogenin and MyoD mRNA levels precede the rapid accumulation of nAChR alpha-subunit transcripts; receptor mRNAs begin to accumulate significantly after 1 day of denervation. The effects of denervation are specific because skeletal alpha-actin mRNA levels are not affected by denervation. In addition, we found that the repression of myogenin and MyoD expression by innervation is due, at least in part, to "electrical activity." Direct stimulation of soleus muscle with extracellular electrodes repressed the increase of myogenin and MyoD transcripts after denervation by 4- to 3-fold, respectively. In view of these results, it is interesting to speculate that myogenin and/or MyoD may regulate a repertoire of skeletal muscle genes that are down-regulated by electrical activity.