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A multiscale mathematical model is presented to describe de novo granulation, and the evolution of multispecies granular biofilms, in a continuously fed bioreactor. The granule is modelled as a spherical free boundary domain with radial symmetry. The equation governing the free boundary is derived from global mass balance considerations and takes into account the growth of sessile biomass as well as exchange fluxes with the bulk liquid. Starting from a vanishing initial value, the expansion of the free boundary is initiated by the attachment process, which depends on the microbial species concentrations within the bulk liquid and their specific attachment velocity. Nonlinear hyperbolic PDEs model the growth of the sessile microbial species, while quasi-linear parabolic PDEs govern the dynamics of substrates and invading species within the granular biofilm. Nonlinear ODEs govern the evolution of soluble substrates and planktonic biomass within the bulk liquid. The model is applied to an anaerobic, granular-based bioreactor system, and solved numerically to test its qualitative behaviour and explore the main aspects of de novo anaerobic granulation: ecology, biomass distribution, relative abundance, dimensional evolution of the granules and soluble substrates, and planktonic biomass dynamics within the bioreactor. The numerical results confirm that the model accurately describes the ecology and the concentrically layered structure of anaerobic granules observed experimentally, and that it can predict the effects on the process of significant factors, such as influent wastewater composition; granulation properties of planktonic biomass; biomass density; detachment intensity; and number of granules., (© 2021. The Author(s).)

Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell-cell and cell-matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell-ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement.

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

A model for fluid and drug transport through the leaky neovasculature and porous interstitium of a solid tumour is developed. The transport problems are posed on a micro-scale characterized by the inter-capillary distance, and the method of multiple scales is used to derive the continuum equations describing fluid and drug transport on the length scale of the tumour (under the assumption of a spatially periodic microstructure). The fluid equations comprise a double porous medium, with coupled Darcy flow through the interstitium and vasculature, whereas the drug equations comprise advection-reaction equations; in each case the dependence of the transport coefficients on the vascular geometry is determined by solving micro-scale cell problems.

Most models of COVID-19 are implemented at a single micro or macro scale, ignoring the interplay between immune response, viral dynamics, individual infectiousness and epidemiological contact networks. Here we develop a data-driven model linking the within-host viral dynamics to the between-host transmission dynamics on a multilayer contact network to investigate the potential factors driving transmission dynamics and to inform how school closures and antiviral treatment can influence the epidemic. Using multi-source data, we initially determine the viral dynamics and estimate the relationship between viral load and infectiousness. Then, we embed the viral dynamics model into a four-layer contact network and formulate an agent-based model to simulate between-host transmission. The results illustrate that the heterogeneity of immune response between children and adults and between vaccinated and unvaccinated infections can produce different transmission patterns. We find that school closures play a significant effect on mitigating the pandemic as more adults get vaccinated and the virus mutates. If enough infected individuals are diagnosed by testing before symptom onset and then treated quickly, the transmission can be effectively curbed. Our multiscale model reveals the critical role played by younger individuals and antiviral treatment with testing in controlling the epidemic. [ABSTRACT FROM AUTHOR]

Understanding disease transmission in the workplace is essential for protecting workers. To model disease outbreaks, the small populations in many workplaces require that stochastic effects are considered, which results in higher uncertainty. The aim of this study was to quantify and interpret the uncertainty inherent in such circumstances. We assessed how uncertainty of an outbreak in workplaces depends on i) the infection dynamics in the community, ii) the workforce size, iii) spatial structure in the workplace, iv) heterogeneity in susceptibility of workers, and v) heterogeneity in infectiousness of workers. To address these questions, we developed a multiscale model: A deterministic model to predict community transmission, and a stochastic model to predict workplace transmission. We extended this basic workplace model to allow for spatial structure, and heterogeneity in susceptibility and infectiousness in workers. We found a non-monotonic relationship between the workplace transmission rate and the coefficient of variation (CV), which we use as a measure of uncertainty. Increasing community transmission, workforce size and heterogeneity in susceptibility decreased the CV. Conversely, increasing the level of spatial structure and heterogeneity in infectiousness increased the CV. However, when the model predicts bimodal distributions, for example when community transmission is low and workplace transmission is high, the CV fails to capture this uncertainty. Overall, our work informs modellers and policy makers on how model complexity impacts outbreak uncertainty. In particular: workforce size, community and workplace transmission, spatial structure and individual heterogeneity contribute in a specific and individual manner to the predicted workplace outbreak size distribution. [ABSTRACT FROM AUTHOR]

Known as one of the hallmarks of cancer (Hanahan and Weinberg in Cell 100:57-70, 2000) cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases or the urokinase plasminogen activator (uPA), the cancer cell population alters the configuration of the surrounding ECM composition and overcomes the physical barriers to ultimately achieve local cancer spread into the surrounding tissue. The active interplay between the tissue-scale tumour dynamics and the molecular mechanics of the involved proteolytic enzymes at the cell scale underlines the biologically multiscale character of invasion and raises the challenge of modelling this process with an appropriate multiscale approach. In this paper, we present a new two-scale moving boundary model of cancer invasion that explores the tissue-scale tumour dynamics in conjunction with the molecular dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in Trucu et al. (Multiscale Model Simul 11(1):309-335, 2013), the modelling that we propose here allows us to study the changes in tissue-scale tumour morphology caused by the cell-scale uPA microdynamics occurring along the invasive edge of the tumour. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as the tumour infiltrative growth patterns discussed in Ito et al. (J Gastroenterol 47:1279-1289, 2012). [ABSTRACT FROM AUTHOR]

In this study, we present a mathematical model for plasmid spread in a growing biofilm, formulated as a nonlocal system of partial differential equations in a 1-D free boundary domain. Plasmids are mobile genetic elements able to transfer to different phylotypes, posing a global health problem when they carry antibiotic resistance factors. We model gene transfer regulation influenced by nearby potential receptors to account for recipient-sensing. We also introduce a promotion function to account for trace metal effects on conjugation, based on literature data. The model qualitatively matches experimental results, showing that contaminants like toxic metals and antibiotics promote plasmid persistence by favoring plasmid carriers and stimulating conjugation. Even at higher contaminant concentrations inhibiting conjugation, plasmid spread persists by strongly inhibiting plasmid-free cells. The model also replicates higher plasmid density in biofilm's most active regions., (© 2024. The Author(s).)

This paper presents a theoretical investigation of the multiphysical phenomena that govern cortical bone behaviour. Taking into account the piezoelectricity of the collagen-apatite matrix and the electrokinetics governing the interstitial fluid movement, we adopt a multiscale approach to derive a coupled poroelastic model of cortical tissue. Following how the phenomena propagate from the microscale to the tissue scale, we are able to determine the nature of macroscopically observed electric phenomena in bone. [ABSTRACT FROM AUTHOR]

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., (© 2023. The Author(s), under exclusive licence to Society for Mathematical Biology.)

A multiscale mathematical model describing the metals biosorption on algal-bacterial photogranules within a sequencing batch reactor (SBR) is presented. The model is based on systems of partial differential equations (PDEs) derived from mass conservation principles on a spherical free boundary domain with radial symmetry. Hyperbolic PDEs account for the dynamics of sessile species and their free sorption sites, where metals are adsorbed. Parabolic PDEs govern the diffusion, conversion and adsorption of nutrients and metals. The dual effect of metals on photogranule ecology is also modelled: metal stimulates the production of EPS by sessile species and negatively affects the metabolic activities of microbial species. Accordingly, a stimulation term for EPS production and an inhibition term for metal are included in all microbial kinetics. The formation and evolution of the granule domain are governed by an ordinary differential equation with a vanishing initial value, accounting for microbial growth, attachment and detachment phenomena. The model is completed with systems of impulsive differential equations describing the evolution of dissolved substrates, metals, and planktonic and detached biomasses within the granular-based SBR. The model is integrated numerically to examine the role of the microbial species and EPS in the adsorption process, and the effect of metal concentration and adsorption properties of biofilm components on the metal removal. Numerical results show an accurate description of the photogranules evolution and ecology and confirm the applicability of algal-bacterial photogranule technology for metal-rich wastewater treatment., (© 2023. The Author(s).)

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. During the complex interaction between cancer cell population and the extracellular matrix (ECM), of key importance is the role played by both bulk two-scale dynamics of ECM fibres within collective movement of the tumour cells and the multiscale leading edge dynamics driven by proteolytic activity of the matrix-degrading enzymes (MDEs) that are secreted by the cancer cells. As these two multiscale subsystems share and contribute to the same tumour macro-dynamics, in this work we develop further the model introduced in Shuttleworth and Trucu (Bull Math Biol 81:2176–2219, 2019. 10.1007/s11538-019-00598-w) by exploring a new aspect of their interaction that occurs at the cell scale. Specifically, here we will focus on understanding the cell-scale cross talk between the micro-scale parts of these two multiscale subsystems which get to interact directly in the peritumoural region, with immediate consequences both for MDE micro-dynamics occurring at the leading edge of the tumour and for the cell-scale rearrangement of the naturally oriented ECM fibres in the peritumoural region, ultimately influencing the way tumour progresses in the surrounding tissue. To that end, we will propose a new modelling that captures the ECM fibres degradation not only at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour. [ABSTRACT FROM AUTHOR]

Bladder cancer poses a significant global health burden with high incidence and recurrence rates. This study addresses the therapeutic challenges in advanced bladder cancer, focusing on the competitive mechanisms of ligand or drug binding to receptors. We developed a refined mathematical model that integrates the dynamics of tumor cells and immune responses, particularly targeting fibroblast growth factor receptor 3 (FGFR3) and immune checkpoint inhibitors (ICIs). This study contributes to understanding combination therapies by elucidating the competitive binding dynamics and quantifying the synergistic effects. The findings highlight the importance of personalized immunotherapeutic strategies, considering factors such as drug dosage, dosing schedules, and patient-specific parameters. Our model further reveals that ligand-independent activated-state receptors are the most essential drivers of tumor proliferation. Moreover, we found that PD-L1 expression rate was more important than PD-1 in driving the dynamic evolution of tumor and immune cells. The proposed mathematical model provides a comprehensive framework for unraveling the complexities of combination therapies in advanced bladder cancer. As research progresses, this multidisciplinary approach contributes valuable insights toward optimizing therapeutic strategies and advancing cancer treatment paradigms., (© 2024. The Author(s), under exclusive licence to the Society for Mathematical Biology.)

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell’s cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction–advection–diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells. [ABSTRACT FROM AUTHOR]

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours’ microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included. [ABSTRACT FROM AUTHOR]

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods., (© 2024. The Author(s), under exclusive licence to the Society for Mathematical Biology.)

A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization. [ABSTRACT FROM AUTHOR]

We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial-Mesenchymal Transition and Mesenchymal-Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites., (© 2024. The Author(s).)

Mathematical modeling is a tool used for understanding diseases dynamics. The discrete-time model is an especial case in modeling that satisfactorily describes the epidemiological dynamics because of the discrete nature of the real data. However, discrete models reduce their descriptive and fitting potential because of assuming a homogeneous population. Thus, in this paper, we proposed contagion probability functions according to two infection paradigms that consider factors associated with transmission dynamics. For example, we introduced probabilities of establishing an infectious interaction, the number of contacts with infectious and the level of connectivity or social distance within populations. Through the probabilities design, we overcame the homogeneity assumption. Also, we evaluated the proposed probabilities through their introduction into discrete-time models for two diseases and different study zones with real data, COVID-19 for Germany and South Korea, and dengue for Colombia. Also, we described the oscillatory dynamics for the last one using the contagion probabilities alongside parameters with a biological sense. Finally, we highlight the implementation of the proposed probabilities would improve the simulation of the public policy effect of control strategies over an infectious disease outbreak. [ABSTRACT FROM AUTHOR]

We developed a framework based on the software Unstructured Reaction-Diffusion Master Equation (URDME) to address tumor cells’ proliferation and migration in a heterogeneous space, herein a 2D percolation cluster. A mitogenic paracrine signaling pathway is utilized phenomenologically to reveal how cells cooperate with one another. We modeled the emerging Allee effect using low seeding density culture (LSDC) assays to fit the model parameters. A Finite time scaling (FTS) function has been formulated to quantitatively analyze invasiveness of a virtual Growth-Migration (GM) system in mimicking the cancer cell growth. Through such simulation, we analyzed the GM dynamics of virtual model in mimicking the growth of BT-474 cancer cell populations in vitro in a 2D percolation cluster and calculated the successful penetration rate (SPR). By analyzing the temporal trajectories of the SPR, we could determine the critical exponents of the critical SPR scaling relation. The SPR transition point ( p c ), which is a fundamentally different from a conventional percolation transition point, is found to be negatively correlated with the invasiveness of this cancer cell. The p c of the three variations of the virtual GM system distinctly designated by varying paracrine-regulated Allee (PAllee) model phenotypes is 0.3408, 0.3675, and 0.4454, respectively. FTS algorithm thereon may serve as an approach to quantify invasiveness of tumor cells. Through a phenomenological paracrine model, inter-cell cooperation and mutual mitogenic boosting are enabled to elicit the Allee effect in the GM systems. The rationale behind such computationally tunable virtual mechanism can be applied to other circumstances concerning emerging processes. [ABSTRACT FROM AUTHOR]

The clustering of membrane-bound proteins facilitates their transport by cortical actin flow in early Caenorhabditis elegans embryo cell polarity. PAR-3 clustering is critical for this process, yet the biophysical processes that couple protein clusters to cortical flow remain unknown. We develop a discrete, stochastic agent-based model of protein clustering and test four hypothetical models for how clusters may interact with the flow. Results show that the canonical way to assess transport characteristics from single-particle tracking data used thus far in this area, the Péclet number, is insufficient to distinguish these hypotheses and that all models can account for transport characteristics quantified by this measure. However, using this model, we demonstrate that these different cluster–cortex interactions may be distinguished using a different metric, namely the scalar projection of cluster displacement on to the flow displacement vector. Our results thus provide a testable way to use existing single-particle tracking data to test how endogenous protein clusters may interact with the cortical flow to localize during polarity establishment. To facilitate this investigation, we also develop both improved simulation and semi-analytic methodologies to quantify motion summary statistics (e.g., Péclet number and scalar projection) for these stochastic models as a function of biophysical parameters. [ABSTRACT FROM AUTHOR]

We formulate a multiscale mathematical model that describes the bioreduction of uranium in porous media. On the mesoscale we describe the bioreduction of uranium [VI] to uranium [IV] using a multispecies one-dimensional biofilm model with suspended bacteria and thermodynamic growth inhibition. We upscale the mesoscopic (colony scale) model to the macroscale (reactor scale) and investigate the behavior of substrate utilization and production, attachment and detachment processes, and thermodynamic effects not usually considered in biofilm growth models. Simulation results of the reactor model indicate that thermodynamic inhibition quantitatively alters the dynamics of the model and neglecting thermodynamic effects may over- or underestimate chemical concentrations in the system. Furthermore, we numerically investigate uncertainties related to the specific choice of attachment and detachment rate coefficients and find that while increasing the attachment rate coefficient or decreasing the detachment rate coefficient leads to thicker biofilms, performance of the reactor remains largely unaffected. [ABSTRACT FROM AUTHOR]

We combine a systems pharmacology approach with an agent-based modelling approach to simulate LoVo cells subjected to AZD6738, an ATR (ataxia–telangiectasia-mutated and rad3-related kinase) inhibiting anti-cancer drug that can hinder tumour proliferation by targeting cellular DNA damage responses. The agent-based model used in this study is governed by a set of empirically observable rules. By adjusting only the rules when moving between monolayer and multi-cellular tumour spheroid simulations, whilst keeping the fundamental mathematical model and parameters intact, the agent-based model is first parameterised by monolayer in vitro data and is thereafter used to simulate treatment responses in in vitro tumour spheroids subjected to dynamic drug delivery. Spheroid simulations are subsequently compared to in vivo data from xenografts in mice. The spheroid simulations are able to capture the dynamics of in vivo tumour growth and regression for approximately 8 days post-tumour injection. Translating quantitative information between in vitro and in vivo research remains a scientifically and financially challenging step in preclinical drug development processes. However, well-developed in silico tools can be used to facilitate this in vitro to in vivo translation, and in this article, we exemplify how data-driven, agent-based models can be used to bridge the gap between in vitro and in vivo research. We further highlight how agent-based models, that are currently underutilised in pharmaceutical contexts, can be used in preclinical drug development. [ABSTRACT FROM AUTHOR]

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells. [ABSTRACT FROM AUTHOR]

Saliva is secreted from the acinar cells of the salivary glands, using mechanisms that are similar to other types of water-transporting epithelial cells. Using a combination of theoretical and experimental techniques, over the past 20 years we have continually developed and modified a quantitative model of saliva secretion, and how it is controlled by the dynamics of intracellular calcium. However, over approximately the past 5 years there have been significant developments both in our understanding of the underlying mechanisms and in the way these mechanisms should best be modelled. Here, we review the traditional understanding of how saliva is secreted, and describe how our work has suggested important modifications to this traditional view. We end with a brief description of the most recent data from living animals and discuss how this is now contributing to yet another iteration of model construction and experimental investigation. [ABSTRACT FROM AUTHOR]

Invasion of the surrounding tissue is one of the recognised hallmarks of cancer (Hanahan and Weinberg in Cell 100: 57–70, 2000. ), which is accomplished through a complex heterotypic multiscale dynamics involving tissue-scale random and directed movement of the population of both cancer cells and other accompanying cells (including here, the family of tumour-associated macrophages) as well as the emerging cell-scale activity of both the matrix-degrading enzymes and the rearrangement of the cell-scale constituents of the extracellular matrix (ECM) fibres. The involved processes include not only the presence of cell proliferation and cell adhesion (to other cells and to the extracellular matrix), but also the secretion of matrix-degrading enzymes. This is as a result of cancer cells as well as macrophages, which are one of the most abundant types of immune cells in the tumour micro-environment. In large tumours, these tumour-associated macrophages (TAMs) have a tumour-promoting phenotype, contributing to tumour proliferation and spread. In this paper, we extend a previous multiscale moving-boundary mathematical model for cancer invasion, by considering also the multiscale effects of TAMs, with special focus on the influence that their directional movement exerts on the overall tumour progression. Numerical investigation of this new model shows the importance of the interactions between pro-tumour TAMs and the fibrous ECM, highlighting the impact of the fibres on the spatial structure of solid tumour. [ABSTRACT FROM AUTHOR]

The disordered network of blood vessels that arises from tumour angiogenesis results in variations in the delivery of oxygen into the tumour tissue. This brings about regions of chronic hypoxia (i.e. sustained low oxygen levels) and regions with alternating periods of low and relatively higher oxygen levels, and makes it necessary for cancer cells to adapt to fluctuating environmental conditions. We use a phenotype-structured model to dissect the evolutionary dynamics of cell populations exposed to fluctuating oxygen levels. In this model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its metabolic phenotype, ranging from fully oxidative to fully glycolytic, and cells are grouped into two competing populations that undergo heritable, spontaneous phenotypic variations at different rates. Model simulations indicate that, depending on the rate at which oxygen is consumed by the cells, dynamic nonlinear interactions between cells and oxygen can stimulate chronic hypoxia and cycling hypoxia. Moreover, the model supports the idea that under chronic-hypoxic conditions lower rates of phenotypic variation lead to a competitive advantage, whereas higher rates of phenotypic variation can confer a competitive advantage under cycling-hypoxic conditions. In the latter case, the numerical results obtained show that bet-hedging evolutionary strategies, whereby cells switch between oxidative and glycolytic phenotypes, can spontaneously emerge. We explain how these results can shed light on the evolutionary process that may underpin the emergence of phenotypic heterogeneity in vascularised tumours. [ABSTRACT FROM AUTHOR]

There are many challenges to coupling the macroscale to the microscale in temporal or spatial contexts. In order to examine effects of an individual movement and spatial control measures on a disease outbreak, we developed a multiscale model and extended the semi-stochastic simulation method by linking individual movements to pathogen’s diffusion, linking the slow dynamics for disease transmission at the population level to the fast dynamics for pathogen shedding/excretion at the individual level. Numerical simulations indicate that during a disease outbreak individuals with the same infection status show the property of clustering and, in particular, individuals’ rapid movements lead to an increase in the average reproduction number R 0 , the final size and the peak value of the outbreak. It is interesting that a high level of aggregation the individuals’ movement results in low new infections and a small final size of the infected population. Further, we obtained that either high diffusion rate of the pathogen or frequent environmental clearance lead to a decline in the total number of infected individuals, indicating the need for control measures such as improving air circulation or environmental hygiene. We found that the level of spatial heterogeneity when implementing control greatly affects the control efficacy, and in particular, an uniform isolation strategy leads to low a final size and small peak, compared with local measures, indicating that a large-scale isolation strategy with frequent clearance of the environment is beneficial for disease control. [ABSTRACT FROM AUTHOR]

We construct a three-dimensional anatomically accurate multicellular model of a parotid gland acinus to investigate the influence that the topology of its lumen has on primary fluid secretion. Our model consists of seven individual cells, coupled via a common lumen and intercellular signalling. Each cell is equipped with the intracellular calcium ( Ca 2 + )-signalling model developed by Pages et al, Bull Math Biol 81: 1394–1426, 2019. 10.1007/s11538-018-00563-z and the secretion model constructed by Vera-Sigüenza et al., Bull Math Biol 81: 699–721, 2019. 10.1007/s11538-018-0534-z. The work presented here is a continuation of these studies. While previous mathematical research has proven invaluable, to the best of our knowledge, a multicellular modelling approach has never been implemented. Studies have hypothesised the need for a multiscale model to understand the primary secretion process, as acinar cells do not operate on an individual basis. Instead, they form racemous clusters that form intricate water and protein delivery networks that join the acini with the gland’s ducts-questions regarding the extent to which the acinus topology influences the efficiency of primary fluid secretion to persist. We found that (1) The topology of the acinus has almost no effect on fluid secretion. (2) A multicellular spatial model of secretion is not necessary when modelling fluid flow. Although the inclusion of intercellular signalling introduces vastly more complex dynamics, the total secretory rate remains fundamentally unchanged. (3) To obtain an acinus, or better yet a gland flow rate estimate, one can multiply the output of a well-stirred single-cell model by the total number of cells required. [ABSTRACT FROM AUTHOR]

Malaria is an infectious disease with an immense global health burden. Plasmodium vivax is the most geographically widespread species of malaria. Relapsing infections, caused by the activation of liver-stage parasites known as hypnozoites, are a critical feature of the epidemiology of Plasmodium vivax. Hypnozoites remain dormant in the liver for weeks or months after inoculation, but cause relapsing infections upon activation. Here, we introduce a dynamic probability model of the activation-clearance process governing both potential relapses and the size of the hypnozoite reservoir. We begin by modelling activation-clearance dynamics for a single hypnozoite using a continuous-time Markov chain. We then extend our analysis to consider activation-clearance dynamics for a single mosquito bite, which can simultaneously establish multiple hypnozoites, under the assumption of independent hypnozoite behaviour. We derive analytic expressions for the time to first relapse and the time to hypnozoite clearance for mosquito bites establishing variable numbers of hypnozoites, both of which are quantities of epidemiological significance. Our results extend those in the literature, which were limited due to an assumption of collective dormancy. Our within-host model can be embedded readily in multiscale models and epidemiological frameworks, with analytic solutions increasing the tractability of statistical inference and analysis. Our work therefore provides a foundation for further work on immune development and epidemiological-scale analysis, both of which are important for achieving the goal of malaria elimination. [ABSTRACT FROM AUTHOR]

Cell polarization requires redistribution of specific proteins to the nascent front and back of a eukaryotic cell. Among these proteins are Rac and Rho, members of the small GTPase family that regulate the actin cytoskeleton. Rac promotes actin assembly and protrusion of the front edge, whereas Rho activates myosin-driven contraction at the back. Mathematical models of cell polarization at many levels of detail have appeared. One of the simplest based on “wave-pinning” consists of a pair of reaction–diffusion equations for a single GTPase. Mathematical analysis of wave-pinning so far is largely restricted to static domains in one spatial dimension. Here, we extend the analysis to cells that change in size, showing that both shrinking and growing cells can lose polarity. We further consider the feedback between mechanical tension, GTPase activation, and cell deformation in both static, growing, shrinking, and moving cells. Special cases (spatially uniform cell chemistry, the absence or presence of mechanical feedback) are analyzed, and the full model is explored by simulations in 1D. We find a variety of novel behaviors, including “dilution-induced” oscillations of Rac activity and cell size, as well as gain or loss of polarization and motility in the model cell. [ABSTRACT FROM AUTHOR]

Angiogenesis, or capillary growth from pre-existing vasculature, is an essential component of several physiological processes, both vital and pathological. These include dermal wound healing and tumour growth that together pose some of the most significant challenges to healthcare systems worldwide. Over the last few decades, mathematical modelling has proven to be a valuable tool for unravelling the complex network of interactions that underlie such processes. Moreover, theoretical frameworks that describe some of the mechanical and chemical aspects of angiogenesis inherent in wound healing and tumour growth have revealed intriguing similarities between the two processes. In this review, we highlight some of the significant contributions made by mathematical models of tumour-induced and wound healing angiogenesis and illustrate how advances in each field have been made using insights from the other. We also detail some open problems that could be addressed through a combination of theoretical and experimental approaches. [ABSTRACT FROM AUTHOR]

Most models of COVID-19 are implemented at a single micro or macro scale, ignoring the interplay between immune response, viral dynamics, individual infectiousness and epidemiological contact networks. Here we develop a data-driven model linking the within-host viral dynamics to the between-host transmission dynamics on a multilayer contact network to investigate the potential factors driving transmission dynamics and to inform how school closures and antiviral treatment can influence the epidemic. Using multi-source data, we initially determine the viral dynamics and estimate the relationship between viral load and infectiousness. Then, we embed the viral dynamics model into a four-layer contact network and formulate an agent-based model to simulate between-host transmission. The results illustrate that the heterogeneity of immune response between children and adults and between vaccinated and unvaccinated infections can produce different transmission patterns. We find that school closures play a significant effect on mitigating the pandemic as more adults get vaccinated and the virus mutates. If enough infected individuals are diagnosed by testing before symptom onset and then treated quickly, the transmission can be effectively curbed. Our multiscale model reveals the critical role played by younger individuals and antiviral treatment with testing in controlling the epidemic., (© 2022. The Author(s), under exclusive licence to Society for Mathematical Biology.)

In this paper, we use multiple scale homogenisation to derive a set of averaged macroscale equations that describe the movement of nutrients in partially saturated soil that contains growing potato tubers. The soil is modelled as a poroelastic material, which is deformed by the growth of the tubers, where the growth of each tuber is dependent on the uptake of nutrients via a sink term within the soil representing root nutrient uptake. Special attention is paid to the reduction in void space, resulting change in local water content and the impact on nutrient diffusion within the soil as the tubers increase in size. To validate the multiple scale homogenisation procedure, we compare the system of homogenised equations to the original set of equations and find that the solutions between the two models differ by ≲ 2 % . However, we find that the computation time between the two sets of equations differs by several orders of magnitude. This is due to the combined effects of the complex three-dimensional geometry and the implementation of a moving boundary condition to capture tuber growth. [ABSTRACT FROM AUTHOR]

The epidermal growth factor receptor (EGFR) signalling cascade is one of the main pathways that regulate the survival and division of mammalian cells. It is also one of the most altered transduction pathways in cancer. Acquired mutations in the EGFR/ERK pathway can cause the overexpression of EGFR on the surface of the cell, while others downregulate the inactivation of switched on intracellular proteins such as Ras and Raf. This upregulates the activity of ERK and promotes cell division. We develop a 3D multiscale model to explore the role of EGFR overexpression on tumour initiation. In this model, cells are described as individual objects that move, interact, divide, proliferate, and die by apoptosis. We use Brownian Dynamics to describe the extracellular and intracellular regulations of cells as well as the spatial and stochastic effects influencing them. The fate of each cell depends on the number of active transcription factors in the nucleus. We use numerical simulations to investigate the individual and combined effects of mutations on the intracellular regulation of individual cells. Next, we show that the distance between active receptors increase the level of EGFR/ERK signalling. We demonstrate the usefulness of the model by quantifying the impact of mutational alterations in the EGFR/ERK pathway on the growth rate of in silico tumours. [ABSTRACT FROM AUTHOR]

A two-dimensional model for red blood cell motion is adapted to consider the dynamics of breast cancer cells in a microfluidic channel. Adjusting parameters to make the membrane stiffer, as is the case with breast cancer cells compared with red blood cells, allows the model to produce reasonable estimates of breast cancer cell trajectories through the channel. In addition, the model produces estimates of quantities not as easily obtained from experiment such as velocity and stress field information throughout the fluid and on the cell membrane. This includes locations of maximum stress along the membrane wall. A sensitivity analysis shows that the model is capable of producing useful insights into various systems involving breast cancer cells. Current results suggest that dynamics taking place when cells are near other objects are most sensitive to membrane and cytoplasm elasticity, dynamics taking place when cells are not near other objects are most sensitive to cytoplasm viscosity, and dynamics are significantly affected by low membrane bending elasticity. These results suggest that continued calibration and application of this model can yield useful predictions in other similar systems. [ABSTRACT FROM AUTHOR]

The objective of the present study is to mathematically model the integrated kinetics of drug release in a polymeric matrix and its ensuing drug transport to the encompassing biological tissue. The model embodies drug diffusion, dissolution, solubilization, polymer degradation and dissociation/recrystallization phenomena in the polymeric matrix accompanied by diffusion, advection, reaction, internalization and specific/nonspecific binding in the biological tissue. The model is formulated through a system of nonlinear partial differential equations which are solved numerically in association with pertinent set of initial, interface and boundary conditions using suitable finite difference scheme. After spatial discretization, the system of nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations which is subsequently solved by the fourth-order Runge-Kutta method. The model simulations deal with the comparison between a drug delivery from a biodegradable polymeric matrix and that from a biodurable polymeric matrix. Furthermore, simulated results are compared with corresponding existing experimental data to manifest the efficaciousness of the advocated model. A quantitative analysis is performed through numerical computation relied on model parameter values. The numerical results obtained reveal an estimate of the effects of biodegradable and biodurable polymeric matrices on drug release rates. Furthermore, through graphical representations, the sensitized impact of the model parameters on the drug kinetics is illustrated so as to assess the model parameters of significance. [ABSTRACT FROM AUTHOR]

Neuroblastoma is the leading cause of cancer death in young children. Although treatment for neuroblastoma has improved, the 5-year survival rate of patients still remains less than half. Recent studies have indicated that bevacizumab, an anti-VEGF drug used in treatment of several other cancer types, may be effective for treating neuroblastoma as well. However, its effect on neuroblastoma has not been well characterized. While traditional experiments are costly and time-consuming, mathematical models are capable of simulating complex systems quickly and inexpensively. In this study, we present a model of vascular tumor growth of neuroblastoma IMR-32 that is complex enough to replicate experimental data across a range of tumor cell properties measured in a suite of in vitro and in vivo experiments. The model provides quantitative insight into tumor vasculature, predicting a linear relationship between vasculature and tumor volume. The tumor growth model was coupled with known pharmacokinetics and pharmacodynamics of the VEGF blocker bevacizumab to study its effect on neuroblastoma growth dynamics. The results of our model suggest that total administered bevacizumab concentration per week, as opposed to dosage regimen, is the major determining factor in tumor suppression. Our model also establishes an exponentially decreasing relationship between administered bevacizumab concentration and tumor growth rate. [ABSTRACT FROM AUTHOR]

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state. [ABSTRACT FROM AUTHOR]