Heterogeneity in the onwards transmission risk between local and imported cases affects practical estimates of the time-dependent reproduction number.

During infectious disease outbreaks, inference of summary statistics characterizing transmission is essential for planning interventions. An important metric is the time-dependent reproduction number (Rt), which represents the expected number of secondary cases generated by each infected individual over the course of their infectious period. The value of Rt varies during an outbreak due to factors such as varying population immunity and changes to interventions, including those that affect individuals' contact networks. While it is possible to estimate a single population-wide Rt, this may belie differences in transmission between subgroups within the population. Here, we explore the effects of this heterogeneity on Rt estimates. Specifically, we consider two groups of infected hosts: those infected outside the local population (imported cases), and those infected locally (local cases). We use a Bayesian approach to estimate Rt, made available for others to use via an online tool, that accounts for differences in the onwards transmission risk from individuals in these groups. Using COVID-19 data from different regions worldwide, we show that different assumptions about the relative transmission risk between imported and local cases affect Rt estimates significantly, with implications for interventions. This highlights the need to collect data during outbreaks describing heterogeneities in transmission between different infected hosts, and to account for these heterogeneities in methods used to estimate Rt. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

A mathematical model of antibody-dependent cellular cytotoxicity (ADCC).

Immunotherapies exploit the immune system to target and kill cancer cells, while sparing healthy tissue. Antibody therapies, an important class of immunotherapies, involve the binding to specific antigens on the surface of the tumour cells of antibodies that activate natural killer (NK) cells to kill the tumour cells. Preclinical assessment of molecules that may cause antibody-dependent cellular cytotoxicity (ADCC) involves co-culturing cancer cells, NK cells and antibody in vitro for several hours and measuring subsequent levels of tumour cell lysis. Here we develop a mathematical model of such an in vitro ADCC assay, formulated as a system of time-dependent ordinary differential equations and in which NK cells kill cancer cells at a rate which depends on the amount of antibody bound to each cancer cell. Numerical simulations generated using experimentally-based parameter estimates reveal that the system evolves on two timescales: a fast timescale on which antibodies bind to receptors on the surface of the tumour cells, and NK cells form complexes with the cancer cells, and a longer time-scale on which the NK cells kill the cancer cells. We construct approximate model solutions on each timescale, and show that they are in good agreement with numerical simulations of the full system. Our results show how the processes involved in ADCC change as the initial concentration of antibody and NK-cancer cell ratio are varied. We use these results to explain what information about the tumour cell kill rate can be extracted from the cytotoxicity assays.

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues.

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification.

Hydrodynamic dispersion within porous biofilms.

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport.

Computational modelling of cardiac electrophysiology: explanation of the variability of results from different numerical solvers.

A recent verification study compared 11 large-scale cardiac electrophysiology solvers on an unambiguously defined common problem. An unexpected amount of variation was observed between the codes, including significant error in conduction velocity in the majority of the codes at certain spatial resolutions. In particular, the results of the six finite element codes varied considerably despite each using the same order of interpolation. In this present study, we compare various algorithms for cardiac electrophysiological simulation, which allows us to fully explain the differences between the solvers. We identify the use of mass lumping as the fundamental cause of the largest variations, specifically the combination of the commonly used techniques of mass lumping and operator splitting, which results in a slightly different form of mass lumping to that supported by theory and leads to increased numerical error. Other variations are explained through the manner in which the ionic current is interpolated. We also investigate the effect of different forms of mass lumping in various types of simulation.

Rabbit-specific ventricular model of cardiac electrophysiological function including specialized conduction system.

The function of the ventricular specialized conduction system in the heart is to ensure the coordinated electrical activation of the ventricles. It is therefore critical to the overall function of the heart, and has also been implicated as an important player in various diseases, including lethal ventricular arrhythmias such as ventricular fibrillation and drug-induced torsades de pointes. However, current ventricular models of electrophysiology usually ignore, or include highly simplified representations of the specialized conduction system. Here, we describe the development of an image-based, species-consistent, anatomically-detailed model of rabbit ventricular electrophysiology that incorporates a detailed description of the free-running part of the specialized conduction system. Techniques used for the construction of the geometrical model of the specialized conduction system from a magnetic resonance dataset and integration of the system model into a ventricular anatomical model, developed from the same dataset, are described. Computer simulations of rabbit ventricular electrophysiology are conducted using the novel anatomical model and rabbit-specific membrane kinetics to investigate the importance of the components and properties of the conduction system in determining ventricular function under physiological conditions. Simulation results are compared to panoramic optical mapping experiments for model validation and results interpretation. Full access is provided to the anatomical models developed in this study.

Integrated approach for the study of anatomical variability in the cardiac Purkinje system: from high resolution MRI to electrophysiology simulation.

The ordered electrical stimulation of the ventricles is achieved by a specialized network of fibres known as the Purkinje system. The gross anatomy and basic functional role of the Purkinje system is well understood. However, very little is known about the detailed anatomy of the Purkinje system, its inter-individual variability and the implications of the variability in ventricular function, in part due to limitations in experimental techniques. In this study, we aim to provide new insight into the inter-individual variability of the free running Purkinje system anatomy and its impact on ventricular electrophysiological function. As a first step towards achieving this aim, high resolution magnetic resonance imaging (MRI) datasets of rat and the rabbit ventricles are obtained and analysed using a novel semi-automatic image processing algorithm for segmentation of the free-running Purkinje system. Segmented geometry from the MRI datasets is used to construct a computational model of the Purkinje system, which is incorporated in to an anatomically-based ventricular geometry to simulate ventricular electrophysiological activity.

A hybrid approach to multi-scale modelling of cancer.

In this paper, we review multi-scale models of solid tumour growth and discuss a middle-out framework that tracks individual cells. By focusing on the cellular dynamics of a healthy colorectal crypt and its invasion by mutant, cancerous cells, we compare a cell-centre, a cell-vertex and a continuum model of cell proliferation and movement. All models reproduce the basic features of a healthy crypt: cells proliferate near the crypt base, they migrate upwards and are sloughed off near the top. The models are used to establish conditions under which mutant cells are able to colonize the crypt either by top-down or by bottom-up invasion. While the continuum model is quicker and easier to implement, it can be difficult to relate system parameters to measurable biophysical quantities. Conversely, the greater detail inherent in the multi-scale models means that experimentally derived parameters can be incorporated and, therefore, these models offer greater scope for understanding normal and diseased crypts, for testing and identifying new therapeutic targets and for predicting their impacts.

The systems biology approach to drug development: application to toxicity assessment of cardiac drugs.

Side effects account for most of the instances of failure of candidate drugs at late stages of development. These development failures contribute to the exorbitant cost of bringing new compounds to market: a single withdrawal can represent a loss of more than $1 billion. Many unwanted actions of drugs affect the heart, resulting in potentially proarrhythmic alteration of ion channel function. Because these can be fatal, potential electrophysiological cardiotoxicity is among the most stringent exclusion criteria in the licensing process.

An integrative computational model for intestinal tissue renewal.

OBJECTIVES: The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation. METHODS: At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole. RESULTS: We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data. CONCLUSIONS: We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis.

Synchronization between arterial blood pressure and cerebral oxyhaemoglobin concentration investigated by wavelet cross-correlation.

Wavelet cross-correlation (WCC) is used to analyse the relationship between low-frequency oscillations in near-infrared spectroscopy (NIRS) measured cerebral oxyhaemoglobin (O(2)Hb) and mean arterial blood pressure (MAP) in patients suffering from autonomic failure and age-matched controls. Statistically significant differences are found in the wavelet scale of maximum cross-correlation upon posture change in patients, but not in controls. We propose that WCC analysis of the relationship between O(2)Hb and MAP provides a useful method of investigating the dynamics of cerebral autoregulation using the spontaneous low-frequency oscillations that are typically observed in both variables without having to make the assumption of stationarity of the time series. It is suggested that for a short-duration clinical test previous transfer-function-based approaches to analyse this relationship may suffer due to the inherent nonstationarity of low-frequency oscillations that are observed in the resting brain.

Modulation of shock-end virtual electrode polarisation as a direct result of 3D fluorescent photon scattering.

Due to the large transmural variation in transmembrane potential following the application of strong electric shocks, it is thought that fluorescent photon scattering from depth plays a significant role in optical signal modulation at shock-end. For the first time, a model of photon scattering is used to accurately synthesize fluorescent signals over the irregular geometry of the rabbit ventricles following the application of such strong shocks. A bidomain representation of electrical activity is combined with finite element solutions to the photon diffusion equation, simulating both the excitation and emission processes, over an anatomically-based model of rabbit ventricular geometry and fiber orientation. Photon scattering from within a 3D volume beneath the epicardial optical recording site is shown to transduce differences in transmembrane potential within this volume through the myocardial wall. This leads directly to a significantly modulated optical signal response with respect to that predicted by the bidomain simulations, distorting epicardial virtual electrode polarization produced at shock-end. Furthermore, we show that this degree of distortion is very sensitive to the optical properties of the tissue, an important variable to consider during experimental mapping set-ups. These findings provide an essential first-step in aiding the interpretation of experimental optical mapping recordings following strong defibrillation shocks.

Transmural electrophysiological heterogeneities in action potential duration increase the upper limit of vulnerability.

Transmural dispersion in action potential duration (APD) has been shown to contribute to arrhythmia induction in the heart. However, its role in termination of lethal arrhythmias by defibrillation shocks has never been examined. The goal of this study is to investigate how transmural dispersion in APD affects cardiac vulnerability to electric shocks, in an attempt to better understand the mechanisms behind defibrillation failure. This study used a three- dimensional, geometrically accurate finite element bidomain rabbit ventricular model. Transmural heterogeneities in ionic currents were incorporated based on experimental data to generate the transmural APD profile recorded in adult rabbits during pacing. Results show that the incorporation of transmural APD heterogeneities in the model causes an increase in the upper limit of vulnerability from 26.7 V/cm in the homogeneous APD ventricles to 30.5 V/cm in the ventricles with heterogeneous transmural APD profile. Examination of shock-end virtual electrode polarisation and postshock electrical activity reveals that the higher ULV in the heterogeneous model is caused by increased dispersion in postshock repolarisation within the LV wall, which increases the likelihood of the establishment of intramural re-entrant circuits.

Towards a Grid infrastructure to support integrative approaches to biological research.

This paper discusses the scientific rationale behind the e-Science project, Integrative Biology, which is developing mathematical modelling tools, HPC-enabled simulations and an underpinning Grid infrastructure to provide an integrative approach to the modelling of complex biological systems. The project is focusing on two key applications to validate the approach: the modelling of heart disease and cancer, which together are responsible for over 60% of deaths in the United Kingdom. This paper provides an overview of the project, describes the initial prototype architecture and discusses the long-term scientific aims.

Digital mammography: a world without film?

OBJECTIVES: eDiaMoND is a next generation Internet ("Grid") multidisciplinary research project funded by the UK e-Science Programme with the following objectives; the development of a next generation Internet enabled prototype to demonstrate the potential benefits of a national infrastructure to support digital mammography; the exploration of potential benefits for digital mammography systems, with particular emphasis being placed on selected applications, namely, screening, training, computer-aided detection and appropriate support for epidemiological studies. METHODS: EDiaMoND has worked in conjunction with selected clinical partners to enable the collection of valuable mammography information and the design of applications based upon extensive requirements gathering exercises. The clinical partners validated both the immediate needs and assisted with defining future needs of such an architecture to support the UK Health Service. RESULTS: The project has succeeded in invoking the interest of clinical partners and representatives of the UK NHS Breast Screening Programme in our vision of a world without film, albeit a long way off. The project has also succeeded in identifying the barriers to adopting this approach with the current limitations within the NHS, and has developed a blueprint for working towards this strategy. CONCLUSIONS: A UK national digital mammography archive has the potential to provide major benefits for the UK. For example, such an archive could: ensure that previous mammograms are always available, and could link up seamlessly the screening, assessment and symptomatic clinics; it could provide a huge teaching and training resource; it could be a huge resource for epidemiological studies.

Periodic breathing induced by arterial oxygen partial pressure oscillations.

The Grodins model of respiratory control (Grodins et al., 1967) describes cardio-respiratory control for a lung with homogeneous gas concentrations. In this study we modify the Grodins model to take account of the inhomogeneities in gas concentration within the lung that are seen in many subjects with respiratory illnesses. This modification has the effect of lowering arterial oxygen partial pressure significantly. We investigate the effect on cardio-respiratory control of this low arterial oxygen signal and find that the governing equations may be reduced to a single delay-differential equation. This reduced model is found to be a good approximation to the full model and gives predictions that are similar to reported clinical data.

The effect of diffusion in the respiratory tree on the alveolar amplitude response technique (AART).

Theoretical data for the alveolar amplitude response technique (AART) (J. Appl. Physiol. 41 (1976) 419-424) for assessing lung function was simulated using a single path lung model. This model takes account of stratified inhomogeneities in gas concentrations within the respiratory tree. The data was inserted into previously published parameter recovery techniques that may be used to estimate dead-space volume, alveolar volume and cardiac output. These parameter recovery techniques are based on much simpler mathematical models that do not allow stratified inhomogeneities in gas concentrations. It was found that: (i) recovered dead-space volume depended significantly on the ventilation pattern and on the distribution of volume within of the conducting airways; (ii) alveolar volume was recovered to a good degree of accuracy; and (iii) the recovered value of cardiac output was highly dependent on both the choice of inert gas and parameter recovery technique.

A tidal ventilation model for oxygenation in respiratory failure.

We develop tidal-ventilation pulmonary gas-exchange equations that allow pulmonary shunt to have different values during expiration and inspiration, in accordance with lung collapse and recruitment during lung dysfunction (Am. J. Respir. Crit. Care Med. 158 (1998) 1636). Their solutions are tested against published animal data from intravascular oxygen tension and saturation sensors. These equations provide one explanation for (i) observed physiological phenomena, such as within-breath fluctuations in arterial oxygen saturation and blood-gas tension; and (ii) conventional (time averaged) blood-gas sample oxygen tensions. We suggest that tidal-ventilation models are needed to describe within-breath fluctuations in arterial oxygen saturation and blood-gas tension in acute respiratory distress syndrome (ARDS) subjects. Both the amplitude of these oxygen saturation and tension fluctuations, and the mean oxygen blood-gas values, are affected by physiological variables such as inspired oxygen concentration, lung volume, and the inspiratory:expiratory (I:E) ratio, as well as by changes in pulmonary shunt during the respiratory cycle.

Variation of venous admixture, SF6 shunt, PaO2, and the PaO2/FIO2 ratio with FIO2.

BACKGROUND: Measures of impairment of oxygenation can be affected by the inspired oxygen fraction. METHODS: We used a mathematical model of an inhomogenous lung to predict the effect of increasing inspired oxygen concentration (FIO2) on: (1) venous admixture (Qva/Qt); (2) arterial oxygen partial pressure (PaO2); (3) the PaO2/FIO2 index of hypoxaemia; and (4) sulphur hexafluoride (SF6) retention (often taken to be true right-to-left shunt). This model predicts whether or not atelectasis will occur. RESULTS: For lungs with regions of low V/Q, increasing the inspired oxygen concentration can cause these regions to collapse. In the absence of atelectasis, the model predicts that Qva/Qt will decrease and arterial oxygen partial pressure increase as FIO2 is increased. However, when atelectasis occurs, Qva/Qt rises to a constant value, whilst PaO2 falls at first, but then begins to rise again, with increasing FIO2. The SF6 retention increased markedly in some cases at high FIO2. CONCLUSIONS: Venous admixture will estimate true right-to-left shunt at high FIO2, even when oxygen consumption is raised. This model can explain the way that the Pa/Fl ratio changes with increasing inspired oxygen concentration.

The effects of ventilation pattern on carbon dioxide transfer in three computer models of the airways.

We investigate the effects on arterial P(CO(2)) and on arterial-end tidal P(CO(2)) difference of six different ventilation patterns of equal tidal volume, and also of various combinations of tidal volume and respiratory rate that maintain a constant alveolar ventilation. We use predictions from three different mathematical models. Models 1 (distributed) and 2 (compartmental) include combined convection and diffusion effects. Model 3 incorporates a single well-mixed alveolar compartment and an anatomical dead-space in which plug flow occurs. We found that: (i) breathing patterns with longer inspiratory times yield lower arterial P(CO(2)); (ii) varying tidal volume and respiratory rate so that alveolar ventilation is kept constant may change both PA(CO(2)) and the PA(CO(2))-PET(CO(2)) difference; (iii) the distributed model predicts higher end-tidal and arterial P(CO(2)) than the compartmental models under similar conditions; and (iv) P(CO(2)) capnograms predicted by the distributed model exhibit longer phase I and steeper phase II than other models.