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'.

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.

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.

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs.

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries.

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.

A tidal breathing model of the inert gas sinewave technique for inhomogeneous lungs.

The tidal breathing model conservation of mass equations for the sinewave technique have been described for a homogeneous alveolar compartment by Gavaghan and Hahn, 1996 [Gavaghan, D.J., Hahn, C.E.W., 1996. A tidal breathing model of the forced inspired gas sinewave technique. Respir. Physiol. 106, 209-221]. We develop these equations first to a multi-discrete alveolar compartment lung model and then to a lung model with a continuous distribution of volume, ventilation and perfusion. The effect on the output parameters of a multi-compartment model is discussed, and the results are compared to those derived from the conventional continuous-ventilation model. Using the barely soluble gas argon as the tracer gas, an empirical index of alveolar inhomogeneity is presented which uses the end-expired and mixed-expired partial pressures on each breath. This index distinguishes between a narrow unimodal distribution of ventilation-volume, a wide unimodal distribution of ventilation-volume and a bimodal distribution of ventilation-volume. By using Monte Carlo simulations, this index is shown to be stable to experimental error of realistic magnitude.

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation.

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.

The effect of inspired oxygen concentration on the ventilation-perfusion distribution in inhomogeneous lungs.

The coupled conservation of mass equations for oxygen, carbon dioxide and nitrogen are written down for a lung model consisting of two homogeneous alveolar compartments (with different ventilation-perfusion ratios) and a shunt compartment. As inspired oxygen concentration and oxygen consumption are varied, the flux of oxygen, carbon dioxide and nitrogen across the alveolar membrane in each compartment varies. The result of this is that the expired ventilation-perfusion ratio for each compartment becomes a function of inspired oxygen concentration and oxygen consumption as well as parameters such as inspired ventilation and alveolar perfusion. Another result is that the "inspired ventilation-perfusion ratio and the "expired ventilation-perfusion ratio differ significantly, under some conditions, for poorly ventilated lung compartments. As a consequence, we need to distinguish between the "inspired ventilation-perfusion distribution, which is independent of inspired oxygen concentration and oxygen consumption, and the "expired ventilation-perfusion distribution, which we now show to be strongly dependent on inspired oxygen concentration and less dependent oxygen consumption. Since the multiple inert gas elimination technique (MIGET) estimates the "expired ventilation-perfusion distribution, it follows that the distribution recovered by MIGET may be strongly dependent on inspired oxygen concentration.

The effect of the width of the ventilation-perfusion distribution on arterial blood oxygen content.

We investigate the effect of the width of ventilation-perfusion distributions on arterial blood oxygen content. We assume that the perfusion within the alveolar volume is a continuous function of ventilation-perfusion ratio, known as the continuous ventilation-perfusion distribution, and then write down the conservation of mass equations in the lung incorporating the nonlinear relationship between oxygen concentration in the gas phase and blood oxygen content. We solve these equations for various unimodal and bimodal ventilation-perfusion distributions believed to occur in practice and calculate the arterial blood oxygen content in each case. When a subject has a unimodal ventilation-perfusion distribution we show that the fraction of cardiac output to that mode (i.e. the fraction of non-shunted blood) has a large effect on arterial oxygen blood content. However, the width of the distribution has only a negligible effect on arterial oxygen blood content. For a bimodal ventilation-perfusion distribution the location and fraction of cardiac output to each mode has a large effect on arterial oxygen blood content. Again, the width of each mode of the distribution has little effect on arterial oxygen blood content. As a result there is little point, from a clinical perspective, in developing techniques for investigating the width of modes of these distributions since all relevant clinical information is contained in the nature (i.e. unimodal or bimodal) and in the location of the modes.

A tidal breathing model for the multiple inert gas elimination technique.

The tidal breathing lung model described for the sine-wave technique (D. J. Gavaghan and C. E. W. Hahn. Respir. Physiol. 106: 209-221, 1996) is generalized to continuous ventilation-perfusion and ventilation-volume distributions. This tidal breathing model is then applied to the multiple inert gas elimination technique (P. D. Wagner, H. A. Saltzman, and J. B. West. J. Appl. Physiol. 36: 588-599, 1974). The conservation of mass equations are solved, and it is shown that 1) retentions vary considerably over the course of a breath, 2) the retentions are dependent on alveolar volume, and 3) the retentions depend only weakly on the width of the ventilation-volume distribution. Simulated experimental data with a unimodal ventilation-perfusion distribution are inserted into the parameter recovery model for a lung with 1 or 2 alveolar compartments and for a lung with 50 compartments. The parameters recovered using both models are dependent on the time interval over which the blood sample is taken. For best results, the blood sample should be drawn over several breath cycles.