Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis

Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification.     

Modelling the Role of Angiogenesis and Vasculogenesis in Solid Tumour Growth

Recent experimental evidence suggests that vasculogenesis may play an important role in tumour vascularisation. While angiogenesis involves the proliferation and migration of endothelial cells (ECs) in pre-existing vessels, vasculogenesis involves the mobilisation of bone-marrow-derived endothelial progenitor cells (EPCs) into the bloodstream. Once blood-borne, EPCs home in on the tumour site, where subsequently they may differentiate into ECs and form vascular structures. In this paper, we develop a mathematical model, formulated as a system of nonlinear ordinary differential equations (ODEs), which describes vascular tumour growth with both angiogenesis and vasculogenesis contributing to vessel formation. Submodels describing exclusively angiogenic and exclusively vasculogenic tumours are shown to exhibit similar growth dynamics. In each case, there are three possible scenarios: the tumour remains in an avascular steady state, the tumour evolves to a vascular equilibrium, or unbounded vascular growth occurs. Analysis of the full model reveals that these three behaviours persist when angiogenesis and vasculogenesis act simultaneously. However, when both vascularisation mechanisms are active, the tumour growth rate may increase, causing the tumour to evolve to a larger equilibrium size or to expand uncontrollably. Alternatively, the growth rate may be left unaffected, which occurs if either vascularisation process alone is able to keep pace with the demands of the growing tumour. To clarify further the effects of vasculogenesis, the full model is also used to compare possible treatment strategies, including chemotherapy and antiangiogenic therapies aimed at suppressing vascularisation. This investigation highlights how, dependent on model parameter values, targeting both ECs and EPCs may be necessary in order to effectively reduce tumour vasculature and inhibit tumour growth.     

Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth,Pre-Existing Vessel Co-Option,Angiogenesis and Blood Perfusion

We propose a coupled mathematical modelling system to investigate glioblastoma growth in response to dynamic changes in chemical and haemodynamic microenvironments caused by pre-existing vessel co-option, remodelling, collapse and angiogenesis. A typical tree-like architecture network with different orders for vessel diameter is designed to model pre-existing vasculature in host tissue. The chemical substances including oxygen, vascular endothelial growth factor, extra-cellular matrix and matrix degradation enzymes are calculated based on the haemodynamic environment which is obtained by coupled modelling of intravascular blood flow with interstitial fluid flow. The haemodynamic changes, including vessel diameter and permeability, are introduced to reflect a series of pathological characteristics of abnormal tumour vessels including vessel dilation, leakage, angiogenesis, regression and collapse. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. The simulation focuses on the avascular phase of tumour development and stops at an early phase of angiogenesis. The model is able to demonstrate the main features of glioblastoma growth in this phase such as the formation of pseudopalisades, cell migration along the host vessels, the pre-existing vasculature co-option, angiogenesis and remodelling. The model also enables us to examine the influence of initial conditions and local environment on the early phase of glioblastoma growth.     

Oxygen-Driven Tumour Growth Model: A Pathology-Relevant Mathematical Approach

Xenografts -as simplified animal models of cancer- differ substantially in vasculature and stromal architecture when compared to clinical tumours. This makes mathematical model-based predictions of clinical outcome challenging. Our objective is to further understand differences in tumour progression and physiology between animal models and the clinic.To achieve that, we propose a mathematical model based upon tumour pathophysiology, where oxygen -as a surrogate for endocrine delivery- is our main focus. The Oxygen-Driven Model (ODM), using oxygen diffusion equations, describes tumour growth, hypoxia and necrosis. The ODM describes two key physiological parameters. Apparent oxygen uptake rate (kR′) represents the amount of oxygen cells seem to need to proliferate. The more oxygen they appear to need, the more the oxygen transport. kR′ gathers variability from the vasculature, stroma and tumour morphology. Proliferating rate (k _p) deals with cell line specific factors to promote growth. The K _H,K _N describe the switch of hypoxia and necrosis. Retrospectively, using archived data, we looked at longitudinal tumour volume datasets for 38 xenografted cell lines and 5 patient-derived xenograft-like models.Exploration of the parameter space allows us to distinguish 2 groups of parameters. Group 1 of cell lines shows a spread in values of kR′ and lower k _p, indicating that tumours are poorly perfused and slow growing. Group 2 share the value of the oxygen uptake rate (kR′) and vary greatly in k _p, which we interpret as having similar oxygen transport, but more tumour intrinsic variability in growth.However, the ODM has some limitations when tested in explant-like animal models, whose complex tumour-stromal morphology may not be captured in the current version of the model. Incorporation of stroma in the ODM will help explain these discrepancies. We have provided an example. The ODM is a very simple -and versatile- model suitable for the design of preclinical experiments, which can be modified and enhanced whilst maintaining confidence in its predictions.     

A Robust Bayesian Random Effects Model for Nonlinear Calibration Problems

Summary In the context of a bioassay or an immunoassay, calibration means fitting a curve, usually nonlinear, through the observations collected on a set of samples containing known concentrations of a target substance, and then using the fitted curve and observations collected on samples of interest to predict the concentrations of the target substance in these samples. Recent technological advances have greatly improved our ability to quantify minute amounts of substance from a tiny volume of biological sample. This has in turn led to a need to improve statistical methods for calibration. In this article, we focus on developing calibration methods robust to dependent outliers. We introduce a novel normal mixture model with dependent error terms to model the experimental noise. In addition, we propose a reparameterization of the five parameter logistic nonlinear regression model that allows us to better incorporate prior information. We examine the performance of our methods with simulation studies and show that they lead to a substantial increase in performance measured in terms of mean squared error of estimation and a measure of the average prediction accuracy. A real data example from the HIV Vaccine Trials Network Laboratory is used to illustrate the methods.     

Prediction of Growth: A Hierarchical Bayesian Approach

A nonparametric hierarchical growth curve model is proposed. Different levels in the model hierarchy are intended to correspond to different sources of variation in an individual's growth. The nonparametric character of the model offers considerable flexibility in fitting the growth curves to empirical data. Here the emphasis is on prediction, and for this purpose the adopted Bayesian inferential approach seems particularly natural and efficient. A Markov chain Carlo method is used to perform the numerical computations. As an illustration of the techniques, we consider the growth of children, during their first two years.     

Extrapolation in Risk Assessment: Improving the Quantification of Uncertainty,and Improving Information to Reduce the Uncertainty

Risk assessments inevitably extrapolate from the known to the unknown. The resulting calculation of risk involves two fundamental kinds of uncertainty: uncertainty owing to intrinsically unpredictable (random) components of the future events, and uncertainty owing to imperfect prediction formulas (parameter uncertainty and error in model structure) that are used to predict the component that we think is predictable. Both types of uncertainty weigh heavily both in health and ecological risk assessments. Our first responsibility in conducting risk assessments is to ensure that the reported risks correctly reflect our actual level of uncertainty (of both types). The statistical methods that lend themselves to correct quantification of the uncertainty are also effective for combining different sources of information. One way to reduce uncertainty is to use all the available data. To further sharpen future risk assessments, it is useful to partition the uncertainty between the random component and the component due to parameter uncertainty, so that we can quantify the expected reduction in uncertainty that can be achieved by investing in a given amount of future data. An example is developed to illustrate the potential for use of comparative data, from toxicity testing on other species or other chemicals, to improve the estimates of low-effect concentration in a particular case with sparse case-specific data.     

A Hybrid Optimization Method for Solving Bayesian Inverse Problems under Uncertainty

In this paper, we investigate the application of a new method, the Finite Difference and Stochastic Gradient (Hybrid method), for history matching in reservoir models. History matching is one of the processes of solving an inverse problem by calibrating reservoir models to dynamic behaviour of the reservoir in which an objective function is formulated based on a Bayesian approach for optimization. The goal of history matching is to identify the minimum value of an objective function that expresses the misfit between the predicted and measured data of a reservoir. To address the optimization problem, we present a novel application using a combination of the stochastic gradient and finite difference methods for solving inverse problems. The optimization is constrained by a linear equation that contains the reservoir parameters. We reformulate the reservoir model’s parameters and dynamic data by operating the objective function, the approximate gradient of which can guarantee convergence. At each iteration step, we obtain the relatively ‘important’ elements of the gradient, which are subsequently substituted by the values from the Finite Difference method through comparing the magnitude of the components of the stochastic gradient, which forms a new gradient, and we subsequently iterate with the new gradient. Through the application of the Hybrid method, we efficiently and accurately optimize the objective function. We present a number numerical simulations in this paper that show that the method is accurate and computationally efficient.     

Macrophage-Based Anti-Cancer Therapy: Modelling Different Modes of Tumour Targeting

Tumour hypoxia is associated with poor drug delivery and low rates of cell proliferation, factors that limit the efficacy of therapies that target proliferating cells. Since macrophages localise within hypoxic regions, a promising way to target hypoxic tumour cells involves engineering macrophages to express therapeutic genes under hypoxia. In this paper we develop mathematical models to compare the responses of avascular tumour spheroids to two modes of action: either the macrophages deliver an enzyme that activates an externally applied prodrug (bystander model), or they deliver cytotoxic factors directly (local model). The models we develop comprise partial differential equations for a multiphase mixture of tumour cells, macrophages and extracellular fluid, coupled to a moving boundary representing the spheroid surface. Chemical constituents, such as oxygen and drugs, diffuse within the multiphase mixture. Simulations of both models show the spheroid evolving to an equilibrium or to a travelling wave (multiple stable solutions are also possible). We uncover the parameter dependence of the wave speed and steady-state tumour size, and bifurcations between these solution forms. For some parameter sets, adding extra macrophages has a counterintuitive deleterious effect, triggering a bifurcation from bounded to unbounded tumour growth. While these features are common to the bystander and local models, the crucial difference is where cell death occurs. The bystander model is comparable to traditional chemotherapy, with poor targeting of hypoxic tumour cells; however, the local mode of action is more selective for hypoxic regions. We conclude that effective targeting of hypoxic tumour cells may require the use of drugs with limited mobility or whose action does not depend on cell proliferation.     

A Multiscale Mathematical Model of Tumour Invasive Growth

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).     

Traceable Calibration,Performance Metrics,and Uncertainty Estimates of Minirhizotron Digital Imagery for Fine-Root Measurements

Even though fine-root turnover is a highly studied topic, it is often poorly understood as a result of uncertainties inherent in its sampling, e.g., quantifying spatial and temporal variability. While many methods exist to quantify fine-root turnover, use of minirhizotrons has increased over the last two decades, making sensor errors another source of uncertainty. Currently, no standardized methodology exists to test and compare minirhizotron camera capability, imagery, and performance. This paper presents a reproducible, laboratory-based method by which minirhizotron cameras can be tested and validated in a traceable manner. The performance of camera characteristics was identified and test criteria were developed: we quantified the precision of camera location for successive images, estimated the trueness and precision of each camera''s ability to quantify root diameter and root color, and also assessed the influence of heat dissipation introduced by the minirhizotron cameras and electrical components. We report detailed and defensible metrology analyses that examine the performance of two commercially available minirhizotron cameras. These cameras performed differently with regard to the various test criteria and uncertainty analyses. We recommend a defensible metrology approach to quantify the performance of minirhizotron camera characteristics and determine sensor-related measurement uncertainties prior to field use. This approach is also extensible to other digital imagery technologies. In turn, these approaches facilitate a greater understanding of measurement uncertainties (signal-to-noise ratio) inherent in the camera performance and allow such uncertainties to be quantified and mitigated so that estimates of fine-root turnover can be more confidently quantified.     

A Bayesian Modelling Approach with Balancing Informative Prior for Analysing Imbalanced Data

When a dataset is imbalanced, the prediction of the scarcely-sampled subpopulation can be over-influenced by the population contributing to the majority of the data. The aim of this study was to develop a Bayesian modelling approach with balancing informative prior so that the influence of imbalance to the overall prediction could be minimised. The new approach was developed in order to weigh the data in favour of the smaller subset(s). The method was assessed in terms of bias and precision in predicting model parameter estimates of simulated datasets. Moreover, the method was evaluated in predicting optimal dose levels of tobramycin for various age groups in a motivating example. The bias estimates using the balancing informative prior approach were smaller than those generated using the conventional approach which was without the consideration for the imbalance in the datasets. The precision estimates were also superior. The method was further evaluated in a motivating example of optimal dosage prediction of tobramycin. The resulting predictions also agreed well with what had been reported in the literature. The proposed Bayesian balancing informative prior approach has shown a real potential to adequately weigh the data in favour of smaller subset(s) of data to generate robust prediction models.     

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

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-\(\upkappa \)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-\(\upkappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.     

A Dynamic Model for Plant Growth: Validation Study Under Changing Temperatures

A dynamic simulation model to describe vegetative growth ofplants, for which some functions and parameter values have beenestimated previously by optimization search techniques and numericalexperimentation based on data from constant temperature experiments,is validated under conditions of changing temperatures. To testthe predictive capacity of the model, dry matter accumulationin the leaves, stems, and roots of tobacco plants (Nicotianatabacum L.) was measured at 2- or 3-day intervals during a 5-weekperiod when temperatures in controlled-environment rooms wereprogrammed for changes at weekly and daily intervals and inascending or descending sequences within a range of 14 to 34°C. Simulations of dry matter accumulation and distributionwere carried out using the programmed changes for experimentaltemperatures and compared with the measured values. The agreementbetween measured and predicted values was close and indicatesthat the temperature-dependent functional forms derived fromconstant-temperature experiments are adequate for modellingplant growth responses to conditions of changing temperatureswith switching intervals as short as 1 day. Nicotiana tabacum, tobacco, translocation, partitioning, root growth, shoot growth     

Bayesian Confidence Intervals for Multiplexed Proteomics Integrate Ion-statistics with Peptide Quantification Concordance,

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Highlights

• •Bayesian Beta-Binomial model integrates ion statistics with peptide ratio agreement.
• •Model appropriately interprets information from low signal peptides.
• •Confidence can be assigned even without replicates.
• •Model adds sensitivity to detection of small changes.

     

Bayesian Inference of Sampled Ancestor Trees for Epidemiology and Fossil Calibration

Phylogenetic analyses which include fossils or molecular sequences that are sampled through time require models that allow one sample to be a direct ancestor of another sample. As previously available phylogenetic inference tools assume that all samples are tips, they do not allow for this possibility. We have developed and implemented a Bayesian Markov Chain Monte Carlo (MCMC) algorithm to infer what we call sampled ancestor trees, that is, trees in which sampled individuals can be direct ancestors of other sampled individuals. We use a family of birth-death models where individuals may remain in the tree process after sampling, in particular we extend the birth-death skyline model [Stadler et al., 2013] to sampled ancestor trees. This method allows the detection of sampled ancestors as well as estimation of the probability that an individual will be removed from the process when it is sampled. We show that even if sampled ancestors are not of specific interest in an analysis, failing to account for them leads to significant bias in parameter estimates. We also show that sampled ancestor birth-death models where every sample comes from a different time point are non-identifiable and thus require one parameter to be known in order to infer other parameters. We apply our phylogenetic inference accounting for sampled ancestors to epidemiological data, where the possibility of sampled ancestors enables us to identify individuals that infected other individuals after being sampled and to infer fundamental epidemiological parameters. We also apply the method to infer divergence times and diversification rates when fossils are included along with extant species samples, so that fossilisation events are modelled as a part of the tree branching process. Such modelling has many advantages as argued in the literature. The sampler is available as an open-source BEAST2 package (https://github.com/CompEvol/sampled-ancestors).     

“Wrong,but Useful”: Negotiating Uncertainty in Infectious Disease Modelling

For infectious disease dynamical models to inform policy for containment of infectious diseases the models must be able to predict; however, it is well recognised that such prediction will never be perfect. Nevertheless, the consensus is that although models are uncertain, some may yet inform effective action. This assumes that the quality of a model can be ascertained in order to evaluate sufficiently model uncertainties, and to decide whether or not, or in what ways or under what conditions, the model should be ‘used’. We examined uncertainty in modelling, utilising a range of data: interviews with scientists, policy-makers and advisors, and analysis of policy documents, scientific publications and reports of major inquiries into key livestock epidemics. We show that the discourse of uncertainty in infectious disease models is multi-layered, flexible, contingent, embedded in context and plays a critical role in negotiating model credibility. We argue that usability and stability of a model is an outcome of the negotiation that occurs within the networks and discourses surrounding it. This negotiation employs a range of discursive devices that renders uncertainty in infectious disease modelling a plastic quality that is amenable to ‘interpretive flexibility’. The utility of models in the face of uncertainty is a function of this flexibility, the negotiation this allows, and the contexts in which model outputs are framed and interpreted in the decision making process. We contend that rather than being based predominantly on beliefs about quality, the usefulness and authority of a model may at times be primarily based on its functional status within the broad social and political environment in which it acts.