Some Mathematical Properties of Cumulative Damage Models Regarding Their Application in Cancer Epidemiology

In an earlier paper, cumulative damage models (CD models) were proposed for modelling the epidemiological aspects of carcinogenesis. In the present paper, further, mainly mathematical support is given for the adequacy of this approach. In the first place, this concerns the aspect that the cumulative damage process is a compound Poisson process. Secondly, it will be demonstrated that the CD models can be considered as a formal generalization of certain well-known special carcinogenesis models. A more intensive investigation of these models themselves makes it evident that, on account of their mathematical qualities, they will possibly place very efficient new measures at the disposal of epidemiology. A diffusion approximation, however, does, after first experiments, not appear to make the handling of the models any easier but, on the contrary, to lead to a loss of certain pleasant qualities.     

A new, vapour-phase mechanism for stomatal responses to humidity and temperature

A new mechanism for stomatal responses to humidity and temperature is proposed. Unlike previously-proposed mechanisms, which rely on liquid water transport to create water potential gradients within the leaf, the new mechanism assumes that water transport to the guard cells is primarily through the vapour phase. Under steady-state conditions, guard cells are assumed to be in near-equilibrium with the water vapour in the air near the bottom of the stomatal pore. As the water potential of this air varies with changing air humidity and leaf temperature, the resultant changes in guard cell water potential produce stomatal movements. A simple, closed-form, mathematical model based on this idea is derived. The new model is parameterized for a previously published set of data and is shown to fit the data as well as or better than existing models. The model contains mathematical elements that are consistent with previously-proposed mechanistic models based on liquid flow as well as empirical models based on relative humidity. As such, it provides a mechanistic explanation for the realm of validity for each of these approaches.     

Additive Damage Models for Cellular Pharmacodynamics of Radiation–Chemotherapy Combinations

Many cancer patients receive combination treatments with radiation and chemotherapy. Available mathematical models for cellular pharmacodynamics have limited ability to represent observed in vitro responses to radiochemotherapy. Here, a family of additive damage models is proposed to describe cell kill resulting from radiochemotherapy with fixed schedule and variable doses. The pathways by which the agents produce cellular damage are assumed to converge in a single cell death process, so that survival depends on total damage, which can be represented as a sum of contributions from the various damage pathways. Heterogeneity in response across the cell population is ascribed to variations in the damage threshold for cell kill. The family of proposed models includes effects of one or two pathways of damage for each agent, saturation in drug responses, and cooperative or antagonistic interactions between agents. Models from this family with 4–7 unknown parameters are tested for their ability to fit 218 in vitro literature data sets for a range of drugs and cell lines. Overall, the additive damage models are found to outperform models based on the existing concept of independent cell kill, according to the corrected Akaike Information Criterion. The results are used to assess the importance of the various effects included in the models. These additive damage models have potential applications to the optimization of treatment and to the analysis and interpretation of in vitro screening data for new drug–radiation combinations.     

The Consideration of Latency Periods in Cumulative Damage Models

The cumulative damage model as outlined in previous papers is unable to reproduce latency periods, particularly after short-term exposures. A refinement being straightforward within the present frame of point process theory is proposed using displacement processes applied to the single damages. For the displacement distribution, the lognormal distribution has been suggested. As in the previous model, exact formula for the behaviour of the survival function after the onset and the cessation of an additional exposure can be derived. Qualitative examinations demonstrate that the thus extended model shows the relevant features known from reports on shortterm exposures. Fitting the model to data of follow-up studies should provide a measure for the impact of an environmental exposure and the parameters for the distribution of the related latency period.     

A novel mathematical model of activation and sensitization of platelets subjected to dynamic stress histories

Blood recirculating devices, such as ventricular assist devices and prosthetic heart valves, are burdened by thromboembolic complications requiring complex and lifelong anticoagulant therapy with its inherent hemorrhagic risks. Pathologic flow patterns occurring in such devices chronically activate platelets, and the optimization of their thrombogenic performance requires the development of flow-induced platelet activation models. However, existing models are based on empirical correlations using the well-established power law paradigm of constant levels of shear stress during certain exposure times as factors for mechanical platelet activation. These models are limited by their range of application and do not account for other relevant phenomena, such as loading rate dependence and platelet sensitization to high stress conditions, which characterize the dynamic flow conditions in devices. These limitations were addressed by developing a new class of phenomenological stress-induced platelet activation models that specifies the rate of platelet activation as a function of the entire stress history and results in a differential equation that can be directly integrated to calculate the cumulative levels of activation. The proposed model reverts to the power law under constant shear stress conditions and is able to describe experimental results in response to a diverse range of highly dynamic stress conditions found in blood recirculating devices. The model was tested in vitro under emulated device flow conditions and correlates well with experimental results. This new model provides a reliable and robust mathematical tool that can be incorporated into computational fluid dynamic studies in order to optimize design, with the goal of improving the thrombogenic performance of blood recirculating devices.     

Second-order analysis of inhomogeneous spatial point processes using case-control data

Summary Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case–control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica 54 , 329–350) concerning estimation of the second‐order properties of a nonstationary spatial point process. First, we show how case–control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second‐order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.     

A methodology for performing global uncertainty and sensitivity analysis in systems biology

Accuracy of results from mathematical and computer models of biological systems is often complicated by the presence of uncertainties in experimental data that are used to estimate parameter values. Current mathematical modeling approaches typically use either single-parameter or local sensitivity analyses. However, these methods do not accurately assess uncertainty and sensitivity in the system as, by default, they hold all other parameters fixed at baseline values. Using techniques described within we demonstrate how a multi-dimensional parameter space can be studied globally so all uncertainties can be identified. Further, uncertainty and sensitivity analysis techniques can help to identify and ultimately control uncertainties. In this work we develop methods for applying existing analytical tools to perform analyses on a variety of mathematical and computer models. We compare two specific types of global sensitivity analysis indexes that have proven to be among the most robust and efficient. Through familiar and new examples of mathematical and computer models, we provide a complete methodology for performing these analyses, in both deterministic and stochastic settings, and propose novel techniques to handle problems encountered during these types of analyses.     

Identifying Predator–Prey Processes from Time-Series

The functional response is a key element in predator–prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator–prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

Point process models of single-neuron discharges

In most neural systems, neurons communicate via sequences of action potentials. Contemporary models assume that the action potentials' times of occurrence rather than their waveforms convey information. The mathematical tool for describing sequences of events occurring in time and/or space is the theory of point processes. Using this theory, we show that neural discharge patterns convey time-varying information intermingled with the neuron's response characteristics. We review the basic techniques for analyzing single-neuron discharge patterns and describe what they reveal about the underlying point process model. By applying information theory and estimation theory to point processes, we describe the fundamental limits on how well information can be represented by and extracted from neural discharges. We illustrate applying these results by considering recordings from the lower auditory pathway.     

Biologically Based Prediction of Empirical Nonlinearity in Lung Cancer Risk vs. Residential/ Occupational Radon Exposure

Ecologic U.S. county data suggest negative associations between residential radon exposure and lung cancer mortality (LCM) that are inconsistent with clearly positive ones revealed by individual data on underground miners. If this inconsistency is due to competing effects of induced cell killing vs. mutations in alpha-radiation exposed bronchial epithelium, then linear extrapolation from miner data may overestimate typical residential radon risks. To investigate the plausibility of this hypothesis, a biologically based “cytodynamic 2-stage” (CD2) cancer-risk model was fit to combined 1950 to 1954 age-specific person-year data on white females of age 40+ y in 2821 U.S. counties (～90% never-smokers), and on five cohorts of underground miners who never smoked, conditional on a realistic rate of alpha-radiation-induced killing of human lung cells, and on linear-no-threshold dose-response relations for both processes assumed to affect cancer risk (alpha-induced mutations and cell killing). As summarized previously (Bogen, K.T., Hum. Exper. Toxicol. 17:691-6, 1998), a good CD2 fit was obtained that involved biologically plausible parameter values and (without further optimization) also predicted inverse dose-rate effects observed in the nonsmoking miners. The present paper reports mathematical details of the CD2 model used, as well as additional modeling results involving the same combined data set. The results obtained are consistent with the hypotheses that low-level radon exposure is nonlinearly related to LCM risk, and that current linear no-threshold extrapolation models overestimate LCM risk associated with relatively low residential radon concentrations (<～200?Bq m^?3). Testing this hypothesis would require more extensive individual-level epidemiological data relating residential radon exposures to LCM than are currently available.     

Data-Based Analysis of Winner-Loser Models of Hierarchy Formation in Animals

We review winner-loser models, the currently popular explanation for the occurrence of linear dominance hierarchies, via a three-part approach. (1) We isolate the two most significant components of the mathematical formulation of three of the most widely-cited models and rigorously evaluate the components’ predictions against data collected on hierarchy formation in groups of hens. (2) We evaluate the experimental support in the literature for the basic assumptions contained in winner-loser models. (3) We apply new techniques to the hen data to uncover several behavioral dynamics of hierarchy formation not previously described. The mathematical formulations of these models do not show satisfactory agreement with the hen data, and key model assumptions have either little or no conclusive support from experimental findings in the literature. In agreement with the latest experimental results concerning social cognition, the new behavioral dynamics of hierarchy formation discovered in the hen data suggest that members of groups are intensely aware both of their own interactions as well as interactions occurring among other members of their group. We suggest that more adequate models of hierarchy formation should be based upon behavioral dynamics that reflect more sophisticated levels of social cognition.     

Bayesian nonparametric estimation of continuous monotone functions with applications to dose-response analysis

Summary .  In this article, we consider monotone nonparametric regression in a Bayesian framework. The monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions, and a general random probability measure is assumed as the prior for the mixing distribution. We investigate the choice of the underlying parametric distribution function and find that the two-sided power distribution function is well suited both from a computational and mathematical point of view. The model is motivated by traditional nonlinear models for dose–response analysis, and provides possibilities to elicitate informative prior distributions on different aspects of the curve. The method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose–response analysis.     

Stochastic modeling of aphid population growth with nonlinear, power-law dynamics

This paper develops a deterministic and a stochastic population size model based on power-law kinetics for the black-margined pecan aphid. The deterministic model in current use incorporates cumulative-size dependency, but its solution is symmetric. The analogous stochastic model incorporates the prolific reproductive capacity of the aphid. These models are generalized in this paper to include a delayed feedback mechanism for aphid death. Whereas the per capita aphid death rate in the current model is proportional to cumulative size, delayed feedback is implemented by assuming that the per capita rate is proportional to some power of cumulative size, leading to so-called power-law dynamics. The solution to the resulting differential equations model is a left-skewed abundance curve. Such skewness is characteristic of observed aphid data, and the generalized model fits data well. The assumed stochastic model is solved using Kolmogrov equations, and differential equations are given for low order cumulants. Moment closure approximations, which are simple to apply, are shown to give accurate predictions of the two endpoints of practical interest, namely (1) a point estimate of peak aphid count and (2) an interval estimate of final cumulative aphid count. The new models should be widely applicable to other aphid species, as they are based on three fundamental properties of aphid population biology.     

Do U.S. county data disprove linear no‐threshold predictions of lung cancer risk for residential radon?‐A preliminary assessment of biological plausibility

Lung‐cancer mortality (LCM) is elevated in underground miners who chronically inhaled the mutagenic, cytotoxic α‐decay products of radon gas. Epidemiologie studies of LCM rates vs. residential‐radon concentration levels are generally considered inconclusive. However, Cohen (Health Physics 68, 157–174, 1995) has hypothesized that data on LCM vs. residential radon concentrations at the U.S. county level are clearly inconsistent with a linear no‐threshold (LN) dose‐response model, and rather are consistent with threshold or hormesis model. Cohen's hypothesis has been criticized as “ecological fallacy,”; particularly because LN (but not threshold or hormesis) models are generally considered biologically plausible for agents like α radiation that damage DNA in linear proportion to dose. To assess the biological plausibility of Cohen's hypothesis, a preliminary study was made of whether a biologically realistic, cytodynamic 2‐stage (CD2) cancer model can provide a good, joint fit to Cohen's set of U.S. county data as well as to underground‐miner data. The CD2 model used adapts a widely applied, mechanistic, 2‐stage stochastic model of carcinogenesis to realistically account for interrelated cell killing and mutation (both assumed to have a LN dose‐response), cell turnover, and incomplete exposure of stem cells. A CD2 fit was obtained to combined summary data on LCM vs. radon‐exposure in white males in 1, 601 U.S. counties (from Cohen) and in white male Colorado Plateau (CP) uranium miners (from the National Research Council's “BEIRIV”; report). The CD2 fit is shown to: (i) be consistent with the combined data; (ii) have parameter values all consistent with biological data; and (iii) predict inverse dose‐rate‐effects data for CP and other radon‐exposed miners, despite the fact that optimization had not involved any of these dose‐rate data. The latter data were not predicted by a simplified CD2 model in which all stem cells were presumed to be exposed. It is concluded that this study provides preliminary evidence that Cohen's hypothesis is biologically plausible.     

Population models with singular equilibrium

A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sector. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. An algorithmic approach to analyze system behavior with parameter changes is presented. The developed methods and algorithm are applied to existing mathematical models of biological systems. In particular, we analyze a model of anticancer treatment with oncolytic viruses, a parasite-host interaction model, and a model of Chagas' disease.     

Effect of needle geometry on flow rate and cell damage in the dispensing-based biofabrication process

Biodispensing techniques have been widely applied in biofabrication processes to deliver cell suspensions and biomaterials to create cell-seeded constructs. Under identical operating conditions,two types of dispensing needles—tapered and cylindrical—can result in different flow rates of material and different cell damage percent induced by the mechanical forces. In this work, mathematical models of both flow rate and cell damage percent in biodispensing systems using tapered and cylindrical needles, respectively, were developed, and experiments were carried out to verify the effectiveness of the developed models. Both simulations and experiments show tapered needles produce much higher flow rates under the same pressure conditions than cylindrical needles. Use of a lower pressure in a tapered needle can therefore achieve the same flow rate as that in a cylindrical needle. At equivalent flow rates, cell damage in a tapered needle is lower than that in a cylindrical one. Both Schwann cells and 3T3 fibroblasts, which have been widely used in tissue engineering, were used to validate the cell damage models. Application of the developed models to specify the influence of process parameters, including needle geometry and air pressure, on the flow rate and cell damage percent represents a significant advance for biofabrication processes.The models can be used to optimize process parameters to preserve cell viability and achieve the desired cell distribution in dispensing-based biofabrication.     

Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments.

Optimal design of experiments as well as proper analysis of data are dependent on knowledge of the experimental error. A detailed analysis of the error structure of kinetic data obtained with acetylcholinesterase showed conclusively that the classical assumptions of constant absolute or constant relative error are inadequate for the dependent variable (velocity). The best mathematical models for the experimental error involved the substrate and inhibitor concentrations and reflected the rate law for the initial velocity. Data obtained with other enzymes displayed similar relationships between experimental error and the independent variables. The new empirical error functions were shown superior to previously used models when utilized in weighted non-linear-regression analysis of kinetic data. The results suggest that, in the spectrophotometric assays used in the present study, the observed experimental variance is primarily due to errors in determination of the concentrations of substrate and inhibitor and not to error in measuring the velocity.     

同龄纯林自然稀疏过程的经验模型研究

应用-3/2法则及广义Schumacher生长方程导出同龄纯林自然稀疏过程中密度随时间变化规律新模型,采用遗传算法对非线性模型参数进行最优估计.以山杨、云南松、杉木等树种同龄纯林自疏过程中密度随时间变化资料对新模型进行了验证,并与前人提出的主要森林自疏过程密度随时间变化规律模型进行了对比.结果表明,所提出的同龄纯林自疏规律模型能很好地拟合实际观测资料,具有良好的使用价值;新模型拟合效果较前人提出的自疏规律模型效果均更佳,说明新模型是一个描述同龄纯林自疏过程密度随时间变化规律的理想经验模型,可在森林自疏规律研究中应用.杉木林自疏过程密度变化规律的研究可为南方林区杉木林经营管理提供参考.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.