On the effect of a therapy able to modify both the growth rates in a Gompertz stochastic model

A Gompertz-type diffusion process characterized by the presence of exogenous factors in the drift term is considered. Such a process is able to describe the dynamics of populations in which both the intrinsic rates are modified by means of time-dependent terms. In order to quantify the effect of such terms the evaluation of the relative entropy is made. The first passage time problem through suitable boundaries is studied. Moreover, some simulation results are shown in order to capture the dependence of the involved functions on the parameters. Finally, an application to tumor growth is presented and simulation results are shown.     

Applying Life Cycle Tools and Process Engineering to Determine the Most Adequate Treatment Process Conditions. A Tool in Environmental Policy (12 pp)

Background The analysis of a wastewater treatment technology, under a expanded boundaries system which includes both the technology and the inputs required for its operation, quantifies the overall environmental impact that may result from the treatment of a wastewater stream. This is particularly useful for environmental policy makers being that a expanded boundaries system tends to provide a holistic view. The former view can be highly enriched with the use of process engineering tools, such as mathematical process modelling, process design, performance assessment and cost optimised models. Main Features The traditional approach used to assess waste treatment technologies is contrasted with a life cycle analysis (LCA) approach. The optimal design of a granular activated carbon adsorption (GAC) process is used as a model system to demonstrate the advantages of LCA approaches over traditional approaches. Further sections of the paper describe a mathematical framework for the assessment of technologies, design considerations applied in the cost optimised carbon adsorption model, the use of LCA techniques to perform an inventory of all emissions associated to the process system and, some of its environmental impacts. Results Economic and environmental considerations regarding the optimum process design are introduced as a basis for decision towards the selection and operating conditions of wastewater treatment technologies. Moreover, the use of LCA has revealed that the environmental burden associated with the wastewater treatment may produce a higher environmental impact than one that can be caused by untreated discharges. Conclusion The paper highlights the string advantages that environmental policy makers may have by combining LCA and process engineering tools. Furthermore, this approach can be incorporated into other existing treatment processes or for process designers.     

A multi-step directional generalized gradient vector flow snake for target tumor segmentation in US-guided high-intensity focused ultrasound ablation

Getting precise locations of target tumors can help to ensure ablation of cancerous tissues and avoid unwanted destruction of healthy tissues in high-intensity focused ultrasound (HIFU) treatment system. Because of speckle noise and spurious boundaries in ultrasound images, traditional image segmentation methods are not suitable for achieving the precise locations of target tumors in HIFU ablation. In this paper, a multi-step directional generalized gradient vector flow snake model is introduced for target tumor segmentation. In the first step, the traditional generalized gradient vector flow (GGVF) snake is used to obtain an approximate contour of the tumor. According to the approximate contour, a new distance map is generated. Subsequently, a new directional edge map is created by calculating a scalar product of the gradients of the distance map and the initial image. In this process, the gradient directional information and the magnitude information of the distance map are used to attenuate unwanted edges and highlight the real edges in the new directional edge map. Finally, a refined GGVF field is derived from a diffusion operation of the gradient vectors of the directional edge map. The GGVF field is used to refine the tumor's contour, by directing the approximate contour to edges with the desired gradient directionality. Based on the newly developed snake model, the influences of the spurious boundaries and the speckle noise are significantly reduced in the ultrasound image segmentation. Experimental results indicate that this technique is greatly useful for target tumor segmentation in HIFU treatment system     

Chromosome aberrations in solid tumors have a stochastic nature

An important question nowadays is whether chromosome aberrations are random events or arise from an internal deterministic mechanism, which leads to the delicate task of quantifying the degree of randomness. For this purpose, we have defined several Shannon information functions to evaluate disorder inside a tumor and between tumors of the same kind. We have considered 79 different kinds of solid tumors with 30 or more karyotypes retrieved from the Mitelman Database of Chromosome Aberrations in Cancer. The Kaplan–Meier cumulative survival was also obtained for each solid tumor type in order to correlate data with tumor malignance. The results here show that aberration spread is specific for each tumor type, with high degree of diversity for those tumor types with worst survival indices. Those tumor types with preferential variants (e.g. high proportion of a given karyotype) have shown better survival statistics, indicating that aberration recurrence is a good prognosis. Indeed, global spread of both numerical and structural abnormalities demonstrates the stochastic nature of chromosome aberrations by setting a signature of randomness associated to the production of disorder. These results also indicate that tumor malignancy correlates not only with karyotypic diversity taken from different tumor types but also taken from single tumors. Therefore, by quantifying aberration spread, we could confront diverse models and verify which of them points to the most likely outcome. Our results suggest that the generating process of chromosome aberrations is neither deterministic nor totally random, but produces variations that are distributed between these two boundaries.     

A new Gompertz-type diffusion process with application to random growth

Stochastic models describing growth kinetics are very important for predicting many biological phenomena. In this paper, a new Gompertz-type diffusion process is introduced, by means of which bounded sigmoidal growth patterns can be modeled by time-continuous variables. The main innovation of the process is that the bound can depend on the initial value, a situation that is not provided by the models considered to date. After building the model, a comprehensive study is presented, including its main characteristics and a simulation of sample paths. With the aim of applying this model to real-life situations, and given its possibilities in forecasting via the mean function, discrete sampling based inference is developed. The likelihood equations are not directly solvable, and because of difficulties that arise with the usual numerical methods employed to solve them, an iterative procedure is proposed. The possibilities of the new process are illustrated by means of an application to real data, concretely, to growth in rabbits.     

Bayesian Wombling for Spatial Point Processes

Summary In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point‐referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point‐process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point‐level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point‐referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point‐referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.     

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Modelling logistic growth by a new diffusion process: Application to biological systems

The present paper introduces a new diffusion process for the purpose of modelling logistic-type behaviour patterns. Unlike other processes in the same context, this one verifies that its mean function is a logistic curve. In addition, its transition density can be found explicitly, which allows to analyse inference from the discrete sampling of trajectories. The main features of the process will be analysed and the maximum likelihood estimation of parameters will be carried out through discrete sampling. Regarding the numerical problems found to solve the likelihood equations, several strategies are developed for obtaining initial solutions for the usual numerical procedures. Such strategies are compared by means of a simulation example. Also, another simulation study is carried out in order to compare the estimation in this process to that developed by means of continuous sampling in the logistic diffusion model considered by Giovanis and Skiadas (1999). Finally an example is given for the growth of a microorganism culture. This example illustrates the predictive possibilities of the new process, as well as its ability to study time variables formulated as first-passage-times.     

Inferring the location of tumor suppressor genes by modeling frequency of allelic loss

Allelic loss is often part of a multistep process leading to tumorigenesis. Analysis of genomic markers highlights regions of elevated allelic loss, which in turn suggests a nearby tumor suppressor. Furthermore, pooling published analyses to combine evidence can increase the power to detect a tumor suppressor gene. If the pattern of loss for each tumor, or allelotype, is known, a stochastic model proposed by Newton et al. (1998, Statistics in Medicine 17, 1425-1445) can be used to analyze the correlated binary data. Many studies report only incomplete allelotypes, augmented with frequencies of allelic loss (FAL) at each marker, in which the number of informative tumors showing allelic loss is provided along with the number of informative tumors. We describe an extension of the allelotype model to handle FAL data, using a hidden Markov model or a normal approximation to compute the likelihood. The FAL model is illustrated using data from a study of colorectal cancer.     

Shot noise perturbations and mean first passage times between stable states

Predicting crossings between stable states is a central issue in population biology. Crossings from low-density to high-density equilibria are often associated with pest outbreaks, while the opposite crossings are often associated with population collapse of harvested species. Here I use a simple, bistable model to demonstrate a technique for estimating mean first passage times (MFPT) of thresholds, including boundaries between stable equilibria. The approach is based on stochastic "shot-noise" perturbations to the population and the MFPTs compare favorably with mean crossing times from Monte Carlo numerical solutions of the stochastically perturbed model. This agreement suggests that MFPT approximations can be used to quantify expected effects of species manipulations, whether the goal is pest control or sustainable harvest.     

Recommended Practice Regarding Selection of Sensitivity Analysis Methods Applied to Microbial Food Safety Process Risk Models

A guideline is presented for selection of sensitivity analysis methods applied to microbial food safety process risk (MFSPR) models. The guideline provides useful boundaries and principles for selecting sensitivity analysis methods for MSFPR models. Although the guideline is predicated on a specific branch of risk assessment models related to food-borne diseases, the principles and recommendations provided are typically generally applicable to other types of risk models. Applicable situations include: prioritizing potential critical control points; identifying key sources of variability and uncertainty; and refinement, verification, and validation of a model. Based on the objective of the analysis, characteristics of the model under study, amount of detail expected from sensitivity analysis, and characteristics of the sensitivity analysis method, recommendations for selection of sensitivity analysis methods are provided. A decision framework for method selection is introduced. The decision framework can substantially facilitate the process of selecting a sensitivity analysis method.     

Bayesian nonparametric inference for panel count data with an informative observation process

In this paper, the panel count data analysis for recurrent events is considered. Such analysis is useful for studying tumor or infection recurrences in both clinical trial and observational studies. A bivariate Gaussian Cox process model is proposed to jointly model the observation process and the recurrent event process. Bayesian nonparametric inference is proposed for simultaneously estimating regression parameters, bivariate frailty effects, and baseline intensity functions. Inference is done through Markov chain Monte Carlo, with fully developed computational techniques. Predictive inference is also discussed under the Bayesian setting. The proposed method is shown to be efficient via simulation studies. A clinical trial dataset on skin cancer patients is analyzed to illustrate the proposed approach.     

The Minimal, Phase-Transition Model for the Cell-Number Maintenance by the Hyperplasia-Extended Homeorhesis

Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the well-known Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescent-cell-duplication time. The continuous tumor morphogeny is described by a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis- and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.     

Implementation of a Markov model for phylogenetic trees

A recently developed mathematical model for the analysis of phylogenetic trees is applied to comparative data for 48 species. The model represents a return to fundamentals and makes no hypothesis with respect to the reversibility of the process. The species have been analysed in all subsets of three, and a measure of reliability of the results is provided. The numerical results of the computations on 17,296 triples of species are made available on the Internet. These results are discussed and the development of reliable tree structures for several species is illustrated. It is shown that, indeed, the Markov model is capable of considerably more interesting predictions than has been recognized to date.     

A Biased Competition Theory of Cytotoxic T Lymphocyte Interaction with Tumor Nodules

The dynamics of the interaction between Cytotoxic T Lymphocytes (CTL) and tumor cells has been addressed in depth, in particular using numerical simulations. However, stochastic mathematical models that take into account the competitive interaction between CTL and tumors undergoing immunoediting, a process of tumor cell escape from immunesurveillance, are presently missing. Here, we introduce a stochastic dynamical particle interaction model based on experimentally measured parameters that allows to describe CTL function during immunoediting. The model describes the competitive interaction between CTL and melanoma cell nodules and allows temporal and two-dimensional spatial progression. The model is designed to provide probabilistic estimates of tumor eradication through numerical simulations in which tunable parameters influencing CTL efficacy against a tumor nodule undergoing immunoediting are tested. Our model shows that the rate of CTL/tumor nodule productive collisions during the initial time of interaction determines the success of CTL in tumor eradication. It allows efficient cytotoxic function before the tumor cells acquire a substantial resistance to CTL attack, due to mutations stochastically occurring during cell division. Interestingly, a bias in CTL motility inducing a progressive attraction towards a few scout CTL, which have detected the nodule enhances early productive collisions and tumor eradication. Taken together, our results are compatible with a biased competition theory of CTL function in which CTL efficacy against a tumor nodule undergoing immunoediting is strongly dependent on guidance of CTL trajectories by scout siblings. They highlight unprecedented aspects of immune cell behavior that might inspire new CTL-based therapeutic strategies against tumors.     

A Bayesian semiparametric joint hierarchical model for longitudinal and survival data

This article proposes a new semiparametric Bayesian hierarchical model for the joint modeling of longitudinal and survival data. We relax the distributional assumptions for the longitudinal model using Dirichlet process priors on the parameters defining the longitudinal model. The resulting posterior distribution of the longitudinal parameters is free of parametric constraints, resulting in more robust estimates. This type of approach is becoming increasingly essential in many applications, such as HIV and cancer vaccine trials, where patients' responses are highly diverse and may not be easily modeled with known distributions. An example will be presented from a clinical trial of a cancer vaccine where the survival outcome is time to recurrence of a tumor. Immunologic measures believed to be predictive of tumor recurrence were taken repeatedly during follow-up. We will present an analysis of this data using our new semiparametric Bayesian hierarchical joint modeling methodology to determine the association of these longitudinal immunologic measures with time to tumor recurrence.     

The rate of multi-step evolution in Moran and Wright-Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

On the behaviour of fluidic material at molecular dynamics boundary conditions used in hybrid molecular-continuum simulations

This paper presents an investigation of the behaviour of the density profile of fluidic material confined by a force field, as it occurs across molecular-continuum mechanics boundaries in multiscale, hybrid molecular-continuum simulations. A theoretical model for the density profile across the boundary is derived. Furthermore, numerical experiments to validate the density profile and thickness of the relaxation zone are performed using molecular dynamics for a Lennard-Jones fluid in gaseous, liquid and supercritical state conditions. The simulation results show excellent agreement with the theoretical model.