From a discrete to a continuum model of cell dynamics in one dimension

Multiscale modeling is emerging as one of the key challenges in mathematical biology. However, the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions. For example, at the cellular scale, how can the appropriate discrete cell-level model be identified in a given context? Additionally, how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior? In order to begin to address such questions, we consider a discrete one-dimensional cell-based model in which cells are assumed to interact via linear springs. From the discrete equations of motion, the continuous Rouse [P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)] model is obtained. This formalism readily allows the definition of a cell number density for which a nonlinear "fast" diffusion equation is derived. Excellent agreement is demonstrated between the continuum and discrete models. Subsequently, via the incorporation of cell division, we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail. In the limit of stiff springs, where cells can be considered to be incompressible, we show that cell velocity can be directly related to cell production. This assumption is frequently made in the literature but our derivation places limits on its validity. Finally, the model is compared with a model of a similar form recently derived for a different discrete cell-based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models.

Classifying general nonlinear force laws in cell-based models via the continuum limit

though discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

Estimation of Gaussian process regression model using probability distance measures

A new class of parameter estimation algorithms is introduced for Gaussian process regression (GPR) models. It is shown that the integration of the GPR model with probability distance measures of (i) the integrated square error and (ii) Kullback–Leibler (K–L) divergence are analytically tractable. An efficient coordinate descent algorithm is proposed to iteratively estimate the kernel width using golden section search which includes a fast gradient descent algorithm as an inner loop to estimate the noise variance. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

A Generalized Log-Normal Model for Grouped Survival Data

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Toda and Volterra lattice equations from discrete symmetries of KP hierarchies

The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry.

Predicting biological parameters of estuarine benthic communities using models based on environmental data

Este trabalho objetivou predizer parâmetros da estrutura de associações macrobentônicas (composição específica, abundância, riqueza, diversidade e equitatividade) em estuários do Sul do Brasil, utilizando modelos baseados em dados ambientais (características dos sedimentos, salinidade, temperaturas do ar e da água, e profundidade). As amostragens foram realizadas sazonalmente em cinco estuários entre o inverno de 1996 e o verão de 1998. Em cada estuário as amostras foram coletadas em áreas não poluídas, com características semelhantes quanto a presença ou ausência de vegetação, profundidade e distância da desenbocadura. Para a obtenção dos modelos de predição, foram utilizados dois métodos: o primeiro baseado em Análise Discriminante Múltipla (ADM) e o segundo em Regressão Linear Múltipla (RLM). Os modelos baseados em ADM apresentaram resultados melhores do que os baseados em regressão linear. Os melhores resultados usando RLM foram obtidos para diversidade e riqueza. É possível então, concluir que modelos como aqui derivados podem representar ferramentas muito úteis em estudos de monitoramento ambiental em estuários.

Predicting smear negative pulmonary tuberculosis with classification trees and logistic regression: a cross-sectional study

Abstract Background Smear negative pulmonary tuberculosis (SNPT) accounts for 30% of pulmonary tuberculosis cases reported yearly in Brazil. This study aimed to develop a prediction model for SNPT for outpatients in areas with scarce resources. Methods The study enrolled 551 patients with clinical-radiological suspicion of SNPT, in Rio de Janeiro, Brazil. The original data was divided into two equivalent samples for generation and validation of the prediction models. Symptoms, physical signs and chest X-rays were used for constructing logistic regression and classification and regression tree models. From the logistic regression, we generated a clinical and radiological prediction score. The area under the receiver operator characteristic curve, sensitivity, and specificity were used to evaluate the model's performance in both generation and validation samples. Results It was possible to generate predictive models for SNPT with sensitivity ranging from 64% to 71% and specificity ranging from 58% to 76%. Conclusion The results suggest that those models might be useful as screening tools for estimating the risk of SNPT, optimizing the utilization of more expensive tests, and avoiding costs of unnecessary anti-tuberculosis treatment. Those models might be cost-effective tools in a health care network with hierarchical distribution of scarce resources.

Advantages and Applications of Transforming Empirical Model Input Space with Dimensional Models

Model-based calibration of steady-state engine operation is commonly performed with highly parameterized empirical models that are accurate but not very robust, particularly when predicting highly nonlinear responses such as diesel smoke emissions. To address this problem, and to boost the accuracy of more robust non-parametric methods to the same level, GT-Power was used to transform the empirical model input space into multiple input spaces that simplified the input-output relationship and improved the accuracy and robustness of smoke predictions made by three commonly used empirical modeling methods: Multivariate Regression, Neural Networks and the k-Nearest Neighbor method. The availability of multiple input spaces allowed the development of two committee techniques: a 'Simple Committee' technique that used averaged predictions from a set of 10 pre-selected input spaces chosen by the training data and the "Minimum Variance Committee" technique where the input spaces for each prediction were chosen on the basis of disagreement between the three modeling methods. This latter technique equalized the performance of the three modeling methods. The successively increasing improvements resulting from the use of a single best transformed input space (Best Combination Technique), Simple Committee Technique and Minimum Variance Committee Technique were verified with hypothesis testing. The transformed input spaces were also shown to improve outlier detection and to improve k-Nearest Neighbor performance when predicting dynamic emissions with steady-state training data. An unexpected finding was that the benefits of input space transformation were unaffected by changes in the hardware or the calibration of the underlying GT-Power model.

Structuring of Ranked Models

Prognostic procedures can be based on ranked linear models. Ranked regression type models are designed on the basis of feature vectors combined with set of relations defined on selected pairs of these vectors. Feature vectors are composed of numerical results of measurements on particular objects or events. Ranked relations defined on selected pairs of feature vectors represent additional knowledge and can reflect experts' opinion about considered objects. Ranked models have the form of linear transformations of feature vectors on a line which preserve a given set of relations in the best manner possible. Ranked models can be designed through the minimization of a special type of convex and piecewise linear (CPL) criterion functions. Some sets of ranked relations cannot be well represented by one ranked model. Decomposition of global model into a family of local ranked models could improve representation. A procedures of ranked models decomposition is described in this paper.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Cell invasion with proliferation mechanisms motivated by time-lapse data

Cell invasion involves a population of cells which are motile and proliferative. Traditional discrete models of proliferation involve agents depositing daughter agents on nearest- neighbor lattice sites. Motivated by time-lapse images of cell invasion, we propose and analyze two new discrete proliferation models in the context of an exclusion process with an undirected motility mechanism. These discrete models are related to a family of reaction- diffusion equations and can be used to make predictions over a range of scales appropriate for interpreting experimental data. The new proliferation mechanisms are biologically relevant and mathematically convenient as the continuum-discrete relationship is more robust for the new proliferation mechanisms relative to traditional approaches.

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.

Prognostic significance of IGF and ECM induced signalling proteins in breast cancer patients

Breast cancer is a leading contributor to the burden of disease in Australia. Fortunately, the recent introduction of diverse therapeutic strategies have improved the survival outcome for many women. Despite this, the clinical management of breast cancer remains problematic as not all approaches are sufficiently sophisticated to take into account the heterogeneity of this disease and are unable to predict disease progression, in particular, metastasis. As such, women with good prognostic outcomes are exposed to the side effects of therapies without added benefit. Furthermore, women with aggressive disease for whom these advanced treatments would deliver benefit cannot be distinguished and opportunities for more intensive or novel treatment are lost. This study is designed to identify novel factors associated with disease progression, and the potential to inform disease prognosis. Frequently overlooked, yet common mediators of disease are the interactions that take place between the insulin-like growth factor (IGF) system and the extracellular matrix (ECM). Our laboratory has previously demonstrated that multiprotein insulin-like growth factor-I (IGF-I): insulin-like growth factor binding protein (IGFBP): vitronectin (VN) complexes stimulate migration of breast cancer cells in vitro, via the cooperative involvement of the insulin-like growth factor type I receptor (IGF-IR) and VN-binding integrins. However, the effects of IGF and ECM protein interactions on the dissemination and progression of breast cancer in vivo are unknown. It was hypothesised that interactions between proteins required for IGF induced signalling events and those within the ECM contribute to breast cancer metastasis and are prognostic and predictive indicators of patient outcome. To address this hypothesis, semiquantitative immunohistochemistry (IHC) analyses were performed to compare the extracellular and subcellular distribution of IGF and ECM induced signalling proteins between matched normal, primary cancer, and metastatic cancer among archival formalin-fixed paraffin-embedded (FFPE) breast tissue samples collected from women attending the Princess Alexandra Hospital, Brisbane. Multivariate Cox proportional hazards (PH) regression survival models in conjunction with a modified „purposeful selection of covariates. method were applied to determine the prognostic potential of these proteins. This study provides the first in-depth, compartmentalised analysis of the distribution of IGF and ECM induced signalling proteins. As protein function and protein localisation are closely correlated, these findings provide novel insights into IGF signalling and ECM protein function during breast cancer development and progression. Distinct IGF signalling and ECM protein immunoreactivity was observed in the stroma and/or in subcellular locations in normal breast, primary cancer and metastatic cancer tissues. Analysis of the presence and location of stratifin (SFN) suggested a causal relationship in ECM remodelling events during breast cancer development and progression. The results of this study have also suggested that fibronectin (FN) and ¥â1 integrin are important for the formation of invadopodia and epithelial-to-mesenchymal transition (EMT) events. Our data also highlighted the importance of the temporal and spatial distribution of IGF induced signalling proteins in breast cancer metastasis; in particular, SFN, enhancer-of-split and hairy-related protein 2 (SHARP-2), total-akt/protein kinase B 1 (Total-AKT1), phosphorylated-akt/protein kinase B (P-AKT), extracellular signal-related kinase-1 and extracellular signal-related kinase-2 (ERK1/2) and phosphorylated-extracellular signal-related kinase-1 and extracellular signal-related kinase-2 (P-ERK1/2). Multivariate survival models were created from the immunohistochemical data. These models were found to fit well with these data with very high statistical confidence. Numerous prognostic confounding effects and effect modifications were identified among elements of the ECM and IGF signalling cascade and corroborate the survival models. This finding provides further evidence for the prognostic potential of IGF and ECM induced signalling proteins. In addition, the adjusted measures of associations obtained in this study have strengthened the validity and utility of the resulting models. The findings from this study provide insights into the biological interactions that occur during the development of breast tissue and contribute to disease progression. Importantly, these multivariate survival models could provide important prognostic and predictive indicators that assist the clinical management of breast disease, namely in the early identification of cancers with a propensity to metastasise, and/or recur following adjuvant therapy. The outcomes of this study further inform the development of new therapeutics to aid patient recovery. The findings from this study have widespread clinical application in the diagnosis of disease and prognosis of disease progression, and inform the most appropriate clinical management of individuals with breast cancer.

Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions

We define a pair-correlation function that can be used to characterize spatiotemporal patterning in experimental images and snapshots from discrete simulations. Unlike previous pair-correlation functions, the pair-correlation functions developed here depend on the location and size of objects. The pair-correlation function can be used to indicate complete spatial randomness, aggregation or segregation over a range of length scales, and quantifies spatial structures such as the shape, size and distribution of clusters. Comparing pair-correlation data for various experimental and simulation images illustrates their potential use as a summary statistic for calibrating discrete models of various physical processes.

Analysis of acute-phase proteins, AHSG, C3, CLI, HP and SAA, reveals distinctive expression patterns associated with breast, colorectal and lung cancer

Early detection, clinical management and disease recurrence monitoring are critical areas in cancer treatment in which specific biomarker panels are likely to be very important in each of these key areas. We have previously demonstrated that levels of alpha-2-heremans-schmid-glycoprotein (AHSG), complement component C3 (C3), clusterin (CLI), haptoglobin (HP) and serum amyloid A (SAA) are significantly altered in serum from patients with squamous cell carcinoma of the lung. Here, we report the abundance levels for these proteins in serum samples from patients with advanced breast cancer, colorectal cancer (CRC) and lung cancer compared to healthy controls (age and gender matched) using commercially available enzyme-linked immunosorbent assay kits. Logistic regression (LR) models were fitted to the resulting data, and the classification ability of the proteins was evaluated using receiver-operating characteristic curve and leave-one-out cross-validation (LOOCV). The most accurate individual candidate biomarkers were C3 for breast cancer [area under the curve (AUC) = 0.89, LOOCV = 73%], CLI for CRC (AUC = 0.98, LOOCV = 90%), HP for small cell lung carcinoma (AUC = 0.97, LOOCV = 88%), C3 for lung adenocarcinoma (AUC = 0.94, LOOCV = 89%) and HP for squamous cell carcinoma of the lung (AUC = 0.94, LOOCV = 87%). The best dual combination of biomarkers using LR analysis were found to be AHSG + C3 (AUC = 0.91, LOOCV = 83%) for breast cancer, CLI + HP (AUC = 0.98, LOOCV = 92%) for CRC, C3 + SAA (AUC = 0.97, LOOCV = 91%) for small cell lung carcinoma and HP + SAA for both adenocarcinoma (AUC = 0.98, LOOCV = 96%) and squamous cell carcinoma of the lung (AUC = 0.98, LOOCV = 84%). The high AUC values reported here indicated that these candidate biomarkers have the potential to discriminate accurately between control and cancer groups both individually and in combination with other proteins. Copyright © 2011 UICC.