Mathematical correlation between biomaterial and cellular parameters--critical reflection of statistics

For mathematical modelling of the biomaterial-cell contact, it is necessary to find both parameters characterizing physical and chemical properties of the material surface and also such describing the reaction of the adhering cells. Only those material and cell parameters that correlate with each other are applicable to model this contact mathematically. Only few papers are dealing with this special problem. The aim of this paper is to present results of physical/chemical and biological investigations made on differently modified rough titanium implant surfaces in order to find out only the correlating parameters. Furthermore we discuss several ways to apply statistical methods to the correlation problem. Only few ones of all investigated parameters both on material and on cellular side were applicable for correlation. For example we found in our studies that fractal structure parameter topothesy has influence on the spreading behaviour of the osteoblastic cells. However the value of the correlation coefficient and its statistical significance heavily depend on the method of averaging the available data. Especially the biological data (spreading area) were afflicted with relatively high error up to 30%. Averaging of this data masks the true facts. That is why the correlation coefficient considerably decreases if the biological parameters are not averaged. On the other hand, the statistical reliability increases due to the higher number of investigated cases. Critical error discussion is necessary in statistical correlation between material and biological parameters. Often the results are heavily influenced by the statistical handling of data, especially if only few data are available. May be that new unconventional methods like bootstrap method can show a way out of this dilemma.     

Estimating material parameters of human skin in vivo

An accurate mathematical representation of the mechanical behaviour of human skin is essential when simulating deformations occurring in the skin during body movements or clinical procedures. In this study constitutive stress–strain relationships based on experimental data from human skin in vivo were obtained. A series of multiaxial loading experiments were performed on the forearms of four age- and gender matched subjects. The tissue geometry, together with recorded displacements and boundary forces, were combined in an analysis using finite element modelling. A non-linear optimization technique was developed to estimate values for the material parameters of a previously published constitutive law, describing the stress–strain relationship as a non-linear anisotropic membrane. Ten sets of material parameters where estimated from the experiments, showing considerable differences in mechanical behaviour both between individual subjects as well as mirrored body locations on a single subject. The accuracy of applications that simulate large deformations of human skin could be improved by using the parameters found from the in vivo experiments as described in this study.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

A mathematical and statistical framework for modelling dispersal

Mechanistic and phenomenological dispersal modelling of organisms has long been an area of intensive research. Recently, there has been an increased interest in intermediate models between the two. Intermediate models include major mechanisms that affect dispersal, in addition to the dispersal curve of a phenomenological model. Here we review and describe the mathematical and statistical framework for phenomenological dispersal modelling. In the mathematical development we describe modelling of dispersal in two dimensions from a point source, and in one dimension from a line or area source. In the statistical development we describe applicable observation distributions, and the procedures of model fitting, comparison, checking, and prediction. The procedures are also demonstrated using data from dispersal experiments. The data are hierarchically structured, and hence, we fit hierarchical models. The Bayesian modelling approach is applied, which allows us to show the uncertainty in the parameter estimates and in predictions. Finally, we show how to account for the effect of wind speed on the estimates of the dispersal parameters. This serves as an example of how to strengthen the coupling in the modelling between the phenomenon observed in an experiment and the underlying process – something that should be striven for in the statistical modelling of dispersal.     

Recombination frequency in plasmid DNA containing direct repeats--predictive correlation with repeat and intervening sequence length

In this study, a simple non-linear mathematical function is proposed to accurately predict recombination frequencies in bacterial plasmid DNA harbouring directly repeated sequences. The mathematical function, which was developed on the basis of published data on deletion-formation in multicopy plasmids containing direct-repeats (14-856 bp) and intervening sequences (0-3872 bp), also accounts for the strain genotype in terms of its recA function. A bootstrap resampling technique was used to estimate confidence intervals for the correlation parameters. More than 92% of the predicted values were found to be within a pre-established +/-5-fold interval of deviation from experimental data. The correlation does not only provide a way to predict, with good accuracy, the recombination frequency, but also opens the way to improve insight into these processes.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

The Routine Fitting of Kinetic Data to Models: A Mathematical Formalism for Digital Computers

A mathematical formalism is presented for use with digital computers to permit the routine fitting of data to physical and mathematical models. Given a set of data, the mathematical equations describing a model, initial conditions for an experiment, and initial estimates for the values of model parameters, the computer program automatically proceeds to obtain a least squares fit of the data by an iterative adjustment of the values of the parameters. When the experimental measures are linear combinations of functions, the linear coefficients for a least squares fit may also be calculated. The values of both the parameters of the model and the coefficients for the sum of functions may be unknown independent variables, unknown dependent variables, or known constants. In the case of dependence, only linear dependencies are provided for in routine use. The computer program includes a number of subroutines, each one of which performs a special task. This permits flexibility in choosing various types of solutions and procedures. One subroutine, for example, handles linear differential equations, another, special non-linear functions, etc. The use of analytic or numerical solutions of equations is possible.     

Associative clustering for exploring dependencies between functional genomics data sets

High-throughput genomic measurements, interpreted as cooccurring data samples from multiple sources, open up a fresh problem for machine learning: What is in common in the different data sets, that is, what kind of statistical dependencies are there between the paired samples from the different sets? We introduce a clustering algorithm for exploring the dependencies. Samples within each data set are grouped such that the dependencies between groups of different sets capture as much of pairwise dependencies between the samples as possible. We formalize this problem in a novel probabilistic way, as optimization of a Bayes factor. The method is applied to reveal commonalities and exceptions in gene expression between organisms and to suggest regulatory interactions in the form of dependencies between gene expression profiles and regulator binding patterns.     

Regional vegetation mapping in Australia: a case study in the practical use of statistical modelling

Conservation evaluation of large areas ( > 10 000 km²) in Australia requires detailed mapping of vegetation types. Predicting the original vegetation cover of extensive cleared areas in an explicit, consistent and repeatable manner necessitates the use of statistical modelling. This paper describes an integrated approach to vegetation mapping in a region of New South Wales, Australia. The approach uses separate statistical models for each tree and shrub species to predict the vegetation composition in each grid cell in a geographic information system (GIS). Allocation of these grid cells to communities allows communities that no longer exist in the remaining remnants of woodland to be defined. Examples of use of this information for management are presented. This paper addresses the practical considerations which constrain the way statistical modelling can be used for vegetation mapping in an applied project. Constraints include: (1) data availability (use of sampling to fill gaps in existing data), (2) the effects of cover abundance values, (3) availability of GIS predictors, (4) data management, (5) current generalised additive model methods and (6) prediction methods. Careful attention to the practicality of all components of a vegetation mapping study is essential if modern methods are to be applied in regional studies which must provide functional products for land managers with limited resources, skills and finances at their disposal.     

Determination of hydrophobicity of non-homologous series of anticancer drugs by reversed-phase high-performance liquid chromatography

The hydrophobiciy and specific hydrophobic surface area of 21 commercial anticancer drugs were determined by reversed-phase high-performance liquid chromatography on an octadecyl-silica column using methanol-water mixtures as eluents. Linear correlations were calculated between the log k′ values and the methanol concentration of the eluent, the intercept and slope were considered as the best estimation of the hydrophobicity and specific hydrophobic surface area. The relationship between retention characteristics and physicochemical parameters of drugs was evaluated by multivariate mathematical statistical methods, such as principal component analysis followed by two-dimensional non-linear mapping, varimax rotation and by cluster analysis. Anticancer drugs can be well separated by reversed-phase HPLC. Various multivariate mathematical statistical calculations indicate that the retention of the investigated drugs is mainly governed by hydrophobic and steric parameters. The results suggest that the use of principal component analysis followed by two-dimensional non-linear mapping is superior to cluster analysis for the evaluation of large retention data matrices.     

Material properties of the placenta under dynamic loading conditions

Trauma during pregnancy especially occurring during car crashes leads to many foetal losses. Numerical modelling is widely used in car occupant safety issue and injury mechanisms analysis and is particularly adapted to the pregnant woman. Material modelling of the gravid uterus tissues is crucial for injury risk evaluation especially for the abruption placentae which is widely assumed as the leading cause of foetal loss. Experimental studies on placenta behaviour in tension are reported in the literature, but none in compression to the authors' knowledge. This lack of data is addressed in this study. To complement the already available experimental literature data on the placenta mechanical behaviour and characterise it in a compression loading condition, 80 indentation tests on fresh placentae are presented. Hyperelastic like mean experimental stress versus strain and corridors are exposed. The results of the experimental placenta indentations compared with the tensile literature results tend to show a quasi-symmetrical behaviour of the tissue. An inverse analysis using simple finite element models has permitted to propose parameters for an Ogden material model for the placenta which exhibits a realistic behaviour in both tension and compression.     

A General Framework for Longitudinal Data through Marked Point Processes

This paper reviews a general framework for the modelling of longitudinal data with random measurement times based on marked point processes and presents a worked example. We construct a quite general regression models for longitudinal data, which may in particular include censoring that only depend on the past and outside random variation, and dependencies between measurement times and measurements. The modelling also generalises statistical counting process models. We review a non-parametric Nadarya-Watson kernel estimator of the regression function, and a parametric analysis that is based on a conditional least squares (CLS) criterion. The parametric analysis presented, is a conditional version of the generalised estimation equations of LIANG and ZEGER (1986). We conclude that the usual nonparametric and parametric regression modelling can be applied to this general set-up, with some modifications. The presented framework provides an easily implemented and powerful tool for model building for repeated measurements.     

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.     

Optimization and scale up of industrial fermentation processes

To increase product yields and to ensure consistent product quality, key issues of industrial fermentations, process optimization and scale up are aimed at maintaining optimum and homogenous reaction conditions minimizing microbial stress exposure and enhancing metabolic accuracy. For each individual product, process and facility, suitable strategies have to be elaborated by a comprehensive and detailed process characterization, identification of the most relevant process parameters influencing product yield and quality and their establishment as scale-up parameters to be kept constant as far as possible. Physical variables, which can only be restrictedly kept constant as single parameters, may be combined with other pertinent parameters to appropriate mathematical groups or dimensionless terms. Process characterization is preferably based on real-time or near real-time data collected by in situ and on-line measurements and may be facilitated by supportive approaches and tools like neural network based chemometric data analysis and modelling, clarification of the mixing and stream conditions through computational fluid dynamics and scale-down simulations. However, as fermentation facilities usually are not strictly designed according to scale-up criteria and the process conditions in the culture vessels thus may differ significantly and since any strategy and model can only insufficiently consider and reflect the highly complex interdependence and mutual interaction of fermentation parameters, successful scale up in most cases is not the result of a conclusive and straight-lined experimental strategy, but rather will be the outcome of a separate process development and optimization on each scale. This article gives an overview on the problems typically coming along with fermentation process optimization and scale up, and presents currently applied scale-up strategies while considering future technologies, with emphasis on Escherichia coli as one of the most commonly fermented organisms.     

Formulation,construction and analysis of kinetic models of metabolism: A review of modelling frameworks

Kinetic models are critical to predict the dynamic behaviour of metabolic networks. Mechanistic kinetic models for large networks remain uncommon due to the difficulty of fitting their parameters. Recent modelling frameworks promise new ways to overcome this obstacle while retaining predictive capabilities. In this review, we present an overview of the relevant mathematical frameworks for kinetic formulation, construction and analysis. Starting with kinetic formalisms, we next review statistical methods for parameter inference, as well as recent computational frameworks applied to the construction and analysis of kinetic models. Finally, we discuss opportunities and limitations hindering the development of larger kinetic reconstructions.     

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.     

Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Absence of desmin in skeletal muscle was found to induce an increase in passive stiffness. The present study aimed at developing rheological models of passive muscle to explain this stiffening. Models were elaborated by using experimental data depicting muscle viscoelastic behaviour. The experimental protocol included stepwise extension tests applied on control and desmin knockout soleus muscles from mice. Linear and non-linear models were composed of elastic and viscous elements. They were constructed with the aim at taking the presence or absence of desmin into account by simulating desmin as an elastic element. Furthermore, associated adaptation of connective tissues in absence of desmin was modelled as an additional elastic element. Differences in passive behaviour induced by absence of desmin were predicted by using a linear model and a non-linear one. The non-linear model was selected because: (1) it is able to predict experimental viscoelastic kinetics accounting for the increase in passive stiffness in muscles lacking desmin, (2) its design is consistent with morphological data, and (3) stiffness characteristics of its elements are in accordance with the literature. Finally, this modelling approach demonstrates that both absence of desmin and adaptation of connective tissue are required to explain the increase in passive stiffness in desmin knockout muscles.     

Summation of afterhyperpolarization after a doublet in fast motoneurones of the rat spinal cord

Motoneurones located in lumbar segments (L4-L5) of the rat spinal cord were activated antidromically by stimulation of the sciatic nerve. Records following a single pulse and double stimuli were collected from fast motoneurones and in each case voltage and time parameters of the action potential and afterhyperpolarization (AHP) were compared to respective values calculated for a simulated doublet - obtained by mathematical summation of two individual single potentials. The most prominent differences between experimentally recorded and simulated potentials concerned spike duration and the AHP half-decay times, with significantly lower values obtained for the simulated doublets. Tendencies for a higher AHP amplitude, shorter AHP peak time and higher AHP area were observed in mathematically modeled doublets, though no statistical differences were present in comparison to experimental data. Results indicated a non-linear summation of the AHP after double stimuli running in a short interval. These observations suggest that alterations in the generation of the AHP, mainly based on the sustained activity of K(Na+) channels, influence time-intervals in motoneuronal discharge pattern and this way force development of individual motor units during muscle activity.     

Statistical modelling of growth using a mixed model with orthogonal polynomials

In statistical modelling, the effects of single-nucleotide polymorphisms (SNPs) are often regarded as time-independent. However, for traits recorded repeatedly, it is very interesting to investigate the behaviour of gene effects over time. In the analysis, simulated data from the 13th QTL-MAS Workshop (Wageningen, The Netherlands, April 2009) was used and the major goal was the modelling of genetic effects as time-dependent. For this purpose, a mixed model which describes each effect using the third-order Legendre orthogonal polynomials, in order to account for the correlation between consecutive measurements, is fitted. In this model, SNPs are modelled as fixed, while the environment is modelled as random effects. The maximum likelihood estimates of model parameters are obtained by the expectation–maximisation (EM) algorithm and the significance of the additive SNP effects is based on the likelihood ratio test, with p-values corrected for multiple testing. For each significant SNP, the percentage of the total variance contributed by this SNP is calculated. Moreover, by using a model which simultaneously incorporates effects of all of the SNPs, the prediction of future yields is conducted. As a result, 179 from the total of 453 SNPs covering 16 out of 18 true quantitative trait loci (QTL) were selected. The correlation between predicted and true breeding values was 0.73 for the data set with all SNPs and 0.84 for the data set with selected SNPs. In conclusion, we showed that a longitudinal approach allows for estimating changes of the variance contributed by each SNP over time and demonstrated that, for prediction, the pre-selection of SNPs plays an important role.