Probability distributed time delays: integrating spatial effects into temporal models

Background In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive. Results We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation. Conclusions Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.

The impact of nonlinear exposure risk relationships on seasonal time series data: modelling danish neonatal birth anthropometric data

Background Birth weight and length have seasonal fluctuations. Previous analyses of birth weight by latitude effects identified seemingly contradictory results, showing both 6 and 12 monthly periodicities in weight. The aims of this paper are twofold: (a) to explore seasonal patterns in a large, Danish Medical Birth Register, and (b) to explore models based on seasonal exposures and a non-linear exposure-risk relationship. Methods Birth weight and birth lengths on over 1.5 million Danish singleton, live births were examined for seasonality. We modelled seasonal patterns based on linear, U- and J-shaped exposure-risk relationships. We then added an extra layer of complexity by modelling weighted population-based exposure patterns. Results The Danish data showed clear seasonal fluctuations for both birth weight and birth length. A bimodal model best fits the data, however the amplitude of the 6 and 12 month peaks changed over time. In the modelling exercises, U- and J-shaped exposure-risk relationships generate time series with both 6 and 12 month periodicities. Changing the weightings of the population exposure risks result in unexpected properties. A J-shaped exposure-risk relationship with a diminishing population exposure over time fitted the observed seasonal pattern in the Danish birth weight data. Conclusion In keeping with many other studies, Danish birth anthropometric data show complex and shifting seasonal patterns. We speculate that annual periodicities with non-linear exposure-risk models may underlie these findings. Understanding the nature of seasonal fluctuations can help generate candidate exposures.

Higher-Order spectra of nonlinear polynomial models for Chua's circuit

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

Velocity-jump models with crowding effects

Velocity jump processes are discrete random walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity jump models are often used to represent a type of persistent motion, known as a “run and tumble”, which is exhibited by some isolated bacteria cells. All previous velocity jump processes are non-interacting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high cell density problems using a velocity jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.

Models of collective cell behaviour with crowding effects : comparing lattice-based and lattice-free approaches

Individual-based models describing the migration and proliferation of a population of cells frequently restrict the cells to a predefined lattice. An implicit assumption of this type of lattice based model is that a proliferative population will always eventually fill the lattice. Here we develop a new lattice-free individual-based model that incorporates cell-to-cell crowding effects. We also derive approximate mean-field descriptions for the lattice-free model in two special cases motivated by commonly used experimental setups. Lattice-free simulation results are compared to these mean-field descriptions and to a corresponding lattice-based model. Data from a proliferation experiment is used to estimate the parameters for the new model, including the cell proliferation rate, showing that the model fits the data well. An important aspect of the lattice-free model is that the confluent cell density is not predefined, as with lattice-based models, but an emergent model property. As a consequence of the more realistic, irregular configuration of cells in the lattice-free model, the population growth rate is much slower at high cell densities and the population cannot reach the same confluent density as an equivalent lattice-based model.

Development of zonal cartilage constructs : effects of chondrocyte subpopulation, compressive stimulation, and culture models

Articular cartilage is a highly resilient tissue located at the ends of long bones. It has a zonal structure, which has functional significance in load-bearing. Cartilage does not spontaneously heal itself when damaged, and untreated cartilage lesions or age-related wear often lead to osteoarthritis (OA). OA is a degenerative condition that is highly prevalent, age-associated, and significantly affects patient mobility and quality of life. There is no cure for OA, and patients usually resort to replacing the biological joint with an artificial prosthesis. An alternative approach is to dynamically regenerate damaged or diseased cartilage through cartilage tissue engineering, where cells, materials, and stimuli are combined to form new cartilage. However, despite extensive research, major limitations remain that have prevented the wide-spread application of tissue-engineered cartilage. Critically, there is a dearth of information on whether autologous chondrocytes obtained from OA patients can be used to successfully generate cartilage tissues with structural hierarchy typically found in normal articular cartilage. I aim to address these limitations in this thesis by showing that chondrocyte subpopulations isolated from macroscopically normal areas of the cartilage can be used to engineer stratified cartilage tissues and that compressive loading plays an important role in zone-dependent biosynthesis of these chondrocytes. I first demonstrate that chondrocyte subpopulations from the superficial (S) and middle/deep (MD) zones of OA cartilage are responsive to compressive stimulation in vitro, and that the effect of compression on construct quality is zone-dependent. I also show that compressive stimulation can influence pericelluar matrix production, matrix metalloproteinase secretion, and cytokine expression in zonal chondrocytes in an alginate hydrogel model. Subsequently, I focus on recreating the zonal structure by forming layered constructs using the alginate-released chondrocyte (ARC) method either with or without polymeric scaffolds. Resulting zonal ARC constructs had hyaline morphology, and expressed cartilage matrix molecules such as proteoglycans and collagen type II in both scaffold-free and scaffold-based approaches. Overall, my findings demonstrate that chondrocyte subpopulations obtained from OA joints respond sensitively to compressive stimulation, and are able to form cartilaginous constructs with stratified organization similar to native cartilage using the scaffold-free and scaffold-based ARC technique. The ultimate goal in tissue engineering is to help provide improved treatment options for patients suffering from debilitating conditions such as OA. Further investigations in developing functional cartilage replacement tissues using autologous chondrocytes will bring us a step closer to improving the quality of life for millions of OA patients worldwide.

Effects of mixed-method cooling on recovery of medium-fast bowling performance in hot conditions on consecutive days

This investigation examined physiological and performance effects of cooling on recovery of medium-fast bowlers in the heat. Eight, medium-fast bowlers completed two randomised trials, involving two sessions completed on consecutive days (Session 1: 10-overs and Session 2: 4-overs) in 31 ± 3°C and 55 ± 17% relative humidity. Recovery interventions were administered for 20 min (mixed-method cooling vs. control) after Session 1. Measures included bowling performance (ball speed, accuracy, run-up speeds), physical demands (global positioning system, counter-movement jump), physiological (heart rate, core temperature, skin temperature, sweat loss), biochemical (creatine kinase, C-reactive protein) and perceptual variables (perceived exertion, thermal sensation, muscle soreness). Mean ball speed was higher after cooling in Session 2 (118.9 ± 8.1 vs. 115.5 ± 8.6 km · h−1; P = 0.001; d = 0.67), reducing declines in ball speed between sessions (0.24 vs. −3.18 km · h−1; P = 0.03; d = 1.80). Large effects indicated higher accuracy in Session 2 after cooling (46.0 ± 11.2 vs. 39.4 ± 8.6 arbitrary units [AU]; P = 0.13; d = 0.93) without affecting total run-up speed (19.0 ± 3.1 vs. 19.0 ± 2.5 km · h−1; P = 0.97; d = 0.01). Cooling reduced core temperature, skin temperature and thermal sensation throughout the intervention (P = 0.001–0.05; d = 1.31–5.78) and attenuated creatine kinase (P = 0.04; d = 0.56) and muscle soreness at 24-h (P = 0.03; d = 2.05). Accordingly, mixed-method cooling can reduce thermal strain after a 10-over spell and improve markers of muscular damage and discomfort alongside maintained medium-fast bowling performance on consecutive days in hot conditions.

Towards Bayesian experimental design for nonlinear models that require a large number of sampling times

The use of Bayesian methodologies for solving optimal experimental design problems has increased. Many of these methods have been found to be computationally intensive for design problems that require a large number of design points. A simulation-based approach that can be used to solve optimal design problems in which one is interested in finding a large number of (near) optimal design points for a small number of design variables is presented. The approach involves the use of lower dimensional parameterisations that consist of a few design variables, which generate multiple design points. Using this approach, one simply has to search over a few design variables, rather than searching over a large number of optimal design points, thus providing substantial computational savings. The methodologies are demonstrated on four applications, including the selection of sampling times for pharmacokinetic and heat transfer studies, and involve nonlinear models. Several Bayesian design criteria are also compared and contrasted, as well as several different lower dimensional parameterisation schemes for generating the many design points.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

Exploring new and emerging models for nonlinear performative works

This dissertation seeks to define and classify potential forms of Nonlinear structure and explore the possibilities they afford for the creation of new musical works. It provides the first comprehensive framework for the discussion of Nonlinear structure in musical works and provides a detailed overview of the rise of nonlinearity in music during the 20th century. Nonlinear events are shown to emerge through significant parametrical discontinuity at the boundaries between regions of relatively strong internal cohesion. The dissertation situates Nonlinear structures in relation to linear structures and unstructured sonic phenomena and provides a means of evaluating Nonlinearity in a musical structure through the consideration of the degree to which the structure is integrated, contingent, compressible and determinate as a whole. It is proposed that Nonlinearity can be classified as a three dimensional space described by three continuums: the temporal continuum, encompassing sequential and multilinear forms of organization, the narrative continuum encompassing processual, game structure and developmental narrative forms and the referential continuum encompassing stylistic allusion, adaptation and quotation. The use of spectrograms of recorded musical works is proposed as a means of evaluating Nonlinearity in a musical work through the visual representation of parametrical divergence in pitch, duration, timbre and dynamic over time. Spectral and structural analysis of repertoire works is undertaken as part of an exploration of musical nonlinearity and the compositional and performative features that characterize it. The contribution of cultural, ideological, scientific and technological shifts to the emergence of Nonlinearity in music is discussed and a range of compositional factors that contributed to the emergence of musical Nonlinearity is examined. The evolution of notational innovations from the mobile score to the screen score is plotted and a novel framework for the discussion of these forms of musical transmission is proposed. A computer coordinated performative model is discussed, in which a computer synchronises screening of notational information, provides temporal coordination of the performers through click-tracks or similar methods and synchronises the audio processing and synthesized elements of the work. It is proposed that such a model constitutes a highly effective means of realizing complex Nonlinear structures. A creative folio comprising 29 original works that explore nonlinearity is presented, discussed and categorised utilising the proposed classifications. Spectrograms of these works are employed where appropriate to illustrate the instantiation of parametrically divergent substructures and examples of structural openness through multiple versioning.

Model-predictive control and closed-loop stability considerations for nonlinear plants described by local ARX-type models

In this paper, a model-predictive control (MPC) method is detailed for the control of nonlinear systems with stability considerations. It will be assumed that the plant is described by a local input/output ARX-type model, with the control potentially included in the premise variables, which enables the control of systems that are nonlinear in both the state and control input. Additionally, for the case of set point regulation, a suboptimal controller is derived which has the dual purpose of ensuring stability and enabling finite-iteration termination of the iterative procedure used to solve the nonlinear optimization problem that is used to determine the control signal.

The lag effects and vulnerabilities of temperature effects on cardiovascular disease mortality in a subtropical climate zone in China

This research quantifies the lag effects and vulnerabilities of temperature effects on cardiovascular disease in Changsha—a subtropical climate zone of China. A Poisson regression model within a distributed lag nonlinear models framework was used to examine the lag effects of cold- and heat-related CVD mortality. The lag effect for heat-related CVD mortality was just 0–3 days. In contrast, we observed a statistically significant association with 10–25 lag days for cold-related CVD mortality. Low temperatures with 0–2 lag days increased the mortality risk for those ≥65 years and females. For all ages, the cumulative effects of cold-related CVD mortality was 6.6% (95% CI: 5.2%–8.2%) for 30 lag days while that of heat-related CVD mortality was 4.9% (95% CI: 2.0%–7.9%) for 3 lag days. We found that in Changsha city, the lag effect of hot temperatures is short while the lag effect of cold temperatures is long. Females and older people were more sensitive to extreme hot and cold temperatures than males and younger people.

Nonlinear effects of ionization on surface waves on a plasma-metal interface

Nonlinear effects associated with density modulation caused by wave-induced ionization in magnetized plasmas were studied. The ionizing surface waves propagate at the interface between the plasma and a metallic surface. It is shown that the ionization nonlinearity can be important for typical experimental conditions.