Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Exact calculations of survival probability for diffusion on growing lines, disks, and spheres: The role of dimension

Unlike standard applications of transport theory, the transport of molecules and cells during embryonic development often takes place within growing multidimensional tissues. In this work, we consider a model of diffusion on uniformly growing lines, disks, and spheres. An exact solution of the partial differential equation governing the diffusion of a population of individuals on the growing domain is derived. Using this solution, we study the survival probability, S(t). For the standard nongrowing case with an absorbing boundary, we observe that S(t) decays to zero in the long time limit. In contrast, when the domain grows linearly or exponentially with time, we show that S(t) decays to a constant, positive value, indicating that a proportion of the diffusing substance remains on the growing domain indefinitely. Comparing S(t) for diffusion on lines, disks, and spheres indicates that there are minimal differences in S(t) in the limit of zero growth and minimal differences in S(t) in the limit of fast growth. In contrast, for intermediate growth rates, we observe modest differences in S(t) between different geometries. These differences can be quantified by evaluating the exact expressions derived and presented here.

Exact solutions of linear reaction-diffusion on a uniformly growing domain: criteria for successful colonization

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0growing domain. Comparing our exact solutions with numerical approximations confirms the veracity of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near x = 0, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the boundary where x = L(t).

Cells Tissues Organs : preface

This volume stems from the 1st International Conference on Epithelial-Mesenchymal Transitions (EMT), which was convened by the editors on October 5–8, 2003 in the beautiful setting of Port Douglas, Queensland, Australia. EMT, the name given to the transformation of cells arranged in a coherent layer – epithelial cells – to more individualistic and potential motile cells – mesenchymal cells – was recognized decades ago by Prof. Elisabeth (Betty) Hay (Harvard Medical School, Boston, Mass., USA) as a primary mechanism in embryogenesis for remodelling tissues. More recently EMT has been seen as crucial to the spread and invasion of carcinoma, and more recently still, EMT-like changes have been detected in various pathologies marked by fi brosis. Despite the basic and clinical importance of EMT, this extremely rapidly growing fi eld had never previously had a conference devoted to it, and indeed the disciplines of developmental biology, cancer and pathology rarely interact although they have much to share. The chapters assembled for this volume encompass these three major themes of the meeting, development, pathology and cancer, and further highlight the commonality in terms of mechanisms and outcomes among them...

A biomechanical analysis of growing rods used in the management of early onset scoliosis

INTRODUCTION Managing spinal deformities in young children is challenging, particularly early onset scoliosis (EOS). Surgical intervention is often required if EOS has been unresponsive to conservative treatment particularly with rapidly progressive curves. An emerging treatment option for EOS is fusionless scoliosis surgery. Similar to bracing, this surgical option potentially harnesses growth, motion and function of the spine along with correcting spinal deformity. Dual growing rods are one such fusionless treatment, which aims to modulate growth of the vertebrae. The aim of this study was to ascertain the extent to which semiconstrained growing rods (Medtronic, Sofamor, Danek, Memphis, TN) with a telescopic sleeve component, reduce rotational constraint on the spine compared with standard "constrained / rigid" rods and hence potentially provide a more physiological mechanical environment for the growing spine. METHODS Six 40-60kg English Large White porcine spines served as a model for the paediatric human spine. Each spine was dissected into a 7 level thoracolumbar multi-segment unit (MSU), removing all non-ligamentous soft tissues and leaving 3cm of ribs either side. Pure nondestructive axial rotation moments of ±4Nm at a constant rotation rate of 8deg.s-1 were applied to the mounted MSU spines using a biaxial Instron testing machine. Displacement of each vertebral level was captured using a 3D motion tracking system (Optotrak 3020, Northern Digital Inc, Waterloo, ON). Each spine was tested in an un-instrumented state first and then with appropriately sized semi-constrained growing rods and rigid rods in alternating sequence. The rods were secured by multi-axial pedicle screws (Medtronic CD Horizon) at levels 2 and 6 of the construct. The range of motion (ROM), neutral zone (NZ) size and stiffness (Nm.deg-1) were calculated from the Instron load-displacement data and intervertebral ROM was calculated through a MATLAB algorithm from Optotrak data. RESULTS Irrespective of the order of testing, rigid rods significantly reduced the total ROM compared with semi-constrained rods (p<0.05) with in a significantly stiffer spine for both left and right axial rotation (p<0.05). Analysing the intervertebral motion within the instrumented levels 2-6, rigid rods showed reduced ROM compared with semi-constrained growing rods and compared with un-instrumented motion segments. CONCLUSION Semi-constrained growing rods maintain similar stiffness in axial rotation to un-instrumented spines, while dual rigid rods significantly reduce axial rotation. Clinically the effect of semi-constrained growing rods as observed in this study is that they would be expected to allow growth via the telescopic rod components while maintaining the axial flexibility of the spine, which may reduce occurrence of the crankshaft phenomenon.

A biomechanical analysis of growing rods used in the management of early onset scoliosis

INTRODUCTION Managing spinal deformities in young children is challenging, particularly early-onset scoliosis (EOS). Any progressive spinal deformity particularly in early life presents significant health risks for the child and a challenge for the treating surgeon. Surgical intervention is often required if EOS has been unresponsive to conservative treatment particularly with rapidly progressive curves. An emerging treatment option particularly for EOS is fusionless scoliosis surgery. Similar to bracing this surgical option potentially harnesses growth, motion and function of the spine along with correcting spinal deformity. Dual growing rods is one such fusionless treatment, which aims to modulate growth of the vertebrae. The aim of this study was to ascertain the extent to which semi-constrained growing rods (Medtronic, Memphis, TN) with a telescopic sleeve component, reduce rotational constraint on the spine compared with standard rigid rods and hence potentially provide a more physiological mechanical environment for the growing spine. METHODS Six 40-60kg English Large White porcine spines served as a model for the paediatric human spine. Each spine was dissected into 7 level thoracolumbar multi-segment unit (MSU) spines, removing all non-ligamentous soft tissues. Appropriately sized semi-constrained growing rods and rigid rods were secured by multi-axial screws (Medtronic) prior to testing in alternating sequences for each spine. Pure nondestructive moments of +/4Nm at a constant rotation rate of 8deg/s was applied to the mounted MSU spines. Displacement of each level was captured using an Optotrak (Northern Digital Inc, Waterloo, ON). The range of motion (ROM), neutral zone (NZ) size and stiffness (Nm/deg) were calculated from the Instron load-displacement data and intervertebral ROM was calculated through a MATLAB algorithm from Optotrak data. RESULTS Irrespective of sequence order rigid rods significantly reduced the total ROM (deg) than compared to semi-constrained rods (p<0.05) and resulted in a significantly stiffer (Nm/deg) spine for both left and right axial rotation testing (p<0.05). Analysing the intervertebral motion within the instrumented levels, rigid rods showed reduced ROM (Deg) than compared to semi-constrained growing rods and the un-instrumented (UN-IN) test sequences. CONCLUSION The semi-constrained growing rods maintained rotation similar to UN-IN spines while the rigid rods showed significantly reduced axial rotation across all instrumented levels. Clinically the effect of semi-constrained growing rods evaluated in this study is that they will allow growth via the telescopic rod components while maintaining the axial rotation ability of the spine, which may also reduce the occurrence of the crankshaft phenomenon.

Percutaneous insertion of Dual Growth Rods using a new mobile bearing implant

Growth rods are commonly used for the treatment of scoliosis in the immature spine. Many variations have been proposed but breakage of implants is a common problem. Growth rod insertion commonly involves large exposures at initial insertion followed by multiple smaller procedures for lengthening. We present our early experiences using a percutaneous technique of insertion of a new titanium mobile bearing implant (Medtronic Inc). The implant allows some rotatory motion in the middle of the construct thus reducing construct stresses and thus possibly reducing rod breakage risk. Based on this small initial series with 12 months follow-up, percutaneous insertion of growth rods using the new implant is a safe and reliable technique although the infection rate in our sample was of note. This may be related to the titanium wear and inflammation seen in the soft tissues at time of operation and visualised on histology. No implants have required removal due to infection, and all infections were treated with debridement at next lengthening and suppressive antibiotics. Propionibacterium is one of the commonest infections seen with spinal implants and sometimes does not respond to simple antibiotic suppression. The technique allows preservation of the soft tissues until definitive fusion is needed and may lead to a decrease in hospital stay. The implant is low profile and seems to offer advantages over other systems on the market. Further follow up is needed to look at longer term outcomes with this new implant type.

Promoting sustainable urban development in fast growing city-regions : practices from Kuala Lumpur and Hong Kong

The Asia‐Pacific region is characterised by rapid population growth and urbanisation. These trends often result in an increasing consumption of land, which in turn lead to spatially expansive and discontinuous urban development. As a consequence, local communities and the environment face strong pressures. Many cities in the region have developed policies to tackle the issue of rapid growth and its associated consequences, for example climate change. The broad aim of this paper is to identify the nature, trends and strategies of growth management in major Asia‐Pacific city‐regions, and their implications for natural resource management and infrastructure provision. More specifically, this research seeks to provide insights on sustainable urban development practice, particularly on the promotion of compact urbanisation within the Asia‐Pacific’s fastest growing regions. The methodology of the paper includes a detailed literature review and a comparative analysis of existing strategies and policies. The literature review focuses on the key concepts related to sustainable urban growth management. It also includes existing applications of urban growth management approaches and planning information system in managing growth. Following the literature review, the paper undertakes a comparative analysis of the strategies of major Asia‐Pacific city‐regions of Kuala Lumpur and Hong Kong in terms of their approaches to sustainable urban development. The findings of the paper provide a clear understanding of the necessity of sustainable urban development practices. It contributes to the development of a substantial base for further research. Ultimately, this research aims to shed light on sustainable urban development by providing insights on the management of growth, natural resources and urban infrastructures.

Growing demand for emergency health services in Queensland, Australia

Introduction: The demand for emergency health services (EHS), both in the prehospital (ambulance) and hospital (emergency departments) settings, is growing rapidly in Australia. Broader health system changes have reduced available health infrastructure, particularly hospital beds, resulting in reduced access to and congestion of the EHS as demonstrated by longer waiting times and ambulance “ramping”. Ambulance ramping occurring when patients have a prolonged wait on the emergency vehicle due to the unavailability of hospital beds. This presentation will outline the trends in EHS demand in Queensland compared with the rest of Australia and factors that appear to be contributing to the growth in demand. Methods: Secondary analysis was conducted using data from publicly available sources. Data from the Queensland Ambulance Service and Queensland Health Emergency Department Information System (EDIS) also were analyzed. Results: The demand for ambulance services and emergency departments has been increasing at 8% and 4% per year over the last decade, respectively; while accessible hospital beds have reduced by almost 10% contributing to the emergency department congestion and possibly contributing to the prehospital demand. While the increase in the proportion of the elderly population seems to explain a great deal of the demand for EHS, other factors also influence this growth including patient characteristics, institutional and societal factors, economic, EHS arrangements, and clinical factors. Conclusions: Overcrowding of facilities that provide EHS are causing considerable community concern. This overcrowding is caused by the growing demand and reduced access. The causes of this growing demand are complex, and require further detailed analysis in order to quantify and qualify these causes in order to provide a resilient foundation of evidence for future policy direction.

Growing up

‘Growing Up’ is the key concept as well as the ideology for modernism. For modernism, ‘Growing Up’ has been regarded as ‘good’, ‘advance’, ‘power’ and ‘positive’; while, postmodernists pay attention to its negatives that it brings about abuse of natural resources, war, suppression of human rights, environmental pollution, and institutionalisation. The artwork, Growing Up illustrates the positive and negative aspects of ‘Growing Up’ by using three images of a flower, a bee and a devil. The flower represents flourish of modernism, the bee does prosperity (spread) of modernism, and a devil image generated from the flower (modernism) is negative aspects of modernism. The message of the artwork is that modernism itself determines its own destiny from ‘Growing Up’ that may jeopardise the bee.