Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Intrinsic kinetics of titania photocatalysis : simplified models for their investigation

Even though titanium dioxide photocatalysis has been promoted as a leading green technology for water purification, many issues have hindered its application on a large commercial scale. For the materials scientist the main issues have centred the synthesis of more efficient materials and the investigation of degradation mechanisms; whereas for the engineers the main issues have been the development of appropriate models and the evaluation of intrinsic kinetics parameters that allow the scale up or re-design of efficient large-scale photocatalytic reactors. In order to obtain intrinsic kinetics parameters the reaction must be analysed and modelled considering the influence of the radiation field, pollutant concentrations and fluid dynamics. In this way, the obtained kinetic parameters are independent of the reactor size and configuration and can be subsequently used for scale-up purposes or for the development of entirely new reactor designs. This work investigates the intrinsic kinetics of phenol degradation over titania film due to the practicality of a fixed film configuration over a slurry. A flat plate reactor was designed in order to be able to control reaction parameters that include the UV irradiance, flow rates, pollutant concentration and temperature. Particular attention was paid to the investigation of the radiation field over the reactive surface and to the issue of mass transfer limited reactions. The ability of different emission models to describe the radiation field was investigated and compared to actinometric measurements. The RAD-LSI model was found to give the best predictions over the conditions tested. Mass transfer issues often limit fixed film reactors. The influence of this phenomenon was investigated with specifically planned sets of benzoic acid experiments and with the adoption of the stagnant film model. The phenol mass transfer coefficient in the system was calculated to be km,phenol=8.5815x10-7Re0.65(ms-1). The data obtained from a wide range of experimental conditions, together with an appropriate model of the system, has enabled determination of intrinsic kinetic parameters. The experiments were performed in four different irradiation levels (70.7, 57.9, 37.1 and 20.4 W m-2) and combined with three different initial phenol concentrations (20, 40 and 80 ppm) to give a wide range of final pollutant conversions (from 22% to 85%). The simple model adopted was able to fit the wide range of conditions with only four kinetic parameters; two reaction rate constants (one for phenol and one for the family of intermediates) and their corresponding adsorption constants. The intrinsic kinetic parameters values were defined as kph = 0.5226 mmol m-1 s-1 W-1, kI = 0.120 mmol m-1 s-1 W-1, Kph = 8.5 x 10-4 m3 mmol-1 and KI = 2.2 x 10-3 m3 mmol-1. The flat plate reactor allowed the investigation of the reaction under two different light configurations; liquid and substrate side illumination. The latter of particular interest for real world applications where light absorption due to turbidity and pollutants contained in the water stream to be treated could represent a significant issue. The two light configurations allowed the investigation of the effects of film thickness and the determination of the catalyst optimal thickness. The experimental investigation confirmed the predictions of a porous medium model developed to investigate the influence of diffusion, advection and photocatalytic phenomena inside the porous titania film, with the optimal thickness value individuated at 5 ìm. The model used the intrinsic kinetic parameters obtained from the flat plate reactor to predict the influence of thickness and transport phenomena on the final observed phenol conversion without using any correction factor; the excellent match between predictions and experimental results provided further proof of the quality of the parameters obtained with the proposed method.

Evaluation of fibroin-based scaffolds for ocular tissue reconstruction

The epithelium of the corneolimbus contains stem cells for regenerating the corneal epithelium. Diseases and injuries affecting the limbus can lead to a condition known as limbal stem cell deficiency (LSCD), which results in loss of the corneal epithelium, and subsequent chronic inflammation and scarring of the ocular surface. Advances in the treatment of LSCD have been achieved through use of cultured human limbal epithelial (HLE) grafts to restore epithelial stem cells of the ocular surface. These epithelial grafts are usually produced by the ex vivo expansion of HLE cells on human donor amniotic membrane (AM), but this is not without limitations. Although AM is the most widely accepted substratum for HLE transplantation, donor variation, risk of disease transfer, and rising costs have led to the search for alternative biomaterials to improve the surgical outcome of LSCD. Recent studies have demonstrated that Bombyx mori silk fibroin (hereafter referred to as fibroin) membranes support the growth of primary HLE cells, and thus this thesis aims to explore the possibility of using fibroin as a biomaterial for ocular surface reconstruction. Optimistically, the grafted sheets of cultured epithelium would provide a replenishing source of epithelial progenitor cells for maintaining the corneal epithelium, however, the HLE cells lose their progenitor cell characteristics once removed from their niche. More severe ocular surface injuries, which result in stromal scarring, damage the epithelial stem cell niche, which subsequently leads to poor corneal re-epithelialisation post-grafting. An ideal solution to repairing the corneal limbus would therefore be to grow and transplant HLE cells on a biomaterial that also provides a means for replacing underlying stromal cells required to better simulate the normal stem cell niche. The recent discovery of limbal mesenchymal stromal cells (L-MSC) provides a possibility for stromal repair and regeneration, and therefore, this thesis presents the use of fibroin as a possible biomaterial to support a three dimensional tissue engineered corneolimbus with both an HLE and underlying L-MSC layer. Investigation into optimal scaffold design is necessary, including adequate separation of epithelial and stromal layers, as well as direct cell-cell contact. Firstly, the attachment, morphology and phenotype of HLE cells grown on fibroin were directly compared to that observed on donor AM, the current clinical standard substrate for HLE transplantation. The production, transparency, and permeability of fibroin membranes were also evaluated in this part of the study. Results revealed that fibroin membranes could be routinely produced using a custom-made film casting table and were found to be transparent and permeable. Attachment of HLE cells to fibroin after 4 hours in serum-free medium was similar to that supported by tissue culture plastic but approximately 6-fold less than that observed on AM. While HLE cultured on AM displayed superior stratification, epithelia constructed from HLE on fibroin maintained evidence of corneal phenotype (cytokeratin pair 3/12 expression; CK3/12) and displayed a comparable number and distribution of ÄNp63+ progenitor cells to that seen in cultures grown on AM. These results confirm the suitability of membranes constructed from silk fibroin as a possible substrate for HLE cultivation. One of the most important aspects in corneolimbal tissue engineering is to consider the reconstruction of the limbal stem cell niche to help form the natural limbus in situ. MSC with similar properties to bone marrow derived-MSC (BM-MSC) have recently been grown from the limbus of the human cornea. This thesis evaluated methods for culturing L-MSC and limbal keratocytes using various serum-free media. The phenotype of resulting cultures was examined using photography, flow cytometry for CD34 (keratocyte marker), CD45 (bone marrow-derived cell marker), CD73, CD90, CD105 (collectively MSC markers), CD141 (epithelial/vascular endothelial marker), and CD271 (neuronal marker), immunocytochemistry (alpha-smooth muscle actin; á-sma), differentiation assays (osteogenesis, adipogenesis and chrondrogenesis), and co-culture experiments with HLE cells. While all techniques supported to varying degrees establishment of keratocyte and L-MSC cultures, sustained growth and serial propagation was only achieved in serum-supplemented medium or the MesenCult-XF„¥ culture system (Stem Cell Technologies). Cultures established in MesenCult-XF„¥ grew faster than those grown in serum-supplemented medium and retained a more optimal MSC phenotype. L-MSC cultivated in MesenCult-XFR were also positive for CD141, rarely expressed £\-sma, and displayed multi-potency. L-MSC supported growth of HLE cells, with the largest epithelial islands being observed in the presence of L-MSC established in MesenCult-XF„¥ medium. All HLE cultures supported by L-MSC widely expressed the progenitor cell marker £GNp63, along with the corneal differentiation marker CK3/12. Our findings conclude that MesenCult-XFR is a superior culture system for L-MSC, but further studies are required to explore the significance of CD141 expression in these cells. Following on from the findings of the previous two parts, silk fibroin was tested as a novel dual-layer construct containing both an epithelium and underlying stroma for corneolimbal reconstruction. In this section, the growth and phenotype of HLE cells on non-porous versus porous fibroin membranes was compared. Furthermore, the growth of L-MSC in either serum-supplemented medium or the MesenCult-XFR culture system within fibroin fibrous mats was investigated. Lastly, the co-culture of HLE and L-MSC in serum-supplemented medium on and within fibroin dual-layer constructs was also examined. HLE on porous membranes displayed a flattened and squamous monolayer; in contrast, HLE on non-porous fibroin appeared cuboidal and stratified closer in appearance to a normal corneal epithelium. Both constructs maintained CK3/12 expression and distribution of £GNp63+ progenitor cells. Dual-layer fibroin scaffolds consisting of HLE cells and L-MSC maintained a similar phenotype as on the single layers alone. Overall, the present study proposed to create a three dimensional limbal tissue substitute of HLE cells and L-MSC together, ultimately for safe and beneficial transplantation back into the human eye. The results show that HLE and L-MSC can be cultivated separately and together whilst maintaining a clinically feasible phenotype containing a majority of progenitor cells. In addition, L-MSC were able to be cultivated routinely in the MesenCult-XF® culture system while maintaining a high purity for the MSC characteristic phenotype. However, as a serum-free culture medium was not found to sustain growth of both HLE and L-MSC, the combination scaffold was created in serum-supplemented medium, indicating that further refinement of this cultured limbal scaffold is required. This thesis has also demonstrated a potential novel marker for L-MSC, and has generated knowledge which may impact on the understanding of stromal-epithelial interactions. These results support the feasibility of a dual-layer tissue engineered corneolimbus constructed from silk fibroin, and warrant further studies into the potential benefits it offers to corneolimbal tissue regeneration. Further refinement of this technology should explore the potential benefits of using epithelial-stromal co-cultures with MesenCult-XF® derived L-MSC. Subsequent investigations into the effects of long-term culture on the phenotype and behaviour of the cells in the dual-layer scaffolds are also required. While this project demonstrated the feasibility in vitro for the production of a dual-layer tissue engineered corneolimbus, further studies are required to test the efficacy of the limbal scaffold in vivo. Future in vivo studies are essential to fully understand the integration and degradation of silk fibroin biomaterials in the cornea over time. Subsequent experiments should also investigate the use of both AM and silk fibroin with epithelial and stromal cell co-cultures in an animal model of LSCD. The outcomes of this project have provided a foundation for research into corneolimbal reconstruction using biomaterials and offer a stepping stone for future studies into corneolimbal tissue engineering.

ACE research vignette 018 : experiments in the MBA classroom : the case of time frames in project teams

This series of research vignettes is aimed at sharing current and interesting research findings from our team of international Entrepeneurship researchers. In this vignette, Dr Rene Bakker considers project team dynamics and how executive education can be enriched by studying them in the classroom.

Scenes, quarters and clusters : new experiments in the formation and governance of creative places in China

This thesis examines the formation and governance patterns of the social and spatial concentration of creative people and creative business in cities. It develops a typology for creative places, adding the terms 'scene' and 'quarter' to 'clusters', to fill in the literature gap of partial emphasis on the 'creative clusters' model as an organising mechanism for regional and urban policy. In this framework, a cluster is the gathering of firms with a core focus on economic benefits; a quarter is the urban milieu usually driven by a growth coalition consisting of local government, real estate agents and residential communities; and a scene is the spontaneous assembly of artists, performers and fans with distinct cultural forms. The framework is applied to China, specifically to Hangzhou – a second-tier city in central eastern China that is ambitious to become a 'national cultural and creative industries centre'. The thesis selects three cases – Ideal & Silian 166 Creative Industries Park, White-horse Lake Eco-creative City and LOFT49 Creative Industries Park – to represent scene, quarter and cluster respectively. Drawing on in-depth interviews with initiators, managers and creative professionals of these places, together with extensive documentary analysis, the thesis investigates the composition of actors, characteristics of the locality and the diversity of activities. The findings illustrate that, in China, planning and government intervention is the key to the governance of creative space; spontaneous development processes exist, but these need a more tolerant environment, a greater diversity of cultural forms and more time to develop. Moreover, the main business development model is still real estate based: this model needs to incorporate more mature business models and an enhanced IP protection system. The thesis makes a contribution to literature on economic and cultural geography, urban planning and creative industries theory. It advocates greater attention to self-management, collaborative governance mechanisms and business strategies for scenes, quarters and clusters. As intersections exist in the terms discussed, a mixed toolkit of the three models is required to advance the creative city discourse in China.

Removal of toxic metals in a multi metal system using sorbents for potential application to urban stormwater treatment

In the context of increasing demand for potable water and the depletion of water resources, stormwater is a logical alternative. However, stormwater contains pollutants, among which metals are of particular interest due to their toxicity and persistence in the environment. Hence, it is imperative to remove toxic metals in stormwater to the levels prescribed by drinking water guidelines for potable use. Consequently, various techniques have been proposed, among which sorption using low cost sorbents is economically viable and environmentally benign in comparison to other techniques. However, sorbents show affinity towards certain toxic metals, which results in poor removal of other toxic metals. It was hypothesised in this study that a mixture of sorbents that have different metal affinity patterns can be used for the efficient removal of a range of toxic metals commonly found in stormwater. The performance of six sorbents in the sorption of Al, Cr, Cu, Pb, Ni, Zn and Cd, which are the toxic metals commonly found in urban stormwater, was investigated to select suitable sorbents for creating the mixtures. For this purpose, a multi criteria analytical protocol was developed using the decision making methods: PROMETHEE (Preference Ranking Organisation METHod for Enrichment Evaluations) and GAIA (Graphical Analysis for Interactive Assistance). Zeolite and seaweed were selected for the creation of trial mixtures based on their metal affinity pattern and the performance on predetermined selection criteria. The metal sorption mechanisms employed by seaweed and zeolite were defined using kinetics, isotherm and thermodynamics parameters, which were determined using the batch sorption experiments. Additionally, the kinetics rate-limiting steps were identified using an innovative approach using GAIA and Spearman correlation techniques developed as part of the study, to overcome the limitation in conventional graphical methods in predicting the degree of contribution of each kinetics step in limiting the overall metal removal rate. The sorption kinetics of zeolite was found to be primarily limited by intraparticle diffusion followed by the sorption reaction steps, which were governed mainly by the hydrated ionic diameter of metals. The isotherm study indicated that the metal sorption mechanism of zeolite was primarily of a physical nature. The thermodynamics study confirmed that the energetically favourable nature of sorption increased in the order of Zn < Cu < Cd < Ni < Pb < Cr < Al, which is in agreement with metal sorption affinity of zeolite. Hence, sorption thermodynamics has an influence on the metal sorption affinity of zeolite. On the other hand, the primary kinetics rate-limiting step of seaweed was the sorption reaction process followed by intraparticle diffusion. The boundary layer diffusion was also found to limit the metal sorption kinetics at low concentration. According to the sorption isotherm study, Cd, Pb, Cr and Al were sorbed by seaweed via ion exchange, whilst sorption of Ni occurred via physisorption. Furthermore, ionic bonding is responsible for the sorption of Zn. The thermodynamics study confirmed that sorption by seaweed was energetically favourable in the order of Zn < Cu < Cd < Cr . Al < Pb < Ni. However, this did not agree with the affinity series derived for seaweed suggesting a limited influence of sorption thermodynamics on metal affinity for seaweed. The investigation of zeolite-seaweed mixtures indicated that mixing sorbents have an effect on the kinetics rates and the sorption affinity. Additionally, the theoretical relationships were derived to predict the boundary layer diffusion rate, intraparticle diffusion rate, the sorption reaction rate and the enthalpy of mixtures based on that of individual sorbents. In general, low coefficient of determination (R2) for the relationships between theoretical and experimental data indicated that the relationships were not statistically significant. This was attributed to the heterogeneity of the properties of sorbents. Nevertheless, in relative terms, the intraparticle diffusion rate, sorption reaction rate and enthalpy of sorption had higher R2 values than the boundary layer diffusion rate suggesting that there was some relationship between the former set of parameters of mixtures and that of sorbents. The mixture, which contained 80% of zeolite and 20% of seaweed, showed similar affinity for the sorption of Cu, Ni, Cd, Cr and Al, which was attributed to approximately similar sorption enthalpy of the metal ions. Therefore, it was concluded that the seaweed-zeolite mixture can be used to obtain the same affinity for various metals present in a multi metal system provided the metal ions have similar enthalpy during sorption by the mixture.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.

Langevin dynamics modeling of the water diffusion tensor in partially aligned collagen networks

In this work, a Langevin dynamics model of the diffusion of water in articular cartilage was developed. Numerical simulations of the translational dynamics of water molecules and their interaction with collagen fibers were used to study the quantitative relationship between the organization of the collagen fiber network and the diffusion tensor of water in model cartilage. Langevin dynamics was used to simulate water diffusion in both ordered and partially disordered cartilage models. In addition, an analytical approach was developed to estimate the diffusion tensor for a network comprising a given distribution of fiber orientations. The key findings are that (1) an approximately linear relationship was observed between collagen volume fraction and the fractional anisotropy of the diffusion tensor in fiber networks of a given degree of alignment, (2) for any given fiber volume fraction, fractional anisotropy follows a fiber alignment dependency similar to the square of the second Legendre polynomial of cos(θ), with the minimum anisotropy occurring at approximately the magic angle (θMA), and (3) a decrease in the principal eigenvalue and an increase in the transverse eigenvalues is observed as the fiber orientation angle θ progresses from 0◦ to 90◦. The corresponding diffusion ellipsoids are prolate for θ < θMA, spherical for θ ≈ θMA, and oblate for θ > θMA. Expansion of the model to include discrimination between the combined effects of alignment disorder and collagen fiber volume fraction on the diffusion tensor is discussed.

Evaluation of methods for cultivating limbal mesenchymal stromal cells

Background: Mesenchymal stromal cells (MSC) with similar properties to bone marrow derived mesenchymal stromal cells (BM-MSC) have recently been grown from the limbus of the human cornea. We presently contribute to this novel area of research by evaluating methods for culturing human limbal MSC (L-MSC). Methods: Four basic strategies are compared: serum-supplemented medium (10% foetal bovine serum; FBS), standard serum-free medium supplemented with B-27, epidermal growth factor, and fibroblast growth factor 2, or one of two commercial serum-free media including Defined Keratinocyte Serum Free Medium (Invitrogen), and MesenCult-XF (Stem Cell Technologies). The phenotype of resulting cultures was examined using photography, flow cytometry (for CD34, CD45, CD73, CD90, CD105, CD141, CD271), immunocytochemistry (α-sma), differentiation assays (osteogenesis, adipogenesis, chrondrogenesis), and co-culture experiments with human limbal epithelial (HLE) cells. Results: While all techniques supported to varying degrees establishment of cultures, sustained growth and serial propagation was only achieved in 10% FBS medium or MesenCult-XF medium. Cultures established in 10% FBS medium were 70-80% CD34-/CD45-/CD90+/CD73+/CD105+, approximately 25% α-sma+, and displayed multi-potency. Cultures established in MesenCult-XF were >95% CD34-/CD45-/CD90+/CD73+/CD105+, 40% CD141+, rarely expressed α-sma, and displayed multi-potency. L-MSC supported growth of HLE cells, with the largest epithelial islands being observed in the presence of MesenCult-XF-grown L-MSC. All HLE cultures supported by L-MSC widely expressed the progenitor cell marker ∆Np63, along with the corneal differentiation marker cytokeratin 3. Conclusions: We conclude that MesenCult-XF® is a superior culture system for L-MSC, but further studies are required to explore the significance of CD141 expression in these cells.

Analysis of genome survey sequences and SSR marker development for Siamese Mud Carp, Henicorhynchus siamensis, using 454 Pyrosequencing

Siamese mud carp (Henichorynchus siamensis) is a freshwater teleost of high economic importance in the Mekong River Basin. However, genetic data relevant for delineating wild stocks for management purposes currently are limited for this species. Here, we used 454 pyrosequencing to generate a partial genome survey sequence (GSS) dataset to develop simple sequence repeat (SSR) markers from H. siamensis genomic DNA. Data generated included a total of 65,954 sequence reads with average length of 264 nucleotides, of which 2.79% contain SSR motifs. Based on GSS-BLASTx results, 10.5% of contigs and 8.1% singletons possessed significant similarity (E value < 10–5) with the majority matching well to reported fish sequences. KEGG analysis identified several metabolic pathways that provide insights into specific potential roles and functions of sequences involved in molecular processes in H. siamensis. Top protein domains detected included reverse transcriptase and the top putative functional transcript identified was an ORF2-encoded protein. One thousand eight hundred and thirty seven sequences containing SSR motifs were identified, of which 422 qualified for primer design and eight polymorphic loci have been tested with average observed and expected heterozygosity estimated at 0.75 and 0.83, respectively. Regardless of their relative levels of polymorphism and heterozygosity, microsatellite loci developed here are suitable for further population genetic studies in H. siamensis and may also be applicable to other related taxa.