Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Cell migration and proliferation on homogeneous and non-homogeneous domains : modelling on the scale of individuals and populations

Cell migration is a behaviour critical to many key biological effects, including wound healing, cancerous cell invasion and morphogenesis, the development of an organism from an embryo. However, given that each of these situations is distinctly different and cells are extremely complicated biological objects, interest lies in more basic experiments which seek to remove conflating factors and present a less complex environment within which cell migration can be experimentally examined. These include in vitro studies like the scratch assay or circle migration assay, and ex vivo studies like the colonisation of the hindgut by neural crest cells. The reduced complexity of these experiments also makes them much more enticing as problems to mathematically model, like done here. The primary goal of the mathematical models used in this thesis is to shed light on which cellular behaviours work to generate the travelling waves of invasion observed in these experiments, and to explore how variations in these behaviours can potentially predict differences in this invasive pattern which are experimentally observed when cell types or chemical environment are changed. Relevant literature has already identified the difficulty of distinguishing between these behaviours when using traditional mathematical biology techniques operating on a macroscopic scale, and so here a sophisticated individual-cell-level model, an extension of the Cellular Potts Model (CPM), is been constructed and used to model a scratch assay experiment. This model includes a novel mechanism for dealing with cell proliferations that allowed for the differing properties of quiescent and proliferative cells to be implemented into their behaviour. This model is considered both for its predictive power and used to make comparisons with the travelling waves which result in more traditional macroscopic simulations. These comparisons demonstrate a surprising amount of agreement between the two modelling frameworks, and suggest further novel modifications to the CPM that would allow it to better model cell migration. Considerations of the model’s behaviour are used to argue that the dominant effect governing cell migration (random motility or signal-driven taxis) likely depends on the sort of invasion demonstrated by cells, as easily seen by microscopic photography. Additionally, a scratch assay simulated on a non-homogeneous domain consisting of a ’fast’ and ’slow’ region is also used to further differentiate between these different potential cell motility behaviours. A heterogeneous domain is a novel situation which has not been considered mathematically in this context, nor has it been constructed experimentally to the best of the candidate’s knowledge. Thus this problem serves as a thought experiment used to test the conclusions arising from the simulations on homogeneous domains, and to suggest what might be observed should this non-homogeneous assay situation be experimentally realised. Non-intuitive cell invasion patterns are predicted for diffusely-invading cells which respond to a cell-consumed signal or nutrient, contrasted with rather expected behaviour in the case of random-motility-driven invasion. The potential experimental observation of these behaviours is demonstrated by the individual-cell-level model used in this thesis, which does agree with the PDE model in predicting these unexpected invasion patterns. In the interest of examining such a case of a non-homogeneous domain experimentally, some brief suggestion is made as to how this could be achieved.

Mid- and near-infrared spectroscopic investigation of homogeneous cation distribution in MgxZnyAl(x+y)/2-layered double hydroxide (LDH)

In this report, a detailed FTIR fitting analysis was used to recognize Mg, Zn and Al homogeneous distribution in MgxZnyAl(x+y)/2-Layered double hydroxide (LDH) hydroxyl layer. In detail, OH-Mg2Al:OH-Mg3 ratios decreased from 95.2:4.8 (MIR) and 94.2:5.8 (NIR) to 58.9:41.1 (MIR) and 61.8:38.2 (NIR), when Mg:Al increased from 2.2:1.0 to 4.1:1.0 in MgAl-LDHs. These fitting results were similar with theoretical calculations of 94.3:5.7 and 59.0:41.0. In a further analysis of MgxZnyAl(x+y)/2-LDHs, OH bonded Zn2Mg, Zn2Al, MgZnAl, Mg2Al and Mg2Zn peaks were identified at 3420, 3430, 3445–3450, 3454 and 3545 cm-1, respectively. With the decrease of Mg:Zn from 3:1 to 1:3, metal-hydroxyl bands changed from OH-Mg2Al and MgZnAl (with a ratio of 49.4:50.6) to OH-MgZnAl and Zn2Al (with a ratio of 55.0:45.0). They were also similar with theoretical calculations of 47.6:52.4 and 54.6:45.4. As a result, these results show that there is an ordered cation distribution in MgxZnyAl(x+y)/2-LDH, and FTIR is feasible in recognizing this structure.

Homogeneous bent functions of degree n in 2n variables do not exist for n>3

We prove that homogeneous bent functions f:GF(2)^2n --> GF(2) of degree n do not exist for n>3. Consequently homogeneous bent functions must have degree 3.

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.

One-step homogeneous C-reactive protein assay for saliva

Background: Cardiovascular disease is the leading cause of death in the world. Human C-reactive protein (CRP) has been used in the risk assessment of coronary events. Human saliva mirrors the body's health and well-being and is non-invasive, easy to collect and ideal for third world countries as well as for large patient screening. The aim was to establish a saliva CRP reference range and to demonstrate the clinical utility of salivary CRP levels in assessing the coronary events in a primary health care setting. Methods: We have used a homogeneous bead based assay to detect CRP levels in human saliva. We have developed a rapid 15 min (vs 90 min), sequential, one-step assay to detect CRP in saliva. Saliva was collected from healthy volunteers (n = 55, ages 20-70 years) as well as from cardiac patients (n = 28, ages 43-86 years). Results: The assay incubation time was optimised from 90 min to 15 mm and generated a positive correlation (n = 29, range 10-2189 pg/mL, r2 = 0.94; Passing Bablok slope 0.885. Intercept 0, p>0.10), meaning we could decrease the incubation time and produce equivalent results with confidence. The mean CRP level in the saliva of healthy human volunteers was 285 pg/mL and in cardiac patients was 1680 pg/mL (p<0.01). Analysis of CRP concentrations in paired serum and saliva samples from cardiac patients gave a positive correlation (r2 = 0.84, p<0.001) and the salivary CRP concentration capable of distinguishing healthy from diseased patients. Conclusions: The results suggest that this minimally invasive, rapid and sensitive assay will be useful in large patient screening studies for risk assessment of coronary events. (C) 2011 Elsevier B.V. All rights reserved.

NT-ProBNP levels in saliva and its clinical relevance to heart failure

Background: Current blood based diagnostic assays to detect heart failure (HF) have large intra-individual and inter-individual variations which have made it difficult to determine whether the changes in the analyte levels reflect an actual change in disease activity. Human saliva mirrors the body's health and well being and similar to 20% of proteins that are present in blood are also found in saliva. Saliva has numerous advantages over blood as a diagnostic fluid which allows for a non-invasive, simple, and safe sample collection. The aim of our study was to develop an immunoassay to detect NT-proBNP in saliva and to determine if there is a correlation with blood levels. Methods: Saliva samples were collected from healthy volunteers (n = 40) who had no underlying heart conditions and HF patients (n = 45) at rest. Samples were stored at -80 degrees C until analysis. A customised homogeneous sandwich AlphaLISA((R)) immunoassay was used to quantify NT-proBNP levels in saliva. Results: Our NT-proBNP immunoassay was validated against a commercial Roche assay on plasma samples collected from HF patients (n = 37) and the correlation was r(2) = 0.78 (p<0.01, y = 1.705 x +1910.8). The median salivary NT-proBNP levels in the healthy and HF participants were <16 pg/mL and 76.8 pg/mL, respectively. The salivary NT-proBNP immunoassay showed a clinical sensitivity of 82.2% and specificity of 100%, positive predictive value of 100% and negative predictive value of 83.3%, with an overall diagnostic accuracy of 90.6%. Conclusion: We have firstly demonstrated that NT-proBNP can be detected in saliva and that the levels were higher in heart failure patients compared with healthy control subjects. Further studies will be needed to demonstrate the clinical relevance of salivary NT-proBNP in unselected, previously undiagnosed populations.

A rapid and cost-effective method of producing recombinant proBNP and NT-proBNP variants in Escherichia coli for immunoassay of heart failure

The measurements of plasma natriuretic peptides (NT-proBNP, proBNP and BNP) are used to diagnose heart failure but these are expensive to produce. We describe a rapid, cheap and facile production of proteins for immunoassays of heart failure. DNA encoding N-terminally His-tagged NT-proBNP and proBNP were cloned into the pJexpress404 vector. ProBNP and NT-proBNP peptides were expressed in Escherichia coli, purified and refolded in vitro. The analytical performance of these peptides were comparable with commercial analytes (NT-proBNP EC50 for the recombinant is 2.6 ng/ml and for the commercial material is 5.3 ng/ml) and the EC50 for recombinant and commercial proBNP, are 3.6 and 5.7 ng/ml respectively). Total yield of purified refolded NT-proBNP peptide was 1.75 mg/l and proBNP was 0.088 mg/l. This approach may also be useful in expressing other protein analytes for immunoassay applications. To develop a cost effective protein expression method in E. coli to obtain high yields of NT-proBNP (1.75 mg/l) and proBNP (0.088 mg/l) peptides for immunoassay use.

Validation of an immunoassay to measure plasminogen-activator inhibitor-1 concentrations in human saliva

Introduction We have previously shown that the concentrations of D-dimer are significantly elevated in saliva compared with plasma. Saliva offers several advantages compared with blood analysis. We hypothesised that human saliva contains plasminogen activator inhibitor-1 (PAI-1) and that the concentrations are not affected by the time of saliva collection. The aim was to adopt and validate an immunoassay to quantify PAI-1 concentrations in saliva and to determine whether saliva collection time has an influence in the measurement. Materials and methods Two saliva samples (morning and afternoon) from the same day were collected from healthy subjects (N = 40) who have had no underlying heart conditions. A customized AlphaLISA® immunoassay (PerkinElmer®, MA, USA) was adopted and used to quantify PAI-1 concentrations. We validated the analytical performance of the customized immunoassay by calculating recovery of known amount of analyte spiked in saliva. Results: The recovery (95.03%), intra- (8.59%) and inter-assay (7.52%) variations were within the acceptable ranges. The median salivary PAI-1 concentrations were 394 pg/mL (interquartile ranges (IQR) 243.4-833.1 pg/mL) in the morning and 376 (129.1-615.4) pg/mL in the afternoon and the plasma concentration was 59,000 (24,000-110,000) pg/mL. Salivary PAI-1 did not correlate with plasma (P = 0.812). Conclusions The adopted immunoassay produced acceptable assay sensitivity and specificity. The data demonstrated that saliva contains PAI-1 and that its concentration is not affected by the time of saliva collection. There is no correlation between salivary and plasma PAI-1 concentrations. Further studies are required to demonstrate the utility of salivary PAI-1 in CVD risk factor studies.

A thermal expulsion approach to homogeneous large-volume methacrylate monolith preparation; enabling large-scale rapid purification of biomolecules

Numerous efforts have been dedicated to the synthesis of large-volume methacrylate monoliths for large-scale biomolecules purification but most were obstructed by the enormous release of exotherms during preparation, thereby introducing structural heterogeneity in the monolith pore system. A significant radial temperature gradient develops along the monolith thickness, reaching a terminal temperature that supersedes the maximum temperature required for structurally homogenous monoliths preparation. The enormous heat build-up is perceived to encompass the heat associated with initiator decomposition and the heat released from free radical-monomer and monomer-monomer interactions. The heat resulting from the initiator decomposition was expelled along with some gaseous fumes before commencing polymerization in a gradual addition fashion. Characteristics of 80 mL monolith prepared using this technique was compared with that of a similar monolith synthesized in a bulk polymerization mode. An extra similarity in the radial temperature profiles was observed for the monolith synthesized via the heat expulsion technique. A maximum radial temperature gradient of only 4.3°C was recorded at the center and 2.1°C at the monolith peripheral for the combined heat expulsion and gradual addition technique. The comparable radial temperature distributions obtained birthed identical pore size distributions at different radial points along the monolith thickness.

A homogeneous surface-enhanced Raman scattering platform for ultra-trace detection of trinitrotoluene in the environment

A facile and sensitive surface-enhanced Raman scattering substrate was prepared by controlled potentiostatic deposition of a closely packed single layer of gold nanostructures (AuNS) over a flat gold (pAu) platform. The nanometer scale inter-particle distance between the particles resulted in high population of ‘hot spots’ which enormously enhanced the scattered Raman photons. A renewed methodology was followed to precisely quantify the SERS substrate enhancement factor (SSEF) and it was estimated to be (2.2 ± 0.17) × 105. The reproducibility of the SERS signal acquired by the developed substrate was tested by establishing the relative standard deviation (RSD) of 150 repeated measurements from various locations on the substrate surface. A low RSD of 4.37 confirmed the homogeneity of the developed substrate. The sensitivity of pAu/AuNS was proven by determining 100 fM 2,4,6-trinitrotoluene (TNT) comfortably. As a proof of concept on the potential of the new pAu/AuNS substrate in field analysis, TNT in soil and water matrices was selectively detected after forming a Meisenheimer complex with cysteamine.