An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

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.

Measuring mechanical properties of tendon in vivo

Introduction: Understanding the mechanical properties of tendon is an important step to guiding the process of improving athletic performance, predicting injury and treating tendinopathies. The speed of sound in a medium is governed by the bulk modulus and density for fluids and isotropic materials. However, for tendon,which is a structural composite of fluid and collagen, there is some anisotropy requiring an adjustment for Poisson’s ratio. In this paper, these relationships are explored and modelled using data collected, in vivo, on human Achilles tendon. Estimates for elastic modulus and hysteresis based on speed of sound data are then compared against published values from in vitro mechanical tests. Methods: Measurements using clinical ultrasound imaging, inverse dynamics and acoustic transmission techniques were used to determine dimensions, loading conditions and longitudinal speed of sound for the Achilles tendon during a series of isometric plantar flexion exercises against body weight. Upper and lower bounds for speed of sound versus tensile stress in the tendon were then modelled and estimates derived for elastic modulus and hysteresis. Results: Axial speed of sound varied between 1850 to 2090 m.s−1 with a non-linear, asymptotic dependency on the level of tensile stress in the tendon 5–35 MPa. Estimates derived for the elastic modulus ranged between 1–2 GPa. Hysteresis derived from models of the stress-strain relationship, ranged from 3–11%. These values agree closely with those previously reported from direct measurements obtained via in vitro mechanical tensile tests on major weight bearing tendons. Discussion: There is sufficiently good agreement between these indirect (speed of sound derived) and direct (mechanical tensile test derived) measures of tendon mechanical properties to validate the use of this non-invasive acoustic transmission technique. This non-invasive method is suitable for monitoring changes in tendon properties as predictors of athletic performance, injury or therapeutic progression.

The molecular structure of the phosphate mineral väyrynenite : a vibrational spectroscopic study

We have studied the mineral väyrynenite from the Viitaniemi pegmatite, located in the Eräjärvi area, Finland using a combination of electron microscopy electron microprobe and vibrational spectroscopic techniques. Chemical analysis shows the formula of the mineral to be (Mn0.88,Fe0.08,Mg0.01)∑0.97Be1.02(PO4)1.00(OH)1.02. Vibrational spectroscopy enables an assessment of the molecular structure of väyrynenite to be assessed. An intense Raman band at 1004 cm−1 is to the ν1 symmetric stretching mode. The observation of multiple bands in the phosphate stretching region, offers support for the concept of different phosphate units in the väyrynenite structure. Infrared spectroscopy confirms this multiplicity of vibrational bands. Multiple bands are observed in the phosphate bending region.

Vibrational spectroscopy of the mineral meyerhofferite CaB3O3(OH)5·H2O – An assessment of the molecular structure

Meyerhofferite is a calcium hydrated borate mineral with ideal formula: CaB3O3(OH)5�H2O and occurs as white complex acicular to crude crystals with length up to �4 cm, in fibrous divergent, radiating aggregates or reticulated and is often found in sedimentary or lake-bed borate deposits. The Raman spectrum of meyerhofferite is dominated by intense sharp band at 880 cm�1 assigned to the symmetric stretching mode of trigonal boron. Broad Raman bands at 1046, 1110, 1135 and 1201 cm�1 are attributed to BOH in-plane bending modes. Raman bands in the 900–1000 cm�1 spectral region are assigned to the antisymmetric stretching of tetrahedral boron. Distinct OH stretching Raman bands are observed at 3400, 3483 and 3608 cm�1. The mineral meyerhofferite has a distinct Raman spectrum which is different from the spectrum of other borate minerals, making Raman spectroscopy a very useful tool for the detection of meyerhofferite in sedimentary and lake bed deposits.

Vibrational spectroscopy of the phosphate mineral kovdorskite – Mg2PO4(OH)⋅3H2O

The mineral kovdorskite Mg2PO4(OH)�3H2O was studied by electron microscopy, thermal analysis and vibrational spectroscopy. A comparison of the vibrational spectroscopy of kovdorskite is made with other magnesium bearing phosphate minerals and compounds. Electron probe analysis proves the mineral is very pure. The Raman spectrum is characterized by a band at 965 cm�1 attributed to the PO3� 4 m1 symmetric stretching mode. Raman bands at 1057 and 1089 cm�1 are attributed to the PO3�4 m3 antisymmetric stretching modes. Raman bands at 412, 454 and 485 cm�1 are assigned to the PO3�4 m2 bending modes. Raman bands at 536, 546 and 574 cm�1 are assigned to the PO3�4 m4 bending modes. The Raman spectrum in the OH stretching region is dominated by a very sharp intense band at 3681 cm�1 assigned to the stretching vibration of OH units. Infrared bands observed at 2762, 2977, 3204, 3275 and 3394 cm�1 are attributed to water stretching bands. Vibrational spectroscopy shows that no carbonate bands are observed in the spectra; thus confirming the formula of the mineral as Mg2PO4(OH)�3H2O.

The molecular structure of the phosphate mineral senegalite Al2(PO4)(OH)3⋅3H2O – A vibrational spectroscopic study

We have studied the mineral senagalite, a hydrated hydroxy phosphate of aluminium with formula Al2(PO4)(OH)3⋅3H2O using a combination of electron microscopy and vibrational spectroscopy. Senegalite crystal aggregates shows tabular to prismatic habitus and orthorhombic form. The Raman spectrum is dominated by an intense band at 1029 cm−1 assigned to the PO43- ν1 symmetric stretching mode. Intense Raman bands are found at 1071 and 1154 cm−1 with bands of lesser intensity at 1110, 1179 and 1206 cm−1 and are attributed to the PO43- ν3 antisymmetric stretching vibrations. The infrared spectrum shows complexity with a series overlapping bands. A comparison is made with spectra of other aluminium containing phosphate minerals such as augelite and turquoise. Multiple bands are observed for the phosphate bending modes giving support for the reduction of symmetry of the phosphate anion. Vibrational spectroscopy offers a means for the assessment of the structure of senagalite.

Vibrational spectroscopy of the phosphate mineral lazulite – (Mg, Fe)Al2(PO4)2·(OH)2 found in the Minas Gerais, Brazil

This research was done on lazulite samples from the Gentil mine, a lithium bearing pegmatite located in the municipality of Mendes Pimentel, Minas Gerais, Brazil. Chemical analysis was carried out by electron microprobe analysis and indicated a magnesium rich phase with partial substitution of iron. Traces of Ca and Mn, (which partially replaced Mg) were found. The calculated chemical formula of the studied sample is: (Mg0.88, Fe0.11)Al1.87(PO4)2.08(OH)2.02. The Raman spectrum of lazulite is dominated by an intense sharp band at 1060 cm-1 assigned to PO stretching vibrations of of tetrahedral [PO4] clusters presents into the HPO2/4- units. Two Raman bands at 1102 and 1137 cm-1 are attributed to both the HOP and PO antisymmetric stretching vibrations. The two infrared bands at 997 and 1007 cm-1 are attributed to the m1 PO3/4- symmetric stretching modes. The intense bands at 1035, 1054, 1081, 1118 and 1154 cm-1 are assigned to the v3PO3/4- antisymmetric stretching modes from both the HOP and tetrahedral [PO4] clusters. A set of Raman bands at 605, 613, 633 and 648 cm-1 are assigned to the m4 out of plane bending modes of the PO4, HPO4 and H2PO4 units. Raman bands observed at 414, 425, 460, and 479 cm-1 are attributed to the m2 tetrahedral PO4 clusters, HPO4 and H2PO4 bending modes. The intense Raman band at 3402 and the infrared band at 3403 cm-1 are assigned to the stretching vibration of the OH units. A combination of Raman and infrared spectroscopy enabled aspects of the molecular structure of the mineral lazulite to be understood.

How long does it take for aquifer recharge or aquifer discharge processes to reach steady state?

Groundwater flow models are usually characterized as being either transient flow models or steady state flow models. Given that steady state groundwater flow conditions arise as a long time asymptotic limit of a particular transient response, it is natural for us to seek a finite estimate of the amount of time required for a particular transient flow problem to effectively reach steady state. Here, we introduce the concept of mean action time (MAT) to address a fundamental question: How long does it take for a groundwater recharge process or discharge processes to effectively reach steady state? This concept relies on identifying a cumulative distribution function, $F(t;x)$, which varies from $F(0;x)=0$ to $F(t;x) \to \infty$ as $t\to \infty$, thereby providing us with a measurement of the progress of the system towards steady state. The MAT corresponds to the mean of the associated probability density function $f(t;x) = \dfrac{dF}{dt}$, and we demonstrate that this framework provides useful analytical insight by explicitly showing how the MAT depends on the parameters in the model and the geometry of the problem. Additional theoretical results relating to the variance of $f(t;x)$, known as the variance of action time (VAT), are also presented. To test our theoretical predictions we include measurements from a laboratory–scale experiment describing flow through a homogeneous porous medium. The laboratory data confirms that the theoretical MAT predictions are in good agreement with measurements from the physical model.

Metabolomics coupled with proteomics advancing drug discovery toward more agile development of targeted combination therapies

To enhance the therapeutic efficacy and reduce the adverse effects of traditional Chinese medicine, practitioners often prescribe combinations of plant species and/or minerals, called formulae. Unfortunately, the working mechanisms of most of these compounds are difficult to determine and thus remain unknown. In an attempt to address the benefits of formulae based on current biomedical approaches, we analyzed the components of Yinchenhao Tang, a classical formula that has been shown to be clinically effective for treating hepatic injury syndrome. The three principal components of Yinchenhao Tang are Artemisia annua L., Gardenia jasminoids Ellis, and Rheum Palmatum L., whose major active ingredients are 6,7-dimethylesculetin (D), geniposide (G), and rhein (R), respectively. To determine the mechanisms underlying the efficacy of this formula, we conducted a systematic analysis of the therapeutic effects of the DGR compound using immunohistochemistry, biochemistry, metabolomics, and proteomics. Here, we report that the DGR combination exerts a more robust therapeutic effect than any one or two of the three individual compounds by hitting multiple targets in a rat model of hepatic injury. Thus, DGR synergistically causes intensified dynamic changes in metabolic biomarkers, regulates molecular networks through target proteins, has a synergistic/additive effect, and activates both intrinsic and extrinsic pathways.

Thyroxine and reserpine-induced changes in metabolic profiles of rat urine and the therapeutic effect of Liu Wei Di Huang Wan detected by UPLC-HDMS

The promise of metabonomics, a new "omics" technique, to validate Chinese medicines and the compatibility of Chinese formulas has been appreciated. The present study was undertaken to explore the excretion pattern of low molecular mass metabolites in the male Wistar-derived rat model of kidney yin deficiency induced with thyroxine and reserpine as well as the therapeutic effect of Liu Wei Di Huang Wan (LW) and its separated prescriptions, a classic traditional Chinese medicine formula for treating kidney yin deficiency in China. The study utilized ultra-performance liquid chromatography/electrospray ionization synapt high definition mass spectrometry (UPLC/ESI-SYNAPT-HDMS) in both negative and positive electrospray ionization (ESI). At the same time, blood biochemistry was examined to identify specific changes in the kidney yin deficiency. Distinct changes in the pattern of metabolites, as a result of daily administration of thyroxine and reserpine, were observed by UPLC-HDMS combined with a principal component analysis (PCA). The changes in metabolic profiling were restored to their baseline values after treatment with LW according to the PCA score plots. Altogether, the current metabonomic approach based on UPLC-HDMS and orthogonal projection to latent structures discriminate analysis (OPLS-DA) indicated 20 ions (14 in the negative mode, 8 in the positive mode, and 2 in both) as "differentiating metabolites".

Metabolic urinary profiling of alcohol hepatotoxicity and intervention effects of Yin Chen Hao Tang in rats using ultra-performance liquid chromatography/electrospray ionization quadruple time-of-flight mass spectrometry

This paper was designed to study metabonomic characters of the hepatotoxicity induced by alcohol and the intervention effects of Yin Chen Hao Tang (YCHT), a classic traditional Chinese medicine formula for treatment of jaundice and liver disorders in China. Urinary samples from control, alcohol- and YCHT-treated rats were analyzed by ultra-performance liquid chromatography/electrospray ionization quadruple time-of-flight mass spectrometry (UPLC/ESI-QTOF-MS) in positive ionization mode. The total ion chromatograms obtained from the control, alcohol- and YCHT-treated rats were easily distinguishable using a multivariate statistical analysis method such as the principal components analysis (PCA). The greatest difference in metabolic profiling was observed from alcohol-treated rats compared with the control and YCHT-treated rats. The positive ions m/z 664.3126 (9.00 min) was elevated in urine of alcohol-treated rats, whereas, ions m/z 155.3547 (10.96 min) and 708.2932 (9.01 min) were at a lower concentration compared with that in urine of control rats, however, these ions did not indicate a statistical difference between control rats and YCHT-treated rats. The ion m/z 664.3126 was found to correspond to ceramide (d18:1/25:0), providing further support for an involvement of the sphingomyelin signaling pathway in alcohol hepatotoxicity and the intervention effects of YCHT.

Simultaneous determination of 6,7-dimethylesculetin and geniposide in rat plasma and its application to pharmacokinetic studies of Yin Chen Hao Tang preparation

A method for the rapid and simultaneous determination of 6,7-dimethylesculetin (CAS 120-08-1) and geniposide (CAS 24512-63-8) in rat plasma has been developed, using validated high performance liquid chromatography (HPLC) with solid phase extraction (SPE). The HPLC analysis was performed on a commercially available column (200 mm x 4.6 mm, 5 microm) with acetonitrile-methanol-0.1% aqueous formic acid as mobile phase and the UV detection at 343 nm and 238 nm for 6,7-dimethylesculetin and geniposide, respectively. The calibration curves for 6,7-dimethylesculetin and geniposide were linear over the range 0.4-25.6 microg/mL and 1.12-71.68 microg/mL, respectively. The lower limits of quantitation were 0.40 microg/ mL and 1.12 microg/mL, and the lower limits of detection were 0.06 microg/mL and 0.09 microg/ mL, respectively. The intra-day and inter-day precision for 6,7-dimethylesculetin and geniposide were < 5%, whereas the absolute recovery percentages were > 74%. A successful application of the developed HPLC analysis was demonstrated for the pharmacokinetic study of a Traditional Chinese Medicine formula of Yin Chen Hao Tang preparation.

Analysis of the constituents in the rat plasma after oral administration of Yin Chen Hao Tang by UPLC/Q-TOF-MS/MS

A UPLC/Q-TOF-MS/MS method for analyzing the constituents in rat plasma after oral administration of Yin Chen Hao Tang (YCHT), a traditional Chinese medical formula, has been established. The UPLC/MS fingerprints of the samples were established first in vitro and in vivo, with 45 compounds in YCHT and 21 compounds in rat plasma after oral administration of YCHT were detected. Of the 45 detected compounds in vitro, 30 were identified, and all of the 21 compounds detected in rat plasma were identified either by comparing the retention time and mass spectrometry data with that of reference compounds or by mass spectrometry analysis and retrieving the reference literatures. Of the identified 21 compounds in rat plasma, 19 were the original form of compounds absorbed from the 45 detected compounds in vitro, 2 were the metabolites of the compounds existed in YCHT. It is concluded that a rapid and validated method has been developed based on UPLC-MS/MS, which shows high sensitivity and resolution that is more suitable for identifying the bioactive constituents in plasma after oral administration of Chinese herbal medicines, and provides helpful chemical information for further pharmacology and active mechanism research on the Chinese medical formula.

Infrared and Raman spectroscopic characterization of the carbonate mineral huanghoite - and in comparison with selected rare earth carbonates

Raman spectroscopy complimented with infrared spectroscopy has been used to study the rare earth based mineral huanghoite with possible formula given as BaCe(CO3)2F and compared with the Raman spectra of a series of selected natural halogenated carbonates from different origins including bastnasite, parisite and northupite. The Raman spectrum of huanghoite displays three bands are at 1072, 1084 and 1091 cm−1 attributed to the symmetric stretching vibration. The observation of three symmetric stretching vibrations is very unusual. The position of symmetric stretching vibration varies with mineral composition. Infrared spectroscopy of huanghoite show bands at 1319, 1382, 1422 and 1470 cm−1. No Raman bands of huanghoite were observed in these positions. Raman spectra of bastnasite, parisite and northupite show a single band at 1433, 1420 and 1554 cm−1 assigned to the ν3 (CO3)2− antisymmetric stretching mode. The observation of additional Raman bands for the ν3 modes for some halogenated carbonates is significant in that it shows distortion of the carbonate anion in the mineral structure. Four Raman bands for huanghoite are observed at 687, 704, 718 and 730 cm−1and assigned to the (CO3)2− ν2 bending modes. Raman bands are observed for huanghoite at around 627 cm−1 and are assigned to the (CO3)2− ν4 bending modes. Raman bands are observed for the carbonate ν4 in phase bending modes at 722 cm−1 for bastnasite, 736 and 684 cm−1 for parisite, 714 cm−1 for northupite. Raman bands for huanghoite observed at 3259, 3484 and 3589 cm−1 are attributed to water stretching bands. Multiple bands are observed in the OH stretching region for bastnasite and parisite indicating the presence of water and OH units in their mineral structure. Vibrational spectroscopy enables new information on the structure of huanghoite to be assessed.