In this paper, we develop a comprehensive mathematical model to describe the phosphorylation of glucose by the enzyme hexokinase I. Glucose phosphorylation is the first step of the glycolytic ...pathway, and as such, it is carefully regulated in cells. Hexokinase I phosphorylates glucose to produce glucose-6-phosphate, and the cell regulates the phosphorylation rate by inhibiting the action of this enzyme. The cell uses three inhibitory processes to regulate the enzyme: an allosteric product inhibitory process, a competitive product inhibitory process, and a competitive inhibitory process. Surprisingly, the cellular regulation of hexokinase I is not yet fully resolved, and so, in this study, we developed a detailed mathematical model to help unpack the behaviour. Numerical simulations of the model produced results that were consistent with the experimentally determined behaviour of hexokinase I. In addition, the simulations provided biological insights into the abstruse enzymatic behaviour, such as the dependence of the phosphorylation rate on the concentration of inorganic phosphate or the concentration of the product glucose-6-phosphate. A global sensitivity analysis of the model was implemented to help identify the key mechanisms of hexokinase I regulation. The sensitivity analysis also enabled the development of a simpler model that produced an output that was very close to that of the full model. Finally, the potential utility of the model in assisting experimental studies is briefly indicated.
Voltage controlled dielectric membranes exhibit two fundamental types of instability, strongly affecting their performances: the occurrence of wrinkling, which is due to membranal compressive ...stresses, and the onset of pull-in, a catastrophic thinning localisation that preludes electrical breakdown. In this manuscript we provide a unifying energetic description of both instabilities for large, out-of-plane and inhomogeneous deformations. By using the ideas of relaxation and regularisation of the energy, originally proposed by Pipkin (1986) and Hilgers and Pipkin (1992) for purely elastic membranes, we show that the onset and development of wrinkling can be effectively described by the relaxed electroelastic energy. For axially symmetric membranes and neo-Hookean materials, we show that pull-in corresponds to failure of the strong ellipticity condition of the regularised electroelastic energy, thus extending to out-of-plane deformations the validity of a previous estimate for planar systems (Zurlo et al., 2017). In agreement with ubiquitous experimental evidence, we also show that wrinkled states are always stable below the pull-in voltage. Our theoretical findings are assessed by the comparison with experiments on out-of-plane, voltage-actuated annular membranes, showing good agreement both in terms of description of wrinkled states, and for the prediction of the pull-in instability.
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Much work has been devoted to analysing thermodynamic models for solid dispersions with a view to identifying regions in the phase diagram where amorphous phase separation or drug ...recrystallization can occur. However, detailed partial differential equation non-equilibrium models that track the evolution of solid dispersions in time and space are lacking. Hence theoretical predictions for the timescale over which phase separation occurs in a solid dispersion are not available. In this paper, we address some of these deficiencies by (i) constructing a general multicomponent diffusion model for a dissolving solid dispersion; (ii) specializing the model to a binary drug/polymer system in storage; (iii) deriving an effective concentration dependent drug diffusion coefficient for the binary system, thereby obtaining a theoretical prediction for the timescale over which phase separation occurs; (iv) calculating the phase diagram for the Felodipine/HPMCAS system; and (iv) presenting a detailed numerical investigation of the Felodipine/HPMCAS system assuming a Flory-Huggins activity coefficient. The numerical simulations exhibit numerous interesting phenomena, such as the formation of polymer droplets and strings, Ostwald ripening/coarsening, phase inversion, and droplet-to-string transitions. A numerical simulation of the fabrication process for a solid dispersion in a hot melt extruder is also presented.
Solid dispersions are products that contain mixtures of drug and other materials e.g. polymer. These are liable to separate-out over time – a phenomenon known as phase separation. This means that it is possible the product differs both compositionally and structurally between the time of manufacture and the time it is taken by the patient, leading to poor bioavailability and so ultimately the shelf-life of the product has to be reduced.
Theoretical predictions for the timescale over which phase separation occurs are not currently available. Also lacking are detailed partial differential equation non-equilibrium models that track the evolution of solid dispersions in time and space. This study addresses these issues, before presenting a detailed investigation of a particular drug-polymer system.
We consider a mathematical model that describes the leakage of heparin-binding growth factors from an affinity-based delivery system. In the delivery system, heparin binds to a peptide which has been ...covalently cross-linked to a fibrin matrix. Growth factor in turn binds to the heparin, and growth factor release is governed by both binding and diffusion mechanisms, the purpose of the binding being to slow growth factor release. The governing mathematical model, which in its original formulation consists of six partial differential equations, is reduced to a system of just two equations. It is usually desirable that there be no passive release of growth factor from a device, with all of the growth factor being held in place via binding until such time as it is actively released by invading cells. However, there will inevitably be some passive release, and so it is of interest to identify conditions that will make this release as slow as possible. In this paper, we identify a parameter regime that ensures that at least a fraction of the growth factor will release slowly. It is found that slow release is assured if the matrix is prepared with the concentration of cross-linked peptide greatly exceeding the dissociation constant of heparin from the peptide, and with the concentration of heparin greatly exceeding the dissociation constant of the growth factor from heparin. Also, for the first time, in vitro experimental release data are directly compared with the theoretical release profiles generated by the model. We propose that the two stage release behaviour frequently seen in experiments is due to an initial rapid out-diffusion of free growth factor over a diffusion time scale (typically days), followed by a much slower release of the bound fraction over a time scale depending on both diffusion and binding parameters (frequently months).
A controlled drug delivery system fabricated from a thermoresponsive polymer was designed to obtain a pulsatile release profile which was triggered by altering the temperature of the dissolution ...medium. Two stages of release behaviour were found: fast release for a swollen state and slow (yet significant and non-negligible) release for a collapsed state. Six cycles of pulsatile release between 4°C and 40°C were obtained. The dosage of drug (rhodamine B) released in these cycles could be controlled to deliver approximately equal doses by altering the release time in the swollen state. However, for the first cycle, the swollen release rate was found to be large, and the release time could not be made short enough to prevent a larger dose than desired being delivered. A model was developed based on Fick's law which describes pulsatile release mathematically for the first time, and diffusion coefficients at different temperatures (including temperatures corresponding to both the fully swollen and collapsed states) were estimated by fitting the experimental data with the theoretical release profile given by this model. The effect of temperature on the diffusion coefficient was studied and it was found that in the range of the lower critical solution temperature (LCST), the diffusion coefficient increased with decreasing temperature. The model predicts that the effective lifetime of the system lies in the approximate range of 1–42h (95% of drug released), depending on how long the system was kept at low temperature (below the LCST). Therefore this system can be used to obtain a controllable pulsatile release profile for small molecule drugs thereby enabling optimum therapeutic effects.
A mathematical model for the release of peptide-binding drugs from affinity hydrogels is analyzed in detail. The model is not specific to any particular peptide/drug/gel system, and can describe drug ...release from a large class of affinity systems. In many cases, it is shown that the model can be reduced to a coupled pair of nonlinear partial differential equations for the total drug and peptide. Quantitative information relating the rate of drug release to the values of the model parameters is presented. Numerical solutions are displayed that illustrate the rich variety of release behaviors the system is capable of exhibiting. Theoretical release profiles generated by the model are compared with experimental release data from three different studies, and good agreement is found. The development of reliable mathematical models for affinity hydrogels will provide useful design tools for these systems.
When carrying out any numerical modelling it is useful to have an analytical approximation available to provide a check on the accuracy of the numerical results and to give insight into the ...underlying physics of the system. The numerical modelling of wave energy converters is an efficient and inexpensive method of undertaking initial optimisation and experimentation. Therefore, the main objective of this paper is to determine an analytical approximation for the wave excitation forces on a floating truncated vertical cylinder in water of infinite depth. The approximation is developed by solving appropriate boundary value problems using the method of separation of variables. A graphical representation of the analytical approximation for the truncated vertical cylinder and the cylinder of infinite depth are presented and are compared to the results from a computational fluid dynamics analysis, using a commercial boundary element package. The presented analytical approximation and the computational fluid dynamics analysis results were found to be in good agreement. Furthermore, the presented analytical approximation was found to be in good agreement with independent experimental data.
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Traditional coronary drug-eluting stents (DES) are made from metal and are coated with a permanent polymer film containing an anti-proliferative drug. Subsequent to stent deployment ...in a diseased coronary artery, the drug releases into the artery wall and helps prevent restenosis by inhibiting the proliferation of smooth muscle cells. Although this technology has proven to be remarkably successful, there are ongoing concerns that the presence of a polymer in the artery can lead to deleterious medical complications, such as late stent thrombosis. Polymer-free DES may help overcome such shortcomings. However, the absence of a rate-controlling polymer layer makes optimisation of the drug release profile a particular challenge. The use of microporous stent surfaces to modulate the drug release rate is an approach that has recently shown particularly promising clinical results. In this study, we develop a mathematical model to describe drug release from such stents. In particular, we develop a mathematical model to describe drug release from microporous surfaces. The model predicts a two-stage release profile, with a relatively rapid initial release of most of the drug, followed by a slower release of the remaining drug. In the model, the slow release phase is accounted for by an adsorption/desorption mechanism close to the stent surface. The theoretical predictions are compared with experimental release data obtained in our laboratory, and good agreement is found. The valuable insights provided by our model will serve as a useful guide for designing the enhanced polymer-free stents of the future.
•We develop a model describing the degradation of hyaluronan by SpnHL.•The model parameters are estimated with the aid of literature data.•Good agreement between theory and experiment is ...found.•Literature data suggests that the model may be applicable to other enzymes.•The key model parameters are identified using a global sensitivity analysis.
Hyaluronic acid (Hyaluronan) is a linear, high molecular weight polysaccharide that forms an important component of the extracellular matrix. It is an excellent biomaterial, and it is increasingly being used in biotechnology, biomedical applications, and drug delivery. Polymer chains of hyaluronan occur in many different lengths in nature, and can be as large as multiples of ten thousand. Since the biological function of a hyaluronan chain often depends on its molecular weight, it is of value for applications to develop reliable quantitative descriptions of the degradation processes of hyaluronan. In particular, the development of such models should assist with the rational design of production processes to create polymer chains in a given molecular weight category for a specific application. In this paper, we propose a new mathematical model for the degradation of hyaluronan by the enzyme streptococcus pneumoniae hyaluronate lyase. The model is based on a processive kinetic mechanism and consists of a coupled system of nonlinear ordinary differential equations for the species of interest. The model parameters are estimated using published experimental data, and good agreement between theory and experiment is found. Numerical experimentation and a Sobol global sensitivity analysis reveal that the key model parameters are the initial enzyme concentration and the rate constants for enzyme adsorption and catalysis.
Abstract
This paper is concerned with a diffusion-controlled moving boundary problem in drug dissolution, in which the moving front passes from one medium to another for which the diffusivity is many ...orders of magnitude smaller. The classical Neumann similarity solution holds while the front is passing through the first layer, but this breaks down in the second layer. Asymptotic methods are used to understand what is happening in the second layer. Although this necessitates numerical computation, one interesting outcome is that only one calculation is required, no matter what the diffusivity is for the second layer.
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