Effective stability of the Trojan asteroids

We study the spatial circular restricted problem of three bodies in the light of Nekhoroshev theory of stability over large time intervals. We consider in particular the Sun-Jupiter model and the Trojan asteroids in the neighborhood of the Lagrangian point $L_4$. We find a region of effective stability around the point $L_4$ such that if the initial point of an orbit is inside this region the orbit is confined in a slightly larger neighborhood of the equilibrium (in phase space) for a very long time interval. By combining analytical methods and numerical approximations we are able to prove that stability over the age of the universe is guaranteed on a realistic region, big enough to include one real asteroid. By comparing this result with the one obtained for the planar problem we see that the regions of stability in the two cases are of the same magnitude.Comment: 9 pages, 2 figures, Astronomy & Astrophysics in pres

Fractional compartmental models and multi-term Mittag–Leffler response functions

Systems of fractional differential equations (SFDE) have been increasingly used to represent physical and control system, and have been recently proposed for use in pharmacokinetics (PK) by (J Pharmacokinet Pharmacodyn 36:165–178, 2009) and (J Phamacokinet Pharmacodyn, 2010). We contribute to the development of a theory for the use of SFDE in PK by, first, further clarifying the nature of systems of FDE, and in particular point out the distinction and properties of commensurate versus non-commensurate ones. The second purpose is to show that for both types of systems, relatively simple response functions can be derived which satisfy the requirements to represent single-input/single-output PK experiments. The response functions are composed of sums of single- (for commensurate) or two-parameters (for non-commensurate) Mittag–Leffler functions, and establish a direct correspondence with the familiar sums of exponentials used in PK

Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method

The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile. The solution was performed with Riemann -Liouville time-fractional derivative since the integral approach avoids the definition of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed through the M-Write function. The solutions demonstrate some basic issues of the suggested integral approach, among them: a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical results allowing comparisons to the known solutions expressed to the M-Write function and error estimations.Comment: 15 pages, 7 figures, 3 table

Priprava i evaluacija mukoadhezivnih filmova glipizida

Glipizide is mainly absorbed in the proximal areas of the gastrointestinal tract. The purpose of this study was formulation and evaluation of mucoadhesive films to prolong the stay of drug in its absorption area. Glipizide was formulated in a mucoadhesive film that could be retained in the stomach for prolonged time intervals. Polymeric films were designed with various compositions of hydroxypropyl cellulose and polyethylene glycol 400 (PEG 400). Properties of the mucoadhesive film such as tensile strength, percentage elongation, swelling index, moisture content, pH and viscosity of polymeric dispersion, film thickness, drug concentration, uniformity and mucoadhesion in a simulated gastric environment were evaluated. In addition, percentage drug retained in stomach mucosa was estimated using a simulated dynamic stomach system as a function of time. Increase in hydroxypropyl cellulose concentration resulted in a higher tensile strength and elongation at break, while increase in concentration of PEG 400 was reflected in a decrease of tensile strength and increase of elongation at break. Glipizide/hydroxypropyl cellulose/PEG 400 (2.5:1:0.5) (GF5) was found to be the optimal composition for a novel mucoadhesive stomach formulation that showed good peelability, relatively high swelling index, moderate tensile strength, and stayed on rat stomach mucosa up to 8 h. In vivo testing of the mucoadhesive films with glipizide demonstrated a potential hypoglycemic effect.Glipizid se pretežno apsorbira u proksimalnom dijelu gastrointestinalnog trakta. Cilj rada je priprava i evaluacija mukoadhezivnih filmova s kojima bi se produljilo zadržavanje lijeka u predjelu apsorpcije. Pripravljeni su mukoadhezivni filmovi glipizida koji se produljeno zadržavaju u želucu. Polimerni filmovi sadržavali su različite količine hidroksipropil celuloze i polietilen glikola 400 (PEG 400). Evaluirana su sljedeća svojstva mukoadhezivnih filmova: čvrstoća, postotak elongacije, indeks bubrenja, sadržaj vlage, pH i viskoznost polimerne disperzije, debljina filma, koncentracija lijeka, jednolikost i mukoadhezivnost u simuliranom želučanom soku. Na dinamičkom modelu želuca određivan je i postotak lijeka koji se zadržava u sluznici želuca u ovisnosti o vremenu. Povećanjem koncentracije hidroksipropil celuloze povećavaju se čvrstoća i elongacija, dok se povećanje koncentracije PEG 400 reflektira na smanjenje čvrstoće i povećanje elongacije kod loma. Omjer glipizid/hidroksipropil celuloza/PEG 400 (2,5:1:0,5) (GF5) bio je optimalan za pripravu mukoadhezivnih formulacija, s dobrom kalavošću, relativno visokim indeksom bubrenja, umjerenom čvrstoćom te zadržavanjem u sluznici želuca štakora do 8 h. U in vivo testiranjima mukoadhesivni filmovi s glipizidom pokazali su potencijalni hipoglikemijski učinak

Fractional dynamics pharmacokinetics–pharmacodynamic models

While an increasing number of fractional order integrals and differential equations applications have been reported in the physics, signal processing, engineering and bioengineering literatures, little attention has been paid to this class of models in the pharmacokinetics–pharmacodynamic (PKPD) literature. One of the reasons is computational: while the analytical solution of fractional differential equations is available in special cases, it this turns out that even the simplest PKPD models that can be constructed using fractional calculus do not allow an analytical solution. In this paper, we first introduce new families of PKPD models incorporating fractional order integrals and differential equations, and, second, exemplify and investigate their qualitative behavior. The families represent extensions of frequently used PK link and PD direct and indirect action models, using the tools of fractional calculus. In addition the PD models can be a function of a variable, the active drug, which can smoothly transition from concentration to exposure, to hyper-exposure, according to a fractional integral transformation. To investigate the behavior of the models we propose, we implement numerical algorithms for fractional integration and for the numerical solution of a system of fractional differential equations. For simplicity, in our investigation we concentrate on the pharmacodynamic side of the models, assuming standard (integer order) pharmacokinetics

A combined model reduction algorithm for controlled biochemical systems

Background: Systems Biology continues to produce increasingly large models of complex biochemical reaction networks. In applications requiring, for example, parameter estimation, the use of agent-based modelling approaches, or real-time simulation, this growing model complexity can present a significant hurdle. Often, however, not all portions of a model are of equal interest in a given setting. In such situations methods of model reduction offer one possible approach for addressing the issue of complexity by seeking to eliminate those portions of a pathway that can be shown to have the least effect upon the properties of interest. Methods: In this paper a model reduction algorithm bringing together the complementary aspects of proper lumping and empirical balanced truncation is presented. Additional contributions include the development of a criterion for the selection of state-variable elimination via conservation analysis and use of an ‘averaged’ lumping inverse. This combined algorithm is highly automatable and of particular applicability in the context of ‘controlled’ biochemical networks. Results: The algorithm is demonstrated here via application to two examples; an 11 dimensional model of bacterial chemotaxis in Escherichia coli and a 99 dimensional model of extracellular regulatory kinase activation (ERK) mediated via the epidermal growth factor (EGF) and nerve growth factor (NGF) receptor pathways. In the case of the chemotaxis model the algorithm was able to reduce the model to 2 state-variables producing a maximal relative error between the dynamics of the original and reduced models of only 2.8% whilst yielding a 26 fold speed up in simulation time. For the ERK activation model the algorithm was able to reduce the system to 7 state-variables, incurring a maximal relative error of 4.8%, and producing an approximately 10 fold speed up in the rate of simulation. Indices of controllability and observability are additionally developed and demonstrated throughout the paper. These provide insight into the relative importance of individual reactants in mediating a biochemical system’s input-output response even for highly complex networks. Conclusions: Through application, this paper demonstrates that combined model reduction methods can produce a significant simplification of complex Systems Biology models whilst retaining a high degree of predictive accuracy. In particular, it is shown that by combining the methods of proper lumping and empirical balanced truncation it is often possible to produce more accurate reductions than can be obtained by the use of either method in isolation