Analytics and numerics of drug release tracking

The study of the dependence of fluxes, concentrations and response times, on the characteristic properties of drug delivery polymeric devices, plays an important role in the design of drug release platforms. The aim of this paper is to develop mathematical tools for an in-depth understanding of drug release tracking. The mathematical model presented takes into account the viscoelastic properties of the polymer and the state of the dispersed drug: free or chemically bound to the matrix. For nonlinear chemical bounds the process is described by a nonlinear integro-differential system and the drug release tracking is treated numerically. For linear chemical bounds closed formulas for the fluxes and response times are established in terms of the parameters that characterize the drug and the platform. These formulas provide a set of a priori estimations for the variables of the model. Numerical examples which show the effectiveness of the approach are included.     

药物从柱型高聚物基体内扩散释放的活动边界问题

采用修正积分法,得到了药物从柱形高聚物基体内扩散释放的活动边界问题的近似解析解,给出了扩散边界和药物释放分数的计算公式及其在不同初装浓度下的计算结果。计算结果与实验结果是一致的。进而,对药物从片形、柱形、球形等不同几何形状的基体内的释放问题进行了对照,分析了基体的不同几何形状对药物释放的影响。还给出了药物从柱形基体内扩散释放的有效时间的近似计算公式。这对临床实验具有一定的指导意义,也为进一步研究药物从柱型高聚物基体内扩散释放问题以及缓释制剂的设计提供了理论工具。     

Iterative solver approach for turbine interactions: application to wind or marine current turbine farms

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach.     

Disrupted capacitated vehicle routing problem with order release delay

With the popularity of the just-in-time system, more and more companies are operating with little or no inventories, which make them highly vulnerable to delays on supply. This paper discusses a situation when the supply of the commodity does not arrive at the depot on time, so that not enough of the commodity is available to be loaded on all vehicles at the start of the delivery period. New routing plans need to be developed in such a case to reduce the impact the delay of supply may have on the distribution company. The resulting vehicle routing problem is different from other types of vehicle routing problems as it involves waiting and multiple trips. Two approaches have been developed to solve the order release delay problem, both of which involve a Tabu Search algorithm. Computational results show the proposed approaches can largely reduce the disruption costs that are caused by the delayed supply and they are especially effective when the length of delay is long.     

Analysis and Optimization of Stability Robustness Bounds for Discretized Systems

In this paper, robustness bounds for the perturbations of continuous-time systems to ensure the stability of their discretized counterparts are developed. Both zero-order hold and P-step matrix integrators are considered. The effect of the sampling time on the robustness bounds is studied via examples. To determine how well a simulated system will retain the robustness properties of the continuous-time system being simulated, a new criterion for the selection of the simulation method and time step is introduced. Both implicit and explicit robustness measures for sampled-data systems are obtained.     

One-dimensional drug release from finite Menger sponges: In silico simulation

The purpose of this work was to evaluate the consequences of the spatial distribution of components in pharmaceutical matrices type Menger sponge on the drug release kinetic from this kind of platforms by means of Monte Carlo computer simulation. First, six kinds of Menger sponges (porous fractal structures) with the same fractal dimension, d_f=2.727, but with different random walk dimension, d_w[2.149,3.183], were constructed as models of drug release device. Later, Monte Carlo simulation was used to describe drug release from these structures as a diffusion-controlled process. The obtained results show that drug release from Menger sponges is characterized by an anomalous behavior: there are important effects of the microstructure anisotropy, and porous structures with the same fractal dimension but with different topology produce different release profiles. Moreover, the drug release kinetic from heteromorphic structures depends on the axis used to transport the material to the external medium. Finally, it was shown that the number of releasing sites on the matrix surface has a significant impact on drug release behavior and it can be described quantitatively by the Weibull function.     

非对称稀疏线性方程组的快速外存解法及其在无网格法计算中的应用

针对局部Petrov-Galerkin无网格法(MLPG)等无网格方法的计算所产生的大型非对称稀疏线性方程组,介绍了一种新的直接解法．与一般非对称求解过程不同,该解法从现有的对称正定解法中演变出来,其分解过程在矩阵的上、下三角阵中对称进行．新的矩阵分解算法可以通过修改对称矩阵分解算法的代码来实现,这提供了从对称解法到非对称解法的快捷转换．还针对MLGP法以及有限元法所产生的方程组开发了多块外存算法(multi-blocked out-of-core strategy)来扩大求解规模．测试结果证明该方法大幅度提高了大型非对称稀疏线性方程组的求解速度．     

A heuristic for scheduling general job shops to minimize maximum lateness

Based on the various priority dispatching rules, three dominant rules were formulated. Heuristic approaches were developed based on these dominant rules and dominant matrix. These heuristics can solve problems where a job may need to be processed by the same machine more than once in its operation sequence and different release times are allowed for different machines. The proposed heuristics appear to be fairly effective procedures, and various numerical results are obtained and compared with the various dispatching rules and with a branch and bound approach.     

Flux tracking in drug delivery

The dynamics of diffusive and stress-induced transport in polymeric delivery systems was investigated. Partial and ordinary differential equations were first written to describe drug release behaviors in Maxwell and Maxwell–Voigt materials. The time constants governing the flux and concentration responses of a permeating species were determined from a Laplace transform solution of the original model. A “tracking strategy”, based on the estimated characteristic times, was proposed to estimate the delivery rate and the concentration near the exit side of the membrane. The methodology was more efficient at times greater than the time constant and the prediction error decreased further as the process approached steady state. Numerical illustrations and comparisons made with published data show the effectiveness of the proposed approach in describing the influence of the Young modulus, viscosity and relaxation time on the transient regime.     

硝苯地平缓释片释放过程的灰色数学模型

采用高分子材料制成硝苯地平缓释片 ,测定其释放度 ,以灰色数学模型进行分析 .结果表明 ,用灰色数学模型对药物释放过程进行拟合 ,拟合精度高 ,结果可靠 .灰色数学模型可用于描述其释药行为 .     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

Structural theory of elastomeric composites based on fiber systems — Invariant description

A three-dimensional theory of elastomeric composites with elastomeric matrices reinforced by systems of fibers is presented. The theory is based on a structural approach in which the matrix and the reinforcement of the composite are considered separately without reduction to a medium having continuously changing characteristics. The approach is based on the idea of a vector field of macroscopic displacements given by the positions of the axial lines of the fibers in the curret (deformed) configuration of the composite. The vector field determines the current macroscopic configuration, the tensor fields of the measures of macroscopic strain, and the field of the macroscopic stress tensor in the composite. The displacement, strain, and stress fields in the elastomeric matrix and the fibers of the reinforcing systems are regarded as derivatives of the field of macroscopic displacements of the medium. Relations are presented to describe the kinematics of the fibers in the current configuration of the composite, including the evolution of their orientation and the frequency of their planar and spatial distribution. Equations are obtained for the macroscopic motion of the fiber-reinforced matrix, and the dynamic variational principle that governs this motion is established. The elastic macroscopic potential of the matrix is found and related to the components of the macroscopic stress tensor. The procedure to be followed in constructing the constitutive equations of the composite is described. The proposed system of equations, relations, and algorithms is closed and can be used to solve problems involving the deformation of products made of fiber-reinforced elastomers and the creation of elastomeric composite products, based on fiber systems, that possess the requisite properties.     

Deformative properties of contractile polymeric structures — Artificial muscles

A dynamic theory is proposed for the longitudinal deformation of contractile polymeric structures, known also under the name of "artificial muscles." The research was carried out to a linear approximation for small deformations. Elasto-osmotic polymeric contractile structures, which represent dispersed systems of the type of more or less diluted gels and in which the deformation depends on the degree of swelling, were examined. The viscoelastic properties of the gel matrix are represented by the classical Kelvin-Voigt model. The exchange of the substance between the surrounding medium and the polymeric body is taken into account in the form of a flow of particles (of the solvent) penetrating the membrane surface of an equivalent elasto-osmotic cell.     

Multilevel computations of dispersed drug release

We study a mathematical model of drug release from controlled delivery systems with initial drug loading higher than solubility. The model combines dissolution, diffusion, swelling, and erosion mechanisms of drug delivery. Multilevel methods are introduced to solve the governing system of diffusion equations numerically with better accuracy and lower computational costs compared with the finite element methods. Numerical examples are given to demonstrate the advantages of the multilevel methods. Numerical solutions are compared to exact and approximate solutions of the reduced models. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical methods for the stability of time-periodic hybrid time-delay systems with applications

This paper extends two numerical methods for the stability analysis of a class of time-periodic hybrid time-delay systems. In particular, the pseudospectral tau and spectral element methods are extended to hybrid systems. The analyzed delay-differential equation involves delayed terms with both continuous and piecewise constant arguments, in other words, it involves delays both without and with zero-order hold, respectively. The analyzed class of hybrid systems can be used to describe time-periodic hereditary processes subjected to digital feedback control. The proposed numerical algorithms are applied to the mathematical models of haptic systems and milling processes subjected to digital feedback control.     

A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter.     

输入采用标准模糊分划的模糊控制系统性质及稳定性分析

具体定义模糊控制系统输入变量的标准模糊分划，研究和证明输入采用标准模糊分划的模糊控制系统的有关性质。在此基础上，采用Lyapunov直接法研究该类模糊控制系统的稳定性，提出一个判定模糊控制系统稳定性的充分条件。该条件将以往方法要在所有的子系统中寻找一个公共的正定矩阵满足Lyapunov不等式，放宽为在各最大交叠规则组内分别寻找公共的正定矩阵，从而减小稳定性判定的保守性和难度。     

Discretization of nonlinear input-driven dynamical systems using the Adomian Decomposition Method

A numerical decomposition method proposed by Adomian provides solutions to nonlinear, or stochastic, continuous time systems without the usual restrictive restraints. It is applicable to differential, delay differential, integro-differential, and partial differential equations without the need for linearization or other restrictions. It also works through both uncoupled boundary conditions as well as delay systems. In the following paper, a new time discretization method for the development of a sample-data representation of a nonlinear continuous-time input-driven dynamical system is proposed. The proposed method is based on both the zero-order hold (ZOH) assumption as well as the Adomian Decomposition Method which exhibit unique algorithmic and computational advantages. To take advantage of this method, the following steps must be followed. First, the method is applied to a linear input-driven dynamical system to explicitly derive an exact sample-data representation, producing proper results. Second, the method is then applied to a nonlinear input-driven dynamical system, which thereby derives exact and approximate sample-data representations, the latter being most suited for practical applications. To evaluate the performance, the proposed discretization procedure was tested using simulations in a case study which involved an illustrative two-degree-of-freedom mechanical system that exhibited nonlinear behavior considering various control and input variable profiles. In conclusion, the suggested algorithm, in comparison to the results of a Taylor-Lie series expansion method, demonstrated increased performance and efficiency.