Modelling of reversible tissue electroporation and its thermal effects in drug delivery.

Electroporation is a very useful tool for drug delivery into various diseased tissues of the human body. This technique helps to improve the clinical treatment by transferring drugs into the targeted cells rapidly. In electroporation, drug particles enter easily into the intracellular compartment through the temporarily permeabilized cell membrane due to the applied electric field. In this work, a mathematical model of drug delivery focusing on reversible tissue electroporation is presented. In addition, the thermal effects on the tissue, which is an outcome of Joule heating, are also considered. This model introduces a time-dependent mass transfer coefficient, which is significant to drug transport. Multiple pulses with low voltage are applied to reach sufficient drugs into the targeted cells. Based on the physical circumstances, a set of differential equations are considered and solved. The changes in drug concentration with different parameters (e.g., diffusion coefficient, drug permeability, pulse length, and pulse number) are analyzed. The model optimizes the electroporation parameters to uptake sufficient drugs into the cells with no thermal damage. This model can be used in clinical experiments to predict drug uptake into the infected cells by controlling the model parameters according to the nature of infections.

Enhanced Drug Uptake on Application of Electroporation in a Single-Cell Model.

Electroporation method is a useful tool for delivering drugs into various diseased tissues in the human body. As a result of an applied electric field, drug particles enter the intracellular compartment through the temporarily permeabilized cell membrane. Consequently, electroporation method allows better penetration of the drug into the diseased tissue and improves treatment clinically. In this study, a more generalized model of drug transport in a single cell is proposed. The model is able to capture non-homogeneous drug transport in the cell due to non-uniform cell membrane permeabilization. Several numerical experiments are conducted to understand the effects of electric field and drug permeability on drug uptake into the cell. Through investigation, the appropriate electric field and drug permeability are identified, which lead to sufficient drug uptake into the cell. This model can be used by experimentalists to get information prior to conduct any experiment, and it may help reduce the number of actual experiments that might be conducted otherwise.

A mathematical model of drug dynamics in an electroporated tissue.

In order to overcome the obstruction of cell membranes in the path of drug delivery to diseased cells, the applications of electric pulses of adequate potency are designated as electroporation. In the present study, a mathematical model of drug delivery into the electroporated tissue is advocated, which deals with both reversibly and irreversibly electroporated cells. This mathematical formulation is manifested through a set of differential equations, which are solved analytically, and numerically, according to the complexity, with a pertinent set of initial and boundary conditions. The time-dependent mass transfer coefficient in terms of pores is used to find the drug concentrations through reversibly and irreversibly electroporated cells as well as extracellular space. The effects of permeability of drug, electric field and pulse period on drug concentrations in extracellular and intracellular regions are discussed. The threshold value of an electric field (E>100 V cm-1) to initiate drug uptake is identified in this study. Special emphasis is also put on two cases of electroporation, drug dynamics during ongoing electroporation and drug dynamics after the electric pulse period is over. Furthermore, all the simulated results and graphical portrayals are discussed in detail to have a transparent vision in comprehending the underlying physical and physiological phenomena. This model could be useful to various clinical experiments for drug delivery in the targeted tissue by controlling the model parameters depending on the tissue condition.

A Nonlinear Mathematical Model of Drug Delivery from Polymeric Matrix.

The objective of the present study is to mathematically model the integrated kinetics of drug release in a polymeric matrix and its ensuing drug transport to the encompassing biological tissue. The model embodies drug diffusion, dissolution, solubilization, polymer degradation and dissociation/recrystallization phenomena in the polymeric matrix accompanied by diffusion, advection, reaction, internalization and specific/nonspecific binding in the biological tissue. The model is formulated through a system of nonlinear partial differential equations which are solved numerically in association with pertinent set of initial, interface and boundary conditions using suitable finite difference scheme. After spatial discretization, the system of nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations which is subsequently solved by the fourth-order Runge-Kutta method. The model simulations deal with the comparison between a drug delivery from a biodegradable polymeric matrix and that from a biodurable polymeric matrix. Furthermore, simulated results are compared with corresponding existing experimental data to manifest the efficaciousness of the advocated model. A quantitative analysis is performed through numerical computation relied on model parameter values. The numerical results obtained reveal an estimate of the effects of biodegradable and biodurable polymeric matrices on drug release rates. Furthermore, through graphical representations, the sensitized impact of the model parameters on the drug kinetics is illustrated so as to assess the model parameters of significance.

Mathematical modelling of liposomal drug release to tumour.

The primary aim of liposomal drug delivery is to wisely modulate the drug delivery system in order to target diseased tissues. Temperature-sensitive liposomes function as a prospective weapon to combat toxic side effects corresponding to direct infusion of anticancer drugs. The main objective of the present study is to model liposomal drug release, subsequent drug transport in solid tumour along with integrated actions of tumour cell surface and endosomal events. Generalized mathematical model for liposomal drug delivery is proposed in which vital physical phenomena, such as kinetics of liposome-encapsulated drug, free drug release from liposomes, transport of both liposomal drug and free drug into the tumour compartment, plasma clearance, protein-drug interactions, drug-tumour cell receptor interactions, internalization of drug through endocytosis along with corresponding endosomal events. The model is expressed through a system of coupled partial differential equations along with appropriate set of initial, interface and boundary conditions which is solved numerically. Simulated results are compared with respective existing experimental data to demonstrate the potency and reliability of the proposed model. Graphical representations of time variant concentration profiles are illustrated to understand the underlying phenomena in details. Moreover, the model speaks for the sensitivity of important drug kinetic parameters, such as advection coefficients, drug release coefficient, plasma clearance rate and internalization parameters through graphical portrayals. The proposed model and the simulated results act as a tool in designing a more effective drug delivery system for cancerous tumours.

A two-phase model for drug release from microparticles with combined effects of solubilisation and recrystallisation.

The present study aims to provide a comprehensive mathematical model for drug release from microparticles to the adjacent tissues. In the elucidation of drug release mechanisms, the role of mathematical modelling has been depicted thereby facilitating the development of new therapeutic drug by a systematic approach, rather than expensive experimental trial-and-error methods. In order to study the whole process, a two-phase mathematical model describing the dynamics of drug transport in two coupled media is presented. Drug release is described taking into consideration both solubilisation dynamics of drug crystallites and diffusion of the solubilised drug through the microparticle. In the coupled media, reversible dissociation/recystallisation processes are taking place. The model has led to a system of partial differential equations that are solved analytically. The model points out the important roles played by the diffusion, mass-transfer and reaction parameters, which are the main architects behind drug kinetics across two layers. The dependence of drug masses on the main parameters is also analysed.

Fully compact higher-order computation of steady-state natural convection in a square cavity.

The flow in a thermally driven square cavity with adiabatic top and bottom walls and differentially heated vertical walls for a wide range of Rayleigh numbers (10(3)< or =Ra< or =10(7)) has been computed with a fourth-order accurate higher-order compact scheme, which was used earlier only for the stream-function vorticity (psi-omega) form of the two-dimensional steady-state Navier-Stokes equations. The boundary conditions used are also compact and of identical accuracy. In particular, a compact fourth-order accurate Neumann boundary condition has been developed for temperature at the adiabatic walls. The treatment of the derivative source term is also compact and has been done in such a way as to give fourth-order accuracy and easy assimilation with the solution procedure. As the discretization for the psi-omega formulation, boundary conditions, and source term treatment are all fourth-order accurate, highly accurate solutions are obtained on relatively coarser grids. Unlike other compact solution procedure in literature for this physical configuration, the present method is fully compact and fully higher-order accurate. Also, use of conjugate gradient and hybrid biconjugate gradient stabilized algorithms to solve the symmetric and nonsymmetric algebraic systems at every outer iteration, ensures good convergence behavior of the method even at higher Rayleigh numbers.