MATHEMATICAL MODELING FOR MAGNETIC DRUG TARGETING

Abstract

The delivery of anticancer agents to the specific target sites with minimum side effects is an important challenge in chemo, radio and gene-therapy. Magnetic Drug Targeting (MDT) is one of the promising methods for effective targeting and delivery of drugs to a specific site with aid of a local magnetic field [2, 6]. In this method, magnetic carrier particles loaded with drug molecules are injected into the microvasculature upstream from the malignant tissue and attracted towards the targeted region in the body with help of a local magnetic field [76, 168]. MDT is growing due to speedy progress in the growth of functionalized magnetic nanoparticles, which are used for chemo, radio, and gene-therapy at a tumour site [16]. It is also shown by various studies that MDT is relatively safe and effective method for targeting drugs to a specific site [128, 134]. Thus present thesis entitled "Mathematical Modeling for Magnetic Drug Targeting" describes the transport and capturing of magnetic nanoparticles flowing within a fluid in a cylindrical tube under magnetic field for magnetic drug targeting (MDT) applications. The trajectories and capture efficiency of magnetic nanoparticles flowing in a fluid under magnetic field are studied through mathematical models. Additionally, in-vitro experiments are performed to validate the mathematical models. The present thesis is divided into six chapters and the chapter wise description is given below. Chapter 1 contains an introductory aspects and survey of the field and describes an overview on the theoretical and experimental studies on magnetic nanoparticles transported under magnetic field for magnetic drug targeting. The general features and challenges in the field of magnetic drug targeting are also reviewed. Chapter 2 describes the effect of external uniform magnetic field on flow parameters of fluid and magnetic particles in a cylindrical tube (simulated blood vessel) through a - mathematical model based on magnetohydrodynamics (MHD) approach. The governing nonlinear partial differential equations are solved numerically using finite difference scheme. Model results show that the velocity of blood and magnetic particles is appreciably reduced under the influence of magnetic field. Further, artificial blood (75 % water + 25 % glycerol) along with iron oxide magnetic particles were transported into a glass tube with help of a peristaltic pump and experimentally measured their velocity at different magnetic fields ranging from 100 to 600 mT. An experimental result of the velocity of magnetic particles flowing within artificial blood in a cylindrical tube supports the mathematical model result. Chapter 3 elaborates a mathematical model developed to predict the trajectories of a cluster of magnetic nanoparticles in a cylindrical tube under the influence of a permanent magnet positioned outside the tube. All forces, including magnetization, drag and buoyancy expected to significantly affect the transport of nanoparticles are incorporated. The coupled mathematical equations are solved numerically using classical fourth order Runge-Kutta method. The results show that all particles are captured either before or at the centre of the magnet when magnet is very close proximity to the blood vessel. This is due to the influence of strong magnetic force, experienced by the magnetic particles, which is responsible to attract them towards the magnet. It is optimized by this study that magnetic particles are captured up to 4.5 cm distance (d) between the blood vessel and magnet. Further increase in d value (above 4.5 cm) results the free movement of magnetic particles. In addition, the present model results are validated through simulations performed using COMSOL software. Chapter 4 presents a mathematical model to study the effect of magnetic field and inlet velocity on capture efficiency of magnetic nanoparticles flowing in a cylindrical tube under magnetic field. The dominant magnetization and drag forces, expected to significantly affect the capturing of magnetic particles, are incorporated in the model. The mathematical equations are solved analytically and numerically using classical fourth order Runge-Kutta method. Enhancement in capture efficiency from 23 to 51 % has been observed through mathematical model by increasing the magnetic field from 0.1 to 0.5 T, respectively. However, decrease in capture efficiency is noticed from 51 to 25 % by increasing the inlet velocity from 2 to 8 mm/s, respectively. In-vitro experiments were performed to study the capture efficiency of magnetic particles at various magnetic fields and inlet velocities. Experimental and mathematical model results are compared, which show good agreement between them and hence validate the mathematical model. Chapter 5 describes a mathematical model to predict the capture efficiency of magnetic nanoparticles flowing in a fluid under magnetic field through inserting a ferromagnetic stent coil in a cylindrical tube. The ferromagnetic SS-430 coil is used as a stent and placed perpendicular to applied magnetic field inside the tube to further enhance the capture efficiency. The dominant magnetization and drag forces, which significantly affect the capturing of magnetic particles, are incorporated in the model and mathematical equations are solved analytically. It is observed through the results the capture efficiency increases from 36 to 81 % as we increase the magnetic field from 0.1 to 0.5 T, respectively. It can be noticed that the capture efficiency increases 1.6 times by using the ferromagnetic stent as compared to without stent geometry (reported in chapter 4). Furthermore, decrease in capture efficiency from 81 to 41 % is observed by increasing the inlet velocity from 2 to 8 mm/s despite of 1.6 times larger values of capture efficiency in respect of without stent geometry. In-vitro experiments to measure the capture efficiency were also performed to validate the mathematical model. An agreement in trend between experimental and mathematical model results is observed, however, the value of capture efficiency in case of experimental is less as compared to model results. Interestingly, this study shows that the capture efficiency can be increased 1.6 times by using ferromagnetic SS-430 as a stent inside the blood vessel. Finally, Chapter 6 presents the summary and concluding remarks of this thesis and the possible directions of the future scope.