Mathematical model of effect of drug delivery on blood flow in external magnetic field by Magnetic Nanoparticles Somna Mishra * , V. K. Katiyar ** , V. Arora *** and Gaurav Varshney **** * Gurukula Kangri Vishwavidyalaya, Haridwar, Uttarakhand, India, mishra_somna@yahoo.com ** Department of Mathematics, Indian Institute of Technology Roorkee, India, vktmafma@iitr.ernet.in *** Gurukula Kangri Vishwavidyalaya, Haridwar, Uttarakhand, India **** Department of Mathematics, Indian Institute of Technology Roorkee, India, gauvadma@iitr.ernet.in ABSTRACT A mathematical model is proposed to study the velocity profiles i.e. velocity of magnetic nanoparticles (diameter 20 nm) as drug carrier and blood in the presence of uniform external magnetic field inside the capillary region. The volumetric flow rate and skin friction are also taken into account and the analytic expressions are also developed. Keywords: magnetic nanoparticles, nano-drug targeting. 1. INTRODUCTION Magnetic drug targeting is one of the various possibilities of drug targeting, which aims at concentrating magnetic drugs at a target site with the aid of magnetic field, so that the drug concentration is enhanced at the target and reduces the toxicity and side effects in normal tissue [1]. Magnetic and hydrodynamic interactions between magnetic beads in micro-fluidic field gradient filters are compared theoretically and it is found that hydrodynamic interactions are of a longer range and dominate the magnetic ones [2]. An in vitro model was developed to study and demonstrate the potential and feasibility of magnetically targeted deposition of aerosols for potential applications in lung cancer treatment [3]. High-pressure liquid chromatography analyses after magnetic drug targeting showed an increasing concentration of the chemotherapeutic agent in tumor region compared to regular systemic chemotherapy [4]. In this paper, we have considered the velocity profiles of magnetic nanoparticles (diameter 20 nm) and blood in the presence of uniform external magnetic field relative to each other inside the capillary region, in cylindrical polar coordinates. The electrical conductivity of blood is not taken into account. The volumetric flow rate and skin friction are also considered and the analytic expressions are also developed. 2. MATHEMATICAL FORMULATION The magnetic force on the magnetic nanoparticles is [6] ( ) . 2 0 B V F M M G G ∇ = µ χ (1) Where 0 µ is the permeability of free space, χ is the magnetic susceptibility of the particle, ) 3 4 ( 3 M M R V π = is the volume of the magnetic nanoparticle and B G is the magnetic induction. The magnetostatic field equations are 0 = ⋅ ∇ B G , . 0 = × ∇ H G (2) where H G is the magnetic field intensity. Using the expression H B G G 0 µ = and the magnetostatic field equations, equation (1) can be transformed into ( ) . 0 H M V F M M G G G ∇ ⋅ = µ Where H M G G χ = , M G is the magnetization of the particle. Assuming that the fluid is non-conducting and that the displacement current is negligible so that 0 = × ∇ H G as defined in [7]. Hence the magnetic force on a nanoparticle (in the form of sphere) can be calculated as . 0 H M V F M M ∇ = µ G (3) In this paper, we have considered the magnetite nanoparticles as drug carrier and also the saturation value of magnetization ( 1 ) is taken for these particles [8]. Here we have considered only those portions of blood vessels, which are oriented perpendicularly to the direction of the magnetization. The intensity of the magnetic field in the direction of magnetization for the cylindrical magnet is given by [9] 000 , 450 − = Am . 2 2 1 y a C H y = Where ‘ a ’ is the radius of the cylindrical magnet. Oy is the direction of magnetization oriented perpendicular to the direction of blood vessel feeding the tumor (Figure 1). The value of the constant is defined in TABLE I. 1 C Since the magnetic field varies over a length scale, typically O( ). The diameter of the capillary in which targeting takes place is much smaller than the above mentioned scale. So the magnetic force across a capillary diameter is assumed constant. m 1 2 10 10 − − − The force that counteracts the magnetic force on the particle in blood stream is due to the blood flow and can be calculated as using the Stokes’ expression for drag force on a sphere as [5] ). ( 6 υ πµ − = u R F M D (4) Where µ is the viscosity of blood, is the radius of the magnetic nanoparticle, u is the velocity of blood and M R υ is the velocity of magnetic nanoparticles. For the present problem, the blood flow has been considered as laminar axi-symmetric flow of a viscous, homogeneous, incompressible Newtonian fluid in cylindrical blood vessel (capillary). The flow in capillary is assumed to NSTI-Nanotech 2008, www.nsti.org, ISBN 978-1-4200-8504-4 Vol. 2 45