A model for magnetic drug targeting in a permeable microvessel with spherical porous carrier particles

We present a mathematical model for targeting a drug at malignant tissue cells in a permeable microvessel. The drug molecules are transported in carrier particles which are assumed to be porous spheres. This mode of drug delivery is non-invasive and has less toxic effects on healthy cells and tissues. The microvessel tube (see Figure 1) is subdivided into three regions, the outer endothelial glycocalyx layer where the blood has a Newtonian character, and a core and plug regions where the blood flow is described using a non-Newtonian Casson fluid model which is suitable for microvessels of radius 5 m μ . Targeting is achieved through a locally applied magnetic field using a cylindrical magnet positioned outside the body near the tumour position so that the carrier particles, bound with nanoparticles and drug molecules are captured at the tumour site. The study seeks to understand, inter alia, the effects of the size and permeability of the carrier particle, the volume fraction of embedded magnetic nanoparticles and the placement of the external magnetic field on the magnetic targeting of the carrier particles. INTRODUCTION Blood flow in microvessels has different characteristics compared to blood flow through large vessels. The microscopic properties of the blood and interactions among plasma, cells and blood vessels, in particular blood-vessel wall interactions affect the nature of the blood flow through micro vessels. Due to deformation and rotation of red blood cells (RBCs), these cells accumulate near the axis of the microvessel producing a layer that moves with a constant velocity. A cell-depleted layer appears at the outer region, near the wall of the microvessel. In micro vessels or channels, blood flow represents a remarkable two-phase nature, with a peripheral layer of plasma (Newtonian fluid) and a core region of suspensions of erythrocytes which has a nonNewtonian character, Bugliarello and Sevilla [1]. Seshadri and Jaffrin [2] considered a two phase fluid model in which the outer cell-depleted layer has a lower hematocrit than the core region. The concentration of RBCs in the celldepleted layer was assumed to be 50% of that in the core region. Gupta et al. [3] divided the outer layer into a cell-free plasma layer and cell-depleted layer. Sankar and Lee [4] investigated a two-phase fluid flow through a stenosed blood vessel. The endothelium layer of the microvessels covered by a glycocalyx layer contains a gel like layer of membrane-bound glycoproteins and plasma proteins. Liu and Yang [5] studied the eloctrokinetic effect of the endothelial glycocalyx layer in small blood vessels. The influence of the glycocalyx layer on the blood flow has been studied by many researchers, see [6, 7]. Shaw and Murthy [8] considered a two-phase fluid model and studied the significant effect of the glycocalyx layer on the magnetic targeting of a carrier particle in an impermeable microvessel. It was observed that the glycocalyx layer caused additional resistance to micro-vessel flow [9]. This was assumed to be due to its high negative charge. A magnetic targeted drug delivery system is an attractive delivery strategy due to its non-invasiveness, high targeting efficiency and minimal toxic side effects on healthy cells and tissues [10, 11]. Mathematicals models for predicting the magnetic targeting of multifunctional carrier particles designed to deliver therapeutic agents to malignant tissue in vivo have been studied in [12]. Recently, a number of studies on magnetic drug targeting in a microvessel have used a two-phase non-Newtonian fluid model [13, 14]. Porous spheres have several, extremely valuable therapeutic and biotechnological applications, including cell immobilization, drug delivery, and as packing material in chromatography [15]. Porosity is important in improving the performance of spheres [16]. Large pores increases the permeability of the spheres, significantly increasing their surface area, allowing them to be used as culture systems for growing adherent cells, to be used for water remediation at high diffusion rates, or be used in the separation of large biomolecules, etc. [17]. The purpose of the present investigation is to explore the effect of a porous carrier particle on magnetic drug targeting in permeable microvessel using a two-phase fluid model. In the peripheral layer, the blood has a