Supplementary information

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1 Supplementary information Theoretic approach of magnetic aerosol drug targeting. The following assumptions were made for computer-aided simulation of inhaled SPION aerosol trajectories: (1) Structure of an aerosol droplet It is assumed that an aerosol droplet is of spherical shape and the average geometric mean diameter is of 3.5 µm. The aerosol droplet comprises SPIONs of 80 nm hydrodynamic diameter with a 50 nm core diameter comprising 5 nm single domain magnetite nanoparticles with a packaging density of 30% and an adsorbed 15 nm PEI 25 kda layer dispersed in water. It is assumed that 2930 SPIONs are loaded into an aerosol droplet. (2) Gravity force acting on a SPION aerosol droplet The density of the SPION is ρ=1.56 g/cm -3 which results in a mass of a SPION (NP) m NP =10-19 kg. The mass of an aerosol droplet is the sum of all of the SPIONs per droplet m NP,droplet = kg and the additional water, which amounts to m H2O = kg. Thus the water dominates the complete mass of an aerosol droplet () which is m, NP = kg and the gravity of an aerosol droplet is G= N. (3) Magnetic moment of a SPION aerosol droplet If direct particle-particle attractions of the core particles are neglected the maximal magnetic moment of a 50 nm SPION multi core particle can be calculated according to μ NP = M SAT V (1) 1

2 where µ NP is the magnetic moment of a SPION, M SAT is the saturation magnetization of magnetite (480 ka/m), and V is the total volume of magnetite within the core of the SPION assuming that the particles are magnetically saturated which is only valid for sufficiently high magnetic field intensity. Thus, the magnetic moment of a single SPION particle can be calculated to µ NP = Am 2 and the maximal magnetic moment of an aerosol droplet can be estimated at µ = Am 2 when particle-particle interactions within an aerosol droplet are neglected. In principle, such particle-particle interactions would result in an increase of the magnetic moment. Therefore, the calculations rather underestimate the magnetic moment of the aerosol droplet (22). Assuming that all of the magnetizable vectors are in parallel orientation to the external field, i.e. the particle is magnetically saturated; the force acting on a magnetizable particle in an external magnetic field can be calculated according to F M, = µ B where µ is the magnetic moment of the SPION aerosol droplet and B (2) is the magnetic flux gradient. For this reason, successful capture of the aerosol droplet will require an inhomogenous magnetic field. The electromagnet used in this study results in a magnetic flux gradient of >100 T/m in close proximity of the tip (Figure 1c). Therefore, the magnetic forces are approximately F M, 3x10-12 N which is about one order of magnitude higher than gravity. Anatomical model of the mouse trachea and computer-aided simulation of inhaled SPION aerosol droplet trajectories The following calculations are based on the geometry of the first generation of the respiratory tract of mice ((23) see Supplementary figure S1). Site-specific enrichment of 2

3 inhaled SPION droplets within the lungs primarily consists of two processes (i) transport of the aerosol droplets from the center of the airway to the airway wall and (ii) capture on its surface. Once the aerosol droplets are deposited on the airway surface the viscous drag forces which act on the particles are negligible as the air flow velocity approaches zero at the airway surface. The transport of the aerosol droplets to the airway surface is facilitated because (i) the airflow velocity is zero for a short period of time between inspiration and expiration, and (ii) the aerosol droplets have to pass the same way during expiration as after inspiration, i.e. have to pass the magnetic field twice. However, latter two aspects are not considered for computer-aided simulations and the trajectories are calculated for only one passage through the airway model. The airflow profile at the main bifurcation was calculated with a finite-element program (Comsol Multiphysics) which completely resolves the Navier-Stokes-equations. The calculations were performed for only the inhalation process. The geometric parameters for the simulation model are given in Supplementary figure S1. In addition the distance between the magnet s tip and the lung lobe was estimated based on the experimental set up. For the computer-aided simulations airflow parameters of mice were assumed based on a breathing frequency of 120/min (2/s) and a tidal volume of 200 µl which resulted in a volume flow of 800 µl/s ( m 3 /s). Assuming a diameter of the trachea of 1.4 mm, the inflow velocity is V/A=0.52 m/s. For the mathematical approach to calculate the particle trajectories the assumption was made that the particles are evenly distributed over the cross section area and the inflow profile is laminar of average velocity of 0.52 m/s. This assumption is justified because the length of the ventilation tube is high compared with its diameter (critical inflow length). The maximal particle velocity is approximately v max ~1.04 m/s. The viscosity of air was assumed to be η air =17 µpas. 3

4 The magnetic force field was approximated through a rotation-symmetric configuration of similar intensities because the magnet s tip was located very close to the target area. Based on this model, the calculations can be performed much faster than by using the original slightly asymmetric data set. The magnetic and the aerodynamic field which are required for the simulation of the aerosol droplet trajectories are shown in Supplementary figure S3 both in horizontal (x-direction) and vertical (y-direction) orientation to the flow direction. The tool for calculating the particle trajectories in dependence of the magnetic field gradient was programmed using MATLAB according to the Nassi-Shneiderman diagram displayed in Supplementary figure S4. Starting from the initial values, the magnetic flux density as well as the magnetic force, the viscous drag force, and gravity are calculated for every point of space within the anatomical model. All values were stored and then plotted. (4) Calculation of the "magnetic" velocity In addition to the flow velocity caused by viscous drag forces and gravity, the velocity due to the magnetic force must be considered to be able to calculate the trajectories of the particles. The force acting on a particle with the magnetic dipole moment μ inhomogeneous static magnetic flux density field B is (24). F ( B) within an external M, = μ (3) If we assume that all magnetic moments of the particles are aligned to the external magnetic field the vector of the magnetic moment is defined by: μ μ = B B (4) 4

5 where μ = μ and B = B. Combining (3) and (4) the magnetic force on a particle with the scalar magnetic moment m can be expressed by: F M, μ μ 2 = B B = B = μ B B B (5) Assuming that the air flow is laminar during entry and exit after inspiration and expiration, the viscous drag forces can be calculated as F 6πηr v A, = A (6) where η is the viscosity of air (η air =17 µpas), r is the radius of the SPION aerosol droplet, and v A is the air flow velocity. The "magnetic" velocity can be calculated using (7). v M, = 1 μ FM, = 6πη r 6πη r B (7) where r denotes the radius diameter of the particle and η the viscosity of the air. The resulting velocity v res of the aerosol droplet can be calculated using (8) v res μ B v = + A, 6πη r + v G (8) where v A, is the flow velocity and v G is the sedimentation velocity. 5

6 Supporting Figures Figure S1 6

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12 12 Figure legends supplementary information Figure S1 Geometry of the mouse trachea and the main bifurcation. Schematic representation of the dimensions of the mouse trachea and the main bifurcation and their branching angle. This model was used for the computer-aided simulations. Figure S2 Magnetic flux density and air flow velocity profile. Shown is the magnetic flux density in the same (a) or perpendicular (b) orientation to the flow direction according to the rotation-symmetric tip model and the air flow velocity profile at the main bifurcation in the same (c) or perpendicular (d) orientation to the flow direction. For the computer-aided simulation a laminar flow of the air-aerosol mixture was assumed. Figure S3 Representation of the field gradients acting on a SPION aerosol droplet in the same (a) or perpendicular (b) orientation to the flow direction. Figure S4 Simplified flow chart of the simulation tool. 12

13 13 Figure S5 Computer-based simulations of SPION deflection in a magnetic gradient field. In this simulation the magnetic moment was set to 9.4x10-18 Am² which corresponds to a single magnetic nanoparticle. The magnet s tip was placed at x = 13 mm, y = 0.8 mm and z = 2.5 mm (above the left airway). None of the droplets were deposited during inspiration on the surface of each of the airways. Twenty-five trajectories were calculated in the simulation. Figure S6 Schematic representation of an improvement of the site-specificity of magnetic lung drug targeting: The magnetic field of a by-pass-magnet positioned near the mainbifurcation could direct the major part of the particles to the desired lung lobe. A second field of a target magnet could retain the particles in the target area as described. 22. Huke, B.M., Thesis, "Einfluss der Teilchen-Wechselwirkung auf die Magnetisierung in Ferrofluiden" (Universität des Saarlandes, Saarbrücken, 2002). 23. Oldham, M.J. & Phalen, R.F. Dosimetry implications of upper tracheobronchial airway anatomy in two mouse varieties. Anat. Rec. 268, (2002). 24. Lominadze, D., Mchedlishvilli, G. Red blood cell behavior at low flow rate in microvesseles. Microvasuclar. Research 56, (1999). 13