Extracorporeal blood purification by means of the adsorption system MDS is based on high specific microparticle adsorbent for toxin removal. A thin-wall hollow-fiber membrane filter separates the microparticle-plasma suspension from the bloodstream. For patient safety, it is necessary to have a means to detect membrane ruptures that could lead to a release of microparticles into the patient's bloodstream. A non invasive optical detection system was developed to monitor the extracorporeal venous bloodstream for the presence of released microparticles. For detection, fluorescence labeled microspheres are suspended with the adsorbent particles. In the case of a membrane rupture, the labeled particles would be released together with the microadsorbent. A simulation model based on fluidic, gravitational and magnetic forces was developed to simulate the motion and sedimentation of the labeled particles in a magnetic trap. The simulation results show excellent accordance to the laboratory experiments.
Ferromagnetic microparticles, particle sedimentation, particle trap, magnetic separation
In the adsorption system MDS (Microspheres based Detoxification System) , high specific micro adsorbent particles in the size of 1-10 micrometers circulate in an extracorporeal filtrate circle. The MDS is especially used for acute liver dysfunctions. The system consists of two circuits: An extracorporeal blood circuit including a hollow fiber plasma filter and a secondary filtrate circuit with suspended high specific adsorbent microparticles. The fluid exchange through the membrane filter between the blood circuit and the filtrate circuit is depending on the circuit transmembrane pressure. To avoid particle contamination of the patient in case of a broken filter membrane (i.e. too high transmembrane pressure) or a broken filter embedding we add a small amount of labeled particles to the micro adsorbent circuit. The used particles are commercial mono-disperse and ferromagnetic polystyrene microparticles, covered with polyurethane (Dynabeads M-280; Invitrogen Corporation, USA) - and in our lab, covalently labeled with the fluorescence dye cresylviolet .
In case of a filter rupture, the labeled beads are released together with the microadsorbent and can be detected by a high sensitive optical sensor including a magnetic trap. The magnetic trap consists of two permanent magnets which accumulate and focus the ferromagnetic beads in the optical beam of the fluorescence detector device. The fluorescence detector illuminates the control volume of the magnetic trap with light at the absorption band of the dye label. In case of trapped labeled particles, the fluorescence dye emit light shifted to the fluorescence wavelength .
Functionalized magnetic microspheres
In biomedical applications, the most commonly used magnetic microparticles are monodisperse spherically shaped polymers comprised of an inner core containing well-dispersed superparamagnetic nanoparticles and a functionalized outer layer for coupling biochemical receptors.
In general, a single particle type cannot be used for all applications. The size and production process of the magnetic micro- or nanoparticles is determined by the target application. Superparamagnetic, ferro- and ferrimagnetic particles can be used for magnetic drug carrier applications - where superparamagnetic particles are favored for biomedical use, due to the fact that they are non-magnetically when they are not under the influence of an external magnetic field. Therefore undesired magnetic agglomeration is prevented.
Ferromagnetic micro- and nanospheres are currently used for biomolecule and cell separation -, cell sorting -, hyperthermia treatment - and controlled drug delivery -. There are also some investigations to use injectable magnetic nanoparticles for biodetoxification . One emerging strategy for managing overdose involves injecting nanosized magnetic carriers (<1m) to reduce the free drug concentration in the body by acting as a sink for the toxin. The injected nanocarriers are either in the circulatory system or have diffused in the peripheral organs extract the drug from the intoxicated tissue and then exit the body via the kidney or liver.
The surface characteristics of biodegradable and long-circulating polymeric nanospheres based on biocompatible amphiphilic copolymers are discussed in .
In  the methods for calculating the velocity fields for particles in streaming liquid as well as the influence of magnetic and gravimetrical forces onto streaming particles is discussed. Preliminary simulations results for magnetic particles streaming in liquid under influence of external magnetic forces using a 3D model are presented. In the actual paper the simulation model is substantial enhanced where detailed simulations on a 2D model as well as a comparison of the simulation results to laboratory experiments are shown.
To optimize the accumulation of the labeled beads in the magnetic trap, we have developed a mathematical model, to simulate all relevant acting forces to the particle. For our investigations, we have developed a model including fluidic, gravitational and magnetic forces to the ferromagnetic particle.
Velocity field for streaming liquids
Fluidic forces for incompressible viscose liquids are described by the Navier-Stokes equation defined as:
From eq. 1 the velocity field can be calculated under consideration of the specific density, the local pressure, the kinetic viscosity and the local force density from external sources. For incompressible fluids, the divergence of the velocity vector field is zero.
The magnetic trap consists of two rare earth permanent magnets generating an inhomogeneous magnetic field to deflect the ferromagnetic beads.
Substituting the local material equation to the well known Maxwell equations, the magnetic field distribution can be calculated by solving for the magnetic vector potential. The magnetic field density can be easily calculated from the magnetic vector potential using For isotropic media ( is parallel to the particle magnetization ), the acting magnetic force to the ferromagnetic beads is given by 
For small particles in liquid, lift forces can not be neglected and must be considered in opposite to the acting gravitational force. The resulting force depends on the mass difference of equal particle and fluid volumes.
Results and Discussion
To simulate the deflection and sedimentation of the labeled particles in the magnetic trap, the Lagrange equation for moving particles must be solved.
The Lagrange equation describes the movement of the particles under influence of all acting forces summarized in except of the fluidic forces which are part of the equation itself. The indices f and p stand for fluid and particle, where ? is the specific density, V the volume, A the cross-section area, CD the flow resistance, the velocity vector field and the sum of all acting external forces.
Figure 1. Two dimensional model geometry of the magnetic trap.
For the presented model, the flow resistance is calculated the by the well known formula  where Re is the Reynolds number. For spherical particles with radius, Re is defined as
We run our simulations with the described mathematical model, using the software COMSOL multiphysics. The model geometry is shown in fig. 1 using a tube with 4.8mm diameter, and two disc magnets with a diameter of 24mm, a high of 20mm and a magnetic remanence of Br=1.2Tesla. For the simulation, we used water as liquid inside the tubing with a flow speed of 250ml/min and a parabolic flow profile at the inlet. The ferromagnetic particles in the practical laboratory experiment and in the simulation are spheres with a diameter of 30m and a magnetic permeability of 5.
- Br=1.2T, t=30s, Q0=250ml/min
- Br=1.2T, y=Rtubing, Q0=250ml/min
- Br=1.2T, x=xcmax, Q0=250ml/min
Figure 2. Simulated accumulation of ferromagnetic particles in a magnetic trap (Br=1.2T); a) particle concentration after t=30 sec, b) time dependence of particle concentration in the longitudinal cross section and c) in the radial cross section.
In figure 2 and figure 5 the simulation- and the experimental results of the magnetic trap are shown. In the simulation result, the concentration c of the trapped particles is depicted. There are two dominant particle sedimentations along the x-axis; one is around the left edge of the permanent magnet and the second around the right edge. Around the permanent magnet edges there is the highest magnetic field density and therefore the highest resulting magnetic force acting to the magnetic particles. At the left side of the magnet, the resulting magnetic fore is acting against the fluid flow direction and at the right side the magnetic force acts in direction of the fluid flow. Therefore the amount of particle accumulation differs between the two areas where the most particles are trapped at the right end of the magnetic trap at position xcmax. In fig. 2a the concentration of trapped particles after a simulation run of 30 seconds is shown. The color grading is equivalent to the quantity of accumulated particles. In fig. 2b and 2c detailed results for the amount of trapped particles in longitudinal and radial section after different simulation times are shown. The amount of trapped particles saturates after approximately 30 seconds and is shown for different applied magnetic fields in fig. 3.
Figure 3. Influence of the external magnetic field to the particle accumulation at xcmax.
The relative position of the maximum particle accumulation varies with the intensity of the external magnetic field. In fig. 4 the dependence of xcmax on the applied magnetic field is shown. For an optimal detection of accumulated particles in the magnetic trap by the fluorescence detection device, the position of the particle accumulation maximum dependence due to the external magnetic field must be considered.
Figure 4. Relative position of the particle accumulation maximum xcmax.
The laboratory experiment for particle sedimentation in a magnetic trap is shown in fig. 4 after a flow through of t=30 seconds. For the experiments we used ferromagnetic Fe(II, III) oxide particles (Alfa Aesar GmbH, Germany, Ref. No. 12374) with a measured mean diameter of 30m and a magnetic permeability of 5. The ferromagnetic particles (1mg) are pooled in 250ml water and pumped with a flow rate of 250 ml/min through the magnetic trap (fig. 5). To avoid particle sedimentation in the pool, the vessel is placed onto a laboratory shaker and periodically moved.
Figure 5. Laboratory particle accumulation experiment using the magnetic trap (t=30 sec).
The good accordance between our simulation and the laboratory experiments validate the physical model.
The objective of this study was the implementation of a simulation model to characterize the accumulation of ferromagnetic particles in a magnetic trap. The simulation model shows the concentration of accumulated particles along the magnetic trap and gives detailed information about the particle accumulation over time and the local position of the particle sedimentation within the trap. For the validation of the simulation model, results of practical laboratory experiments are compared with the simulation results and show good accordance.
The authors would like to thank the government of Lower Austria, Fresenius Medical Care Germany and the European Commission for supporting the project. Project ID: WST3-T-91/012-2008.
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