Biomedical Engineering Reference
In-Depth Information
Figure 9.27
Graphical scheme of the first-order predictor-corrector method.
In (9.43), two terms on the right-hand side contribute to the convection of the
particles: the drag force of the velocity field and the magnetic force term. Equation
(9.43) must be solved by finite elements or finite differences/volumes numerical
schemes.
9.10.3 Results and Comparison
We examine the case of an MFFF channel of 100-
μ
m width and 1-mm length. The
fluid carrier is water and its average flow velocity is 0.1 mm/s. Particles have a
hydrulic radius of 1.4
μ
m and a magnetic susceptibility of 0.2, their diffusivity is
1.53
μ
m
2
is and the magnetic force is 3.45 pN.
In this particular case, the magnetic force is assumed constant. The velocity
field is obtained under a closed form by (9.34) and the trajectory by (9.38)—if the
injection point is located at the top plate—or else by a predictor-corrector scheme
(9.42).
Figure 9.28 shows contour plots of the particles concentration compared to
the calculated trajectories. The location of the injection is either at the top of the
channel or at 1/3 of the vertical height. As expected, the concentration contours are
centered on the trajectories: Brownian motion slightly disperses the particles from
their determinist trajectory.
In a second step, the same method has then been applied to the case of the
separation of a colloid mixture containing two different types of submicronic para-
magnetic particles (Figure 9.29) differing by their magnetic susceptibility (0.2 and
0.8). We verify that the characteristic relation (9.40) applies even for trajectories
not starting from the depletion (upper) wall.
9.11 Assembly of Magnetic Beads—Magnetic Beads Chains
Superparamagnetic beads in an external magnetic field are similar to small induced
magnetic dipoles [22, 23]. Their magnetic moment is given by
