# Permanent magnet systems to study the interaction between magnetic
nanoparticles and cells in microslide channels

Leon Abelmann1,2 Eunheui Gwag1,3 Baeckkyoung Sung1 1KIST Europe, Saarbrücken,
Germany
2Currently at Delft University of Technology, The Netherlands
3Currently at School of Chemical and Biological Engineering, Institute of
Chemical Processes, Seoul National University, Republic of Korea

###### Abstract

We optimized designs of permanent magnet systems to study the effect of
magnetic nanoparticles on cell cultures in microslide channels. This produced
two designs, one of which is based on a large cylindrical magnet that applies
a uniform force density of 6MN/m^3 on soft magnetic iron-oxide spherical
nanoparticles at a field strength of over 300mT. We achieved a force
uniformity of better than 14 over the channel area, leading to a concentration
variation that was below our measurement resolution. The second design was
aimed at maximizing the force by using a Halbach array. We indeed increased
the force by more than one order of magnitude at force density values over
400MN/m^3, but at the cost of uniformity. However, the latter system can be
used to trap magnetic nanoparticles efficiently and to create concentration
gradients. We demonstrated both designs by analyzing the effect of magnetic
forces on the cell viability of human hepatoma Hep G2 cells in the presence of
bare Fe2O3 and cross-linked dextran iron-oxide cluster-type particles
(MicroMod). Python scripts for magnetic force calculations and particle
trajectory modeling as well as source files for 3D prints have been made
available so these designs can be easily adapted and optimized for other
geometries.

###### Contents

  1. I Introduction
  2. II Theory
    1. II.1 Field and force calculations
    2. II.2 Trajectory calculations
  3. III Experimental
    1. III.1 Fabrication cylindrical magnet system
    2. III.2 Fabrication Halbach magnet system
    3. III.3 Magnetic field measurements
    4. III.4 Microscopy
    5. III.5 Particles
    6. III.6 Channel slides
    7. III.7 Cell staining
    8. III.8 Mortality assay
  4. IV Results and discussion
    1. IV.1 Magnetic field and forces
      1. IV.1.1 Uniform forces: Cylindrical magnet
      2. IV.1.2 Concentration gradients: Halbach array
    2. IV.2 Cell viability
      1. IV.2.1 Cylindrical Magnet
      2. IV.2.2 Halbach array
  5. V Conclusions
  6. A Supplementary Material
    1. A.1 Optical microscope of stained cells and bare Fe2O3 nanoparticles
    2. A.2 Cades input files
    3. A.3 Python source files
    4. A.4 3D-print source files
    5. A.5 Time-lapse videos
    6. A.6 Gradient filling
    7. A.7 High resolution stiched images

##  I Introduction

Depending on their concentration, micro- and nanoparticles can be cyto-toxic
[1] and can enter our bodies accidentally. They can also be administered
intentionally in biomedical procedures such as drug delivery [2, 3] and in
vivo imaging.[4, 5]

To assess the effect of nanoparticles on cells, most interaction studies start
with tests on in vitro cell cultures in multi-well plates. In the case of
small molecules, diffusion ensures that the molecule concentration is
reasonably constant over the volume of the well. However, micro- and
nanoparticles are subject to sedimentation. This has two implications. First,
sedimentation gradually increases particle concentration at the cell membrane,
the rate of which depends strongly on the particle diameter. Secondly, the
particles exert a force on the cell membrane, which may affect particle
incorporation.[6]

The material composition of the nanoparticles is the prime aspect determining
particle toxicity. In this study we focus on magnetic nanopartices, in
particulare those composed of iron-oxide. Iron-oxide has adverse effects at a
concentration in the order of 50g/mL, far above the 1g/mL of for instance
silver nanoparticles.[7, 8] The advantage of magnetic nanoparticles however is
that they can be manipulated by external magnetic fields. The magnetic forces
that one can apply are several orders of magnitude greater than gravitational
forces. Therefore, by magnetically attracting nanoparticles towards the bottom
of the well, we can accelerate sedimentation and increase particle
incorporation. This study of the relation between force and toxicity can help
us to better understand the toxicity of non-magnetic nanoparticles.

The ability to manipulate magnetic nanoparticles magnetically has made them
useful for targeted drug delivery,[9, 10] mechano-stimulation [11] and
hyperthermia treatment.[12] Unprotected iron-oxide particles have strong
effects on cell viability. Experiments with epithelial cells showed that
viability noticeably declined above 50g/mL.[13] Similarly, fibroblast cell
proliferation strongly decreased in the presence of bare iron-oxide
nanoparticles with concentrations above 50g/mL.[14] Therefore, magnetic
nanoparticles for biomedical treatment are usually embedded in a protective
coating.[15] Their interaction with cells depends strongly on the type of
coating. For instance, starch-coated particles show no reduction of cell
viability up to concentrations of 500mg/mL. In contrast, dextran-sulfide-
coated particles have noticeably decreased cell viability at concentrations
above 50g/mL.[16]

The application of magnetic forces has a strong effect on cell viability. To
study the effect of magnetic forces, permanent magnets are typically
positioned below the cell culture dishes or multi-well plates. Prijic and
colleagues elegantly demonstrated the increased uptake of super-paramagnetic
particles in human melanoma and mesothelial cell cultures placed on top of
magnets.[17] They found that the total iron content in the cell, measured by
inductively coupled plasma atomic emission spectroscopy, increased by a factor
of $3$–$8$. Cell viability decreased by 50 for a concentration of 100g/ mL.

A particularly convincing method to study particle uptake is to use magnetic
particles to transinfect cells, a method called magnetofection.[18, 19]
Pickard and Chari [20] attached green fluorescent protein (GFP) plasmids to
Neuromag SPIONs. When the particles entered the cell, the plasmids were
reproduced. The subsequent generation of GFP determined whether cells are
transfected. The application of force by means of magnetic field gradients
enhanced the uptake by a factor of 5. A slow oscillation appeared to have a
positive effect.

Particles in a cell can be identified with optical microscopy by means of
fluorescence, for which Dejardin and colleagues used ScreenMag-Amine magnetic
particles tagged with fluorescein.[21] The particles were treated with
activated penetratin to increase their uptake. By integrating the intensity of
the emitted light over the sample area, an increase in uptake of about 30 was
observed. A similar approach was taken by Venugopal and colleagues [22] as
well as by Park and colleagues,[23] who used flow cytometry to demonstrate an
increase in uptake ranging from a factor of $0.5$ to $7$, respectively.

In all cases cited above, the magnets were smaller than or of a similar size
as the cell culture area, leading to particle accumulation in the center of
the observation area and subsequent loss of information on particle
concentration. In this work, we designed magnetic systems for maximum
uniformity so that the particle concentration is known more precisely.

If a uniform particle concentration is not important, for instance if the aim
is merely to capture particles, then non-uniform forces are not an issue. In
that case, higher gradients can be achieved by arrays of magnets. For example,
a (circular) Halbach array has been used to capture magnetically labeled
connective tissue progenitor cells from bone marrow [24] or to trap
magnetically labeled cells in the leg [25] or the brain.[26] Using a yoke with
an embedded rotating magnet, so that the field can be removed [27],
magnetically labelled polyclonal antibodies could be captured at efficiencies
of 95. [28]

In a microfluidic system, non-uniform magnetic forces are used to capture
magnetically labeled circulating tumor cells by means of small single magnets
[29] or by linear [30] or rotating cylindrical Halbach arrays.[31]

Most of the magnets used are alloys of NdFeB. These can be manufactured down
to millimeter dimensions. Smaller magnets can be obtained with thin-film
technology by using lithography to pattern the thin film. Examples are
12-$\mu$m-thick, 40x40m permalloy squares, which were used to capture cells
decorated with magnetic particles,[32] or 100-nm-thick 300x50 bars to grow
magnetically labeled neurons directly.[33]

The research mentioned above focused primarily on the effect of magnetic
forces on cells, but less so on the design of the magnetic systems themselves.
In this contribution, we therefore analyzed the magnetic force profiles in
more detail, either to achieve as uniform a field as possible by using bigger
magnets, or to achieve high forces by using Halbach arrays. The designs were
made specifically for microslides dedicated to cell cultures. We used these
combinations of magnetic systems and microslides to study the effect of
magnetic forces on the viability of human liver cells in the presence of
magnetic nanoparticles.

In the following, we introduce models to calculate magnetic fields, forces and
trajectories of magnetic nanoparticles. We compare calculations and
measurements of the fields and forces on a system designed for maximum
uniformity and on a system based on a Halbach array. We will then draw
conclusions regarding the strength of fields and forces of both magnetic
systems and the efficiency of particle capture under flow conditions for a
Halbach array.

To illustrate the application to cell viability studies, we analyzed the
effect of bare and protected iron-oxide particles on the viability of liver
cells with and without an applied magnetic force.

##  II Theory

###  II.1 Field and force calculations

To calculate the magnetic field and forces generated by a single cylindrical
magnet, we integrated magnetic charge densities. In contrast to finite-element
methods, the field is calculated only at the points of interest, which is much
faster at high precision. The resulting equations are generated automatically
by the MagMMEMS package, which is a preprocessor for Cades.[34] The input
files are available in the Supplementary Material.

For Halbach arrays, the calculated field profiles serve as input to particle
trajectory calculations. Therefore, we implemented the integrals in Python
using the dblquad integration method of the scipy package. The source files
are available on github (https://github.com/LeonAbelmann/Trajectory).

The force on a magnetic object, which is small compared to the spatial
variation of the externally applied field $\boldsymbol{B}$ [T], can be
approximated from its total magnetic moment $\boldsymbol{m}$ [A/m]

$\displaystyle\boldsymbol{F}=-\boldsymbol{\nabla}(\boldsymbol{m}\boldsymbol{B})\text{.}$
(1)

The magnetic moment of a magnetic object in a fluid is generally a function of
strength, direction and history of the applied field. The applied field is a
combination of the external field and the field of all other magnetic
particles in the fluid. Moreover, very small particles will be subject to
Brownian motion. Therefore, in principle, the calculation of forces on
magnetic particles in a magnetic field gradient is complex. To obtain first
approximations, we consider a single particle that is either a permanent
magnetic dipole or a soft magnetic sphere.

In the case of the permanent magnetic dipole approximation, we assume a
particle with a permanent magnet moment
$\mu_{0}m_{\text{r}}=I_{\text{r}}V_{\text{p}}$, where $I_{\text{r}}$ [T] is
the remanent magnetization of the particle with volume $V_{\text{p}}$ [m3]. We
further assume that field changes are slow such that particles can rotate
against viscous drag into the direction of the field. In this case, Eq. (1)
reduces to

$\displaystyle\boldsymbol{F}=m_{\text{r}}\boldsymbol{\nabla}B\text{,
}B=|\boldsymbol{B}|\text{.}$ (2)

For the second approximation, we assume a soft magnetic sphere with a
susceptibility of $\chi$, which is the ratio between the magnetization $I$ [T]
in the particle and the internal field $B_{\text{in}}$ [T]. As a sphere has a
demagnetization factor of 1/3, the internal magnetic field is

$B_{\text{in}}=-B+\frac{1}{3}I=-B+\frac{1}{3}\chi B_{\text{in}}\text{,}$

where

$I=\frac{3\chi}{(3+\chi)}B$

and all fields are (anti-)parallel. In this approximation, the energy and
resulting force are

$\displaystyle U$
$\displaystyle=-\frac{1}{2}V_{\text{p}}\frac{3\chi}{\mu_{0}(3+\chi)}B^{2}$ (3)
$\displaystyle F$
$\displaystyle=\frac{1}{2}V_{\text{p}}\frac{3\chi}{\mu_{0}(3+\chi)}\boldsymbol{\nabla}B^{2}$
$\displaystyle=V_{\text{p}}\frac{3\chi}{\mu_{0}(3+\chi)}(\boldsymbol{B}\boldsymbol{\nabla})\boldsymbol{B}\text{.}$

The factor 1/2 originates from integrating from $-\infty$, where the energy is
0, and we used the vector identity
$\boldsymbol{\nabla}B^{2}=2(\boldsymbol{B}\boldsymbol{\nabla})\boldsymbol{B}$.

###  II.2 Trajectory calculations

In our experiments with the Halbach array, we filled the channel with the
nanoparticle suspension when it is on top of the array. The nanoparticles are
taken along by the fluid flow and at the same time dragged down towards the
array. As the particles are small and flow velocities are low, inertial
effects are negligible. In that case, the magnetic force $F_{\text{mag}}$ (N)
is instantly balanced by the drag force

$F_{\text{mag}}=6\pi\eta r(v_{\text{p}}-v_{\text{f}})\text{,}$ (4)

where $\eta$ is the fluid viscosity (set to
$1\text{\,}\mathrm{m}\mathrm{P}\mathrm{a}\mathrm{s}$). From the force balance,
we can obtain the velocity of the particles $v_{\text{p}}$ with respect to the
fluid background velocity $v_{\text{f}}$ (m/s).

The average background fluid velocity $v_{\text{flow}}$ is estimated from the
filling time and length of the channel to be around
$25\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{s}$. As the flow velocity is
zero at the surfaces of the channel, the flow profile is parabolic

$v_{\text{f}}(z)=1.5v_{\text{flow}}\left(1-(\frac{2z}{h}-1)^{2}\right)\text{,}$
(5)

where $h$ is the channel height ($0.8\text{\,}\mathrm{m}\mathrm{m}$). We
assume the flow direction to be uniquely along the channel length ($x$).
However, the magnetic force is allowed to have components in all directions.

The resulting differential equation is solved by using the solve_int routine
of the scipy.integrate package using the RK23 integration method with an
absolute tolerance of $1\text{\times}{10}^{-5}$ and relative tolerance of
$2\text{\times}{10}^{-3}$. [35, 30] Particles at the top of the channel have
the least chance of being captured. In case of incomplete capture, the capture
height and capture efficiency are obtained by integrating backwards in time
from a trajectory that starts exactly at the bottom of the channel at the
channel exit. Source codes are available on github
https://github.com/LeonAbelmann/Trajectory.

##  III Experimental

###  III.1 Fabrication cylindrical magnet system

The aim of the first magnetic system was to obtain as uniform a force as
possible over the area of interest. To maximize uniformity, we used the
largest NdFeB magnet we could readily obtain (Supermagnete.de). This magnet
has a diameter of 70mm, a height of 35mm and is made of N45, which is
specified to have a remanent magnetization of 1.3T. These magnets can
seriously injure the experimentalist if accidentally brought too close
together. Therefore we encased them in PVC (70mm (121605), 100mm (121457) and
125mm (121620) reducer rings from Wildkamp, Netherlands). The position of the
$\mu$-slide channel and the distance to the magnet were accurately fixed using
a 3D-printed nylon plastic top holder, see Figure 1. The holder can be
disinfected by a 70 isopropanol solution. The source files for the 3D-printed
holder are available in the Supplementary Material.

Figure 1: System to investigate the effect of a high magnetic field gradient
from a single magnet on the interaction of magnetic nanoparticles on cells.
The magnet has a diameter of 70mm and a height of 35mm. The cell culture is
located 10.5mm above the magnet in a slide channel.

###  III.2 Fabrication Halbach magnet system

The Halbach array was assembled from $48$ individual 1.5x1.0x5.0mm magnets
(supermagnet.de), see Figure 2. The magnets were placed at a distance on a
thin copper foil on a tapered soft magnetic plate and then carefully pushed
together. After assembly, the array was glued together using a thin
cyanoacrylate glue, and carefully slid off the thin end of the wedge. The
array, including the copper foil, was then glued with the foil on top onto a
3D-printed holder. The design for the holder is available in the Supplementary
Material.

Figure 2: System to investigate the effect of concentration gradients. A
strong field gradient is achieved by means of a Halbach array consisting of
$48$ 1.5x1.0x5.0mm magnets (top). The Halbach array (top) is embedded in a
3D-printed holder (center) that keeps the channel slide in place (bottom).

###  III.3 Magnetic field measurements

The magnetic field above the cylindrical magnet (Figure 3) and Halbach array
(Figure 7) was measured with a MetroLab THM1176 three-axis Hall sensor. The
sensor was placed at the same height as the channel of the $\mu$-slide and
translated using a manual micromanipulator. The location of the probe was
estimated to within 1mm and was then fine-tuned laterally to the symmetry in
the field profile. The orientation of the probe with respect to the vertical
($z$-axis) is difficult to adjust and could deviate up to $None$.

The field profile of the Halbach array was visualized using a magnet viewer,
see Figure 7, which has low contrast when the field is parallel to the sheet
(TRU components 507706).

###  III.4 Microscopy

Overview images of the particle distribution in the $\mu$-slide channel were
taken by a Canon EOS 800D camera with a Canon EFS18-55 lens at a distance of
approximately 20cm, see Figure 8.

The contrast variation over the channel is very low as observed on the
cylindrical magnet. For this reason, we mounted the camera in a light-shielded
box and illuminated it with a home-built LED ring driven by a constant
voltage, see Figure 5. For future experiments, we advise using a constant-
current driver to avoid illumination intensity changes during long
experiments. Time-lapse images were taken at 2-min intervals using the
external shutter input of the Canon camera. The images were then isolated
using an optocoupler and driven by an Arduino Uno.

Images were taken using the raw CR2 format (41MB each) until the 64-GB SD card
was full (approximately $128$ images). A Python script was used to load the
CR2 images and average the pixel intensity $I$ for the green channel over
$160$ slices of 16x400pixel (0.2x3.6mm). A reference image was taken from an
empty channel ($I_{0}$), and the measured average intensity was converted to
absorbance using $\log_{10}(I/I_{0})$.

Higher-resolution images of particle suspension inside the $\mu$-slide channel
on top of the cylindrical magnet were taken with a Zeiss Stemi 508 stereo
microscope, see Figure 6.

When the $\mu$-slides are removed from their magnet, they can be imaged by
transmitted light microscopy. The images in Figure 11 were taken with a Zeiss
Axiovert 100 using a 12V 100W halogen light source, a Zeiss LD A-Plan
10$\times$/0.25 Ph1 lens and a Jenoptik Gryphax Prokyon camera. The microscope
stage is equipped with stepper motors. To image along the channel, images were
taken every 1mm at fixed illumination and combined into a 0.7x36mm image using
a Hugin photo stitcher (hugin.sourceforge.io). The intensity profiles were
extracted from the stitched images by means of Gwyddion’s “extract profile
along arbitrary lines” tool, using the green channel for the bare Fe2O3
particles and the red channel for the core-shell particles and a width of 128
pixels.

The $\mu$-slide channels are dedicated to cell studies. Images of cells
(Figures 13, 14 and 15) with a size of 2048x2048pixel were taken with a Leica
DMi8 fluorescence microscope with a K5 sCMOS camera. Both a $20$$\times$ and a
$40$$\times$ lens were used, calibrated at $324$ and 162nm/pixel,
respectively. To observe the channels on the Halbach array, we 3D-printed a
holder that allowed us to image the location along the channel with a
reproducibility of at least $1\text{\,}\mathrm{m}\mathrm{m}$.

###  III.5 Particles

We used both bare Fe2O3 and core-shell magnetic nanoparticle suspensions. The
bare Fe2O3 suspension was based on iron(III) oxide powder (Alfa Aesar NanoArc,
45007). According to the manufacturer, the particles have a diameter of 2040nm
and are over 98 in the $\gamma$ crystal phase. The powder was mixed with
demineralized water at an iron concentration of 10.6/. From experiments by
Prijic [17] and Rafieepour [13], we esimated that for significant cell
mortality, we needed to apply concentrations well above 250/ml. . Therefore,
for cell studies on the cylindrical magnet, the stock suspension was diluted
$27$ times into a phosphate-buffered saline (PBS) solution with a pH of $7.4$
to achieve an iron concentration of 393(20)/, corresponding to ($2$ to
$18$)$\times$e12particles / . For the experiments on the Halbach array, the
suspension was diluted further to 100(5)/, so that the expected maximum
concentration was approximately equal to the experiment with the cylindrical
magnet.

In addition to the bare Fe2O3 powder, we used crosslinked dextran iron-oxide
cluster-type particles prepared by a core-shell method (Bionized NanoFerrite
(BNF) 94-00-102 from Micromod). These particles are red fluorescent (redF) and
shipped in a PBS suspension. According to the manufacturer, they have a core
of 7580 (w/w) magnetite and a shell of dextran. The particles have a reported
diameter of 100nm with a magnetite crystallite diameter of about 20nm. The
hydrodynamic diameter of these particles lies between $95$ and 140nm.[36] The
original iron concentration as reported by the manufacturer is 6.0/, and
particle concentration is 6e12particles/mL. The original suspension was
diluted into PBS to achieve an iron concentration of 280(10)/, corresponding
to a concentration of 3e12particles / .

Suspensions were kept inside a refrigerator in the dark. Before use, the tubes
with suspensions were vigorously shaken and ultrasonically agitated for
approximately one minute. For studies that did not involve cells, the PBS was
replaced with demineralized water.

###  III.6 Channel slides

All experiments were performed using Ibidi $\mu$-Slide I Luer channels with an
Ibitreat surface coating to promote cell adhesion (Ibidi 80606). The dimension
of the channels is 0.8x5.0x50mm.

###  III.7 Cell staining

Cells were stained for the microscopy images in Figures 13 and 15 with an Iron
Stain Kit (ab150674; Abcam, Cambridge, UK). To identify cellular uptake and
localization of iron-oxide nanoparticles, HepG2 cells in the $\mu$-slide were
washed with PBS and incubated in a mix of potassium ferrocyanide and
hydrochloric acid for 3 min. After being rinsed with deionized water, the
cells were counterstained with a nuclear fast red solution.

###  III.8 Mortality assay

For the cell studies (Figures 12 and 14), human hepatoma HepG2 cells (ATCC,
HB-8065) were cultured in Eagle’s minimum essential medium supplemented with
10 fetal bovine serum and 1 penicillin-streptomycin in an incubator at 37 and
5 CO2 atmosphere. The cell concentration was determined by means of a
hemocytometer and diluted to 5e4cells/. Of this solution, 200 was introduced
into the $\mu$-slide channels. The maximum cell density was 40cells/mm^2.

The Ibidi channels were left inside the incubator for 24hours on top of the
holder with and without a magnet before analysis. To assess cell viability, we
used a live/dead double-staining assay (Sigma-Aldrich, 04511). Ten fluorescent
microscopy images (Leica DMi8) were chosen randomly over the channel area for
analysis. Average and standard deviations were calculated from three
independent experiments ($30$ images in total).

##  IV Results and discussion

###  IV.1 Magnetic field and forces

To verify the calculations of fields and forces, we measured the fields of
both the cylindrical magnet and the Halbach array as well as the concentration
variation over the channel filled with magnetic nanoparticles.

####  IV.1.1 Uniform forces: Cylindrical magnet

The force field above the permanent cylindrical magnet varies with distance to
the magnet surface in both strength and direction. For experiments with
magnetic nanoparticles inside the channel slide, we want the lateral forces in
the plane of the channel to be as small as possible, yet the vertical force to
be as high as possible and very uniform over the area of interest. Using the
soft sphere model of Eq. 3, we determined that the optimal height of the
channel slide for both conditions is 10.5mm above the magnet.

Figure 3: Field strength above the cylindrical magnet showing both calculated
and measured vertical (top) and lateral (bottom) field components at the
height of the channel slide (10.5mm). Calculated field values for 9.5mm and
11.5mm are shown by thinner lines as well. Assuming a remanent magnetization
of 1.30T, the field is predicted within measurement error. Over the length of
the channel (30mm), the vector component of the field in the vertical
direction is dominant, with a strength greater than 300mT.

Figure 3 shows the measured and calculated magnetic field components
perpendicular to channel $B_{\text{z}}$ and along channel $B_{\text{x}}$ at
the optimum height of 10.5mm. The measurements follow the prediction quite
accurately, but the deviations are greater than the field measurement
uncertainty. Additional sources of uncertainty are the angle of the sensor
with respect to the magnet’s surface and the distance between the sensor and
the magnet’s surface. The latter deviation has a greater impact and may vary
over the radial distance. Therefore, we also show predictions that are 1mm
higher and lower than the optimum height, respectively. Assuming a remanent
magnetization of 1.3T as provided by the manufacturer’s data, the measurements
fall in this band within measurement error. Over the length of the channel
(30mm, radial distance 15mm), the perpendicular field component is dominant
with a value in excess of 300mT. The field strength varies only by 5, and the
field angle rotates at a maximum of $\pm$17.

Figure 4: Force densities calculated from the field profiles shown in Figure 3
in vertical (black) and lateral (blue) directions as a function of the
position over the length of the channel slide. We apply two extreme models.
The top curve shows the result for a model in which we assume that the
particle is a permanent magnet with remanent moment $I_{\text{r}}$ (Eq. 2). To
obtain the force, multiply by $V_{\text{p}}I_{\text{r}}$. For the bottom
curve, we assumed that the particle is a sphere of volume $V_{\text{p}}$ and
susceptibility $\chi$ (Eq. 3). To obtain the force, multiply the value on the
vertical axis by $3V_{\text{p}}\chi/(3+\chi)$. The vector component of the
force in the vertical direction is dominant, with a strength in excess of
3MN/m^3. Over the length of the 30mm channel, the force strength remains
within 14 and the force direction varies by less than 1.

Figure 4 shows the calculated forces on particles for both the permanent
dipole magnet (top) and soft sphere (bottom) models. For the permanent dipole
model, the forces are normalized to the particle magnetic moment
$I_{\text{r}}V_{\text{p}}$ [Tm3]. A typical particle has a magnetization in
the range from 0.1 to 1T, so force densities are on the order of 1 to
10MN/m^3. For comparison, the gravitation force density on a typical iron-
oxide particle is only 40kN/m^3 (standard gravity times mass density
difference with water).

For the soft sphere model, the forces are normalized to
$3V_{\text{p}}\chi/(3+\chi)$, which ranges from 0 to 3$V_{\text{p}}$. A
typical iron-oxide particle has a susceptibility greater than 1 [37], so force
densities are in the same range as for the permanent dipole approximation.

Both models show a variation of 14 in the vertical component of the force over
the length of the 30-mm channel. The soft sphere model has a much weaker
lateral force. At the entrance of the channel, the force tilts only about 1
inward, whereas for the permanent dipole model, it tilts at 10.

Figure 5: Channel filled with core-shell magnetic nanoparticles immediately
after filling (top, black) and after 4.3h. The graph shows the absorbance
averaged over a 3.6-mm band in the center of the channel, referenced to an
empty channel. In the Beer–Lambert approximation, the absorbance is
proportional to the concentration. Error bars indicate the standard deviation
of the absorbance over the width of the band. The concentration variation over
the length of the channel is less than the measurement uncertainty of 12. The
reduction in light absorbance with time could also be caused by a gradual
decrease in the intensity of the light source. A time-lapse video of the
256-min process is available in the Supplementary Material.

The large magnet produces a very uniform distribution of magnetic particles.
Figure 5 shows a channel slide filled with a diluted suspension of 5-nm iron-
oxide particles. No concentration gradient can be observed even after more
than 4hours. We measured the intensity of the image over a 3.6-mm band over
the center of the channel. By comparison with an empty channel, we can measure
the concentration variation. Assuming that the logarithm of the ratio of the
filled and unfilled channels is proportional to the concentration
(Beer–Lambert law), the variation in concentration along the channel is less
than the measurement uncertainty of 12. A time-lapse video of the 256-min
process with timesteps of 32s is available in the Supplementary Material
(CoreShell.mp4). There is a gradual reduction in overall intensity. Closer
inspection of the images shows that the magnetic field causes the appearance
of many small spots in the case of core-shell particles, see Figure 6, left.
We assume that the appearance of these spots is caused by particle clustering,
which may explain the reduction in overall reflection of light. Clustering can
also be observed for the bare Fe2O3 particles, see Figure 6, right.

Figure 6: Optical microscopy of bare Fe2O3 and core-shell particles without an
applied magnetic field (top) and on top of the cylindrical magnet. In the
presence of the magnetic field, the particles have a greater tendency to
cluster, leading to a reduction of light brown areas in the case of Fe2O3 and
the appearance of many more small spots in the case of the core-shell
particles.

####  IV.1.2 Concentration gradients: Halbach array

The aim of the large cylindrical magnet described above was to create as
uniform a force as possible. If uniformity is not an issue, then a Halbach
array can create much higher forces.

Figure 7 (top) shows the calculated and measured magnetic field profiles of a
Halbach array of $16$ 1.5x1.0mm magnets. The total width of the array (in
$y$-direction) is 15mm. The values of the field components fit very well with
measurements if we assume a distance between probe and magnet surface of
0.58mm, which is very reasonable. The maximum field strength is on the order
of 400mT, which is comparable to the large cylindrical magnet.

An image of a magnetic viewer film is shown in the background. The light and
dark areas indicate regions where the field is parallel and perpendicular to
the plane of the viewer film, respectively. The measurement and model show
very good qualitative as well as quantitative agreement. There is a small
shift in the pattern, which may be caused by the fact that the distance
between magnet centers is slightly increased because of the glue used to
assemble the array.

Using this model, we can calculate the forces on magnetic nanoparticles. The
bottom two graphs in Figure 7 show the in-plane and perpendicular force
components for a model, assuming that the nanoparticles are either permanent
magnets (center) or soft magnetic spheres (bottom). The perpendicular force is
always attractive, and varies by 25 (permanent dipole) to 30 (soft sphere). In
contrast, the in-plane force can be both positive and negative. Particles will
therefore tend to diffuse to regions where the in-plane force is zero, leading
to very non-uniform particle concentrations. However, the force densities are
approximately $60$ to $90$ times higher than for the cylindrical magnet. This
is the effect of reducing the size of the magnets by a factor of $70$ and
using the Halbach configuration.

Figure 7: Calculated and measured vertical and lateral components of the field
(top) and calculated forces on permanent magnet (center) and soft magnetic
sphere (bottom) particles as a function of distance along the Halbach array.
To obtain the force, multiply by $V_{\text{p}}I_{\text{r}}$ for the permanent
magnet or by $3V_{\text{p}}\chi/(3+\chi)$ for the soft magnetic sphere. In the
background of the top field graph, the field direction is shown by means of a
magnetic viewer (in the light areas, the field is parallel to the viewer
sheet). The circles are measured values. The height was set to 0.58mm in the
calculation to match these values. As can be seen by the shift in the measured
pattern, the dimensions of the magnets are slightly larger than specified. The
field strength is of the same order as was the case for the uniform field in
Figure 3. However, the forces are more than one order of magnitude higher
because the individual magnets in the Halbach array are $70$ times smaller. In
contrast to the cylindrical magnet, the Halbach array has very non-uniform
forces, with lateral forces being both positive and negative and vertical
forces varying by as much as 30.

In the experiment with the large cylindrical magnet, the channel was filled
first with magnetic nanoparticles, so that the concentration distribution was
uniform. The channel slide was then carefully lowered vertically along the
axis of the magnet to avoid disturbing the concentration. In the case of the
Halbach array, we first positioned the empty slide on the array, slightly
titled, and filled it from the lower side, see Figure 8. As the suspension
flows over the array, particles are trapped by the magnetic field gradient,
and the concentration decreases. This achieves a gradient in the
concentration. The filling process can be observed in the video in the
Supplementary Material (GradientFilling.mov).

As the particles are trapped at the transitions between the magnets, the
concentration oscillates slightly. However, the increase in intensity is more
or less linear over a range of 12mm (from 820mm). Assuming a Beer–Lambert law
for the relation between intensity and concentration, the concentration
decreases approximately exponentially. This suggests that a fixed fraction of
the particles is indeed captured as the suspension flows over the array.

Figure 8: As the channel slowly fills, bare Fe2O3 nanoparticles are captured
in the stray field of the Halbach array. Particles are captured as the fluid
moves over the array, leading to a concentration gradient. The contrast of the
image is shown below and increases almost linearly from 5 to 90bit in the
range of 820mm. On the basis of the Beer–Lambert law, we conclude that there
is a logarithmic reduction in concentration in this region. A video of the
filling process (GradientFilling.mov) is available in the Supplementary
Material.

The velocity of the particles with respect to the fluid is determined by the
balance between magnetic and drag force. Using the magnetic force calculated
by the permanent dipole model and assuming a parabolic flow profile, we can
calculate the trajectories of magnetic nanoparticles in suspension, see
Section II.2. Figure 9 shows an example of magnetic nanoparticles with a
radius of 67nm and a remanent magnetization of 1.0T in a flow with an average
velocity of 25mm/s. For this particular situation, about half the particles
are trapped on the array, whereas the other half leaves the channel slide.
Particles at the bottom of the channel are captured more easily. Using this
type of calculation, we can predict a capture height and capture efficiency.
In this particular calculation, the capture height is 0.4mm and the capture
efficiency is 50.

Figure 9: Calculated trajectories of magnetic nanoparticles released at the
left-hand side of the Halbach array. The flow profile in the channel is
assumed to be of parabolic shape with an average flow velocity of 25mm/s, or
6mL/min. Note that the $x$ and $z$ scales of the image are very different. The
particular example we have chosen here illustrates the situation where 50 are
captured (particles with a radius of 67nm and magnetic remanence of 1.0T.)

The magnetic force of the particles in the suspension scales with their
volume, whereas the drag force scales with the radius. The smaller the
particles, the less chance they have of being captured. Figure 10 shows the
estimated capture height and capture efficiency as a function of particle
radius (again assuming particles with a remanent magnetization of
$1.0\text{\,}\mathrm{T}$). Up to a particle radius of about 75nm, the capture
rate increases linearly with particle radius. Owing to the parabolic flow
profile, the capture efficiency increases quadratically. Particles with radii
above 110nm are all captured. Particles with radii of less than 20nm have a
chance of less than 7 of being captured in this specific situation.

Figure 10: Calculated capture height, see Figure 9 (left axis, black) and
resulting capture efficiency (right axis, red) as a function of particle
radius. The capture rate for particles with radii below 20nm is less than 7.
Owing to the parabolic flow profile, the capture efficiency increases
quadratically with increasing particle radius. A capture efficiency of 50 is
reached for a particle radius of about 67nm, which is the example shown in
Figure 9. For small particle radii, the capture height increases linearly with
increasing particle radius. Above this value, the linear relationship between
capture height and radius is lost. The red line is not a fit, but shown merely
to guide the eye. Full capture is achieved for particle radii above 110nm.

Experiments on liver cell viability were performed with bare Fe2O3 particles,
as described above, as well as with core-shell magnetic nanoparticles. Figure
11 shows stitched microscope images over the length of the channel slide for
both types of particles. The gradient in particle concentration is clearly
visible. The black dots at the entry of the channel with the Fe2O3 suspension
are gas bubbles. These bubbles did not form in the suspension of core-shell
particles. The Fe2O3 particles show a less uniform distribution. Rather, the
particles cluster into elongated structures with lengths up to 100. At the
start of the Halbach array, these clusters are irregularly distributed.
Further into the channel, the distribution becomes more regular. Above the in-
plane magnets of the array, the clusters align along the channel; above the
perpendicular magnets, the clusters are oriented more randomly. Not all
particles are captured by the Halbach array. At the end of the array, a number
of larger clusters can be found. In the remaining part of the channel, chains
of particles can be seen aligned very well along the channel length. For the
Fe2O3 particles, this alignment is lost towards the end of the channel. In the
case of the core-shell particles, the alignment turns towards 45 at the end of
the channel.

In the case of bare Fe2O3 particles, we observed the formation of tiny bubbles
at the entry of the channel. We have two hypotheses regarding the origin of
these bubbles. They could be caused by heating of the fluid due to the intense
illumination in the microscope. The increased temperature of the fluid could
trigger the release of dissolved gases (nitrogen, oxygen). Indeed, we have
been able to melt the channel slide by using full-intensity light, so an
increase in temperature is very likely. However, the channel filled with core-
shell particles did not show bubble formation, not even in the very dark, i.e.
light-absorbing, area from 03mm. Bubbles formed also at moderate light
intensity, see Supplementary Materials, video Fe2O3.mp4. Our other hypothesis
is that Fe2O3 causes photo-oxidation of water, releasing hydrogen and
oxygen.[38] Indeed, when a suspension of bare Fe2O3 is placed in sunlight, gas
bubbles are produced continuously for days. In the case of core-shell
particles, the dextran shell may prevent contact between water and the Fe2O3,
which could explain the absence of bubbles. Further research will be required
to determine the origin of the bubbles. In any case, if the concentrations are
low and/or illumination is moderate, bubbles do not form.

Figure 11: Microscopy images of bare Fe2O3 (top) and core-shell particles
(bottom) after the channel slide is removed from the Halbach array. The images
were stitched to visualize a $36$$\times$0.7mm^2 band along the center of the
channel and are available in full resolution in the Supplementary Material
(FullGradient*.png). Typical examples are extracted, and the estimated
location of the Halbach array is shown. The bare Fe2O3 particles show a more
non-uniform distribution. Particles tend to cluster into structures that can
be as long as 100. At the start of the Halbach array, the clusters are
irregularly distributed. Midway through the array, the clusters align along
the channel. Not all particles are captured. Those that escape the Halbach
array tend to align into chains along the channel length. The bottom graph
shows intensity profiles along the stitched images for both types of
particles. The change in intensity along the channel is stronger for the bare
Fe2O3 particles. The strong variation at the start of the array is caused by
the gas bubbles.

###  IV.2 Cell viability

We demonstrated the use of two designs on human hepatoma HepG2 cells in
channel slides. The viability of the cells in the presence of magnetic
nanoparticles was observed over a period of one to six days.

####  IV.2.1 Cylindrical Magnet

Figure 12 shows the viability of HepG2 cells inside channel slides after six
days, with and without application of a uniform magnetic force and with and
without the presence of magnetic nanoparticles. The application of a magnetic
force by itself does not significantly affect cell viability. However, when
there are magnetic nanoparticles in the solution, the application of a
magnetic force strongly reduces cell viability. The effect is similar for bare
Fe2O3 and core-shell nanoparticles. There seems to be a small reduction in
cell death when magnetic nanoparticles are added. However, this effect is only
statistically significant for the core shell particles.

Figure 12: Human hepatoma HepG2 cell mortality after
$6\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$, with (red) and without
(black) application of a uniform magnetic field. Error bars indicate the
standard error, based on ten images per slide and three independent
experiments. The control without magnetic particles shows that the magnetic
field by itself has no statistically significant effect on cell mortality
(confidence less than $95\text{\,}\mathrm{\char 37\relax}$). The presence of
bare Fe2O3 particles at a concentration of
$400\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$
and core-shell particles at
$280\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$
increases cell mortality up to approximately $50\text{\,}\mathrm{\char
37\relax}$ (confidence exceeds $99\text{\,}\mathrm{\char 37\relax}$).

Closer inspection of the cells and nanoparticles in the channel slides shows
that, on the magnet, the nanoparticles tend to cluster more strongly, see
Figure 13. The effect is strongest for the bare Fe2O3 particles, which form
elongated structures with lengths of up to 100. However, cell morphology does
not seem to be strongly affected by the application of a magnetic force.

Figure 13: Optical microscopy image of HepG2 cells cultivated for six days in
the presence of bare Fe2O3 (left,
$400\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$)
and core-shell particles (right,
$280\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$).
The top and bottom rows show cells that were cultivated off- and on-magnet,
respectively. The nuclei in the cells are stained red; the iron-oxide is
stained blue. Cell morphology is not noticeably dependent on the type of
particles. In the magnetic field, the particles agglomerate into chains. This
effect is strongest for the bare Fe2O3 nanoparticles.

####  IV.2.2 Halbach array

Figure 14, top, shows the viability of HepG2 cells in the presence of bare
Fe2O3 nanoparticles with an average concentration of 100g/mL. If the channel
slide is not positioned on the Halbach array (“control”), the nanoparticles
have no effect on cell viability. This is in agreement with experimental
results for the cylindrical magnet.

If the channel with HepG2 cells is filled when it is positioned on top of the
Halbach array, a gradient forms. If it is kept there for several days, cell
viability strongly decreases. The decrease is strongest at the start of the
channel where the concentration is highest. From 15mm onwards, there is no
difference in the control.

If we assume that all particles are captured on the array, i.e. on the first
half of the microslide, the average concentration doubles to 200g / mL. Taking
the intensities extracted from Figure 11, we can use the Beer–Lambert model to
estimate the concentration as a function of position. With a single
experiment, we can therefore obtain a first estimate of cell viability versus
concentration. The result is shown in Figure 14 (bottom). Cells in areas where
the concentration is below 75g/mL are not noticeably affected.

Figure 14: Top: Viability of HepG2 cells cultivated in the presence of bare
Fe2O3 nanoparticles with concentration decreasing to zero at about
$20\text{\,}\mathrm{m}\mathrm{m}$ as a function of time. Error bars indicate
the standard error, based on ten images per slide and three independent
experiments. As the control (left), a channel slide without magnetic particles
was analyzed at day 6. Bottom: Viability at day 6 as a function of
nanoparticle concentration, estimated based on intensity gradients as shown in
Figure 11 and the Beer–Lambert law. Error bars indicate the standard error for
the fraction of dead cells and standard deviation for the estimated
concentration. Cell mortality increases with increasing concentration and time
up to $40(5)\text{\,}\mathrm{\char 37\relax}$ at the highest concentration
after 6 days of exposure. Cells at a distance of
$15\text{\,}\mathrm{m}\mathrm{m}$ and above, i.e. with concentrations below
$75\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$,
are not noticeably affected.

##  V Conclusions

We constructed magnetic systems that exert a strong force on magnetic
nanoparticles inside a channel slide with a two-dimensional human liver cell
culture. These systems were used to assess the effect of these magnetic forces
on the mortality of mixtures of liver cells and nanoparticles. The
nanoparticles we used were bare Fe2O3 with diameters of $20$ to 40nm and core-
shell particles of 100nm diameter with a magnetite core and a cross-linked
dextran shell.

The first system we investigated exerts a uniform force over the entire
channel via a cylindrical magnet with a diameter of 70mm. The vector
components of the field and force in the vertical direction are dominant, with
a strength in excess of 300mT and 6MN/m^3, respectively. Over the length of
the 30-mm channel, the force strength remains within 14, and the force
direction varies by less than 1, leading to a concentration gradient that is
less than the measurement uncertainty of 12 after 4.3h exposure. Application
of a magnetic field leads to an increased clustering of particles, both for
bare Fe2O3 and core-shell nanoparticles.

The second system is based on a Halbach array that applies non-uniform fields,
the strength of which is on the same order as the field of the large magnet of
the first system. However, the forces are more than one order of magnitude
higher because the individual magnets in the Halbach array are $70$ times
smaller. Owing to the Halbach configuration, the lateral forces are both
positive and negative and attract particles toward the transitions. The
vertical forces are always downward but vary up to 30.

When a suspension of bare Fe2O3 is passed over the Halbach array, particles
are captured, and the concentration in the flow diminishes as the wavefront
progresses. As a result, a gradient in particle concentration is achieved over
the length of the Halbach array.

Calculations show that particles with a radius of less than 20nm and a
remanent magnetization of 1T are trapped above the Halbach array with an
efficiency of less than 7. However, our results show that we are able to
capture most of the Fe2O3 particles. Therefore, we conclude that clustering
already takes place in suspension. Calculations show that clusters with a
diameter of 67nm and a remanence of 1T are captured with an efficiency of 51\.
For cluster radii below this value, capture height increases linearly with
increasing radius. Above this value, capture efficiency increases rapidly, and
full capture is achieved for cluster radii greater than 110nm.

We observed that particles form clusters of structures with lengths of up to
100 at the bottom of the channel. At the start of the Halbach array, these
clusters are distributed irregularly. Midway through the array, however, the
distribution becomes more regular. Above the in-plane magnets of the array,
the clusters align along the channel, whereas above the perpendicular magnets,
the clusters become oriented more randomly. Not all particles are captured by
the Halbach array. At the end of the array, a number of larger clusters can be
found. In the remainder of the channel, chains of particles can be seen
aligned along the channel length.

Both systems demonstrate cell survival in the presence of magnetic
nanoparticles and strong magnetic field gradients. Application of a uniform
magnetic force reduces the viability of liver cells in the presence of bare
Fe2O3 as well as core-shell nanoparticles by approximately 50. Cell morphology
is not noticeably dependent on the type of particles, but the presence of the
magnetic field has a strong influence on their distribution. In the magnetic
field, the particles agglomerate into chains. This effect is strongest for the
bare Fe2O3 nanoparticles.

The Halbach array allows us to analyze the cell response as a function of
concentration. In a gradient of Fe2O3 nanoparticles, cell mortality depends on
the local concentration, ranging from 40(5) after six days of exposure at the
highest concentration of approximately 500g/mL to indistinguishable from the
control at a concentration of less than 75g/mL.

The models presented in this paper offer a solid basis for designing magnetic
systems for cell culture studies and particle trapping. These design
methodologies, together with the tools available online, can be used to modify
designs for other channel geometries. The simulation tools presented here for
analyzing particle trajectories can serve as a fast alternative to finite-
element simulations.[30] The capability of creating concentration gradients
may speed up the screening of the effect of magnetic particles on cell lines.
It may also allow other applications, such as cell migration studies.[39]

## Acknowledgments

This work was financed by KIST Europe under project number 12008. The authors
would like thank Lilli-Marie Pavka for proofreading the manuscript, Jonathan
O’Connor for design of the microscopy mask and Matthias Altmeyer and Michiel
Stevens for discussions.

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##  Appendix A Supplementary Material

###  A.1 Optical microscope of stained cells and bare Fe2O3 nanoparticles

Figure 15 shows optical microscopy images of bare Fe2O3 nanoparticles in
combination with cells (similar to Figure 13) as a function of the position in
the channel. From left to right, the particle concentration decreases, as
indicated by the positions on the image of the slide channel below. (For
clarity, note that this is the image of Figure 8 that was at higher
concentration and without cells. The actual particle density is more
comparable to Figure 11.) The cell morphology does not vary significantly with
nanoparticle concentration. The increase in nanoparticle concentration is
clearly visible, but the concentration is non-uniform over length scales
comparable to the cell dimensions.

Figure 15:  Optical microscopy of cells (stained red) and bare Fe2O3
nanoparticles (stained blue) inside the channel slides after 4 days of
cultivation.

###  A.2 Cades input files

The compressed folder Cades.zip contains the source files for the MagMMEMS
package. They are used to calculate the magnetic field and forces of the
cylindrical magnet, see Section II.1.

###  A.3 Python source files

The Python source code used to calculate the trajectories of particles above
the Halbach array is available on github:
https://github.com/LeonAbelmann/Trajectory

###  A.4 3D-print source files

The OpenScad source files (*.scad) and print files (*.stl) for the 3D-printed
holders are available in the compressed folder 3Dprints.zip.

###  A.5 Time-lapse videos

The time-lapse video CoreShell.mp4 shows the change in concentration over
time, see Figure 5. Also shown is the same experiment for the bare Fe2O3
suspension, Fe2O3.mp4, where the formation of gas bubbles can be observed.

###  A.6 Gradient filling

The formation of the concentration gradient can be observed in the video
GradientFilling.mp4. The end result is shown in Figure 8.

###  A.7 High resolution stiched images

Figures FullGradientFe2O3.png and FullGradientCoreShell.png show high-
resolution versions of the images in Figure 11.