Supplementary
Figure
1
-
Simulated
dipole
fields.
Simulated
B
x
(
a
),
B
y
(
b
),
and
B
z
(
c
)
field
projections
for
a
point
dipole
oriented
towards
the
top
of
the
image
with
a
magnetic
moment
of
The
x
and
y
coordinates
of
the
dipole
are
fixed
at
the
center
of
the
image
and
the
dipole
is
spaced
two
μm
above
the
plane
of
projection.
As
in
the
main
text,
x
is
defined
along
the
dipole
axis,
z
is
defined
out
of
the
page,
and
y
is
defined
to
complete
the
normal
ba
sis.
Scale
bars
are
2.5
μm.
2
0
-
2
-
4
-
6
-
8
-
1
0
-
1
2
3
2
1
0
-
1
-
2
-
3
1
0
8
6
4
2
0
-
2
-
4
-
6
-
8
-
1
0
D
i
p
o
l
e
O
r
i
e
n
t
a
t
i
o
n
a
b
c
B
x
(
μ
T
)
B
z
(
μ
T
)
B
y
(
μ
T
)
0
5
-
5
2
.
5
-
2
.
5
0
1
-
1
1
0
-
1
-
2
-
3
-
4
F
l
u
o
r
e
s
c
e
n
c
e
M
a
g
n
e
t
i
c
F
i
e
l
d
L
o
c
a
l
i
z
a
t
i
o
n
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
2
2
1
0
-
3
-
2
-
1
-
2
-
1
0
1
Davis, Ramesh et al.
Supple
mentary Information
Page
2
S
upplementary Figure 2
-
Magnetic
–
f
luorescent
c
olocalization
.
Fluorescence and
magnetic images of
fixed cells after the uptake of
200 nm
(iron oxine nanoparticles)
IONs labeled with Alexa 488 fluorescent
dye
.
Magnetic
localization (red circles) show
s
strong fidelity to
the
centroids of fluorescent images (green
circles)
,
with a mean offset of <
800
nm. Circle diameters are fixed to the diffraction limit for the
magnetometry
and Alexa 488 dye fluorescence respectively.
The sole mismatch occurred in the top cell,
where a sec
ond dipole was visible in the nitrogen vacancy (NV)
-
based
image and localization, but there
was no corresponding centroid in the Alexa 488 fluorescent image.
Magnetic
localization of this dipole is
mar
ked with blue. F
itting of this dipole revealed that it pos
sessed the magnetic moment of a single
nanoparticle, perhaps explaining the weak fluorescent signal. Scale bars are 5
μm
.
S
upplementary
Figure
3
-
SQUID
magnetometry
and
saturation
of
IONs
(a)
S
uperconducting
quantum
interference
device
(SQUID)
magnetometry
of
a
100
μg
stock
of
our
IONs
at
300K.
(b)
A
representative
pseudo
-
spherical
cluster
(
N
=100
nanoparticles)
used
in
our
Monte
Carlo
magnetization
simulations.
(c)
A
pproximate
error
of
our
bulk
approximation
for
clusters
containing
varying
numbers
of
nanoparticles.
Each
point
represents
the
mean
value
from
60
random
particle
arrangements.
a
c
b
N
u
m
b
e
r
o
f
N
a
n
o
p
a
r
t
i
c
l
e
s
i
n
C
l
u
s
t
e
r
(
N
)
D
e
v
i
a
t
i
o
n
f
r
o
m
B
u
l
k
M
a
g
n
e
t
i
z
a
t
i
o
n
(
α
)
M
a
g
n
e
t
i
z
a
t
i
o
n
(
E
M
U
x
1
0
-
3
)
A
p
p
l
i
e
d
F
i
e
l
d
(
k
O
e
)
D
i
s
t
a
n
c
e
(
n
m
)
D
i
s
t
a
n
c
e
(
n
m
)
D
i
s
t
a
n
c
e
(
n
m
)
-
6
0
0
-
4
0
0
-
2
0
0
0
2
0
0
4
0
0
6
0
0
5
0
0
0
5
0
0
0
-
5
0
0
-
5
0
0
Davis, Ramesh et al.
Supple
mentary Information
Page
3
S
upplementary
Figure
4
-
Additional
cells
for
Monte
Carlo
library.
Vec
tor
magnetometry
results
from
three
additional
cells.
These
cells
were
measured
as
described
in
Figure
1
with
the
exception
that
the
imaging
time
was
cut
to
2
hours.
Scale
bars
are
2.5
μm.
Supplementary
Figure
5
-
Supplementary
in
silico
models
of
T
2
relaxation.
In
order
to
further
assess
the
predicted
effect
of
spatial
frequency
and
cellular
confinement
on
nanoparticle
relaxivity,
we
simulated
several
additional
particle
distribution
scenarios
using
the
same
Monte
Carlo
algorithm
described
in
the
mai
n
text.
The
scenarios
are
illustrated
on
top,
with
corresponding
T
2
relaxivities
below.
Orange
bars
correspond
to
data
in
the
main
text.
(a)
Diffuse
(unclustered)
nanoparticles
were
randomly
placed
throughout
the
lattice.
As
this
geometry
minimized
cluster
ing,
it
maximized
relaxivity
for
our
IONs.
(b)
U
nclustered
particles
randomly
placed
in
the
extracellular
space
in
the
lattice.
This
is
the
same
as
the
diffuse
condition
that
was
experimentally
verified
in
F
ig
.
2
of
the
main
text.
As
the
particles
are
still
unclustered,
the
partial
refocusing
effect
is
small,
maintaining
the
high
nanoparticle
relaxivity
(c)
C
lusters
from
NV
-
established
cell
library
randomly
dispersed
throughout
the
extracellular
space
of
the
lat
tice.
The
clustering
of
the
particles
significantly
reduces
their
relaxivity
relative
to
the
unclustered
condition
,
but
the
large
distances
between
the
cluster
s
significantly
increase
relaxivity
compared
to
clusters
spatially
confined
in
“host”
cells
,
as
s
hown
in
(d).
(d)
Cells
from
the
NV
library
were
randomly
placed
in
the
lattice,
and
clustered
nanoparticles
were
confined
inside
of
their
host
cells.
This
is
the
same
condition
as
the
“clustered”
case
that
was
experimentally
verified
in
Fig.
2
of
the
main
text.
(e)
In
order
to
determine
the
effect
of
confinement
in
an
intracellular
compartment,
we
added
a
n
impermeable
5
nm
diffusion
barrier
around
the
clusters
and
randomly
placed
them
inside
their
host
cells
as
in
condition
(d).
There
was
a
statistically
in
significant
decrease
in
the
nanoparticle
relaxivity,
supporting
the
hypothesis
that
the
majority
of
the
relaxivity
of
these
particles
comes
from
is
outer
-
sphere
effects
on
aqueous
protons
.
Error
bars
represent
S.E.M.
C
e
l
l
4
C
e
l
l
5
C
e
l
l
6
0
.
4
0
-
0
.
4
-
0
.
8
-
0
.
6
-
1
.
0
-
0
.
8
-
0
.
6
-
0
.
4
-
0
.
4
-
0
.
2
-
0
.
2
0
.
2
0
.
2
0
0
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
a
b
e
c
d
0
5
1
0
1
5
2
0
2
5
3
0
a
b
c
d
e
R
e
l
a
x
i
v
i
t
y
(
m
M
-
1
s
-
1
)
Davis, Ramesh et al.
Supple
mentary Information
Page
4
S
upplementary
Figure
6
-
Additional
tissue
sections.
(
a
)
Fluorescent
image
of
a
wide
field
of
view
of
a
representative
liver
tissue
section
from
an
iron
-
injected
mouse.
Punctate
fluorescent
spots
from
the
fluorescently
labeled
900
nm
ION
are
sparsely
visible
in
the
fluorescent
image.
(
b
-
c
)
F
ield
profile
of
two
additional
clusters
measured
using
our
NV
microscope,
measured
as
in
Figure
2.
Scale
bars
are
20
μm.
S
uppl
ementary
Figure
7
-
Live
cell
imaging
with
extended
time
course.
(a
–
b
)
Two
additional
live
cell
replicates
matching
Figure
3b.
Cells
were
confirmed
alive
with
trypan
blue
after
NV
imaging.
(
c
–
e
)
0
.
5
5
0
-
5
-
1
0
-
1
5
1
0
0
-
0
.
5
-
1
.
0
F
l
u
o
r
e
s
c
e
n
c
e
W
i
d
e
F
i
e
l
d
F
l
u
o
r
e
s
c
e
n
c
e
F
l
u
o
r
e
s
c
e
n
c
e
a
b
c
B
(
μ
T
)
B
(
μ
T
)
1
5
5
-
5
-
1
5
2
h
r
s
5
h
r
s
1
0
h
r
s
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
B
(
μ
T
)
1
0
0
-
1
0
-
2
0
2
h
r
s
5
h
r
s
1
0
h
r
s
e
c
5
h
r
s
1
0
h
r
s
2
h
r
s
2
0
1
0
0
-
1
0
-
2
0
1
0
-
1
-
2
2
-
5
h
r
s
9
-
1
2
h
r
s
d
f
1
0
1
0
5
5
0
0
-
5
-
5
-
1
0
-
1
0
-
1
5
-
1
5
1
0
5
0
-
5
-
1
0
-
1
5
a
b
T
i
m
e
(
m
i
n
)
1
2
0
1
3
0
1
4
0
1
5
0
1
6
0
1
7
0
1
8
0
1
9
0
2
0
0
2
1
0
T
i
m
e
(
m
i
n
)
1
2
0
1
3
0
1
4
0
1
5
0
1
6
0
1
7
0
1
8
0
1
9
0
2
0
0
2
1
0
B
(
μ
T
)
B
(
μ
T
)
Davis, Ramesh et al.
Supple
mentary Information
Page
5
Bright
field
and
magnetic
images
of
ION
endocytosis
in
RAW
cells
acquired
2,
5,
and
10
hours
after
initial
nanoparticle
exposure
to
279
ng
ml
-
1
200
nm
IONs.
Trypan
blue
assay
revealed
an
~70%
viability
for
these
imaging
studies.
All
displayed
cells
were
still
alive
after
imaging.
Bright
field
illumination
was
provided
by
a
hand
-
positioned
light
emitting
diode
(
LED
)
source
that
was
repositioned
between
images.
(
f
)
Magnetic
field
map
s
from
a
single
fixed
cell
acquired
7
hours
apart
to
show
the
absence
of
dynamic
changes
in
the
magnetic
field
as
a
negative
control
for
dynamic
rearrangements
seen
in
live
cell
experiments
.
Scale
bars
are
5
μm.
S
upplementary
Figure 8
-
T
2
quantification. (a
–
b)
The
T
2
decay of a representative sample of ION
-
supplemented RAW cells with (a) the first 10 echoes used in fitting
and (b) the same sample fitted with
only the even echoes.
(c
–
d)
The
T
2
decay of a representative sample of RAW cells with (c) the first 20
echoes used in the fitting and (d) the same sample fitted with only the even echoes. The
T
2
value obtained
from each fit is displayed on the plot.
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
1
0
0
1
1
0
2
4
6
8
1
0
E
v
e
n
e
c
h
o
e
s
f
i
t
t
e
d
T
i
m
e
(
m
s
)
f
i
t
t
e
d
c
u
r
v
e
T
2
=
2
7
.
6
6
m
s
b
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
1
0
0
1
1
0
2
4
6
8
1
0
1
2
A
l
l
e
c
h
o
e
s
f
i
t
t
e
d
T
i
m
e
(
m
s
)
f
i
t
t
e
d
c
u
r
v
e
T
2
=
2
7
.
6
5
m
s
a
f
i
t
t
e
d
c
u
r
v
e
T
2
=
1
2
4
.
4
m
s
0
5
0
1
0
0
1
5
0
2
0
0
2
5
0
1
1
.
5
2
2
.
5
3
3
.
5
T
i
m
e
(
m
s
)
A
l
l
e
c
h
o
e
s
f
i
t
t
e
d
S
i
g
n
a
l
(
x
1
0
5
)
S
i
g
n
a
l
(
x
1
0
5
)
S
i
g
n
a
l
(
x
1
0
4
)
S
i
g
n
a
l
(
x
1
0
4
)
c
2
0
6
0
1
0
0
1
4
0
1
8
0
2
2
0
1
2
3
f
i
t
t
e
d
c
u
r
v
e
T
2
=
1
2
4
.
2
m
s
E
v
e
n
e
c
h
o
e
s
f
i
t
t
e
d
T
i
m
e
(
m
s
)
1
.
5
2
.
5
d
R
A
W
w
i
t
h
I
O
N
R
A
W
w
i
t
h
o
u
t
I
O
N
Davis, Ramesh et al.
Supple
mentary Information
Page
6
Supplementary
Note
1
:
SQUID
Magnetometry
and
Saturation
Field
Scaling
S
trong
off
-
axis
fields
shift
the
eigenbasis
of
the
Nitrogen
Vacancy
(
NV
)
spin
Hamiltonian
from
along
the
NV
axis
to
along
the
applied
field.
In
this
condition,
m
s
is
no
longer
an
eigenstate
of
the
spin
Hamiltonian,
leading
to
mixing
of
the
m
s
=0
and
m
s
=+/
-
1
states.
This
effect
significantly
reduces
the
sensitivity
of
NV
vector
magnetometry
at
bias
fields
above
10
mT
1
.
Therefore
,
all
our
vector
magnetometry
experiments
were
conducted
with
a
10
mT
bias
field.
To
translate
these
measure
ments
to
the
7
T
field
strength
of
magnetic
resonance
imaging
(
MRI
)
in
Monte
Carlo
simulations,
we
scaled
the
measured
magnetic
moments
from
the
10
mT
bias
field
to
7
T
using
the
results
of
superconducting
quantum
interference
device
(
SQUID
)
magnetometry
performed
on
a
dried
sample
conta
ining
~
IONs
(
Supplementary
Fig.
3
a
).
This
scaling
works
well
for
large
pseudo
-
spherical
clusters,
but
does
not
fully
account
for
the
difference
in
inter
-
particle
effects
between
small
clusters
of
nanoparticles
and
the
dried
SQUID
sample
in
a
non
-
satu
rated
field.
As
has
been
previously
demonstrated,
bulk
mass
magnetization
of
continuum
nanoparticle
assemblies
is
reduced
from
the
mass
magnetization
of
a
single
nanoparticle
or
a
small
nanoparticle
cluster
(
M
)
due
to
magnetic
dipole
coupling
2
such
that:
(
)
(1)
To
assess
the potential impact of dipole
-
dipole interactions on the accuracy of our
dipole scaling,
we
estimated it using a Monte Carlo model of
magnetic coupling in nanoparticle clusters
(
Supplementary
Fig.
3
b
)
.
Since we used multi
-
core particles, w
e assume
d
that
each nanoparticle has many domains and is
in thermal equilibrium, allowing us to neglect the complex time
-
dependence of Neel relaxatio
n
for single
domains. We spline
-
interpolated the SQUID magnetization curve in MATLAB
,
and solved the following
many
-
body problem employing a similar method to one used previously to simulate magnetic dipole
coupling
3
. Our governing equations are as follows:
[
]
∑
(2)
(
(
)
)
(3)
Here
is
the
magnetic
moment
of
the
ith
nanoparticle
in
the
cluster,
S[
]
is
the
splined
approximation
of
the
SQUID
M
vs
H
curve
for
the
effective
field
at
the
location
of
the
i
th
nanoparticle
,
is
the
initial
bias
field
applied
to
all
nanoparticles
in
the
lattice,
and
separating
the
i
th
and
the
j
th
nanoparticle.
The
effect
on
the
mass
magnetization
of
clusters
due
to
dipole
coupling
is
calculated
as
follows:
First,
n
anoparticles
are
randomly
dispersed
into
a
pseudo
-
spherical
arrangement
with
packing
fraction
.
This
value
is
equal
to
the
packing
fraction
from
our
Monte
Carlo
simulation
and
is
within
the
range
of
measured
values
in
literature
4
,
5
.
Next,
t
he
magnetic
moment
of
each
nanoparticle
is
calculated
based
on
of
the
bias
field
.
Next,
t
he
field
at
each
nanoparticle
is
calculated
as
the
superposition
of
the
dipole
fields
from
the
other
nanoparticles
in
the
cluster.
Next,
i
n
order
to
enforce
a
smooth
process,
the
magnetic
moment
magnitude
and
orientation
of
each
nanoparticle
are
adjusted
to
a
weighted
average
of
their
value
in
the
previous
step
and
the
value
calculated
from
(
Supplementary
Equation
2
).
Next,
the
field
calculation
and
magnetic
moment
adjustment
steps
are
repeated
until
the
effective
applied
field
and
the
magnetic
moment
of
each
nanoparticle
are
aligned
such
that
‖
‖
‖
‖
and
the
fractional
change
of
each
nanoparticle’s
magnetic
moment
is
less
than
.
Finally,
t
he
dipo
le
-
coupling
induced
magnetization’s
deviation
from
the
bulk
measurement
is
quantified
as:
(
)
(
(
)
(
)
)
(
)
(4)
Davis, Ramesh et al.
Supple
mentary Information
Page
7
Here
(
)
is
the
simulated
magnetic
moment
of
a
cluster
with
N
particles
and
(
)
is
the
predicted
magnetic
moment
for
that
cluster
applying
the
bulk
mass
-
magnetization.
We
as
sume
that
a
100
-
nanoparticle
cluster
behaves
as
bulk.
The
results
of
this
simulation
are
shown
in
Supplementary
Figure
3
c
.
The
expected
difference
relative
to
the
bulk
measurement
is
largest
for
small
clusters,
where
the
dipole
interaction
is
not
in
the
continuum
limit.
However,
in
all
cases
it
is
below
25%,
an
d
for
the
mean
cluster
size
of
approximately
28
nanoparticles
measured
with
our
NV
magnetometer
it
is
7.5%.
The
curve
flattens
above
N
=80,
validating
our
treatment
of
N
=100
as
a
bulk
material.
Overall,
this
represents
a
modest
under
-
estimation
of
the
mass
magnetization,
meaning
that
NV
measurements
w
ould
slightly
over
-
estimate
the
mass
of
particles
in
a
given
cluster
at
10
mT,
which
in
turn
would
cause
an
over
-
estimation
of
the
magnetic
moment
of
a
given
cluster
at
saturation
(7
T).
This
in
turn
could
help
to
account
for
our
simulation’s
approximately
3%
over
-
estim
ation
of
the
relaxivity
of
our
clustered
samples.
Future
work
mapping
magnetic
fields
of
nanoparticles
could
use
the
presented
simulations
to
better
estimate
the
relaxivity
from
pseudo
-
spherical
and
anisotropic
particle
clusters.
Supplemen
tary Note 2
: Packing and Distribution Effects on
T
2
Relaxivity
In
Figure 2
of the main text we
evaluated
the ability of NV measurement
-
based Monte Carlo modeling to
predict the
effect of nanoparticle clustering
patterns in cells
on
T
2
relaxivity
compared to unclustered
particles distributed
in the extracellular space. While these two cases
enabled
experimental
validation of
our method
, we performed
additional
in silico
trials of hypothetical particle geometries to better
understand the
parameters
driving the measured difference
in
relaxation
(Supplementary Fig.
5
)
.
One
hypothetical condition
(
Supplementary Fig. 5a
)
addresses the significance of extracellular
confinement for unclustered nanoparticles by randomly placing unclustered nanoparticles
throughout the
whole lattice
, including intracellular space
.
D
ispersing the particles throughout the entire lattice slightly
increases their relaxivity
compared to extracellular confinement,
from 25.6
±
0
.3 mM
-
1
s
-
1
to 27.8
±
0
.8
mM
-
1
s
-
1
. However, this
effe
ct
is small compared to th
at caused by endocytosis and clustering
(
Supplementary Fig. 5
d
).
Two additional hypothetical conditions utilized
clusters drawn from the NV
-
measured cell
library
described in the main text.
One
condition
(
Supplementary Fig
. 5
c
)
examined
clustered nanoparticles
placed
in the extracellular
, rather than intracellular,
space.
Clusters obtained from the NV measurement
library were randomly distributed throughout the extracellular space of the cell lattice.
This increase
d
T
2
relaxivity
from
4.1
± 0
.2
0
mM
-
1
s
-
1
to 6.7
± 0
.3 mM
-
1
s
-
1
compared to the cell
-
confined intracellular clusters
analyzed in the main text. This
63% increase can be understood as arising from a more homogeneous
distribution of particles in the lattice, compared to confi
nement within a subset of cells. This result
supports the significance of
using
NV magnetometry
to visualize the sub
-
tissue and sub
-
cellular
distribution of magnetic fields
.
The final condition
analyzed the effect of confining intracellular clusters in a lipid compartment
(
Supplementary Fig. 5
e
)
. We simulated the effect of such a compartment by creating a
n impermeable
5
nm diffusion barrier surrounding the nanoparticle clusters.
This d
ecreas
ed
t
he relaxivity from 4.1
± 0
.2
0
mM
-
1
s
-
1
to 3.8
± 0
.16 mM
-
1
s
-
1
,
within statistical error
, indicating that the majority of the contrast from
these large nanoparticle clusters does not come from water molecules in close proximity to the cluster
surface
.
Suppl
ementary Note 3
:
Uniqueness of
Fit for Dipole Magnetization and Height off the Diamond
Here, w
e seek to demonstrate that for a given (
z
,
M
) value pair, there does not exist another (
z’
,
M’
) value
pair such that
(
)
(
)
for all values of
x
and
y
. This
can
be proven by
contradiction. The coordinate system is set such that the point dipole is at the origin and the measurement
plane is below the point dipole and is parallel to the
xy
plane.
Davis, Ramesh et al.
Supple
mentary Information
Page
8
Assume there exists
(
)
(
)
(
)
(
)
(
)
Let
(
)
(
)
with
. From the equation for
provided in the main text,
̂
̂
(5)
Where
√
√
. Simplifying
(6)
Now take
(
)
(
)
with
. By similar logic:
(7)
Where
√
√
. Substituting from equation S6
(8)
Plugging in the definitions and sim
plifying gives
(9)
Cross
-
Multiplying and Simplifying gives
(
)
(
)
(10)
As the measurement plane is always below the magnetic source in our system, this implies either
, both of which violate assumptions in the proof
.
Thus, sampling any two
points
with
y
>0
along
x
=0 on the measurement plane uniquely specifies both
M
and
z
.
(It is trivial to demonstrate that this
also holds for any two points with
y
<0.
The degeneracy from
makes sense given the symmetric
shape of the dipolar field.
)
Due to
signal
-
to
-
noise ratio (
SNR
)
constraints and the need to localize the (
x
,
y
)
position of the dipole source, we fit to many more than two points per dipole source
.
Davis, Ramesh et al.
Supple
mentary Information
Page
9
Supplementary References
1.
Tetienne, J.P.
et al.
Magnetic
-
field
-
dependent photodynamics of single NV defects in diamond:
an application to qualitative all
-
optical magnetic imaging.
New Journal of Physics
14
, 103033
(2012).
2.
Sanchez, F.H.e.a. Dipolar interaction and demagnetizing effects in magnetic nanoparticle
dispersions: introducing the Mean Field Interacting Superparamagnet Model (MFISP Model).
arXiv:1507.05192
(2015).
3.
Varón, M.
et al.
Dipolar Magnetism in Ordered and
Disordered Low
-
Dimensional Nanoparticle
Assemblies.
Scientific Reports
3
, 1234 (2013).
4.
Wilhelm, C., Cebers, A., Bacri, J.C. & Gazeau, F. Deformation of intracellular endosomes under
a magnetic field.
European Biophysics J
ournal
32
, 655
-
660 (2003).
5.
Au
bertin, K.
et al.
Impact of photosensitizers activation on intracellular trafficking and viscosity.
PLoS One
8
, e84850 (2013).