Specific absorption rate parameter model in magnetic hyperthermia

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Specific Absorption Rate Parameter Model in
Magnetic Hyperthermia
Arkadiusz Miaskowski
University of Life Sciences in Lublin
Department of Applied Mathematics
and Computer Science
Akademicka 13, 20-950 Lublin, Poland
Email: arek.miaskowski@up.lublin.pl
Mahendran Subramanian
Imperial College London
Department of Computing and Department of Bioengineering
Royal School of Mines
South Kensington Campus, London SW7 2AZ
Email: subramanian.mahendran@yahoo.co.uk
Abstract—This study presents the results of specific absorption
rate (SAR) analysis of polydisperse water based ferrofluid with
median F e3O4particle diameter of 15.2nm. The time dependent
temperatures were measured in alternating magnetic field of
various frequencies and at fixed amplitude. On the base of the
corrected slope method maximum SARs were acquired for the
sample. Next, the linear response theory (LRT) with magnetic
field dependent relaxation times was employed in order to cal-
culate SAR distribution in the sample. Taking into consideration
the polydisperse character of the ferrofluid the Zemann energy
condition was used in order to delimit the area where LRT is
valid. For the larger single-domain particles the Stoner-Wohlfarth
based theory was applied.
I. INTRODUCTION
Targeted hyperthermia treatment using magnetic nanoparti-
cles (MNs) is a promising cancer therapy [1] [2]. However,
from practical point of view, it is very important to deliver
therapeutically sufficient concentration of MNPs to a tumour
in order to receive the required effects like, for example,
apoptosis. That is why, the mechanism of heat dissipation,
when the MNs are being injected into a tumor in the alternating
magnetic field (AMF) used during the treatment, has to be
clarified.
The most commonly quoted measure for magnetic nanopar-
ticles (MNs) heating ability is the specific absorption rate
(SAR), which is defined as the heating power generated per
unit mass of magnetic material and expressed in W/g. In order
to acquire the SAR parameter calorimetic experiments (i.e.
measurements under non-adiabatic conditions) are conducted
and a time dependent temperature relations are received.
There are few analytical models and methods used to
extract the SAR parameter from ”magnetic heating” in the
literature [3][4] [5]. In general, all of them are based on the
polynominal fit including suggestions required for making the
most accurate measurements. However, none of them includes
efficient computer modelling as the helpful tool before the
experiment planning.
In this paper, it was shown how to combine the laboratory
measurement with a computer-aid modelling. The corrected
slope method [3] was used to extract SAR parameters from the
measurements and the low-frequency quasi-static algorithm
[6][7] was used in order to evaluate alternating magnetic field
in magneTherm system.
II. CALORIMETRIC EXPERIMENTS
In this work, 2 ml of 5 mg/ ml concentrations (φ) of DMSA
stabilised magnetite nanoparticles were used for calorimetric
experiments. The same type of 2 ml graduated temperature
and pressure resistant vials (Nalgene, Thermo Fisher scientific,
New York, USA) were used for calorimetric experiments.
MNPs were subjected to vortex and ultra-sonication prior to
calorimetry. Temperature sensor was positioned in the center
of the sample. Osensa FTX-200-LUX+ optical temperature
sensor system (British Columbia, Canada) was used for real-
time temperature measurement, while subjecting the sample
to a 17-turn coil, 50 mm ID attached to the mageTherm
system (nanoTherics, Staffordshire, UK). The MNPs sample
was exposed to the same magnetic field Hmax = 10.3kA/m
in the middle of the coil but for different frequencies f1=
171 kHz and f2= 257 kH z. To experimentally estimate
the magnetic field strength in the coil, the Rogowski current
waveform transducer CWT3B/4/100M/5 (Power Electronic
Measurements, Nottingham, UK) was used to measure the
real-time current flow through the coil in the 100 kHz to
1MHz frequency range [7].
The amount of heat generated by MNPs is usually quantified
in terms of specific absorption rate (W/g of particles) and can
be expressed as:
SAR =C
φ
∆T
∆t(1)
where specific heat capacity of the MNPs (J/K/ml) is C,
concentration of MNPs (mg/mL) is φ, and the rate of change
of temperature over time is ∆T/∆t. An appropriate region of
the graph was used for our calculations by using the corrected
slope method. This method corrects the value determined by
the (initial) slope method for any linear losses already apparent
at that temperature (P=C(∆T/∆t) + L∆T).
When the value of thermal loss Lof the system is known,
SAR is calculated using:
SARcorrected−slope−method =C∆T
∆t+L∆T/mMNP
(2)

In this equation ∆Tis the mean temperature difference
between the sample and baseline, which will be within the
bounds of the linear-loss regime. When the loss Lis not known
as it is hard to measure all the parameters involved accurately,
it is possible to estimate the linear loss parameter from the
temperature difference over time slope based on fitting interval
and the number of fits [3]. In our case the sample was exposed
to magnetic field for 1500 seconds in order to to observe
when the change of temperature over time curve plateaus. The
measurements were repeated three times for the sample.
The sample studied in this work was supplied by Liquids
Research Ltd [8] and the MNPs size distribution was fitted by
a log-normal function g(D)(see. Fig. 1) [9].
Fig. 1. Number weighted distribution approximated as log-normal density
function g(D)(solid line).
III. POWE R LO SS ES
The magnetic nanoparticles used for hyperthermia appli-
cations are usually in the superparamagnetic state at room
temperature. When they are exposed to an alternating magnetic
field (AMF) with given parameters i.e. magnetic field strength
amplitude and frequency (Hmax, f ) the heating mechanism
from such MNPs can be explained in terms of the linear
response theory (LRT). Assuming that the angle between the
easy axis and the magnetic field is zero, the SAR can be
computed as
SARLRT =πµ0χ00 H2
maxf w−1(3)
where wis the weight of the magnetic core of the MNPs (kg),
χ00 is the average out-of-phase component of AC susceptibility
from each MNP and is given by
χ00 =µ0µ2
3kBT·2πf τ
1 + (2πf τ )2(4)
where µ=MsVstands for the magnetic moment of the MNP
and Msfor the magnetic saturation.
Furthermore, in the ferrofluid sample, two relaxation times
coexist i.e. the Neel (τN) and the Brownian (τB). The first one
is connected with particle’s magnetic moment due to its weak
coupling to the crystal structure and the second one is con-
nected with rotation of particles due to the random mechanical
forces acting on the particle inside the fluid [10]. Hence, the
Brownian relaxation time depends on a hydrodynamic volume
(VH) of MNP and the viscosity of the medium (η). According
to [11] the relaxation times depend on the AMF and can be
expressed as
1
τN(Hmax)=1
τ0
(1 −h2)(1 + h)exp −KV
kbT(1 + h)2
+ (1 −h)exp −KV
kbT(1 −h)2 (5)
and
1
τB(Hmax)=τ−1
B"1+0.07 µ0MsV H
kBT2#0.5
(6)
where h=HmaxMs
2K,Kis the anisotropy constant of the MNPs
and τB=3VHη
kBT. Finally, the effective relaxation time (τ) is
then defined as
1
τ(Hmax)=1
τN(Hmax)+1
τB(Hmax)(7)
Since MNPs in the sample are polydisperse, for larger
single-domain particles, the relaxation time lengthens and such
particles cannot be considered as superparamagnetic but as the
blocked ones. Behavior of these MNPs is usually explained
within Stoner-Wohlfarth based theories [12] [13].
In the case of randomly oriented MNPs, the following
analytical expression for the coercive field was obtained [12]:
µ0Hc≈0.48µ0Hk"1−kBT
KV lnτm
τ0
3
4#(8)
where Hk= 0.96K/Ms/µ0,τmis the ”measurement time”
(τm= 1/f), and τ0= 10−9sis the frequency factor of the
Neel-Brown relaxation time. In this case SAR can be written
as
SARS−W= 2Msµ0HcV f w−1(9)
Finally, taking into account the Zeeman energy condition
together with log-normal particles size distribution, one can
obtain effective SARs as follows:
SARef f
LRT =ZD0
0
SARLRT ·g(V)dV (10)
and
SARef f
S−W=Z∞
D0
SARS−W·g(V)dV (11)
where g(V)is the volume weighted distribution define for
spheres as
g(V) = g(D)D3
Rg(D)D3dD (12)

One should notice that the number weighted distribution
(g(D)), which is usually acquired from TEM analysis, was
changed to volume weighted distribution g(V)because the
investigated effects and properties are volume dependent in
general.
IV. RESULTS AND DISCUSSION
The heating properties of the sample were measured using
calorimetric experiments (see section II). The sample was
exposed to the same magnetic field Hmax = 10.3kA/m but
two different frequencies f1= 171 kHz and f2= 257 kHz,
respectively. After 1500 seconds of heating, the magnetic field
was switched off and the sample was cooled down, as it is
shown in Fig. 2. In order to validate the measurements the
calorimetric experiments were repeated three times for each
frequency.
Fig. 2. Heating curves for frequencies f1= 171 kHz and f2= 257 kHz
and Hmax = 10.3kA/m.
Taking into account the measured heating curves and (2)
the average SAR was calculated. The following results were
obtained SAR = 16.27 ±0.80 W/g and SAR = 25.23 ±
0.70 W/g for f1= 171 kH z and f2= 257 kH z, respectively.
Next, on the base of (10) and (11) the total power losses
in the sample were computed. In our case the calculations
were done for ferrofluid which has magnetic saturation Ms=
92.0kA/m, anisotropy constant K= 30.0kJ/m3, the Boltz-
mann constant kB= 1.38 ·10−23m2kg/s2/K, thickness of a
sorbed surfactant layer δ= 2nm, density ρ= 5180kg/m3and
viscosity η= 0.86 ·10−3kg/m/s [14]. All these parameters
were equivalent to physical properties of magnetite (F e3O4)
dispersed in water.
The total power density in XZ cross-section distribution in
2ml sample can be seen in Fig. 3. Furthermore, in order to
improve visualization the minimum power was set on 5.0·
104W/m3and on 10 ·105W/m3for f1= 171 kHz and
f2= 257 kHz, respectively.
Apart from the cross-sections, the total power density distri-
bution along z-axis is shown in Fig. 4, where ”distance zero”
indicates the middle of the sample.
Fig. 3. Total power density distribution in XZ plane passing through the
center of the sample for f1= 171 kHz (left) and f2= 257 kHz (right)
and Hmax = 10.3kA/m.
Fig. 4. Total power density along z-axis for f1= 171 kHz and f2=
257 kHz and Hmax = 10.3kA/m
It can be seen that the power is not homogeneous in the
sample volume and it is varying from about 80.0kW/m3to
117kW/m3for f1= 171 kHz and from about 120kW/m3to
176kW/m3for f2= 257 kHz. In the case of f2= 257 kH z
there is 170kW/m3in the middle middle of the sample.
The power densities seen in Figs. 3-4 were introduced in
watt per cubic meter unit and one can easily convert volume
power density to SAR as
PRMS =√2·φ·SAR (13)
where φis the magnetic fluid concentration.
In our case, taking into account minimum and maximum
power densities one can get SAR range from 11.31W/g
to 16.55W/g and from 16.97W/g to 24.89W/g for f1=
171 kHz and f2= 257 kH z, respectively. In the fist
case the maximum SAR value is overestimated, whereas in
the second one, it is underestimated when comparing with
the experimental SARs. The interpretation of this result is
not trivial due to the complexity of system. The diffusion
equation analysis could be helpful in this case but it should

be underlined that dealing with the temperature distribution
the heat losses like, for example, conduction, convection and
radiation should be considered apart from physical properties
of the ferrofluid [15] (this task will be approached in the
near future). Nevertheless, the results clearly indicate that
temperature sensor positioning can play a crucial role during
calorimetic SAR measurements.
Finally, one can conclude that maximum SARs received
from the our measurements are very close to the ones received
from the mathematical model. It is worth underlining that
our model was simplified in some points. First of all, it was
assumed that there were no dipole-dipole interactions in the
ferrofluid. Moreover, the calculations were based on a constant
value of Kwhich could give an overestimate of the expected
heating power.
V. CONCLUSION
The results of specific absorption rate (SAR) analysis
in the polydisperse ferrofluid were presented. The ferrofluid
sample was exposed to two different frequencies, but the same
magnetic field amplitude.
At one point, in the middle of the sample, the temperature
over time was recorded and on the base of it the SAR values
were obtained. Next, linear response theory (LRT) together
with Stoner-Wohlfarth one were successfully applied in order
to validate the experiment. As the boundary condition for LRT
validity the Zeeman energy formula was used.
Theoretical and experimental investigation carried out in
this study showed that the power density is not homogeneous
in the sample volume. During calorimetric experiments the
temperature sensor positioning can lead to significant errors
in reporting experimental SAR values. Indirectly, on this base
one can also conclude that temperature distribution is not
uniform in the sample either. During calorimetric experiments
the sensor should be positioned in the warmest region of
the sample in order to avoid significant errors. It was shown
that using computer-aid simulation it is possible to find such
regions and plan the experiments.
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