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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

kBT2#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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