Specific Absorption Rate Determination of Magnetic Nanoparticles Through Hyperthermia Measurements in Non-Adiabatic Conditions

8/17/2019 Specific Absorption Rate Determination of Magnetic Nanoparticles Through Hyperthermia Measurements in Non-Ad…

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both samples, solutions containing 10 mg/ml of particles in deio-

nised water have been prepared. Room temperature hysteresis

loops of as-received dried particles have been measured with a

LakeShore 7410 VSM: the results are reported in Fig. 1.

Sample A is constituted by larger particles, with a lower surface

to volume ratio with respect to sample P; therefore, as expected,

their saturation magnetisation is higher, and almost equal to the

saturation of bulk magnetite. Sample P, instead, is characterised by

smaller particles, with a higher surface to volume ratio leading to a

reduced saturation, that display a superparamagnetic behaviour,

with zero coercive eld and a much slower approach to saturation.

These two samples are representative of two different classes of

particles often studied for MPH applications.

3. Hyperthermia setup: calibration and modelling

The hyperthermia setup used in our experiments consists in a

water-cooled copper coil made of 4 turns, connected to a rf gen-

erator through a matching network, and able to generate elec-

tromagnetic elds with intensity up to 100 mT at a frequency of

100 kHz. Inside the coil, a PTFE sample holder hosts an Eppendorf

test tube containing 1 ml of water into which known concentra-

tions of magnetic nanoparticles are dispersed. At equilibrium and

with no rf eld applied, the sample remains at a constant tem-

perature T a¼23.5 °C thanks to the water-cooling of the coil. In

order to measure the temperature of the water solution, a type-T

(wires diameter 0.127 mm) thermocouple is available, that doesnot allow the measurement of the sample temperature during the

rf irradiation process. However, it can be inserted in the test tube

immediately after the rf eld is switched off, in order to measure

the time-dependent temperature decay of the sample towards T a.

As we will show, if a suitable calibration of the experimental setup

is available, an accurate determination of the specic absorption

rate (SAR), and consequently of intrinsic loss power (ILP), can be

obtained by just knowing the two temperatures T a and T M , the

latter being the equilibrium temperature reached by the nano-

particles solution at steady state during irradiation.

An accurate determination of SAR would require a hy-

perthermia setup operating in ideally adiabatic conditions [10,18],

i.e. a calorimeter would be the optimum choice for this kind of

measurements. However, such complex setups are often not

available and SAR is therefore determined by measuring the initial

slope of the sample temperature vs. time when the rf eld isswitched on [11]. This procedure leads to approximate results that

may be affected by how the temperature vs. time curve is analysed

and by the accuracy of the adiabatic hypothesis in the actual ex-

perimental setup, and requires suitable corrections for compen-

sating the heat exchange with the surrounding environment

[12,13].

To avoid these problems, we opted to keep simple the hy-

perthermia experiment, while taking into account the heat ex-

change of the sample with the surrounding environment in the

physical model that describes its thermodynamics. In this way,

after a suitable calibration procedure, the SAR can be accurately

measured even in non-adiabatic conditions.

The hyperthermia system can be modelled as described in

Fig. 2. A heat source S (e.g. the nanoparticles inside the solution)provides energy Q s in, to the system (e.g. through the power lossesof the magnetic particles excited by the rf eld), and exchanges

heat with the water W , Q s out , and Q w in, representing the sameamount of heat, respectively, owing out of the S subsystem and

into the W one. Finally, the water exchanges heat Q w out , with theenvironment (at temperature T a). At each instant of time, the heat

exchange equations for the S and W subsystems are the following:

δ δ δ = +

( )

Q

dt

Q

dt

Q

dt 1as s in s out , ,

δ δ δ = +

( )

Q

dt

Q

dt

Q

dt 1bw w in w out , ,

where each term is taken with its proper sign (positive for heatentering the subsystem, negative for heat released by the

subsystem).

The rst term δ Q

dt

s in, is given by the power P that is provided

through the source. The two terms δ Q dt

s out , and δ Q

dt

w in, are of opposite

sign and given by the following expression:

δ δ

τ = − = − [ ( ) − ( )]

( )

Q

dt

Q

dt

c mT t T t

2s out w in s s

ss w

, ,

where c s is the specic heat of the source, ms is its mass, τ s is the

time constant of the heat exchange process between the source

and the water, T s is the source temperature and T w is the tem-

perature of the water. Both quantities T s and T w depend on time,

but in typical hyperthermia experiments involving magnetic

Fig. 1. Room temperature hysteresis loops of sample A (blue curve) and sample P

(red curve). Inset: magnication at low elds to better show the coercivity. (For

interpretation of the references to colour in this gure caption, the reader is re-

ferred to the web version of this paper.)

Fig. 2. Thermodynamic scheme of the hyperthermia system. S is the source sub-

system (i.e. the magnetic nanoparticles). W is the water subsystem in which the

nanoparticles are dispersed. P is the exciting power. Q terms represent the heat

exchanged by the two subsystems with each other and with the environment.

M. Coïsson et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

Please cite this article as: M. Coïsson, et al., Journal of Magnetism and Magnetic Materials (2015), http://dx.doi.org/10.1016/j. jmmm.2015.11.044i

http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044http://dx.doi.org/10.1016/j.jmmm.2015.11.044


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particles only T w is accessible.

Finally, the term δ Q dt

w out , needs to be evaluated. To this purpose, a

rst calibration procedure has been carried on, consisting in put-

ting in the test tube 1 ml of water that has been pre-heated on a

hot plate. The test tube is in its sample holder and the cooling

water is let circulate in the rf coil, even though no rf eld is gen-

erated. The thermocouple logs the time dependence of the tem-

perature of water as it gradually recovers to the temperature T a of

the surrounding environment. Fig. 3 reports the results of one of the several tests that have been performed on water that has been

pre-heated at different temperatures. An exponential decay of the

temperature over time is expected. However, the experimental

data could not be tted with a single exponential function,

whereas two were required to reproduce the data. The time con-

stants of the two exponential functions turned out to be fairly

reproducible within the whole set of measurements; an average

value has then been calculated, giving rise to the nal results:

τ = 137.5 sw1 and τ = 498.1 sw2 . The presence of two time constants

indicates that the dissipation of heat towards the external en-

vironment is constituted by two mechanisms: the rst possibly

involves heat transfer through the test tube walls and sample

holder, and the second through the top surface of the liquid in the

test tube.

Therefore, it is necessary to express δ Q dt

w out , as the sum of two

terms in the form:

δ

τ τ = − [ ( ) − ] − [ ( ) − ]

( )

Q

dt

c mT t T

c mT t T

3aw out w w

ww a

w w

ww a

,

1 2

τ τ

τ τ = −

( + )[ ( ) − ]

( )

c mT t T

3bw w w w

w ww a

1 2

1 2

where c w and mw are the specic heat and mass of the water,

respectively. τ w1 and τ w2 are the time constants of the two heat

transfer mechanisms to the surrounding environment.

Therefore, Eq. (1) can be written in the following way:

δ

τ = − [ ( ) − ( )]

( )

Q

dt P

c mT t T t

4as s s

ss w

δ

τ

τ τ

τ τ = [ ( ) − ( )] −

( + )[ ( ) − ]

( )

Q

dt

c mT t T t

c mT t T

4bw s s

ss w

w w w w

w ww a

1 2

1 2

The term P appearing in Eq. (4a) is the power that is released to

the system by the source. In a typical hyperthermia experiment,

this term is the power generated by the nanoparticles that aresubmitted to the rf eld (this is not the power emitted by the rf

generator driving the coil, that is usually much higher). From the

rst principle of thermodynamics, Δ = Δ − ΔU Q L, with U being the

internal energy of the system, and Q and L the heat and work

exchanged by the system with the outside, respectively. If we as-

sume that in the studied temperature range the water does not

appreciably change its volume, then Δ =L 0. The internal energy of

both subsystems S and W can then be expressed in terms of

temperature through the expressions =Δ

Δ c m

Q

T s w s w, ,s w

s w

,

,. As a con-

sequence, the thermodynamic behaviour of the hyperthermia

setup can be modelled with the following set of coupled differ-

ential equations:

τ − [ ( ) − ( )] =

( )

( )P

c m

T t T t c m

dT t

dt 5as s

ss w s s

s

τ

τ τ

τ τ [ ( ) − ( )] −

( + )[ ( ) − ]

= ( )

( )

c mT t T t

c mT t T

c m dT t

dt 5b

s s

ss w

w w w w

w ww a

w ww

1 2

1 2

In order to solve the system (5) for the two unknowns T s(t ) and

T w(t ), values for c s, ms and τ s should be provided. Instead, c w is

known (4187 J kg1 K1), as well as mw, since the amount of water

put in the test tube can be measured. However, the specic heat,

mass and time constant of the source are not easily determined for

a typical hyperthermia sample consisting on magnetic nano-

particles. Therefore, we performed a second calibration step where

the source term is replaced by a known resistor having a small sizeand a value of 10.2 Ω. When a known current of constant intensityis let pass through the resistor, it produces a known power =P RI 2

that is used to heat the water. This is also the power term that

enters Eq. (4a) and following. To simplify the system and ease the

calibration process, the resistor has been immersed in a solution of

water and ice, that is therefore at temperature ≈ °T 0 Cice ; this

temperature will not change when the resistor heats up due to the

current that ows through it, leading to a much simplied ther-

modynamic description of the system. The thermocouple has been

placed in direct contact with the resistor, in order to measure its

temperature as it heats up when the current ows through it. In

this way, the system (5) reduces to a single equation with the

temperature of the source (resistor) being the only unknown:

τ

( )= − [ ( ) − ]

( )c m dT t

dt P c m T t T

6s s

s s s

ss ice

where the quantities with suf x s now refer to the resistor that

acts as heat source in this particular case. Eq. (6) can be rearranged

as:

τ τ

( )= − ( ) + +

( )

⎡

⎣⎢

⎤

⎦⎥

dT t

dt T t

P

c m

T 1

7

s

ss

s s

ice

s

that is in the form ′ = − y ay b that, for the initial condition

( ) = y y0 0, has the analytical solution = + [ − ] y y eb

a

b

aat

0 . Therefore,

Eq. (6) can be solved as:

τ ( ) = + [ − ]

( )

τ −T t T P

c m

e1

8

s ices

s s

t / s

0 500 1000 1500 200020

25

30

35

40

45

time (s)

experimental data

Fig. 3. Symbols: time dependence of the temperature of 1 ml of pre-heated watercooling down in the hyperthermia setup. Red line: tting with the sum of two

exponential functions. (For interpretation of the references to colour in this gure

caption, the reader is referred to the web version of this paper.)

M. Coïsson et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎ ∎∎–∎∎∎ 3




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for the initial condition ( ) =T T 0s ice. The experimental results of the

time dependence of the temperature of the resistor immersed in

water and ice for a value of P ¼0.41 W are reported in Fig. 4, that is

representative of the several measurements that have been per-

formed with different heating powers up to E1 W (higher powers

would have led to an excessive temperature increase of the water,

well beyond the range that is interesting for MPH applications).

The data reported in Fig. 4, as well as all the other available

data, have been tted with Eq. (8) leaving τ s and the product c ms sas free parameters. The obtained values have been averaged and

the nal results have been obtained: τ = 89.0 ss and =c m 7.0 J/Ks s .

We will assume that these values describe the properties of the

resistor when it exchanges heat with water, both in the conditions

exploited for calibration purposes and in an actual hyperthermia

experiment represented in Fig. 2 and modelled by Eq. (5), when

the water will effectively heat up.

It is therefore now possible to validate the thermodynamic

model by performing a measurement of the temperature of the

water in the test tube in its sample holder with the cooling water

circulating in the rf coil but without the rf eld, as the heat source

will be the power provided by the calibration resistor. In practice,

an experiment in the same conditions as a normal hyperthermia

measurement is performed, with the difference that the heatsource is not the magnetic nanoparticles excited by the rf eld, but

the calibration resistor that heats the water by means of the Joule

effect with a known power. Since the rf eld is off, in this case it is

possible to use the thermocouple to measure T w(t ) during both

heating and cooling. The results, for heating powers of 0.41 W and

0.10 W (taken as examples) applied for 7200 s and 1800 s, re-

spectively, are reported in Fig. 5 (black symbols).

Eq. (5) is solved numerically using the values for c ms s, τ s, τ w1and τ w2 obtained in the two calibration steps, and with a volume

of 0.7 ml (mw¼0.7 g) of water. The solutions, calculated for the

same heating powers and heating times adopted in the experi-

ment, are plotted in Fig. 5 as the green lines. The agreement with

the experimental results is excellent. It is important to point out

that no free parameter has been adjusted to obtain this

agreement: all the quantities involved in Eq. (5) have been either

measured during the two calibration processes (time constants of

the heat transfer process from the test tube to the environment,

heat capacity of the calibration resistor, time constant of the heat

transfer process from the calibration resistor to water, power), or

are otherwise known (volume of water, its specic heat).

As a result, we can therefore consider the system of (5) as an

adequate thermodynamic representation of our hyperthermia

setup. We can then exploit them to determine the SAR of an un-

known source (magnetic nanoparticles). If the system is brought to

thermal equilibrium during the heating process, the energy en-

tering the system through the rf eld exciting the nanoparticles

and the energy released to the environment are equal and oppo-

site in sign; as a consequence, both ( )dT t dt s and ( )dT t

dt w are equal to

zero. Therefore, from Eq. (5):

τ τ

τ τ =

( + )[ − ]

( )P

c mT T

9w w w w

w wM a

1 2

1 2

where T M is the maximum (equilibrium) temperature reached

during the heating process. With this approach, it is not necessary

to measure the temperature of the water during the heating pro-

cess and to correct its time derivative for the residual heat ex-

change with the surrounding environment. Instead, it is suf cient

to know T M , that can be measured in stationary conditions with an

optical thermometer or immediately after the rf eld is switched

off with just a thermocouple, to determine P , as all the other

quantities entering Eq. (9) are now known and the model Eq. (9)

belongs to has been validated. With P obtained in this way, by

knowing the mass of particles used in the experiment, their spe-

cic absorption rate is obtained accurately for the given rf eld

intensity and frequency.

4. SAR measurements on magnetic nanoparticles

Magnetic hyperthermia has been measured on both samples A

0 100 200 300 400 500 6000

2

4

6

8

10

t (s)

experimental data

thermodynamic model

s> = 89.0 s

= 7 J / K

<

Fig. 4. Symbols: time dependence of the temperature of the calibration resistor

immersed in iced water when an electrical current ows through it giving a heating

power of 0.41 W. Blue line: tting with Eq. (8) with τ s and the product c ms s left as

free parameters. (For interpretation of the references to colour in this gure cap-

tion, the reader is referred to the web version of this paper.)

0 2 4 6 8 1020

25

30

35

40

t (103 s)

experimental data

thermodynamic model

P = 0.41 W

P = 0.10 W

Fig. 5. Symbols: time dependence of the temperature of the water heated by

means of the calibration resistor with a power of 0.41 W. and 0.10 W. Green lines:

numerical solutions of Eq. (5) with the values obtained in the course of the two

calibration processes, for the same heating powers. (For interpretation of the re-

ferences to colour in this gure caption, the reader is referred to the web version of

this paper.)





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and P under three different values of rf eld (30, 60, and 90 mT)

applied for 1800 s. According to our preliminary investigations,

carried on by performing several measurements at different

heating times, 1800 s are suf cient to reach the equilibrium stable

state. The thermocouple is inserted in the test tube immediately

after switching off the rf eld, therefore the rst measured tem-

perature is assumed to be equal to T M . The samples are then left

cool down to T a. Fig. 6 shows the three temperature vs. time curvesfor both samples A and P; for convenience, the abscissa is relative

to the switching on of the rf eld; as only the cooling part of the

curve is acquired, the horizontal axis in Fig. 6 starts at 1800 s.

The equilibrium temperatures T M for the three applied rf eld

intensities are shown in the inset: as expected, the system heats at

higher temperatures when higher rf elds are applied. With the

measured values of T M for both samples A and P it is then possible

to calculate SAR using Eq. (9):

=( )

SAR P

m 10 particles

where m particles is the mass of the particles dispersed in the solu-

tion. The results are reported in Fig. 7a. As expected, SAR increases

with increasing eld value; for rf elds below 30 mT the power

losses of the studied particles become negligible.

However, the intrinsic loss power (ILP) has been proposed as a

quantity more suitable for comparing hyperthermia results ob-

tained on different families of particles, since the power losses are

normalised not only to the sample mass, but also to eld intensity

and frequency [19]:

ν=

( )ILP

SAR

H 112

where ν is the frequency of the rf exciting eld H . In Fig. 7b values of

intrinsic loss power are reported for both samples A and P. A cleardistinction between the two samples can be observed: sample A,

which is constituted by larger particles that display static hysteresis

(see Fig. 1), has a non-monotonous dependence of ILP with the rf

eld, indicating non-linear magnetisation dynamics giving rise to

power losses that depend on eld intensity. Conversely, sample P,

that is constituted by smaller particles having a superparamagnetic

behaviour, is characterised by an almost constant ILP as a function

of H , having a lower value with respect to sample A.

5. Conclusions

A thermodynamic model and a calibration procedure have been

proposed for a magnetic hyperthermia setup that does not need to

Fig. 6. Time dependence of the temperature of 1 ml of water containing 10 mg of

particles of samples A (top) and P (bottom), submitted to three values of rf eld for

1800 s. The origin of the abscissa is at the instant of switching on of the rf eld.

Insets: dependence of the equilibrium temperature T M as a function of the rf eld

intensity.

Fig. 7. Specic absorption rate (SAR, panel (a)) and intrinsic loss power (ILP, panel

(b)) as a function of rf exciting eld of samples A and P.

M. Coïsson et al. / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎ ∎∎–∎∎∎ 5




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operate in adiabatic conditions. Accurate determination of the

specic absorption rate is possible without measuring the initial

slope of the temperature vs. time curve; instead, by measuring the

equilibrium temperature at the stationary state, SAR can be ob-

tained. The thermodynamic model is validated on the actual ex-

perimental setup through a proper calibration procedure, and the

method is nally applied on a set of samples consisting in water

solutions of magnetic nanoparticles allowing the experimental

determination of SAR and ILP.

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