Enhancing hyperthermia through magnetic nanoparticle

clustersDina Niculaes,†,‡,¶ Aidin Lak,†,¶ George C. Anyfantis,† Sergio Marras,† Oliver Laslett,§

Sahitya K. Avugadda,†,‡ Marco Cassani,†,‡ David Serantes,∥ Ondrej Hovorka,§ Roy Chantrell,⊥and Teresa Pellegrino*,†

†Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy

‡Dipartimento di Chimica e Chimica Industriale, Universita di Genova, Via Dodecaneso 31, 16146 Genova, Italy

§Engineering and the Environment, University of Southampton, Southampton SO16 7QF, United Kingdom

∥Applied Physics Department and Instituto de Investigacions Tecnoloxicas, Universidade de Santiago de Compostela, 15782 Santiagode Compostela, Spain

⊥Department of Physics, University of York, York YO10 5DD, United Kingdom

Niculaes et al., “Asymmetric Assembling of Iron Oxide Nanocubes for Improving Magnetic Hyperthermia Performance”, ACS Nano 2017, 11, 12121-12133.

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Outline:

• Magnetic nanoparticles in medicine (theranostics):– Magnetic particle hyperthermia cancer therapy

– Bio-detection based on Magnetorelaxometry (MRX)

– Magnetic particle imaging (MPI)

• Examples of the effects of interactions– Particle aggregates

• Modeling thermal activation in aggregates of interacting nanoparticle in the long time scale range– Landau-Lifshitz-Gilbert dynamics – inefficient at the required timescales

– Master Equation approach (kinetic Monte-Carlo)

• Results - effects of interactions on:– thermal magnetization decay

– hysteresis loop behaviour (hyperthermia, MPI)

• Conclusions and Outlook

Hyperthermia – main goals:

• Optimize heating efficiency by controlling properties of magnetic nanoparticles

– Material choice (γ-Fe2O3, Fe3O4, Fe-MgO core shell, …)– Variation in size (20-100 nm; single domain/multi-domain)– Crystalline anisotropy (uniaxial, cubic)– Shape (spherical, cubic, …) – Inter-particle interactions– Sweep rate (frequency) of the external applied field (f ~ 0.5-1.2 MHz)

• Relative to natural time scales of particle system (ferromagnetism/superparamagnetism)

– AC field range (H ~ 0-300 Oe, such that Hf < 6.3x106 Oe s-1 )– Viscosity of the surrounding fluid

Périgo et. al., Applied Physics Reviews 2, 041302 (2015)

Specific heat absorption (loss) rate (SAR / SLP):

• Calorimetric measurements• Temperature rise• Hysteresis loop based

SAR = f ×MS ×AhystLoop r ILP = SAR f ×H 2

M / MS

H

f

3

Experiments (Niculaes et al)

4Preparation results in clusters of different numbers and configurations

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Main experimental finding from Niculaes et al.State of aggregation, determined by the number of polymer molecules/nm2 determined clustering which governs the SAR.Can we understand this in terms of a theoretical model?(inset says we can!)

• Magnetic nanoparticles serve as the primary functional element for heat generation and detection (therapy, diagnosis, diagnostics)

– Require simultaneous optimization of nanoparticle properties

• Computer guided magnetic nanoparticle design:

– Accurate models of thermal relaxation and Hysteresis loop behavior (time & frequency domain)

– Forward problem -> prediction of optimal behavior

– Inverse problem -> accurate estimation of parameters and statistical properties from the data

• Thermal activation & long time scales (μs to s)

• Importance of interactions7

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• 1, 2, 3 in the order of increasing value of (uniaxial) anisotropy• Interactions decrease SAR but in some cases can give rise to increasing trend

H

M

f

S. Ruta et al, Sci Rep 5, 9090 (2015)

Effects of dipolar interactions:

0

100

200

300

400

10 20 30 40

SAR (W/g)

D (nm)

(b)

1

2

3

e=0.00e=0.05e=0.10

ε – particle concentration

Energybarriers Δe

• Relaxation rates (Arrhenius law) w21 12 = f0 exp -KV

kBTDe21 12

æ

èç

ö

ø÷

Magnetic nanoparticles & thermal relaxation: Néel relaxation

state 1 state 2state 1state 2

De12 De21

e

a

• Thermal fluctuations of magnetic moment

• Master Equation:

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M(H(t); f ,K,V )

E =KV sin2 a -MSVH cos a -qH( )• Stoner-Wohlfarth model:

• Solution methods for systems of ordinary differential equations– Direct integration

– Eigenvalue techniques

• Kinetic Monte-Carlo method– Probabilistic approach based on Monte-Carlo moves consistent with

the choice of a transition paths and relaxation rates wij

– Natural time scales as opposed to Metropolis Monte-Carlo

– Iterative approach

In multi-dimensions: Brown’s discrete orientation model

wij =1/t ij = t 0

-1 exp -DEij kBT( )d

dtPi = w jiPj -wijPi( )

i¹ j

å Pi t = 0( ) = P0i

Chantrell, et. al., Phys. Rev. B 63, 024410 (2000)Tan et al, PRB 90, 214421 (2014)

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W. F. Brown, IEEE TMAG 15, 1196 (1979)

Typical simulated clusters

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Calculations were done by averaging over different realisations of the clusters

Calculations – effect of anisotropy

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LH column gives SAR as a function of the cluster size

RH column gives the calculated SAR after averaging over the cluster concentrations for each experimental sample

D and E give close agreement with experiment.

Best fit for low anisotropy of the particles – interactions dominate

Strong dependence on particle separation

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SAR enhancement peaks with dimers. Trimers and beyond tend to form flux closure structures leading to more narrow hysteresis loopsNB calculations for comparison with experiment were made with a 1nm separation corresponding to experimental observations.

Representation by fractal dimension

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Fractal dimension Df is a measure of asymmetry of the statistical arrangement of particles. As the Dfincreases from 1 to 3 the statistical arrangement changes from chain-like (Df ~ 1), through planar (Df ~ 2), to spherical geometry (Df ~ 3). The values of SAR are the largest for chain-like structures, and decrease with the increasing degree of geometrical symmetry. Spherical cluster geometries leadnto the lowest values of SAR.

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Magnetic particle imaging: Gleich & Weizenecker, Nature 435, 1214 (2005)Weizenecker et al, Phys in Med and Bio 54, L1 (2009)Goodwill et al., Adv. Mater. 24, 3870 (2012)

- An alternative to MRI (?)- Detect signal from magnetic particles instead of the water protons

• SNR is proportional to the amplitude of 3rd harmonics

• Study time dependence of magnetization (hysteresis loops)

M(t)

t

M. R. Ferguson et al., J. Magn. Magn. Mater. 321, 1548 (2009)

F. Wiekhorst, U. Steinhoff, D. Eberbeck, L. Trahms, Pharmaceutical research 29, 1189 (2012).

MRX – Magnetorelaxometry (Magnetic bio-sensing)

1

t eff=

1

t N+

1

t B

t

H

M

t

RelaxationMagnetizingstage

t eff

• Reconstructing the magnetic nanoparticle distribution in biological tissue• Quantification of particle accumulations in tissue• Interactions of magnetic nanoparticles with their biological environment 15

t B =3hVhkBT

t N = t 0 expKV

kBT

æ

èç

ö

ø÷

Conclusions & Outlook:

• Modeling statistical clusters of magnetic nanoparticles, including thermal fluctuations and inter-particle interactions

• Directional dependence of dipolar interactions

– clusters containing the same number of particles undergo different relaxation and hysteresis depending on the details of particle arrangement within (fractal dimension)

• Aggregation seems to lead to down-graded theranosticperformance

• Future: We have not considered studying the actual aggregation process and therefore we cannot answer the questions related to how likely are the different cluster structures to form within living cells – study particle-cell interactions

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Part. diam: 20 nm; MS = 400 emu/cc; K = 2x105 erg/cc; T = 300K; random anisotropy axes, 100 clusters

Magnetization thermal decay for different Df :

Df = 1.25 Df = 1.5 Df = 1.75 Df = 2

Df = 2.25 Df = 2.5 Df = 2.75 Df = 3

Non-interactingcase

From stretchedexponential

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Laslett et al., APL 106,

012407 (2015)

𝜏𝑓

𝜏𝑓

D = 30nm, Dm = 20 nm; MS = 400 emu/cc; K = 2x105 erg/cc;T = 300K; random anisotropy axes; 100 clusters

Hysteresis loops for different Df (f = 100 kHz):Df = 1.25 Df = 1.5 Df = 1.75 Df = 2

Df = 2.25 Df = 2.5 Df = 2.75 Df = 3

f = 100 kHz, sinusoidalRed loop: H0 = 300 Oe 18

Fourier analysis of M(t) corresponding to the ‘red loop’ :

Df = 1.25 Df = 1.5 Df = 1.75 Df = 2

Df = 2.25 Df = 2.5 Df = 2.75 Df = 3

D = 30nm, Dm = 20 nm; MS = 400 emu/cc; K = 2x105 erg/cc;T = 300K; random anisotropy axes; 100 clusters

f = 100 kHz, sinusoidalRed loop: H0 = 300 Oe 16

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Comparison:

𝜏𝑓0 𝑆𝐴𝑅0 𝑆𝑁𝑅0

• No interactions

SAR SNR

• With interactions

𝜏𝑓

• Presence of interactions within aggregates degrades SAR and SNR in MPI.

Acknowledgements

O. Laslett1

S. Ruta & R. W. Chantrell2

C. Menendez & D. Serantes3

& D. Baldomir3

O. Chubykalo – Fesenko4

K. Livesey5

1 FEE, University of Southampton, Southampton, UK

2 Department of Physics, University of York, York, UK

3 Instituto de Investigacións Tecnolóxicas and Departamento de Física Aplicada, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

4 Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, Madrid, Spain

5 Department of Physics and Energy ScienceUniversity of Colorado, Colorado Springs