Optimisation of properties for magnetic hyperthermia beyond LRT

S. Ruta1, E. Rannala1, D. Serantes1,2 , O. Hovorka3 , R. Chantrell1

1Department of Physics, University of York, York YO10 5DD, U.K.2IIT and Appl. Phys. Dept., Universidade de Santiago de Compostela, 15703, Spain.

3Faculty of Engineering and the Environment, University of Southampton, U.K.

15 July 2019, Santiago de Compostela

Outline

1. Magnetic nanoparticle hyperthermia (MNH)1. Motivation2. Basic ideas3. Problems/Aims

2. LRT and kMC model1. How to calculate the heating

2. Limitation of LRT

3. Beyond LRT1. High efficiency of transition region

2. Using kMC to extrapolate beyond LRT

4. Conclusions

Magnetic hyperthermia

Magnetic particles will heat up

Apply an AC magnetic field

ΔU = Q − L

∫ M⃗⋅⃗dH

SAR (Specific absorption rate)=Energytime⋅mass

Magnetic particles will heat up

Cancer cells are more sensitive to heat

Possibility of developing a non-invasive cancer treatment

Biomedical limitation [1]:

f Hmax<6⋅107Oe / s

[1] Hergt, R., & Dutz, S. (2007). Magnetic particle hyperthermia—biophysical limitations of a visionary tumour therapy. Journal of Magnetism and Magnetic Materials, 311(1), 187–192.

Magnetic hyperthermia

ΔU = Q − L

∫ M⃗⋅⃗dH

SAR (Specific absorption rate)=Energytime⋅mass

Apply an AC magnetic field

Motivation – cancer treatment● Magnetic hyperthermia is a promising methodology for cancer treatment.

Clinical application since 2013 glioblastoma multiforme

Basic ideas

Clinical requisites: Accurate ΔT: Ttreatment ~ 43º - 47ºC Biocompatibility (composition; coating; dose) Size ~ 10 -100 nm

Limited HAC(f*Hmax

< 6*107 Oe/s): Hmax~[5-200] Oe; f~[0.1-1] MHz

Characterization: Specific Absorption Rate (SAR)

Basic ideas

• Experiment: • Theory: SAR= HL∙fSAR= cpΔT/Δt

M/M

S

H/HA

Clinical requisites: Accurate ΔT: Ttreatment ~ 43º - 47ºC Biocompatibility (composition; coating; dose) Size ~ 10 -100 nm

Limited HAC(f*Hmax

< 6*107 Oe/s): Hmax~[5-200] Oe; f~[0.1-1] MHz

Characterization: Specific Absorption Rate (SAR)

Aims: Identification of optimum condition

● Magnetic hyperthermia is a promising methodology for cancer treatment.

● The ability to predict particle heating is crucial for:● Controlling the heating inside the human body.● synthesizing the particles with optimal properties.

● Study of:● Intrinsic properties and their distribution

(particle size, anisotropy value, easy axis orientation).

● Extrinsic properties (AC magnetic field amplitude, AC field frequency).

● The role of dipole interactions.● Environment effects (heat difuzion, Brownian

rotation, change of particle properties). (not considered here)

Outline

1. Magnetic nanoparticle hyperthermia (MNH)1. Motivation2. Basic ideas3. Problems/Aims

2. LRT and kMC model1. How to calculate the heating

2. Limitation of LRT

3. Beyond LRT1. High efficiency of transition region

2. Using kMC to extrapolate beyond LRT

4. Conclusions

How to calculate the heating?

Properties LRT kMC Other(Metropolis MC, LLG)

Intrinsic properties and their distribution (particle size, anisotropy value, easy axis orientation)

Yes Yes Compromise

Extrinsic properties (AC magnetic field amplitude, AC field frequency)

No Yes Compromise

The role of dipole interactions No Yes Compromise

Calculate the heating: Master equation

[2] E. Stoner and E. Wohlfarth, “A mechanism of magnetic hysteresis in heterogeneous alloys,” Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 240, no. 826, pp. 599–642, 1948.

E tot=−KVcos2 (θ )−M sVH ap cos (θ−θ0 )

Anisotropy energy Zeeman energy

● Consider SW theory for mono-domain particle with uniaxial anisotropy at 0K [2].

Hap

is the applied field

M is the magnetization vector

e.a is the easy axis

θ is the angle between magnetization vector and easy axis

Φ is the angle between the field

direction and easy axis

Calculate the heating: Master equation

Angle between easy axis and magnetic moment

dP1

dt=−W 12 P1+W 21P2

dP2

dt=−W 21P2+W 12P1

dP1

dt=

1τ (W 21 τ−P1)

dM ( t)dt

=1τ (M 0(t )−M ( t) )

P1+P2=1

W 12 ,W 21=f 0e

−ΔE1,2

K B T

τ=1

(W 12+W 21)

Calculate the heating: LRT

dM ( t)dt

=1τ (M 0(t )−M ( t) )

● Power dissipation:

● Imaginary susceptibility:

● Neel Relaxation time:

● Equilibrium susceptibility:

P=∫MdH=f H 02 2π f∫0

1/ f

χ' ' sin2

(2π f t )dt

χ' '=χ0

1+(2π f τ)22π f τ

τ=1

2 f 0

eKVK bT

χ0=[ M s L(α)

H ]H=0

f 0=109Hz

α=M sV H

K bT

P=πχ' ' f Hmax

2

Linear Response Theory

SAR=P

mass

Hmax

=300 Oe

For a particular set of parameters (f,H,K,Ms,V);

L

f*Hmax< 6*107 Oe/s

Limitation of LRT

τ=1

2 f 0

eKVKbT

χ0= [ dL(α)

dH ]H=0

α=M sV H

KbT

● Neel Relaxation time:

● Equilibrium susceptibility (*):

W 12(H (t )= 1f 0

exp (−KVkbT

(1+H (t )H k

)n

))

W 21(H (t ))=1f 0

exp(−KVk bT

(1−H (t)H k

)n

)

τ (H ( t ))=1

W 12(H (t ))+W 21(H ( t ))

H /Hk≪1

χ' '=

χ0

1+(2 π f τ)2 2π f τ Where

L(α)=1/3.0 α−1/45α3+2/945α

5+... χ0=1 /3.0

α<0.5

H <0.5KbT

M sV

Calculate the heating: LRT and kMC

LRT-approximation Exact behaviour

Imaginary susceptibility

Néel Relaxation time

Equilibrium susceptibility

Power dissipation

χ' '(τ ,χ0 )

τ(K ,V ,T )

χ0(M s ,V ,T )

χ' ' (τ (H (t )) ,χ0(H (t )))

τ(K ,V ,T , H (t ))

χ0(M s ,V ,T ,H (t ))

P=πχ ' ' f H 02

P=?

H=Hmaxcos (2π f t)

M=H max[ χ' cos(2π f t )+χ

' ' sin (2π f t)]

dM ( t)dt

=1τ (M 0(t )−M ( t) )

H /Hk≪1

H <0.5KbT

M sV

Conditions for LRT

Calculate the heating: LRT and kMC

dM ( t)dt

=1τ (M 0(t )−M ( t) )

Kinetic Monte Carlo (kMC)

LRT-approximation Exact behaviour

Imaginary susceptibility

Néel Relaxation time

Equilibrium susceptibility

Power dissipation

χ' '(τ ,χ0 )

τ(K ,V ,T )

χ0(M s ,V ,T )

χ' ' ( τ (H (t )) ,χ0(H (t )))

τ(K ,V ,T , H (t ))

χ0(M s ,V ,T ,H (t ))

P=πχ ' ' f H 02

P=?

Calculate the heating: kMC

The model [3] includes all the complexity of a real system:

Distributions of particle volumes. Distributions of particle anisotropy value. Random distributions of uniaxial

anisotropy vectors. Thermal activation is included in the

Kinetic Monte-Carlo model, allowing capturing both superparamagnetic and hysteretic regimes.

Inter-particle interaction are modelled as dipole-dipole interactions.

Various spatial arrangements of nanoparticles are considered.

M. L. Etheridge, K. R. Hurley, J. Zhang, S. Jeon, H. L. Ring, C. Hogan, C. L. Haynes, M. Garwood, and J. C. Bischof, “Accounting for biological aggregation in heating and imaging of magnetic nanoparticles.,” Technology, vol. 2, no. 3, pp. 214–228, Sep. 2014.

● We analyze the optimum conditions:

– The maximum SAR and

– The diameter corresponding to the maximum SAR.

L

Deviation from LRT

LRT LRT

● For small magnetic field the kMC is in agreements with RT.

● With increasing H the RT overestimates the maximum SAR and the optimal particle size.

● SAR vs H is quadratic ( in agreement with RT) in small fields, but becomes linear with increasing H.

L

Deviation from LRT

LRT LRT

Investigated system

● Spherical magnetite nanoparticles

● Ms=400 emu/cm3

● K=(0.5-3) x105 erg/cm3

● f=105 Hz; H0=300 Oe

● A real system will have:

– Distribution of easy axis

– distribution of particle size and anisotropy

– Magnetostatic interaction

K=1.5

K=0.5

K=3.0

K=3.0 K=1.5 K=0.5

3 main regions

3 magnetic regions:

1) Low field (linear approximation) regime: H/Hk<<1.

2) Intermediate (transition) regime.

3) Large field ( full hysteretic) regime.

3

2

1

123 123

High efficiency of transition regime

3 magnetic regions:

1) Low field (linear approximation) regime: H/Hk<<1.

2) Intermediate (transition) regime.

3) Large field ( full hysteretic) regime.

3

21

13 13 2

Transition regime close to fully hysteretic regime is ideal for magnetic hyperthermia

Transition regime close to fully hysteretic regime is ideal for magnetic hyperthermia

Transition regime close to fully hysteretic regime is ideal for magnetic hyperthermia

0.9 0.8 0.75

Transition regime close to fully hysteretic regime is ideal for magnetic hyperthermia

0.9 0.8 0.75

For SAR>0.9 of max SARD: >5nm tolerance

For SAR: >0.9 of max SARD: <2nm tolerance

Effect of interactions

Effect of interactions

Conclusions (part 1)● Magnetic behaviour can be categorize in

3 regions in terms of the applied field:

– a) low field region: linear approximation theory can be used.

– b) large field region: where full hysteresis models are applicable.

– c) transition region: ideal for magnetic hyperthermia: large SAR, less sensitive to size.

● Inter-particle interaction must be also considered.

Outline

1. Magnetic nanoparticle hyperthermia (MNH)1. Motivation2. Basic ideas3. Problems/Aims

2. LRT and kMC model1. How to calculate the heating

2. Limitation of LRT

3. Beyond LRT1. High efficiency of transition region

2. Using kMC to extrapolate beyond LRT

4. Conclusions

P=πχ ' ' f Hmax2

Linear Response Theory

SAR=P

mass

Hmax

=300 Oe

For a particular set of parameters (f,H,K,Ms,V);

L

f*Hmax< 6*107 Oe/s

● Magnetic hyperthermia is a promising methodology for cancer treatment.

● The ability to predict particle heating is crucial for:

– synthesizing the particles with optimal properties.

● Linear Response Theory (LRT) is used for prediction of SAR.

Can we extend the LRT prediction?

P=πχ' ' f H max

2

Optimum condition

L

Optimum SAR=?Optimum D=?

Beyond LRT

● LRT is limited to small field for which SAR is also small;

● kMC can be used to predict SAR beyond LRT limit;

● Disadvantage of kMC is that it depends on a large set of parameters (f,H

max,K,M

s,V);

HmaxHmax

Beyond LRT

● LRT is limited to small field for which SAR is also small;

● kMC can be used to predict SAR beyond LRT limit;

● Disadvantage of kMC is that it depends on a large set of parameters (f,H

max,K,M

s,V);

Hmax

Normalise parameters

Etot=−KV [ cos2( θ )+2

HapHK

cos (θ−θ0 ) ]

E tot=−KVcos2 (θ )−M sVH apcos (θ−θ0 )

For a particular set of parameters (f,H,K,Ms,V):

SAR→SAR

SAR (KV )

D→KVkBT

H→HH K

General picture

SAR→SAR

SAR(KV )

D→KVk B T

H→HH K

General picture

H max

H K

For a particular value of frequency (f);

Extrapolation of optimum SAR from LRT

SARSAR (KV )

∝[ H /Hk ]2

SARSAR (KV )

∝[ H /Hk ]

SAR=SAR( LRT )∝ [H /Hk ]2, H / Hk< 0.1

SARSAR (KV )

=SAR(LRT ) HHK

=0.1+g2 ( f ) [H /Hk−0.1 ] ,H /Hk∈[0.1,0 .3 ]

Extrapolation of optimum SAR from LRT

KVkBT

=KVkBT

(LRT )+ g1( f ) [ H /Hk ]2

Prediction beyond LRT

L

KVkBT

=8.22+47 [ H /Hk ]2⇒D=13.85nm

SAR=SAR(KV ) [0.2+3.89 (H /Hk−0.1 ) ]⇒ SAR=676W / g

Hmax

=300 Oe

Prediction beyond LRT

L

KVkBT

=8.22+47 [ H /Hk ]2⇒D=13.85nm

SAR=SAR(KV ) [0.2+3.89 (H /Hk−0.1 ) ]⇒ SAR=676W / g

Conclusion (part 2)

Can we extend the LRT prediction? Yes

● kMC can be used to predict SAR beyond LRT limit;

● For a given frequency:● We have a general picture● The optimum condition can be analytical predicted:

KV ∝ [H /Hk ]2

SAR∝ [H /Hk ] 2 ,H /Hk<0.1

SAR∝ [H /Hk ] ,H /Hk∈[0.1,0 .3 ]

H max

H K