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2) Analytical approach: Rosensweig theory (LRT) Solving the master equation in the linear regime: Energy loss per cycle: Imaginary susceptibility: Equilibrium susceptibility: 3) Computational methods: kinetic Monte- Carlo model ● Individual particles are described by the Stoner-Wohlfarth theory: ● distributions of particle volumes; ● distributions of particle anisotropy value; ● random distributions of uniaxial anisotropy vectors; ● Thermal bath is included in the models, allowing capturing both superparamagnetic and hysteretic regimes. ● Inter-particle interaction are modelled as dipole-dipole interactions. ● Various spatial arrangements of nanoparticles can be considered. 4) High efficiency in transition region between RT and fully hysteretic regimes System of spherical nanoparticles with: ● Random spherical distribution of easy axis ● Log-normal distribution of size ( σ d =0.1) and anisotropy ( σ K =0.1) 5) Role of interactions 1) Motivation Magnetic hyperthermia is a promising methodology for cancer treatment. Clinical requisites: Accurate ΔT: T treatment ~ 42º - 45ºC Biocompatibility (composition; coating; dose) Size ~ 10 -100 nm Limited H AC (f*H max < 6*10 7 Oe/s): H max ~[5-200] Oe; f~[0.1-1] MHz Unified model of hyperthermia via hysteresis heating in systems of interacting magnetic nanoparticles Sergiu Ruta 1 , Ondrej Hovorka 2 , Roy Chantrell 1 1 Physics Department, University of York, York, UK 2 Faculty of Engineering and the Environment, U. of Southampton, Southampton, UK Δ U = H max 2 2 π f χ '' χ '' = χ 0 1 +( 2 π f τ eff ) 2 2 π f τ eff τ N = 1 2 f 0 e KV K b T χ 0 = [ M s L (α) H ] H = 0 α= M s VH K b T Analytical approach: ● Rosensweig theory (RT) Simulations: ● Minor hysteresis cycle ? H=300 Oe f= 100 KHz Size and anisotropy are optimised 0.9 0.8 0.75 For SAR>0.9 of max SAR D: >5nm tolerance For SAR: >0.9 of max SAR D: <2nm tolerance Magnetic behaviour can be categorize in 3 regions in terms of the applied field: 1) low field region: linear approximation theory can be used. 2) large field region: where full hysteresis models are applicable. 3) transition region: ideal for magnetic hyperthermia: large SAR, less sensitive to size. dM ( t ) dt = 1 τ eff ( M 0 ( t )− M ( t ) ) τ eff = τ N τ B τ N +τ B τ B = 3 η V K b T Brownian relaxation time Neel relaxation time Magnetic particles will heat up Cancer cells are more sensitive to heat Possibility of developing a non-invasive cancer treatment Biomedical limitation [2]: fH max < 6 ⋅ 10 7 Oe / s Apply an AC magnetic field Δ U = Q − L ∫ M⋅ dH SAR ( Specific absorption rate )= Energy time ⋅mass Néel H AC Brown H AC Inmobilized (tumour tissue) Rotating (fluid) • Experiment : SAR= c p ΔT/Δt • Theory: SAR= HL∙f/V t M/M S H/H A ● The ability to predict particle heating is crucial for: ● Controling 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) H max =300 Oe f=100 kHz LRT kMC LRT 3 2 1 3 1 2 kMC calculations of SAR as a function of the mean particle size D assuming non-interacting system, in comparison with the predictions based on the RT (solid green lines). Considered are three values of mean anisotropy constant K: 3 × 10 5 erg/cm 3 (circles, curve set 1 peaking at low D), 1.5 × 10 5 erg/cm 3 (triangles, curve set 2 peaking in the intermediate D range), and 0.5 × 10 5 erg/cm 3 (squares, curve set 3 saturating for large D). Maximum SAR value (a) and peak position (b) as function packing fraction. 6) Conclusion We have developed a kinetic Monte-Carlo model of the underlying heating mechanisms associated with the hyperthermia phenomenon used in cancer therapy. We show that the magnetic behaviour can be categorized in 3 regions in terms of the applied field: 1) the low field region where the linear response theory approximation, developed in previous studies, can be used, 2) the large field region where full hysteresis els are applicable and 3) an intermediate region where the transition between the two behaviour occurs and the conventional approaches no longer apply. Magnetostatic interaction must be also considered. Maximum SAR value (a) and the corresponding particle size (b) as function of anisotropy for: RT (black line), kMC non- interacting case (red circles) and kMC with interaction (green squares). In the fully hysteretic regime the SAR dependence on D does not present a peak, but is reaching a saturation value with increasing D. ● Magnetostatic interaction (random position of particle in a 3D configuration with volumetric packing fraction of 0.0, 0.05 and 0.10) 7) Reference [1] R. Rosensweig, “Heating magnetic fluid with alternating magnetic field,” Journal of Magnetism and Magnetic Materials, vol. 252, pp. 370–374, Nov. 2002. [2] 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. [3] 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. [4] R. Chantrell, N. Walmsley, J. Gore, and M. Maylin, “Calculations of the susceptibility of interacting superparamagnetic particles,” Physical Review B, vol. 63,p. 024410, Dec. 2000. [5] S. Ruta, R. Chantrell, and O. Hovorka, “Unified model of hyperthermia via hysteresis heating in systems of interacting magnetic nanoparticles.,” Sci. Rep., vol. 5, p. 9090, Jan. 2015. LRT KMC ε=0.0 KMC ε=0.1 Contact: [email protected] More information at:
