Quantitative comparison of delta P1 versus optical diffusion approximations for modeling near‐infrared gold nanoshell heating

Abstract

Laser induced thermal therapy combined with the wavelength dependent optical absorption and heating power of gold-coated silica nanoshells can achieve therapeutic heating localized to a tumor volume. Accurate modeling of the spatiotemperal thermal distribution associated with this heating is essential for accurate thermal therapy treatment planning. The optical diffusion approximation (ODA), used in numerous applications of laser fluence in biology, is compared to the delta P1 optical approximation in phantoms containing different concentrations of nanoshells for several laser powers. Results are compared with temperature maps generated by magnetic resonance temperature imaging techniques and show that the delta P1 approximation is more effective than ODA at modeling the thermal distribution. The discrepancy between the two is especially prominent in phantoms with higher nanoshell concentrations where ODA was shown to give unsatisfactory results.

Estimating nanoparticle optical absorption with magnetic resonance temperature imaging and bioheat transfer simulation

“…[17] Of the harmonic expansion methods we chose the delta-P1 (δ-P1) approximation because it is more accurate for higher-absorbing media compared to the most basic harmonic expansion method, the standard diffusion approximation, and has been previously demonstrated to work well for the NS at these concentrations. [19,20] Under the (δ-P1) approximation the photon fluence is parameterized by three values: the absorption and scattering coefficients, μ a and μ s , which represent the probability of a photon being absorbed or scattering over an infinitely small distance and the anisotropy factor, g, which is the average cosine of the scattering angle of the photons. Thus, an anisotropy factor of 0 represents isotropic scattering while an anisotropy factor of 1 represents pure forward scattering.…”

“…For the two regions in our phantom simulations, this represents 12 total parameters. The thermal conductivity, specific heat, physical density, and anisotropy factors are assumed to remain unperturbed in the presence of nanoparticles, [20,21] reducing the number of parameters needed to eight. The absorption coefficient of agar was assumed to be the same as that of water [22,23], and the reduced scattering coefficient was estimated by measuring the extinction coefficient, μ tr , with the spectrophotometer and by assuming the relation ${\mu }_{tr}={\mu }_s^{\prime }+{\mu }_a$.…”

Remote Hyperthermia, Drug Delivery and Thermometry: The Multifunctional Platform Provided by Nanoparticles

“…The fluence (F) is then obtained from the spatial integration of the radiance (L) given the characteristics of the light source, absorption coefficient, scattering coefficient, and the scattering phase function. Exact solutions of the RTE can be accomplished in very simple cases only whereas several approximations have been proposed to allow comparison between numerical RTE's solutions and experimental data recorded from laser illumination of GNPs [68,69].…”

Fluence rate distribution in laser-induced interstitial thermotherapy by mesh free collocation

“…Numerical techniques such as finite difference method (FDM), finite volume method (FVM) or finite element method (FEM) are typically used for the solution of bio-heat equation. On the other hand, the diffusion approximation, P1 [9,10] approximation and the delta-P1 approximation [7,11] are computationally efficient methods that may be used in conjunction with the mentioned numerical techniques but may not be accurate enough.…”

“…The second stage is the bio-heat transportation modeling, which simulates biological tissues' thermal response to the absorbed laser energy. The Monte Carlo method [3][4][5] and the diffusion approximation [6][7][8][9][10][11] are the most commonly used computational tools for the optical modeling. The Monte Carlo method is well known for its accuracy and its flexibility of handling complex computational geometries.…”