International Journal of Heat and Mass Transfer
A combined transient thermal model for laser hyperthermia of tumors with embedded gold nanoshells
Introduction
The use of laser light directly or via minimally invasive fiber optics for induced thermal treatment (hyperthermia) of tumors is one of the present-day tools to fight cancer [1], [2], [3], [4]. The normal human tissues are highly scattering but weakly absorbing (semi-transparent) media in the wavelength range form about 0.6 to 1.4 μm providing a “therapeutic window”. The absorption of laser light leading to a targeted heating of the tumor cells can be strongly increased by embedding silver or gold nanoparticles in the tumor. These nano-sized noble metal particles are characterized by strong resonance absorption and relatively weak scattering in the therapeutic window. Particularly, one can chose geometrical parameters of silica-core gold nanoshells to make these particles very good absorbers for the light of a widely used helium–neon laser with the radiation wavelength 0.6328 μm.
Nanoscale thermal therapy of targeted cancer cells is a promising new weapon in the battle against cancer. A combination of hyperthermia with radiotherapy and chemotherapy is also considered by many researchers. Therefore, many studies have been published on different aspects of the complicated and multi-faceted problem of photothermal therapy. Theoretical studies were focused on unusual resonance optical properties of various gold nanoparticles in the visible and near infrared spectral ranges, modeling of propagation of laser radiation in various human tissues with embedded nanoparticles, and developing of specific heat transfer models taking into account heat conduction, blood perfusion, metabolic heat generation, and radiative power absorbed in the processed tissues. A reader is directed to recent key publications [5], [6], [7], [8], [9], [10] to learn about the state-of-the-art in this research field. Laser irradiation in combination with hollow gold nanoparticles has been recently used in the treatment of melanoma in mice [11].
It is known that no measurable effects are observed while heating part of the human body up to about 42 °C. The first mechanism by which biological tissue is thermally affected can be attributed to some changes of molecules. These effects, accompanied by bond destruction and membrane alterations, are summarized in the term hyperthermia ranging from about 42 to 50 °C. If such a hyperthermia lasts for several minutes, a significant percentage of the tissue will undergo necrosis [12]. Strictly speaking, the thermal problem of hyperthermia should be considered by taking into account the kinetics of the cell destruction processes as described by the Arrhenius-type equation. In the present paper, we consider only the initial period of the so-called soft thermal treatment when the thermal damage is not so important either for the tissue physical properties or the temperature field in the body. Of course, a complete analysis of thermal processes during long-term treatment should include the effect of tissue damage but this is beyond the scope of this paper.
In the present paper, a combined transient heat transfer model for laser thermal treatment of tumors with embedded gold nanoshells is developed. At every stage of the problem solving, the simplest approach was used as it enabled us to take into account various physical processes without considerable mathematical difficulties. In the section concerning the radiative properties of single silica-core gold nanoshells, a possible use of simple relations of the Rayleigh approximation instead of the general Mie theory was examined. In radiation heat transfer modeling, a modified two-flux approximation was employed instead of the complete radiative transfer equation (RTE). Mathematically, it means that we considered a boundary-value problem for the second-order ordinary differential equation which is much simpler than the integro-differential RTE formulated for the radiation intensity depending on both spatial and angular variables. We also simplified the combined heat transfer problem by using a one-dimensional model of the process. Specific conditions of the thermal problem under consideration make it possible to employ a simplified equilibrium model for the temperature field. This model leads to a transient single energy equation instead of two coupled equations for tissue and arterial blood. As was mentioned above, we do not consider these assumptions valid for long-term hyperthermia.
The papers by Tjahjono and Bayazitoglu [13] and Vera and Bayazitoglu [14], [15] can be considered as prototype studies of our work. The objective of the present paper was to take into account the effect of the human tissue index of refraction on both radiative properties of gold nanoshells and radiative transfer in a tissue layer and also to suggest a more accurate formulation for the transient combined heat transfer problem taking into account blood perfusion and metabolic heat generation.
The complete calculations for a model problem with realistic parameters of human tissues are also given in the paper. It should be emphasized that presented example problem is still far from current medical practice and should be considered only as one of the first steps towards the computational modeling of complex processes in human body during laser thermal treatment of tumors with embedded gold nanoparticles.
Section snippets
Spectral optical properties of a human tissue with embedded gold nanoparticles
It is known that spectral absorption and scattering coefficients of a composite medium which can be treated as a matrix (host medium) containing a not too high volume fraction, fv, of small spherical particles of the same radius a can be calculated using the following simple relations [16]:where αλ is the absorption coefficient and is the transport scattering coefficient (σλ is the ordinary scattering coefficient, is the asymmetry
Radiative transfer modeling
In our study, we employed a continuum approach to model the radiative transfer in a complex medium containing scattering (and weakly absorbing) tissue and absorbing (and weakly scattering) nanoparticles. The so-called radiative transfer equation (RTE) is considered in the traditional continuum theory. In the case of a negligible emission of the radiation by a scattering and absorbing medium, the RTE can be written as follows [16], [42], [43]:
Transient energy equation for combined heat transfer in human tissues
The energy transport in a biological system is usually expressed by the so-called bioheat equation. The bioheat equation developed by Pennes [54] is one of the earliest models for energy transport in tissues. Pennes assumed that the arterial blood temperature, Tb, is uniform throughout the tissue while the vein blood temperature is equal to the local tissue temperature T. The resulting transient energy equation is as follows:where the second term on the right-hand
Analysis of computational results for the model problem
The main physical parameters of the model problem for a tumor close to the human body surface (see Fig. 3) are specified in Table 1. The geometrical parameters, tissue density, specific heat capacity, thermal conductivity, and the metabolic heat generation rates were taken from papers by Çetingül and Herman [49], [50]. The values of absorption and transport scattering coefficients were partially taken from paper by Vera and Bayazitoglu [14] and completed using some data reported in [18], [19],
Conclusions
A combined thermal model for the transient temperature field in a tumor and surrounding tissue during laser heating of embedded gold nanoparticles in conditions of cancer hyperthermia is developed. The approach considered is based on the coupling particular models for the absorbed radiation power and transient temperature field.
The spectral properties of nanoshells with a silica core and gold coating are calculated using the Mie theory. It was shown that spectral position of the plasmon
Acknowledgements
The first author is grateful for the partial financial support of this work by the University of New South Wales (Sydney) and Russian Foundation for Basic Research (Grant No. 10-08-00218a).
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