 Original Paper
 Open Access
 Published:
Thermooptical analysis and selection of the properties of absorbing nanoparticles for laser applications in cancer nanotechnology
Cancer Nanotechnology volume 1, pages 35–46 (2010)
Abstract
Applications of nanoparticles (NPs) as photothermal (PT) and photoacoustic (PA) labels and agents for diagnosis and therapy of cancer and other diseases in laser medicine are fast growing areas of research. Many potential benefits include possibility for imaging with higher resolution and treatment of deeper tissues containing NPs, killing of individual abnormal cells, etc. Nevertheless, despite successful results, there is a lack of focused analysis of requirements to NPs for optimization of PT/PA applications, especially with pulsed lasers. Here, we present a platform for analysis of NP properties (e.g., optical, thermal, acoustic, structural, and geometric), allowing to select their parameters in the presence of different ambient tissues. The several types of NPs are described, which provide significant increased conversion of laser pulse energy in PT/PA phenomena. These NPs make it possible to use them with maximal efficiency for detection and killing single malignant cells labeled with minimal amount of NPs and in laser nanomedicine.
1 Introduction
Recent advances in photothermal (PT) and photoacoustic (PA) techniques based on nonradiative conversion of absorbed energy by nanoparticles (NPs) and following thermal and accompanied phenomena demonstrated its great potential. The PT/PA techniques may use NPs as exogenous contrast agents for therapy of cancer and infection or imaging tumor, blood vessels in deeper tissue in living organisms with higher resolution and sensitivity compared to other optical methods (Zharov et al. 2005a, b). Recently, various NPs demonstrated advantages as PT/PA agents for clinical use (Hirsch et al. 2003, 2006; Pitsillides et al. 2003; Zharov et al. 2003; Pissuwan et al. 2006; Huang et al. 2006; Jain et al. 2006; Blaber et al. 2009) because of their extremely high absorption for visible and nearinfrared radiation with relatively deep penetration into most tissues, low toxicity, photostability, absence of photobleaching or blinking effects, and capacity for molecular targeting using appropriate bioconjugation with antibodies, proteins, and other ligands. Two gold NPs (GNPs) have already been approved for cancerrelated clinical trials (Nanospectra Biosciences 2008). High absorption of radiation by NPs can be used for conversion of absorbed energy into NP thermal energy, heating of NPs itself and ambient tissue, and following PT/PA phenomena. These phenomena can be used in selective PT therapy when NPs are conjugated to antibodies (antiEGFR) specifically targeted to malignant cells. This includes (but not limited) gold nanospheres (Pitsillides et al. 2003; Zharov et al. 2003), nanoshells (Hirsch et al. 2003, 2006), nanorods (Huang et al. 2006; Eghtedari et al. 2007), and nanocages (Chen et al. 2007) among other NPs. Our experimental contribution includes first pioneer application of GNPs for detection and killing of individual tumor cells (Zharov et al. 2003), bacteria (Zharov et al. 2006), viruses (Everts et al. 2006), synergistic enhancement of PT/PA contrasts (Zharov et al. 2005a, b; Khlebtsov et al. 2006), the use ethanol (Kim et al. 2007), ultrasensitive PA/PT detection of single NPs, and cells labeled with NPs (Zharov et al. 2007).
However, despite long history of NP development and its application, it is still lack of systematic analysis of optimal parameters of NPs for using them as PT/PA agents in laser nanomedicine. Here, we propose a platform for analysis and optimization of properties of NPs as diagnostic tools and cell killers.
2 Phenomenological parameters and properties of laser–nanoparticle–tissue interactions
Optimization of different NP types is based on the investigation of the influence of different parameters of NP itself, laser pulses, and ambient tissues on efficiency of NP applications for laser diagnostics and therapy of cancer. The NPs have two basic geometries: spherical and cylindrical with various compositions including spherical homogeneous and core–shell twolayered NPs, gold nanorods.
Different parameters of laser radiation, NPs, and ambiences can influence on thermooptical properties of absorbing NPs and determine the achievement of maximal efficacy of transformation of absorbed energy into PT/PA phenomena, including the increase of NP temperature T_{0} and arising pressure p in ambient tissue. Among these parameters, we can note the next ones:

1.
Laser radiation

(a)
Pulse duration t_{P}

(b)
Wavelength

(c)
Energy density E_{0} (intensity I_{0})

(a)

2.
Nanoparticle

(a)
Material of NP with values of density, heat capacity, and optical properties

(b)
Size

(c)
Concentration of NPs in tissue

(d)
Shape (spherical and cylindrical)

(e)
Structure (homogeneous and core–shell)

(a)

3.
Ambient tissue

(a)
Coefficient of thermal conductivity, density, and heat capacity

(b)
Coefficient of absorption, scattering, and extinction

(a)
When NPs are irradiated by short laser pulses with duration t_{P}, excitation and relaxation processes in the NPs lead eventually to conversion of absorbed laser energy into heat and subsequent PT and PA phenomena. To provide maximal efficiency of PT/PA process parameters of laser radiation, NP and ambient tissue should meet several requirements referred to as conditions and “confinements.”
3 Influence of pulse duration on photothermal processes
3.1 Thermal confinement
To provide efficient heating of NPs without heat loss, in analogy to selective photothermolysis (Anderson and Parrish 1983), the pulse duration t_{P} should be less than the characteristic thermal relaxation time τ_{T} of NP cooling (Pustovalov et al. 2008):
For nanosphere with radius r_{0}, τ_{T} ∼ r ^{2}_{0} с_{0}ρ_{0}/3k_{∞}, where с_{0} and ρ_{0} are the heat capacity and density of NP material, and k_{∞} is the coefficient of thermal conductivity of ambient tissue. For gold nanosphere with r_{0} = 30 nm in ambient water with k_{∞} = 6 × 10^{−3} W/cmK, τ_{T} ∼ 1.25 ns. Under thermal confinement, the absorbed energy is almost instantaneously (characteristic time, ∼10^{−12} s) transformed in thermal energy leading to immediate increase NP temperature. The fulfillment of thermal confinement means achievement of maximal value of NP temperature T_{max} = T_{0}(t_{P}) practically without heat exchange with ambience for t_{P} < τ_{T}. Case t_{P} > τ_{T} can be used for heat exchange of NPs with ambient tissue and its heating.
3.2 Acoustic (stress) confinement
The most efficient transformation of thermal energy into acoustic energy occurs under the condition:
where τ_{A} = 2r_{0}/c_{s} is the transit time of the acoustic wave traveling through distance of 2r_{0}, and c_{s} is the speed of sound in tissue. For nanosphere with r_{0} = 30 nm in water with c_{s} = 1.5 × 10^{5} cm/s, τ_{A} ∼ 40 ps. The PA response under Eq. 2 includes component associated with thermal expansion of NPs into ambient soft tissue (biofluid). We have to note that τ_{A} ≪ τ_{T}, and it is not possible to create PA response from fluid around NPs heated by heat diffusion from NPs during long laser pulse action (at t_{P} > τ_{T}) when Eq. 1 is not valid.
We estimate the fulfillment of thermal and acoustic confinements (Eqs. 1 and 2) for some values of pulse duration t_{P} and characteristic radii r_{0} ∼ 10–40 nm of spherical NPs:
 t_{P} < τ_{A}, τ_{T}:

Pulse duration meets both thermal (Eq. 1) and acoustic (Eq. 2) confinements for pico ∼1–10 ps and femtosecond ∼100 fs pulse duration range
 τ_{A} < t_{P} < τ_{T}:

Pulse duration meets thermal confinement (Eq. 1), but does not meet acoustic confinement (Eq. 2) for subnanosecond range of pulse duration, t_{P} ∼ 0.1 ns
 t_{P} > τ_{A}, τ_{T}:

Pulse duration does not meet both acoustic and thermal confinements (Eqs. 1 and 2) for nanosecond range of pulse duration, 1–10 ns and more
4 Analysis of the parameters of homogeneous spherical gold nanoparticles placed in different ambiences and optimization by selection of their thermooptical properties
4.1 Optical properties
We investigate optical properties and conditions of optical confinement of NPs in tissues. In most medical applications, NPs are surrounded by bioliquids such as blood, lymph, or protein. We will investigate the influence of different liquid ambiences on parameters of spherical homogeneous GNPs.
4.1.1 Optical NPs confinement
The absorption of laser radiation by NPs should be greater than absorption of radiation by ambient tissue to enhance contrast of NPs and should be greater than scattering of radiation by NPs because of harmful action of scattered radiation on tissue. Extinction of laser radiation by NPs should be smaller than extinction of radiation by ambient tissue for effective use of NPs for PT therapy in deeper tissue. This difference between coefficients of absorption ά_{abs}, scattering ά_{sca}, and extinction ά_{ext} of laser radiation by NPs and the coefficients of absorption β_{abs} and extinction β_{ext} of laser radiation by ambient tissue should provide optical confinement:
where N_{0} is the concentration of NPs in tissue; r_{0} is the radius of spherical NP or equivolume sphere for nanorod; K_{abs}, K_{sca}, and K_{ext} are the efficiency factors of absorption, scattering, and extinction of laser radiation, respectively, by NP (Bohren and Huffman 1983; Pustovalov and Babenko 2004). Analysis of optical properties of NPs and tissues, concentration, and sizes of NPs can give us appropriate types of NPs.
4.1.2 Gold nanoparticle in water
Water is the main component of soft tissues, blood, etc., and so this one was chosen for calculation. Figures 1 and 2 present efficiency factors of absorption K_{abs}, scattering K_{sca}, and extinction K_{ext} of laser radiation with wavelengths λ = 532 and 800 nm by spherical GNPs in the range of radii 5–100 nm in water (lines 1) calculated on the base of Mie theory (Bohren and Huffman 1983). Optical parameters for gold and water (indexes of refraction and absorption) were taken from Johnson and Christy (1972) and Zuev (1970).
For wavelength λ = 532 nm, K_{abs} for gold spheres in the range 5 < r_{0} < 50 nm has maximal values K_{abs} ∼ 3.9–3.6 for r_{0} ∼ 25–40 nm. Factor K_{sca} for radiation 532 nm lies in the limits K_{sca} ∼ 0.1–2.5 and K_{abs} > K_{sca} for the range 5 < r_{0} < 50 nm. Values of K_{abs} < K_{sca} for the range 50 < r_{0} < 100 nm. Taking into account the correlation between the values of K_{abs}, K_{sca}, and K_{ext} and the possibility to select the values of NP concentration N_{0}, we can achieve the fulfillment of optical confinement (Eq. 3) for the range 5 < r_{0} < 50 nm.
For λ = 800 nm, K_{abs} has lower values in the limit ∼1 × 10^{−2}–2 × 10^{−1} and K_{sca} ∼ 1 × 10^{−2}–4 for the range r_{0} ∼ 5–100 nm and K_{sca} ≥ K_{abs}. GNPs are bad absorbers for wavelength 800 nm and cannot be used for our purposes.
4.1.3 Gold nanoparticle in blood, protein, and ethanol
GNPs can be placed in blood ambience and used for thermal action in blood vessels, hemorrhages, etc. Normal human whole blood consists of about 55 vol.% plasma (90% water and 10% proteins) and 45 vol.% cells (erythrocytes, leucocytes, and thrombocytes). Figures 1 and 2 present efficiency factors of K_{abs}, K_{sca}, and K_{ext} for spherical GNPs in the range of radii 5–100 nm placed in blood for laser wavelengths λ = 532 and 800 nm (lines 2). Optical (Ivanov et al. 1988) and thermophysical (Welch and van Gemert 1995) properties of blood are close to water ones, and factors K_{abs}, K_{sca}, and K_{ext} for GNPs in blood are close to analogous values for water ambience (compare lines 1 and 2 in Figs. 1 and 2). Results of laser action on GNPs in blood will be close to results of analogous action on GNPs in water.
Figures 1 and 2 present factors of K_{abs}, K_{sca}, and K_{ext} for laser wavelengths λ = 532 and 800 nm and for spherical GNPs in the range of radii 5–100 nm placed in protein (lines 3). Optical and thermophysical properties of protein (egg white; Arakawa et al. 2001; Opielinski 2007) are close to properties of water (Zuev 1970) because normal hen's white egg consists of about 80–90% water. For wavelength 532 nm, maximal values of K_{abs} lie in the range r_{0} ∼ 20–40 nm, and they are approximately equal to maximal values of K_{abs} for water ambience. Consequently, the heating and maximal temperature of GNPs under laser action with λ = 532 nm in protein will be approximately equal to values for NP in water. For wavelength λ = 800 nm, values of K_{abs} and K_{sca} are greater than analogous parameters for water and blood ambiences. Heating of GNPs and scattering of radiation in protein will be higher in comparison with mentioned ambiences for λ = 800 nm.
Possible variant of substitution of ambient water for GNPs could be ethanol. Figures 1 and 2 present efficiency factors of K_{abs}, K_{sca}, and K_{ext} for spherical GNPs in the range of radii 5–100 nm placed in ethanol for laser wavelengths λ = 532 and 800 nm (lines 4). Optical parameters of ethanol were taken from Rheims et al. (1997). Maximal values of K_{abs} lie in the range K_{abs} ∼ 3.5–3.7 for λ = 532 nm and r_{0} ∼ 20–35 nm. Values of K_{sca} are lower for 40 < r_{0} < 100 nm in comparison with the ones for GNP in water for λ = 532 nm. Maximal values of K_{abs}, K_{sca}, and K_{ext} for λ = 800 nm are greater than the ones for GNPs in other ambiences.
4.2 Thermal and acoustic properties
4.2.1 Thermal properties
We investigate the thermal and acoustic properties of spherical homogeneous NPs in liquid ambience. Characteristic time τ_{T} is equal to τ_{T} ∼ 3.2 × 10^{−11}–3.2 × 10^{−9} s for the range r_{0} = 5–50 nm and for water k_{∞} = 6 × 10^{−3} W/cmK, τ_{T} ∼ 1.25 ns for r_{0} = 30 nm. The fulfillment of thermal confinement t_{P} < τ_{T} (Eq. 1) for most interesting range of r_{0}, 25 < r_{0} < 40 nm, means that the value of t_{P} will be in the range of pulse durations, t_{P} < 1 × 10^{−9} s. Parameter ΔT_{0}/E_{0} can be used for determination of thermooptical properties of NPs, and it is equal (Pustovalov et al. 2008):
where ΔT_{0} = T_{max} − T_{∞}, T_{ ∞ } is the initial temperature, T_{max} = T_{0}(t_{P}). Equation 4 may be viewed as NP heating efficacy depending on r_{0}, K_{abs} (λ), ρ_{0}, c_{0}, t_{P}, and k_{∞} under action of radiation energy density E_{0}. This parameter determines the increase of NP temperature under action of laser radiation with energy density value equal to 1 J/cm^{2}. Heating efficacy parameter ∆T_{0}/E_{0} under conditions t_{P} < τ_{T} and t_{P} > τ_{T} will be approximately determined by (see Eq. 4)
E_{0} = I_{0}t_{P} is the laser energy density. This parameter determines the heating of NP and depends on t_{P} and combination K_{abs}/r_{0} under fixed values c_{0} and ρ_{0} for gold. The selection of mentioned parameters in Eqs. 4 and 5 can provide maximal values of ∆T_{0} for concrete E_{0}.
Figures 1 and 2 present dependencies of parameter ΔT_{0}/E_{0} (Eq. 4) for pulse duration t_{P} = 1 × 10^{−8}, 1 × 10^{−10}, and 1 × 10^{−12} s for laser wavelength λ = 532 (Fig. 1) and 800 nm (Fig. 2) on radius r_{0} of spherical GNPs. The condition of “short” pulses t_{p} < τ_{T} is applicable for t_{P} = 1 × 10^{−12} s for all range of r_{0}, 5 < r_{0} < 100 nm, and for t_{P} = 1 × 10^{−10} s in the range r_{0} > 30 nm. The values of ΔT_{0}/E_{0} (lines 9 and 10 in Fig. 1) for r_{0} > 30 nm coincide to each other for t_{P} = 1 × 10^{−10} and 1 × 10^{−12} s because for the case of “short” pulses parameter, ΔT_{0}/E_{0} does not depend on t_{P} (see Eq. 5). Only for r_{0} < 30 nm, these curves are different ones. Under condition of “short” pulses t_{P} < τ_{T} parameter, ΔT_{0}/E_{0} depends on combination K_{abs}/r_{0}, accordingly Eq. 5, describing the increasing and decreasing of the value of ΔT_{0}/E_{0}. Maximal value of ∆T_{0}/E_{0} ∼ 4 × 10^{5} for r_{0} ∼ 30 nm and for t_{P} < 1 × 10^{−9} s under laser energy density E_{0} = 0.005 J/cm^{2}, heating of such NP could achieve 2 × 10^{3} K.
For “long” pulses, t_{P} = 1 × 10^{−8} s > τ_{T}, for all range of r_{0} = 5–100 nm, and value of ΔT_{0}/E_{0} is much smaller than value of this one for the case of short pulses because of dependence ΔT_{0}/E_{0} ∼ 1/t_{P} (see Eq. 5). Behavior of ∆T_{0}/E_{0} (see Figs. 1 and 2) depends on combination of K_{abs}r_{0} accordingly (Eq. 5).
For λ = 800 nm, the values of ΔT_{0}/E_{0} are much smaller than the ones for λ = 532 nm because of low values of K_{abs}, and combination of K_{abs}r_{0} describes the behavior of ΔT_{0}/E_{0}.
4.2.2 Acoustic properties
PA signal excited in a medium around NP under action of short laser pulse consists of pressure wave. The most important case for effective destruction of ambient tissue around NP is determined by the following conditions. (1) The thickness of the heated layer of the ambient tissue is small compared to NP radius r_{0}: \( {r_0} > \sqrt {{\chi {t_{\rm{P}}}}} \). (2) All volume of NP was heated during laser pulse action: \( {r_0} < \sqrt {{{\chi_0}{t_{\rm{P}}}}} \), χ, and χ_{0} are coefficients of thermal diffusivity of ambient tissue and NP material, respectively. The pressure amplitude p of the spherical acoustic wave excited is determined by the thermal expansion of NP (Karabutov et al. 1996):
ρ is the density of ambient tissue, β_{0} is the effective thermal expansion coefficient of the NP material, r is the radius of observation point, and f(t) function defines the time dependence of the laser radiation intensity. Maximal efficacy of transformation of heat energy into acoustic pressure will be determined by parameter p/I_{0} or p/E_{0} (see Eq. 6):
4.2.3 Analysis of thermal and acoustic NP properties in water and ethanol
Compare some thermophysical parameters of water and ethanol. Heating of fixed volume of liquid to some value of temperature will be determined by parameter ρc (see Eq. 8), and for ethanol and water, it is equal to 1.92 and 4.18 J/cm^{3} K (Kreith and Black 1980; Grigor'ev and Meilikhov 1991). We need to spend energy for heating of fixed volume (mass) of water up to 2.2 times greater in comparison with ethanol. Heat conduction and diffusivity coefficients for ethanol are smaller up to 3.5 and 1.5 times than these ones for water (Kreith and Black 1980; Grigor'ev and Meilikhov 1991), and as a result, the thickness of heated layer around NP in ethanol will be smaller than in water. Substitution of water by ethanol leads to increasing of value τ_{T} up to 3.5 times and easier fulfillment of thermal confinement (Eq. 1). Coefficient of thermal volume expansion for ethanol β_{0} is up to five times greater compared to water (1.1 × 10^{−3} 1/K vs. 2.1 × 10^{−4} 1/K; Kreith and Black 1980; Grigor'ev and Meilikhov 1991) that can lead to formation of stronger pressure wave in ambient tissue (see Eq. 3) and facilitation of acoustic confinement. As a result of all comparisons, the level of laser energy required to produce the PT and PA effects around GNP in ethanol is dramatically decreased up to oneorder magnitude in comparison with water.
It was experimentally found that replacement of water with ethanol led to an increase in both PT and PA signals from NPs of about five to sevenfold at the same level of laser energy (Kim et al. 2007). This approach can also be applied for PT laser cancer therapy with GNP because particular percutaneous ethanol injection is already used for disinfection purposes and to treat liver tumor.
Characteristic time τ_{A} for r_{0} = 25–40 nm, and water with c_{s} = 1.5 × 10^{5} cm/s is equal to τ_{A} ∼ 3.3–5.5 × 10^{−11} s. The fulfillment of acoustic confinement t_{P} < τ_{A} (Eq. 2) for the range of r_{0}, 25 < r_{0} < 40 nm, means in this case, t_{P} < 3 × 10^{−11} s; value of t_{P} is thus in the picosecond ranges. Parameter p/I_{0} (see Eq. 3) determines the dependence of efficacy of transformation of heat energy into acoustic energy on parameters of NP: r_{0}, ρ_{0}, c_{0}, β_{0}, K_{abs}, and density of ambient liquid ρ. We estimate the value of p for GNP, r_{0} = 40 nm, t_{P} = 200 ps, parameters of ambient tissue are equal parameters of water, K_{abs} is from Fig. 1 for wavelength 532 nm, thermophysical parameters are from Kreith and Black (1980), I_{0} = I_{max}t for time interval 0, t_{P}/2 with I_{max} = 1 × 10^{18} W/s cm^{2} and I_{0} = 1 × 10^{8} W/cm^{2} at t = t_{P}/2, and as a result, p ∼ 25 atm. We see real possibility to use PA mode for our purposes.
Spherical GNPs with sizes in the range 25 < r_{0} < 40 nm for wavelength 532 nm can be used for laser thermal regimes for values of pulse durations t_{P} < 1 × 10^{−9} s and for acoustic regime for t_{P} < 3 × 10^{−11} s.
Spherical homogeneous GNPs with sizes in the range 20 < r_{0} < 40 nm can be used for laser thermal and acoustical regimes for λ = 532 nm under fulfillment of all confinement conditions with approximate accuracy in water, protein, and blood ambiences. The use of GNPs in ethanol ambience leads to increase of efficacy. The use of infrared wavelengths (λ = 800 nm) and spherical homogeneous GNPs leads to significant decrease of efficacy.
5 Analysis and optimization of the properties of spherical twolayered core–shell nanoparticles by selection of their parameters
Spherical core–shell NPs have great potential for diagnostics and therapeutic applications due to strongly enhanced surface plasmon resonance for scattering and absorption and tuning of absorption band in visible and nearinfrared region (Hirsch et al. 2003, 2006) by varying the relative core size and shell thickness. They include various compositions including solid absorbing core with nonabsorbing shell, dielectric core with absorbing coating (silica core and gold shells), etc. The results of analysis and optimization of twolayered core–shell NPs placed in water are presented on the base of selection of optical, structural, and thermophysical properties for some types of NPs. Optical properties of twolayered core–shell NPs were calculated on the base of extended Mie theory (Pustovalov et al. 2009).
5.1 Heating of spherical twolayered core–shell nanoparticles by short laser pulses
A twolayered particle consists of a spherical homogeneous core of radius r_{0} enveloped by the spherically symmetric homogeneous shell of radius r_{1}. We should take into account that lines in Figs. 3, 4, and 5 are presented for the values of radius r_{0} + ∆r_{0} with thickness of shell ∆r_{0} = r_{1} − r_{0}. Process of laser heating of twolayered core–shell NP and its cooling after the end of laser pulse action is described by Eq. 8:
with the initial condition:
T_{10} is uniform temperature over the particle volume, ρ_{0}, c_{0}, and ρ_{1}, c_{1} are the heat capacity and density of core and shell materials accordingly; J_{ C } is the energy flux density removed from the particle surface by heat conduction. Volumes V_{0} and V_{1} of core and shell are respectively equal: \( {V_0} = \frac{4}{3}\pi r_0^3,\,{V_1} = \frac{4}{3}\pi \left( {r_1^3  r_0^3} \right);\,{S_{10}} = \pi r_1^2 \) is the square of NP crosssection, and S_{1} = 4πr ^{2}_{1} is the surface area of a spherical particle of radius r_{1}.
Maximal value of spherical NP temperature T_{max} at the end of laser pulse action with pulse duration t_{P} under constant radiation intensity I_{0} = const during t_{P}, we find from Eq. 8, taking into account the methodology of Pustovalov (2005):
Characteristic thermal time for cooling of core–shell NP from Eq. 10 is equal:
and it is determined by core ρ_{0}, c_{0}, r_{0} and shell ρ_{1}, c_{1}, r_{1} parameters, where \( {\tau_T} = \frac{{{c_0}{\rho_0}r_0^2}}{{3{k_\infty }}} \).
For core–shell NPs heating efficacy parameter, ∆T_{0}/E_{0} = (T_{max} − T_{∞})/E_{0} is determined by
For “short” laser pulses with pulse duration t_{P} < τ_{T1}, the loss of heat from the NP by heat conduction during the time t_{P} can be ignored. For “long” laser pulses t_{P} > τ_{T1}, the loss of heat from the particle by heat conduction will be significant. For “short” and “long” pulses from Eq. 12, we can get
These parameters (Eqs. 12 and 13) are determined by core and shell geometrical, optical, and material characteristics. Mutual feature for core–shell NPS is the approximation of their properties to the properties of homogeneous NPs from shell material when mass of shell will be greater than the mass of core.
5.2 Core–shell liquid–gold nanoparticles
Core–shell liquid (water)–gold NPs can be used for laser release of different liquid drugs on the target when drug can be placed inside NP in its core. Figure 3 presents factors of K_{abs}, K_{sca}, and K_{ext} of laser radiation with wavelengths 532 and 800 nm by water–gold spherical NPs in the range of core radii r_{0} = 5–100 nm and thicknesses of shell Δr_{0} = 10, 20, and 40 nm. Calculation of the absorption, scattering, and extinction factors of twolayered spherical NPs was made on the base of extended Mie theory (Kattawar and Hood 1976; Bhandari 1985). Maximal values of K_{abs} ∼ 2.7 for λ = 532 nm lie close to r_{0} ∼ 5 nm and Δr_{0} = 20 nm, and for λ = 800 nm, value K_{abs} ∼ 1.8 lies in the range r_{0} ∼ 50–60 nm and Δr_{0} = 10 nm. The increase of shell thickness Δr_{0} increases the values of K_{abs}, K_{sca}, and K_{ext} for λ = 532 nm. In the range of NP sizes, r_{0} ∼ 5–100 nm, Δr_{0} = 10–40 nm K_{abs} > K_{sca} for 532 nm, and the conditions of optical confinement can be fulfilled by variation of concentration N_{0}. In this case, NPs could be viewed as strong absorbers and weak scatterers. For wavelength 800 nm in the range r_{0} ∼ 10–100 nm, Δr_{0} = 10–40 nm K_{abs} < K_{sca}, and condition of optical confinement (Eq. 3) is not fulfilled.
Optical properties of nanoshells with silica core and gold shell were investigated (Hirsch et al. 2003, 2006). Efficiency factors of water–gold and silica–GNPs show analogous dependencies of factors K_{abs}, K_{sca}, and K_{ext} on r_{0} because optical parameters of silica for wavelengths 532 and 800 nm (Grigor'ev and Meilikhov 1991; Palik 1985) are close to optical parameters of water (Zuev 1970). Main advantage of silica–gold and water–GNPs is the possibility to tune their optical properties in visible and nearinfrared regions between 500 and 1,000 nm. Under NP overheating because of absorption of laser energy, vapor bubble can be formed inside NP core (Pustovalov et al. 2008) with subsequent explosion of NP and release of drug on target.
Characteristic thermal relaxation time τ_{T1} will be defined by Eq. 11, taking into account thermophysical parameters both core and shell, for example, for r_{0} = 30 nm and Δr_{0} = 10, 20 nm τ_{T1} is equal accordingly τ_{T1} = 2.1; 3.5 ns. The condition of acoustic confinement t_{P} < 2r_{1}/c_{s} is the same one as for homogeneous NP with radius r_{1}. Parameter c_{0}ρ_{0} for water is bigger than for gold, and it needs to spend additional energy to heat such NP in comparison with pure GNP.
5.3 Core–shell air–gold nanoparticles
Figure 4 presents factors of K_{abs}, K_{sca}, and K_{ext} of laser radiation with wavelengths 532 and 800 nm by air core gold shell spherical NPs in the range of radii r_{0} = 5–100 nm and thicknesses of shell Δr_{0} = 10, 20, and 40. Optical properties of air were taken from Grigor'ev and Meilikhov (1991). Maximal value of K_{abs} is equal to K_{abs} ∼ 4.0 for λ = 532 nm and r_{0} ∼ 10–20 nm, Δr_{0} = 20 nm. Values of K_{abs} are approximately equal to values for homogeneous GNPs, but values of K_{sca} are bigger than for pure GNPs for r_{0} ∼ 5–40 nm. For λ = 800 nm K_{abs} ∼ 1.5 in the range r_{0} ∼ 50–60 nm, Δr_{0} = 10 nm. The increase of shell thickness Δr_{0} > 20 nm decreases the values of K_{abs} and approximates their properties to properties for homogeneous GNPs. In the range of NP sizes r_{0} ∼ 5–60 nm, Δr_{0} = 10 and 20 nm K_{abs} > K_{sca} for 532 nm and by variation of concentration of N_{0} conditions of optical confinement (Eq. 3) can be fulfilled.
Figure 5 presents parameters ∆T_{0}/E_{0} for spherical twolayered air–gold particles in water for t_{P} = 1 × 10^{−8}, 1 × 10^{−10}, and 1 × 10^{−12} s, λ = 532 and 800 nm for the range of radii r_{0} = 5–100 nm and thicknesses of shell Δr_{0} = 10, 20, and 40 nm. The condition of “short” pulses t_{P} < τ_{T1} is applicable for t_{P} = 1 × 10^{−10} and 1 × 10^{−12} s for the ranges of radii r_{0} > 20 nm and thicknesses of shell Δr_{0} = 10 and 20 nm. Increasing of t_{P} leads to decreasing the value of ∆T_{0}/E_{0} accordingly, Eq. 13, for t_{P} > τ_{T1}.
Energy spent for air (gas)–gold NP heating can be decreased up to a few times in comparison with pure GNP with equal outer radius, because of lower value of c_{0}ρ_{0} for air core, for example, up two times for Δr_{0} ∼ 0.2 r_{1}. The feature of the practical use of air (gas)–GNPs is the possibility of the NP destruction because of increase of gas pressure with increase of NP temperature under absorption of laser energy and pressure can be higher than the durability limit of shell. This mode could be used for fragmentation of NPs and nanophotothermolysis of cancer cells (Pustovalov et al. 2008; Letfullin et al. 2006) under much lower value of laser intensity I_{0} in comparison with fragmentation of homogeneous GNP because of optical breakdown or other nonlinear mechanisms.
5.4 Gold core–protein shell nanoparticles
For molecular targeting, external NP surface is functionalized with shell from different ligands including DNA, antibodies, proteins, etc. Figure 6 presents factors of K_{abs}, K_{sca}, and K_{ext} of laser radiation with wavelengths 532 and 800 nm by gold core–protein shell spherical NPs placed in water in the range of radii r_{0} = 5–100 nm and Δr_{0} = 2, 5, 10, and 20 nm. Optical properties of protein were taken from Opielinski (2007). Maximal value K_{abs} ∼ 3.7 for λ = 532 nm for r_{0} ∼ 30–35 nm and Δr_{0} = 2 nm. Factors K_{abs} and K_{sca} are decreasing with increasing of Δr_{0} up to 20 nm. Maximal value K_{abs} is equal to K_{abs} ∼ 10^{−1}–10^{−2}, and K_{sca} increases up to ∼3–4 in the range r_{0} ∼ 80–100 nm, and these NPs are bad absorbers and scatterers in the range r_{0} ∼ 5–50 nm for λ = 800 nm. Nonabsorbing layer of protein with different index of refraction in comparison with ambient bioliquid can lead to decreasing of absorption efficiency factor of GNP.
5.5 Gold core–polymer shell nanoparticles
To reduce toxicity and prolong circulation time, NPs are coated with thin polymer layer (e.g., PEG or Dextran). Thermal confinement of twolayered NPs could be improved by using a material of external layer with low thermal conductivity like some polymer materials. The heat flux density J_{ C } from NP will be determined by the equation: . Decreasing the value of k_{1} means decreasing J_{ C } and conservation energy inside NP. Typical values of k_{1} for polymer are approximately equal for photoroplast k_{1} ∼ 2.3 × 10^{−3} W/cmK, polystirol k_{1} ∼ 1.6 × 10^{−3} W/cmK (Grigor'ev and Meilikhov 1991), and much smaller in comparison with value k_{1} = 6 × 10^{−3} W/cmK for water. Decreasing the value of k_{1} leads to increasing the values of τ_{T}, τ_{T1}, and increasing the range of time for t_{P} < τ_{T}, τ_{T1} (Grigor'ev and Meilikhov 1991). Decreasing the thermal diffusivity χ of ambient medium will lead to decreasing of thickness of heated layer during laser pulse action Δr ∼ (χt_{P})^{1/2}. The advantage of the use of such twolayered NP could be the promotion of the fulfillment of thermal and acoustic confinements with increasing of t_{P}.
6 Influence of nanoparticle shape on its properties
6.1 Gold nanorods in water
Figures 2 and 3 present factors K_{abs}, K_{sca}, and K_{ext} for laser radiation with wavelength λ = 532 (Fig. 2) and λ = 800 nm (Fig. 3) for infinite gold nanorods (GNRs) with angles between the direction of laser radiation propagation and main axis of nanorod: 0°, 45°, and 90° (lines 5–7). Angle 0° means the propagation of laser radiation along the main axis of nanorod, angle 90° means the direction of propagation is perpendicular to the main axis, and angle 45° means intermediate position of GNR. For infinite rod, its length L is much greater than radius r_{0}, L ≫ r_{0}. For λ = 532 nm, K_{abs} ∼ 0.3–0.5 and K_{sca} ∼ 2–1.2 in the range 5–100 nm for angles of irradiation 45° and 90°, and maximal values for angle 90°. Factors K_{abs} and K_{sca} for angle 0° are very small and close to 0. For wavelength 800 nm for GNRs, values of K_{abs} are small, but values of K_{sca} for angles 45° and 90° are equal to 1.5–2.5.
For homogeneous nanorod from Eqs. 8 and 9 under r_{1} = r_{0}, ρ_{1} = ρ_{0}, c_{1} = c_{0}, V_{0} = πr ^{2}_{0} L, S_{1} = 2πr_{0}L, and S_{10} = 2πr_{0}L, we have for T_{max} and ΔT_{0}/I_{0}:
For “short” pulses t_{P} < τ_{T1}, we can get a simple equation for heating efficacy ∆T_{0}/E_{0} from Eq. 15, neglected by the loss of energy from nanorod J_{ C } = 0:
It is interesting to note that for the case of “short” pulses, parameter ΔT_{0}/E_{0} depends on thermophysical, optical, and geometrical parameters of nanorod. This parameter is close to parameter (Eq. 5) for homogeneous spherical NPs, but different geometry of GNR was taken into account. GNRs with suitable aspect ratios (length divided by width) can absorb and scatter strongly in the region 700–900 nm (Huang et al. 2008), where transmissivity of tissue is maximal. Their optical resonance can be linearly tuned across the hearinfrared region by changing the effective size or the aspect ratio of the nanorods (Huang et al. 2008).
The nanorods in bioliquid (tissue) show arbitrary orientations relative to the direction of laser radiation propagation. From the other side, the optical properties of long GNRs show great dependence on angle of orientation of main axis of nanorods upon laser radiation propagation leading to decrease absorption and scattering up to ten or more times (Figs. 1 and 2). It means that some parts of GNRs with small angles of orientation will not actually take part in the processes of absorption and scattering of laser radiation. The rest of the parts of GNRs will take part in the processes of absorption and scattering of laser radiation with variable efficacy in the range 0–100%. Moreover, in some regions, collection of nanorods can have identical orientation (Huang et al. 2006), and this will be a possible situation when in some macroscopic regions, thermal effect will be realized with absorption laser energy by GNRs, but in some regions, thermal effects will be absent. In any case, these situations should be taken into account for the purposes of clinical use of GNRs.
7 Conclusion
We carried out analysis and selection of PT and PA properties of NPs using some special materials, shapes, sizes, and compositions of NPs placed in different tissues (ambiences) for laser wavelengths 532 and 800 nm on the base of the results of computer and analytical modeling. Selection and optimization of different NP types and their properties are based on investigation of the influence of different parameters of NP itself, laser pulses, and ambient tissues and fulfillment of some conditions and “confinements” on efficacies of NP applications for cancer nanotechnology.
Thermal (Eq. 1), acoustic (Eq. 2), and optical (Eq. 3) confinements should be fulfilled for the selection of the properties of NPs. Optical confinement (Eq. 3) can be realized and improved for many types of NPs by selection of optical parameters, material, sizes, and concentrations of NP. Thermal (Eq. 1) and acoustic (Eq. 2) confinements can meet three possible situations:
 t_{P} < τ_{A}, τ_{T} and τ_{A} < t_{P} < τ_{T}:

These cases allow further NP optimization by the increase of laser heating ∆T_{0}/E_{0} (Eqs. 4 and 12) and acoustic p/E_{0} (Eq. 7) efficacies and selection of thermooptical and acoustic parameters (by increasing NP size or using ambiences with low sound speed)
 t_{P} > τ_{A}, τ_{T}:

This case allows NP optimization by improvement of acoustic and thermal parameters of ambiences and efficacy of laser heating ∆T_{0}/E_{0} (Eqs. 4 and 12)
To provide penetration through small physiological pores in cell membrane and wall vessels, the radius of NP should be small enough in the range r_{0} < 30–40 nm. The thermal (Eq. 1) and more strict acoustic confinement (Eq. 2) are satisfied for these sizes at the short subnanosecond and picosecond laser pulses. The use of expensive femtosecond lasers in therapeutic applications can be limited because most laser energy can be converted in ionization of NPs with subsequent plasma formation and decrease NP heating. Nanosecond lasers with t_{P} ∼ 5–8 ns are broadly used in laser medicine because they are simpler, less expensive than other lasers, and less harmful to healthy tissue. The condition t_{P} < τ_{T} (Eq. 1) for nanosecond lasers can be achieved for GNP by using bigger values of r_{0} and smaller values of k_{∞} for different ambient tissues, for example, for water t_{P} ∼ 5 × 10^{−9} s < τ_{T} for the range r_{0} > 70 nm.
NPs could be placed in different ambiences (water containing tissue, blood, protein, ethanol, etc.). Optical and thermophysical properties of blood and protein are very close to water properties because these ones contain up to 60–90% of water. Spherical GNPs in the range 25 < r_{0} < 40 nm for wavelength 532 nm can be used for laser thermal regimes for pulse durations t_{P} < 1 × 10^{−9} s and acoustical regime for t_{P} < 3 × 10^{−11} s in water containing tissue, blood, and protein ambiences. The use of GNPs in ethanol leads to increase of thermal and acoustic efficacies. The use of infrared wavelengths with λ = 800 nm and spherical homogeneous GNPs leads to significant decrease of efficacy.
Selection of twolayered spherical NPs (core–shell: liquid–gold, silica–gold, air–gold, gold–protein, and gold–polymer) influences on efficacies of NP applications in laser medicine. Main advantage of silica–gold and water (liquid drug)–gold NPs is the possibility to tune their optical properties in visible and nearinfrared regions between 500 and 1,000 nm. Under liquid–gold NP overheating because of absorption of laser energy, vapor bubble can be formed inside NP core containing liquid drug with subsequent explosion of NP and release of liquid drug on target. The feature of the practical use of air (gas)–GNPs is the possibility of the NP destruction because of increase of gas pressure with increase of NP temperature under absorption of laser energy, and pressure can be higher than the durability limit of shell. This mode could be used for fragmentation of NPs and nanophotothermolysis of cancer cells (Letfullin et al. 2006; Pustovalov et al. 2008) under much lower laser intensity I_{0} in comparison with fragmentation of homogeneous GNP because of optical breakdown. Nonabsorbing layer of protein with different index of refraction in comparison with ambient bioliquid can lead to increasing of scattering efficiency factor of GNP and decreasing of heating efficacy and possibility to satisfy the optical confinement. The advantage of the use gold core–polymer shell NP could be the promotion of the fulfillment of thermal and acoustic confinements with increasing of t_{P}.
The gold nanorods (GNRc) placed in liquid media (waterrich tissue) show arbitrary orientations relatively on the direction of laser radiation propagation. The optical properties of long GNRs show great dependence on angle of orientation of main axis of nanorods upon direction of radiation propagation (Figs. 1 and 2) leading to decrease of absorption and scattering up to ten or more times. These situations should be taken into account for the purposes of clinical use of GNRs.
The final goal is to identify the ways to improve the increase of laser energy conversion into PT and PA phenomena by selection of the NP and ambience properties. These effects should be analyzed for different NPs using homogenous GNPs as “gold standards” for comparison.
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Acknowledgements
This work was supported by the National Institutes of Health/Institute of Biomedical Imaging and Bioengineering under grants EB000873 and EB0005123, and by the Arkansas Biosciences Institute for E.G. and V.P.Z.
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Pustovalov, V.K., Astafyeva, L.G., Galanzha, E. et al. Thermooptical analysis and selection of the properties of absorbing nanoparticles for laser applications in cancer nanotechnology. Cancer Nano 1, 35–46 (2010). https://doi.org/10.1007/s1264501000051
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DOI: https://doi.org/10.1007/s1264501000051
Keywords
 Nanoparticle
 Properties
 Laser
 Analysis
 Optimization
 Cancer
PACS
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 87.54.Br
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