# Thermo-optical analysis and selection of the properties of absorbing nanoparticles for laser applications in cancer nanotechnology

- Victor K. Pustovalov
^{1}Email author, - L. G. Astafyeva
^{2}, - E. Galanzha
^{3}and - V. P. Zharov
^{3}

**1**:5

**DOI: **10.1007/s12645-010-0005-1

© Springer-Verlag 2010

**Received: **9 April 2010

**Accepted: **8 August 2010

**Published: **7 September 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.

### Keywords

Nanoparticle Properties Laser Analysis Optimization Cancer### PACS

42.62.Be 87.54.Br 78.67.Bf## 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 near-infrared 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 cancer-related 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 (anti-EGFR) 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 two-layered NPs, gold nanorods.

*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

*t*

_{P}should be less than the characteristic thermal relaxation time

*τ*

_{T}of NP cooling (Pustovalov et al. 2008):

*r*

_{0},

*τ*

_{T}

*∼ r*

_{0}

^{2}

*с*

_{0}

*ρ*

_{0}/3

*k*

_{∞}, 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

*τ*

_{A}= 2

*r*

_{0}/

*c*

_{s}is the transit time of the acoustic wave traveling through distance of 2

*r*

_{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.

*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

## 4 Analysis of the parameters of homogeneous spherical gold nanoparticles placed in different ambiences and optimization by selection of their thermo-optical 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

*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:

*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

*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

*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 thermo-optical properties of NPs, and it is equal (Pustovalov et al. 2008):

*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

*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 one-order 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 two-layered 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 near-infrared 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 two-layered 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 two-layered core–shell NPs were calculated on the base of extended Mie theory (Pustovalov et al. 2009).

### 5.1 Heating of spherical two-layered core–shell nanoparticles by short laser pulses

*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 two-layered core–shell NP and its cooling after the end of laser pulse action is described by Eq. 8:

*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 cross-section, and *S*_{1} = *4πr*
_{1}
^{2}
is the surface area of a spherical particle of radius *r*_{1}.

*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):

*ρ*

_{0},

*c*

_{0},

*r*

_{0}and shell

*ρ*

_{1},

*c*

_{1},

*r*

_{1}parameters, where \( {\tau_T} = \frac{{{c_0}{\rho_0}r_0^2}}{{3{k_\infty }}} \).

*∆T*

_{0}/

*E*

_{0}= (

*T*

_{max}−

*T*

_{∞})/

*E*

_{0}is determined by

*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 two-layered 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 near-infrared 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} < 2*r*_{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 two-layered 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

*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 two-layered 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 two-layered 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.

*r*

_{1}=

*r*

_{0},

*ρ*

_{1}=

*ρ*

_{0},

*c*

_{1}=

*c*

_{0},

*V*

_{0}

*= πr*

_{0}

^{2}

*L*,

*S*

_{1}

*=*2

*πr*

_{0}

*L*, and

*S*

_{10}= 2

*πr*

_{0}

*L*, we have for

*T*

_{max}and

*ΔT*

_{0}/

*I*

_{0}:

*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 hear-infrared 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.

*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 thermo-optical and acoustic parameters (by increasing NP size or using ambiences with low sound speed)

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 two-layered 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 near-infrared 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 (water-rich 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.

## Declarations

### 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.

## Authors’ Affiliations

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