3.1.

Evaluation of GNP Heating Efficiency via Absorbed Power

As the initial evaluation criterion for GNP heating efficiency, the parameter $Q_{\mathrm{abs}}$ can be taken.³⁸ In the framework of the Mie theory, calculations over a wide range of wavelengths of the incident light $\lambda$ from 320 to 1400 nm and nanoparticle radius $R$ from 2 to 100 nm were performed. The results of calculations are shown in Fig. 1 in the form of topograms. This graphic representation allows one to understand specific features of interaction of the incident light with GNPs, which appear in the form of nanoparticle absorbed power and correspondingly in the local temperature increase around the particle.

Fig. 1

Distribution of the efficiency of absorption $Q_{\mathrm{abs}}$ (a), scattering $Q_{\mathrm{sca}}$ (b), and extinction $Q_{\mathrm{ext}}$ (c) of the gold nanoparticle (GNP) versus wavelength $\lambda$ and nanoparticle radius $R$.

The overall behavior of absorption efficiency $Q_{\mathrm{abs}}$ demonstrates the sensitivity to change both the light wavelength and the particle size [Fig. 1(a)]. This characteristic has a sharp maximum at the wavelength of 520 nm and nanoparticle radius of 38 to 40 nm. This is a result of existence of surface plasmon resonance (SPR) for GNPs induced by collective oscillatory motion of conduction electrons in the nanoparticle (see, for example, Refs. 2 and 38).

Parameters, $Q_{\mathrm{abs}}$, $Q_{\mathrm{sca}}$, and $Q_{\mathrm{ext}}$, calculated for spherical GNPs at a few laser wavelengths can be found in literature.²⁹ The obtained dependences that account for the size effects are also in good agreement with the data on $Q_{\mathrm{ext}}$ presented in Ref. 2. The SPR phenomenon is one of the fundamental bases of laser-induced hyperthermia of tissues doped by nanoparticles.^{1,2,5–7,18,25,27–33,41–45} Namely, SPR absorption of laser radiation caused by collective oscillations of electrons in the nanoparticle material, described by Eqs. (1)–(3), provides a broad use of GNPs in different fields of biology and medicine.^{1,2,5–7,10–18,21–33,41–45}

At the same time, we draw attention to the existence of a wide plateau around the exact resonance at its half-maximum. This plateau has a rather large area in coordinates $\lambda –R$: for wavelength range from less than 320 nm to 520 nm and for GNP radius from $\sim 40\mathrm{nm}$ to more than 100 nm. At the wavelengths above 720 nm in the entire NIR region and throughout the considered range of nanoparticle radius, 2 to 100 nm, the absorption efficiency $Q_{\mathrm{abs}}$ is of more than two orders of magnitude lower.

The distribution of volumetric density of heating power $U_{\mathrm{abs}}$ can be considered as an alternative criterion of nanoparticle heating efficiency. Figure 2 shows a plot for normalized (dimensionless) parameter $U_{\mathrm{abs}}/U_{\mathrm{abs}0}$ in $\lambda \times R$-coordinates. The calculations were executed at $I=100\mathrm{kW}/{\mathrm{cm}}^2$ with $U_{\mathrm{abs}0}=1.0\times {10}^{17}\mathrm{W}·{\mathrm{m}}^{-3}$ as a scaling factor equal to the maximum of $U_{\mathrm{abs}}$ in the considered region of ($\lambda \times R$)-space. The behavior of $U_{\mathrm{abs}}/U_{\mathrm{abs}0}$ maximal value takes the form of “ridge,” stretched along $R$-axis.

Fig. 2

Distribution of the normalized volumetric density of the heating power $U_{\mathrm{abs}}/U_{\mathrm{abs}0}$ of the nanoparticle versus the light wavelength $\lambda$ and the nanoparticle radius $R$.

The third criterion of the heating efficiency is the normalized integral thermal power of a single nanoparticle loading (dimensionless) $W_{\mathrm{abs}}/W_{\mathrm{abs}0}$ (Fig. 3). Here, $W_{\mathrm{abs}0}=5.2\times {10}^{-5}\mathrm{W}$ is a scaling factor equal to the maximum of $W_{\mathrm{abs}}$ in the ($\lambda \times R$)-space.

Fig. 3

Distribution of the normalized integral heating power of the nanoparticle $W_{\mathrm{abs}}/W_{\mathrm{abs}0}$ versus the light wavelength $\lambda$ and the nanoparticle radius $R$.

From the $W_{\mathrm{abs}}/W_{\mathrm{abs}0}$-topogram, it follows that this criterion gives a distinct advantage of large nanoparticles (to 100 nm) in a wide range of wavelengths less than 530 nm. In the region corresponding in Fig. 1 to the SPR (530 nm), now only poorly expressed signs of the local maximum, about three times lower than the global maximum, are seen. It becomes clear that the volume of nanoparticle is the predominant factor for heating efficiency, as SPR plays a leading role only for relatively small nanoparticles.

The further calculations of temperature increment will clarify the origin of the correlation of the considered criteria with the photothermal efficiency.

3.2.

Evaluating the GNP Heating Efficiency by Induced Temperature Field

From the calculations of the temperature increment on the nanoparticle surface regarding $T_0$ (temperature of tissue far away from the GNP, for living body $T_0=310\mathrm{K}$), $\mathrm{\Delta }T=T-T_0$ given in Fig. 4, it follows that $\mathrm{\Delta }T$ in two areas, where SPR exists and where large nanoparticles contributed are substantially equal. The likelihood that temperature generated by a large NP at a certain wavelength far from SPR may exceed temperature of a smaller NP with a strong SPR is also noted in Refs. 27 and 28. General view of topogram $\mathrm{\Delta }T$ in Fig. 4 differs markedly from the topograms in Figs. 1–3. Evidently, the distribution $Q_{\mathrm{abs}}$ in Fig. 1(a) has a bimodal maximum also; however, character of the peaks differs greatly from that in Fig. 4, where the plateau transforms into an extended rather steep slope with a maximum at large $R$. The ratio of the two maxima in Fig. 4 is opposite to that in Fig. 1(a). The comparison of topograms is presented in Figs. 2 and 4, and Figs. 3 and 4 allow one to reveal similar differences in the form of maxima and at their altitudes. The observed conversion of topograms is determined by the obvious relationships:

Fig. 4

Distribution of the temperature increment $\mathrm{\Delta }T$ on the surface of the nanoparticle versus wavelength $\lambda$ and nanoparticle radius $R$.

Within the SPR region, the integrated power absorbed by a nanoparticle $W_{\mathrm{abs}}$ is about three times lower than the power absorbed by a big nanoparticle (see Fig. 3). However, $\mathrm{\Delta }T$ is inversely proportional to $R$ for a given $W_{\mathrm{abs}}$. As a result, we get approximately an equivalent thermal effect (in terms of maximal temperature achievement) for these two practically important conditions (Fig. 4).

The temperature increment at a distance apart from nanoparticle surface in surrounding medium depends on nanoparticle size as well (Fig. 5). Near the surface of the nanoparticle within the distance, $\mathrm{\Delta }r=10\mathrm{nm}$ plasmon-resonant nanoparticle provides $\mathrm{\Delta }T$ about 10% less than a large one. As the distance from the nanoparticle increases, this difference also became bigger. At the distance $\mathrm{\Delta }r=100\mathrm{nm}$, $\mathrm{\Delta }T$ is 40% and at $\mathrm{\Delta }r=1000\mathrm{nm}$—60% less. Therefore, plasmon-resonant nanoparticles are less effective in heating of the surrounding medium. In general, with increase of distance $r$, $\mathrm{\Delta }T$-distribution (see Fig. 5) becomes similar to distribution of the normalized integral heating power of a single nanoparticle $W_{\mathrm{abs}}/W_{\mathrm{abs}0}$ in Fig. 3.

Fig. 5

Distribution of the temperature increment $\mathrm{\Delta }T$ in the medium versus wavelength $\lambda$ and nanoparticle radius $R$ at a distance $\mathrm{\Delta }r=10\mathrm{nm}$ (a); $\mathrm{\Delta }r=100\mathrm{nm}$ (b); $\mathrm{\Delta }r=1000\mathrm{nm}$ (c) from the particle surface.

3.3.

Evaluation of the Efficiency of GNPs by the Criterion of Arrhenius Damage Function

Thermal impact on tissues or cells is often used to provide therapeutic or biological effects, such as ablation of a tumor or coagulation a gastric ulcer, or cell laser optoporation and DNA transfection using lasers.^{1,2,5–7,10–18,21–33,41–45} In all these and many other optothermal technologies, we are able to estimate the time-temperature exposure that will cause a particular thermal transition.^39,40,46 The variety of thermal transitions includes very slow low-temperature exposures, such as thermally triggered heat shock protein expression, to the fast high-temperature exposures required to coagulate collagen fibers.⁴⁶ Van der Waals, hydrogen, ionic, disulfide, and covalent molecular bonds may be broken during these thermal transitions. As it was shown in Ref. 46, the rate of such transitions is characterized by the number of bonds being cooperatively broken, and for the most irreversible thermal transitions, simultaneous breakage of many bonds is characteristic. Exposure time versus exposure temperature to achieve thermal damage for different types of transitions, such as induced heat shock protein expression, cell membrane damage, cell apoptosis, tissue necrosis, dermal shrinkage, collagen birefringence loss, collagen coagulation, etc., are in a wide range of time (from ${10}^{-8}$ to ${10}^5\mathrm{s}$) and temperature (from 40°C to 140°C) exposures.^39,40,44,46

If the concentration of GNPs is high and other nanoparticles are in the area of thermal impact of the nanoparticle under consideration, then the summation of thermal fields is necessary with the contribution of each of these GNPs. The area around the nanoparticle of radius $r_{\mathrm{a}}$, within which impact is negligible, for example $\mathrm{\Delta }T(r_{\mathrm{a}})/\mathrm{\Delta }T(r_{\mathrm{a}}=R)\le 0.01$, can be introduced as “the zone of thermal impact of a nanoparticle.” From Eq. (8), it follows that for GNP of $R=2\mathrm{nm}$, $r_{\mathrm{a}}=0.2\mu \mathrm{m}$, and for GNP of $R=100\mathrm{nm}$ $r_{\mathrm{a}}=10\mu \mathrm{m}$, i.e., in the limits of a single cell or its organelles. For simplicity of consideration, but without loss of generality, we assume that in the zone of thermal impact of the nanoparticle, other nanoparticles are absent. This assumption in quite reasonable for many biomedical applications.⁴⁴

The possibility of applying the stationary Eq. (8) to consider laser exposures $\tau$ of tens to hundreds seconds follows from numerical estimates of transient temperature field during heating of nanoparticle and its environment.^30–32 It was found that for a pulse duration ${\tau }_{\mathrm{p}}\ge 100\mathrm{ns}$, temperature difference within the area of thermal impact of the GNP becomes stationary throughout the investigated range of particle size $R=2÷100\mathrm{nm}$.

The irreversible thermal damage of a particular type is described by the Arrhenius rate process (Arrhenius function or integral) with the corresponding parameters ^39,40,46

The dependence of the damage accumulation rate, defined as time derivative of the Arrhenius function $d\mathrm{\Omega }/d\tau$, on temperature is shown in Fig. 6(a). For these calculations, parameters $A=3.1·{10}^{98}{\mathrm{s}}^{-1}$ and $E_{\mathrm{a}}=6.3·{10}^5\mathrm{J}/\mathrm{mol}$ that characteristic for pig skin damage were used.⁴⁰ The damage critical temperature $T_{\mathrm{crit}}$, defined as temperature at $d\mathrm{\Omega }/d\tau =1$, is equal to 59.7°C.⁴⁰ The $d\mathrm{\Omega }/d\tau$ function represents the rate of growth of tissue damage with temperature. The corresponding exposure time needed for 100%-damage of pig skin versus stationary temperature exposure is presented in Fig. 6(b). These data illustrate the criticality of thermal impact (threshold effect) relative to the temperature exposure.

Fig. 6

Dependence of the rate of Arrhenius function change $d\mathrm{\Omega }/d\tau$ (a) and exposure time needed for 100%-damage of pig skin versus stationary temperature exposure (b).

As it was already pointed out, various photothermal processes can be described using Arrhenius function,⁴⁶ including the increased permeability of the cell membrane at its local thermal damage provided by hot metallic nanoparticles that are heated by laser irradiation.^{10–13,31,33,47,48} The temperature exposure providing this damage is determined by the time exposure during which the irreversible phase transitions are distributed throughout the entire thickness of the cell membrane and some area along membrane around the nanoparticle connecting with the cell membrane. Therefore, it is advisable to transform Eq. (10) for the damage integral into inequality as

Knowing the temperature distribution in the vicinity of the nanoparticle that is irradiated by a laser light, one can determine on the basis of Eq. (11) the exposure time $\tau$, required to damage locally cell membrane or tissue. As the temperature decreases monotonically with distance from the heated nanoparticle, the execution of the inequality described by Eq. (11) at a distance $\mathrm{\Delta }r$ from the particle will mean automatically that within the layer of thickness $\mathrm{\Delta }r=r-R$ of the biological medium around the nanoparticle, the damage condition is obviously achieved. Targeted delivery of nanoparticles to the cell within a tissue allows for selective nanoparticle heating and corresponding thermal impact on the cell in accordance with Arrhenius damage function. The precision of the light/temperature exposure needed to induce a particular action is determined by the ability to create the conditions, when smoothly controlled interaction of laser radiation with tissues doped by nanoparticles is provided.

The exposure time $\tau$ is the most widespread and robust controlling parameter used for implementation of laser hyperthermia techniques. The calculated topograms of the isoline distribution for $\tau$ in the range of 60 to 600 s for different distances $\mathrm{\Delta }r$ apart from the nanoparticle are shown in Fig. 7.

Fig. 7

($\lambda \times R$)-Topograms for the exposure time $\tau$ that provides a local damage in a tissue (pig skin): on the surface of the nanoparticle (a), at a distance of 30 nm apart nanoparticle (b), 120 nm (c), and 300 nm (d).

Obtained results show with evidence that in practice, the ensuring compliance of the specific requirements to exposure time is a nontrivial task. The range of admissible values of parameters $\lambda$ and $R$ occupies only a small area in the coordinate space ($\lambda \times R$). Often it is similar to a line of small thickness, which indicates a high criticality of providing of laser hyperthermia. Small changes in any of the parameters $\lambda$ or $R$ will lead to considerable changes in exposure time τ.

Thus, the results shown in Fig. 7(a) (on the surface of the nanoparticle) demonstrate that the admissible variability of nanoparticle size is only in the limits of 2 to 4 nm for the wide range of laser wavelengths from 320 to 600 nm. If size of the nanoparticle is outside these limits, the local damage in a cell or tissue contacting with the nanoparticle comes very quickly at decreasing of nanoparticle radius (subsecond range instead of predetermined minute range), or very slowly within hours of exposure at radius increase. Transition from overheating to underheating becomes practically not controllable. Thus, noncriticality in relation to the wavelength is accompanied by criticality with respect to the radius of the GNP. This is characteristic for curve fragments with small inclination to the horizontal axis.

However, as can be seen in Fig. 7(a), there is a region of parameters: $80<R<100\mathrm{nm}$ and $615<\lambda <660\mathrm{nm}$, where $\tau$ isolines rarefied and distributed relatively uniformly over the area of interest. This indicates a lack of criticality with respect to the radius of nanoparticle as well as the laser wavelength. There is also a second area of noncriticality that corresponds to the wavelength close to 670 nm. The change of nanoparticle radius $R$ from 60 to 75 nm does not result in catastrophic change of exposure $\tau$, while remaining within the prescribed limits. This vertical fragment of curve corresponds to the zone of noncriticality in relation to $R$ and simultaneously to the zone of criticality with respect to $\lambda$.

These statements are also important for the problem of polydispersity of used nanoparticles. For better control of photothermal action mediated by plasmonic nanoparticles, monodisperse nanoparticle solutions are preferable. However, polydisperse systems allow one to activate differently sized nanoparticles by the wavelength tuning.

If we set thickness of a layer in surrounding medium with hyperthermia as $\mathrm{\Delta }r=30\mathrm{nm}$ (thrice larger than the thickness of cell membrane that complies requirements of cell optoporation),^11–13 the two zones of noncriticality (optimal exposure parameters) are converted to the form shown in Fig. 7(b). The size of the zone corresponding to the 60 to 600 s exposure time is reduced by coordinate $R$ on 12 to 15 nm and decreased in the wavelength range on 20 to 40 nm. At the same time, there is a tendency for optimal $R$ and $\lambda$ to be decreased.

The further increase of $\mathrm{\Delta }r$ to 120 nm leads to the appearance of the new zone of noncriticality (optimality) in Fig. 7(c), the center of which corresponds to coordinates of SPR. Its special feature is the ability to support exposure time $\tau$ in a very narrow range of 300 to 360 s with using nanoparticles of radius $45\pm 5\mathrm{nm}$ and the laser wavelength of $525\pm 6\mathrm{nm}$. The fulfillment of damage conditions in the space around the nanoparticle within 300 nm from its surface may be achievable only for large nanoparticles with a radius of no less than 95 nm and the irradiating wavelength of 520 nm [see Fig. 7(d)]. The general trend of parameter transformation to provide the desired exposure time is as follows: with increasing distance from the nanoparticle, the necessary heating/impact conditions are provided only with nanoparticle radius increase and use of a shorter wavelength.

The described technique can be also applicable to study local laser hyperthermia at short pulse irradiation where integration of Eq. (10) should be done for nonstationary temperature fields.⁴⁵ To account for more complex nanoparticle shape, such as composite nanoshells, nanorods, nanocages, and nanostars, the adaptive numerical models for calculating temperature fields should be used while the other main stages of the simulation will be retained.