Laser heating of dielectric particles for medical and biological applications Michael I. Tribelsky1,2,∗ and Yasuhide Fukumoto3,4 1 Faculty
of Physics, M. V. Lomonosov Moscow State University, Moscow, 119991, Russia 2 Moscow Technological University MIREA, Moscow, 119454, Russia 3 Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan 4 [email protected] ∗ [email protected]
Abstract: We consider the general problem of laser pulse heating of a spherical dielectric particle embedded in a liquid. The discussed range of the problem parameters is typical for medical and biological applications. We focus on the case, when the heat diffusivity in the particle is of the same order of magnitude as that in the fluid. We perform quantitative analysis of the heat transfer equation based on interplay of four characteristic scales of the problem, namely the particle radius, the characteristic depth of light absorption in the material of the particle and the two heat diffusion lengths: in the particle and in the embedding liquid. A new quantitative characteristic of the laser action, that is the cooling time, describing the temporal scale of the cooling down of the particle after the laser pulse is over, is introduced and discussed. Simple analytical formulas for the temperature rise in the center of the particle and at its surface as well as for the cooling time are obtained. We show that at the appropriate choice of the problem parameters the cooling time may be by many orders of magnitude larger the laser pulse duration. It makes possible to minimize the undesirable damage of healthy tissues owing to the finite size of the laser beam and scattering of the laser radiation, simultaneously keeping the total hyperthermia period large enough to kill the pathogenic cells. An example of application of the developed approach to optimization of the therapeutic effect at the laser heating of particles for cancer therapy is presented. © 2016 Optical Society of America OCIS codes: (350.4990) Particles; (170.1610) Clinical applications.
References and links 1. A.N. Volkov, C. Sevilla, and L. Zhigilei,“Numerical modeling of short pulse laser interaction with Au nanoparticle surrounded by water,” Appl. Surface Sci. 253, 6394–6399 (2007). 2. H.H. Richardson, M.T. Carlson, P.J. Tandler, P. Hernandez, and A.O. Govorov, “Experimental and theoretical studies of light-to-heat conversion and collective heating effects in metal nanoparticle solutions,” Nano Lett. 9, 1139–1146 (2009). 3. G.W. Hanson and S.K. Patch, “Optimum electromagnetic heating of nanoparticle thermal contrast agents at rf frequencies,” J. Appl. Phys. 106, 054309 (2009). 4. E. Sassaroni, K.C.P. Li, and B.E. O’Neill, “Numerical investigation of heating of a gold nanoparticle and the surrounding microenvironment by nanosecond laser pulses for nanomedicine applications,” Phys. Med. Biol. 54, 5541–5560 (2009). 5. S. Bruzzone and M. Malvaldi, “Local field effects on laser-induced heating of metal nanoparticles,” J. Phys. Chem. C 113, 15805–15810 (2009).
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2781
6. L.I. Sedov, Similarity and Dimensional Methods in Mechanics (CRC Press, 1993) pp. 493. 7. Y. Fukumoto and T. Izuyama, “Thermal attenuation and dispersion of sound in a periodic emulsion,” Phys. Rev. A, 46, 4905–4921 (1992). 8. M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Luk’yanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011). 9. J.M. Geffrin, B. Garc´ıa-C´amara, R. G´omez-Medina, P. Albella, L.S. Froufe-Perez, C. Eyraud, A. Litman, R. Vaillon, F. Gonzalez, M. Neito-Vesperinas, J.J. Saenz, and F. Moreno, “Magnetic and electric coherence in forwardand back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nature Comm. 3, 1171 (2012). 10. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Comm. 4, 1527 (2013). 11. L.D. Landau, L.P. Pitaevskii, and E.M. Lifshitz Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), pp. 460. 12. H.S. Carslow and J.C. Jaeger Conduction of Heat in Solids (Oxford Univeristy Press, Oxford, 1959). 13. M. Born and E. Wolf Principles of Optics (Cambridge University Press, Cambridge, 1999). 14. E.D. Palik Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985). 15. M.I. Tribelsky, A. E. Miroshnichenko, “Giant in-particle field concentration and Fano resonances at light scattering by high-refractive index particles,” Phys. Rev. A, 93, 053837 (2016).
Introduction The problem of the laser pulse heating of absorbing particles embedded in a transparent liquid medium is important for different applications, including stimulation of chemical reactions, laser sintering, selective killing of pathogenic bacteria or cancer cells, etc. A metal nanoparticle heated by a laser pulse with the frequency close to its plasmon resonance efficiently converts electromagnetic energy into thermal energy, with dramatic temperature rise in the surrounding medium. As a result, this problem or its substantial parts were a subject of many recent papers in physics, biology, and chemistry (see, e.g., Refs. [1–5]). Among the variety of these publications, the most close to the subject of our present study is ref. [8]. In this paper the problem of the laser heating of the particles is attacked with the help of dimensional analysis. The power of dimensional analysis is well known, see, e.g. book [6]. Changes of the relation between a scale of a problem and the corresponding characteristic scale often results in changes of the corresponding regime(s). An example of this effect in the thermal attenuation and dispersion of sound in an emulsion is presented in ref. [7]. Unfortunately, in most cases the dimensional methods can result just in rather rough estimates with the accuracy up to the order of magnitude. In contrast, in ref. [8] it was shown that dimensional analysis of the non-steady heat diffusion problem based upon the concept of the characteristic scales gives rise to simple formulas for the temperature rise at the surface of the particle Ts . These formulas are in surprisingly good agreement with more sophisticated study, as well as with results of the direct numerical simulation of the problem. However, the analysis performed in ref. [8] was restricted by the case, when the heat diffusivity of the particle (χ p ) is much larger than that in the embedded fluid (χ f ). Such a condition holds for metal particles. On the other hand, recently dielectric particles, especially those with high refractive index, have been moved to the very frontier of the general light scattering problem [9, 10]. Then, extension of the results of ref. [8] to the case of dielectrics with low values of the heat diffusivity is required. This extension is presented below. There are a few important points to be made in this connection: First, the approach developed in ref. [8] and employed in the present paper is valid in a very broad range of the problem parameters. In particular, it may be applied for the duration of the laser pulses varying from femtoseconds to CW regimes and for the laser wavelengths lying from UV to far IR diapasons of the spectrum. Second, though the basic ideas employed in the present paper are the same as those in ref. [8], the different relations between the characteristic scales of the problem give rise to quite different heating regimes of the particle. Third, in the present paper we introduce and
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2782
discuss a new quantity — the cooling time of the irradiated particle. This quantity describes the characteristic temporal scale of the cooling down of the particle after the laser pulse is over. In our view this characteristic of the process is extremely important, especially for biomedical applications. Since the total time of the hyperthermia of a biological tissue in the vicinity the particle is the sum of the heating time and cooling one, a large cooling time may extend the hyperthermia period by many orders of magnitudes, making it much larger than the laser pulse duration τ (see the corresponding discussion below). The latter may be vital to the minimization of laser damage of healthy tissues far from the particle, inevitably occurring during the laser action because of a finite size of the laser beam and scattering of the incident radiation. This damage does not occur in the cooling period, while the hyperthermia does, since it lasts as long as the particle remains heated. Last but not least, the results obtained should not be overvalued. The toy spherically symmetric models adopted in our approach do not take into account many fine effects, such as the essentially non-uniform field distribution in the optically thick dielectric particle with pronounced foci, interference patterns, etc., see, e.g., the recent publication of one of the authors [15]. However, as a rule, these peculiarities do not change the order of magnitude of the temperature achieved at the surface of the particle. Then, the simple results discussed in the present paper can help to select from a broad domain of the possible values of the problem parameters rather narrow subdomains of a special interest. Next, within these subdomains the heating problem may be studied with the help of more precise methods, if such a study is required. Characteristic scales Let us consider the heating with a laser pulse with duration τ of a single spherical, spatially uniform particle, embedded in a liquid. We will focus on medico-biological applications of the problem, when the thermometric properties of the embedded liquid are close to those for water. In this case particles with heat diffusivity much smaller than that in water look hypothetical — the lowest actual heat diffusivity of a dielectric particle should be of the same order of magnitude as that of the fluid, i.e., χ p ∼ χ f . It is also supposed that the embedding medium is transparent at the frequency of the irradiating laser beam. In other words the laser radiation is absorbed by the particle solely. Regarding the heating of the surrounding particle medium, it occurs just because the latter is in a heat contact with the particle. Since in what follows we are interested in estimations rather than in the exact expressions, we may suppose that the particle is a sphere with radius R. Then, under the specified conditions the problem in question is characterized by the three quantities with dimension of length, namely R, the depth of penetration of the laser radiation into the particle material δ and heat diffusion √ length 2 τ χ p, f [8]. Interplay of these three scales defines the possible regimes of the laser heating. Obviously, there are just six possible cases of the hierarchy of these scales, namely: √ τ χ p, f √ τ χ p, f
√ √ R δ , R τ χ p, f δ , R δ τ χ p, f ; √ √ δ R, δ τ χ p, f R, δ R τ χ p, f .
(1) (2)
The case with R δ [Eq. (1)] we will name the case of a small particle and indicate with letter S, the case Eq. (2) will be named the one of a large particle and indicated with letter L. Let us discuss all the cases one by one. To begin with, we focus on the power dissipated in the particle P = σabs I, where I stands for the intensity of the laser beam [W/cm2 ] and σabs is the absorption cross section. In the case of a small particle (R δ ) the incident light penetrates into the entire particle, and the energy release should be proportional to the particle volume, that is to say, the absorption cross section
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2783
should be proportional to R3 . Then, following the approach developed in ref. [8] it is convenient to present σabs in the form: σabs = αkR3 , (3) where α is a dimensionless quantity and k = n f ω/c stands for the wave number of the incident light in the fluid. Here n f is the (purely real) refractive index of the fluid and c designates the speed of light in a vacuum. In the case of the Rayleigh scattering α has the following simple R-independent form [8]: α=
12πε 00 , (ε 0 + 2)2 + ε 002
(4)
where ε 0 and ε 00 stand for the real and imaginary parts of the complex relative dielectric permittivity of the particle (ε = ε p /n2f ). For a large particle (R δ ) σabs = πR2 Qabs . (5) Apart the points of the so called Mie resonances the R-dependence of the dimensionless quantity Qabs (the scattering efficiency) is weak, so that the main part of the R-dependence of σabs is reduced to R2 , explicitly singled out in Eq. (5). Now we are ready to discuss each of the cases, Eqs. (1), (2) in detail. Specific cases √ Case S1: τ χ p, f R δ . Owing the smallness of the particle the energy of the laser beam is released homogeneously across the particle. On the other hand, the heat transfer occurs just in narrow layers (one in the particle and another in the liquid) in the proximity of the particle surface. For the sake of simplicity here and in what follows we suppose that the laser pulse has a rectangular shape, so that I at 0 ≤ t ≤ τ, I(t) = (6) 0 at t < 0; t > τ. Then, from the energy conservation law, neglecting the heat transfer, we obtain: 4 3 πR C p ρ p Tmax = σabs It = αkR3 It, 3
(7)
where C p is the specific heat of the particle, ρ p stands for its density and Tmax designates the maximal temperature rise in the particle, which in this case is achieved in its center. According to Eq. (7) 3 σabs It 3 αkIt Tmax = = . (8) 4 πR3C p ρ p 4 πC p ρ p Thus, in this case Tmax is, practically, R-independent. However, if we could neglect the heat transfer in the bulk of the particle, we cannot do that in the proximity of the particle surface. To simplify the problem we approximate the actual temperature field in this region at 0 ≤ t ≤ τ by linear profiles, supposing that √ Tmax at x ≤ −2 tχ p , √ √ −Ts x at − 2 tχ p < x ≤ 0, Ts − T2max tχ p (9) T= √ Ts 1 − 2√xtχ at 0 < x < 2 tχ f , f 0 at x ≥ 2√tχ , f #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2784
where Ts stands for yet unknown temperature rise at the surface of the particle and x = r − R. √ √ Then, the energy balance at −2 tχ p ≤ x ≤ 2 tχ f reads as follows: ! Z 0 Z 2√tχ f 3 σabs It 2 p 2 4πR 2 tχ p T dx . (10) = 4πR C p ρ p T dx +C f ρ f √ 4 πR3C p ρ p 0 −2 tχ p Substituting Eq. (9) into Eq. (10) and solving the resulting equation with respect to Ts , one obtains 1 √ . Ts = Tmax (11) Cf ρf χf 1 + C p ρ p √χ p As it already has been pointed out above, an important characteristic of the laser heating of the particle embedded into a biological tissue for medical, or biological applications is the cooling time τcool , i.e., the time required for a particle to cool down to the initial temperature after the end of the laser pulse heating. In some cases (see below) τcool may be much larger than the duration of the laser pulse τ, so that just τcool determines the actual time of the hyperthermia in the proximity of the particle. For the case in question we have exactly this situation: 2 R ∼ τcool = Max τ. (12) 4χ p, f √ √ Cases S2, S3: R τ χ p, f δ and R δ τ χ p, f . In these cases there is a quasi-steady temperature field both within and outside the particle. Within the particle this field is described by the the following quasi-steady version of the heat diffusion equation with a spatially homogeneous volume source: 1 d 3 σabs I 2 dT κp 2 r + = 0 at r ≤ R, (13) r dr dr 4 πR3 where κ p is the thermal conductivity of the particle. Regarding the liquid, the temperature field there is governed by the homogeneous Laplace equation. The well known solution of the latter, satisfying the boundary condition T → 0 at r → ∞ is R T = Ts . (14) r Owing to the quasi-steadiness of the temperature field in this case, the energy balance means that at any moment of time the entire power of the laser beam released in the particle is transferred to the surrounding liquid and then, by heat diffusion in the liquid the heat flux is conveyed to “infinity”, which plays the role of a sink in this problem. Then, the heat flux through any spherical surface with radius r, larger than, or equal to R has one and the same value, equal to the power of the laser beam, released in the particle. These arguments give rise to the following equation: dT −4πr2 κ f = σabs I, (15) dr r≥R where κ f stands for the heat conductivity of the fluid. Eqs. (14), (15) yield Ts =
σabs I αR2 kI = . 4πRκ f 4πκ f
(16)
The temperature profile within the particle is obtained by integration of Eq. (13). A finite at r = 0 solution of this equation, satisfying the boundary condition T (R) = Ts is κf r2 T = Ts 1 + 1− 2 . (17) 2κ p R #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2785
The cooling time in this case is still defined by Eq. (12). It is much smaller than τ. √ Case L1: τ χ p, f δ R. For a large particle the absorption cross section is described by Eq. (5). For the case under consideration the heat diffusion length is the smallest scale in the problem. Then, we can suppose that the energy is released in spatially homogenous manner within the absorption layer δ lying just below the particle surface, while the bulk of the particle remains cool. For the temperature rise Tmax within the heated layer δ and at the very surface of the particle Ts the arguments analogous to those in Case S1 are applied. The only difference between the present case and Case S1 is in the expression for the volume density of the energy release, which now is given by the formula: σabs It/4πR2 δ . Thus, now in Eqs. (8), (10) we must replace 4πR3 /3 by 4πR2 δ . It yields Tmax =
Qabs It 1 σabs It ≡ . 2 4πR δ C p ρ p 4C p ρ p δ
(18)
Regarding Ts , formally it is given by the same Eq. (11), though now in this expression Tmax is defined by Eq. (18). The cooling time is the one required for broadening of a heated layer with thickness δ due to heat diffusion. The broadening goes faster in the medium with higher value of χ. Then, based on the dimensional analysis we may conclude that 2 δ ∼ τ. (19) τcool = Min 4χ p, f √ Case L2: δ τ χ p, f R. Since for the case in question the temperature diffusion length is much smaller than the radius of the particle, but much larger than the energy release scale, the heating problem may be approximated by heating of two contacting along a plane semi-infinite spaces with one possessing the thermometric properties of the particle and the other with those of the fluid, while the energy is released at the boundary plane between the semispaces. This problem is exactly solvable [12]. The solution yields the following expression for the surfacetemperature rise: √ √ tχ p χ f tχ p χ f σabs I Qabs I Ts = 2 3/2 √ (20) √ ≡ √ √ . √ 2 π κp χ f + κ f χp 2R π κ p χ f + κ f χ p √ The temperature profile is practically R-independent. and the temperature rise increases as t, so that its maximum is achieved at the very end of the laser pulse (t = τ). In this limit the heating – cooling problem does not have any characteristic timescale but τ. Thus, the heating and cooling times are the same and both equal τ. √ Case L3: δ R τ χ p, f . In this case the temperature field is quasi-steady. It is described by the homogeneous Laplace equation both in the liquid and in the particle, while the energy may be supposed releasing at the surface of the particle. Then, in the fluid the temperature profile is described by Eq. (14), where now Ts =
σabs I RQabs I = , 4πRκ f 4κ f
(21)
see Eq. (5). Regarding the temperature within the particle, the only non-singular solution of the homogeneous Laplace equation in the spherically-symmetric case is a constant. Owing to continuity of the temperature field at the surface of the particle, this constant must be equal to Ts . The cooling time is given by Eq. (12). In the case in question it is much shorter than τ. #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2786
Fig. 1. (a) The maximal temperature rise at the surface of a dielectric, high absorbing particle with δ = 10−5 cm, R = 10−4 cm as a function of the laser pulse duration τ. For different curves (from right to left) the intensity of the laser beam equals 105 , 106 , 107 , 108 , 109 , 1010 W/cm2 , respectively. The discontinuity at τ = 1.79 10−8 s corresponds to the transition from Eq. (18) to Eq. (20). (b) The cooling time normalized over the pulse duration τ, as a function of τ. Note that at τ = 10−14 s the total hypothermal exposure is more than a million times larger the pulse duration, see the text for details.
Example of optimization of the laser heating Let us give an example of application of the developed approach to the optimization of the laser treatment of cancer. It is supposed that the cancer cells are killed selectively owing to their hyperthermia caused by heating of light absorbing particles localized close to the cells, or inside them. In this case it is highly desirable to maximize (i) the temperature at the surface of the particles and (ii) the time of the exposure of the cells to high temperature. In contrast, the duration of the exposure of the healthy tissues to the laser beam should be minimized to prevent harmful side-effects of the treatment. Bearing it in mind, we inspect the dependance of Ts , the heating and cooling time on τ for, e.g., a large particle with R δ . In this case, increasing τ from τ δ 2 /χ p, f to τ R2 /χ p, f we sequentially move from Case L1 to Case L3. Accordingly, at τ ∼ δ 2 /(4χ)√the linear growth of Ts with an increase in τ is replaced by much slower ones, proportional to τ, which, saturates at τ ∼ R2 /(4χ). Here χ stands for the heat diffusivity of both the fluid and the particle, since for the problem in question these quantities are of the same order of magnitude. Regarding the cooling time, it is given by the condition: 2 δ τcool ∼ ,τ . (22) = Max 4χ To give an impression about the discussed dependences in Fig. 1(a) the maximal surface temperature rise for a particle with δ = 10−5 cm, R = 10−4 cm and the thermometric constants identical to those of water are presented as functions of the pulse duration τ at several values of the laser beam intensity. In accord with the specified application the range of variations of the temperature rise is selected from zero to hundred degrees Celsius. For the estimate we suppose that σabs = πR2 , i.e., the absorption cross section equals the geometric one. The cooling time as a function of the pulse duration is shown in Fig 1(b). Note the considerable decrease of the contribution of the pulse duration to the total exposure time at small τ due to the overwhelming role which the cooling time plays in this region. Thus, to optimize the action of the laser beam according to the aforementioned principles one has to remain within the range of Case L1. For small particles the same is true in Case S1,
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2787
see Eqs. (8), (11), (12). More concrete recommendations may be formulated if the desired value of Ts and the maximal duration of the laser exposure to the healthy tissues are specified. Conclusions Our analysis results in simple analytical expressions, describing the heating with a laser pulse of a dielectric particle embedded in a transparent liquid and its cooling down after the pulse is over. The obtained formulas cover any possible case of the hierarchy of the characteristic scales of the problem, provided the heat diffusivity of the particle and that of the fluid are of the same order of magnitude. Quite different dependence of the maximal temperature rise achieved during the laser pulse within the particle and on it surface on the problem parameters at different relations between the characteristic scales should be stressed. The results obtained may be employed for the optimization of the laser heating in order to maximize the positive effects of the laser treatment at various medical and/or biological applications of the problem in question and in a wide variety of other problems. Acknowledgment This study has been conducted during a visit of MT to the Institute of Mathematics for Industry. Taking the opportunity, MT expresses his gratitude to the inure staff of IMI for the kind hospitality and provision of excellent conditions to carry outs research. The financial support of the Ministry of Education, Culture, Sport, Science, and Technology of Japan, which has made the visit possible is greatly appreciated. The study has been partially supported by the Center of Innovation Program from Japan Science and Technology Agency, JST. YF was also partially supported by a Grant-in-Aid for Scientific Research from the Japan Society of Promotion of Science (Grant No. 16K05476).
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2788
of Physics, M. V. Lomonosov Moscow State University, Moscow, 119991, Russia 2 Moscow Technological University MIREA, Moscow, 119454, Russia 3 Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan 4 [email protected] ∗ [email protected]
Abstract: We consider the general problem of laser pulse heating of a spherical dielectric particle embedded in a liquid. The discussed range of the problem parameters is typical for medical and biological applications. We focus on the case, when the heat diffusivity in the particle is of the same order of magnitude as that in the fluid. We perform quantitative analysis of the heat transfer equation based on interplay of four characteristic scales of the problem, namely the particle radius, the characteristic depth of light absorption in the material of the particle and the two heat diffusion lengths: in the particle and in the embedding liquid. A new quantitative characteristic of the laser action, that is the cooling time, describing the temporal scale of the cooling down of the particle after the laser pulse is over, is introduced and discussed. Simple analytical formulas for the temperature rise in the center of the particle and at its surface as well as for the cooling time are obtained. We show that at the appropriate choice of the problem parameters the cooling time may be by many orders of magnitude larger the laser pulse duration. It makes possible to minimize the undesirable damage of healthy tissues owing to the finite size of the laser beam and scattering of the laser radiation, simultaneously keeping the total hyperthermia period large enough to kill the pathogenic cells. An example of application of the developed approach to optimization of the therapeutic effect at the laser heating of particles for cancer therapy is presented. © 2016 Optical Society of America OCIS codes: (350.4990) Particles; (170.1610) Clinical applications.
References and links 1. A.N. Volkov, C. Sevilla, and L. Zhigilei,“Numerical modeling of short pulse laser interaction with Au nanoparticle surrounded by water,” Appl. Surface Sci. 253, 6394–6399 (2007). 2. H.H. Richardson, M.T. Carlson, P.J. Tandler, P. Hernandez, and A.O. Govorov, “Experimental and theoretical studies of light-to-heat conversion and collective heating effects in metal nanoparticle solutions,” Nano Lett. 9, 1139–1146 (2009). 3. G.W. Hanson and S.K. Patch, “Optimum electromagnetic heating of nanoparticle thermal contrast agents at rf frequencies,” J. Appl. Phys. 106, 054309 (2009). 4. E. Sassaroni, K.C.P. Li, and B.E. O’Neill, “Numerical investigation of heating of a gold nanoparticle and the surrounding microenvironment by nanosecond laser pulses for nanomedicine applications,” Phys. Med. Biol. 54, 5541–5560 (2009). 5. S. Bruzzone and M. Malvaldi, “Local field effects on laser-induced heating of metal nanoparticles,” J. Phys. Chem. C 113, 15805–15810 (2009).
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2781
6. L.I. Sedov, Similarity and Dimensional Methods in Mechanics (CRC Press, 1993) pp. 493. 7. Y. Fukumoto and T. Izuyama, “Thermal attenuation and dispersion of sound in a periodic emulsion,” Phys. Rev. A, 46, 4905–4921 (1992). 8. M. I. Tribelsky, A. E. Miroshnichenko, Y. S. Kivshar, B. S. Luk’yanchuk, and A. R. Khokhlov, “Laser pulse heating of spherical metal particles,” Phys. Rev. X 1, 021024 (2011). 9. J.M. Geffrin, B. Garc´ıa-C´amara, R. G´omez-Medina, P. Albella, L.S. Froufe-Perez, C. Eyraud, A. Litman, R. Vaillon, F. Gonzalez, M. Neito-Vesperinas, J.J. Saenz, and F. Moreno, “Magnetic and electric coherence in forwardand back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nature Comm. 3, 1171 (2012). 10. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Comm. 4, 1527 (2013). 11. L.D. Landau, L.P. Pitaevskii, and E.M. Lifshitz Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984), pp. 460. 12. H.S. Carslow and J.C. Jaeger Conduction of Heat in Solids (Oxford Univeristy Press, Oxford, 1959). 13. M. Born and E. Wolf Principles of Optics (Cambridge University Press, Cambridge, 1999). 14. E.D. Palik Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985). 15. M.I. Tribelsky, A. E. Miroshnichenko, “Giant in-particle field concentration and Fano resonances at light scattering by high-refractive index particles,” Phys. Rev. A, 93, 053837 (2016).
Introduction The problem of the laser pulse heating of absorbing particles embedded in a transparent liquid medium is important for different applications, including stimulation of chemical reactions, laser sintering, selective killing of pathogenic bacteria or cancer cells, etc. A metal nanoparticle heated by a laser pulse with the frequency close to its plasmon resonance efficiently converts electromagnetic energy into thermal energy, with dramatic temperature rise in the surrounding medium. As a result, this problem or its substantial parts were a subject of many recent papers in physics, biology, and chemistry (see, e.g., Refs. [1–5]). Among the variety of these publications, the most close to the subject of our present study is ref. [8]. In this paper the problem of the laser heating of the particles is attacked with the help of dimensional analysis. The power of dimensional analysis is well known, see, e.g. book [6]. Changes of the relation between a scale of a problem and the corresponding characteristic scale often results in changes of the corresponding regime(s). An example of this effect in the thermal attenuation and dispersion of sound in an emulsion is presented in ref. [7]. Unfortunately, in most cases the dimensional methods can result just in rather rough estimates with the accuracy up to the order of magnitude. In contrast, in ref. [8] it was shown that dimensional analysis of the non-steady heat diffusion problem based upon the concept of the characteristic scales gives rise to simple formulas for the temperature rise at the surface of the particle Ts . These formulas are in surprisingly good agreement with more sophisticated study, as well as with results of the direct numerical simulation of the problem. However, the analysis performed in ref. [8] was restricted by the case, when the heat diffusivity of the particle (χ p ) is much larger than that in the embedded fluid (χ f ). Such a condition holds for metal particles. On the other hand, recently dielectric particles, especially those with high refractive index, have been moved to the very frontier of the general light scattering problem [9, 10]. Then, extension of the results of ref. [8] to the case of dielectrics with low values of the heat diffusivity is required. This extension is presented below. There are a few important points to be made in this connection: First, the approach developed in ref. [8] and employed in the present paper is valid in a very broad range of the problem parameters. In particular, it may be applied for the duration of the laser pulses varying from femtoseconds to CW regimes and for the laser wavelengths lying from UV to far IR diapasons of the spectrum. Second, though the basic ideas employed in the present paper are the same as those in ref. [8], the different relations between the characteristic scales of the problem give rise to quite different heating regimes of the particle. Third, in the present paper we introduce and
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Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2782
discuss a new quantity — the cooling time of the irradiated particle. This quantity describes the characteristic temporal scale of the cooling down of the particle after the laser pulse is over. In our view this characteristic of the process is extremely important, especially for biomedical applications. Since the total time of the hyperthermia of a biological tissue in the vicinity the particle is the sum of the heating time and cooling one, a large cooling time may extend the hyperthermia period by many orders of magnitudes, making it much larger than the laser pulse duration τ (see the corresponding discussion below). The latter may be vital to the minimization of laser damage of healthy tissues far from the particle, inevitably occurring during the laser action because of a finite size of the laser beam and scattering of the incident radiation. This damage does not occur in the cooling period, while the hyperthermia does, since it lasts as long as the particle remains heated. Last but not least, the results obtained should not be overvalued. The toy spherically symmetric models adopted in our approach do not take into account many fine effects, such as the essentially non-uniform field distribution in the optically thick dielectric particle with pronounced foci, interference patterns, etc., see, e.g., the recent publication of one of the authors [15]. However, as a rule, these peculiarities do not change the order of magnitude of the temperature achieved at the surface of the particle. Then, the simple results discussed in the present paper can help to select from a broad domain of the possible values of the problem parameters rather narrow subdomains of a special interest. Next, within these subdomains the heating problem may be studied with the help of more precise methods, if such a study is required. Characteristic scales Let us consider the heating with a laser pulse with duration τ of a single spherical, spatially uniform particle, embedded in a liquid. We will focus on medico-biological applications of the problem, when the thermometric properties of the embedded liquid are close to those for water. In this case particles with heat diffusivity much smaller than that in water look hypothetical — the lowest actual heat diffusivity of a dielectric particle should be of the same order of magnitude as that of the fluid, i.e., χ p ∼ χ f . It is also supposed that the embedding medium is transparent at the frequency of the irradiating laser beam. In other words the laser radiation is absorbed by the particle solely. Regarding the heating of the surrounding particle medium, it occurs just because the latter is in a heat contact with the particle. Since in what follows we are interested in estimations rather than in the exact expressions, we may suppose that the particle is a sphere with radius R. Then, under the specified conditions the problem in question is characterized by the three quantities with dimension of length, namely R, the depth of penetration of the laser radiation into the particle material δ and heat diffusion √ length 2 τ χ p, f [8]. Interplay of these three scales defines the possible regimes of the laser heating. Obviously, there are just six possible cases of the hierarchy of these scales, namely: √ τ χ p, f √ τ χ p, f
√ √ R δ , R τ χ p, f δ , R δ τ χ p, f ; √ √ δ R, δ τ χ p, f R, δ R τ χ p, f .
(1) (2)
The case with R δ [Eq. (1)] we will name the case of a small particle and indicate with letter S, the case Eq. (2) will be named the one of a large particle and indicated with letter L. Let us discuss all the cases one by one. To begin with, we focus on the power dissipated in the particle P = σabs I, where I stands for the intensity of the laser beam [W/cm2 ] and σabs is the absorption cross section. In the case of a small particle (R δ ) the incident light penetrates into the entire particle, and the energy release should be proportional to the particle volume, that is to say, the absorption cross section
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Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2783
should be proportional to R3 . Then, following the approach developed in ref. [8] it is convenient to present σabs in the form: σabs = αkR3 , (3) where α is a dimensionless quantity and k = n f ω/c stands for the wave number of the incident light in the fluid. Here n f is the (purely real) refractive index of the fluid and c designates the speed of light in a vacuum. In the case of the Rayleigh scattering α has the following simple R-independent form [8]: α=
12πε 00 , (ε 0 + 2)2 + ε 002
(4)
where ε 0 and ε 00 stand for the real and imaginary parts of the complex relative dielectric permittivity of the particle (ε = ε p /n2f ). For a large particle (R δ ) σabs = πR2 Qabs . (5) Apart the points of the so called Mie resonances the R-dependence of the dimensionless quantity Qabs (the scattering efficiency) is weak, so that the main part of the R-dependence of σabs is reduced to R2 , explicitly singled out in Eq. (5). Now we are ready to discuss each of the cases, Eqs. (1), (2) in detail. Specific cases √ Case S1: τ χ p, f R δ . Owing the smallness of the particle the energy of the laser beam is released homogeneously across the particle. On the other hand, the heat transfer occurs just in narrow layers (one in the particle and another in the liquid) in the proximity of the particle surface. For the sake of simplicity here and in what follows we suppose that the laser pulse has a rectangular shape, so that I at 0 ≤ t ≤ τ, I(t) = (6) 0 at t < 0; t > τ. Then, from the energy conservation law, neglecting the heat transfer, we obtain: 4 3 πR C p ρ p Tmax = σabs It = αkR3 It, 3
(7)
where C p is the specific heat of the particle, ρ p stands for its density and Tmax designates the maximal temperature rise in the particle, which in this case is achieved in its center. According to Eq. (7) 3 σabs It 3 αkIt Tmax = = . (8) 4 πR3C p ρ p 4 πC p ρ p Thus, in this case Tmax is, practically, R-independent. However, if we could neglect the heat transfer in the bulk of the particle, we cannot do that in the proximity of the particle surface. To simplify the problem we approximate the actual temperature field in this region at 0 ≤ t ≤ τ by linear profiles, supposing that √ Tmax at x ≤ −2 tχ p , √ √ −Ts x at − 2 tχ p < x ≤ 0, Ts − T2max tχ p (9) T= √ Ts 1 − 2√xtχ at 0 < x < 2 tχ f , f 0 at x ≥ 2√tχ , f #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2784
where Ts stands for yet unknown temperature rise at the surface of the particle and x = r − R. √ √ Then, the energy balance at −2 tχ p ≤ x ≤ 2 tχ f reads as follows: ! Z 0 Z 2√tχ f 3 σabs It 2 p 2 4πR 2 tχ p T dx . (10) = 4πR C p ρ p T dx +C f ρ f √ 4 πR3C p ρ p 0 −2 tχ p Substituting Eq. (9) into Eq. (10) and solving the resulting equation with respect to Ts , one obtains 1 √ . Ts = Tmax (11) Cf ρf χf 1 + C p ρ p √χ p As it already has been pointed out above, an important characteristic of the laser heating of the particle embedded into a biological tissue for medical, or biological applications is the cooling time τcool , i.e., the time required for a particle to cool down to the initial temperature after the end of the laser pulse heating. In some cases (see below) τcool may be much larger than the duration of the laser pulse τ, so that just τcool determines the actual time of the hyperthermia in the proximity of the particle. For the case in question we have exactly this situation: 2 R ∼ τcool = Max τ. (12) 4χ p, f √ √ Cases S2, S3: R τ χ p, f δ and R δ τ χ p, f . In these cases there is a quasi-steady temperature field both within and outside the particle. Within the particle this field is described by the the following quasi-steady version of the heat diffusion equation with a spatially homogeneous volume source: 1 d 3 σabs I 2 dT κp 2 r + = 0 at r ≤ R, (13) r dr dr 4 πR3 where κ p is the thermal conductivity of the particle. Regarding the liquid, the temperature field there is governed by the homogeneous Laplace equation. The well known solution of the latter, satisfying the boundary condition T → 0 at r → ∞ is R T = Ts . (14) r Owing to the quasi-steadiness of the temperature field in this case, the energy balance means that at any moment of time the entire power of the laser beam released in the particle is transferred to the surrounding liquid and then, by heat diffusion in the liquid the heat flux is conveyed to “infinity”, which plays the role of a sink in this problem. Then, the heat flux through any spherical surface with radius r, larger than, or equal to R has one and the same value, equal to the power of the laser beam, released in the particle. These arguments give rise to the following equation: dT −4πr2 κ f = σabs I, (15) dr r≥R where κ f stands for the heat conductivity of the fluid. Eqs. (14), (15) yield Ts =
σabs I αR2 kI = . 4πRκ f 4πκ f
(16)
The temperature profile within the particle is obtained by integration of Eq. (13). A finite at r = 0 solution of this equation, satisfying the boundary condition T (R) = Ts is κf r2 T = Ts 1 + 1− 2 . (17) 2κ p R #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2785
The cooling time in this case is still defined by Eq. (12). It is much smaller than τ. √ Case L1: τ χ p, f δ R. For a large particle the absorption cross section is described by Eq. (5). For the case under consideration the heat diffusion length is the smallest scale in the problem. Then, we can suppose that the energy is released in spatially homogenous manner within the absorption layer δ lying just below the particle surface, while the bulk of the particle remains cool. For the temperature rise Tmax within the heated layer δ and at the very surface of the particle Ts the arguments analogous to those in Case S1 are applied. The only difference between the present case and Case S1 is in the expression for the volume density of the energy release, which now is given by the formula: σabs It/4πR2 δ . Thus, now in Eqs. (8), (10) we must replace 4πR3 /3 by 4πR2 δ . It yields Tmax =
Qabs It 1 σabs It ≡ . 2 4πR δ C p ρ p 4C p ρ p δ
(18)
Regarding Ts , formally it is given by the same Eq. (11), though now in this expression Tmax is defined by Eq. (18). The cooling time is the one required for broadening of a heated layer with thickness δ due to heat diffusion. The broadening goes faster in the medium with higher value of χ. Then, based on the dimensional analysis we may conclude that 2 δ ∼ τ. (19) τcool = Min 4χ p, f √ Case L2: δ τ χ p, f R. Since for the case in question the temperature diffusion length is much smaller than the radius of the particle, but much larger than the energy release scale, the heating problem may be approximated by heating of two contacting along a plane semi-infinite spaces with one possessing the thermometric properties of the particle and the other with those of the fluid, while the energy is released at the boundary plane between the semispaces. This problem is exactly solvable [12]. The solution yields the following expression for the surfacetemperature rise: √ √ tχ p χ f tχ p χ f σabs I Qabs I Ts = 2 3/2 √ (20) √ ≡ √ √ . √ 2 π κp χ f + κ f χp 2R π κ p χ f + κ f χ p √ The temperature profile is practically R-independent. and the temperature rise increases as t, so that its maximum is achieved at the very end of the laser pulse (t = τ). In this limit the heating – cooling problem does not have any characteristic timescale but τ. Thus, the heating and cooling times are the same and both equal τ. √ Case L3: δ R τ χ p, f . In this case the temperature field is quasi-steady. It is described by the homogeneous Laplace equation both in the liquid and in the particle, while the energy may be supposed releasing at the surface of the particle. Then, in the fluid the temperature profile is described by Eq. (14), where now Ts =
σabs I RQabs I = , 4πRκ f 4κ f
(21)
see Eq. (5). Regarding the temperature within the particle, the only non-singular solution of the homogeneous Laplace equation in the spherically-symmetric case is a constant. Owing to continuity of the temperature field at the surface of the particle, this constant must be equal to Ts . The cooling time is given by Eq. (12). In the case in question it is much shorter than τ. #263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2786
Fig. 1. (a) The maximal temperature rise at the surface of a dielectric, high absorbing particle with δ = 10−5 cm, R = 10−4 cm as a function of the laser pulse duration τ. For different curves (from right to left) the intensity of the laser beam equals 105 , 106 , 107 , 108 , 109 , 1010 W/cm2 , respectively. The discontinuity at τ = 1.79 10−8 s corresponds to the transition from Eq. (18) to Eq. (20). (b) The cooling time normalized over the pulse duration τ, as a function of τ. Note that at τ = 10−14 s the total hypothermal exposure is more than a million times larger the pulse duration, see the text for details.
Example of optimization of the laser heating Let us give an example of application of the developed approach to the optimization of the laser treatment of cancer. It is supposed that the cancer cells are killed selectively owing to their hyperthermia caused by heating of light absorbing particles localized close to the cells, or inside them. In this case it is highly desirable to maximize (i) the temperature at the surface of the particles and (ii) the time of the exposure of the cells to high temperature. In contrast, the duration of the exposure of the healthy tissues to the laser beam should be minimized to prevent harmful side-effects of the treatment. Bearing it in mind, we inspect the dependance of Ts , the heating and cooling time on τ for, e.g., a large particle with R δ . In this case, increasing τ from τ δ 2 /χ p, f to τ R2 /χ p, f we sequentially move from Case L1 to Case L3. Accordingly, at τ ∼ δ 2 /(4χ)√the linear growth of Ts with an increase in τ is replaced by much slower ones, proportional to τ, which, saturates at τ ∼ R2 /(4χ). Here χ stands for the heat diffusivity of both the fluid and the particle, since for the problem in question these quantities are of the same order of magnitude. Regarding the cooling time, it is given by the condition: 2 δ τcool ∼ ,τ . (22) = Max 4χ To give an impression about the discussed dependences in Fig. 1(a) the maximal surface temperature rise for a particle with δ = 10−5 cm, R = 10−4 cm and the thermometric constants identical to those of water are presented as functions of the pulse duration τ at several values of the laser beam intensity. In accord with the specified application the range of variations of the temperature rise is selected from zero to hundred degrees Celsius. For the estimate we suppose that σabs = πR2 , i.e., the absorption cross section equals the geometric one. The cooling time as a function of the pulse duration is shown in Fig 1(b). Note the considerable decrease of the contribution of the pulse duration to the total exposure time at small τ due to the overwhelming role which the cooling time plays in this region. Thus, to optimize the action of the laser beam according to the aforementioned principles one has to remain within the range of Case L1. For small particles the same is true in Case S1,
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2787
see Eqs. (8), (11), (12). More concrete recommendations may be formulated if the desired value of Ts and the maximal duration of the laser exposure to the healthy tissues are specified. Conclusions Our analysis results in simple analytical expressions, describing the heating with a laser pulse of a dielectric particle embedded in a transparent liquid and its cooling down after the pulse is over. The obtained formulas cover any possible case of the hierarchy of the characteristic scales of the problem, provided the heat diffusivity of the particle and that of the fluid are of the same order of magnitude. Quite different dependence of the maximal temperature rise achieved during the laser pulse within the particle and on it surface on the problem parameters at different relations between the characteristic scales should be stressed. The results obtained may be employed for the optimization of the laser heating in order to maximize the positive effects of the laser treatment at various medical and/or biological applications of the problem in question and in a wide variety of other problems. Acknowledgment This study has been conducted during a visit of MT to the Institute of Mathematics for Industry. Taking the opportunity, MT expresses his gratitude to the inure staff of IMI for the kind hospitality and provision of excellent conditions to carry outs research. The financial support of the Ministry of Education, Culture, Sport, Science, and Technology of Japan, which has made the visit possible is greatly appreciated. The study has been partially supported by the Center of Innovation Program from Japan Science and Technology Agency, JST. YF was also partially supported by a Grant-in-Aid for Scientific Research from the Japan Society of Promotion of Science (Grant No. 16K05476).
#263677 (C) 2016 OSA
Received 20 Apr 2016; revised 30 May 2016; accepted 7 Jun 2016; published 23 Jun 2016 1 Jul 2016 | Vol. 7, No. 7 | DOI:10.1364/BOE.7.002781 | BIOMEDICAL OPTICS EXPRESS 2788
