Computer Practicum 5 (Lecture 24): Simulating Temperature Fields Around the Nanoparticle Contents: Computer Code Practice Problem 7: Thermal Fields Around a Gold Nanoparticle in Aqueous Solution Abstract Method, Model and Input Data Discussion and Results Internal Temperature Distribution in a Single Gold Nanoparticle Outside Temperature Distribution Around a Single Gold Nanoparticle Conclusions Practice Example 8: Thermal Ablation of Cancerous Cell Nuclei in Cytoplasm Model and Input Data

Statement of The Research Task The research task for this computer practicum is to numerically solve the heat transfer equations below to determine the special behavior of temperature fields around a single nanoparticle. To perform the heat diffusion simulations at the nanoscale, we have developed an effective Maple code, shown in Appendices E and F. Appendix E contains a Maple code for 2D simulations of thermal fields, and Appendix F provides a code for 3D simulations of the temperature distributions around the nanoparticle (third “dimension” is a temperature).

Computer Code The Maple code begins with introducing the energy density of the radiation, radius of the gold nanoparticle, and absorption efficiency of the gold nanoparticle at a given size, and using 530-nm wavelength, thermal conductivities and diffusivities of the particle’s material and surrounding media (water and blood in the provided example), as shown below.

Computer Code (continued) Then we define all constants and the rate of heat generation used in the thermal model

Computer Code (continued) Introducing the integral solutions for the temperature distributions within and outside the nanoparticle.

Computer Code (continued) Defining the heat generation rate and introducing the integral solutions for the temperature distributions around the nanoparticle.

Computer Code (continued) Simulating the temperature distributions inside the nanoparticle.

Practice Example 7: Thermal Fields Around a Gold Nanoparticle in Aqueous Solution The method used in this practice example consists of two steps: (1) computing the optimal radius and maximum absorption efficiency of gold nanoparticles in surrounding water and blood media using the Mie diffraction theory, and (2) plugging these absorption values into the heat diffusion model to simulate the spatial distribution of temperature fields around the nanoparticle. The absorption efficiency for the optimal radius of the gold nanoparticle is computed by using the MiePlot software demonstrated in the Computer Practicum 1.

Practice Example 7: Input Data Medium   Density ρ (kg/m3) Specific heat C (J/K kg) Interval of T (K) Phase transitio n point Tph (K) Thermal Conductivity µ (W/mK) Thermal Diffusivity χ (m2/s) Blood 1060 3645 - 38 97 273-373 (323) 373 0.45-0.6 (0.525) 1.6E-07 Fat 900 2975 0.185-0.233 (0.209) 7.81E-08 Cytoplasm/ water 1000 4181.6- 4215.6 0.597-0.682 (0.640) 1.52E-07 Cell Membrane 3000 0.3 1.11E-07 Collagen 1900 440 0.16-0.34 (0.25) 2.99E-07 Nanoparticle type Dimension Wavelength Absorption efficiency Gold nanospheres   35 nm 532 nm 4.02 40 nm 550 nm 3.50 Silica-gold nanoshells 120 nm 1100 nm 750 nm 3.75 Gold nanorods 11.4 nm 800 nm 14.0 17.9 nm 825 nm 14.5 Property of Au Values Thermal conductivity 3.18102 W/m·K Density 1.93104 kg/m3 Specific heat 1.29102 J/kg·K Thermal diffusivity 1.2710-4 m2/s

Internal Temperature Distribution in a Single Gold Nanoparticle As can be seen from the figure, the gold nanoparticle reaches its maximum temperature Tmax ~ 500 K in water and Tmax = 622.9 K in blood at the center of the sphere. Then we observe a small decrease of the inside temperature by 0.6 K to the particle’s boundaries, establishing the stationary surface temperature of 622.3 K in blood and about 500 K in water. Since the temperature variation inside the nanoparticle occurs over a small range of 0.6 K, the internal particle’s temperature can be assumed to be constant. Thus, the quasistationary and homogeneous approach can be applied to modeling the laser heating and evaporation of the nanoparticles in the biological media.

Outside Temperature Distribution Around a Single Gold Nanoparticle As follows from the figure, there is a significant heat loss from the surface of a nanoparticle to the surrounding medium, suggesting that even lower-energy pulses are enough to achieve true and large thermal damage of the biological surroundings. The calculations show that at the level of the threshold temperature (425 K), the damage area produced by a heated 30-nm gold particle in blood irradiated by 5 J/cm2 can reach a size of ~ 0.5 µm in diameter, which exceeds by 17 times the size of the nanoparticle. This size of a damage area requires the nanoparticle to be heated to over 1000 K. Thus, under the chosen conditions, the damage area produced by a 30-nm particle heated by a single laser pulse of 4-5 J/cm2 is comparable to the size of the laser focal spot. Figure. Spatial distribution of the temperature outside the 30-nm gold particle in the surrounding water (blue curve) and blood (red curve) media irradiated by a single laser pulse of energy density 4.0 J/cm2 and duration 8 ns.

Practice Example 7 Conclusions The temperature variation inside the nanoparticle during the laser heating by a 8 ns pulse occurs in a small temperature range and can be assumed to be constant. Thus, the quasistationary and homogeneous approach can be applied for modeling the laser heating and evaporation of the nanoparticles in the biological media. The damage area produced by a 30-nm particle heated by a single laser pulse of 5 J/cm2 is about 0.5 µm.

Practice Example 8: Thermal Ablation of Cancerous Cell Nuclei in Cytoplasm The main purpose of this practice example is to determine the feasibility of destroying cancerous cells using pulsed laser radiation. Specifically, we explore the possibility of heating the cancerous nucleus to the ablation temperatures without damaging the surrounding tissue by applying low-energy density radiation. The nucleus itself may actually act as an absorbing particle, which would mean cancer cells could be destroyed without inserting any nanoparticles into the body. This is accomplished by using the heat transfer model, which takes into account the size and heat capacity of the particle as well as the thermal conductivity of the medium in which it is suspended.

Practice Example 8: Input Data Working from the Maple worksheet presented in Appendix E or F, the first step is plugging in the input data for healthy and cancerous nuclei and cytoplasm. The MiePlot software is also used to determine the absorption efficiencies of normal and cancerous nuclei along with the optimum wavelengths of radiation for treatment purposes. Organelle and cytoplasm properties. Organelle Diameter, d (µm) Specific heat, C (J/kg-K) Thermal conductivity, µ (W/m-K) Density, p (kg/m3) Nucleus 5-10 3000 0.3 1050-1400 Mitochond ria 0.85-3.57 1050-1100 Ribosome 0.02-0.025 1050-1600 Microtubul es 0.02-2 Lysosomes 0.1-0.25 Cytoskelet on 0.004- 0.012 Cytoplasm - 4180 0.59   Nucleus Qabs opt (nm) Normal (r0 = 5.0 μm) 0.9859 942.18 Cancero us (r0 = 6.5 μm) 1.00836 5 899.99 Cancero us (r0 = 7.0 μm) 1.0125

Practice Example 8 Discussion and Results Simulations have been performed for heating healthy (r0 = 5.0 μm, blue curve) and cancerous (r0 = 7.0 μm, red curve) nuclei in cytoplasm. It is evident from this figure that a cancerous nucleus heats up to much higher temperature than a normal nucleus, reaching the critical temperature for ablation of 425 K. With the significant increase in temperature of the cancerous nucleus, the cancer cells undergo cell death via the thermal ablation of the nucleus. At the same time, the change in temperature for the normal nucleus in the cytoplasm is only 10 K above the normal body temperature of 310 K. Figure. Spatial distribution of the temperature around healthy (blue curve) and cancerous (red curve) cell nuclei.

Practice Example 8 Conclusions Heating the largest cell organelle, i.e., the cancerous nucleus, can effectively kill a cancer cell. The size difference between normal and abnormal nuclei provides a significant temperature difference with equal incident energy density which could be used to kill cancer cells while leaving healthy cells unharmed. In this simulation, the cancerous nucleus temperature reaches the damage threshold (425 K), while the healthy nucleus heats up to only 320 K, which is a little above the normal body temperature. Thus, at high enough energy density, heating of the nuclei can be used to kill cancer cells which have enlarged nuclei while keeping the healthy cells undamaged.

Appendix E Maple Code for Simulating Heat Diffusion Around a Nanoparticle

Appendix F Maple Code for 3D Simulation of Heat Diffusion Around a Nanoparticle