1. 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

2. 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).

3. 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.

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

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

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

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

8. 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.

9. 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
(323)
373
(0.525)
1.6E-07
Fat
900
2975
(0.209)
7.81E-08
Cytoplasm/ water
1000
(0.640)
1.52E-07
Cell Membrane
3000
0.3
1.11E-07
Collagen
1900
440
(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

10. 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 = 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 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.

11. 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.

12. 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.

13. 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.

14. 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
Mitochond ria
Ribosome
Microtubul es
0.02-2
Lysosomes
Cytoskelet on
Cytoplasm -
4180
0.59
Nucleus
Qabs
opt (nm)
Normal (r0 = 5.0 μm)
0.9859
942.18
Cancero us (r0 = 6.5 μm)
899.99
Cancero us (r0 = 7.0 μm)
1.0125

15. 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.

16. 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.

17. Appendix E Maple Code for Simulating Heat Diffusion Around a Nanoparticle

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