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Contents:
• Introduction
• Practice Problem 4: Time Dynamics of the Gold Nanoparticle Temperature in a
Single Pulse Heating Mode
 Abstract
 Model
 Time Profile of the Laser Pulse
 Heat Transfer to the Surrounding Medium
 Effect of the Energy Density on the Time Dynamics of Nanoparticle Heating in
the Single-Pulse Mode
 Effect of the Particle Size and Shape on the Time Dynamics of Nanoparticle
Heating
 Effect of the Pulse Duration in the Single-Pulse Mode of Nanoparticle Heating
 Conclusion
• Practice Problem 5: Thermal Dynamics of Normal and Cancerous Ribosome in
Cytoplasm Heated by a Single Pulse of Laser Radiation
 Abstract
 Procedure and Results
One-Temperature Model
 1


 Ts 
dTs 3Qabs I 0 f (t )
3 Ts
3L dr0
   1 


2
dt
4r0Cs (Ts )  s (  1)r0 Cs (Ts )  s  T 
 r0Cs (Ts ) dt

• The absorption efficiency Qabs of the nanoparticles in various
biomedia have been simulated in the Computer Practicum 1
(Section 5.1) by using a generalized Lorenz-Mie diffraction
theory.
• Using the complex indices of refraction for the medium and
particle (which depend on wavelength as well as the particle
size), we have found the absorption efficiency of spherical
particles at a given wavelength within the bounds of the
refractive index values provided.
• The results of these calculations for absorption efficiency of
gold nanospheres, silica-gold nanoshells and gold nanorods are
listed in Table 5.4.
Assumptions and Limitations
• We assume here that the possible vapor bubble formation around the
nanotarget in the surrounding aqueous medium due to heating of the
nanoparticles does not change the absorption efficiency of the
nanoparticles, since the bubble is transparent for the laser
wavelength that is used, and the bubble could not change
considerably the plasmon resonance absorption of the metal
nanoparticles. This is a phenomenon which is related to electron
subsystem oscillations in the particle materials rather than in the
surrounding medium.
• We also limit the maximum temperature of the theoretical
calculations to the melting point of the material (for example, TM ~
1336 K for bulk gold material and remaining above 1100 K for gold
particles larger than 5 nm in diameter), so that the particles will not
undergo advanced phenomena of heating (evaporation, melting or
explosion), but require that the nanoparticles surpass 433 K as
required for protein denaturing.
Radiation Pulse Profile, f(t)
 1


 Ts 
dTs 3Qabs I 0 f (t )
3 Ts
   1


2
dt
4r0Cs (Ts )  s (  1)r0 Cs (Ts )  s  T 


• The function f(t) determines the pulse shape, pulse duration and time
interval between the pulses in the case of a multipulse heating mode.
• A few examples of laser pulse profiles as shown below include rectangular,
triangle and Gaussian shapes.
• In OTM simulations, we use a Gaussian profile, which is a most accurate
representative of the pulse shape for most laser sources.
f(t)
f(t)
f(t)
t
t
Rectangular
shape
Triangle
shape
t
Gaussian
shape
Modeling the Radiation Pulse Profile
• A single pulse can be described by the Gaussian-like curve
2
−
𝑏𝑡−𝑑
𝑓 𝑡 = 𝑒
,
where b and d are constants defining a pulse duration.
• The first step in simulations is to compute a Gaussian pulse profile
by using MAPLE:
Input Data for OTM
• The next step in developing a computer code is to introduce input
data from Tables 6.1, 6.2 and 6.3, as required for the simulations,
listed below with names and units.
Material
Specific
heat
C (J/Kkg)
Phase
transition
point
Tph (K)
Thermal Thermal
Density
conductivit diffusivity

1
y
(kg/m3)
0 (W/mK)  (m2/s)
Silver
(Ag)
187
225
235
239
250
262
277
1234.93
444
430
429
425
412
396
379
.000226
.000182
.000174
.000169
.000157
.000144
.00013
10500
Fulleren
e (C60)
Polystyre
ne
1600
800-1073
(sublimes)
670
1.45e-7
1720
1170
463-533
.13
1.06e-7
1050
Modeling OTM Differential Equations
• Then we type the OTM differential equations without a last
evaporation term for each nanoparticle, as shown below:
Numerical Solutions
• To solve these equations we use dsolve command, which finds a numerical solution for an
ordinary differential equation using a Fehlberg fourth-fifth order Runge-Kutta
(rkf45) method, as shown below
Arranging the Output Results
• The output results of these simulations can be arranged in numerical or graphical
formats.
• To record results numerically , we create a matrix zz, which contains the time and
temperature values, as shown below.
• For graphical presentation of the output results, we use the odeplot command specifying
a time interval and temperature frame for the output graph, as shown below.
Plots of Temperature Profiles for
Nanoparticles
• To see the actual graph, we use display command, which plots the graph in a defined time
interval and temperature frame, as shown below.
• The results of these simulations provide a time dynamics of heating and cooling of
nanoparticles in different surrounding biological media.
Practice Example 4: Time Dynamics of the Gold
Nanoparticle Temperature in a Single-Pulse Heating
Mode
Practice Example 4: Time Dynamics of the Gold
Nanoparticle Temperature in a Single-Pulse Heating
Mode (continued)
In conclusion, we have performed timedependent simulations and detailed analyses of
different non-stationary laser-nanoparticle
interactions in a single-ulse mode of heating.
Practice Example 5: Thermal Dynamics of Normal
and Cancerous Ribosome in Cytoplasm Heated by a
Single Pulse of Laser Radiation
Appendix B
Maple Code for Nanoparticles Heating in a
Single-Pulse Mode
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Pulse shape
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Specific heat of a Blood in J/(Kkg)
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Display graphs
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Appendix C
Maple Code for Cell Organelle Heating in a
Single-Pulse Mode
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Specific heat of Cytoplasm in J/(Kkg)
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Radius of the Healthy Ribosome in cm
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Radius of the Cancerous Ribosomele in cm
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Absorption efficiency of Healthy Ribosome
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Absorption efficiency of Cancerous Ribosome
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Initial temperature in K
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