Influence of natural convection on gold nanorods-assisted photothermal treatment of bladder cancer in mice

Figures & data

Figure 1. Geometry of the mouse bladder and the three orientations investigated. (a) The 3D computational domain developed in the present study. The bladder is shown in cyan, while the tumor is shown in red. (b) View of the computational model across the z = 0 plane. The cyan domain represents the urine. (c) The three different bladder orientations considered in the present study with respect to direction of gravity and laser irradiation. All dimensions are in mm.

Table 1. Thermal properties employed in this study.

Parameter	Value	Source
Bladder
Thermal conductivity, $k$ (W/m^2K)	0.52	[31]
Density, $\rho$ (kg/m^3)	1086	[31]
Specific heat, $c$ (J/kg K)	3581	[32]
Blood perfusion rate, ${\omega }_{\mathrm{b}}$ (1/s)	0.0014	[31]
Frequency factor, $A$ (1/s)	3.3 × 1038	[33]
Activation energy, $\Delta E$ (J/mol)	2.57 × 105	[33]
Tumor
Thermal conductivity, $k$ (W/m^2K)	0.49	[32]
Density, $\rho$ (kg/m^3)	1090	[32]
Specific heat, $c$ (J/kgK)	3421	[32]
Blood perfusion rate, ${\omega }_{\mathrm{b}}$ (1/s)	0.00165	[32]
Frequency factor, $A$ (1/s)	1.46 × 1052	[34]
Activation energy, $\Delta E$ (J/mol)	3.43 × 105	[34]
Surrounding tissue
Thermal conductivity, $k$ (W/m^2K)	0.21	[31]
Density, $\rho$ (kg/m^3)	911	[31]
Specific heat, $c$ (J/kg K)	2348	[31]
Blood perfusion rate, ${\omega }_{\mathrm{b}}$ (1/s)	0.0005	[31]
Frequency factor, $A$ (1/s)	1.61 × 1045	[35]
Activation energy, $\Delta E$ (J/mol)	3.06 × 105	[36]
Urine
Thermal conductivity, $k$ (W/m^2K)	0.59	[31]
Density, $\rho$ (kg/m^3)	1024	[31]
Specific heat, $c$ (J/kg K)	4178	[31]
Thermal expansion coefficient, ${\beta }_{\mathrm{f}}$ (1/K)	3.374 × 10^-4	[37]
Dynamic viscosity, ${\mu }_{\mathrm{f}}$ (Pa s)	0.0007	[37]
Constants
Universal gas constant, $R_{\mathrm{c}}$ (J/mol K)	8.314
Ambient temperature, $T_{\mathrm{amb}}$(^oC)	20
Ambient convection coefficient, $h_{\mathrm{amb}}$ (W/m^2K)	3

Table 2. Optical parameters and the peak absorption and scattering coefficients for $\varphi$ = 0.001, 0.005 and 0.01%.

Parameter	Value	Source
Free electron density, $n_o$ (1/m^3)	5.9 × 1028	^a
Permittivity of vacuum, ${\epsilon }_o$ (F/m)	8.85 × 10^-12	^a
Electron charge, $e$ (C)	1.6 × 10^-19	^a
Mass of electron, $m_{\mathrm{e}}$ (kg)	9.1 × 10^-31	^a
Fermi velocity, $v_{\mathrm{F}}$ (m/s)	1.39 × 106	^a
Absorption coefficient, ${\mu }_{\mathrm{a}}$(1/mm)
Bladder	0.02	[41]
Surrounding tissue	0.105	[42]
Urine	0.002	[43]
Tumor embedded with GNR
$\varphi$ = 0.001%	4.29	^b
$\varphi$ = 0.005%	21.47	^b
$\varphi$= 0.01%	42.93	^b
Scattering coefficient, ${\mu }_{\mathrm{s}}$ (1/mm)
Bladder	4.41	[41]
Surrounding tissue	11.16	[42]
Urine	1	[43]
Tumor embedded with GNR
$\varphi$ = 0.001%	0.052	^b
$\varphi$ = 0.005%	0.261	^b
$\varphi$ = 0.01%	0.521	^b

^aConstants.

^bEstimated using Mie–Gans theory.

Figure 2. Plots of the (a) absorption and (b) scattering coefficients at different wavelengths calculated for ϕ = 0.001, 0.005 and 0.01%.

Figure 3. Fluence distribution across the z = 0 plane. Simulation was performed for three GNR volume fractions (a) $\varphi$ = 0.001%, (b) $\varphi$ = 0.005% and (c) $\varphi$ = 0.01%. The black arrows, indicating low fluence, point to the position of the tumor. An increase in volume fraction leads to lower fluence magnitude due to higher photon absorption.

Figure 4. Contours of the temperature distribution at x = 0 and z = 0 planes 10 min after laser exposure. (a) Case I, (b) Case II, (c) Case III, and (d) Case IV. The temperature distribution inside the bladder is visualized using isothermal surfaces in order to show the flow-induced variation. Results were obtained for $\varphi$ = 0.001% and $P_{\mathrm{laser}}$ = 0.2 W. The yellow arrows show the direction of gravity and the red arrows show the flow field. Figures indicate significant changes in the isotherms inside the bladder due to natural convection under different bladder orientation.

Table 3. Results for the different configurations for $\varphi$ = 0.001% and P_laser = 0.2 W .

Parameter	Maximum tumor temperature (°C)	Minimum tumor temperature (°C)	Maximum skin temperature (°C)	Average convective heat transfer (W/m^2)	Heterogeneity coefficient, $HC$
Case I	55.20	52.27	56.83	3840	0.527
Case II	49.95	46.18	51.14	34,882	0.339
Case III	52.02	48.04	54.25	29,490	0.465
Case IV	55.70	52.76	57.26	—	0.535

Figure 5. Contours of the velocity magnitude in mm/s and the velocity vector inside the bladder plotted for z < 0 at t = 10, 20, 30, 60, and 600 s. (a) Case I, (b) Case II, and (c) Case III.

Figure 6. Contours of PD across the tumor, the bladder and the surrounding tissue at the z = 0 plane, the skin surface (for z < 0) and the inner wall of the bladder, 10 min after the start of laser irradiation. (a) Case I, (b) Case II, (c) Case III, and (d) Case IV, for ϕ = 0.001% and laser power of 0.2 W. Inset: enlarged view of the tumor and the tissue above it.

Figure 7. Plots of maximum PD values against time inside the tumor and the bladder for (a) $\varphi$ = 0.001%, (b) $\varphi$ = 0.005% and (c) $\varphi$ = 0.01% for $P_{\mathrm{laser}}$ = 0.2 W.

Figure 8. Contours of PD across the tumor, the bladder and the surrounding tissue across the z = 0 plane, the skin surface (for z < 0) and the inner wall of the bladder after 10 min of 0.2 W laser irradiation. (a) ϕ = 0.005% and (b) ϕ = 0.01%. Inset: enlarged view of the tumor and the tissue above it.

Figure 9. Contours of PD across the tumor, the bladder and the surrounding tissue at the z = 0 plane, the skin surface (for z < 0) and the inner wall of the bladder after 10 min of 0.3 W laser irradiation. (a) ϕ = 0.001%, (b) ϕ = 0.005% and (c) ϕ = 0.01. Inset: enlarged view of the tumor and the tissue above it.

Table 4. Values of $R_{\mathrm{T}},$ $R_{\mathrm{B}},$ and $R_{\mathrm{ST}}$ obtained for Cases I, II, and III for different laser power and GNR volume fraction.

Parameters	Case I			Case II			Case III
	$R_{\mathrm{T}}$	$R_{\mathrm{B}}$	$R_{\mathrm{ST}}$	$R_{\mathrm{T}}$	$R_{\mathrm{B}}$	$R_{\mathrm{ST}}$	$R_{\mathrm{T}}$	$R_{\mathrm{B}}$	$R_{\mathrm{ST}}$
Plaser = 0.2 W
$\varphi =0.001\%$	0	0	0	0	0	0	0	0	0
$\varphi =0.005\%$	0.125	0	0	0	0	0	0	0	0
$\varphi =0.01\%$	1^a	0.001	0	0	0	0	0.066	0	0
Plaser = 0.3 W
$\varphi =0.001\%$	1	0.005	0	0	0	0	0.002	0	0
$\varphi =0.005\%$	1	0.049	0.004	0.389	0	0	0.783	0.001	0
$\varphi =0.01\%$	1	0.112	0.012	0.97	0.005	0	1^a	0.017	0.001

^aValues greater than 0.998 and thus rounded to 1.

Figure B1. Results from mesh convergence study, where the different bars represent the values obtained at different sampled points.

Figure C1. Results from the model verification study for GNR concentration of 100 µg/ml.

Table C1. Results from the model verification study against the work of Terentyuk et al. [45].

Time (min)	Maximum temperature (^oC)
	$h_{\mathrm{amb}}$ (W/m^2K)			Terentyuk et al. [45]
	15	10	5
1	35.94 (1.2)	36.21 (0.4)	36.51 (0.4)	36.36
2	40.83 (6.4)	41.56 (4.8)	42.34 (3.0)	43.64
3	44.28 (9.8)	45.63 (7.0)	47.02 (4.2)	49.09
4	45.15 (16.2)	48.90 (9.3)	50.98 (5.4)	53.86
5	45.90 (20.9)	51.59 (11.1)	54.42 (6.2)	58.00

Numbers in brackets indicate the percentage difference between the numerical and experimental results.

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