Heating of metallic biliary stents during magnetic hyperthermia of patients with pancreatic ductal adenocarcinoma: an in silico study

Figures & data

Table 1. Relevant physical properties of the bile duct tissues and the stent alloy.

Tissue	Muscularis propria	Submucosa layer	Mucosa layer	Connective tissue	Bile	Nitinol (austenite)	Nitinol (martensite)	Elgiloy^®
Electrical conductivity (S/m)	0.202	0.251	0.511	0.39	1.4	10^6	1.25 × 10^6	10^6
Density (kg/m^3)	1090	1027	1088	1027	928	6450	6450	8300
Heat capacity (J/(kg·K))	3421	2372	3690	2372	4037	837	837	430
Thermal conductivity (W/(m·K))	0.49	0.39	0.53	0.39	0.58	18	8.6	12.5
Heat transfer rate (ml/(min·kg))	37	37	460	37	0	–	–	–
Heat generation rate (W/kg)	0.91	0.58	7.13	0.58	0	–	–	–
Reference	[22]	[22]	[22]	[21]	[21]	[23]	[23]	[24]

Figure 1. (a) Model of the bile duct (green) and tumor along with the detail of the 1D structure of the metallic stent. The angles identifying the stent axis are: ψ = 72.5° and θ = 0°. The orientation reference for admissible field directions is indicated in the inset. (b) Full computational model domain.

Figure 2. Time evolution of the spatial distribution of ΔT during the exposure to the B_max field orientation upon exposure after 10 s (a), 60 s (b), 360 s (c), and 1800 s (d). The field amplitude is 5 mT, and the wire diameter is 0.3 mm. The maps refer to the duct full of bile. Two different color maps are used to represent the temperature increase in the tissues and the power density in the stent wires. The black arrow in (a) shows the direction of B_max (ψ = 52° and θ = 130°).

Figure 3. Influence of the magnetic field orientation on the power deposition in the stent and the consequent tissue heating after 30 min of exposure. From the upper plot to the bottom: (a) total power dissipated into the stent, (b) maximum power density in the stent, (c) corresponding maximum temperature increase (ΔT_max) in tissues for an empty duct, (d) corresponding ΔT_max in tissues for a duct full of bile. All the results, obtained at 300 kHz, are reported in p.u. values, where the values of zero correspond to the minimum of the physical quantity, while the value of one correspond to the maximum of the physical quantity.

Table 2. ΔT_max (°C) after 30 min of exposure to a 10 mT field calculated for the tissues that make up the bile duct, with or without bile in the duct.

Direction	Mucosa Layer		Submucosa layer		Muscularis propria		Connective tissue
	Duct full of bile	Empty duct	Duct full of bile	Empty duct	Duct full of bile	Empty duct	Duct full of bile	Empty duct
x	43.0	58.2	43.6	47.8	42.5	45.5	42.2	44.4
y	44.9	58.8	40.2	44.1	33.1	35.3	25.8	30.7
z	40.1	50.0	39.2	44.1	35.8	39.3	28.9	34.1
Bmax	55.6	72.1	57.3	70.4	53.3	64.3	45.5	59.3

As indicated before, B_max is the field direction for which a maximum heat is obtained.

Table 3. ΔT_max (°C) after 30 min of exposure.

Field amplitude (mT)	Wire diameter (mm)	Mucosa layer	Submucosa layer	Muscularis propria	Connective tissue
5	0.1	2.91 (3.76)	2.89 (3.60)	2.70 (3.39)	2.58 (3.34)
	0.2	7.53 (8.73)	7.83 (8.76)	7.44 (8.19)	6.47 (7.43)
	0.3	14.5 (17.9)	15.0 (17.8)	14.1 (16.4)	12.1 (14.9)
	0.4	24.8 (31.2)	25.7 (30.8)	24.0 (28.2)	20.4 (25.8)
10	0.1	7.54 (8.73)	7.83 (8.76)	7.44 (8.19)	6.46 (7.43)
	0.2	25.0 (31.5)	25.9 (31.1)	24.2 (28.5)	20.6 (26.0)
	0.3	55.6 (72.1)	57.3 (70.4)	53.3 (64.3)	45.4 (59.3)
	0.4	>100	>100	>100	>100

The field is oriented along the B_max direction, with amplitudes equal to 5 mT and 10 mT. For comparison purposes, the calculated ΔT_max values considering an empty duct are indicated in brackets. Values are not reported for the cases where ΔT_max exceeds 100 °C.

Table 4. Amount of tissue mass (milligrams) of a given tissue where the heating exceeds 80% of the maximum temperature increase ΔT_max found in each tissue (ΔT_lim = 0.8·ΔT_max).

B (mT)	Wire diameter (mm)	Mucosa layer		Submucosa layer		Muscularis propria		Connective tissue
		ΔTmax (°C)	Mass (mg)	ΔTmax (°C)	Mass (mg)	ΔTmax (°C)	Mass (mg)	ΔTmax (°C)	Mass (mg)
5	0.1	2.91	3.93	2.89	2.56	2.70	6.83	2.58	8.27
	0.2	7.53	1.93	7.83	1.08	7.44	2.21	6.47	4.98
	0.3	14.5	0.88	15.0	0.64	14.13	1.40	12.05	2.47
	0.4	24.8	0.71	25.7	0.58	24.0	1.33	20.43	2.25
10	0.1	7.54	1.93	7.83	1.08	7.44	2.21	6.47	4.96
	0.2	25.0	0.71	25.9	0.58	24.2	1.33	20.6	2.26
	0.3	55.6	0.69	57.3	0.55	53.3	1.26	45.5	2.19
	0.4	–	–	–	–	–	–	–	–

Values are not reported for the cases where ΔT_max exceeds 100 °C. For each tissue, the amount of mass was computed extracting the number of voxels whose temperature increase is higher than ΔT_lim; the number of voxels defines the volume that is multiplied by the tissue mass density to get the total mass.

Table 5. Calculated ΔT_max (°C) for each of the bile duct tissue layers after 30 min exposure to different field intensities along four different orientations.

B (mT)	Direction	Stent power (W)	Mucosa layer	Submucosa layer	Muscularis propria	Connective tissue
5	x	0.548	11.6	11.9	11.6	11.7
	y	0.540	12.1	11.0	9.5	7.5
	z	0.614	10.9	10.8	10.0	8.0
	Bmax	0.586	14.5	15.0	14.1	12.1
10	x	2.191	43.0	43.6	42.5	42.2
	y	2.159	44.9	40.2	33.1	25.8
	z	2.454	40.1	39.2	35.8	28.9
	Bmax	2.342	55.6	57.3	53.3	45.5
15	x	4.930	>100	>100	>100	>100
	y	4.858	>100	>100	>100	>100
	z	5.522	95.5	90.9	82.4	71.0
	Bmax	5.270	>100	>100	>100	>100

The duct is full of bile.

Table 6. Amount of tissue mass (grams) where the heating exceeds 3 °C, 6 °C, and 9 °C.

ΔTmax (°C)	B (mT)	Mucosa layer		Submucosa layer		Muscularis propria		Connective tissue
	5	0.122	22.1 %	0.080	21.6 %	0.134	21.2 %	1.210	2.7 %
3	10	0.175	31.6 %	0.116	31.2 %	0.197	30.9 %	4.472	9.8 %
	15	–	–	–	–	–	–	–	–
	5	0.068	12.3 %	0.043	11.6 %	0.063	9.9 %	0.077	0.17 %
6	10	0.126	22.8 %	0.083	22.4 %	0.140	22.1 %	1.475	3.2 %
	15	–		–		–		–
	5	0.0085	1.54 %	0.0061	1.65 %	0.0076	1.19 %	0.0042	0.01 %
9	10	0.107	19.4 %	0.070	18.9 %	0.118	18.5 %	0.771	1.70 %
	15	–	–	–	–	–	–	–	–

The percentage with respect to the entire tissue mass in the considered duct tract is also reported. Values are not reported for cases where ΔT_max exceeds 100 °C. For each tissue, the amount of mass was computed extracting the number of voxels whose ΔT_max exceeds the given 3 °C, 6 °C, and 9 °C thresholds; the number of voxels defines the volume that is multiplied by the tissue mass density to obtain the sought total mass.

Table 7. ΔT_max (°C) after 30 min of exposure.

Alloy type	Overall max	Mucosa layer	Submucosa layer	Muscularis propria	Connective tissue
Nitinol (Austenite)	57.3 (>100)	55.6 (72.1)	57.3 (70.4)	53.3 (64.3)	45.4 (59.3)
Nitinol (Martensite)	71.6 (>100)	69.7 (>100)	71.6 (>100)	66.6 (>100)	57.0 (>100)
Elgiloy	57.3 (>100)	55.6 (72.1)	57.3 (70.4)	53.3 (64.3)	45.4 (59.3)
Ti–6Al–4V (*)	32.3 (>100)	31.1 (39.1)	32.2 (39.4)	29.5 (35.1)	25.6 (32.5)

The field is oriented along B_max direction and its amplitude is equal to 10 mT. The representative wire diameter is 0.3 mm in all the cases. The values obtained considering an empty bile duct are shown in brackets for comparison.

(*) Electrical conductivity = 0.56 10⁶ S/m, thermal conductivity = 6.7 W/(m·K), heat capacity = 526.3 J/(kg·K), density = 4430 kg/m³.

Ren F, Li Q, Gao X, et al. Electrical and thermal analyses of catheter-based irreversible electroporation of digestive tract. Int J Hyperthermia. 2019;36(1):853–866.

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