Cell death, perfusion and electrical parameters are critical in models of hepatic radiofrequency ablation

Figures & data

Figure 1. A graphical representation of the sampling strategy in the Morris method for three parameters. Four random starting points are selected and then trajectories are formed from these so that three main effects can be computed for each starting location.

Figure 2. Axisymmetric geometry adapted from Trujillo et al. [4,5].

Table 1. Values of the specific heat capacity of liver ± standard deviation with references.

c	± SD	Species	T	Sample	Tissue	Reference
3411		Bovine	298	In vitro	Normal	[17]
3690	70	Bovine	310	Ex vivo	Normal	[18]
3650	90	Bovine	310	Ex vivo	F-Co	[18]
4187		Bovine	357	In vitro	Normal	[17]
3598		Human	273–298	Ex vivo	Normal	[19]
3758	66	Human	310	Ex vivo	Tumour	[18]
3617	78	Human	310	Ex vivo	Normal	[18]
3730	150	Porcine	310	Ex vivo	Normal	[18]
3720	180	Porcine	310	Ex vivo	F-Co	[18]
3660	100	Porcine	310	Ex vivo	F-T	[18]
3480	100	Porcine	310	Ex vivo	F-T-Co	[18]
3628		Porcine	293–343	Ex vivo	Normal	[20,21]
3858		Porcine	353–373	Ex vivo	Normal	[20,21]
3986		Porcine	383–393	Ex vivo	Normal	[20,21]
3690						[22]
3629						[23,24]
3600						[25]

Blank spaces denote incomplete information in the source. SD, standard deviation; F, freeze; T, thaw; Co, coagulated).

Table 2. Values for the density of liver with references.

ρ	Species	T	Sample	Tissue	Reference
1050	Human		Ex vivo	Normal	[19]
1020	Porcine	293–343	Ex vivo	Normal	[20,21]
1050					[23,24]
1080					[22]
1060					[25,26]
1050					[25,27,28]

Blank spaces denote incomplete information in the source.

Table 3. Baseline values of thermal conductivity k₀ in W/m/K.

k0	Species	T	Sample	Tissue	Reference
0.565	Human	278	Ex vivo	Normal	[19]
0.512	Human	276–318	Ex vivo	2 days saline	[29]
0.465	Porcine	293–343	Ex vivo	Normal	[20,21]
0.867	Porcine	353–373	Ex vivo	Normal	[20,21]
1.46	Porcine	383–393	Ex vivo	Normal	[20,21]
0.46–0.49					[22]
0.565					[23,24]

Blank spaces denote incomplete information in the source.

Table 4. Rate of change of thermal conductivity Δ k in W/m/K².

Δ k	Species	T	Sample	Tissue	Ref
0.001161	Human	276–318	Ex vivo	Normal	[29]
0.0033	Bovine	298–353	Ex vivo	Normal	[30]

Table 5. Electrical conductivity σ in S/m of liver tissue.

σ	± SD	Species	T	f (MHz)	Sample	Tissue	Reference
0.141		Human	310	0.1	Ex vivo	Normal	[25]
0.227		Human	310	1.0	Ex vivo	Normal	[25]
0.19–0.45		Human	296–298	0.1	Ex vivo	Normal	[32]
0.24–0.28		Human	296–298	1.0	Ex vivo	Normal	[32]
0.277	15–25%	Human		0.5	Ex vivo	Normal	[33]
0.26	0.06	Human		0.46	Ex vivo	Normal	[34]
0.5	0.2	Human		0.46	Ex vivo	MCT	[34]
0.333				0.5			[35,36]
0.148				0.5			[33,37]

Blank spaces denote incomplete information in the source. SD, standard deviation; MCT, metastatic colorectal tumour.

Table 6. Rate of change of the electrical conductivity in %/K.

Δ σ	± SD	Species	Organ	T	f (MHz)	Reference
1.62	0.04	Porcine	Kidney	321–351	0.46	[38]
0.6–1.3		Mixed	Liver		0.1–1	[25]
2		Mixed				[25,39]
2		Mixed				[40,41]

Blank spaces denote incomplete information in the source. SD, standard deviation.

Table 7. Blood perfusion (ω) values for the liver converted to consistent units of s^-1.

Value	± SD	Unit	Conversion	ω (s^−1)	± SD	Species	Tissue	Reference
108	34	mL/100 mL/min	10^−2/60	0.018	0.006	Human	Normal	[42]
98	36	mL/100 mL/min	10^−2/60	0.016	0.006	Human	d	[42]
70	22	mL/100 mL/min	10^−2/60	0.012	0.004	Human	ca	[42]
69	30	mL/100 mL/min	10^−2/60	0.012	0.005	Human	cb	[42]
56	13	mL/100 mL/min	10^−2/60	0.009	0.002	Human	cc	[42]
1000/1100		mL/kg/min	1/10^6/60 × ρ	0.018/0.019		Human m/f	Normal	[43]
100	20	mL/100 mL/min	10 × ρ/60/10^6	0.018	0.004			[22]
9.19		kg/m^3/s	1/ρb	0.009				[23,24]

Blank spaces denote incomplete information in the source. SD, standard deviation; m, male; f, female; d, diseased non-cirrhotic; ca, child a; cb, child b; cc, child c.

Table 8. Water content C of liver tissue.

C	± SD	Units	Species	Reference
0.73		MF	Bovine	[44]
0.71	0.045	MF	Bovine	[45]
0.77		MF	Human	[19]
0.72–0.76		VF		[25,26]
0.68		VF		[46]
0.78		MF		[47]

Blank spaces denote incomplete information in the source. MF, mass fraction; VF, volume fraction.

Table 9. Summary of parameter value ranges used in the sensitivity study given as a range. These are contrasted against the IT’IS database [16].

Parameter	Range	IT’IS mean ± SD	IT’IS range	IT’IS no. of studies
c (J/kg/K)	3600–4000	3540 ± 119	3332–3617	5
ρ (kg/m^3)	1020–1080	1079 ± 53	1050–1158	4
k (W/m/K)	0.46–0.57	0.52 ± 0.03	0.48–0.57	9
Δ k (W/m/K^2)	0–0.0033
σ (S/m)	0.14–0.28	0.144
Δ σ (%/K)	1–2
ω (s^-1)	0.009–0.018	0.015 ± 0.003	0.007–0.022	50
C	0.71–0.76
Δ T (K)	1–10
ρvap cvap (J/m^3/K)	440,000–800,000
ξ	0–1
σvap/σ	0.01–0.0001
G0 (%)	70–90

Blank spaces denote information unavailable in the IT’IS database.

Table 10. Final set of parameters included in the sensitivity analysis.

Parameter	Units	Lower	Upper
ρc	J/m^3/K	3.7 × 10^6	4.3 × 10^6
ρvap cvap	J/m^3/K	0.44 × 10^6	0.80 × 10^6
k0	W/m/K	0.46	0.57
Δ k	W/m/K^2	0	0.0033
σ0	S/m	0.14	0.28
Δ σ	%/K	1	2
ω b	J/m^3/K/s	34,020	68,040
C		0.71	0.76
Δ T	K	1	10
ξ		0	1
σvap/σ		0.01	0.0001
G0	%	70	90

Figure 3. Results of the Morris method when considering the ablation zone size at the end of the procedure. The means of the main effects are plotted along the x axis and the standard deviations along the y axis. The solid line is plotted for y = ± x and for points below the line the mean is larger than the standard deviation and therefore the expected value of the mean is non-zero.

Figure 4. Sensitivity measures plotted versus time and contrasted against the mean ablation zone area. The long-term behaviour is of interest here and it can be seen that σ₀, ω_b, ξ and G₀ have the greatest impact on the ablation zone area due to their relatively large mean. The standard deviations are also larger for some of these parameters, indicating parameter interactions or non-linear responses. A, B, and C denote initial, steady-state, and transient periods respectively.

Figure 5. Sensitivity measures for the 50 °C isotherm versus time contrasted against the mean area. The changes in the area of the isotherm are complex but are separated into regions of (A) active heating and (B) cooling. During the initial heating phase σ₀ and dominates. During cooling the thermal parameters perfusion ω_b, baseline thermal conductivity k₀ and rate of change of thermal conductivity Δk are the most important.

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