Review of the mathematical functions used to model the temperature dependence of electrical and thermal conductivities of biological tissue in radiofrequency ablation

Figures & data

Table I. Studied cases according to the different mathematical functions used to model the temperature dependence of electrical (σ) and thermal (k) conductivities.

		σ
		T < 100 °C
		Linear growth (%/°C)		Exponential growth (%/°C)		T > 100 °C		k
Cases		1.5	2	1.5	2	σ drops rate 2 orders	σ drops rate 4 orders	Constant	Linear growth +  constant	Different value for each phase (liquid–gas)	Linear drop +  constant
Set 1	1	X				X		X
	2		X			X		X
	3			X		X		X
	4				X	X		X
	5	X					X	X
	6		X				X	X
	7			X			X	X
	8				X	X		X
Set 2	9				X			X	X
	10	X				X			X
	11	X				X					X
	12				X		X		X
	13				X		X			X
	14				X		X				X

Figure 1. Representation of the functions used for modelling the thermal dependence of σ and k. In the case of σ was showed only a two order drop for T > 105 °C. Legends specify some references in which these kinds of functions of σ and k are used.

Figure 2. Geometry of the two-dimensional theoretical model (out of scale and dimensions in mm) and electrode detail. The domain is divided into three zones: plastic portion of the electrode, metallic electrode and hepatic tissue.

Table II. Characteristics of the materials used in the theoretical model [12,19,29,30].

Tissue/material	σ (S/m)	k (W/m K)	ρ(kg/m^3)	c (J/kg K)
Electrode	7.4 × 10^6	15	8 × 10^3	480
Plastic	10^−5	0.026	70	1045
Liver	0.132^a	0.502^b	1080^c	3455^c
			370^d	2156^d

σ, electric conductivity; k, thermal conductivity; ρ, density; c, specific heat.

^aAssessed at 17 °C. This allows initial impedance (≈100 Ω) to match the experimental mean value obtained from [30].

^bAssessed at 37 °C [12].

^cTissue in liquid phase.

^dTissue in gas phase.

Figure 3. (A) Impedance evolution to the first roll-off in cases 1 to 4. (B) Impedance evolution to the first roll-off in cases 1, 4, 5 and 8. The only characteristic which varies in both figures is σ, in accordance with Table I.

Figure 4. (A) Impedance evolution in cases 1, 9, 10 and 11. (B) Impedance evolution in cases 4, 12, 13 and 14. A different mathematical function for k was used in each case, in accordance with Table I.

Figure 5. Evolution of the lesion short diameter (a) throughout 360 s for cases 1 to 4 (see Table I for details). Dashed lines represent the time in which the first roll-off was achieved in each case.

Figure 6. Two-dimensional tissue temperature distribution during RFA and impedance evolution. The impedance progress can be divided into three intervals: 1) initial impedance drop; 2) impedance plateau; and 3) abrupt increase of impedance. These zones are related to the physical conditions of the tissue during the RFA process [30].

Duck F. Physical properties of tissue – A comprehensive reference book. New York: Academic Press; 1990

 Google Scholar

Pätz T, Kröger T, Preusser T. Simulation of radiofrequency ablation including water evaporation. IFMBE Proceedings 2009;25/IV:1287–90

 Google Scholar

Berjano EJ, Burdío F, Navarro AC, Burdío JM, Güemes A, Aldana O, et al. Improved perfusion system for bipolar radiofrequency ablation of liver: Preliminary findings from a computer modeling study. Physiol Meas 2006;27:N55–66

 PubMed Web of Science ®Google Scholar

Trujillo M, Alba J, Berjano E. Relation between roll-off occurrence and spatial distribution of dehydrated tissue during RF ablation with cooled electrodes. Int J Hyperthermia 2012;28:62–8

 PubMed Web of Science ®Google Scholar