Computer modelling of RF ablation in cortical osteoid osteoma: Assessment of the insulating effect of the reactive zone

Abstract

Purpose: The aim was to study by computer simulations the insulating role of the reactive zone surrounding a cortical osteoid osteoma (OO) in terms of electrical and thermal performance during radiofrequency ablation (RFA). Material and methods: We modelled a cortical OO consisting of a nidus (10 mm diameter) enclosed by a reactive zone. The OO was near a layer of cortical bone 1.5 mm thick. Trabecular bone partially surrounds the OO and there was muscle around the cortical bone layer. We modelled RF ablations with a non-cooled-tip 17-gauge needle electrode (300 s duration and 90 °C target temperature). Sensitivity analyses were conducted assuming a reactive zone electrical conductivity value (σ_rz) within the limits of the cortical and trabecular bone, i.e. 0.02 S/m and 0.087 S/m, respectively. In this way we were really modelling the different degrees of osteosclerosis associated with the reactive zone. Results: The presence of the reactive zone drastically reduced the maximum temperature reached outside it. The temperature drop was proportional to the thickness of the reactive zone: from 68 °C when it was absent to 44 °C when it is 7.5 mm thick. Higher nidus conductivity values (σ_n) implied higher temperatures, while lower temperatures meant higher σ_rz values. Changing σ_rz from 0.02 S/m to 0.087 S/m reduced lesion diameters from 2.4 cm to 1.8 cm. Conclusions: The computer results suggest that the reactive zone plays the role of insulator in terms of reducing the temperature in the surrounding area.

• Computer modelling
• cortical bone
• finite element method
• osteoid osteoma
• radiofrequency ablation
• reactive zone

Introduction

Osteoid osteoma (OO) is a common bone tumour, representing approximately 10% of all benign skeletal lesions, that involves progressively worsening pain [1]. Although OO is benign and spontaneous remission occasionally occurs, an intervention for definitive relief of pain is sometimes required [2]. Among other techniques [3], radiofrequency ablation (RFA) is a minimally invasive technique which allows thermal destruction of the OO [4–6]. The most common location of OO is in the long bones [5]. Most OOs are cortical and macroscopically consist of a nidus of active bone surrounded by a reactive zone. The nidus, usually 5–10 mm in diameter [2], is the actual tumour, and hence the target of the RFA. The reactive zone of a cortical OO is partially surrounded by the cortex (cortical bone) while the rest is enclosed by trabecular bone (Figure 1). It has been suggested that the presence of cortical bone during RFA could play the role of a thermal insulator and hence grant protection to the adjacent soft tissues [7–9]. We have already observed this performance in a previous modelling study [10], which could be explained by the lower electrical conductivity of the cortical bone than the surrounding tissues.

Figure 1. (a) Cortical osteoid osteoma consisting of a nidus surrounded by cortical and trabecular bone. The RF applicator is introduced into the nidus through a drill hole prior to RF application. (b) Theoretical model of cortical OO consisting of a nidus enclosed by a reactive zone.

The reactive zone is histologically characterised by very dense reactive sclerotic bone [1] and can in fact be radiographically observed as a cortical thickening. For these reasons, and even though there are no experimental data available on electrical and thermal characteristics of the reactive zone, one could assume that its characteristics could somehow be similar to those of cortical bone. With all these issues in mind, we hypothesised that the reactive zone could also play an important role as thermal insulation during RFA of a cortical OO. As a detailed assessment of this issue has not been carried out to date, our aim was to build computer models to study the temperature distributions during RFA in the tissues involved in a cortical OO, with a special focus on assessing the possible insulating effect of the reactive zone and identifying the factors involved. Computer modelling technique is a methodological approach broadly employed to study the impact of factors on temperature distributions during RF heating of biological tissues [11].

Materials and methods

Physical situation

We considered a spherical nidus surrounded by a reactive zone, more or less next to the cortex of a long bone. Access to the OO to conduct an ablation is not a straightforward task. Firstly, a biopsy trochar is advanced under image guidance to the cortex (cortical layer). Then the cortex is drilled to within 1 mm or less of the tumour [6]. The RF applicator, which includes a metallic electrode on its distal part, is introduced into the nidus through the drill hole [12] (Figure 1a). The result of this approach is a physical arrangement as sketched in Figure 1b, in which the RF applicator passes through the cortical bone, and is exactly the physical situation modelled in this study. We considered a non-cooled tip needle-like electrode with 17-gauge diameter and a 5 mm exposed tip (including a sharp zone 1 mm long).

Geometry of the model

Figure 1b shows the geometry of the cortical OO model, consisting of a nidus (10 mm diameter) enclosed by a reactive zone of variable thickness (2.5, 5.0 and 7.5 mm). We considered two relative positions of the OO and the cortical bone layer, as shown in Figure 2 (Positions 1 and 2). Both positions are relevant because they represent the two limits between which the cortical OO can be located in relation to the cortical bone layer, which was assumed to be 1.5 mm thick. Position 2 represents a case in which the presence of the OO has caused a periosteal reaction, i.e. an elevation of the periosteum from the cortex [13]. The model includes muscle beyond this layer. Likewise, we defined three relevant directions in which temperature profiles were computed: −45°, 0° and 45° (all with the origin at the midpoint of the electrode surface). In this way three points (T₄₅, T₀ and T₋₄₅) were defined to assess the temperature beyond the reactive zone and composed the intersection points between these directions and the outer layer of the reactive zone.

Figure 2. Relative positions of the OO and the cortical bone layer. We defined three directions in which temperature profiles were computed: −45°, 0° and 45° (origin at the midpoint of the electrode surface). Note that in Position 2, the 45° direction, has muscle tissue outside the reactive zone, while 0° and −45° have trabecular muscle. The intersection points between these directions and the outer layer of the reactive zone defined three locations at which the temperature was assessed (T₄₅, T₀ and T₋₄₅).

Governing equations, numerical method and boundary conditions

The model was based on a coupled electric-thermal problem in which the governing equation for the thermal problem was the bioheat equation, which was modified by the enthalpy method to take into account the vaporisation phenomenon in the tissues [14]: (1) where ρ is tissue density, h enthalpy, k thermal conductivity, T temperature and t time. q_RF refers to the heat source from RF power, Q_p refers to the blood perfusion heat and Q_met refers to the metabolic heat, which is negligible in RFA. For biological tissues enthalpy is related to tissue temperature by the following expression [14]. (2) where ρ_i and c_i are density and specific heat of tissue respectively at temperatures below 100 °C (i = l) and at temperatures above 100 °C (i = g), h_fg is the product of water latent heat of vaporisation and water density at 100 °C, and C is tissue water content inside the tissue. The blood perfusion heat Q_p is computed from: (3) where ρ_b is the density of blood, c_b specific heat of blood, T_b blood temperature (37 °C), ω_i perfusion coefficient of the tissue i, and β is a coefficient which took the values of 0 and 1, according to the value of the temperature reached: β = 0 for T ≥ 50 °C, and β = 1 for T < 50 °C. Although some models included a slight increase in blood perfusion flow with initial temperature elevation [15], it has been demonstrated that the most important effect is the ceasing/decreasing due to the thermal coagulation.

The heat source q_RF is the distributed heat source due to the Joule effect (loss) which is given by: (4) where σ (S/m) is electrical conductivity at 500 kHz, and |E| the magnitude of the vector electric field (V/m), which is given by E = −∇V. The electrical problem is governed by the following equation: (5)

At RF frequencies (≈500 kHz) and over the distance of interest, the biological medium can be considered almost totally resistive and a quasi-static approach is therefore possible to solve the electrical problem [16].

The coupled electrical-thermal problem was numerically solved using COMSOL Multiphysics (Comsol, Burlington, MA, USA). We performed convergence tests to determine the adequate size of domain limits, and spatial and temporal resolution. Spatial resolution was achieved by refining the mesh so that the maximum temperature reached in the tissue after 300 s (T_max) was within 0.5% of the value obtained from the previous refinement step. When an adequate spatial resolution was achieved, we decreased the time step until T_max was within 0.5% of the value obtained from the previous time step.

There are several protocols for delivering electrical power in RFA of OO. In general, a constant temperature mode is clinically used with a target temperature of 90 °C and duration ranging from 240 s [4,17–19] to 360 s [20]. We considered this same mode with a target temperature of 90 °C and duration of 300 s. To implement the constant temperature mode we used the same approach as in Haemmerich and Webster [21], in which the authors considered a dynamic system for control purposes. The input variable of the system was the voltage applied to the electrode (V_e) and the output variable was the temperature measured at the tip of the electrode (T_e). There are different ways of controlling tip temperature, such as a proportional–integral–derivative (PID) controller, adaptive, neural network prediction, and fuzzy logic control [21]. We considered a proportional–integral (PI) defined by the parameters K_p and K_i, with values of 1.15 and 0.06, respectively.

Electrical boundary conditions were zero current density at the axis, zero voltage at the lower limit (to mimic the dispersive electrode) and voltage at the active electrode. Initial temperature and the temperature for the surfaces away from the active electrode were assumed to be 37 °C.

Physical characteristics of the model

Table 1 shows the values of the physical characteristics of the materials used in the model [22,23]. The thermal conductivity and specific heat of cortical bone and muscle were taken from a previous sensitivity analysis [10]. A change in σ with temperature was considered +1.5%/°C (with an abrupt drop when temperature reaches 100 °C) and also a change in k with temperature of +0.003 K⁻¹ (with a constant value above 100 °C) for the tissues [24]. The internal tissue water content, C in equation (2(2) ), was assumed to be 68% [25].

Table 1. Physical properties of the materials used in the model.

	Thermal parameters (@37 °C)
Material/tissue	Density (kg/m^3)	Electrical conductivity (S/m)	Specific heat (J/kg K)	Thermal conductivity (W/m K)	Blood perfusion coefficient (×10^−4s^−1)
Cortical bone	1908	0.022	1026^a	0.30^a	0^b
Trabecular bone	1178	0.087^f	2274	0.31	5.9
Nidus	1046	0.08^d–0.5^d	2726^e	0.56^e	26^c–70^c
Reactive zone	1908	0.02–0.087	1026	0.30	0^c–5.9
Muscle	1090	0.446	2824^a	0.52^a	6.7
Blood	1046	–	3639	–	–
Plastic (insulated trocar)	70	10−5	1045	0.026	0
Electrode	6450	10^8	840	18	0

All tissue data from Hasgall et al. [22] except: ^a[10], ^b[30], ^c[31], ^d[28], ^e[26], ^f[32]. Data for insulated trocar (plastic) and electrode from Tungjitkusolmun et al. [23].

As far as we can ascertain, there are no data available on the electrical and thermal characteristics of the nidus and the reactive zone. Most of the physical data on tumour tissue is taken from human and animal models of breast carcinoma [26–28]. In this study, electrical conductivity (σ) of both tissues was considered to be within reasonable ranges, and computer simulations were conducted to assess the sensitivity of each parameter. To be more precise, since the nidus is composed of vascularised connective tissue [29], we assumed that its electric conductivity (σ_n) would not be very different from that measured in human breast carcinoma (within 0.08 S/m and 0.5 S/m [28]). We considered σ_n values of 0.08, 0.22, 0.36 and 0.50 S/m.

The reactive zone corresponds histologically with a very dense reactive sclerotic bone [1], so that we can assume a value of σ_rz within these limits coinciding with the σ of cortical and trabecular bone, i.e. 0.02 S/m and 0.087 S/m, respectively. In this way we are really modelling the different degrees of osteosclerosis associated with the reactive zone. We considered 0.02, 0.042, 0.065 and 0.087 S/m as σ_rz values. The thermal characteristics of the reactive zone were assumed to be identical to those of cortical bone. Rossmanna and Haemmerich [26] reviewed the thermal properties of adenocarcinoma in the human breast and we took these values for the specific heat and the thermal conductivity of the nidus.

The indicator for heat-sink strength by blood perfusion is the blood perfusion rate (m), usually expressed in mL/min/100 g [30]. In order to consider this parameter in equation (3(3) ), the perfusion coefficient (ω_i) must be computed. The perfusion coefficient of the tissue i is related to m_i as follows: ω_i = m_i ρ_i 10⁻⁵/60. For instance, the mean value for trabecular bone in humans is m = 1 mL/min/100 g [30] which corresponds to a ω_i value of approximately 1.967 × 10⁻⁴s⁻¹. In a study by Mcintosh and Anderson [30] the authors reported a perfusion coefficient of zero for cortical bone, while in a study by Hasgall et al. [22] a mean value of 3.18 × 10⁻⁴s⁻¹ was reported. Due to the lack of data on perfusion in the reactive zone, and in accordance with the histological criterion mentioned above, we assumed a range of values of ω_i within these limits, coinciding with cortical and trabecular bone, i.e. 1.967 × 10⁻⁴, 3.933 × 10⁻⁴ and 5.9 × 10⁻⁴s⁻¹ (see Table 1). In contrast, since the nidus is a well-vascularised tissue [29], we assumed values between 26 × 10⁻⁴s⁻¹ and 70 × 10⁻⁴s⁻¹, which are the minimum and maximum values measured by Song et al. [31]. The following values were thus considered: 26 × 10⁻⁴, 41 × 10⁻⁴, 55 × 10⁻⁴ and 70 × 10⁻⁴s⁻¹.

Outcomes

The only target of the RFA is the nidus and not the surrounding tissue. The situation is not critical if the adjacent tissue is reactive bone. On the other hand, when there is not a sufficient degree of sclerosis, soft tissues such as vessels and nerves can suffer thermal injury. For this reason, and also to broadly assess the insulating role of the reactive zone, we studied the maximum temperature reached at the outer boundary of the reactive zone, and in particular, in two very dissimilar areas: muscle (45° direction, see Figure 2) and trabecular bone (directions 0° and −45°, see Figure 2). We also studied other variables involved in RFA, such as impedance and the distributed heat source (q_RF), which could provide a physical explanation of the computer results. The 50 °C isotherm was used to represent the thermal lesion contour.

Results

The appropriate size of the model domain limits was a width of 21.5 mm and a height of 33 mm. The models had around 8000 triangular elements and a time step of 0.01 s was used for each simulation.

Effect of perfusion coefficient and electrical conductivity

In the first set of simulations we considered an OO in Position 2 with a reactive zone 5 mm thick. We analysed the maximum temperatures reached at the outer boundary of the reactive zone, in particular at the interface with the trabecular bone (T₀) and muscle (T₄₅). In these simulations a sensitivity analysis was conducted by simultaneously varying the electrical conductivity and perfusion coefficient of the nidus and reactive zone. As shown in Figure 3, temperatures at the reactive zone−trabecular bone (T₀) were slightly higher (around 1–2 °C) than at the interface reactive zone-muscle (T₄₅). Overall, the perfusion coefficients of nidus and reactive zone did not appear to have an influence on the temperatures reached at the outer limit of the reactive zone (Figure 3a and b).

Figure 3. Maximum temperatures reached after 300 s of RFA at the outer limit of the reactive zone, in particular in the interface with trabecular bone (T₀) and with muscle (T₄₅) for different values of perfusion coefficient of nidus (ω_n) and reactive zone (ω_rz) (a, b), and of electrical conductivity of nidus (σ_n) and reactive zone (σ_rz) (c, d). A total of 64 simulations were conducted for each specific value of one of these parameters, varying the other three parameters (four values within a range). We then calculated and plotted the mean value of the 64 simulations. The maximum value of the difference with respect to the mean value was also plotted as a dispersion bar. These simulations were conducted with the OO in Position 2 (see Figure 2).

In contrast, the effect of σ_n and σ_rz was important, and surprisingly, was the opposite in each, i.e. while an increase in σ_n involved an increase in both T₀ and T₄₅, an increase in σ_rz involved a decrease. Both T₀ and T₄₅ varied by around 5 °C when σ_n and σ_rz were varied. Once more, temperatures at the outer layer of the reactive zone on the trabecular bone side (T₀) were 1–2 °C higher than on the muscle side (T₄₅).

In order to check that the behaviour shown in Figure 3c and d is not due to the entire parameter being changed simultaneously, in Figure 4 we plotted the change of temperature profile along the 0° direction with the change in σ_rz (keeping σ_n constant with a mean value of 0.29 S/m) and with the change in σ_n (for a mean value of σ_rz of 0.0535 S/m). Figure 4 confirms that for higher values of σ_n, temperatures at the end of the RFA are higher. In contrast, temperatures at the end of the RFA are lower for higher values of σ_rz. In terms of thermal lesion, a change in σ_rz from 0.02 S/m (reactive zone with high degree of osteosclerosis) to 0.087 S/m (reactive zone with characteristics very similar to trabecular bone) reduces lesion diameter from 24 mm to 18 mm. Although the effect of changing σ_n is the opposite, it is more or less similar in absolute values of change in the lesion diameter.

Figure 4. Temperature profiles (at 300 s) computed in the 0° direction (see Figure 2) for different electrical conductivity values of the reactive zone (a) and nidus (b). The 50 °C line represents the thermal lesion contour. These simulations were conducted with the OO in Position 2 (see Figure 2).

Figure 4 also shows that in general the temperature drop with distance inside the reactive zone is steeper than in the nidus zone, which suggests that the presence of the reactive zone somehow limits the lesion volume (this issue is analysed below).

To determine the reasons for the different behaviour of the nidus and reactive zone in terms of electrical conductivity, we analysed the spatial distribution of the resistive heating q_RF at end of ablation (300 s). Figure 5 shows q_RF distribution of the tissues involved for different values of σ_n and σ_rz. As can be expected, q_RF drops drastically around the electrode. The differences of q_RF in the nidus when σ_n and σ_rz vary do not seem to be relevant; however, there are interesting differences in the reactive zone. The differences shown in Figure 5a are in agreement with the temperature profiles seen in Figure 4a. An increase in σ_rz implies a decrease in q_RF (and consequently in temperature). The differences in Figure 5b are rather more subtle and cannot be explained so easily.

Figure 5. Distributed heat source q_RF due to RF power in the involved tissues for different values of electrical conductivity of reactive zone σ_rz (a) and nidus σ_n (b). These simulations were conducted with the OO in Position 2 (see Figure 2).

Figure 6 shows the progress of the applied voltage throughout RFA for different values of σ_n and σ_rz. Note that the change of σ_rz causes larger changes in the applied voltage than changes of σ_n. In general, an increase in the electrical conductivity implies a decrease in the applied voltage.

Figure 6. Progress of the applied voltage throughout RF ablation for different values of electrical conductivity of the reactive zone σ_rz (a) and nidus σ_n (b). These simulations were conducted with the OO in Position 2 (see Figure 2).

Effect of reactive zone (RZ) thickness

Figure 7 shows the temperature distributions at the end of RFA and the thermal lesion contour evolution for different RZ thicknesses. These simulations were conducted with OO in Position 1 (see Figure 2). The results show that RZ thickness has little effect on the temperature distributions across the mathematical domain, regardless of tissue type. Obviously, the thinner the reactive zone, the faster the thermal lesion contour reaches the area beyond it. Also, the thermal lesion enlarges almost homogeneously, i.e. growth is identical regardless of direction (α).

Figure 7. Temperature distributions (in °C) after 300 s of RFA for three RZ thicknesses: 2.5 mm (a), 5.0 mm (b) and 7.5 mm (c). Dashed line is the 50 °C isoline which represents thermal lesion contour. Progress of the thermal lesion radius throughout 300 s of RFA at the outer limit of the reactive zone for three RZ thicknesses: 2.5 mm (d), 5.0 mm (e) and 7.5 mm (f), and for three directions (α = 0, 45° and −45°). These simulations were conducted with the OO in Position 1 (see Figure 2).

Table 2 and Figure 8 show the electrical impedance (Ω) recorded at the start of RFA for different RZ thicknesses and electrical conductivity (σ_rz). Note that the effect of RZ thickness is especially important when σ_rz is very low, which is found when there is a high degree of osteosclerosis (reactive zone with characteristics similar to cortical bone). For medium and high σ_rz values, the thickness has very little effect on impedance. Table 3 shows the electrical impedance for different RZ thickness and electrical conductivity in the nidus (σ_n).

Figure 8. Effect of electrical conductivity of nidus (σ_n) and reactive zone (σ_rz) on electrical impedance (Ω) registered at the start of RFA. The effect of RZ thickness and OO position are also considered. Black and red colours correspond with Positions 1 and 2, respectively. Trend lines have been added as a guide.

Table 2. Electrical impedance (Ω) at the start of RFA for different RZ thickness and electrical conductivity (σ_rz).

	Thickness (mm)
	2.5		5.0		7.5
σrz (S/m)	Position 1	Position 2	Position 1	Position 2	Position 1	Position 2
0.02	511.1	490.3	616.8	598.1	680.1	694.1
0.042	363.4	338.2	400.1	377.4	422.2	427.7
0.065	314.6	285.4	328.6	302.5	337.4	328.2
0.087	289.7	257.2	291.6	262.8	293.3	287.1

Table 3. Electrical impedance (Ω) at the start of RFA for different RZ thickness and electrical conductivity of the nidus (σ_n).

	Thickness (mm)
	2.5		5.0		7.5
σn (S/m)	Position 1	Position 2	Position 1	Position 2	Position 1	Position 2
0.08	703.3	677.2	727.3	697.6	742.0	773.6
0.22	376.6	348.9	400.3	374.7	414.8	421.2
0.36	308.6	281.7	331.4	307.7	345.4	346.3
0.50	279.0	252.8	301.2	278.5	314.8	310.9

Effect of presence of the reactive zone

Although a marked reactive sclerosis is commonly found around the nidus, some lesions may have little or no reactive sclerosis [33], which led us to conduct computer simulations without a reactive zone, considering the mean values of σ_n and σ_rz (0.29 and 0.0535 S/m, respectively). Figure 9 shows the temperature profiles at the end of RFA for different RZ thicknesses and with no RZ. In the latter case, since we were modelling Position 1, the nidus is in direct contact with the trabecular bone. Note that the case with no RZ in Position 2 is fictitious, since a bone layer has to necessarily surround the nidus.

Figure 9. Temperature profiles (at 300 s) computed in 0° direction (see Figure 2) for different RZ thicknesses. The case without a reactive zone is also plotted (solid line). The 50 °C line represents the thermal lesion contour. All simulations were conducted with OO in Position 1 (see Figure 2).

Figure 9 suggests that the presence of a reactive zone drastically reduces the maximum temperature reached in the trabecular bone. The drop in the maximum temperature is proportional to RZ thickness: from 68 °C when it is absent to 44 °C at a thickness of 7.5 mm. However, surprisingly, the presence of a reactive zone around the nidus involves an augmentation in the temperature profiles (see Figure 9). This difference may be up to ≈5 °C between the absence of an RZ and a thickness of 7.5 mm. From this it would seem that the reactive zone does not itself provide an insulating effect.

Initial impedance in the absence of a reactive zone was ≈45 Ω lower than an RZ thickness of 2.5 mm: 656.8 Ω, 330.7 Ω, 265.2 Ω, and 237.1 Ω, for σ_n of 0.08, 0.22, 0.36 and 0.5 S/m, respectively.

Discussion

We conducted computer simulations aimed at assessing the electrical and thermal performance of the reactive zone during RFA of a cortical OO. In this assessment we assumed that the electrical conductivity of the reactive zone depends on the degree of osteosclerosis of the RZ: from 0.02 S/m for severe osteosclerosis to 0.087 S/m for mild. We also analysed the effect of blood perfusion in the nidus and reactive zone on the temperature reached beyond the RZ. Although tissue perfusion is known to be a crucial parameter in determining lesion size during RFA, we did not find this parameter to have a strong effect on the temperature reached in the RZ outer layer, which could be explained by the relatively narrow range of values considered for nidus perfusion (26−70 × 10⁻⁴s⁻¹).

On the other hand, we did find the effect of the electrical conductivity of the nidus and reactive zone to be considerable. To account for these results, we first considered an analytical model of the distributed heat q_RF, which is a key factor in the temperature increase. Although the geometry considered here is too complex to allow an analytical mathematical expression of q_RF, it can be assumed that when a voltage constant V_e is applied, q_RF is somehow proportional to , and also proportional to the electrical conductivity of the tissue (σ). It is also evident that an increase in σ implies a decrease in the measured electrical impedance (Z). In our study we observed that the change from the minimum to maximum value of either σ_n or σ_rz reduced Z by about 50% (see Tables 2 and 3).

We can also assume that in order to keep the electrode tip temperature constant, a variation in Z will imply a variation in V_e. The power required to keep the temperature on target is . Hence, a decrease in Z (due to increased σ) should imply a drop in V_e in order to balance the required power (see Figure 6). Unlike the relation between σ and Z (which is more or less constant at 50%), the relation between Z and V_e depends on the tissue type, as can be seen in Figure 6. While an increase in σ_rz drastically reduces V_e (see Figure 6a), the change in V_e is minor when σ_n is changed (see Figure 6b). This could explain the distribution of q_RF in the reactive zone (see Figure 5), since an increase in σ_rz considerably reduces the applied voltage, which implies that × σ_rz will be globally reduced in the reactive zone, as shown in Figure 5a. On the other hand, an increase in σ_n does not seriously reduce V_e (see Figure 6a), and consequently the product × σ_rz will probably be more affected by σ_rz than V_e, i.e. the increase in σ_rz could be mainly responsible for the increase in q_RF in the reactive zone shown in Figure 5b. This could be the explanation for the higher temperature in the reactive zone when σ_rz is raised (see Figure 4b). Additionally, the temperature increase in the reactive zone also induces higher σ_rz, which further contributes to increasing q_RF.

In any case, this explanation is based on the simplification that q_RF in each tissue is proportional to × σ_rz, V_e being the total applied voltage. The real situation of the model is probably more complex in terms of geometry (note that the electrode has a cylindrical central part but a sharp distal part) and the phenomena involved (thermal conduction has been ignored in this theoretical explanation).

The effect of RZ thickness was also analysed. As expected, the results suggest that the thicker the reactive zone, the lower the temperature outside it. In the limiting case of RZ absent, the temperature at the outer limit of the nidus reached 68 °C, as opposed to 44 °C at the outer limit of a 7.5 mm thick RZ. Surprisingly, increasing RZ thickness involved a slight increase in the temperature profiles. This could be explained in the same way as the changes in σ_rz, i.e. since the increased thickness implies an increase in Z (see Table 2), this could bring about much higher V_e (similar to what happened when reducing σ_rz, see Figure 6a), and consequently an increase in q_RF in the reactive zone, which would be responsible for the higher temperature observed predominantly in this zone (see Figure 9).

In short, the computer results confirm that the presence of a reactive zone minimises the thermal lesion contour in tissues beyond the reactive zone. This is reasonable: the key is to take away the point that must not be thermally damaged. However, what is surprising is the temperature increase at the boundary of the reactive zone, when this zone is less electrically conductive, as the opposite effect could be expected, i.e. more protective effect in a less conductive zone. Besides the possible explanation for this performance given above, we also believe it could be due to the fact that σ_rz is much smaller than σ_n. However, we conducted additional computer simulations with similar values for σ_rz and σ_n, and the results confirmed the general performance described: in an arrangement with two tissues (internal and external), temperature increases when electrical conductivity of the external tissue is reduced and decreases when electrical conductivity of the internal tissue is reduced.

Although the computer results can be explained thus, we should not forget the physical significance of having reactive zones with different σ_rz values. If a RZ such as that defined in the theoretical model has a high σ_rz value, it probably means that its properties are similar to those of trabecular bone, and is therefore a less reactive zone and could possibly be considered as trabecular bone. This could have the following clinical implication: although our results suggest that an increase in σ_rz implies a decrease in the temperature in the outer reactive zone, i.e. a protective effect, the truth is that the area with a high σ_rz value represents a zone with a low degree of osteosclerosis and hence should be more thermally protected than another zone with a low σ_rz value.

The number of experimental studies reporting lesion sizes after RFA of OO that we could find was very limited. Furthermore, there are no experimental studies on the effect of the reactive zone characteristics on the thermal lesion. In spite of this, the computed lesion diameters in this study (1.8–2.4 cm) were in agreement with the value of 1.85 cm reported in [9] for an electrode and ablation protocol similar to our model.

We believe this to be the first computer modelling on RFA in OO. Its main limitation is probably the lack of an accurate experimental validation. Unfortunately, no previous data on temperature mapping around the OO during RFA are available for validation purposes. It also seems somewhat complex to carry out experiments in which the thickness and electrical conductivity of the reactive zone are kept under control. We suggest that future studies could use imaging techniques to accurately assess the geometry of the reactive zone, and even to try to infer its characteristics from the degree of osteosclerosis derived from the images.

Conclusions

The computer results suggest that the reactive zone plays the role of insulator in terms of reducing the temperature beyond it. However, they also showed an unexpected relationship between the electrical conductivity of the reactive zone, which could be related to its degree of osteosclerosis, and the ability to thermally protect the tissues beyond.

Declaration of interest

This work was supported by a grant from the Agencia Nacional de Promoción Científica y Tecnológica de Argentina (Ref. PICT-2012-1201), and by the Spanish Programa Estatal de Investigación, Desarrollo e Innovación Orientada a los Retos de la Sociedad under grant number TEC2014-52383-C3-R (TEC2014-52383-C3-1-R). The authors alone are responsible for the content and writing of the paper.