Modelling Of Heat Conduction Along Vein Walls In Endovenous Laser Treatment

Abstract

Laser ablation of the saphenous vein uses laser-tipped probes to produce photothermal effects in the vein. In this study we construct a mathematical model of the effects of laser-induced thermal heating and conduction along the vein wall. We formulate and solve the relevant two-dimensional heat conduction problem. Simulations of the solution to the problem resolve an aspect of a debated question of the procedure. INTRODUCTION Since its inception, controversy has surrounded the exact etiology of thermal damage to the saphenous vein [1-5]. There are proponents of both conduction energy (laser tip contact) and convective energy (steam bubbles) as being the etiology of resultant tissue damage. This controversy has been debated not only from the scientific aspect, but in the legal arena as well. As reported in previous publications [1, 5], the hallmark of acute damage at the time of thermal ablation is complete loss of endothelium. This finding is present when laser firing at 12-15 watts is done at intervals of 3-4 mm apart. If it is assumed that the laser point contacts the surface of the vein, which occurs when a conventional bare tip fiber is used, then the question arises of whether the contact and the resultant conduction by tissue is responsible for the histologic findings. The absence of endothelium immediately following laser thermal ablation is a constant finding regardless of wavelength and pulse duration. Based on our histologic observations it is our hypothesis that heat conduction would not be the primary etiology of these observed luminal changes. The purpose of this article is to present the results of testing our hypothesis by simulations based on the construction of a relevant mathematical model which will accurately calculate both temperature and distance traveled of energy from the laser application site. CONSTRUCTION OF THE MODEL The key parameter of the model is the initial temperature to which the laser probe elevates the affected region of the vein wall. In the model we will keep this temperature as a free parameter, but when we present results of simulations based on the model, we will assume that at the time of laser firing the initial temperature at the laser tip is 100 C. This assumption is reasonable since (1) steam bubbles occur at this point, and (2) we have measured the temperature of these bubbles impinging on the vein wall. We have run simulations at other initial temperatures, such as 700 C and the associated results do not significantly change our final conclusions. The first step in building a valid mathematical model is the enunciation of the assumptions on which the model rests. We base our model on five postulates derived from both experience and theory. Postulate 1: We assume that the probe tip contacts the partially collapsed vein wall in a rectangular region. In this region the transfer of a short burst of energy raises the temperature to an initial value T0. Discussion: The rectangular shape is chosen so as to model the region where the probe tip contacts the partially collapsed vein wall. Our analysis will indicate that the time evolution of the heat conduction smooths out the boundary of the initial region so that the assumption of a rectangular shape is not of particular importance in determining the region of cell death. As indicated above, we will present the results of simulations for a value of T of 100 C. Postulate 2: We assume that no transport of mass occurs along the vein wall during the process, so that the only Modelling Of Heat Conduction Along Vein Walls In Endovenous Laser Treatment 2 of 8 transport involved is the transport of energy in the form of heat. Postulate 3: We assume that the vein wall is a right circular cylinder of infinite length. Discussion: That the saphenous vein is a uniform right circular cylinder is a simplifying assumption that models the actual vein as it appears in ultrasounds of the procedure and in clinical experience. The tumescent solution outside of the vein certainly exerts no tensile forces on the vein wall, and any compression it might provide has minimal effect on the surface area. The assumption of infinite length merely indicates that the regions of laser application are far enough from the ends of the vein that the longitudinal boundaries play no role in the local heat conduction process. Postulate 4: We assume that after the initial thermal contact the vein wall remains a closed system. Discussion: This postulate, along with Postulate 2, restricts the phenomenon at hand to one of heat conduction along the vein wall. The assumption that heat conduction is the primary mechanism causing endothelial necrosis thus requires Postulates 2 and 4 as its underlying support. From this point of view the effect of energy transfer through the blood and into the vein wall is a separate problem. It is true that dissipation of energy into the ambient anesthetic and tissue occurs. Experimental results [6] indicate that such heat loss is minimal. In any case assumption of Postulate 4 ensures that whatever region of cell death we obtain in this model will be an upper bound for the actual region. Any model incorporating other modes of energy loss will certainly produce smaller regions of cell death. Postulate 5: We assume Fourier's law of heat conduction: Heat flows from regions of higher temperature to lower temperature along the thermal gradient -T, where T = T(x, y, z, t) is the temperature at the point with coordinates x = (x, y, z) at time t. Discussion: Postulates 4 and 5 imply that heat flows along the vein wall in such a way that energy is conserved. Standard energy conservation arguments indicate that the local behavior of the heat flow is described by the heat conduction equation [7]. Figure 1 Here ΔT is the Laplacian of T and the constant k is the thermal diffusivity of the vein wall. We take for the value of thermal diffusivity that of human myocardium tissue: k = 1.289 x 10-7 m2/sec. THE HEAT CONDUCTION PROBLEM: THE PLANAR APPROXIMATION Consider a right-handed coordinate system with the z-axis along the central longitudinal axis of the vein and the x-axis through the center of the rectangle initially raised to a temperature T on the vein wall. In cylindrical coordinates equation (1) becomes