Modelling Of Heat Conduction Along Vein Walls In Endovenous Laser Treatment

W. R. Fuller, Maria Raiti, R. Bush
{"title":"Modelling Of Heat Conduction Along Vein Walls In Endovenous Laser Treatment","authors":"W. R. Fuller, Maria Raiti, R. Bush","doi":"10.5580/1e73","DOIUrl":null,"url":null,"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","PeriodicalId":22514,"journal":{"name":"The Internet journal of microbiology","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2010-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Internet journal of microbiology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5580/1e73","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1

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
静脉内激光治疗中沿静脉壁热传导的模拟
隐静脉的激光消融使用激光探针在静脉中产生光热效应。在本研究中,我们建立了激光诱导的沿静脉壁加热和传导效应的数学模型。我们提出并求解了相关的二维热传导问题。问题解决方案的模拟解决了程序中有争议问题的一个方面。从一开始,热损伤隐静脉的确切病因就存在争议[1-5]。有支持者认为传导能(激光尖端接触)和对流能(蒸汽气泡)是导致组织损伤的病因。这一争议不仅从科学方面进行了辩论,而且在法律领域也是如此。正如以前的出版物所报道的[1,5],热消融时急性损伤的标志是内皮细胞的完全丧失。当以3-4毫米的间隔进行12-15瓦的激光发射时,存在这一发现。如果假设激光点与静脉表面接触,当使用传统的裸端光纤时就会发生这种情况,那么问题就出现了,这种接触和由此产生的组织传导是否对组织学结果负责。无论波长和脉冲持续时间如何,激光热消融后内皮细胞立即消失是一个恒定的发现。根据我们的组织学观察,我们的假设是热传导不会是这些观察到的腔内变化的主要病因。本文的目的是在建立相关数学模型的基础上,通过模拟来验证我们的假设,该模型可以准确地计算激光应用点的温度和能量传播距离。模型的关键参数是激光探头将静脉壁受影响区域升高到的初始温度。在模型中,我们将这个温度作为一个自由参数,但是当我们给出基于模型的模拟结果时,我们将假设激光发射时激光尖端的初始温度为100℃。这个假设是合理的,因为(1)此时出现了蒸汽气泡,(2)我们已经测量了这些气泡撞击脉壁的温度。我们在其他初始温度下进行了模拟,比如700摄氏度,相关结果并没有显著改变我们的最终结论。建立一个有效的数学模型的第一步是阐明模型所依据的假设。我们的模型建立在从经验和理论中得出的五个假设的基础上。假设1:我们假设探头尖端在一个矩形区域内接触部分塌陷的静脉壁。在这一区域,能量的短暂爆发使温度升高到初始值T0。讨论:选择矩形是为了模拟探针尖端接触部分塌陷静脉壁的区域。我们的分析将表明,热传导的时间演化使初始区域的边界变得平滑,因此矩形的假设在确定细胞死亡区域时并不特别重要。如上所述,我们将给出T值为100℃时的模拟结果。假设2:我们假设在此过程中没有沿着静脉壁发生质量传输,因此,在静脉内激光治疗中,唯一的沿静脉壁热传导模型2所涉及的8种传输是热量形式的能量传输。假设3:我们假设静脉壁是一个无限长的右圆柱体。讨论:隐静脉是一个均匀的右圆柱体,这是一个简化的假设,它模拟了实际的静脉,因为它在超声检查和临床经验中出现。静脉外的肿胀溶液当然不会对静脉壁施加张力,它可能提供的任何压缩对表面面积的影响都很小。无限长假设仅仅表明激光作用的区域离叶脉的末端足够远,纵向边界在局部热传导过程中不起作用。假设4:我们假设在初始热接触后,静脉壁仍然是一个封闭的系统。讨论:这个假设与假设2一起,将手边的现象限制为沿静脉壁热传导的现象。因此,热传导是导致内皮细胞坏死的主要机制的假设需要假设2和假设4作为其基本支持。从这个角度来看,能量通过血液转移到静脉壁的效果是一个单独的问题。确实,能量会耗散到周围的麻醉剂和组织中。 实验结果[6]表明,这种热损失是最小的。在任何情况下,假设4确保我们在这个模型中得到的细胞死亡区域将是实际区域的上界。任何包含其他能量损失模式的模型肯定会产生较小的细胞死亡区域。假设5:我们假设傅里叶热传导定律:热量沿着热梯度-T从温度较高的区域流向温度较低的区域,其中T = T(x, y, z, T)是坐标x = (x, y, z)在时间T点的温度。讨论:假设4和5暗示热量沿脉壁流动,能量守恒。标准的能量守恒论证表明,热流的局部行为可以用热传导方程来描述[7]。图1其中ΔT为T的拉普拉斯式,常数k为静脉壁的热扩散系数。我们取人体心肌组织的热扩散系数k = 1.289 × 10-7 m2/sec。考虑一个右手坐标系,z轴沿着静脉的中心纵轴,x轴穿过矩形的中心,最初在静脉壁上升高到温度T。在柱坐标下,方程(1)变为
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信