雷竞技rebat前沿|点源模型代表热量分布不计算焦耳热在射频消融术</t我tle> <style> .async-hide { opacity: 0 !important } </style> <link href="https://209cd62b0febf2a55d40-715eb384bc6027b876e93ad33d5c5ec3.ssl.cf3.rackcdn.com/font-awesome-4.3.0/css/font-awesome.min.css" rel="stylesheet"> <link href="https://9d0dd7a648345f19af83-877d2ecaf11b88d5e17327c758e17ef6.ssl.cf2.rackcdn.com/museo-sans-1.0.1/css/museo-sans.css" rel="stylesheet"> <style type="text/css"> </style> </head> <body class="journal-thermal-engineering section-heat-transfer-mechanisms-and-applications"> <a href="#main-content" class="bypassBlock-wrapper" id="bypass"><span class="bypassBlock-button">跳转到主要内容</年代pan></a>  <noscript> <iframe src="https://www.thespel.com/ns.html?id=GTM-TFKV632>m_auth=NsJH3Ts1-89I8lZURwQYmw>m_preview=env-1>m_cookies_win=x" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <noscript> <iframe src="https://www.thespel.com/ns.html?id=GTM-N279F7B>m_auth=jfKETtlWamfZ4l02YiFCQw>m_preview=env-1>m_cookies_win=x" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <frontiers-ibar main-domain="frontiersin.org" loop-url="https://loop.frontiersin.org" journal-id="2105"></frontiers-ibar> <div class="page-container sidemenu-collapsed" id="main-content"> <div id="article" class="boxed white no-padding"> <div class="container-fluid main-container-xxl"> <div class="row" id="similar-articles"> <div class="new-wrapper"> <style> .disabled-supplementary-btn { cursor: not-allowed; pointer-events: none; opacity: .65; filter: alpha(opacity=65); -webkit-box-shadow: none; box-shadow: none; } </style> <aside class="right-container"> <div class="side-article-download clearfix"> <ul class="list-unstyled list-inline clearfix"> <li class="dropdown text-center paper"><button data-target="#" class="dropdown-toggle" data-test-id="download-button" data-toggle="dropdown" role="button" aria-expanded="false"><span class="icon-label" data-event="download-button-click">下载文章</年代pan></button> <div class="dropdown-menu-wrapper"> <div class="dropdown-menu-mobile-title"> 下载文章</d我v> <ul class="dropdown-menu" role="menu"> <li><a class="download-files-pdf action-link" href="//www.thespel.com/articles/10.3389/fther.2022.982768/pdf" data-test-id="article-downloadpdf" data-event="downloadpdf-button-click" id="download_article">下载</一个></li> <li><a class="download-files-readcube" href="http://www.readcube.com/articles/10.3389/fther.2022.982768" data-test-id="article-viewenhancedpdf" data-event="downloadreadcube-button-click" id="download_article">ReadCube</一个></li> <li><a class="download-files-epub" href="//www.thespel.com/articles/10.3389/fther.2022.982768/epub" data-test-id="article-epub" data-event="downloadepub-button-click" id="download_article">EPUB</一个></li> <li><a class="download-files-nlm" href="//www.thespel.com/articles/10.3389/fther.2022.982768/xml/nlm" data-test-id="article-nlm" data-event="downloadnlm-button-click" id="download_article">XML (NLM)</一个></li> <li><a data-test-id="supplementary-button" class="supplemental-data-disabled" data-event="supplementary-button-click" id="supplementary_view">补充<br>材料</一个></li> </ul> <button class="dropdown-menu-mobile-close-button" aria-label="Close" data-toggle="dropdown"></button> </div></li> <li><button class="btn btn-open-supplemental" data-event="supplementary-button-click" id="supplementary_view">补充数据</button></li> <li class="dropdown"><button type="button" class="navbar-toggle navbar-share" data-toggle="dropdown" data-target="#" aria-expanded="false"></button> <div class="dropdown-menu-wrapper"> <div class="dropdown-menu-mobile-title"> 分享</d我v> <ul class="dropdown-menu" role="menu"> <li class="social_block"> <div class="atButton"> <a class="addthis_button_twitter at300b" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" title="" target="_blank" href="#"><span class="at-icon-wrapper"></span></a> </div></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_linkedin at300b" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" title="" target="_blank" href="#"><span class="at-icon-wrapper"></span></a> </div></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_facebook at300b" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" title="" target="_blank" href="#"><span class="at-icon-wrapper"></span></a> </div></li> </ul> <button class="dropdown-menu-mobile-close-button" aria-label="Close" data-toggle="dropdown"></button> </div></li> <li class="dropdown"><button type="button" class="navbar-toggle navbar-citations" data-event="citation-button-click" data-toggle="dropdown" data-target="#" aria-expanded="false"></button> <div class="dropdown-menu-wrapper"> <div class="dropdown-menu-mobile-title"> 出口的引用</d我v> <ul class="dropdown-menu" role="menu"> <li><a data-test-id="article-endnote" href="//www.thespel.com/articles/10.3389/fther.2022.982768/endNote" data-event="endnotedownload-button-click">尾注</一个></li> <li><a data-test-id="article-referencemanager" href="//www.thespel.com/articles/10.3389/fther.2022.982768/reference" data-event="referencemanagerdownload-button-click">引用管理器</一个></li> <li><a data-test-id="article-simpletextfile" href="//www.thespel.com/articles/10.3389/fther.2022.982768/text" data-event="simpletextfiledownload-button-click">简单的文本文件</一个></li> <li><a data-test-id="article-bibtex" href="//www.thespel.com/articles/10.3389/fther.2022.982768/bibTex" data-event="bibtexdownload-button-click">助理</一个></li> </ul> <button class="dropdown-menu-mobile-close-button" aria-label="Close" data-toggle="dropdown"></button> </div></li> </ul> </div> <div class="stats-container"> <div class="divide-container"> <div class="flex-container-metrics"> <ul class="nav"> <li class="impact-data" id="views-section" style="display:none"><span class="views-count" id="views-count"></span><span class="title-views">总观点</年代pan></li> <li class="impact-data" id="downloads-section" style="display:none"><span class="views-count" id="downloads-count"></span><span class="title-views">下载</年代pan></li> <li class="impact-data" id="citations-section" style="display:none"><span class="views-count" id="citations-count"></span><span class="title-views">引用</年代pan></li> </ul> <div class="infoPopover" id="infoPopover" style="display:none"> <div role="button" class="infoPopover__trigger"></div> <div class="infoPopover__content"> <p>2023年5月,边界采用计雷竞技rebat数器5兼容的一个新的报告平台,符合行业标准。</p> <a href="https://blog.frontiersin.org/2023/05/16/open-access-publisher-frontiers-moves-to-a-new-article-metric-platform-counter/" target="_blank">阅读更多</一个> <div role="button" class="infoPopover__close"></div> </div> </div> </div> <div class="articleImpact" id="article-impact"> <span class="borderLine"></span> <a class="articleImpact--url" data-test-id="view-article-impact" data-event="impact-button-click" href="http://loop-impact.frontiersin.org/impact/article/982768" target="_blank">查看文章的影响</一个> </div> </div> <div class="side-article-impact"> <ul class="nav"> <li class="altmetric-wrapper"> <div class="altmetric-embed" data-badge-type="donut" data-event="altmetric-button-click" data-badge-popover="bottom" data-doi="10.3389/fther.2022.982768" data-condensed="true" data-link-target="new"></div><a class="altmetricScore_url" href="https://www.altmetric.com/details/doi/10.3389/fther.2022.982768" target="_blank">视图altmetric得分</一个></li> </ul> </div> </div> <div class="side-article-share "> <h5 class="like-h4">分享</h5> <ul class="share-media clearfix" style="padding: 0;"> <li class="social_block"> <div class="atButton"> <a class="addthis_button_facebook at300b" data-event="shareicon-button-click" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="facebook_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_twitter at300b" data-event="shareicon-button-click" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="twitter_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_google at300b" data-event="shareicon-button-click" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="twitter_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_linkedin at300b" data-event="shareicon-button-click" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="linkedin_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_more at300b" data-event="shareicon-button-click" data-event-param="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:title="A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation" addthis:description="Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Qr = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ0 where δ0 is a Dirac-delta function. Therefor..." addthis:url="//www.thespel.com/articles/10.3389/fther.2022.982768" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="total_count"></span></a></li> </ul> </div> <aside id="anchors" class="pull-left table-of-contents side-article"> <div class="side-people hidden-sm hidden-xs"> <section class="side-article-editors"> <header> <h5 class="like-h4">编辑</h5> </header> <a class="authors" href="//www.thespel.com/loop/people/1636862/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="https://loop.frontiersin.org/images/profile/1636862/24"><h6 class="author-link-aside">(Sanjeev索尼</h6></a> <div class="clearfix"></div> <p><span class="notes">CSIR-Central科学仪器组织,印度</年代pan></p> <header> <h5 class="like-h4">看过的</h5> </header> <a class="authors" href="//www.thespel.com/loop/people/1796845/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="https://loop.frontiersin.org/images/profile/1796845/24"><h6 class="author-link-aside">马塞洛Iasiello</h6></a> <div class="clearfix"></div> <p><span class="notes">Dipartimento di Ingegneria年意大利degli研究di那不勒斯费德里科•II,意大利</年代pan></p> <a class="authors" href="//www.thespel.com/loop/people/1830898/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="https://loop.frontiersin.org/images/profile/1830898/24"><h6 class="author-link-aside">Zailan Siri</h6></a> <div class="clearfix"></div> <p><span class="notes">马来西亚马来亚大学</年代pan></p> </section> </div> <nav> <header> <h5 class="like-h4" style="padding-top: 14px; padding-left: 6px; margin-bottom: 6px;">表的内容</h5> </header> <ul class="nav nav-list list-unstyled contents" data-event="toc-link-click"> <li> <ul class="flyoutJournal"> <li><a href="#h1">文摘</一个></li> <li><a href="#h2">介绍</一个></li> <li><a href="#h3">Bioheat方程模型和细胞死亡</一个></li> <li><a href="#h4">针几何</一个></li> <li><a href="#h5">焦耳热模型</一个></li> <li><a href="#h6">分析推导</一个></li> <li><a href="#h7">点源模型</一个></li> <li><a href="#h8">数值评估点源模型和焦耳热模型的二维简化模型</一个></li> <li><a href="#h9">数值评估点源模型和焦耳热模型的3 d模型</一个></li> <li><a href="#h10">限制</一个></li> <li><a href="#h11">结论</一个></li> <li><a href="#h12">数据可用性声明</一个></li> <li><a href="#h13">作者的贡献</一个></li> <li><a href="#h14">资金</一个></li> <li><a href="#h15">确认</一个></li> <li><a href="#h16">的利益冲突</一个></li> <li><a href="#h17">出版商的注意</一个></li> <li><a href="#h18">补充材料</一个></li> <li><a href="#h19">引用</一个></li> </ul></li> </ul> </nav> </aside> <div class="clearfix"></div> <div class="side-article-download side-export clearfix"> <ul class="list-unstyled list-inline clearfix"> <li><button class="btn btn-open-supplemental" data-event="supplementary-button-click" id="supplementary_view">开放补充数据</button></li> <li class="dropdown text-center citation"><button data-target="#" data-test-id="citation-button" data-toggle="dropdown" role="button" aria-expanded="false"><span class="icon-label" data-event="citation-button-click">出口的引用</年代pan></button> <ul class="dropdown-menu" role="menu"> <li><a data-test-id="article-endnote" id="export_citation" href="//www.thespel.com/articles/10.3389/fther.2022.982768/endNote" data-event="endnotedownload-button-click">尾注</一个></li> <li><a data-test-id="article-referencemanager" id="export_citation" href="//www.thespel.com/articles/10.3389/fther.2022.982768/reference" data-event="referencemanagerdownload-button-click">引用管理器</一个></li> <li><a data-test-id="article-simpletextfile" id="export_citation" href="//www.thespel.com/articles/10.3389/fther.2022.982768/text" data-event="simpletextfiledownload-button-click">简单的文本文件</一个></li> <li><a data-test-id="article-bibtex" id="export_citation" href="//www.thespel.com/articles/10.3389/fther.2022.982768/bibTex" data-event="bibtexdownload-button-click">助理</一个></li> </ul></li> </ul> </div> <div class="side-crossmark"> <a data-target="crossmark" class="crossmark"> <figure> <img id="crossmark-icon" style="border: 0; width:64px; height:64px;" src="https://crossmark-cdn.crossref.org/widget/v2.0/logos/CROSSMARK_BW_square_no_text.svg" title="" alt=""> </figure> <div id="crossmark-icon"> 检查更新</d我v></a> </div> <article class="widget-listing people-also-looked-at side-article-related hidden"> <h5 class="like-h4" data-event="peoplelookedat-link-click">人也看了</h5> </article> </aside> <main class=""> <div class="article-section"> <div class="article-container" data-html="True"> <div class="abstract-container"> <div class="article-header-container">   <div class="header-bar-one"> <h2>原始研究的文章</h2> </div> <div class="header-bar-three-container"> <div class="header-bar-three"> 前面。小卡。Eng。,11October 2022<br>秒。传热机制和应用程序<br> <span class="volumeInfo">卷2 - 2022 |</年代pan> <a href="https://doi.org/10.3389/fther.2022.982768">https://doi.org/10.3389/fther.2022.982768</一个> </div> </div> </div> <div class="JournalAbstract"> <a id="h1" name="h1"></a> <h1>点源模型代表热量分布不计算焦耳热在射频消融术</h1> <div class="authors"> <a href="//www.thespel.com/people/u/1631266" class="user-id-1631266"><img class="pr5" src="https://loop.frontiersin.org/images/profile/1631266/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.org">Panchatcharam Mariappan</一个> <sup>1、2</年代up>*,<我米g class="pr5" src="https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg" alt="www.雷竞技rebatfrontiersin.org">Gangadhara B<年代up>1</年代up>和<我米g class="pr5" src="https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg" alt="www.雷竞技rebatfrontiersin.org">罗南·弗拉纳根<年代up>2</年代up> </div> <ul class="notes"> <li><span><sup>1</年代up></span>数学和统计,印度理工学院Tirupati Tirupati,印度</li> <li><span><sup>2</年代up></span>NUMA工程服务有限公司、邓多克、爱尔兰</li> </ul> <p class="mb15">很多肝癌肿瘤专家建议桥接治疗限制肿瘤生长,直到捐助者是可用的。介入放射学包括射频消融术(RFA)就是这样的一个桥接治疗。这个局部区域治疗的目的是产生一个最优的热量杀死癌细胞,产生的热量是一个射频(RF)针。经验较少的介入放射科医师(IRs)需要software-assisted智能解决方案预测的最佳热量分布overkilling和未经处理的肿瘤细胞都有问题的治疗方法。因此,两个三大偏微分方程,1)热方程(人们应用生理学杂志》,1948年,1,93 - 122年)来预测热分布和2)拉普拉斯方程(普拉卡什,打开生物医学。Eng。J。,20.10, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses<e米>问</e米><sub><em>r</e米></sub>=<e米>σ</e米>|∇<e米>V</e米>|<年代up>2</年代up>从拉普拉斯方程获得吗<e米>σ</e米>Δ<e米>V</e米>= 0,它叫做焦耳热模型。传统的焦耳热模型可以取代一个点源模型获得热源项。这个模型背后的想法是解决<e米>σ</e米>Δ<e米>V</e米>=<e米>δ</e米><sub>0</年代ub>在哪里<e米>δ</e米><sub>0</年代ub>是一个狄拉克δ函数。因此,使用拉普拉斯方程的基本解(埃文斯、偏微分方程、2010)我们代表焦耳热模型使用一个替代的解决方案模型称为点源模型的高斯分布。</p> <div class="equationImageholder"> <math id="equ1"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> x</米我> </mrow> </mfenced> <mo> =</米o> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <mo> ∈</米o> <mi mathvariant="normal"> Ω</米我> </mrow> </munder> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mi> K</米我> </mrow> </mfrac> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mi> 我</米我> </mrow> </munder> <msub> <mrow> <mi> c</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mo> −</米o> <mfrac> <mrow> <mo stretchy="false"> |</米o> <mi> x</米我> <mo> −</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> <mrow> <mn> 2</米n> <msup> <mrow> <mi> σ</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </mfrac> </mrow> </msup> </math> <div class="clear"></div> </div> <p class="mb15">在哪里<e米>K</e米>和<e米>c</e米><sub><em>我</e米></sub>通过使用指针参数。这个模型是用于我们的一个软件解决方案称为RFA监护人(Sci Voglreiter et al。代表,2018年,787),预计超过100名患者的治疗效果很好。</p> <div class="clear"></div> </div> <div class="JournalFullText"> <a id="h2" name="h2"></a> <h2>介绍</h2> <p class="mb15">射频消融(RFA)已经被用于焦主要和次要的治疗肝脏恶性肿瘤微创,影像替代标准手术切除(<一个href="#B6">Berjano 2006</一个>;<一个href="#B7">Blachier et al ., 2013</一个>)。计算模拟被用来模拟电子在RFA热过程和预测的结果消融治疗(<一个href="#B23">佩恩et al ., 2010</一个>;<一个href="#B27">舒曼et al ., 2010</一个>)。在射频消融(300使用的频率<e米>千赫</e米>−1<e米>兆赫</e米>)和在感兴趣的领域(电力沉积小范围内活动电极),可以将组织视为一个纯粹的电阻中,位移电流可以忽略不计。使用静态方法,分布式热源<e米>问</e米>(焦耳损失)是由(<一个href="#B10">小客栈,1982</一个>)</p> <div class="equationImageholder"> <math id="e1"> <mi> 问</米我> <mo> =</米o> <mrow> <mover accent="true"> <mrow> <mi> J</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> 。</米o> <mrow> <mover accent="true"> <mrow> <mi> E</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> =</米o> <mi> σ</米我> <mo stretchy="false"> |</米o> <mo> ∇</米o> <mi> V</米我> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 1</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf1"> <math id="m3"> <mrow> <mover accent="true"> <mrow> <mi> J</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> ,</米o> <mrow> <mover accent="true"> <mrow> <mi> E</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> ,</米o> <mi> σ</米我> </math> </inline-formula>和<e米>V</e米>分别表示电流密度、电场强度、导电性(<e米>年代</e米>/<e米>米</e米>)和电压。在文献中,电压是假定为常数表面的射频电极探针针和产生的势场探针使用拉普拉斯方程数值求解(<一个href="#B25">普拉卡什,2010</一个>)</p> <div class="equationImageholder"> <math id="e2"> <mi> σ</米我> <mi mathvariant="normal"> Δ</米我> <mi> V</米我> <mo> =</米o> <mn> 0</米n> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 2</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">在这里<e米>σ</e米>被认为是空间齐次(<一个href="#B22">Panescu et al ., 1995</一个>;<一个href="#B31">Tungjiktkusolmun et al ., 2002</一个>;<一个href="#B1">艾哈迈德et al ., 2008</一个>)。这个模型已经应用在非常薄的调查技巧。因此,在数值模拟的准确性,需要更精细的网格依次增加计算成本。解决<一个href="#e2">方程2</一个>一起bioheat方程(<一个href="#B24">人们1948</一个>),细胞死亡模型(<一个href="#B21">奥尼尔et al ., 2011</一个>)和适当的初始和边界条件为不可行的预测时间。此外,整个电极探头的精确位置并不总是可以在CT扫描图像。相反,计算焦耳热损失非常敏感探测器位置(<一个href="#B17">Khlebnikov Muehl, 2010</一个>)。另外,如果焦耳热模型的解析解是已知的,那么可以减少计算成本。由于焦耳热的解析解模型对肝脏领域并不容易获得,整个电极几何形状的独立组织,估计焦耳热在射频消融是更好的选择,以减少计算成本。</p> <p class="mb15">在这项研究中,提出了另一种模型来代表RFA热沉积不需要计算焦耳热,因为组织附近的探测器接收重要的焦耳热而组织远离探测器接收到的热量可以忽略不计(<一个href="#B26">施拉姆et al ., 2007</一个>;<一个href="#B16">Haemmerich 2010</一个>)。拟议中的高斯分布点源模型使用几个像素点包含电极探针代表RFA功率沉积。这部小说对传统模型进行了验证计算焦耳热。此外,该模型采用回顾性临床数据进行验证。</p> <a id="h3" name="h3"></a> <h2>Bioheat方程模型和细胞死亡</h2> <p class="mb15">RFA治疗需要预测杀死少量的肿瘤细胞。bioheat方程预测肿瘤细胞周围的温度分布。执政bioheat方程(<一个href="#B24">人们1948</一个>)是由</p> <div class="equationImageholder"> <math id="e3"> <mi> ρ</米我> <mi> C</米我> <mfrac> <mrow> <mi> ∂</米我> <mi> T</米我> </mrow> <mrow> <mi> ∂</米我> <mi> t</米我> </mrow> </mfrac> <mo> =</米o> <mo> ∇</米o> <mo> 。</米o> <mi> k</米我> <mo> ∇</米o> <mi> T</米我> <mo> −</米o> <msub> <mrow> <mi> ρ</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <msub> <mrow> <mi> C</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <msub> <mrow> <mi> ω</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> T</米我> <mo> −</米o> <msub> <mrow> <mi> T</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfenced> <mo> +</米o> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 3</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>ρ</e米>和<e米>ρ</e米><sub><em>b</e米></sub>分别表示血液和组织的密度,<e米>C</e米>和<e米>C</e米><sub><em>b</e米></sub>分别表示热容的组织和血液,<e米>k</e米>热导率,<e米>ω</e米><sub><em>b</e米></sub>是血液灌注,<e米>T</e米><sub><em>一个</e米></sub>动脉温度和吗<e米>问</e米><sub><em>r</e米></sub>由射频功率源项生成焦耳热探针可计算的模型或估计的点源模型解释。边界和初始条件。</p> <div class="equationImageholder"> <math id="e4"> <mi> T</米我> <mo> =</米o> <mn> 37</米n> <mo> °</米o> <mi> C</米我> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 圆形/球形</米text> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 4</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e5"> <mfrac> <mrow> <mi> ∂</米我> <mi> T</米我> </mrow> <mrow> <mi> ∂</米我> <mi> n</米我> </mrow> </mfrac> <mo> =</米o> <mn> 0</米n> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 在探针表面</米text> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 5</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e6"> <mi> T</米我> <mfenced open="(" close=")"> <mrow> <mn> 0</米n> </mrow> </mfenced> <mo> =</米o> <mn> 37</米n> <mo> °</米o> <mi> C</米我> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 6</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb15">在解决上述方程,细胞周围的温度分布。基于细胞死亡的温度分布模型(<一个href="#B21">奥尼尔et al ., 2011</一个>)预测,活着和死细胞。</p> <div class="equationImageholder"> <math id="e7"> <mfrac> <mrow> <mi> d</米我> <mi> 一个</米我> </mrow> <mrow> <mi> d</米我> <mi> t</米我> </mrow> </mfrac> <mo> =</米o> <mo> −</米o> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mi> f</米我> </mrow> </msub> <mi> 一个</米我> <mo> +</米o> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mn> 1</米n> <mo> −</米o> <mi> 一个</米我> <mo> −</米o> <mi> D</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 7</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e8"> <mfrac> <mrow> <mi> d</米我> <mi> D</米我> </mrow> <mrow> <mi> d</米我> <mi> t</米我> </mrow> </mfrac> <mo> =</米o> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mi> f</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mn> 1</米n> <mo> −</米o> <mi> 一个</米我> <mo> −</米o> <mi> D</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 8</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e9"> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mi> f</米我> </mrow> </msub> <mo> =</米o> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <msub> <mrow> <mi> f</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> </mrow> </msub> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mfrac> <mrow> <mi> T</米我> </mrow> <mrow> <msub> <mrow> <mi> T</米我> </mrow> <mrow> <mi> k</米我> </mrow> </msub> </mrow> </mfrac> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <mn> 1</米n> <mo> −</米o> <mi> 一个</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 9</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">在这里<e米>一个</e米>活细胞的一部分,<e米>D</e米>的分数是死细胞,<e米>k</e米><sub><em>f</e米></sub>和<e米>k</e米><sub><em>b</e米></sub>表示向前和向后常数描述变化的速度从活细胞和死细胞<e米>反之亦然</e米>。<我nline-formula id="inf2"> <math id="m12"> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <msub> <mrow> <mi> f</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> </mrow> </msub> </math> </inline-formula>比例因子和吗<e米>T</e米><sub><em>k</e米></sub>表示指数随着温度增加的速度。</p> <p class="mb15">除了获取源项的焦耳热模型或点源模型,bioheat模型使用有限元方法数值求解细孔网域,而细胞死亡模型解决了使用传统的有限差分法。</p> <p class="mb15">提供理由点源模型的焦耳热模型的源项,针几何解释。在下一节中解释了针几何然后紧接着焦耳热模型是解释说。</p> <a id="h4" name="h4"></a> <h2>针几何</h2> <p class="mb15">multi-array针(<一个href="#F1">图1</一个>)有不同数量的针<一个href="#B9">AngioDynamics (2022 b</一个>,<一个href="#B8">一)</一个>从3 - 9所示。在这项研究中,我们使用了九个阵列+一个活跃的套管针小费<一个href="#B8">AngioDynamics (2022)</一个>。在治疗期间,机关里的针插入加热,用于肿瘤的中心或肿瘤附近的一个点。这个点被称为目标点或交点。对于数组中的每个针,目标点,另一个端点和这两个之间的中点作为点源模型。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图1</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g001.jpg" name="Figure1" target="_blank"><img id="F1" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g001.jpg"></a> <p><b>图1</b>。RFA针几何:丽塔的亮光XL针。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <a id="h5" name="h5"></a> <h2>焦耳热模型</h2> <p class="mb15">为了验证我们的点源模型,有必要比较电势,获得的热量源项和温度的焦耳热模型和点源模型。我们已经测试了我们的模型在二维和三维模型。为了生成一个适当的点源模型,让我们先模拟焦耳热模型。焦耳热模型分别模拟了单位圆和单位球面上的几何2 d和3 d。针放置在几何中心。对于二维模型,我们有圆盘上的有限元网格生成三个数组指针放置在盘的中心所示<一个href="#F2">图2</一个>。三维模型,有限元网格球面上的几何中心与九针阵列生成球面所示<一个href="#F3">图3</一个>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图2</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g002.jpg" name="Figure2" target="_blank"><img id="F2" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g002.jpg"></a> <p><b>图2</b>。二维几何针为例(2)(左)和自适应有限元网格中使用模拟。网格生成使用Gmsh (<一个href="#B13">Geuzaine Remacle, 2020</一个>)(右)。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图3</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g003.jpg" name="Figure3" target="_blank"><img id="F3" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g003.jpg"></a> <p><b>图3</b>。3 d几何和自适应有限元网格中使用模拟。网格生成使用Gmsh (<一个href="#B13">Geuzaine Remacle, 2020</一个>)。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15">焦耳热模型的控制方程给出</p> <div class="equationImageholder"> <math id="e10"> <mi> σ</米我> <mi mathvariant="normal"> Δ</米我> <mi> V</米我> <mo> =</米o> <mn> 0</米n> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 10</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">电势的边界条件如下。</p> <div class="equationImageholder"> <math id="e11"> <mi> V</米我> <mo> =</米o> <msup> <mrow> <mi> 一个</米我> </mrow> <mrow> <mo> +</米o> </mrow> </msup> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 表面上的技巧</米text> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 11</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e12"> <mi> V</米我> <mo> =</米o> <msup> <mrow> <mi> 一个</米我> </mrow> <mrow> <mo> −</米o> </mrow> </msup> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 圆面</米text> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 12</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e13"> <mfrac> <mrow> <mi> ∂</米我> <mi> V</米我> </mrow> <mrow> <mi> ∂</米我> <mi> n</米我> </mrow> </mfrac> <mo> =</米o> <mn> 0</米n> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 轴表面上</米text> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 13</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb15">在解决这些方程<e米>V</e米>,我们可以获得的源项bioheat模型如下:</p> <div class="equationImageholder"> <math id="e14"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mo> =</米o> <mi> σ</米我> <mo stretchy="false"> |</米o> <mo> ∇</米o> <mi> V</米我> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 14</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>σ</e米>导电性。</p> <a id="h6" name="h6"></a> <h2>分析推导</h2> <p class="mb15">几何、电气和热性能发挥重要作用的能量沉积分布的治疗。不同的制造商产生不同的探针对射频消融术。这里的研究是基于丽塔的亮光XL multi-array针(<一个href="#F1">图1</一个>)和丽塔1500 x丽塔生成器。然而,这里的分析结果也适用于其他类型的消融探针与适当的变化。</p> <h3 class="pt0">矩量法</h3> <p class="mb0">首先分析解决有限长度的导线上的电荷分布,一个简化的电探针的几何。库仑排斥将解除对结束,就像实心导体上的电荷流到表面。然而,显式解析解电荷分布在科学界是一个不适定问题(<一个href="#B15">格里菲思和李,1996年</一个>)。</p> <p class="mb0">在目前的研究中,我们调查针模仿薄,导线的长度<e米>l</e米>和半径<e米>r</e米>面向沿着<e米>x</e米>轴所示<一个href="#F4">图4</一个>。因为线的半径很小,导线的电(库仑)潜力可以表示使用下面的积分</p> <div class="equationImageholder"> <math id="e15"> <msub> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> </mfenced> <mo> =</米o> <munderover accentunder="false" accent="false"> <mrow> <mo> ∫</米o> </mrow> <mrow> <mn> 0</米n> </mrow> <mrow> <mi> l</米我> </mrow> </munderover> <mfrac> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> <mi> d</米我> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> <mrow> <mn> 4</米n> <mi> π</米我> <mi> ε</米我> <mo stretchy="false"> |</米o> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> −</米o> <mrow> <mover accent="true"> <mrow> <msup> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo stretchy="false"> |</米o> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 15</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>问</e米><sub><em>e</e米></sub>电荷密度和吗<e米>ɛ</e米>电常数。导线分为<e米>N</e米>段<我nline-formula id="inf5"> <math id="m21"> <msubsup> <mrow> <mrow> <mo stretchy="false"> {</米o> <mrow> <msub> <mrow> <mi> 我</米我> </mrow> <mrow> <mi> j</米我> </mrow> </msub> <mo> =</米o> <mrow> <mo stretchy="false"> (</米o> <mrow> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> j</米我> <mo> −</米o> <mn> 1</米n> </mrow> </msub> <mo> ,</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> j</米我> </mrow> </msub> </mrow> <mo stretchy="false"> ]</米o> </mrow> </mrow> <mo stretchy="false"> }</米o> </mrow> </mrow> <mrow> <mi> j</米我> <mo> =</米o> <mn> 1</米n> </mrow> <mrow> <mi> N</米我> </mrow> </msubsup> </math> </inline-formula>每个长度Δ<e米>x</e米>每个亚节的电荷密度是假定有一个值<e米>问</e米><sub><em>j</e米></sub>(<e米>j</e米>= 1,2,…<e米>N</e米>)。因此,<一个href="#e15">方程15</一个>就变成了</p> <div class="equationImageholder"> <math id="e16"> <msub> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <mn> 2</米n> <mi> π</米我> <mi> r</米我> </mrow> <mrow> <mn> 4</米n> <mi> π</米我> <mi> ε</米我> </mrow> </mfrac> <munderover accentunder="false" accent="false"> <mrow> <mo> ∑</米o> </mrow> <mrow> <mi> j</米我> <mo> =</米o> <mn> 1</米n> </mrow> <mrow> <mi> N</米我> </mrow> </munderover> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> j</米我> </mrow> </msub> <munder> <mrow> <mo> ∫</米o> </mrow> <mrow> <msub> <mrow> <mi> 我</米我> </mrow> <mrow> <mi> j</米我> </mrow> </msub> </mrow> </munder> <mfrac> <mrow> <mi> d</米我> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> <mrow> <mo stretchy="false"> |</米o> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> −</米o> <mrow> <mover accent="true"> <mrow> <msup> <mrow> <mi mathvariant="bold"> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo stretchy="false"> |</米o> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 16</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent"></p> <div class="DottedLine"></div> <div class="Imageheaders"> 图4</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g004.jpg" name="Figure4" target="_blank"><img id="F4" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g004.jpg"></a> <p><b>图4</b>。细的导线。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">让我们解决源点连接轴和观测点钢丝表面的(<一个href="#B14">吉布森2008年</一个>),确保没有被积函数的奇点。被积函数的分母</p> <div class="equationImageholder"> <math id="e17"> <mo stretchy="false"> |</米o> <mrow> <mover accent="true"> <mrow> <mi> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo> −</米o> <mrow> <mover accent="true"> <mrow> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> <mo> ⃗</米o> </mover> </mrow> <mo stretchy="false"> |</米o> <mo> =</米o> <msqrt> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </msqrt> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 17</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">让我们假设均匀电荷密度除了端点(<一个href="#B5">1997年安德鲁斯</一个>)。因此,我们分配的电荷密度</p> <div class="equationImageholder"> <math id="equ2"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> j</米我> </mrow> </msub> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mspace width="1em"></mspace> <mi> j</米我> <mo> =</米o> <mn> 2、3</米n> <mo> ,</米o> <mo> 。</米o> <mo> 。</米o> <mo> ,</米o> <mi> N</米我> <mo> −</米o> <mn> 1</米n> </mtd> </mtr> <mtr> <mtd columnalign="left"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mspace width="1em"></mspace> <mi> j</米我> <mo> =</米o> <mn> 1</米n> <mo> ,</米o> <mi> N</米我> </mtd> </mtr> </mtable> </mrow> </mfenced> </math> <div class="clear"></div> </div> <p class="indent">替换<e米>问</e米><sub><em>j</e米></sub>价值观和<一个href="#e17">方程17</一个>成<一个href="#e16">方程16</一个>,我们获得</p> <div class="equationImageholder"> <math id="e18"> <msub> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mover accent="true"> <mrow> <mi> x</米我> </mrow> <mo> ⃗</米o> </mover> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mn> 2</米n> <mi> ε</米我> </mrow> </mfrac> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> <munderover accentunder="false" accent="false"> <mrow> <mo> ∫</米o> </mrow> <mrow> <mi mathvariant="normal"> Δ</米我> <mi> x</米我> </mrow> <mrow> <mfenced open="(" close=")"> <mrow> <mi> N</米我> <mo> −</米o> <mn> 1</米n> </mrow> </mfenced> <mi mathvariant="normal"> Δ</米我> <mi> x</米我> </mrow> </munderover> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> <mi> d</米我> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> <mo> +</米o> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <munder> <mrow> <mo> ∫</米o> </mrow> <mrow> <msub> <mrow> <mi> 我</米我> </mrow> <mrow> <mn> 1</米n> </mrow> </msub> <mo> ∪</米o> <msub> <mrow> <mi> 我</米我> </mrow> <mrow> <mi> N</米我> </mrow> </msub> </mrow> </munder> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> <mi> d</米我> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 18</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">很容易验证</p> <div class="equationImageholder"> <math id="e19"> <mo> ∫</米o> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> <mi> d</米我> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> <mo> =</米o> <mi> 日志</米我> <mfenced open="(" close=")"> <mrow> <msqrt> <mrow> <msup> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </msqrt> <mo> +</米o> <msup> <mrow> <mi> x</米我> </mrow> <mrow> <mo> ′</米o> </mrow> </msup> <mo> −</米o> <mi> x</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 19</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb0">根据以下假设:<我nline-formula id="inf6"> <math id="m27"> <mn> 4</米n> <mo> ≪</米o> <mi> N</米我> <mo> ≪</米o> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> </math> </inline-formula>和<我nline-formula id="inf7"> <math id="m28"> <mfrac> <mrow> <mi> r</米我> </mrow> <mrow> <mi mathvariant="normal"> Δ</米我> <mi> x</米我> </mrow> </mfrac> <mo> ≪</米o> <mn> 1</米n> </math> </inline-formula>,我们评估的价值<e米>ϕ</e米><sub><em>e</e米></sub>在<e米>x</e米>=<e米>x</e米><sub>0</年代ub>= 0和<我nline-formula id="inf8"> <math id="m29"> <mi> x</米我> <mo> =</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mfrac> <mrow> <mi> N</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </mfrac> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </mfrac> </math> </inline-formula></p> <div class="equationImageholder"> <math id="e20"> <msub> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mn> 0</米n> </mrow> </mfenced> <mo> ≈</米o> <mfrac> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> <mi> ε</米我> </mrow> </mfrac> <mfenced open="[" close="]"> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mo> ⁡</米o> <mi> 日志</米我> <mfrac> <mrow> <mn> 2</米n> <mi> l</米我> </mrow> <mrow> <mi> N</米我> <mi> r</米我> </mrow> </mfrac> <mo> +</米o> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> <mo> ⁡</米o> <mi> 日志</米我> <mo> ⁡</米o> <mi> N</米我> <mo> +</米o> <mfrac> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <mi> N</米我> </mrow> </mfrac> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 20.</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e21"> <msub> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mi> e</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </mfrac> </mrow> </mfenced> <mo> ≈</米o> <mfrac> <mrow> <mi> r</米我> </mrow> <mrow> <mn> 2</米n> <mi> ε</米我> </mrow> </mfrac> <mfenced open="[" close="]"> <mrow> <mfrac> <mrow> <mn> 2</米n> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <mi> N</米我> </mrow> </mfrac> <mo> +</米o> <mn> 2</米n> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> <mo> ⁡</米o> <mi> 日志</米我> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> +</米o> <mfrac> <mrow> <mn> 2</米n> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <mi> N</米我> </mrow> </mfrac> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 21</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb0">让<e米>γ</e米><sub><em>问</e米></sub>表示电荷密度的比值在导体的两端(<e米>问</e米><sub><em>b</e米></sub>),在剩下的导体(<e米>问</e米><sub><em>一个</e米></sub>)。由于导体的电势是常数,使用<e米>ϕ</e米><sub><em>e</e米></sub>(<e米>x</e米>)=<e米>V</e米>,0≤<e米>x</e米>≤<e米>l</e米>在<一个href="#e20">(20)</一个>和<一个href="#e21">21</一个>,我们获得</p> <div class="equationImageholder"> <math id="e22"> <msub> <mrow> <mi> γ</米我> </mrow> <mrow> <mi> 问</米我> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfrac> <mo> =</米o> <mfrac> <mrow> <mn> 2</米n> <mo> ⁡</米o> <mi> 日志</米我> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> −</米o> <mi> 日志</米我> <mo> ⁡</米o> <mi> N</米我> </mrow> <mrow> <mi> 日志</米我> <mfrac> <mrow> <mn> 2</米n> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> −</米o> <mi> 日志</米我> <mo> ⁡</米o> <mi> N</米我> <mo> −</米o> <mfrac> <mrow> <mn> 3</米n> </mrow> <mrow> <mi> N</米我> </mrow> </mfrac> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 22</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">让<e米>γ</e米><sub><em>P</e米></sub>表示功率沉积的比例在导体的两端(<e米>P</e米><sub><em>b</e米></sub>),在剩下的导体(<e米>P</e米><sub><em>一个</e米></sub>)。由于功率沉积(<e米>P</e米>)成正比的电场,电场(<e米>E</e米>)电荷密度成正比(<e米>问</e米>),使用<e米>P</e米>∝<e米>E</e米><sup>2</年代up>和<e米>E</e米>∝<e米>问</e米>在<一个href="#e22">(22)</一个>收益率</p> <div class="equationImageholder"> <math id="e23"> <msub> <mrow> <mi> γ</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfrac> <mo> =</米o> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mn> 2</米n> <mo> ⁡</米o> <mi> 日志</米我> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> −</米o> <mi> 日志</米我> <mo> ⁡</米o> <mi> N</米我> </mrow> <mrow> <mi> 日志</米我> <mfrac> <mrow> <mn> 2</米n> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> −</米o> <mi> 日志</米我> <mo> ⁡</米o> <mi> N</米我> <mo> −</米o> <mfrac> <mrow> <mn> 3</米n> </mrow> <mrow> <mi> N</米我> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 23</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb0">丽塔的亮光针有14个指标(<我nline-formula id="inf9"> <math id="m34"> <mo> ≈</米o> <mn> 1.6</米n> </math> </inline-formula>毫米)外径和长度10厘米。然而,制造商<一个href="#B9">AngioDynamics (2022 b)</一个>供应各种类型的针的长度从10到25厘米不等。在这项研究中,我们考虑<e米>l</e米>= 10厘米<e米>r</e米>= 0.8毫米。根据我们的假设,<e米>N</e米>必须满足<我nline-formula id="inf10"> <math id="m35"> <mn> 4</米n> <mo> ≪</米o> <mi> N</米我> <mo> ≪</米o> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> =</米o> <mn> 125年</米n> </math> </inline-formula>。的功率比<e米>γ</e米><sub><em>P</e米></sub>当点的数量增加<e米>N</e米>增加了<一个href="#F5">图5</一个>。在附近的电荷变化的针更准确地表示为段的数量增加。因此,功率沉积的探针可以表示为<e米>N</e米>沿着针点光源,同时两个点光源位于针的两端提供更高的权力比其他<e米>N</e米>−2点光源。功率比由针结束和针不能获得直接从中间<e米>γ</e米>(<e米>P</e米>)假设<e米>N</e米>= 3,<e米>N</e米>= 3不满足不平等<我nline-formula id="inf11"> <math id="m36"> <mn> 4</米n> <mo> ≪</米o> <mi> N</米我> <mo> ≪</米o> <mfrac> <mrow> <mi> l</米我> </mrow> <mrow> <mi> r</米我> </mrow> </mfrac> <mo> =</米o> <mn> 125年</米n> </math> </inline-formula>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图5</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g005.jpg" name="Figure5" target="_blank"><img id="F5" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g005.jpg"></a> <p><b>图5</b>。比功率沉积<我nline-formula id="inf12"> <math id="m39"> <msub> <mrow> <mi> γ</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfrac> </math> </inline-formula>。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">因此,当已知点的数量(<e米>k</e米>)是有限的,<e米>k</e米><<e米>N</e米>可以模仿,那么功率比为:</p> <div class="equationImageholder"> <math id="e24"> <msub> <mrow> <mi> η</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> k</米我> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 结束</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 中期</米我> </mrow> </msub> </mrow> </mfrac> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mo> +</米o> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> N</米我> </mrow> <mrow> <mi> k</米我> </mrow> </mfrac> <mo> −</米o> <mn> 1</米n> </mrow> </mfenced> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> <mrow> <mfrac> <mrow> <mi> N</米我> </mrow> <mrow> <mi> k</米我> </mrow> </mfrac> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 24</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">注意,当<e米>k</e米>=<e米>N</e米>,<e米>η</e米><sub><em>P</e米></sub>=<e米>γ</e米><sub><em>P</e米></sub>。</p> <p class="mb0">在二维的研究中,我们考虑三针,把它们分成以下三种情况:1)点光源的数量是四个,三个最后的提示和一个十字路口的三针,2)点光源的数量是7个,三个位于两端的技巧和3位于中间的技巧和一个十字路口的三针,<e米>k</e米>= 3,3)点光源的数量是13日三个位于结束的提示和其余6位于同样放置结束提示和十字路口之间的技巧,<e米>k</e米>= 5。<一个href="#F2">图2</一个>显示了二维几何和针的技巧(2),<e米>k</e米>= 3。</p> <p class="mb0">例(2),<e米>k</e米>= 3和功率比的中点是由端点</p> <div class="equationImageholder"> <math id="e25"> <msub> <mrow> <mi> η</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mo> =</米o> <msub> <mrow> <mi> η</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mn> 3</米n> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 结束</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 中期</米我> </mrow> </msub> </mrow> </mfrac> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mo> +</米o> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> N</米我> </mrow> <mrow> <mn> 3</米n> </mrow> </mfrac> <mo> −</米o> <mn> 1</米n> </mrow> </mfenced> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> <mrow> <mfrac> <mrow> <mi> N</米我> </mrow> <mrow> <mn> 3</米n> </mrow> </mfrac> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi> 一个</米我> </mrow> </msub> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 25</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb0"><a href="#F6">图6</一个>情节的比率<e米>η</e米><sub><em>P</e米></sub>针对的范围<e米>N</e米>。的值<e米>η</e米><sub><em>P</e米></sub>衰变为<e米>N</e米>增加最后收敛于大约1.76,即功率比我们提出点源模型。注意,功率比的变化当针的长度和直径变化。例如,当<e米>r</e米>= 0.25<e米>毫米</e米>和<e米>l</e米>= 5<e米>厘米</e米>,功率比<e米>η</e米><sub><em>P</e米></sub>= 1.5中给出<一个href="#F6">图6</一个>。同样的,当<e米>k</e米>变化,给定的功率比的变化<一个href="#F7">图7</一个>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图6</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g006.jpg" name="Figure6" target="_blank"><img id="F6" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g006.jpg"></a> <p><b>图6</b>。功率比<我nline-formula id="inf3"> <math id="m13"> <msub> <mrow> <mi> η</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 结束</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 中期</米我> </mrow> </msub> </mrow> </mfrac> </math> </inline-formula>。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图7</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g007.jpg" name="Figure7" target="_blank"><img id="F7" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g007.jpg"></a> <p><b>图7</b>。功率比<我nline-formula id="inf4"> <math id="m14"> <msub> <mrow> <mi> η</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <mo> =</米o> <mfrac> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 结束</米我> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi> P</米我> </mrow> <mrow> <mi mathvariant="italic"> 中期</米我> </mrow> </msub> </mrow> </mfrac> </math> </inline-formula>为不同的<e米>k</e米>和<e米>r</e米>= 0.25<e米>毫米</e米>,<e米>l</e米>= 5<e米>厘米</e米>。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <a id="h7" name="h7"></a> <h2>点源模型</h2> <p class="mb15">拉普拉斯方程旋转不变(<一个href="#B11">埃文斯,2010</一个>),因此径向的解决方案是,后来发现以下基本解拉普拉斯方程的推导</p> <div class="equationImageholder"> <math id="e26"> <mrow> <mi> ϕ</米我> </mrow> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> </mrow> </mfenced> </mrow> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mo> −</米o> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mn> 2</米n> <mi> π</米我> </mrow> </mfrac> <mi> 日志</米我> <mo stretchy="false"> |</米o> <mi> x</米我> <mo stretchy="false"> |</米o> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mi> n</米我> <mo> =</米o> <mn> 2</米n> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mi> n</米我> <mfenced open="(" close=")"> <mrow> <mi> n</米我> <mo> −</米o> <mn> 2</米n> </mrow> </mfenced> <mi> α</米我> <mfenced open="(" close=")"> <mrow> <mi> n</米我> </mrow> </mfenced> </mrow> </mfrac> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mo stretchy="false"> |</米o> <mi> x</米我> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mi> n</米我> <mo> −</米o> <mn> 2</米n> </mrow> </msup> </mrow> </mfrac> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mi> n</米我> <mo> ≥</米o> <mn> 3</米n> </mtd> </mtr> </mtable> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 26</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">定义为<我nline-formula id="inf13"> <math id="m41"> <mi> x</米我> <mo> ∈</米o> <msup> <mrow> <mi mathvariant="double-struck"> R</米我> </mrow> <mrow> <mi> n</米我> </mrow> </msup> <mo> ,</米o> <mi> x</米我> <mo> ≠</米o> <mn> 0</米n> </math> </inline-formula>,在那里<e米>α</e米>(<e米>n</e米>)表示单位球的体积<我nline-formula id="inf113"> <math id="m141"> <msup> <mrow> <mi mathvariant="double-struck"> R</米我> </mrow> <mrow> <mi> n</米我> </mrow> </msup> </math> </inline-formula>。然而,拉普拉斯方程有时写成</p> <div class="equationImageholder"> <math id="e27"> <mo> −</米o> <mi mathvariant="normal"> Δ</米我> <mi> V</米我> <mo> =</米o> <msub> <mrow> <mi> δ</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <mtext> 在</米text> <mspace width="0.3333em" class="nbsp"></mspace> <mspace width="0.3333em" class="nbsp"></mspace> <msup> <mrow> <mi mathvariant="double-struck"> R</米我> </mrow> <mrow> <mi> n</米我> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 27</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>δ</e米><sub>0</年代ub>狄拉克表示测量<我nline-formula id="inf14"> <math id="m43"> <msup> <mrow> <mi mathvariant="double-struck"> R</米我> </mrow> <mrow> <mi> n</米我> </mrow> </msup> </math> </inline-formula>。也就是说,<e米>δ</e米><sub>0</年代ub>提供了一个单位质量的0。点源模型开发、单位质量或期望的质量是针的尖端。解决上述方程安慰者定义如下(<一个href="#B11">埃文斯,2010</一个>):</p> <div class="equationImageholder"> <math id="e28"> <mi> η</米我> <mfenced open="(" close=")"> <mrow> <mi> x</米我> </mrow> </mfenced> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mi> C</米我> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mo stretchy="false"> |</米o> <mi> x</米我> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> −</米o> <mn> 1</米n> </mrow> </mfrac> </mrow> </msup> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mi> 我</米我> <mi> f</米我> <mo stretchy="false"> |</米o> <mi> x</米我> <mo stretchy="false"> |</米o> <mo> <</米o> <mn> 1</米n> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn> 0</米n> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mi> 我</米我> <mi> f</米我> <mo stretchy="false"> |</米o> <mi> x</米我> <mo stretchy="false"> |</米o> <mo> ≥</米o> <mn> 1</米n> </mtd> </mtr> </mtable> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 28</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>C</e米>是一个常数选择这样<我nline-formula id="inf15"> <math id="m45"> <msub> <mrow> <mo> ∫</米o> </mrow> <mrow> <msup> <mrow> <mi mathvariant="double-struck"> R</米我> </mrow> <mrow> <mi> n</米我> </mrow> </msup> </mrow> </msub> <mi> η</米我> <mrow> <mo stretchy="false"> (</米o> <mrow> <mi> x</米我> </mrow> <mo stretchy="false"> )</米o> </mrow> <mi> d</米我> <mi> x</米我> <mo> =</米o> <mn> 1</米n> </math> </inline-formula>。</p> <p class="mb15">基于上述讨论,点源模型的高斯分布</p> <div class="equationImageholder"> <math id="e29"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> x</米我> </mrow> </mfenced> <mo> =</米o> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <mo> ∈</米o> <mi mathvariant="normal"> Ω</米我> </mrow> </munder> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mi> K</米我> </mrow> </mfrac> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mi> 我</米我> </mrow> </munder> <msub> <mrow> <mi> c</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mo> −</米o> <mfrac> <mrow> <mo stretchy="false"> |</米o> <mi> x</米我> <mo> −</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi> 我</米我> </mrow> </msub> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> <mrow> <mn> 2</米n> <msup> <mrow> <mi> σ</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> </mfrac> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 29日</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<e米>K</e米>和<e米>c</e米><sub><em>我</e米></sub>通过使用指针参数。这里Ω表示几何域,在我们的案例中圆形域或球形领域。在真正的病人的情况下,这是一个拓扑肝脏领域。由于分析推导,我们分类<e米>c</e米><sub><em>我</e米></sub>分为两类中产技巧(绿色)和结束提示所示(红色)<一个href="#F8">图8</一个>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图8</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g008.jpg" name="Figure8" target="_blank"><img id="F8" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g008.jpg"></a> <p><b>图8</b>。点选择在每个needle-array的中间和结束。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15">二维几何图形,是由点源模型。</p> <div class="equationImageholder"> <math id="e30"> <mi> P</米我> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mn> 9</米n> </mrow> </mfrac> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi> σ</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <msqrt> <mrow> <mn> 2</米n> <mi> π</米我> </mrow> </msqrt> </mrow> </mfenced> </mrow> <mrow> <mn> 3</米n> </mrow> </msup> </mrow> </mfrac> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </munder> <msub> <mrow> <mi> c</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mo> −</米o> <mfrac> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> y</米我> <mo> −</米o> <msub> <mrow> <mi> y</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> <mrow> <mn> 2</米n> <msubsup> <mrow> <mi> σ</米我> </mrow> <mrow> <mi> P</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msubsup> </mrow> </mfrac> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 30.</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e31"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> </mrow> </mfenced> <mo> =</米o> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> </mrow> </mfenced> <mo> ∈</米o> <mi mathvariant="normal"> Ω</米我> </mrow> </munder> <mi> P</米我> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 31日</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">的值<e米>c</e米><sub><em>提示</e米></sub>是获得<e米>γ</e米><sub><em>P</e米></sub>。此外,的值<e米>x</e米><sub><em>提示</e米></sub>和<e米>y</e米><sub><em>提示</e米></sub>获得使用样条方法如下:给出的参数曲线针</p> <div class="equationImageholder"> <math id="e32"> <mi> γ</米我> <mfenced open="(" close=")"> <mrow> <mi> t</米我> </mrow> </mfenced> <mo> =</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msub> <msup> <mrow> <mi> t</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 1</米n> </mrow> </msub> <mi> t</米我> <mo> +</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 32</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="mb15">在哪里</p> <div class="equationImageholder"> <math id="equ3"> <mi> t</米我> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mo> −</米o> <mfrac> <mrow> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 1</米n> </mrow> </msub> <mo stretchy="false"> |</米o> <mo> +</米o> <msqrt> <mrow> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 1</米n> </mrow> </msub> <msup> <mrow> <mo stretchy="false"> |</米o> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <mn> 4</米n> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msub> <mo stretchy="false"> |</米o> <mi> 年代</米我> </mrow> </msqrt> </mrow> <mrow> <mn> 2</米n> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msub> <mo stretchy="false"> |</米o> </mrow> </mfrac> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mtext> 如果</米text> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msub> <mo stretchy="false"> |</米o> <mo> ≠</米o> <mn> 0</米n> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo> −</米o> <mfrac> <mrow> <mi> 年代</米我> </mrow> <mrow> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 1</米n> </mrow> </msub> <mo stretchy="false"> |</米o> </mrow> </mfrac> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mtext> 如果</米text> <mo stretchy="false"> |</米o> <msub> <mrow> <mi> 一个</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msub> <mo stretchy="false"> |</米o> <mo> =</米o> <mn> 0</米n> </mtd> </mtr> </mtable> </mrow> </mfenced> </math> <div class="clear"></div> </div> <p class="indent">三维几何,是由点源模型。</p> <div class="equationImageholder"> <math id="e33"> <mi> P</米我> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> <mo> ,</米o> <mi> z</米我> </mrow> </mfenced> <mo> =</米o> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <mn> 24</米n> </mrow> </mfrac> <mfrac> <mrow> <mn> 1</米n> </mrow> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi> σ</米我> </mrow> <mrow> <mi> P</米我> </mrow> </msub> <msqrt> <mrow> <mn> 2</米n> <mi> π</米我> </mrow> </msqrt> </mrow> </mfenced> </mrow> <mrow> <mn> 3</米n> </mrow> </msup> </mrow> </mfrac> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </munder> <msub> <mrow> <mi> c</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> <msup> <mrow> <mi> e</米我> </mrow> <mrow> <mo> −</米o> <mfrac> <mrow> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> −</米o> <msub> <mrow> <mi> x</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> y</米我> <mo> −</米o> <msub> <mrow> <mi> y</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <msup> <mrow> <mfenced open="(" close=")"> <mrow> <mi> z</米我> <mo> −</米o> <msub> <mrow> <mi> z</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> </mrow> <mrow> <mn> 2</米n> <msubsup> <mrow> <mi> σ</米我> </mrow> <mrow> <mi> P</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msubsup> </mrow> </mfrac> </mrow> </msup> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 33</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e34"> <msub> <mrow> <mi> 问</米我> </mrow> <mrow> <mi> r</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> <mo> ,</米o> <mi> z</米我> </mrow> </mfenced> <mo> =</米o> <munder> <mrow> <mo> ∑</米o> </mrow> <mrow> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> <mo> ,</米o> <mi> z</米我> </mrow> </mfenced> <mo> ∈</米o> <mi mathvariant="normal"> Ω</米我> </mrow> </munder> <mi> P</米我> <mfenced open="(" close=")"> <mrow> <mi> x</米我> <mo> ,</米o> <mi> y</米我> <mo> ,</米o> <mi> z</米我> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 34</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">在这两种情况下,<e米>σ</e米><sub><em>P</e米></sub>表示正态分布的标准偏差<e米>σ</e米><sub><em>P</e米></sub>= 0.003,</p> <div class="equationImageholder"> <math id="e35"> <msub> <mrow> <mi> c</米我> </mrow> <mrow> <mi mathvariant="italic"> 提示</米我> </mrow> </msub> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mn> 1.5</米n> <mo> ∗</米o> <mtext> 权力</米text> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mtext> 对于终端提示</米text> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn> 1.0</米n> <mo> ∗</米o> <mtext> 权力</米text> <mspace width="1em"></mspace> </mtd> <mtd columnalign="left"> <mtext> 中间的技巧</米text> </mtd> </mtr> </mtable> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 35</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">在这里,<e米>k</e米>= 3。为<e米>k</e米>= 5,<e米>c</e米><sub><em>提示</e米></sub>值是不同的。</p> <a id="h8" name="h8"></a> <h2>数值评估点源模型和焦耳热模型的二维简化模型</h2> <p class="mb15">焦耳热模型是解决使用FEniCS (Dolfin 2019.2.0) (<一个href="#B2">Alnæs et al ., 2009</一个>),一个有限元软件工具,获得电势和源项<e米>问</e米><sub><em>r</e米></sub>对于一个二维圆几何半径为3厘米。圆的中心是位于needle-trocar交点。圆是离散使用Gmsh (<一个href="#B13">Geuzaine Remacle, 2020</一个>)。58965个节点的离散网格包含117650个三角形元素。</p> <p class="mb15">为了实现这两个不同的方法之间的直接比较,总功率的点源模型和焦耳热模型设置为150 w在这两种情况下。这相当于设置参数<e米>一个</e米><sup>±</年代up>=±22.75<e米>V</e米>在边界条件的电势焦耳热模型。点源模型的输入功率是150<e米>W</e米>。为<e米>k</e米>= 3,在中间提示输入功率值是16.6<e米>W</e米>最后的力量值提示是25<e米>W</e米>。所有其他参数的值用于bioheat方程给出<一个href="#T1">表1</一个>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 表1</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-t001.jpg" name="Table1" target="_blank"><img id="T1" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-t001.jpg"></a> <p><b>表1</b>。模型参数。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15"><a href="#F9">图9</一个>显示了ParaView电势和热源分布的可视化,更多的热量分布针的尖端附近,减少随着尖端的距离增加。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图9</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g009.jpg" name="Figure9" target="_blank"><img id="F9" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g009.jpg"></a> <p><b>图9</b>。电势(左)和热源项<e米>问</e米><sub><em>r</e米></sub>(右)的焦耳热模型。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15">点源模型,使用简单的python代码相同的网格,计算值并使用ParaView绘制。因为点源模型符合高斯分布,分布的性质是显而易见的<一个href="#F10">图10</一个>。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图10</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g010.jpg" name="Figure10" target="_blank"><img id="F10" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g010.jpg"></a> <p><b>图10</b>。热源项<e米>问</e米><sub><em>r</e米></sub>从点源模型3(左)和结束点<e米>k</e米>= 3(右)。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15">的<e米>问</e米><sub><em>r</e米></sub>值计算使用焦耳热模型和点源模型分别是7.13和4.5年代为每个迭代。当功率值变化取决于温度,计算的<e米>问</e米><sub><em>r</e米></sub>每一步是必要的。因此,对于600年代模拟0.1年代的时间步,需要11.8 h焦耳热模型而点源模型7.5 h<e米>问</e米><sub><em>r</e米></sub>计算。除了<e米>问</e米><sub><em>r</e米></sub>计算,我们必须解决bioheat方程和细胞死亡模型在每个时间步。由于矩阵组装和离散化焦耳热模型类似,对于每一个时间步,需要7.13 bioheat方程和5.34年代的细胞死亡模式。当所有参数的值是常数,然后FEniCS花费更少的时间来计算整个仿真。需要大约1.5 h为600年代消融协议由于线性动力学。然而,实际应用中的电源输入不是常数,因此,仿真所需的时间在使用焦耳热模型是大约13.3 h,而点源模型需要9 h。借助图形处理单元(GPU)并行化(<一个href="#B19">Mariappan et al ., 2017</一个>;<一个href="#B32">Voglreiter et al ., 2018</一个>),它可以在几分钟内计算。</p> <p class="mb15">通过求解bioheat方程获得的热量分布<e米>问</e米><sub><em>r</e米></sub>任期从焦耳热模型和点源模型<一个href="#F11">图11</一个>。因为点源模型是一个近似模型的焦耳热模型,大约3°<e米>C</e米>学位的区别是获得细网格时可以忽略不计。进一步,根据已发现的温度分布和模型之间的病变。点源模型的预测损伤区(293.7毫米<年代up>2</年代up>)是95%的焦耳加热模型(308.51毫米<年代up>2</年代up>100年代)模拟时<e米>k</e米>= 3。当<e米>k</e米>= 5,预测病变区体积减少到217.69<e米>毫米</e米><sup>2</年代up>,这是焦耳热模型体积的70%,因此,得出<e米>k</e米>= 3提供了最佳的病灶大小。<一个href="#F12">数字12</一个>,<一个href="#F13">13</一个>坏死细胞提供了视觉对比。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图11</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g011.jpg" name="Figure11" target="_blank"><img id="F11" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g011.jpg"></a> <p><b>图11</b>。通过使用bioheat温度分布方程的源项焦耳热(左)和点源模型<e米>k</e米>100年代= 3(右)。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图12</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g012.jpg" name="Figure12" target="_blank"><img id="F12" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g012.jpg"></a> <p><b>图12</b>。死细胞从温度分布,获得bioheat方程实现的源项的焦耳热。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图13</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g013.jpg" name="Figure13" target="_blank"><img id="F13" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g013.jpg"></a> <p><b>图13</b>。死细胞从温度分布,获得bioheat方程的源项实现从源模型<e米>k</e米>= 3(中间)和<e米>k</e米>100年代= 5(右)。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <a id="h9" name="h9"></a> <h2>数值评估点源模型和焦耳热模型的3 d模型</h2> <p class="mb15">点源模型显示明显良好的协议与焦耳热模型。点源模型已经在我们的GPU加速的软件工具实现实时预测损伤。最初,在回顾数据集猪模型的结果进行了比较。例如,<一个href="#F14">图14</一个>显示了比较获得的死细胞的焦耳热(8891.1毫米<年代up>3</年代up>)模型和点源模型(7912.4毫米<年代up>3</年代up>),91%的准确率。模型参数是一样的二维几何,而有限元网格与237725点1486412三角形和四面体元素。解决已经GPU加速以来,与点源模型解决了600年代产生焦耳热的死细胞和解决模型花了1000年代产生死细胞。</p> <div class="DottedLine"></div> <div class="Imageheaders"> 图14</d我v> <div class="FigureDesc"> <a href="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g014.jpg" name="Figure14" target="_blank"><img id="F14" alt="www.雷竞技rebatfrontiersin.org" class="lazy" data-src="//www.thespel.com/files/Articles/982768/fther-02-982768-HTML/image_m/fther-02-982768-g014.jpg"></a> <p><b>图14</b>。死细胞,利用焦耳热模型(灰色)和点源模型(红色)bioheat热源项的方程。</p> </div> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb15">点源模型已经实现在我们的ClinicIMPPACT项目模拟温度变化在RFA和预测损伤区(<一个href="#B19">Mariappan et al ., 2017</一个>;<一个href="#B32">Voglreiter et al ., 2018</一个>)。进一步,真正的病变和模拟病变有良好的协议而使用点源模型在这个项目。相同的模型被用于临床试验,发现了一个很好的匹配。很好的定量之间已经达成协议模型仿真和实验测量,这表明点源模型可以用作替代计算焦耳热消融治疗计划。</p> <a id="h10" name="h10"></a> <h2>限制</h2> <p class="mb15">这项研究有一些局限性在模型假设。人们bioheat方程有一些限制(<一个href="#B20">Nakayama和Kuwahara, 2008</一个>;<一个href="#B4">Andreozzi et al ., 2020</一个>)如常数和均匀温度而不是三个温度方程三个媒体(肝脏、血管和肿瘤),其他模型也由研究人员使用。本研究只使用人们bioheat方程。由于本研究的目的是取代焦耳热模型与点源模型,可以使用相同的点源模型的其他修改人们bioheat方程。此外,血液动力学固耦合问题和组织接触力(<一个href="#B3">安娜et al ., 2018</一个>;<一个href="#B34">燕et al ., 2020</一个>;<一个href="#B12">方et al ., 2022</一个>)对热损伤大小是没有考虑到在这个研究。这两项研究可能主要影响由于变形大血管附近的温度分布。此外,给出了模型参数<一个href="#T1">表1</一个>在这个研究中被视为常数。一些模型参数如组织热导率(<一个href="#B30">特鲁希略Berjano, 2013</一个>;<一个href="#B33">吴et al ., 2016</一个>),组织的热容<一个href="#B35">杨et al . (2007</一个>,<一个href="#B36">2006)</一个>和血液灌注(<一个href="#B28">shutt Haemmerich, 2008</一个>;<一个href="#B29">辛格和Melnik, 2019</一个>),可以包含与温度有关的参数。</p> <div class="equationImageholder"> <math id="e36"> <mi> k</米我> <mfenced open="(" close=")"> <mrow> <mi> T</米我> </mrow> </mfenced> <mo> =</米o> <mrow> <mover accent="true"> <mrow> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> </mrow> <mo> ̃</米o> </mover> </mrow> <mo> +</米o> <mi mathvariant="normal"> Δ</米我> <mi> k</米我> <mfenced open="(" close=")"> <mrow> <mi> T</米我> <mo> −</米o> <msub> <mrow> <mi> T</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 36</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e37"> <mi> C</米我> <mo> =</米o> <msub> <mrow> <mi> C</米我> </mrow> <mrow> <mi> t</米我> </mrow> </msub> <mo> −</米o> <mfrac> <mrow> <mi> α</米我> </mrow> <mrow> <mi> ρ</米我> </mrow> </mfrac> <mfrac> <mrow> <mi> d</米我> <mi> W</米我> <mfenced open="(" close=")"> <mrow> <mi> T</米我> </mrow> </mfenced> </mrow> <mrow> <mi> d</米我> <mi> T</米我> </mrow> </mfrac> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 37</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e38"> <mi> W</米我> <mfenced open="(" close=")"> <mrow> <mi> T</米我> </mrow> </mfenced> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mn> 0.778</米n> <mo> −</米o> <mn> 0.778</米n> <mo> ×</米o> <mtext> 经验值</米text> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> T</米我> <mo> −</米o> <mn> 106年</米n> </mrow> <mrow> <mn> 3.42</米n> </mrow> </mfrac> </mrow> </mfenced> <mspace width="1em"></mspace> <mtext> 为</米text> <mspace width="1em"></mspace> <mi> T</米我> <mo> ≤</米o> <mn> 103年</米n> <mo> °</米o> <mi> C</米我> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn> 0.0289</米n> <msup> <mrow> <mi> T</米我> </mrow> <mrow> <mn> 3</米n> </mrow> </msup> <mo> −</米o> <mn> 0.8924</米n> <msup> <mrow> <mi> T</米我> </mrow> <mrow> <mn> 2</米n> </mrow> </msup> <mo> +</米o> <mn> 919.6</米n> <mi> T</米我> <mo> −</米o> <mn> 31573年</米n> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mtext> 为</米text> <mspace width="5.69046pt"></mspace> <mn> 103年</米n> <mo> °</米o> <mi> C</米我> <mo> <</米o> <mi> T</米我> <mo> ≤</米o> <mn> 104年</米n> <mo> °</米o> <mi> c</米我> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn> 0.778</米n> <mo> ×</米o> <mtext> 经验值</米text> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> T</米我> <mo> −</米o> <mn> 80年</米n> </mrow> <mrow> <mn> 34.37</米n> </mrow> </mfrac> </mrow> </mfenced> <mspace width="5.69046pt"></mspace> <mtext> 为</米text> <mspace width="5.69046pt"></mspace> <mi> T</米我> <mo> ≥</米o> <mn> 104年</米n> <mo> °</米o> <mi> C</米我> <mspace width="1em"></mspace> </mtd> </mtr> </mtable> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 38</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e39"> <msub> <mrow> <mi> ω</米我> </mrow> <mrow> <mi> b</米我> </mrow> </msub> <mfenced open="(" close=")"> <mrow> <mi> T</米我> </mrow> </mfenced> <mo> =</米o> <mfenced open="{" close=""> <mrow> <mtable class="cases"> <mtr> <mtd columnalign="left"> <mi> ω</米我> <mspace width="5.69046pt"></mspace> <mtext> 为</米text> <mspace width="5.69046pt"></mspace> <mi> T</米我> <mo> <</米o> <mn> 60</米n> <mo> °</米o> <mi> C</米我> <mspace width="1em"></mspace> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn> 0</米n> <mspace width="19.91684pt"></mspace> <mtext> 为</米text> <mspace width="5.69046pt"></mspace> <mi> T</米我> <mo> ≥</米o> <mn> 60</米n> <mo> °</米o> <mi> C</米我> <mspace width="1em"></mspace> </mtd> </mtr> </mtable> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> (</米o> <mn> 39</米n> <mo stretchy="false"> )</米o> </math> <div class="clear"></div> </div> <p class="indent">在哪里<我nline-formula id="inf16"> <math id="m58"> <mrow> <mover accent="true"> <mrow> <msub> <mrow> <mi> k</米我> </mrow> <mrow> <mn> 0</米n> </mrow> </msub> </mrow> <mo> ̃</米o> </mover> </mrow> </math> </inline-formula>基线是导热系数(W / m°C),Δ吗<e米>k</e米>是变化的<e米>k</e米>由于温度,<e米>T</e米><sub>0</年代ub>参考温度(°C)的吗<e米>k</e米><sub>0</年代ub>被测量,<e米>α</e米>是水潜热的内容。对功率比,还可以使用无量纲数如Pomerantsev号码(<一个href="#B18">Kuneš2012</一个>)规模热源。</p> <a id="h11" name="h11"></a> <h2>结论</h2> <p class="mb15">在这项研究中,我们提出了一个新的点源模型来近似的力量来自针不需要完整的解决方案使用焦耳热模型获得的电场。二维和三维仿真表明,点光源功率模型和焦耳热模型有很好的按照它们之间的偏差最小。同时,点源模型减少了计算时间明显的有限元动力学是完全被点源模型所取代。</p> <a id="h12" name="h12"></a> <h2>数据可用性声明</h2> <p class="mb15">在本文中给出的数据并不容易获得,因为二维数据集可以,但不能提供病人和猪的数据集。请求访问数据集应该指向<一个href="mailto:panch.m@iittp.ac.in">panch.m@iittp.ac.in</一个>。</p> <a id="h13" name="h13"></a> <h2>作者的贡献</h2> <p class="mb0">GB导致了FEniCS代码开发二维模型的比较。点开发了仿真三维模型的一部分,实现同一使用c++和CUDA编程。他也造成了GPU加速和写这手稿。射频,ClinicIMPPACT项目的主要研究者之一,帮助开发和验证的想法点源模型和三维数据集集合。</p> <a id="h14" name="h14"></a> <h2>资金</h2> <p class="mb0">数据集集合和RFA监护人工具开发是由欧盟第七框架计划研究、技术开发和示范赠款协议没有61088。</p> <a id="h15" name="h15"></a> <h2>确认</h2> <p class="mb03">我们感谢Tingyin彭,克莱尔·泊斯德大卫•奥尼尔斯蒂芬•佩恩菲尔堰,菲利普•Voglreiter图Alhonnoro,米卡Pollari,迈克尔•莫奇哈拉尔德会,尤尔根•富特,霍斯特鲁珀特•葡萄牙和滨Kolesnik贡献在项目ClinicIMPPACT,详细讨论了该模型发展。</p> <a id="h16" name="h16"></a> <h2>的利益冲突</h2> <p class="mb0">作者点和射频受雇于公司NUMA工程服务有限公司</p> <p class="mb0">其余作者宣称的研究是在没有进行任何商业或财务关系可能被视为一个潜在的利益冲突。</p> <a id="h17" name="h17"></a> <h2>出版商的注意</h2> <p class="mb15">本文表达的所有索赔仅代表作者,不一定代表的附属组织,或出版商、编辑和审稿人。任何产品,可以评估在这篇文章中,或声称,可能是由其制造商,不保证或认可的出版商。</p> <a id="h18" name="h18"></a> <h2>补充材料</h2> <p class="mb15" id="SM1">本文的补充材料在网上可以找到:<一个href="//www.thespel.com/articles/10.3389/fther.2022.982768/full">https://www.雷竞技rebatfrontiersin.org/articles/10.3389/fther.2022.982768/full补充材料</一个></p> <a id="h19" 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Measurement and analysis of tissue temperature during microwave liver ablation.<e米>IEEE反式。生物医学。Eng。</e米>54岁,150 - 155。doi: 10.1109 / tbme.2006.884647</p> <p class="ReferencesCopy2"><a href="https://pubmed.ncbi.nlm.nih.gov/17260866/">《公共医学图书馆摘要》</一个>|<一个href="https://doi.org/10.1109/tbme.2006.884647">CrossRef全文</一个>|<一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Measurement+and+analysis+of+tissue+temperature+during+microwave+liver+ablation&btnG=">谷歌学术搜索</一个></p> </div> </div> <div class="thinLineM20"></div> <div class="AbstractSummary"> <p><span>关键词:</年代pan>射频消融、热规划、干预支持,焦耳热,点源模型</p> <p><span>引用:</年代pan>Mariappan P, B G和弗拉纳根R(2022)点源模型代表热量分布不计算焦耳热在射频消融术。<e米>前面。前面。小卡。Eng。</e米>2:982768。doi: 10.3389 / fther.2022.982768</p> <p id="timestamps"><span>收到:</年代pan>2022年6月30日;<年代pan>接受:</年代pan>2022年9月13日;<br><span>发表:</年代pan>2022年10月11日。</p> <div> <p>编辑:</p> <a href="//www.thespel.com/loop/people/1636862/overview">(Sanjeev索尼</一个>,印度CSIR-Central科学仪器组织</d我v> <div> <p>审核:</p> <a href="//www.thespel.com/loop/people/1830898/overview">Zailan Siri</一个>马来西亚马来亚大学<br> <a href="//www.thespel.com/loop/people/1796845/overview">马塞洛Iasiello</一个>意大利,意大利degli研究di那不勒斯费德里科•II</d我v> <p><span>版权</年代pan>©2022 Mariappan, B和弗拉纳根。这是一个开放分布式根据文章<一个rel="license" href="http://creativecommons.org/licenses/by/4.0/" target="_blank">知识共享归属许可(CC)。</一个>使用、分发或复制在其他论坛是允许的,提供了原始作者(年代)和著作权人(s)认为,最初发表在这个期刊引用,按照公认的学术实践。没有使用、分发或复制是不符合这些条件的允许。</p> <p><span>*通信:</年代pan>Panchatcharam Mariappan,<一个href="mailto:panch.m@iittp.ac.in">panch.m@iittp.ac.in</一个></p> <div class="clear"></div> </div> <div class="research-topic-container" style="display: none;"> <a href="//www.thespel.com/research-topics/32421" data-test-id="relatedRT-link"> <div class="research-topic-data"> <h5 class="topic-title"><span>本文是研究课题的一部分</年代pan></h5> <p style="margin-bottom:0;">各种程序的计算模型在人类肿瘤的热疗</p> <div class="topic-link-trim" data-event="rt-link-click"> 查看所有3篇文章<年代pan style="position: relative;top: 1px;"></span> </div> </div></a> </div>   <div class="thin-line-dark"></div> </div> </div> </div> </main> </div> <div class="side-article-subjects only-devices widget-listing people-also-looked-at hidden"> <h5 class="like-h4">人也看了</h5> </div> </div> </div> </div> <div id="divTemplates"> <div class="modal fade modal-container impact-modal-container" data-backdrop="static" id="impactModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel" aria-hidden="true"></div> <div class="modal fade modal-container supplementary-modal-container" data-backdrop="static" data-keyboard="false" id="supplementaryFilesModal" tabindex="-1" role="dialog" aria-labelledby="mySupplementaryModalLabel" aria-hidden="true" style="display: none; padding-right: 17px;"></div> <div class="modal fade modal-container notifyme-modal-container" data-backdrop="static" id="notifyModal" tabindex="-1" role="dialog" aria-labelledby="Notify on publication" aria-hidden="true"></div> </div> </div> <a id="floating-download-pdf-btn" class="floating-button" data-event="downloadfloating-button-click" href="//www.thespel.com/articles/10.3389/fther.2022.982768/pdf"><span class="floating-button-text" data-event="downloadfloating-button-click">下载</年代pan></a> <frontiers-footer main-domain="frontiersin.org"></frontiers-footer>   </body> </html>

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