Document Zbl 1051.80002

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Anal. Bound. Elem.</a> <a href="/?q=in%3A103393" title="Articles in this Issue">28, No. 2, 97-108 (2004)</a>. </div> <div class="abstract">Cryosurgery is now applied to delete undesired tissues. One or several cryoprobes are placed on an accessible surface of the undesired tissue. A frozen zone then starts to develop inside the tissue, this frozen zone being further removed by surgery. Following a modelisation by Pennes which consists of two linear reaction-diffusion equations posed in the frozen or in the unfrozen zones, the authors consider a model which consists of a single linear reaction-diffusion equation in the whole region, but with variable coefficients and variable source terms (depending on the underlying zone). On the moving interface between the frozen and the unfrozen zones, the (equal traces of the) temperatures and the difference of the heat fluxes are imposed. On the remaining part of the boundary Dirichlet or Neumann or even Robin type boundary conditions are imposed.<br class="zbmathjax-paragraph">The authors develop a numerical method for the resolution of this problem, using a dual reciprocity boundary elements method, as developped by <span class="zbmathjax-textit">D. Nardini</span> and <span class="zbmathjax-textit">C. A. Brebbia</span> [Appl. Math. Modelling 7, 157–162 (1983; <a href="/0545.73078">Zbl 0545.73078</a>)]. The authors restrict their numerical resolution to the two-dimensional case. They observe that the solution of the linear reaction-diffusion equation with source terms can be written as the sum of a solution of the reaction-diffusion equation without source terms and of a solution of the underlying elliptic equation with these source terms. Because it is difficult to compute such a solution of this elliptic problem, they introduce as many particular solutions of this elliptic equation as the total number of nodes. From these particular solutions, they then build a solution of the elliptic equation, using a boundary element method and the fundamental solution of the Laplace operator. The rest of the paper discusses the computations considering several kinds of tissue. Many figures illustrate the numerical results in the case of a single or of many cryoprobes.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10228">Alain Brillard (Mulhouse)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A2102399">Cited in <strong>13</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 80A20 </td> <td class="space"> Heat and mass transfer, heat flow (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K57" title="MSC2020">35K57</a> </td> <td class="space"> Reaction-diffusion equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R35" title="MSC2020">35R35</a> </td> <td class="space"> Free boundary problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A80A22" title="MSC2020">80A22</a> </td> <td class="space"> Stefan problems, phase changes, etc. </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92C50" title="MSC2020">92C50</a> </td> <td class="space"> Medical applications (general) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Areaction-diffusion+equations">reaction-diffusion equations</a>; <a href="/?q=ut%3Aphase+change">phase change</a>; <a href="/?q=ut%3Acryosurgery">cryosurgery</a>; <a href="/?q=ut%3Adual+reciprocity+boundary+elements">dual reciprocity boundary elements</a>; <a href="/?q=ut%3Aboundary+element+method">boundary element method</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0545.73078">Zbl 0545.73078</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1051.80002" data-ciurl="/ci/02102399" data-biburl="/bibtex/02102399.bib" data-amsurl="/amsrefs/02102399.bib" data-xmlurl="/xml/02102399.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02102399.pdf" title="Zbl 1051.80002 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/S0955-7997(03)00128-0" aria-label="DOI for “Modeling of multidimensional freezing problem during cryosurgery by the dual reciprocity boundary element method”" title="10.1016/S0955-7997(03)00128-0">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Diller, K. 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