{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T11:38:11Z","timestamp":1788349091676,"version":"build-2803163510"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"name":"ERC via EU FP 7","award":["615216"],"award-info":[{"award-number":["615216"]}]},{"name":"German Science Foundation","award":["EXC 1003"],"award-info":[{"award-number":["EXC 1003"]}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["WO1425\/5-1"],"award-info":[{"award-number":["WO1425\/5-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>Transcranial direct current stimulation (tDCS) is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes. The aim of this study is to introduce and analyze a novel optimization method for safe and well-targeted multiarray tDCS. For optimization, we consider an optimal control problem for a Laplace equation with Neumann boundary conditions with control and pointwise gradient state constraints. We prove well-posedness results for the proposed methods and provide computer simulation results in a highly realistic six-compartment geometry-adapted hexahedral head model. For discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown. For numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality, target intensity, and orientation. The numerical results reveal that the optimized current flow fields show significantly higher focality and, in most cases, higher directional agreement to the target vector in comparison to standard bipolar electrode montages.<\/jats:p>","DOI":"10.1137\/15m1026481","type":"journal-article","created":{"date-parts":[[2016,11,1]],"date-time":"2016-11-01T11:36:44Z","timestamp":1478000204000},"page":"2154-2174","source":"Crossref","is-referenced-by-count":54,"title":["An Optimization Approach for Well-Targeted Transcranial Direct Current Stimulation"],"prefix":"10.1137","volume":"76","author":[{"given":"Sven","family":"Wagner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Burger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carsten H.","family":"Wolters","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,11,1]]},"reference":[{"key":"atypb1","unstructured":"B. Agsten,\n                      Comparing the Complete and the Point Electrode Model for Combining tCS and EEG\n                      , Master thesis, University of Mu\u0308nster, Mu\u0308nster, Germany, 2015."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0118753"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0093154"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1038\/196584a0"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jad.2006.10.026"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1561\/2200000016"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0014-4886(62)90056-0"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"M. Dannhauer, D. Brooks, D. Tucker, and R. MacLeod,\n                      A pipeline for the simulation of transcranial direct current stimulation for realistic human head models using SCIRun\/BioMesh3D\n                      , in Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2012;https:\/\/doi.org\/10.1109\/EMBC.2012.6347236.","DOI":"10.1109\/EMBC.2012.6347236"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1088\/1741-2560\/8\/4\/046011"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2013.02.049"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2014.04.085"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1212\/01.wnl.0000317060.43722.a3"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1111\/j.1528-1167.2006.00426.x"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"D. Gilbarg and N. S. Trudinger,\n                      Elliptic Partial Differential Equations of Second Order\n                      , Springer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2013.2264632"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/1037043"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1006\/nimg.2001.0962"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/100815037"},{"key":"atypb20","unstructured":"S. Homo\u0308lle,\n                      Comparison of Optimization Approaches in High-Definition Transcranial Current Stimulation in the Mammalian Brain\n                      , Master thesis, University of Mu\u0308nster, Mu\u0308nster, Germany, 2016."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/53\/11\/N03"},{"key":"atypb22","unstructured":"J.L. Lions,\n                      Optimal Control of Systems Governed by Partial Differential Equations\n                      , Dunod, Gauthier-Villars, Paris, 1968."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/S1567-424X(09)70230-2"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1111\/j.1469-7793.2000.t01-1-00633.x"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-011-0360-9"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/57\/4\/999"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1023\/B:BRAT.0000032859.68959.76"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2013.12.002"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.3389\/fpsyt.2012.00090"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/080742154"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.171473898"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1088\/1741-2560\/11\/1\/016002"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/110836419"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1026481","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:03:11Z","timestamp":1787338991000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1026481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1]]},"references-count":32,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1137\/15M1026481"],"URL":"https:\/\/doi.org\/10.1137\/15m1026481","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1]]}}}