{"id":"https://openalex.org/W4407196002","doi":"https://doi.org/10.3390/sym17020244","title":"Nonlinear Coupled Equations Free-Boundary Problem in Atherosclerosis: Symmetry Analysis of Solutions and Numerical Simulation","display_name":"Nonlinear Coupled Equations Free-Boundary Problem in Atherosclerosis: Symmetry Analysis of Solutions and Numerical Simulation","publication_year":2025,"publication_date":"2025-02-06","ids":{"openalex":"https://openalex.org/W4407196002","doi":"https://doi.org/10.3390/sym17020244"},"language":"en","primary_location":{"id":"doi:10.3390/sym17020244","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020244","pdf_url":"https://www.mdpi.com/2073-8994/17/2/244/pdf?version=1738855757","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/2/244/pdf?version=1738855757","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010903763","display_name":"Yarong Zhang","orcid":"https://orcid.org/0000-0002-4735-7328"},"institutions":[{"id":"https://openalex.org/I148099405","display_name":"Xi'an University of Architecture and Technology","ror":"https://ror.org/04v2j2k71","country_code":"CN","type":"education","lineage":["https://openalex.org/I148099405"]},{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]},{"id":"https://openalex.org/I4210106387","display_name":"Anhui Special Equipment Inspection Institute","ror":"https://ror.org/01nqeer97","country_code":"CN","type":"other","lineage":["https://openalex.org/I4210106387"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yarong Zhang","raw_affiliation_strings":["School of Power and Energy, Northwestern Polytechnical University, Xi\u2019an 710129, China","School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China","Xi\u2019an Special Equipment Inspection Institute, Xi\u2019an 710061, China"],"affiliations":[{"raw_affiliation_string":"School of Power and Energy, Northwestern Polytechnical University, Xi\u2019an 710129, China","institution_ids":["https://openalex.org/I17145004"]},{"raw_affiliation_string":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China","institution_ids":["https://openalex.org/I148099405"]},{"raw_affiliation_string":"Xi\u2019an Special Equipment Inspection Institute, Xi\u2019an 710061, China","institution_ids":["https://openalex.org/I4210106387"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109722381","display_name":"X. Mu","orcid":"https://orcid.org/0000-0001-5036-4855"},"institutions":[{"id":"https://openalex.org/I148099405","display_name":"Xi'an University of Architecture and Technology","ror":"https://ror.org/04v2j2k71","country_code":"CN","type":"education","lineage":["https://openalex.org/I148099405"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaoya Mu","raw_affiliation_strings":["School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"],"affiliations":[{"raw_affiliation_string":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China","institution_ids":["https://openalex.org/I148099405"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101996629","display_name":"Xinyu Shi","orcid":"https://orcid.org/0000-0002-5906-088X"},"institutions":[{"id":"https://openalex.org/I148099405","display_name":"Xi'an University of Architecture and Technology","ror":"https://ror.org/04v2j2k71","country_code":"CN","type":"education","lineage":["https://openalex.org/I148099405"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinyu Shi","raw_affiliation_strings":["School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"],"affiliations":[{"raw_affiliation_string":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China","institution_ids":["https://openalex.org/I148099405"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010513600","display_name":"Ji\u2010Huan He","orcid":"https://orcid.org/0000-0002-1636-0559"},"institutions":[{"id":"https://openalex.org/I148099405","display_name":"Xi'an University of Architecture and Technology","ror":"https://ror.org/04v2j2k71","country_code":"CN","type":"education","lineage":["https://openalex.org/I148099405"]},{"id":"https://openalex.org/I4210108760","display_name":"Hohhot Minzu College","ror":"https://ror.org/01qhcb906","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210108760"]},{"id":"https://openalex.org/I4210127700","display_name":"Zhejiang Shuren University","ror":"https://ror.org/0331z5r71","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210127700"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Jihuan He","raw_affiliation_strings":["School of Jia Yang, Zhejiang Shuren University, Hangzhou 310015, China","School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010010, China","School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"],"affiliations":[{"raw_affiliation_string":"School of Jia Yang, Zhejiang Shuren University, Hangzhou 310015, China","institution_ids":["https://openalex.org/I4210127700"]},{"raw_affiliation_string":"School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010010, China","institution_ids":["https://openalex.org/I4210108760"]},{"raw_affiliation_string":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China","institution_ids":["https://openalex.org/I148099405"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5010513600"],"corresponding_institution_ids":["https://openalex.org/I148099405","https://openalex.org/I4210108760","https://openalex.org/I4210127700"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.03883072,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"2","first_page":"244","last_page":"244"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9889000058174133,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9646000266075134,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.7101656198501587},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.6096471548080444},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5140824913978577},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4538102149963379},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43940412998199463},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42682385444641113},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3413695693016052},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1637585163116455}],"concepts":[{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.7101656198501587},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.6096471548080444},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5140824913978577},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4538102149963379},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43940412998199463},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42682385444641113},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3413695693016052},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1637585163116455},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17020244","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020244","pdf_url":"https://www.mdpi.com/2073-8994/17/2/244/pdf?version=1738855757","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3390/sym17020244","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020244","pdf_url":"https://www.mdpi.com/2073-8994/17/2/244/pdf?version=1738855757","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2082826544","display_name":null,"funder_award_id":"Postdoctoral","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3608315534","display_name":null,"funder_award_id":"2022M722604","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G382887264","display_name":null,"funder_award_id":"12101482","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4020255992","display_name":null,"funder_award_id":"Project","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5939423041","display_name":null,"funder_award_id":"Technology","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320328010","display_name":"Yakult Bio-Science Foundation","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4407196002.pdf","grobid_xml":"https://content.openalex.org/works/W4407196002.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1990844197","https://openalex.org/W2072685235","https://openalex.org/W2324841714","https://openalex.org/W2634715622","https://openalex.org/W2806911615","https://openalex.org/W2808034144","https://openalex.org/W3003299879","https://openalex.org/W3041149422","https://openalex.org/W3102902719","https://openalex.org/W3106783344","https://openalex.org/W3110954381","https://openalex.org/W3154365865","https://openalex.org/W3165793489","https://openalex.org/W3192073598","https://openalex.org/W3194631261","https://openalex.org/W3209790898","https://openalex.org/W4292117329","https://openalex.org/W4303879177","https://openalex.org/W4306148700","https://openalex.org/W4321598362","https://openalex.org/W4324345701","https://openalex.org/W4383532515","https://openalex.org/W4386574037","https://openalex.org/W4386757276","https://openalex.org/W4387857882","https://openalex.org/W4393212360","https://openalex.org/W4396919955","https://openalex.org/W4399138769","https://openalex.org/W4399267427","https://openalex.org/W4399358971","https://openalex.org/W4399980991","https://openalex.org/W4405430810","https://openalex.org/W6752263230","https://openalex.org/W6765722046"],"related_works":["https://openalex.org/W2154204858","https://openalex.org/W3209622820","https://openalex.org/W2382250648","https://openalex.org/W2367363545","https://openalex.org/W2051519054","https://openalex.org/W2488633177","https://openalex.org/W4372049322","https://openalex.org/W2352708639","https://openalex.org/W2083247966","https://openalex.org/W2043702156"],"abstract_inverted_index":{"Atherosclerosis,":[0],"a":[1,14,43,87,192],"chronic":[2],"inflammatory":[3,50,150],"cardiovascular":[4],"disease":[5],"closely":[6],"related":[7],"to":[8,17,166,220],"plaque":[9,153],"formation":[10],"during":[11],"arteriosclerosis,":[12],"poses":[13],"significant":[15],"threat":[16],"global":[18,67],"health.":[19],"To":[20],"deepen":[21,227],"the":[22,25,60,64,80,94,105,110,114,131,136,158,163,177,186,197,201,222,228],"understanding":[23,229],"of":[24,71,96,104,119,139,179,230],"multifaceted":[26],"interactions":[27],"driving":[28],"atherosclerosis":[29],"progression":[30],"and":[31,54,66,69,93,116,122,144,155,170,182,188,215,226],"provide":[32],"theoretical":[33],"support":[34],"for":[35],"designing":[36],"targeted":[37],"therapeutic":[38,146],"strategies,":[39],"this":[40,231],"study":[41],"establishes":[42],"nonlinear":[44],"coupled":[45],"atherosclerotic":[46],"free-boundary":[47],"model":[48,90,108,180,204],"integrating":[49,216],"immune":[51],"cells,":[52],"cytokines,":[53],"oxidized":[55,123],"low-density":[56,124],"lipoprotein.":[57],"By":[58,148],"applying":[59],"compression":[61],"mapping":[62],"principle,":[63],"local":[65],"existence":[68,95],"uniqueness":[70],"solutions":[72],"are":[73],"proven,":[74],"while":[75],"revealing":[76],"certain":[77],"symmetries":[78,184],"in":[79,134,176,185,195],"model\u2019s":[81,132,164,223],"solution":[82,98,198],"structure.":[83],"Under":[84],"specific":[85],"assumptions,":[86],"quasi-steady-state":[88,106],"approximate":[89,107],"is":[91,99,126],"derived,":[92],"its":[97,172],"demonstrated.":[100],"Through":[101],"numerical":[102],"simulations":[103],"using":[109],"finite":[111],"difference":[112],"method,":[113],"temporal":[115],"spatial":[117],"evolution":[118],"pro-inflammatory":[120],"macrophages":[121],"lipoprotein":[125],"analyzed.":[127],"The":[128],"findings":[129],"highlight":[130],"strength":[133],"capturing":[135],"intricate":[137],"dynamics":[138,151],"atherosclerosis,":[140],"uncovering":[141],"underlying":[142],"mechanisms":[143],"identifying":[145],"targets.":[147],"evaluating":[149],"across":[152],"types":[154],"stenosis":[156],"levels,":[157],"experimental":[159],"design":[160],"further":[161],"validated":[162],"ability":[165],"replicate":[167],"clinical":[168,224],"processes":[169],"reinforced":[171],"predictive":[173],"accuracy.":[174],"Notably,":[175],"process":[178],"analysis":[181],"solution,":[183],"equations":[187],"boundary":[189],"conditions":[190],"play":[191],"crucial":[193],"role":[194],"determining":[196],"properties.":[199],"However,":[200],"current":[202],"one-dimensional":[203],"has":[205],"limitations.":[206],"Future":[207],"research":[208],"should":[209],"focus":[210],"on":[211],"developing":[212],"higher-dimensional":[213],"models":[214],"more":[217],"influencing":[218],"factors":[219],"enhance":[221],"applicability":[225],"complex":[232],"disease.":[233]},"counts_by_year":[],"updated_date":"2026-03-22T08:09:32.410652","created_date":"2025-10-10T00:00:00"}