{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T19:32:04Z","timestamp":1791401524992,"version":"4.3.3"},"reference-count":23,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T00:00:00Z","timestamp":1778457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Neurol."],"abstract":"\n Objective<\/jats:title>\n We represent primary headaches within the International Headache Classification (ICHD3) in matrix form and show that this representation allows for automated diagnosis as well as additional insights into headache classification.<\/jats:p>\n <\/jats:sec>\n \n Background<\/jats:title>\n ICHD3 has been the gold standard for clinical trials and research in headache medicine. Given its criteria-based language, ICHD3 can be interpreted as a set of logical statements and as a result can be encoded as the biadjacency matrix of a mathematical graph. This paper implements this interpretation of the classification and explores the clinical and theoretical implications of this approach.<\/jats:p>\n <\/jats:sec>\n \n Methods<\/jats:title>\n \n Each diagnosis in the ICHD3 is defined by a list of characteristics. Combinations of characteristics form phenotypes. Multiple phenotypes may fit a given diagnosis. We first translated all characteristics for primary headache diagnoses in the ICHD3 into true\/false statements. We then generated a matrix of valid ICHD3 diagnoses as follows:\n \n \n Each row of the matrix represents a phenotype.<\/jats:p>\n <\/jats:list-item>\n \n Each column of the matrix represents a characteristic.<\/jats:p>\n <\/jats:list-item>\n \n \n If any phenotype contains a characteristic, then that element is encoded as\n \n \n Otherwise, it is encoded as 0.<\/jats:p>\n <\/jats:list-item>\n <\/jats:list>\n <\/jats:p>\n <\/jats:list-item>\n <\/jats:list>\n <\/jats:p>\n From this matrix, we calculated its bipartite projection and Markov cluster. We also row reduced to derive the basis vectors that span the space of all headache phenotypes.<\/jats:p>\n <\/jats:sec>\n \n Results<\/jats:title>\n Chronic migraine diagnoses as well as the characteristics \u201cgreater than 15 days per month\u201d and \u201cmore than 3 months\u201d have the strongest associations based on bipartite projection. Markov clustering yields 64 clusters. These clusters can be organized by ICHD3 diagnoses and demonstrates the level of fragmentation of individual diagnoses within the classification: Migraine is composed of 1 cluster, for example, whereas paroxysmal hemicrania can be broken down into 9 clusters. Finally, row reduction of our matrix yields 63 basis vectors, implying that all headache diagnoses in the ICHD3 can be represented as linear combinations of 63 characteristics. These 63 characteristics correspond to the following: duration, frequency, aura characteristics, size\/location, laterality, clearly remembered onset, TAC features, total number of episodes, severity, nausea\/vomiting, photophobia, pulsating, alleviation by triptans, and association with awakening, sexual activity, physical activity, temperature, compression or traction, coughing.<\/jats:p>\n <\/jats:sec>\n \n Conclusion<\/jats:title>\n Our result demonstrates that ICHD3 is a mathematical entity and that headache diagnoses can exist in a 63-dimensional vector space. This mathematical embodiment of classification allows us to conduct (1) large scale systematic investigations of relationships between headache and phenotypes, (2) generate a graphical representation of characteristics and phenotypes and (3) potentially improve diagnostic accuracy and efficiency.<\/jats:p>\n <\/jats:sec>","DOI":"10.3389\/fneur.2026.1812996","type":"journal-article","created":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T15:04:15Z","timestamp":1778511855000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Matrix and graphical representation of the primary headache syndromes in the International Classification of Headache Disorders (ICHD3): a basis for automated diagnosis and analysis of criteria"],"prefix":"10.3389","volume":"17","author":[{"given":"Pengfei","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Neurology, Beth Israel Deaconess Medical Center","place":["Boston, MA, United States"]},{"name":"Department of Neurology, Harvard Medical School","place":["Boston, MA, United States"]},{"name":"Department of Neurology, Rutgers Robert Wood Johnson Medical School","place":["New Brunswick, NJ, United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roger","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Neurology, Rutgers Robert Wood Johnson Medical School","place":["New Brunswick, NJ, United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2026,5,11]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1177\/0333102417738202","article-title":"Headache Headache Classification Committee of the International Headache Society (IHS) The International Classification of Headache Disorders, 3rd edition","volume":"38","year":"2018","journal-title":"Cephalalgia"},{"key":"B2","doi-asserted-by":"publisher","first-page":"864","DOI":"10.1017\/S1471068420000290","article-title":"A logic-based decision support system for the diagnosis of headache disorders according to the ICHD-3 international classification","volume":"20","author":"Costabile","year":"2020","journal-title":"Theory Prac Log Prog."},{"key":"B3","doi-asserted-by":"publisher","first-page":"1221209","DOI":"10.3389\/fneur.2023.1221209","article-title":"Which headache disorders can be diagnosed concurrently? 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