{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,13]],"date-time":"2026-09-13T14:04:13Z","timestamp":1789308253495,"version":"build-2803163510"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T00:00:00Z","timestamp":1785196800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T00:00:00Z","timestamp":1785196800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"J\u00f6nk\u00f6ping University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2026,8]]},"abstract":"Abstract<\/jats:title>\n \n Systemic sclerosis (SSc) is an autoimmune disease marked by excessive extracellular matrix (ECM) deposited by myofibroblasts. The disease carries the risk of pathologically progressing to internal organs, particularly the lung. Since abnormally large densities of myofibroblasts are associated with SSc, clinical studies consider trials that reduce the density of myofibroblasts: imatinib treatment (\n \n \n $$N$$<\/jats:tex-math>\n \n N<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ), which promotes apoptosis in myofibroblasts, and SSc by fresolimumab (\n \n \n $$S$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ), which inhibits TGF-\n \n \n $$\\beta $$<\/jats:tex-math>\n \n \u03b2<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , a key growth factor of myofibroblasts. In this paper, we develop a mathematical model of SSc by a system of partial differential equations, and use it to explore a range of treatment strategies with\n \n \n $$N$$<\/jats:tex-math>\n \n N<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$S$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . For example, we considered administering\n \n \n $$S$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in fractions three weeks apart. In this case, we determined the smallest amount of\n \n \n $$S$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n such that\n \n \n $$\\rho (t)$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the density of the ECM at time\n \n \n $$t$$<\/jats:tex-math>\n \n t<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , will continuously and oscillatingly decrease from a disease level\n \n \n $$2\\rho ^0$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$\\rho ^0$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the density\n \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in health. Since SSc has no cure,\n \n \n $$\\rho (t)$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n cannot decrease below\n \n \n $$\\rho ^0$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We found that with the smallest amount of\n \n \n $$S$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n $$\\rho (t)$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n decreases over a few months to\n \n \n $$1.19\\rho ^0$$<\/jats:tex-math>\n \n \n 1.19<\/mml:mn>\n \n \u03c1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and remains nearly stable thereafter. The results of the paper could be useful in the design of future clinical trials aimed to decrease the excessive extracellular matrix in SSc patients.\n <\/jats:p>","DOI":"10.1007\/s11538-026-01700-9","type":"journal-article","created":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T13:38:18Z","timestamp":1785245898000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mathematical Modeling of Systemic Sclerosis and Its Treatment"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7851-8147","authenticated-orcid":false,"given":"Teddy","family":"Lazebnik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Avner","family":"Friedman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,28]]},"reference":[{"issue":"4","key":"1700_CR1","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1097\/FJC.0b013e3182116e39","volume":"57","author":"J Baum","year":"2011","unstructured":"Baum J, Duffy HS (2011) Fibroblasts and myofibroblasts: what are we talking about? 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