{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T20:11:26Z","timestamp":1767211886664,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T00:00:00Z","timestamp":1750636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Processes"],"abstract":"<jats:p>Chitosan hydrogels have gained attention in biomedical and pharmaceutical research due to their biocompatibility, biodegradability, and tunable properties. To enhance mechanical strength and to control swelling and degradation, chitosan is often cross-linked with either bio-based (e.g., genipin) or synthetic (e.g., glutaraldehyde) agents. A comprehensive understanding of the degradation mechanisms of cross-linked chitosan hydrogels is essential, as it directly impacts performance optimization, regulatory compliance, and their integration into personalized medicine. Despite extensive studies, the fundamental mechanisms governing hydrogel degradation remain partially understood. In this work, we introduce a general data-driven framework based on symbolic regression to elucidate the degradation kinetics of hydrogels. Using genipin-cross-linked chitosan hydrogels as a model system, we analyze experimental degradation data to identify governing kinetic laws. Our results suggest that degradation proceeds primarily via a surface-mediated mechanism. The proposed approach provides a robust and interpretable method for uncovering mechanistic insights and is broadly applicable to other hydrogel systems.<\/jats:p>","DOI":"10.3390\/pr13071981","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T07:42:34Z","timestamp":1750664554000},"page":"1981","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Unraveling the Degradation Kinetics of Genipin-Cross-Linked Chitosan Hydrogels via Symbolic Regression"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2550-4320","authenticated-orcid":false,"given":"Belmiro P. M.","family":"Duarte","sequence":"first","affiliation":[{"name":"Departmento de Engenharia Qu\u00edmica e Biol\u00f3gica, Instituto Superior de Engenharia de Coimbra, Instituto Polit\u00e9cnico de Coimbra Rua Pedro Nunes, 3030-199 Coimbra, Portugal"},{"name":"Instituto Nacional de Engenharia de Sistemas e Computadores-Coimbra, Universidade de Coimbra, Rua S\u00edlvio Lima, P\u00f3lo II, 3030-790 Coimbra, Portugal"},{"name":"Chemical Engineering and Renewable Resources for Sustainability Research Center\u2014CERES, Universidade de Coimbra, Rua S\u00edlvio Lima, P\u00f3lo II, 3030-790 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5883-5516","authenticated-orcid":false,"given":"Maria J.","family":"Moura","sequence":"additional","affiliation":[{"name":"Departmento de Engenharia Qu\u00edmica e Biol\u00f3gica, Instituto Superior de Engenharia de Coimbra, Instituto Polit\u00e9cnico de Coimbra Rua Pedro Nunes, 3030-199 Coimbra, Portugal"},{"name":"Chemical Engineering and Renewable Resources for Sustainability Research Center\u2014CERES, Universidade de Coimbra, Rua S\u00edlvio Lima, P\u00f3lo II, 3030-790 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1007\/s10924-016-0865-5","article-title":"Chitin and chitosan: Structure, properties and applications in biomedical engineering","volume":"25","author":"Islam","year":"2017","journal-title":"J. Polym. Environ."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Singh, G., and Chanda, A. (2021). Mechanical properties of whole-body soft human tissues: A review. Biomed. Mater., 16.","DOI":"10.1088\/1748-605X\/ac2b7a"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1276","DOI":"10.1111\/iwj.12797","article-title":"Chitin and chitosan: Biopolymers for wound management","volume":"14","author":"Singh","year":"2017","journal-title":"Int. Wound J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10717540590889781","article-title":"Chitosan-based particles as controlled drug delivery systems","volume":"12","author":"Prabaharan","year":"2004","journal-title":"Drug Deliv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.biotechadv.2007.07.009","article-title":"Chitosan and its derivatives for tissue engineering applications","volume":"26","author":"Kim","year":"2008","journal-title":"Biotechnol. Adv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2525","DOI":"10.1002\/pol.20220154","article-title":"Progress in the mechanical enhancement of hydrogels: Fabrication strategies and underlying mechanisms","volume":"60","author":"Lin","year":"2022","journal-title":"J. Polym. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Yammine, P., El Safadi, A., Kassab, R., El-Nakat, H., Obeid, P.J., Nasr, Z., Tannous, T., Sari-Chmayssem, N., Mansour, A., and Chmayssem, A. (2025). Types of crosslinkers and their applications in biomaterials and biomembranes. Chemistry, 7.","DOI":"10.3390\/chemistry7020061"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109034","DOI":"10.1016\/j.foodhyd.2023.109034","article-title":"Chitosan with modified porosity and crosslinked with genipin: A dynamic system structurally characterized","volume":"144","author":"Siquiera","year":"2023","journal-title":"Food Hydrocoll."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.jconrel.2023.05.042","article-title":"Scaffolds for drug delivery and tissue engineering: The role of genetics","volume":"359","author":"Karczewski","year":"2023","journal-title":"J. Control. Release"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ding, R., Zhu, Y., Jing, L., Chen, S., Lu, J., and Zhang, X. (2024). Sulfhydryl functionalized chitosan-covalent organic framework composites for highly efficient and selective recovery of gold from complex liquids. Int. J. Biol. Macromol., 282.","DOI":"10.1016\/j.ijbiomac.2024.137037"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.carbpol.2010.07.038","article-title":"Controlled gelation temperature, pore diameter and degradation of a highly porous chitosan-based hydrogel","volume":"83","author":"Dang","year":"2011","journal-title":"Carbohydr. Polym."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Jennings, J.A., and Bumgardner, J.D. (2017). 7\u2014Controlling chitosan degradation properties in vitro and in vivo. Chitosan Based Biomaterials Volume 1, Woodhead Publishing.","DOI":"10.1016\/B978-0-08-100230-8.00007-8"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s10971-006-9007-1","article-title":"Gelation time and degradation rate of chitosan-based injectable hydrogel","volume":"42","author":"Ganji","year":"2007","journal-title":"J. Sol-Gel Sci. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3855","DOI":"10.1007\/s10853-021-06757-6","article-title":"The latest advances in biomedical applications of chitosan hydrogel as a powerful natural structure with eye-catching biological properties","volume":"57","author":"Noruzi","year":"2022","journal-title":"J. Mater. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Xu, C., Sun, N., Li, H., Han, X., Zhang, A., and Sun, P. (2024). Stimuli-responsive vesicles and hydrogels formed by a single-tailed dynamic covalent surfactant in aqueous solutions. Molecules, 29.","DOI":"10.3390\/molecules29214984"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0939-6411(03)00160-7","article-title":"Structure and interactions in chitosan hydrogels formed by complexation or aggregation for biomedical applications","volume":"57","author":"Berger","year":"2004","journal-title":"Eur. J. Pharm. Biopharm."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1016\/j.progpolymsci.2011.02.001","article-title":"Chitosan\u2014A versatile semi-synthetic polymer in biomedical applications","volume":"36","author":"Dash","year":"2011","journal-title":"Prog. Polym. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2001","DOI":"10.1271\/bbb.60.2001","article-title":"Oxidative depolymerization of chitosan by hydroxyl radical","volume":"60","author":"Tanioka","year":"1996","journal-title":"Biosci. Biotechnol. Biochem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1080\/00914037.2020.1760274","article-title":"A facile way to synthesize a photocrosslinkable methacrylated chitosan hydrogel for biomedical applications","volume":"70","author":"Samani","year":"2021","journal-title":"Int. J. Polym. Mater. Polym. Biomater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2767","DOI":"10.1002\/bit.28755","article-title":"In vitro degradation, swelling, and bioactivity performances of in situ forming injectable chitosan-matrixed hydrogels for bone regeneration and drug delivery","volume":"121","author":"Kocak","year":"2024","journal-title":"Biotechnol. Bioeng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1016\/j.carbpol.2018.08.102","article-title":"One-pot fabrication of pH\/reduction dual-stimuli responsive chitosan-based supramolecular nanogels for leakage-free tumor-specific DOX delivery with enhanced anti-cancer efficacy","volume":"201","author":"Li","year":"2018","journal-title":"Carbohydr. Polym."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4714","DOI":"10.1021\/acsnano.1c11505","article-title":"Self-powered multifunction ionic skins based on gradient polyelectrolyte hydrogels","volume":"16","author":"Xia","year":"2022","journal-title":"ACS Nano"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1038\/s41392-024-01852-x","article-title":"Harnessing the potential of hydrogels for advanced therapeutic applications: Current achievements and future directions","volume":"9","author":"Lu","year":"2024","journal-title":"Signal Transduct. Target. Ther."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Duarte, A.R.C., Correlo, V.M., Oliveira, J.M., and Reis, R.L. (2016). Recent Developments on Chitosan Applications in Regenerative Medicine. Biomaterials from Nature for Advanced Devices and Therapies, John Wiley & Sons, Ltd.. Chapter 14.","DOI":"10.1002\/9781119126218.ch14"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.addr.2009.09.004","article-title":"Biodegradation, biodistribution and toxicity of chitosan","volume":"62","author":"Kean","year":"2010","journal-title":"Adv. Drug Deliv. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF00175355","article-title":"Genetic programming as a means for programming computers by natural selection","volume":"4","author":"Koza","year":"1994","journal-title":"Stat. Comput."},{"key":"ref_27","unstructured":"Banzhaf, W., Nordin, P., Keller, R.E., and Francone, F.D. (1998). Genetic programming: An Introduction: On the Automatic Evolution of Computer Programs and Its Applications, Morgan Kaufmann Publishers Inc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1126\/science.1165893","article-title":"Distilling free-form natural laws from experimental data","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"133032","DOI":"10.1016\/j.cej.2021.133032","article-title":"Functional-Hybrid modeling through automated adaptive symbolic regression for interpretable mathematical expressions","volume":"430","author":"Narayanan","year":"2022","journal-title":"Chem. Eng. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"120606","DOI":"10.1016\/j.ces.2024.120606","article-title":"Machine learning uncovers analytical kinetic models of bioprocesses","volume":"300","author":"Forster","year":"2024","journal-title":"Chem. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"120580","DOI":"10.1016\/j.ces.2024.120580","article-title":"Integrating knowledge-guided symbolic regression and model-based design of experiments to automate process flow diagram development","volume":"300","author":"Rogers","year":"2024","journal-title":"Chem. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"025004","DOI":"10.1063\/5.0082147","article-title":"Machine learning symbolic equations for diffusion with physics-based descriptions","volume":"12","author":"Papastamatiou","year":"2022","journal-title":"Aip Adv."},{"key":"ref_33","unstructured":"Servia, M.A.D., and del Rio Chanona, E.A. (2023). Interpretable Machine Learning for Kinetic Rate Model Discovery. Machine Learning and Hybrid Modelling for Reaction Engineering: Theory and Applications, Royal Society of Chemistry."},{"key":"ref_34","unstructured":"Bragone, F., Morozovska, K., Laneryd, T., Shukla, K., and Markidis, S. (2025). Discovering partially known ordinary differential equations: A case study on the chemical kinetics of cellulose degradation. arXiv."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"6017","DOI":"10.1021\/cr030441b","article-title":"Chitosan chemistry and pharmaceutical perspectives","volume":"104","author":"Muzzarelli","year":"2004","journal-title":"Chem. Rev."},{"key":"ref_36","first-page":"966","article-title":"Salt-free chitosan solutions: Thermodynamics, structure and intramolecular force balance","volume":"41","author":"Alexeev","year":"1999","journal-title":"Polym. Sci. Ser. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"100168","DOI":"10.1016\/j.sintl.2022.100168","article-title":"Chitosan based injectable hydrogels for smart drug delivery applications","volume":"3","author":"Singha","year":"2022","journal-title":"Sens. Int."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Kildeeva, N., Chalykh, A., Belokon, M., Petrova, T., Matveev, V., Svidchenko, E., Surin, N., and Sazhnev, N. (2020). Influence of genipin crosslinking on the properties of chitosan-based films. Polymers, 12.","DOI":"10.3390\/polym12051086"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1039\/D0BM01403F","article-title":"Genipin-cross-linked hydrogels based on biomaterials for drug delivery: A review","volume":"9","author":"Yu","year":"2021","journal-title":"Biomater. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0144-8617(00)00281-2","article-title":"Rheological characterisation of thermogelling chitosan\/glycerol-phosphate solutions","volume":"46","author":"Chenite","year":"2001","journal-title":"Carbohydr. Polym."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3823","DOI":"10.1021\/bm700762w","article-title":"Rheological study of genipin cross-linked chitosan hydrogels","volume":"8","author":"Moura","year":"2007","journal-title":"Biomacromolecules"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/BF03218435","article-title":"Preparation and biodegradation of thermosensitive chitosan hydrogel as a function of pH and temperature","volume":"12","author":"Han","year":"2004","journal-title":"Macromol. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3941","DOI":"10.1016\/j.biomaterials.2004.10.005","article-title":"Self-cross-linking biopolymers as injectable in situ forming biodegradable scaffolds","volume":"26","author":"Balakrishnan","year":"2005","journal-title":"Biomaterials"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0009-8981(84)90097-4","article-title":"Determination of lysozyme in serum, urine, cerebrospinal fluid and feces by enzyme immunoassay","volume":"142","author":"Brouwer","year":"1984","journal-title":"Clin. Chim. Acta"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Kronberger, G., Burlacu, B., Kommenda, M., Winkler, S.M., and Affenzeller, M. (2024). Symbolic Regression, CRC Press.","DOI":"10.1201\/9781315166407"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"025065","DOI":"10.1088\/2632-2153\/ad513a","article-title":"Symbolic regression as a feature engineering method for machine and deep learning regression tasks","volume":"5","author":"Shmuel","year":"2024","journal-title":"Mach. Learn. Sci. Technol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4492","DOI":"10.1109\/TSMC.2018.2853719","article-title":"Multifactorial genetic programming for symbolic regression problems","volume":"50","author":"Zhong","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1162\/evco_a_00278","article-title":"Improving model-based genetic programming for symbolic regression of small expressions","volume":"29","author":"Virgolin","year":"2021","journal-title":"Evol. Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"22071","DOI":"10.1073\/pnas.1900654116","article-title":"Definitions, methods, and applications in interpretable machine learning","volume":"116","author":"Murdoch","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s10462-023-10622-0","article-title":"Interpretable scientific discovery with symbolic regression: A review","volume":"57","author":"Makke","year":"2024","journal-title":"Artif. Intell. Rev."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Keren, L.S., Liberzon, A., and Lazebnik, T. (2023). A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-28328-2"},{"key":"ref_52","unstructured":"La Cava, W., Orzechowski, P., Burlacu, B., de Fran\u00e7a, F.O., Virgolin, M., Jin, Y., Kommenda, M., and Moore, J.H. (2021). Contemporary symbolic regression methods and their relative performance. arXiv."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Radwan, Y.A., Kronberger, G., and Winkler, S. (2024). A comparison of recent algorithms for symbolic regression to genetic programming. arXiv.","DOI":"10.1007\/978-3-031-82949-9_15"},{"key":"ref_54","unstructured":"Cranmer, M. (2023). Interpretable machine learning for science with PySR and SymbolicRegression.jl. arXiv."},{"key":"ref_55","unstructured":"Stephens, T. (2025, March 26). gplearn: Genetic Programming in Python, with a Scikit-Learn Inspired API. Available online: https:\/\/github.com\/trevorstephens\/gplearn."},{"key":"ref_56","first-page":"2171","article-title":"DEAP: Evolutionary algorithms made easy","volume":"13","author":"Fortin","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Olson, R.S., Bartley, N., Urbanowicz, R.J., and Moore, J.H. (2016, January 20\u201324). Evaluation of a Tree-based Pipeline Optimization Tool for Automating Data Science. Proceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO\u201916, New York, NY, USA.","DOI":"10.1145\/2908812.2908918"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"e103","DOI":"10.7717\/peerj-cs.103","article-title":"SymPy: Symbolic computing in Python","volume":"3","author":"Meurer","year":"2017","journal-title":"PeerJ Comput. Sci."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1021\/ac60214a047","article-title":"Smoothing and differentiation of data by simplified least squares procedures","volume":"36","author":"Savitzky","year":"1964","journal-title":"Anal. Chem."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"123412","DOI":"10.1016\/j.cej.2019.123412","article-title":"A new formulation for symbolic regression to identify physico-chemical laws from experimental data","volume":"387","author":"Neumann","year":"2020","journal-title":"Chem. Eng. J."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.1007\/s00521-017-3208-0","article-title":"New predictive method for estimation of natural gas hydrate formation temperature using genetic programming","volume":"31","author":"Abooali","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1109\/TEVC.2017.2683489","article-title":"Feature selection to improve generalization of genetic programming for high-dimensional symbolic regression","volume":"21","author":"Chen","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Sikorski, D., Gzyra-Jagie\u0142a, K., and Draczy\u0144ski, Z. (2021). The kinetics of chitosan degradation in organic acid solutions. Mar. Drugs, 19.","DOI":"10.3390\/md19050236"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"4845","DOI":"10.1021\/jf001469g","article-title":"Kinetics and products of the degradation of chitosan by hydrogen peroxide","volume":"49","author":"Chang","year":"2001","journal-title":"J. Agric. Food Chem."}],"container-title":["Processes"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-9717\/13\/7\/1981\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:57:03Z","timestamp":1760032623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-9717\/13\/7\/1981"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,23]]},"references-count":64,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["pr13071981"],"URL":"https:\/\/doi.org\/10.3390\/pr13071981","relation":{},"ISSN":["2227-9717"],"issn-type":[{"type":"electronic","value":"2227-9717"}],"subject":[],"published":{"date-parts":[[2025,6,23]]}}}