{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T11:09:41Z","timestamp":1773659381390,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Electric impedance spectrum (EIS) is attracting attention in many areas of science, ranging from electrochemistry and material science to medical diagnosis. Interestingly, theoretical description often stops at material constants and specific physical mechanisms are represented by equivalent circuit elements, which is also motivated by the common use of various bridge methods. This specifically applies to biological samples, which exhibit a rich variety of responses to the electric field. Here, we present a step further from the description that utilizes equivalent circuit elements. We demonstrate how alteration of the mesoscopic structure affects the EIS in a biological sample: a cucumber under thermal treatment that comprises a cooling and warming phase. As the freezing temperature of water is exceeded during the cycle, the cucumber becomes frosted, which leads to unrecoverable changes in the internal structure, with no change of chemical composition. The experimental evidence is complemented by theoretical analysis, based on a novel approach to modeling non-stationary problems, derived from the stationary Poisson\u2013Boltzmann equation. We demonstrate a qualitative agreement between the theoretical and the experimental results, and discuss the procedure for tuning the model. We also demonstrate that, of the temperature variations of the position of the beta dispersion, the one related to the mesoscopic structure, can be used to assess the ionic strength of the material, determine the microscopic diffusion constant, or reflect the changes in mesoscopic structure, depending on experimental protocol.<\/jats:p>","DOI":"10.3390\/e28030291","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T13:11:36Z","timestamp":1772629896000},"page":"291","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Non-Stationary Model for Analysis of Impedance Spectra of Biological Samples"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1649-1671","authenticated-orcid":false,"given":"Gabriela","family":"Janik","sequence":"first","affiliation":[{"name":"Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00662 Warszawa, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-4651-0985","authenticated-orcid":false,"given":"Urszula","family":"Kami\u0144ska","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00662 Warszawa, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5513-8750","authenticated-orcid":false,"given":"Marta","family":"Kasprzyk","sequence":"additional","affiliation":[{"name":"Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00664 Warszawa, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6745-0044","authenticated-orcid":false,"given":"Leszek","family":"Niedzicki","sequence":"additional","affiliation":[{"name":"Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00664 Warszawa, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0030-3194","authenticated-orcid":false,"given":"Teodor","family":"Buchner","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00662 Warszawa, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253702","DOI":"10.1063\/1.4885087","article-title":"Complex permittivities of breast tumor tissues obtained from cancer surgeries","volume":"104","author":"Sugitani","year":"2014","journal-title":"Appl. Phys. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1109\/TBME.2004.836523","article-title":"Skin cancer identification using multifrequency electrical impedance\u2013a potential screening tool","volume":"51","author":"Aberg","year":"2004","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1111\/j.0909-725X.2005.00125.x","article-title":"Non-invasive and microinvasive electrical impedance spectra of skin cancer\u2014A comparison between two techniques","volume":"11","author":"Aberg","year":"2005","journal-title":"Skin Res. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"491","DOI":"10.33963\/KP.a2023.0070","article-title":"Validity of the Pneumonitor for RR intervals acquisition for short-term heart rate variability analysis extended with respiratory data in pediatric cardiac patients","volume":"81","author":"Wieniawski","year":"2023","journal-title":"Kardiol. Pol."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bielecka-D\u0105browa, A., Zasada, I., Olejniczak, W., Sakowicz, A., and \u015alot, M. (2024). Abstract 4116486: Association of assessment of clinical congestion in patients with heart failure compared with assessment using bioimpedance and microwave spectroscopy. Circulation, 150.","DOI":"10.1161\/circ.150.suppl_1.4116486"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gonzalez, C.A., Valencia, J.A., Mora, A., Gonzalez, F., Velasco, B., Porras, M.A., Salgado, J., Polo, S.M., Hevia-Montiel, N., and Cordero, S. (2013). Volumetric Electromagnetic Phase-Shift Spectroscopy of Brain Edema and Hematoma. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0063223"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"15454","DOI":"10.1038\/s41598-021-94600-y","article-title":"A bioimpedance-based monitor for real-time detection and identification of secondary brain injury","volume":"11","author":"Everitt","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"D\u2019Ambrogio, G., Zahhaf, O., Le, M.Q., Gouriou, Y., Josset, L., Pialoux, V., Lermusiaux, P., Capsal, J.F., Cottinet, P.J., and Schiava, N.D. (2022). Investigation of Blood Coagulation Using Impedance Spectroscopy: Toward Innovative Biomarkers to Assess Fibrinogenesis and Clot Retraction. Biomedicines, 10.","DOI":"10.3390\/biomedicines10081833"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1038\/s43856-024-00436-8","article-title":"Realtime monitoring of thrombus formation in vivo using a self-reporting vascular access graft","volume":"4","author":"Hoare","year":"2024","journal-title":"Commun. Med."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3302","DOI":"10.1039\/C7AY00714K","article-title":"Electrochemical impedance spectroscopy of blood for sensitive detection of blood hematocrit, sedimentation and dielectric properties","volume":"9","author":"Zhbanov","year":"2017","journal-title":"Anal. Methods"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bielicka, N., Stankiewicz, A., Misztal, T., Koca\u0144da, S., Chabielska, E., and Gromotowicz-Pop\u0142awska, A. (2023). PECAM-1\/Thrombus Ratio Correlates with Blood Loss during Off-Pump Coronary Artery Bypass Grafting (OPCAB) Surgery: A Preliminary Study. Int. J. Mol. Sci., 24.","DOI":"10.3390\/ijms241713254"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Marcinczyk, N., Misztal, T., Gromotowicz-Poplawska, A., Zebrowska, A., Rusak, T., Radziwon, P., and Chabielska, E. (2021). Utility of Platelet Endothelial Cell Adhesion Molecule 1 in the Platelet Activity Assessment in Mouse and Human Blood. Int. J. Mol. Sci., 22.","DOI":"10.3390\/ijms22179611"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2251","DOI":"10.1088\/0031-9155\/41\/11\/002","article-title":"The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz","volume":"41","author":"Gabriel","year":"1996","journal-title":"Phys. Med. Biol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2271","DOI":"10.1088\/0031-9155\/41\/11\/003","article-title":"The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues","volume":"41","author":"Gabriel","year":"1996","journal-title":"Phys. Med. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Iftime, A., Scheau, C., Babe\u0219, R.M., Ionescu, D., Periferakis, A., and C\u0103linescu, O. (2025). Confounding Factors and Their Mitigation in Measurements of Bioelectrical Impedance at the Skin Interface. Bioengineering, 12.","DOI":"10.3390\/bioengineering12090926"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Goyal, K., Borkholder, D.A., and Day, S.W. (2022). Dependence of Skin-Electrode Contact Impedance on Material and Skin Hydration. Sensors, 22.","DOI":"10.3390\/s22218510"},{"key":"ref_17","unstructured":"von Hippel, A.R. (1995). Dielectrics and Waves, Artech House Microwave Library, Artech House. [illustrated reprint ed.]."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"012073","DOI":"10.1088\/1742-6596\/224\/1\/012073","article-title":"Alpha-dispersion in human tissue","volume":"224","author":"Grimnes","year":"2010","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1134\/S1061933X10050169","article-title":"Theory of broadband dispersion of permittivity of biological cell suspensions","volume":"72","author":"Shilov","year":"2010","journal-title":"Colloid J."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Eisenberg, R.S. (2021). Maxwell equations without a polarization field, using a paradigm from biophysics. Entropy, 23.","DOI":"10.3390\/e23020172"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Shee Da En, Z., Mhd Noor, E.E., Ahmed Kayani, A., Hussin, M.H., and Farrukh Baig, M. (2024). Electrochemical Impedance Spectroscopy in the Determination of the Dielectric Properties of Tau-441 Protein for Dielectrophoresis Response Prediction. Bioengineering, 11.","DOI":"10.3390\/bioengineering11070698"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2661","DOI":"10.1021\/acs.macromol.3c02556","article-title":"Universal Scaling of DC Conductivity with Dielectric Interfacial Polarization in Conjugated Polymers","volume":"57","author":"Drakopoulos","year":"2024","journal-title":"Macromolecules"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1111\/j.1749-6632.1986.tb27143.x","article-title":"Impedance measurements as estimators of the properties of the extracellular space","volume":"481","author":"Eisenberg","year":"1986","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_24","unstructured":"Schwan, H.P. (2002). Electrical properties of tissues and cell suspensions: Mechanisms and models. Proceedings of the 16th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, IEEE."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Schwan, H.P. (1957). Electrical properties of tissue and cell suspensions. Advances in Biological and Medical Physics, Elsevier.","DOI":"10.1016\/B978-1-4832-3111-2.50008-0"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Agarwal, D., Randall, P., White, Z., Bisnette, B., Dickson, J., Allen, C., Chamani, F., Prakash, P., Ade, C., and Natarajan, B. (2022). A Non-Invasive Hydration Monitoring Technique Using Microwave Transmission and Data-Driven Approaches. Sensors, 22.","DOI":"10.3390\/s22072536"},{"key":"ref_27","unstructured":"Martinsen, O.G., and Grimnes, S. (2008). Bioimpedance and Bioelectricity Basics, Academic Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1109\/TBME.2004.836499","article-title":"Cole Electrical Impedance Model\u2014A Critique and an Alternative","volume":"52","author":"Grimnes","year":"2005","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_29","unstructured":"Gabrielli, C. (1984). Identification of Electrochemical Processes by Frequency Response Analysis, Solartron. Technical Report No 004."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Magar, H.S., Hassan, R.Y.A., and Mulchandani, A. (2021). Electrochemical Impedance Spectroscopy (EIS): Principles, Construction, and Biosensing Applications. Sensors, 21.","DOI":"10.3390\/s21196578"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"142939","DOI":"10.1016\/j.electacta.2023.142939","article-title":"Electrochemical impedance spectroscopy beyond linearity and stationarity\u2014A critical review","volume":"466","author":"Hallemans","year":"2023","journal-title":"Electrochim. Acta"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/S0006-3495(61)86894-X","article-title":"Anomalous impedance, a phenomenological property of time-variant resistance. An analytic review","volume":"1","author":"Mauro","year":"1961","journal-title":"Biophys. J."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Huh, S., Kim, H.J., Lee, S., Cho, J., Jang, A., and Bae, J. (2021). Utilization of electrical impedance spectroscopy and image classification for non-invasive early assessment of meat freshness. Sensors, 21.","DOI":"10.3390\/s21031001"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1085\/jgp.25.1.29","article-title":"Rectification and inductance in the squid giant axon","volume":"25","author":"Cole","year":"1941","journal-title":"J. Gen. Physiol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"159","DOI":"10.5604\/01.3001.0054.9268","article-title":"Capacitive coupling between the heart and tissue and its mathematical representation in future problems","volume":"20","author":"Polak","year":"2024","journal-title":"Bio-Algorithms Med-Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.bbe.2025.03.001","article-title":"Physically motivated projection of the electrocardiogram\u2014A feasibility study","volume":"45","author":"Wildowicz","year":"2025","journal-title":"Biocybern. Biomed. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1002\/elps.202200289","article-title":"The surface conductance of red blood cells and platelets is modulated by the cell membrane potential","volume":"44","author":"Hughes","year":"2023","journal-title":"Electrophoresis"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.physa.2019.03.056","article-title":"On the physical nature of biopotentials, their propagation and measurement","volume":"525","author":"Buchner","year":"2019","journal-title":"Phys. A: Stat. Mech. Its Appl."},{"key":"ref_39","unstructured":"Buchner, T. (2022). Molecular Teory of Biopotential, Oficyna Wydawnicza Politechniki Warszawskiej."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Pietak, A., and Levin, M. (2016). Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine. Front. Bioeng. Biotechnol., 4.","DOI":"10.3389\/fbioe.2016.00055"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/S0006-3495(83)84368-9","article-title":"Effect of tortuous extracellular pathways on resistance measurements","volume":"42","author":"Mathias","year":"1983","journal-title":"Biophys. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1606","DOI":"10.1529\/biophysj.103.039495","article-title":"A Model of Effective Diffusion and Tortuosity in the Extracellular Space of the Brain","volume":"87","author":"Hrabe","year":"2004","journal-title":"Biophys. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1039\/tf9575300687","article-title":"The electrical conductivity of ice","volume":"53","author":"Bradley","year":"1957","journal-title":"Trans. Faraday Soc."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Barthel, J., Wachter, R., and Gores, H.J. (1979). Temperature Dependence of Conductance of Electrolytes in Nonaqueous Solutions. Modern Aspects of Electrochemistry, Springer.","DOI":"10.1007\/978-1-4615-7455-2_1"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"462591","DOI":"10.1016\/j.chroma.2021.462591","article-title":"Methodology of measurement of ionic strength based on field-flow fractionation","volume":"1658","author":"Rah","year":"2021","journal-title":"J. Chromatogr. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"291","DOI":"10.21273\/JASHS.135.4.291","article-title":"Transcriptional Profiling of Rapidly Growing Cucumber Fruit by 454-Pyrosequencing Analysis","volume":"135","author":"Ando","year":"2010","journal-title":"J. Am. Soc. Hortic. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Karabaliev, M., Paarvanova, B., Savova, G., Tacheva, B., and Georgieva, R. (2025). In Situ Monitoring of Morphology Changes and Oxygenation State of Human Erythrocytes During Surfactant-Induced Hemolysis. Cells, 14.","DOI":"10.3390\/cells14070469"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"120554","DOI":"10.1016\/j.molliq.2022.120554","article-title":"The true H\u00fcckel equation for electrolyte solutions and its relation with the Born term","volume":"368","author":"Silva","year":"2022","journal-title":"J. Mol. Liq."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/291\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:22:37Z","timestamp":1773656557000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/291"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,4]]},"references-count":48,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030291"],"URL":"https:\/\/doi.org\/10.3390\/e28030291","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,4]]}}}