{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T03:40:10Z","timestamp":1773027610153,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T00:00:00Z","timestamp":1666310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003443","name":"Moscow Center for Fundamental and Applied Mathematics","doi-asserted-by":"publisher","award":["075-15-2019-1624"],"award-info":[{"award-number":["075-15-2019-1624"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003443","name":"Moscow Center for Fundamental and Applied Mathematics","doi-asserted-by":"publisher","award":["075-15-2019-1624"],"award-info":[{"award-number":["075-15-2019-1624"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003443","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["075-15-2019-1624"],"award-info":[{"award-number":["075-15-2019-1624"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003443","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["075-15-2019-1624"],"award-info":[{"award-number":["075-15-2019-1624"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We develop a numerical method for solving three-dimensional problems of fluid filtration and absorption in a piecewise homogeneous medium by means of boundary integral equations. This method is applied to a simulation of the lymph flow in a lymph node. The lymph node is considered as a piecewise homogeneous domain containing porous media. The lymph flow is described by Darcy\u2019s law. Taking into account the lymph absorption, we propose an integral representation for the velocity and pressure fields, where the lymph absorption imitates the lymph outflow from a lymph node through a system of capillaries. The original problem is reduced to a system of boundary integral equations, and a numerical algorithm for solving this system is provided. We simulate the lymph velocity and pressure as well as the total lymph flux. The method is verified by comparison with experimental data.<\/jats:p>","DOI":"10.3390\/a15100388","type":"journal-article","created":{"date-parts":[[2022,10,23]],"date-time":"2022-10-23T20:43:50Z","timestamp":1666557830000},"page":"388","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Computational Modeling of Lymph Filtration and Absorption in the Lymph Node by Boundary Integral Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1173-0976","authenticated-orcid":false,"given":"Alexey","family":"Setukha","sequence":"first","affiliation":[{"name":"Moscow Center of Fundamental and Applied Mathematics at INM RAS, 119333 Moscow, Russia"},{"name":"Research Computing Center, Lomonosov Moscow State University, 119992 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1814-9140","authenticated-orcid":false,"given":"Rufina","family":"Tretiakova","sequence":"additional","affiliation":[{"name":"Moscow Center of Fundamental and Applied Mathematics at INM RAS, 119333 Moscow, Russia"},{"name":"Marchuk Institute of Numerical Mathematics of the RAS, 119333 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1115\/1.3449641","article-title":"Natural convection in enclosed porous media with rectangular boundaries","volume":"92","author":"Chan","year":"1970","journal-title":"Heat Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1115\/1.3244544","article-title":"A numerical study of natural convection in a horizontal porous layer subjected to an end-to-end temperature difference","volume":"103","author":"Hickox","year":"1981","journal-title":"Heat Transf."},{"key":"ref_3","unstructured":"Gartling, D.K., and Hickox, C.E. (1982). MARIAH: A Finite-Element Computer Program for Incompressible Porous Flow Problems. Theoretical Background, NASA. NASA STI\/Recon Technical Report N."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1115\/1.3246629","article-title":"Convective heat transfer in a rectangular porous cavity\u2014effect of aspect ratio on flow structure and heat transfer","volume":"106","author":"Prasad","year":"1984","journal-title":"Heat Transf."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Chen, Z., Huan, G., and Ma, Y. (2006). Computational Methods for Multiphase Flows in Porous Media, SIAM.","DOI":"10.1137\/1.9780898718942"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1137\/S106482750240443X","article-title":"Iterative solution methods for modeling multiphase flow in porous media fully implicitly","volume":"25","author":"Lacroix","year":"2003","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1134\/S1995080216050097","article-title":"Nonlinear finite volume method with discrete maximum principle for the two-phase flow model","volume":"37","author":"Nikitin","year":"2016","journal-title":"Lobachevskii J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/j.jcp.2019.06.009","article-title":"Finite volume method for coupled subsurface flow problems, I: Darcy problem","volume":"395","author":"Terekhov","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s11538-015-0128-y","article-title":"An image-based model of fluid flow through lymph nodes","volume":"78","author":"Cooper","year":"2016","journal-title":"Bull. Math. Biol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1089\/lrb.2015.0028","article-title":"Modeling lymph flow and fluid exchange with blood vessels in lymph nodes","volume":"13","author":"Jafarnejad","year":"2015","journal-title":"Lymphat. Res. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF02120313","article-title":"A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles","volume":"1","author":"Brinkman","year":"1949","journal-title":"Flow Turbul. Combust."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.enganabound.2017.12.006","article-title":"A Stokes\u2013Brinkman model of the fluid flow in a periodic cell with a porous body using the boundary element method","volume":"88","author":"Mardanov","year":"2018","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1007\/s11012-018-0832-4","article-title":"A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation","volume":"53","author":"Nishad","year":"2018","journal-title":"Meccanica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s10665-020-10083-2","article-title":"The method of fundamental solutions for Brinkman flows. Part I. Exterior domains","volume":"126","author":"Karageorghis","year":"2021","journal-title":"J. Eng. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"021902","DOI":"10.1063\/1.4941258","article-title":"Swimming in a two-dimensional Brinkman fluid: Computational modeling and regularized solutions","volume":"28","author":"Leiderman","year":"2016","journal-title":"Phys. Fluids"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1016\/j.camwa.2014.08.002","article-title":"Meshfree methods for nonhomogeneous Brinkman flows","volume":"68","author":"Martins","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_17","unstructured":"Piven, V. (2015). Mathematical Models of Fluid Filtration, Orel State University. (In Russian)."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1515\/rnam-2002-0106","article-title":"Mathematical modelling of the three-dimensional boundary value problem of the discharge of the well system in a homogeneous layer","volume":"17","author":"Lifanov","year":"2002","journal-title":"Russ. J. Numer. Anal. Math. Model."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4725","DOI":"10.1182\/blood-2009-10-250118","article-title":"Global lymphoid tissue remodeling during a viral infection is orchestrated by a B cell\u2013lymphotoxin-dependent pathway","volume":"115","author":"Kumar","year":"2010","journal-title":"Blood"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"16534","DOI":"10.1038\/srep16534","article-title":"Organ-wide 3D-imaging and topological analysis of the continuous microvascular network in a murine lymph node","volume":"5","author":"Kelch","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Harisinghani, M.G. (2012). Atlas of Lymph Node Anatomy, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-9767-8"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1152\/physrev.00037.2011","article-title":"Interstitial fluid and lymph formation and transport: Physiological regulation and roles in inflammation and cancer","volume":"92","author":"Wiig","year":"2012","journal-title":"Physiol. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1182","DOI":"10.1134\/S0012266119090076","article-title":"Methods of potential theory in a filtration problem for a viscous fluid","volume":"55","author":"Setukha","year":"2019","journal-title":"Differ. Equat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2076","DOI":"10.1134\/S0965542520120131","article-title":"Numerical Solution of a Stationary Filtration Problem of Viscous Fluid in a Piecewise Homogeneous Porous Medium by Applying the Boundary Integral Equation Method","volume":"60","author":"Setukha","year":"2020","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_25","unstructured":"Colton, D., and Kress, R. (1983). Integral Equation Methods in Scattering Theory, Wiley."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1134\/S1995080221060305","article-title":"Filtration of Viscous Fluid in Homogeneous Domain with Mixed Boundary Condition","volume":"42","author":"Tretiakova","year":"2021","journal-title":"Lobachevskii J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Lifanov, I.K. (1996). Singular Integral Equations and Discrete Vortices, VSP.","DOI":"10.1515\/9783110926040"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Katz, J., and Plotkin, A. (2001). Low-Speed Aerodynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511810329"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"H351","DOI":"10.1152\/ajpheart.1982.243.3.H351","article-title":"Quantitation of changes in lymph protein concentration during lymph node transit","volume":"243","author":"Adair","year":"1982","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"H616","DOI":"10.1152\/ajpheart.1983.245.4.H616","article-title":"Modification of lymph by lymph nodes. II. Effect of increased lymph node venous blood pressure","volume":"245","author":"Adair","year":"1983","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"H777","DOI":"10.1152\/ajpheart.1985.249.4.H777","article-title":"Modification of lymph by lymph nodes. III. Effect of increased lymph hydrostatic pressure","volume":"249","author":"Adair","year":"1985","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Tretiakova, R., Setukha, A., Savinkov, R., Grebennikov, D., and Bocharov, G. (2021). Mathematical Modeling of Lymph Node Drainage Function by Neural Network. Mathematics, 9.","DOI":"10.3390\/math9233093"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/388\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:00:31Z","timestamp":1760144431000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/388"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,21]]},"references-count":32,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["a15100388"],"URL":"https:\/\/doi.org\/10.3390\/a15100388","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,21]]}}}