{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T12:45:30Z","timestamp":1777293930895,"version":"3.51.4"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,3,10]],"date-time":"2017-03-10T00:00:00Z","timestamp":1489104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Multiple myeloma (MM) is a genetically complex hematological cancer that is characterized by proliferation of malignant plasma cells in the bone marrow. MM evolves from the clonal premalignant disorder monoclonal gammopathy of unknown signi\ufb01cance (MGUS) by sequential genetic changes involving many different genes, resulting in dysregulated growth of multiple clones of plasma cells. The migration, survival, and proliferation of these clones require the direct and indirect interactions with the non-hematopoietic cells of the bone marrow. We develop a hybrid discrete-continuous model of MM development from the MGUS stage. The discrete aspect of the modelisobservedatthecellularlevel: cellsarerepresentedasindividualobjectswhichmove,interact, divide, and die by apoptosis. Each of these actions is regulated by intracellular and extracellular processes as described by continuous models. The hybrid model consists of the following submodels that have been simpli\ufb01ed from the much more complex state of evolving MM: cell motion due to chemotaxis, intracellular regulation of plasma cells, extracellular regulation in the bone marrow, and acquisition of mutations upon cell division. By extending a previous, simpler model in which the extracellular matrix was considered to be uniformly distributed, the new hybrid model provides a more accurate description in which cytokines are produced by the marrow microenvironment and consumed by the myeloma cells. The complex multiple genetic changes in MM cells and the numerous cell-cell and cytokine-mediated interactions between myeloma cells and their marrow microenviroment are simpli\ufb01ed in the model such that four related but evolving MM clones can be studied as they compete for dominance in the setting of intraclonal heterogeneity.<\/jats:p>","DOI":"10.3390\/computation5010016","type":"journal-article","created":{"date-parts":[[2017,3,10]],"date-time":"2017-03-10T09:39:55Z","timestamp":1489138795000},"page":"16","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity"],"prefix":"10.3390","volume":"5","author":[{"given":"Anass","family":"Bouchnita","sequence":"first","affiliation":[{"name":"Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France"},{"name":"Laboratoire de Biom\u00e9trie et Biologie Evolutive, UMR 5558 CNRS, University Lyon 1, Villeurbanne 69622, France"},{"name":"Mohammadia School of Engineering, Universit\u00e9 Mohamed V, Rabat 10080, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fatima-Ezzahra","family":"Belmaati","sequence":"additional","affiliation":[{"name":"Mohammadia School of Engineering, Universit\u00e9 Mohamed V, Rabat 10080, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rajae","family":"Aboulaich","sequence":"additional","affiliation":[{"name":"Mohammadia School of Engineering, Universit\u00e9 Mohamed V, Rabat 10080, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7303-7694","authenticated-orcid":false,"given":"Mark","family":"Koury","sequence":"additional","affiliation":[{"name":"Vanderbilt University Medical Center, Nashville, TN 37232-6307, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vitaly","family":"Volpert","sequence":"additional","affiliation":[{"name":"Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France"},{"name":"Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia"},{"name":"INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne 60603, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1056\/NEJMra1011442","article-title":"Multiple myeloma","volume":"364","author":"Palumbo","year":"2011","journal-title":"N. Engl. J. Med."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1038\/nrc3257","article-title":"The genetic architecture of multiple myeloma","volume":"12","author":"Morgan","year":"2012","journal-title":"Nat. Rev. Cancer"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0025-5564(86)90119-7","article-title":"A simplified mathematical model of tumor growth","volume":"81","author":"Adam","year":"1986","journal-title":"Math. Biosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1115\/1.3426723","article-title":"Instability and mitotic patterns in tissue growth","volume":"95","author":"Glass","year":"1973","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0025-5564(78)90033-0","article-title":"Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth","volume":"39","author":"McElwain","year":"1978","journal-title":"Math. Biosci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0025-5564(94)00117-3","article-title":"Growth of non-necrotic tumours in the presence and absence of inhibitors","volume":"130","author":"Byrne","year":"1995","journal-title":"Math. Biosci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0025-5564(96)00023-5","article-title":"Growth of Necrotic Tumours in the Presence and Absence of Inhibitors","volume":"135","author":"Byrne","year":"1996","journal-title":"Math. Biosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/j.jtbi.2008.03.027","article-title":"Three-dimensional multispecies nonlinear tumor growth\u2014I: Model and numerical method","volume":"253","author":"Wise","year":"2008","journal-title":"J. Theor. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20140079","DOI":"10.1098\/rsif.2014.0079","article-title":"Clonal selection and therapy resistance in acute leukaemias: Mathematical modelling explains different proliferation patterns at diagnosis and relapse","volume":"11","author":"Stiehl","year":"2014","journal-title":"J. R. Soc. Interface"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Walenda, T., Stiehl, T., Braun, H., Fr\u00f6bel, J., Ho, A.D., Schroeder, T., Goecke, T.W., Rath, B., Germing, U., and Marciniak-Czohra, A. (2014). Feedback Signals in Myelodysplastic Syndromes: Increased Self-Renewal of the Malignant Clone Suppresses Normal Hematopoiesis. PLoS Comp. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003599"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/S0025-5564(97)00080-1","article-title":"A mathematical model of drug resistance: Heterogeneous tumors","volume":"147","author":"Panetta","year":"1998","journal-title":"Math. Biosci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1140\/epjb\/e2008-00249-y","article-title":"Studying the emergence of invasiveness in tumours using game theory","volume":"63","author":"Basanta","year":"2008","journal-title":"Eur. Phys. J. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1917","DOI":"10.1038\/sj.bjc.6605071","article-title":"Migration rules: Tumours are conglomerates of self-metastases","volume":"100","author":"Enderling","year":"2009","journal-title":"Br. J. Cancer"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jtbi.2009.02.008","article-title":"A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth","volume":"258","author":"Piotrowska","year":"2009","journal-title":"J. Theor. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1088\/1478-3975\/2\/3\/001","article-title":"A single-cell-based model of tumor growth in vitro: Monolayers and spheroids","volume":"2","author":"Drasdo","year":"2005","journal-title":"Phys. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shirinifard, A., Gens, J.S., Zaitlen, B.L., Poplawski, N.J., Swat, M., and Glazier, J.A. (2009). 3D Multi-Cell Simulation of Tumor Growth and Angiogenesis. PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0007190"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Swat, M.H., Thomas, G.L., Shirinifard, A., Clandenon, S.G., and Glazier, J.A. (2015). Emergent Stratification in Solid Tumors Selects for Reduced Cohesion of Tumor Cells: A Multi-Cell, Virtual-Tissue Model of Tumor Evolution Using CompuCell3D. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0127972"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2326","DOI":"10.1016\/j.camwa.2009.08.041","article-title":"Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion","volume":"59","author":"Hatzikirou","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1088\/1478-3975\/3\/2\/001","article-title":"A cellular automaton model for the migration of glioma cells","volume":"3","author":"Aubert","year":"2006","journal-title":"Phys. Biol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1529\/biophysj.107.114678","article-title":"Modeling the influence of the E-cadherin-\u03b2-catenin pathway in cancer cell invasion: A multiscale approach","volume":"95","author":"Drasdo","year":"2008","journal-title":"Biophys. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"016008","DOI":"10.1088\/1478-3975\/6\/1\/016008","article-title":"Multi-scale modelling of cancer cell intravasation: The role of cadherins in metastasis","volume":"6","author":"Chaplain","year":"2009","journal-title":"Phys. Biol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/s00285-008-0211-1","article-title":"Multiscale agent-based cancer modeling","volume":"58","author":"Zhang","year":"2009","journal-title":"J. Math. Biol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1093\/imammb\/dqi005","article-title":"A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion","volume":"22","author":"Anderson","year":"2005","journal-title":"Math. Med. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.semcancer.2008.03.011","article-title":"Adaptation to hypoxia and acidosis in carcinogenesis and tumor progression","volume":"18","author":"Fang","year":"2008","journal-title":"Semin. Cancer Biol."},{"key":"ref_25","first-page":"729","article-title":"An evolutionary model for initiation, promotion, and progression in carcinogenesis","volume":"32","author":"Vincent","year":"2008","journal-title":"Int. J. Oncol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1158\/0008-5472.CAN-14-2103","article-title":"Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation","volume":"75","author":"Chisholm","year":"2015","journal-title":"Cancer Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1186\/1745-6150-5-28","article-title":"A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease","volume":"5","author":"Ayati","year":"2010","journal-title":"Biol. Direct"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1002\/ajh.24291","article-title":"Bone marrow infiltration by multiple myeloma causes anemia by reversible disruption of erythropoiesis","volume":"91","author":"Bouchnita","year":"2016","journal-title":"Am. J. Hematol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1182\/blood-2012-03-412981","article-title":"Intraclonal heterogeneity and distinct molecular mechanisms characterize the development of t(4;14) and t(11;14) myeloma","volume":"120","author":"Walker","year":"2012","journal-title":"Blood"},{"key":"ref_30","unstructured":"Bouchnita, A., Belmaati, F.E., Aboulaich, R., Ellaia, R., and Volpert, V. (2016, January 11\u201314). Mathematical modelling of intra-clonal heterogeneity in multiple myeloma. Proceedings of the CARI 2016, Hammamet, Tunisia."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1111\/bjh.12805","article-title":"The impact of intra-clonal heterogeneity on the treatment of multiple myeloma","volume":"165","author":"Brioli","year":"2014","journal-title":"Br. J. Haematol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1038\/leu.2014.13","article-title":"Single-cell genetic analysis reveals the composition of initiating clones and phylogenetic patterns of branching and parallel evolution in myeloma","volume":"28","author":"Melchor","year":"2014","journal-title":"Leukemia"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1182\/blood.V88.2.674.bloodjournal882674","article-title":"Dysregulation of cyclin D1 by translocation into an IgH gamma switch region in two multiple myeloma cell lines","volume":"88","author":"Chesi","year":"1996","journal-title":"Blood"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.canlet.2015.11.028","article-title":"Targeting SDF-1 in multiple myeloma tumor microenvironment","volume":"380","author":"Bouyssou","year":"2015","journal-title":"Cancer Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1080\/028418600750063488","article-title":"Homing of the myeloma cell clone","volume":"39","author":"Vanderkerken","year":"2000","journal-title":"Acta Oncol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1038\/nrc2189","article-title":"Understanding multiple myeloma pathogenesis in the bone marrow to identify new therapeutic targets","volume":"7","author":"Hideshima","year":"2007","journal-title":"Nat. Rev. Cancer"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.3324\/haematol.12349","article-title":"A bioluminescence imaging based in vivo model for preclinical testing of novel cellular immunotherapy strategies to improve the graft-versus-myeloma effect","volume":"93","author":"Rozemuller","year":"2008","journal-title":"Haematologica"},{"key":"ref_38","first-page":"157496","article-title":"Bone marrow microenvironment in multiple myeloma progression","volume":"2012","author":"Manier","year":"2012","journal-title":"BioMed Res. Int."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1126\/science.1069492","article-title":"Systems Biology: A brief overview","volume":"295","author":"Kitano","year":"2002","journal-title":"Science"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/5\/1\/16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:30:12Z","timestamp":1760207412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/5\/1\/16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,10]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["computation5010016"],"URL":"https:\/\/doi.org\/10.3390\/computation5010016","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,10]]}}}