Mass concentration in a nonlocal model of clonal selection
Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured...
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| Veröffentlicht in: | Journal of mathematical biology 2016-10, Vol.73 (4), p.1001-1033 |
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| creator | Busse, J.-E. Gwiazda, P. Marciniak-Czochra, A. |
| description | Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations. |
| doi_str_mv | 10.1007/s00285-016-0979-3 |
| format | Article |
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Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). 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Math. Biol</addtitle><addtitle>J Math Biol</addtitle><description>Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations.</description><subject>Applications of Mathematics</subject><subject>Cell Differentiation</subject><subject>Cell Proliferation</subject><subject>Clone Cells</subject><subject>Computer Simulation</subject><subject>Humans</subject><subject>Mathematical and Computational Biology</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Models, Biological</subject><subject>Neoplasms</subject><subject>Stem Cells</subject><issn>0303-6812</issn><issn>1432-1416</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><sourceid>EIF</sourceid><sourceid>BENPR</sourceid><recordid>eNp1kU1LxDAQhoMouq7-AC9S8OKlOpOvJh4EEb9A8aLnkE3TtZJNtNkV_Pd27SoqeMkQ5pl33uElZA_hCAGq4wxAlSgBZQm60iVbIyPkjJbIUa6TETBgpVRIt8h2zs8AWAmNm2SLSs0kMDYiJ3c258Kl6Hycd3bepli0sbBFTDEkZ0MxS7UPRWoKF1Ls_9kH75bcDtlobMh-d1XH5PHy4uH8ury9v7o5P7stnWBy3r-Ka6kryXUlKE6aCSgmlfVCKFqLyjOOSnusOeeVBuFr4HXD3AQ5ZcwhG5PTQfdlMZn5ejAazEvXzmz3bpJtze9ObJ_MNL0ZAaiAs17gcCXQpdeFz3Mza7PzIdjo0yIbVKgYpxxojx78QZ_TouvPHijgn4pjggPlupRz55tvMwhmmYwZkjF9MmaZjFma2P95xffEVxQ9QAcg96049d2P1f-qfgDoOpc_</recordid><startdate>20161001</startdate><enddate>20161001</enddate><creator>Busse, J.-E.</creator><creator>Gwiazda, P.</creator><creator>Marciniak-Czochra, A.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>CGR</scope><scope>CUY</scope><scope>CVF</scope><scope>ECM</scope><scope>EIF</scope><scope>NPM</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7TK</scope><scope>7TM</scope><scope>7U9</scope><scope>7X7</scope><scope>7XB</scope><scope>88A</scope><scope>88E</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FH</scope><scope>8FI</scope><scope>8FJ</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BBNVY</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>BHPHI</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>FYUFA</scope><scope>GHDGH</scope><scope>GNUQQ</scope><scope>H94</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>K9.</scope><scope>L6V</scope><scope>LK8</scope><scope>M0S</scope><scope>M1P</scope><scope>M7N</scope><scope>M7P</scope><scope>M7S</scope><scope>M7Z</scope><scope>P5Z</scope><scope>P62</scope><scope>P64</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>7X8</scope><scope>5PM</scope></search><sort><creationdate>20161001</creationdate><title>Mass concentration in a nonlocal model of clonal selection</title><author>Busse, J.-E. ; 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| subjects | Applications of Mathematics Cell Differentiation Cell Proliferation Clone Cells Computer Simulation Humans Mathematical and Computational Biology Mathematics Mathematics and Statistics Models, Biological Neoplasms Stem Cells |
| title | Mass concentration in a nonlocal model of clonal selection |
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