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January
2018, 23(1): 193-202.
doi: 10.3934/dcdsb.2018013
Free boundary problems arising in biology
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
The present paper describes the general structure of free boundary problems for systems of PDEs modeling biological processes. It then proceeds to review two recent examples of the evolution of a plaque in the artery, and of a granuloma in the lung. Simplified versions of these models are formulated, and rigorous mathematical results and open questions are stated.
Keywords:
Systems of partial differential equations,
wound healing,
atherosclerosis,
cholesterol,
granulomas.
Mathematics Subject Classification:
35Q92, 35R35, 37N25.
Citation:
Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B,
2018, 23
(1)
: 193-202.
doi: 10.3934/dcdsb.2018013
![]()
References:
| [1] |
C. S. Chou and A. Friedman,
Mathematical Introduction to Mathematical Biology, Springer, 2016.
doi: 10.1007/978-3-319-29638-8. |
| [2] |
A. Friedman, Free boundary problems in biology Proceeding Royal Society, 373 (2015). 20140368 (16 pages).
doi: 10.1098/rsta.2014.0368. |
| [3] |
A. Friedman,
Free boundary problems for systems of Stokes equations, Discrete and Continuous Dynamical Systems, 21 (2016), 1455-1468.
doi: 10.3934/dcdsb.2016006. |
| [4] |
A. Friedman and W. Hao,
A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors, Bull. Math. Biololgy, 77 (2015), 758-781.
doi: 10.1007/s11538-014-0010-3. |
| [5] |
A. Friedman, W. Hao and B. Hu,
A free boundary problem for steady small plaques in the artery and their stability, J. Diff. Eqs., 259 (2015), 1227-1255.
doi: 10.1016/j.jde.2015.02.002. |
| [6] |
A. Friedman and W. Hao,
Mathematical modeling of liver fibrosis, Math. Biosc. and Bioengineering, 14 (2017), 143-164.
doi: 10.3934/mbe.2017010. |
| [7] |
A. Friedman, B. Hu and C. Xue,
Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040, arXiv:0910.0039.
doi: 10.1137/090772630. |
| [8] |
A. Friedman, B. Hu and C. Xue,
A three dimensional model of wound healing: Analysis and computation, Discrete and Continuous Dynamical Systems, Ser. B, 17 (2012), 2691-2712.
doi: 10.3934/dcdsb.2012.17.2691. |
| [9] |
A. Friedman and C. Y. Kao,
Mathematical Modeling of Biological Processes, Springer, 2014.
doi: 10.1007/978-3-319-08314-8. |
| [10] |
A. Friedman and K. Y. Lam,
On the stability of steady states in a granuloma model, J. Diff. Eqs., 256 (2014), 3743-3769.
doi: 10.1016/j.jde.2014.02.019. |
| [11] |
A. Friedman and K. Y. Lam,
Analysis of a free boundary tumor model with angiogenesis, J. Diff. Eqs., 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
| [12] |
A. Friedman, R. Leander and C. Y. Kao,
Dynamics of radially symmetric granulomas, J. Math. Anal. Appl, 412 (2014), 776-791.
doi: 10.1016/j.jmaa.2013.11.017. |
| [13] |
W. Hao and A. Friedman, The LDL-HDL profile determine the risk of atherosclerosis: A mathematical model PLoS One, 9 (2014), e90497 (15 pages).
doi: 10.1371/journal.pone.0090497. |
| [14] |
W. Hao, E. Crouser and A. Friedman,
A mathematical model of sarcoidosis, PNAS, 111 (2014), 16065-16070.
doi: 10.1073/pnas.1417789111. |
| [15] |
W. Hao, L. Schlesinger and A. Friedman, Modeling granulomas in response to infection in the lung PLoS ONE, 11 (2016), e0148738.
doi: 10.1371/journal.pone.0148738. |
| [16] |
C. Xue, A. Friedman and C. Sen,
A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.
|
show all references
References:
| [1] |
C. S. Chou and A. Friedman,
Mathematical Introduction to Mathematical Biology, Springer, 2016.
doi: 10.1007/978-3-319-29638-8. |
| [2] |
A. Friedman, Free boundary problems in biology Proceeding Royal Society, 373 (2015). 20140368 (16 pages).
doi: 10.1098/rsta.2014.0368. |
| [3] |
A. Friedman,
Free boundary problems for systems of Stokes equations, Discrete and Continuous Dynamical Systems, 21 (2016), 1455-1468.
doi: 10.3934/dcdsb.2016006. |
| [4] |
A. Friedman and W. Hao,
A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors, Bull. Math. Biololgy, 77 (2015), 758-781.
doi: 10.1007/s11538-014-0010-3. |
| [5] |
A. Friedman, W. Hao and B. Hu,
A free boundary problem for steady small plaques in the artery and their stability, J. Diff. Eqs., 259 (2015), 1227-1255.
doi: 10.1016/j.jde.2015.02.002. |
| [6] |
A. Friedman and W. Hao,
Mathematical modeling of liver fibrosis, Math. Biosc. and Bioengineering, 14 (2017), 143-164.
doi: 10.3934/mbe.2017010. |
| [7] |
A. Friedman, B. Hu and C. Xue,
Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040, arXiv:0910.0039.
doi: 10.1137/090772630. |
| [8] |
A. Friedman, B. Hu and C. Xue,
A three dimensional model of wound healing: Analysis and computation, Discrete and Continuous Dynamical Systems, Ser. B, 17 (2012), 2691-2712.
doi: 10.3934/dcdsb.2012.17.2691. |
| [9] |
A. Friedman and C. Y. Kao,
Mathematical Modeling of Biological Processes, Springer, 2014.
doi: 10.1007/978-3-319-08314-8. |
| [10] |
A. Friedman and K. Y. Lam,
On the stability of steady states in a granuloma model, J. Diff. Eqs., 256 (2014), 3743-3769.
doi: 10.1016/j.jde.2014.02.019. |
| [11] |
A. Friedman and K. Y. Lam,
Analysis of a free boundary tumor model with angiogenesis, J. Diff. Eqs., 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
| [12] |
A. Friedman, R. Leander and C. Y. Kao,
Dynamics of radially symmetric granulomas, J. Math. Anal. Appl, 412 (2014), 776-791.
doi: 10.1016/j.jmaa.2013.11.017. |
| [13] |
W. Hao and A. Friedman, The LDL-HDL profile determine the risk of atherosclerosis: A mathematical model PLoS One, 9 (2014), e90497 (15 pages).
doi: 10.1371/journal.pone.0090497. |
| [14] |
W. Hao, E. Crouser and A. Friedman,
A mathematical model of sarcoidosis, PNAS, 111 (2014), 16065-16070.
doi: 10.1073/pnas.1417789111. |
| [15] |
W. Hao, L. Schlesinger and A. Friedman, Modeling granulomas in response to infection in the lung PLoS ONE, 11 (2016), e0148738.
doi: 10.1371/journal.pone.0148738. |
| [16] |
C. Xue, A. Friedman and C. Sen,
A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.
|
Figure 2.
The plaque occupies the shaded area
Figure 3.
Interplay among cholesterol, macrophages and foam cells
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