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

Received  December 2016 Revised  April 2017 Published  January 2018

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.

Citation: Avner Friedman. Free boundary problems arising in biology. Discrete & 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. FriedmanW. 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. FriedmanB. 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. FriedmanB. 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. FriedmanR. 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. HaoE. 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. XueA. 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. FriedmanW. 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. FriedmanB. 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. FriedmanB. 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. FriedmanR. 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. HaoE. 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. XueA. Friedman and C. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.

Figure 1.  A wound $W(t)$ with partially healed tissue $\Omega(t)$
Figure 2.  The plaque occupies the shaded area
Figure 3.  Interplay among cholesterol, macrophages and foam cells
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