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January  2018, 23(1): 193-202. doi: 10.3934/dcdsb.2018013

Free boundary problems arising in biology

Avner Friedman

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.

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 & 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.  Google Scholar

[2]

A. Friedman, Free boundary problems in biology Proceeding Royal Society, 373 (2015). 20140368 (16 pages). doi: 10.1098/rsta.2014.0368.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Friedman and W. Hao, Mathematical modeling of liver fibrosis, Math. Biosc. and Bioengineering, 14 (2017), 143-164.  doi: 10.3934/mbe.2017010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman and C. Y. Kao, Mathematical Modeling of Biological Processes, Springer, 2014. doi: 10.1007/978-3-319-08314-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

W. Hao, E. Crouser and A. Friedman, A mathematical model of sarcoidosis, PNAS, 111 (2014), 16065-16070.  doi: 10.1073/pnas.1417789111.  Google Scholar

[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.  Google Scholar

[16]

C. Xue, A. Friedman and C. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.   Google Scholar

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.  Google Scholar

[2]

A. Friedman, Free boundary problems in biology Proceeding Royal Society, 373 (2015). 20140368 (16 pages). doi: 10.1098/rsta.2014.0368.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Friedman and W. Hao, Mathematical modeling of liver fibrosis, Math. Biosc. and Bioengineering, 14 (2017), 143-164.  doi: 10.3934/mbe.2017010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman and C. Y. Kao, Mathematical Modeling of Biological Processes, Springer, 2014. doi: 10.1007/978-3-319-08314-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

W. Hao, E. Crouser and A. Friedman, A mathematical model of sarcoidosis, PNAS, 111 (2014), 16065-16070.  doi: 10.1073/pnas.1417789111.  Google Scholar

[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.  Google Scholar

[16]

C. Xue, A. Friedman and C. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.   Google Scholar

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