American Institute of Mathematical Sciences

Advanced
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 $\Omega(t)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The plaque occupies the shaded area
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Interplay among cholesterol, macrophages and foam cells
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Avner Friedman, Bei Hu, Chuan Xue. A three dimensional model of wound healing: Analysis and computation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2691-2712. doi: 10.3934/dcdsb.2012.17.2691
[2]

Haiyan Wang, Shiliang Wu. Spatial dynamics for a model of epidermal wound healing. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1215-1227. doi: 10.3934/mbe.2014.11.1215
[3]

Sophia A. Maggelakis. Modeling the role of angiogenesis in epidermal wound healing. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 267-273. doi: 10.3934/dcdsb.2004.4.267
[4]

Shuai Zhang, L.R. Ritter, A.I. Ibragimov. Foam cell formation in atherosclerosis: HDL and macrophage reverse cholesterol transport. Conference Publications, 2013, 2013 (special) : 825-835. doi: 10.3934/proc.2013.2013.825
[5]

John A. Adam. Inside mathematical modeling: building models in the context of wound healing in bone. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 1-24. doi: 10.3934/dcdsb.2004.4.1
[6]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[7]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[8]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[9]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[10]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[11]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[12]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[13]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[14]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[15]

Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469
[16]

Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165
[17]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[18]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[19]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065
[20]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (134)
  • HTML views (123)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences