-
Previous Article
Some remarks on the Gottman-Murray model of marital dissolution and time delays
- DCDS-B Home
- This Issue
-
Next Article
Multiplicity results for discrete anisotropic equations
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 & 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
| [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
Other articles
by authors
[Back to Top]