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Mathematical Biology: Modeling and Analysis
About this Title
Avner Friedman, Ohio State University, Columbus, OH
Publication: CBMS Regional Conference Series in Mathematics
Publication Year:
2018; Volume 127
ISBNs: 978-1-4704-4715-1 (print); 978-1-4704-4803-5 (online)
DOI: https://doi.org/10.1090/cbms/127
MathSciNet review: MR3793187
MSC: Primary 92D25; Secondary 35Q92, 35R35, 49J10, 92C42, 92C50
ReadershipTable of Contents
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Front/Back Matter
Chapters
- Introductory biology
- Introduction to modeling
- Models of population dynamics
- Cancer and the immune system
- Parameters estimation
- Mathematical analysis inspired by cancer models
- Mathematical model of artherosclerosis: Risk of high cholesterol
- Mathematical analysis inspired by the atherosclerosis model
- Mathematical models of chronic wounds
- Mathematical analysis inspired by the chronic wound model
- Introduction to PDEs
- H M Byrne, M A J Chaplain, D L Evans, and I Hopkinson, Mathematical modelling of angiogenesis in wound healing: comparison of theory and experiment, Computational and Mathematical Methods in Medicine 2 (2000), no. 3, 175–197.
- Duan Chen, Julie M. Roda, Clay B. Marsh, Timothy D. Eubank, and Avner Friedman, Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model, Bull. Math. Biol. 74 (2012), no. 11, 2752–2777. MR 2994755, DOI 10.1007/s11538-012-9776-3
- Xinfu Chen, Shangbin Cui, and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth: asymptotic behavior, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4771–4804. MR 2165387, DOI 10.1090/S0002-9947-05-03784-0
- Xinfu Chen and Avner Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal. 35 (2003), no. 4, 974–986. MR 2049029, DOI 10.1137/S0036141002418388
- Xinfu Chen, Jiaxing Hong, and Fahuai Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1705–1727. MR 1421209, DOI 10.1080/03605309608821243
- B J Coventry, P L Lee, D Gibbs, and D N J Hart, Dendritic cell density and activation status in human breast cancer–cd1a, cmrf-44, cmrf-56 and cd-83 expression, British Journal of Cancer 86 (2002), no. 4, 546–551.
- Elliott D Crouser, Daniel A Culver, Kenneth S Knox, Mark W Julian, Guohong Shao, Susamma Abraham, Sandya Liyanarachchi, Jennifer E Macre, Mark D Wewers, Mikhail A Gavrilin, et al., Gene expression profiling identifies mmp-12 and adamdec1 as potential pathogenic mediators of pulmonary sarcoidosis, American Journal of Respiratory and Critical Care Medicine 179 (2009), no. 10, 929–938.
- Shangbin Cui and Avner Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc. 355 (2003), no. 9, 3537–3590. MR 1990162, DOI 10.1090/S0002-9947-03-03137-4
- Shangbin Cui and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces Free Bound. 5 (2003), no. 2, 159–181. MR 1980470, DOI 10.4171/IFB/76
- Marco A. Fontelos and Avner Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal. 35 (2003), no. 3-4, 187–206. MR 2011787
- Avner Friedman and Fernando Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), no. 3, 262–284. MR 1684873, DOI 10.1007/s002850050149
- Avner Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound. 8 (2006), no. 2, 247–261. MR 2256843, DOI 10.4171/IFB/142
- Avner Friedman, A multiscale tumor model, Interfaces Free Bound. 10 (2008), no. 2, 245–262. MR 2453131, DOI 10.4171/IFB/188
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Avner Friedman and Wenrui Hao, A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors, Bull. Math. Biol. 77 (2015), no. 5, 758–781. MR 3350421, DOI 10.1007/s11538-014-0010-3
- Avner Friedman, Wenrui Hao, and Bei Hu, A free boundary problem for steady small plaques in the artery and their stability, J. Differential Equations 259 (2015), no. 4, 1227–1255. MR 3345849, DOI 10.1016/j.jde.2015.02.002
- Avner Friedman and Bei Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations 227 (2006), no. 2, 598–639. MR 2237681, DOI 10.1016/j.jde.2005.09.008
- Avner Friedman and Bei Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 293–330. MR 2210911, DOI 10.1007/s00205-005-0408-z
- Avner Friedman and Bei Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal. 39 (2007), no. 1, 174–194. MR 2318381, DOI 10.1137/060656292
- Avner Friedman and Bei Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl. 327 (2007), no. 1, 643–664. MR 2277439, DOI 10.1016/j.jmaa.2006.04.034
- Avner Friedman and Bei Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5291–5342. MR 2415075, DOI 10.1090/S0002-9947-08-04468-1
- Avner Friedman, Bei Hu, and Chiu-Yen Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol. 60 (2010), no. 6, 881–907. MR 2606518, DOI 10.1007/s00285-009-0290-7
- Avner Friedman, Bei Hu, and Chuan Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal. 42 (2010), no. 5, 2013–2040. MR 2684309, DOI 10.1137/090772630
- Avner Friedman, Bei Hu, and Chuan Xue, A three dimensional model of wound healing: analysis and computation, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 8, 2691–2712. MR 2959247, DOI 10.3934/dcdsb.2012.17.2691
- Avner Friedman and Chiu-Yen Kao, Mathematical modeling of biological processes, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2014. MR 3290270, DOI 10.1007/978-3-319-08314-8
- Avner Friedman and King-Yeung Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations 259 (2015), no. 12, 7636–7661. MR 3401608, DOI 10.1016/j.jde.2015.08.032
- Avner Friedman and Fernando Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1587–1634. MR 1806728, DOI 10.1090/S0002-9947-00-02715-X
- Avner Friedman and Chuan Xue, A mathematical model for chronic wounds, Math. Biosci. Eng. 8 (2011), no. 2, 253–261. MR 2793481, DOI 10.3934/mbe.2011.8.253
- Avner Friedman and Abdul-Aziz Yakubu, Anthrax epizootic and migration: persistence or extinction, Math. Biosci. 241 (2013), no. 1, 137–144. MR 3019701, DOI 10.1016/j.mbs.2012.10.004
- Avner Friedman and Abdul-Aziz Yakubu, A bovine babesiosis model with dispersion, Bull. Math. Biol. 76 (2014), no. 1, 98–135. MR 3150818, DOI 10.1007/s11538-013-9912-8
- Wenrui Hao, Elliott D. Crouser, and Avner Friedman, Mathematical model of sarcoidosis, Proc. Natl. Acad. Sci. USA 111 (2014), no. 45, 16065–16070. MR 3283037, DOI 10.1073/pnas.1417789111
- Wenrui Hao and Avner Friedman, The ldl-hdl profile determines the risk of atherosclerosis: a mathematical model, PloS ONE 9 (2014), e90497.
- Wenrui Hao, Larry S Schlesinger, and Avner Friedman, Modeling granulomas in response to infection in the lung., PLoS ONE 11 (2016), e0148738.
- Marc Hellerstein, M B Hanley, D Cesar, S Siler, C Papageorgopoulos, E Wieder, D Schmidt, R Hoh, R Neese, D Macallan, et al., Directly measured kinetics of circulating t lymphocytes in normal and hiv-1-infected humans., Nature Medicine 5 (1999), no. 1, 83–89.
- Haomin Huang and Mingxin Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 7, 2039–2050. MR 3423210, DOI 10.3934/dcdsb.2015.20.2039
- Yaodan Huang, Zhengce Zhang, and Bei Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl. 35 (2017), 483–502. MR 3595338, DOI 10.1016/j.nonrwa.2016.12.003
- Kwang Ik Kim, Zhigui Lin, and Lai Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Anal. Real World Appl. 11 (2010), no. 1, 313–322. MR 2570551, DOI 10.1016/j.nonrwa.2008.11.015
- Kang-Ling Liao, Xue-Feng Bai, and Avner Friedman, Mathematical modeling of interleukin-35 promoting tumor growth and angiogenesis, PLoS ONE 9 (2014), no. 10, e110126.
- El Mehdi Lotfi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi, Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations ID 186437 (2014).
- Zhien Ma, Yicang Zhou, and Jianhong Wu (eds.), Modeling and dynamics of infectious diseases, Series in Contemporary Applied Mathematics CAM, vol. 11, Higher Education Press, Beijing; World Scientific Publishing Co. Pte. Ltd., Singapore, 2009. Lecture notes from the China-Canada Joint Program on Infectious Disease Modeling held at Xi’an Jiaotong University, Xi’an, May 10–29, 2006. MR 2555773, DOI 10.1142/7223
- G. Pettet, M. A. J. Chaplain, D. L. S. Mcelwain, and H. M. Byrne, On the role of angiogenesis in wound healing, Proceedings of the Royal Society B: Biological Sciences 263 (1996), 1487–1493.
- G J Pettet, H M Byrne, D L McElwain, and J Norbury, A model of wound-healing angiogenesis in soft tissue, Mathematical Biosciences 136 (1996), 35–63.
- Richard C Schugart, Avner Friedman, Rui Zhao, and Chandan K Sen, Wound angiogenesis as a function of tissue oxygen tension: a mathematical model, Proceedings of the National Academy of Sciences 105 (2008), no. 7, 2628–2633.
- Ying-Bo Shui, Xiaohui Wang, Joan S Hu, Shui-Ping Wang, Claudia M Garcia, Jay D Potts, Yogendra Sharma, and David C Beebe, Vascular endothelial growth factor expression and signaling in the lens, Investigative ophthalmology & visual science 44 (2003), no. 9, 3911–3919.
- Barbara Szomolay, Tim D. Eubank, Ryan D. Roberts, Clay B. Marsh, and Avner Friedman, Modeling the inhibition of breast cancer growth by GM-CSF, J. Theoret. Biol. 303 (2012), 141–151. MR 2912960, DOI 10.1016/j.jtbi.2012.03.024
- Cruz Vargas-De-León, On the global stability of sis, sir and sirs epidemic models with standard incidence, Chaos, Solitons & Fractals 44 (2011), no. 12, 1106–1110.
- Avner Friedman, Bei Hu, and Chuan Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal. 42 (2010), no. 5, 2013–2040. MR 2684309, DOI 10.1137/090772630