Open Access
Volume 58, Number 2, March-April 2024
Page(s) 613 - 638
Published online 09 April 2024
  1. R.P. Araujo and D.L.S. McElwain, A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol. 66 (2004) 1039–1091. [Google Scholar]
  2. P. Bénilan and N. Igbida, Singular limit of perturbed nonlinear semigroups. Comm. Appl. Nonlinear Anal. 3 (1996) 23–42. [Google Scholar]
  3. M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34 (2012) B559–B583. [Google Scholar]
  4. H.M. Byrne, T. Alarcon, M.R. Owen, S.D. Webb and P.K. Maini, Modelling aspects of cancer dynamics: a review. Philos. Trans. A Math. Phys. Eng. Sci. 364 (2006) 1563–1578. [Google Scholar]
  5. V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Cambridge (2010). [Google Scholar]
  6. V. Cristini, E. Koay and Z. Wang, An Introduction to Physical Oncology: How Mechanistic Mathematical Modeling Can Improve Cancer Therapy Outcomes, 1st edition. Chapman and Hall/CRC (2016). [Google Scholar]
  7. V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth. J. Math. Biol. 46 (2003) 191–224. [Google Scholar]
  8. J.A. Cruz and D.S. Wishart, Applications of machine learning in cancer prediction and prognosis. Cancer Inf. 2 (2006) 117693510600200030. [Google Scholar]
  9. M. Dashti and A.M. Stuart, The Bayesian approach to inverse problems, in Handbood of Uncertainty Quantification. Springer, Cham (2017) 311–428. [Google Scholar]
  10. N. David and B. Perthame, Free boundary limit of a tumor growth model with nutrient. J. Math. App. 155 (2021) 62–82. [Google Scholar]
  11. N. David, T. Debiec and B. Perthame, Convergence rate for the incompressible limit of nonlinear diffusion–advection equations. Annales de l’Institut Henri Poincaré C (2022). [Google Scholar]
  12. X. Dou, J.-G. Liu and Z. Zhou, A tumor growth model with autophagy: The reaction-(cross-)diffusion system and its free boundary limit. Discrete Continuous Dyn. Syst. B 28 (2023) 1964–1992. [Google Scholar]
  13. C. Falcó, D.J. Cohen, J.A. Carrillo and R.E. Baker, Quantifying tissue growth, shape and collision via continuum models and Bayesian inference. J. R. Soc. Interface 20 (2023) 20230184. [Google Scholar]
  14. Y. Feng, M. Tang, X. Xu and Z. Zhou, Tumor boundary instability induced by nutrient consumption and supply. Z. Angew. Math. Phys. 74 (2023) 107. [Google Scholar]
  15. S. Friedlander and D. Serre, editors. Handbook of Mathematical Fluid Dynamics. Elsevier (2002). [Google Scholar]
  16. A. Friedman and B. Hu, Stability and instability of Liapunov–Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model. Trans. Am. Math. Soc. 360 (2008) 5291–5342. [Google Scholar]
  17. A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth. Trans. Am. Math. Soc. 353 (2001) 1587–1634. [Google Scholar]
  18. H. Garcke, K.F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26 (2016) 1095–1148. [CrossRef] [MathSciNet] [Google Scholar]
  19. H.P. Greenspan, On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56 (1976) 229–242. [Google Scholar]
  20. N. Guillen, I. Kim and A. Mellet, A Hele–Shaw limit without monotonicity. Arch. Ration. Mech. Anal. 243 (2022) 829–868. [Google Scholar]
  21. Q. He, H.L. Li and B. Perthame, Incompressible limits of Patlak–Keller–Segel model and its stationary state. Acta App. Math. 188 (2023) 11. [Google Scholar]
  22. N. Igbida, L1-theory for incompressible limit of reaction–diffusion porous medium flow with linear drift. Preprint arXiv:2112.10411 (2021). [Google Scholar]
  23. N. Igbida, L1-theory for Hele–Shaw flow with linear drift. Math. Models Methods Appl. Sci. 33 (2023) 1545–1576. [Google Scholar]
  24. M. Jacobs, I. Kim and J. Tong, Tumor growth with nutrients: regularity and stability. Comm. Amer. Math. Soc. 3 (2023) 166–208. [Google Scholar]
  25. C. Kahle and K.F. Lam, Parameter identification via optimal control for a Cahn–Hilliard-chemotaxis system with a variable mobility. Appl. Math. Optim. 82 (2020) 63–104. [Google Scholar]
  26. C. Kahle, K.F. Lam, J. Latz and E. Ullmann, Bayesian parameter identification in Cahn–Hilliard models for biological growth. SIAM/ASA J. Uncertainty Quant. 7 (2019) 526–552. [Google Scholar]
  27. I. Kim and N. Požár, Porous medium equation to Hele–Shaw flow with general initial density. Trans. Am. Math. Soc. 370 (2018) 873–909. [Google Scholar]
  28. I.C. Kim, B. Perthame and P.E. Souganidis, Free boundary problems for tumor growth: a viscosity solutions approach. Non- linear Anal. 138 (2016) 207–228. [Google Scholar]
  29. E.J. Kostelich, Y. Kuang, J.M. McDaniel and N.Z. Moore, Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors. Biol. Direct 6 (2011) 1–20. [Google Scholar]
  30. K. Kourou, T.P. Exarchos, K.P. Exarchos, M.V. Karamouzis and D.I. Fotiadis, Machine learning applications in cancer prog- nosis and prediction. Comput. Struct. Biotechnol. J. 13 (2015) 8–17. [Google Scholar]
  31. J. Lipková, P. Angelikopoulos, S. Wu, E. Alberts, B. Wiestler, C. Diehl, C. Preibisch, T. Pyka, S.E. Combs, P. Hadjidoukas and K. Van Leemput, Personalized radiotherapy design for glioblastoma: integrating mathematical tumor models, multimodal scans, and Bayesian inference. IEEE Trans. Med. Imaging 38 (2019) 1875–1884. [Google Scholar]
  32. Y. Liu, S.M. Sadowski, A.B. Weisbrod, E. Kebebew, R.M. Summers and J. Yao, Patient specific tumor growth prediction using multimodal images. Med. Image Anal. 18 (2014) 555–566. [Google Scholar]
  33. J.-G. Liu, M. Tang, L. Wang and Z. Zhou, An accurate front capturing scheme for tumor growth models with a free boundary limit. J. Comput. Phys. 364 (2018) 73–94. [Google Scholar]
  34. J.-G. Liu, M. Tang, L. Wang and Z. Zhou, Toward understanding the boundary propagation speeds in tumor growth models. SIAM J. Appl. Math. 81 (2021) 1052–1076. [Google Scholar]
  35. J.S. Lowengrub, H.B. Frieboes, F. Jin, Y.L. Chuang, X. Li, P. Macklin, S.M. Wise and V. Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23 (2010) R1–R9. [Google Scholar]
  36. M.J. Lu, C. Liu, J. Lowengrub and S. Li, Complex far-field geometries determine the stability of solid tumor growth with chemotaxis. Bull. Math. Biol. 82 (2020) 1–41. [Google Scholar]
  37. M.J. Lu, W. Hao, C. Liu, J. Lowengrub and S. Li, Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis. J. Comput. Phys. 459 (2022) 111153. [Google Scholar]
  38. J. Nolen, G.A. Pavliotis and A.M. Stuart, Multiscale modelling and inverse problems. Numerical Analysis of Multiscale Problems. Springer (2012) 1–34. [Google Scholar]
  39. B. Perthame, Some mathematical models of tumor growth. Université Pierre et Marie Curie-Paris (2016) 6. [Google Scholar]
  40. B. Perthame, F. Quirós and J.L. Vázquez, The Hele–Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal. 212 (2014) 93–127. [Google Scholar]
  41. K. Pham, E. Turian, K. Liu, S. Li and J. Lowengrub, Nonlinear studies of tumor morphological stability using a two-fluid flow model. J. Math. Biol. 77 (2018) 671–709. [Google Scholar]
  42. T. Roose, S.J. Chapman and P.K. Maini, Mathematical models of avascular tumor growth. SIAM Rev. 49 (2007) 179–208. [Google Scholar]
  43. R. Selvanambi, J. Natarajan, M. Karuppiah, S.H. Islam, M.M. Hassan and G. Fortino, Lung cancer prediction using higher- order recurrent neural network based on glowworm swarm optimization. Neural Comput. App. 32 (2020) 4373–4386. [Google Scholar]
  44. S. Subramanian, K. Scheufele, M. Mehl and G. Biros, Where did the tumor start? An inverse solver with sparse localization for tumor growth models. Inverse Probl. 36 (2020) 045006. [Google Scholar]
  45. C. Villani, A review of mathematical topics in collisional kinetic theory. Handb. Math. Fluid Dyn. 1 (2002) 3–8. [Google Scholar]
  46. E. Weinan, Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011). [Google Scholar]
  47. L. Zhang, L. Lu, X. Wang, R.M. Zhu and M. Bagheri, Spatio-temporal convolutional LSTMs for tumor growth prediction by learning 4D longitudinal patient data. IEEE Trans. Med. Imaging 39 (2019) 1114–1126. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you