References
- Frigeri, S., Grasselli, M., Rocca, E. (2015). On a diffuse interface model of tumour growth. Eur. J. Appl. Math. 26(2):215–243. DOI: https://doi.org/10.1017/S0956792514000436.
- Garcke, H., Lam, K.F., Nürnberg, R., Sitka, E. (2018). A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Models Methods Appl. Sci. 28(3):525–577. DOI: https://doi.org/10.1142/S0218202518500148.
- Hilhorst, D., Kampmann, J., Nguyen, T.N., Van Der Zee, K.G. (2015). Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25(6):1011–1043. DOI: https://doi.org/10.1142/S0218202515500268.
- Oden, J. T., Hawkins, A., Prudhomme, S. (2010). General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20(3):477–517. DOI: https://doi.org/10.1142/S0218202510004313.
- Agosti, A., Cattaneo, C., Giverso, C., Ambrosi, D., Ciarletta, P. (2018). A computational framework for the personalized clinical treatment of glioblastoma multiforme. ZAMM - J. Appl. Math. Mech./Zeitschrift Für Angewandte Mathematik Und Mechanik. 98(12):2307–2327.
- Agosti, A., Ciarletta, P., Garcke, H., Hinze, M. (2020). Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data. Math. Meth. Appl. Sci. 43(15):8945–8979. DOI: https://doi.org/10.1002/mma.6588.
- Bearer, E.L., Lowengrub, J.S., Frieboes, H.B., Chuang, Y.L., Jin, F., Wise, S.M., Ferrari, M., andAgus, V., Cristini, D.B. (2009). Multiparameter computational modeling of tumor invasion. Cancer Res. 69(10):4493–4501. DOI: https://doi.org/10.1158/0008-5472.CAN-08-3834.
- Frieboes, H.B., Lowengrub, J., Wise, S., Zheng, X., Macklin, P., Bearer, E., Cristini, V. (2007). Computer simulation of glioma growth and morphology. NeuroImage. 37(Suppl 1):S59–S70. DOI: https://doi.org/10.1016/j.neuroimage.2007.03.008.
- Colli, P., Gilardi, G., Hilhorst, D. (2015). On a Cahn–Hilliard type phase field system related to tumor growth. Discrete Contin. Dyn. Syst. 35(6):2423–2442. DOI: https://doi.org/10.3934/dcds.2015.35.2423.
- Colli, P., Gilardi, G., Rocca, E., Sprekels, J. (2015). Vanishing viscosities and error estimate for a Cahn–Hilliard type phase field system related to tumor growth. Nonlinear Anal. Real World Appl. 26:93–108. DOI: https://doi.org/10.1016/j.nonrwa.2015.05.002.
- Colli, P., Gilardi, G., Rocca, E., Sprekels, J. (2017). Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth. Discrete Contin. Dyn. Syst. Ser. S. 10(1):37–54.
- Frigeri, S., Lam, K.F., (2017). and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities. In Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Volume 22 of Springer INdAM Ser.. Cham: Springer, pp. 217–254.
- Frigeri, S., Lam, K.F., Signori, A. (2021). Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities. Eur. J. Appl. Math. 1–42. DOI: https://doi.org/10.1017/S0956792521000012
- Garcke, H., Lam, K.F. (2017). Analysis of a Cahn–Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete Contin. Dyn. Syst. 37(8):42–77.
- Garcke, H., Lam, K.F., Signori, A. (2021). On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects. Nonlinear Anal. Real World Appl. 57:103192. DOI: https://doi.org/10.1016/j.nonrwa.2020.103192.
- Scarpa, L., Signori, A. (2021). On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport. Nonlinearity. 34(5):3199–3250. DOI: https://doi.org/10.1088/1361-6544/abe75d.
- Cavaterra, C., Rocca, E., Wu, H. (2021). Long-Time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth. Appl. Math. Optim. 83(2):739–787. DOI: https://doi.org/10.1007/s00245-019-09562-5.
- Colli, P., Gilardi, G., Rocca, E., Sprekels, J. (2017). Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity. 30(6):2518–2546. DOI: https://doi.org/10.1088/1361-6544/aa6e5f.
- Colli, P., Signori, A., Sprekels, J. (2020). Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials. Appl. Math. Optim. 82:517–549.
- Colli, P., Signori, A., Sprekels, J. (2021). Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis. ESAIM Control Optim. Calc. Var. 27:73. DOI: https://doi.org/10.1051/cocv/2021072.
- Garcke, H., Lam, K.F., Rocca, E. (2018). Optimal control of treatment time in a diffuse interface model of tumor growth. Appl. Math. Optim. 78(3):495–544. DOI: https://doi.org/10.1007/s00245-017-9414-4.
- Garcke, H., Lam, K.F., Signori, A. (2021). Sparse optimal control of a phase field tumor model with mechanical effects. SIAM J. Control Optim. 59(2):1555–1580. DOI: https://doi.org/10.1137/20M1372093.
- Kahle, C., Lam, K.F. (2018). Parameter Identification via Optimal Control for a Cahn–Hilliard-Chemotaxis System with a Variable Mobility. Appl. Math. Optim. 82:63–104.
- Rocca, E., Scarpa, L., Signori, A. (2020). Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis. Math. Models Methods Appl. Sci. arXiv:2009.11159 [Math.AP].
- Signori, A. (2020). Optimal distributed control of an extended model of tumor growth with logarithmic potential. Appl. Math. Optim. 82(2):517–549. DOI: https://doi.org/10.1007/s00245-018-9538-1.
- Signori, A. (2020). Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme. Math. Control Relat. Fields. 10:305–331.
- Signori, A. (2020). Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach. Evol. Equ. Control Theory. 9(1):193–217. DOI: https://doi.org/10.3934/eect.2020003.
- Signori, A. (2020). Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete Contin. Dyn. Syst. Ser. A. 41(6):2519–2542.
- Signori, A. (2020). Vanishing parameter for an optimal control problem modeling tumor growth. Asymptot. Anal. 117:43–66.
- Sprekels, J., Tröltzsch, F. (2021). Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth. ESAIM: Control, Optim. Calc. Var. 27:S26.
- Byrne, H.M., King, J.R., McElwain, D.L.S., Preziosi, L. (2003). A two-phase model of solid tumour growth. Appl. Math. Lett. 16(4):567–573. DOI: https://doi.org/10.1016/S0893-9659(03)00038-7.
- Chaplain, M.A.J. (1996). Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development. Math. Comput. Modell. 23(6):47–87. DOI: https://doi.org/10.1016/0895-7177(96)00019-2.
- Franks, S.J., King, J.R. (2009). Interactions between a uniformly proliferating tumour and its surroundings: stability analysis for variable material properties. Internat. J. Engrg. Sci. 47(11–12):1182–1192. DOI: https://doi.org/10.1016/j.ijengsci.2009.07.004.
- Friedman, A. (2009). Free boundary problems associated with multiscale tumor models. Math. Model. Nat. Phenom. 4(3):134–155. DOI: https://doi.org/10.1051/mmnp/20094306.
- Friedman, A. (2016). Free boundary problems for systems of Stokes equations. DCDS-B. 21(5):1455–1468. DOI: https://doi.org/10.3934/dcdsb.2016006.
- Byrne, H.M., Chaplain, M.A.J. (1997). Free boundary value problems associated with the growth and development of multicellular spheroids. Eur. J. Appl. Math. 8(6):639–658. DOI: https://doi.org/10.1017/S0956792597003264.
- Franks, S.J., King, J.R. (2003). Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties. Math. Med. Biol. 20(1):47–89. DOI: https://doi.org/10.1093/imammb/20.1.47.
- Greenspan, H.P. (1976). On the growth and stability of cell cultures and solid tumors. J. Theoret. Biol. 56(1):229–242. DOI: https://doi.org/10.1016/S0022-5193(76)80054-9.
- Donatelli, D., Trivisa, K. (2015). On a nonlinear model for tumour growth with drug application. Nonlinearity. 28(5):1463–1481. DOI: https://doi.org/10.1088/0951-7715/28/5/1463.
- Perthame, B., Vauchelet, N. (2015). Incompressible limit of a mechanical model of tumour growth with viscosity. Phil. Trans. R Soc. A. 373(2050):20140283. DOI: https://doi.org/10.1098/rsta.2014.0283.
- Srinivasan, S., Rajagopal, K.R. (2014). A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations. Int. J. Non-Linear Mech. 58:162–166. DOI: https://doi.org/10.1016/j.ijnonlinmec.2013.09.004.
- Zheng, X., Wise, S.M., Cristini, V. (2005). Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Biol. 67(2):211–259. DOI: https://doi.org/10.1016/j.bulm.2004.08.001.
- Dedè, L., Garcke, H., Lam, K.F. (2018). A Hele–Shaw–Cahn–Hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid Mech. 20(2):531–567. DOI: https://doi.org/10.1007/s00021-017-0334-5.
- Feng, X., Wise, S. (2012). Analysis of a Darcy–Cahn–Hilliard diffuse interface model for the Hele–Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50(3):1320–1343. DOI: https://doi.org/10.1137/110827119.
- Giorgini, A., Grasselli, M., Wu, H. (2018). The Cahn–Hilliard–Hele–Shaw system with singular potential. Ann. Inst. H. Poincaré Anal. Non Linéaire. 35(4):1079–1118. DOI: https://doi.org/10.1016/j.anihpc.2017.10.002.
- Bosia, S., Conti, M., Grasselli, M. (2015). On the Cahn–Hilliard–Brinkman system. Comm. Math. Sci. 13(6):1541–1567. DOI: https://doi.org/10.4310/CMS.2015.v13.n6.a9.
- Conti, M., Giorgini, A. (2020). Well-posedness for the Brinkman–Cahn–Hilliard system with unmatched viscosities. J. Differ. Equ. 268(10):6350–6384. DOI: https://doi.org/10.1016/j.jde.2019.11.049.
- Ebenbeck, M., Garcke, H. (2019). Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis. J. Differ. Equations. 266(9):5998–6036. DOI: https://doi.org/10.1016/j.jde.2018.10.045.
- Ebenbeck, M., Garcke, H. (2019). On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits. SIAM J. Math. Anal. 51(3):1868–1912. DOI: https://doi.org/10.1137/18M1228104.
- Ebenbeck, M., Garcke, H., Nürnberg, R. (2021). Cahn–Hilliard–Brinkman systems for tumour growth. DCDS-S. 0(0):0. DOI: https://doi.org/10.3934/dcdss.2021034.
- Ebenbeck, M., Knopf, P. (2019). Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation. Calc. Var. 58(2019):1–31. DOI: https://doi.org/10.1007/s00526-019-1579-z.
- Ebenbeck, M., Knopf, P. (2020). Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth. ESAIM: COCV. 26:71. DOI: https://doi.org/10.1051/cocv/2019059.
- Ebenbeck, M., Lam, K.F. (2020). Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms. Adv. Nonlinear Anal. 10(1):24–65. DOI: https://doi.org/10.1515/anona-2020-0100.
- Byrne, H., Preziosi, L. (2003). Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20(4):341–366.
- Roose, T., Chapman, S.J., Maini, P.K. (2007). Mathematical models of avascular tumor growth. SIAM Rev. 49(2):179–208. [Database] DOI: https://doi.org/10.1137/S0036144504446291.
- Sherratt, J.A., Chaplain, M.A.J. (2001). A new mathematical model for avascular tumour growth. J. Math. Biol. 43(4):291–312.
- Wallace, D.I., Guo, X. (2013). Properties of tumor spheroid growth exhibited by simple mathematical models. Front Oncol. 3(51):1–9. DOI: https://doi.org/10.3389/fonc.2013.00051.
- Zhang, J.Z., Bryce, N.S., Siegele, R., Carter, E.A., Paterson, D., de Jonge, M.D., Howard, D.L., Ryan, C.G., Hambley, T.W. (2012). The use of spectroscopic imaging and mapping techniques in the characterisation and study of DLD-1 cell spheroid tumour models. Integr Biol (Camb). 4(9):1072–1080. DOI: https://doi.org/10.1039/c2ib20121f.
- Araujo, R.P., McElwain, D.L.S. (2004). A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol. 66(5):1039–1091. DOI: https://doi.org/10.1016/j.bulm.2003.11.002.
- Astanin, S., Preziosi, L. (2008). Multiphase models of tumour growth. In: Selected Topics in Cancer Modeling, Model. Simul. Sci. Eng. Technol. Boston, MA: Birkhäuser, pp. 223–253.
- Escher, J., Matioc, A.-V., Matioc, B.-V. (2011). Analysis of a mathematical model describing necrotic tumor growth. In Modelling, Simulation and Software Concepts for Scientific-Technological Problems, Volume 57 of Lect. Notes Appl. Comput. Mech.. Berlin: Springer, pp. 237–250.
- Frieboes, H.B., Jin, F., Chuang, Y.-L., Wise, S.M., Lowengrub, J.S., Cristini, V. (2010). Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis. J. Theor. Biol. 264(4):1254–1278. DOI: https://doi.org/10.1016/j.jtbi.2010.02.036.
- Fritz, M., Jha, P.K., Köppl, T., Oden, J.T., Wohlmuth, B. (2021). Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1d vascular network. Nonlinear Anal. Real World Appl. 61:103331. DOI: https://doi.org/10.1016/j.nonrwa.2021.103331.
- Sciumè, G., Shelton, S., Gray, W.G., Miller, C.T., Hussain, F., Ferrari, M., Decuzzi, P., Schrefler, B.A. (2013). A multiphase model for three-dimensional tumor growth. New J. Phys. 15(1):015005. DOI: https://doi.org/10.1088/1367-2630/15/1/015005.
- Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V. (2008). Three-dimensional multispecies nonlinear tumor growth-I Model and numerical method. J. Theor. Biol. 253(3):524–543. DOI: https://doi.org/10.1016/j.jtbi.2008.03.027.
- Garcke, H., Lam, K.F. (2016). Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Mathematics. 1(3):318–360. (Math-01-00318): DOI: https://doi.org/10.3934/Math.2016.3.318.
- Garcke, H., Lam, K.F. (2017). Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28(2):284–316. DOI: https://doi.org/10.1017/S0956792516000292.
- Garcke, H., Lam, K.F. (2018). On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms. In Trends in Applications of Mathematics to Mechanics, Volume 27 of Springer INdAM Ser.. Cham: Springer, pp. 243–264.
- Garcke, H., Lam, K.F., Sitka, E., Styles, V. (2016). A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26(6):1095–1148. DOI: https://doi.org/10.1142/S0218202516500263.
- Dai, M., Feireisl, E., Rocca, E., Schimperna, G., Schonbek, M.E. (2017). Analysis of a diffuse interface model of multispecies tumor growth. Nonlinearity. 30(4):1639–1658. DOI: https://doi.org/10.1088/1361-6544/aa6063.
- Frigeri, S., Lam, K.F., Rocca, E., Schimperna, G. (2018). On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials. Comm. in Math. Sci. 16(3):821–856. DOI: https://doi.org/10.4310/CMS.2018.v16.n3.a11.
- Dharmatti, S., Perisetti, L.N.M. (2021). Nonlocal Cahn–Hilliard–Brinkman System with Regular Potential: Regularity and Optimal Control. J. Dyn. Control Syst. 27(2):221–246. DOI: https://doi.org/10.1007/s10883-020-09490-6.
- Galdi, G.P. (2011). An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer Monographs in Mathematics, 2nd ed. New York: Springer, Steady-state problems.
- Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. Amsterdam-New York: North-Holland Publishing Company.
- Constantin, P., Foias, C. (1988). Navier-Stokes Equations. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press.
- Abels, H., Terasawa, Y. (2009). On stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2):381–429. DOI: https://doi.org/10.1007/s00208-008-0311-7.
- Boyer, F., Fabrie, P. (2013). Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Volume 183 of Applied Mathematical Sciences. New York: Springer.
- Alt, H.W. (2016). Linear Functional Analysis - An Application-Oriented Introduction. London: Springer.
- Amann, H. (1995). Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory. Basel: Birkhäuser.