## A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids.(English)Zbl 1178.35387

A nonlinear system of reaction-diffusion-advection equations with free boundary is proposed to model the cell cycle dynamics and chemotactic driven cell movements in a multicellular tumor spheroid. There are two types of tumor cells: proliferating cells and quiescent cells, which have different chemotactic responses to an extracellular nutrient supply. Let $$c(r,t)$$ be the concentration of nutrient, $$p(r,t)$$ and $$q(r,t)$$ the proliferating and quiescent cell densities respectively, $$u_p(r,t)$$ and $$u_q(r,t)$$ the velocities of proliferating and quiescent cells respectively, and $$R(t)$$ the radius of the spheroid. Then for $$0<r<R(t)$$, $$t>0$$, \begin{aligned} {1\over r^2}{\partial \over \partial r}\left(r^2{\partial c\over \partial r}\right) & = \lambda(c)c, \\ {\partial p\over \partial t}+{1\over r^2}{\partial \over \partial r}(r^2(u_pp)) & = D{1\over r^2}{\partial \over \partial r}\left(r^2{\partial p\over \partial r}\right)+(K_b(c)-K_q(c)-K_a(c))p+K_p(c)q, \\ {\partial q\over \partial t}+{1\over r^2}{\partial \over \partial r}(r^2(u_qq)) & = D{1\over r^2}{\partial \over \partial r}\left(r^2{\partial q\over \partial r}\right)+K_q(c)p-(K_d(c)+K_p(c))q. \end{aligned} The unknown functions $$c,p,q,u_p,u_q$$ satisfy the conditions $$p+q=N$$ and $$u_q(r,t)=u_p(r,t)+\chi{\partial c\over \partial r}$$ for some constant $$N$$ (total number of live cells per volume) and a parameter $$\chi$$. In addition, appropriate boundary and initial conditions are assumed. Using a fixed point argument and the $$L^p$$-theory for parabolic equations, the author proves the global existence and uniqueness of solutions to the model.

### MSC:

 35R35 Free boundary problems for PDEs 92C17 Cell movement (chemotaxis, etc.) 35K35 Initial-boundary value problems for higher-order parabolic equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences
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### References:

  Byrne, H.M.; Chaplain, M.A.J., Growth of nonnecrotic tumours in the presence and absence of inhibitors, Math. biosci., 130, 151-181, (1995) · Zbl 0836.92011  Byrne, H.M.; Chaplain, M.A.J., Growth of necrotic tumors in the presence and absence of inhibitors, Math. biosci., 135, 187-216, (1996) · Zbl 0856.92010  Chaplain, M.A.J.; Lolas, G., Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. models methods appl. sci., 15, 1685-1734, (2005) · Zbl 1094.92039  Chen, X.; Cui, S.; Friedman, A., A hyperbolic free boundary problem modeling tumor growth: asymptotic behavior, Trans. amer. math. soc., 357, 4771-4804, (2005) · Zbl 1082.35166  Chen, X.; Friedman, A.; Hu, B., A parabolic – hyperbolic quasilinear system, Comm. partial differential equations, 33, 969-987, (2008) · Zbl 1158.35004  Cui, S., Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors, Interfaces free bound., 7, 147-159, (2005) · Zbl 1078.35139  Cui, S.; Friedman, A., A hyperbolic free boundary problem modeling tumor growth, Interfaces free bound., 5, 159-181, (2003) · Zbl 1040.35143  Cui, S.; Friedman, A., A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. amer. math. soc., 355, 3537-3590, (2003) · Zbl 1036.34018  Friedman, A., Cancer models and their mathematical analysis, (), 223-246  Friedman, A.; Hu, B., Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. ration. mech. anal., 172, 267-294, (2004)  Friedman, A.; Hu, B., Asymptotic stability for a free boundary problem arising in a tumor model, J. differential equations, 227, 598-639, (2006) · Zbl 1136.35106  Friedman, A.; Lolas, G., Analysis of a mathematical model of tumor lymphangiogenesis, Math. models methods appl. sci., 1, 95-107, (2005) · Zbl 1060.92036  Friedman, A.; Reitich, F., Analysis of a mathematical model for the growth of tumors, J. math. biol., 38, 262-284, (1999) · Zbl 0944.92018  Friedman, A.; Reitich, F., Quasi-state motion of a capillary drop, I: the two dimensional case, J. differential equations, 178, 212-263, (2001) · Zbl 0993.76017  Friedman, A.; Reitich, F., Quasi-state motion of a capillary drop, II: the three dimensional case, J. differential equations, 186, 509-557, (2002) · Zbl 1146.76593  Friedman, A.; Tao, Y., Analysis of a model of a virus that replicates selectively in tumor cells, J. math. biol., 47, 391-423, (2003) · Zbl 1052.92027  Friedman, A.; Tian, J.P.; Fulci, G.; Chiocca, E.A.; Wang, J., Glioma virotherapy: effects of innate immune suppression and increased viral replication capacity, Cancer res., 66, 2314-2319, (2006)  Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (2001), Springer-Verlag Berlin, Heidelberg · Zbl 0691.35001  Greenspan, H., Models for the growth of solid tumor by diffusion, Stud. appl. math., 51, 317-340, (1972) · Zbl 0257.92001  Jackson, T.L., Vascular tumor growth and treatment: consequence of polyclonality, competition and dynamic vascular support, J. math. biol., 44, 201-226, (2002) · Zbl 0992.92032  Ladyzenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasi-linear equations of parabolic type, Amer. math. soc. transl., vol. 23, (1968), Amer. Math. Soc. Providence, RI  Lieberman, G.M., Second order parabolic differential equations, (1996), World Scientific · Zbl 0884.35001  Norris, E.S.; King, J.R.; Byrne, H.M., Modelling the response of spatially structured tumours to chemotherapy: drug kinetics, Math. comput. modelling, 43, 820-837, (2006) · Zbl 1130.92033  Pettet, G.; Please, C.; Tindall, M.; McElwain, D., The migration of cells in multicell tumor spheroids, Bull. math. biol., 63, 231-257, (2001) · Zbl 1323.92037  Sherrat, J.; Chaplain, M., A new mathematical model for avascular tumor growth, J. math. biol., 43, 291-312, (2001) · Zbl 0990.92021  Tao, Y.; Chen, M., An elliptic – hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19, 419-440, (2006) · Zbl 1105.35149  Tao, Y.; Guo, Q., The competitive dynamics between tumor cells, a replication-competent virus and an immune response, J. math. biol., 51, 37-74, (2005) · Zbl 1066.92035  Tao, Y.; Guo, Q., A free boundary problem modelling cancer radiovirotherapy, Math. models methods appl. sci., 17, 1241-1259, (2007) · Zbl 1141.35476  Tao, Y.; Yoshida, N.; Guo, Q., Nonlinear analysis of a model of vascular tumour growth and treatment, Nonlinearity, 17, 867-895, (2004) · Zbl 1073.35215  Tao, Y.; Zhang, H., A parabolic – hyperbolic free boundary problem modelling tumor treatment with virus, Math. models methods appl. sci., 17, 63-80, (2007) · Zbl 1115.35137  Tindall, M.; Please, C., Modelling the cell cycle and cell movement in multicellular tumour spheroids, Bull. math. biol., 69, 1147-1165, (2007) · Zbl 1298.92021  Ward, J.P.; King, J.R., Mathematical modelling of avascular tumor growth, IMA J. math. appl. med. biol., 14, 39-69, (1997) · Zbl 0866.92011  Ward, J.P.; King, J.R., Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures, Math. biosci., 181, 177-207, (2003) · Zbl 1014.92021  Wei, X.; Cui, S., Global well-posedness for a drug transport model in tumor multicell spheroids, Math. comput. modelling, 45, 553-563, (2007) · Zbl 1165.35318  Wu, J.T.; Byrne, H.M.; Kirn, D.H.; Wein, L.M., Modeling and analysis of a virus that replicates selectively in tumor cells, Bull. math. biol., 63, 731-768, (2001) · Zbl 1323.92112  Wu, J.T.; Kirn, D.H.; Wein, L.M., Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response, Bull. math. biol., 66, 605-625, (2004) · Zbl 1334.92243
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