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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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