## 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:

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