A mathematical model for the evolution of cell populations under the action of mutagenic agents
The cells of the population are characterized by three state variables: they are in the resting phase r or replication cycle c, they have an amount x, \(0\leq x\leq 1\), of genetic damage; the cells in the replication cycle have the maturity \(\omega\), \(0\leq \omega \leq T\) (replication time) corresponding to the number of operations performed until time t for given damage x. The damage is influenced by some given mutagenic agent s(t) and by a repairing process, such that \[ \dot x=\phi (x)=a_ r(1-x)s(t)-b_ rx\text{ in phase } r, \] and \[ \dot x=\psi (x)=a_ c(1-x)s-b_ cx\text{ in phase } c. \] The maturity \(\omega\) is governed by an equation \(\omega_ t=\rho (x)\), \(0\leq \rho (x)\leq 1\), a known function, describing the retardation of the replication given the damage x. Balance considerations lead to the following equations for u and v, the density of the number of cells in phase r or in phase \(c:\) \[ v_ t+(\psi v)_ x+(\rho v)_{\omega}=\mu \cdot v,\quad u_ t+(\phi u)_ x=(\mu +\nu)u+2\rho v_ 0 \] with suitable initial and compatibility conditions. \(\mu\) and \(\nu\) are the death rates for u and v, and \(v_ 0(t,x)=v(t,x,T)\).
first order partial differential equations
scientific article; zbMATH DE number 3972698
scientific article