We employ the framework of multitype Galton-Watson processes to model a population of dividing cells. The cellular type is represented by its biological age, defined as the count of harmful proteins hosted by the cell. The stochastic evolution of the biological age of a cell is modeled as a discrete Markov chain with a finite state space $\{0,1,\ldots,n\}$, where $n$ signifies the absorbing state corresponding to the senescent state of the cell. Consequently, the set of individual types in the multitype Galton-Watson process becomes $\{0,\ldots,n-1\}$. In our setting, a dividing cell may undergo rejuvenation meaning that its biological ages reduces due to transfer of harmful proteins to the daughter cell. For the proposed model, we define and study several biologically meaningful features, such as rejuvenation states, expected replicative lifespan, population growth rate, stable biological age distribution, and the population average of the biological age.