We will study a mathematical framework for analysis and simulation of development of stem cell based, growing organs with cell self-renewal and differentiation regulated by signalling factors. Considered model consist of PDEs and ODEs and describe concentrations of signals in moving domain $\Omega(t)$. Moreover, movement of $\Omega(t)$ depends on solution of PDEs inside $\Omega(t)$. There is no general approach which allow us to obtain mathematical results for such phenomena. We start with presentation of derivation of general mathematical model and its biological motivation. Later, we will focus on some simplified model and for this simplified model existence and uniqueness of solution will be proved. Moreover, we present numerical simulations which show that our mathematical model fit to its biological origin.