This study is concerned with a monoclonal growth with piecewise constant arguments, where [t] and [t - 1] are embedded as coefficients to equation (A) to emphasize the population growth for specific times such as dx(t)/dt=x(t)r(1 - alpha x(t) - beta(0)[t]x([t]) - beta(1)[t - 1]x([t - 1])). The parameters alpha,beta(0),beta(1) and r belongs to R+ and [t] is the integer part of t is an element of [0, infinity). The parameter r is the population growth rate of the monoclonal tumor, alpha, beta(0) and beta(1) are rates for the delayed tumor volume that are based on the logistic population model. Two models are constructed in this work; in the first one we consider a growth around the positive equilibrium point (without Allee effect), in the second one, we analyzed the growth in case of early detection of the tumor (with Allee effect). Examination of the characterization of the stability shows that increase of the population growth rate decreases the local stability of the positive equilibrium point of equation (A). Simulations in this work give a detailed description of the behavior in (A) with and without Allee effect.