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We model the physical parameters of Solar Cycles 23 and 24 using a nonlinear dynamical mean-field dynamo model that includes the formation and evolution of bipolar magnetic regions (BMR). The Parker-type dynamo model consists of a complete MHD system in the mean-field formulation: the 3D magnetic induction equation, and 2D momentum and energy equations in the anelastic approximation. The initialization of BMR is modeled in the framework of Parker's magnetic buoyancy instability. It defines the depths of BMR injections, which are typically located at the edge of the global dynamo waves. The distribution with longitude and latitude and the size of the initial BMR perturbations are taken from the NOAA database of active regions. The tilt of the perturbations is modeled by random function, and the mean tilt is modeled as a near-surface helicity (alpha-effect) term. The data-driven models are compared with the models calculated for random longitudinal and latitudinal distributions of the initial perturbation. The modeling results are compared with various observed characteristics of the solar cycles, including the magnetic butterfly diagram, the polar magnetic and basal magnetic fluxes, and the probability distributions of the BMR flux on the surface. Our results show that BMR can play a substantial role in the dynamo processes, and affect the strength of the solar cycle. However, the data-driven model shows that the BMR effect alone cannot explain the weak Cycle 24. This weak cycle and the prolonged preceding minimum of magnetic activity were probably caused by a decrease of the turbulent helicity in the bulk of the convection zone during the decaying phase of Cycle 23.