Multiband (MB) excitation and in-plane acceleration of first-pass perfusion imaging has the potential to provide a high aggregate acceleration rate. Our recent slice-SPIRiT work formulated MB reconstruction as a constrained optimization problem that jointly uses in-plane and through-plane coil information and MB data consistency. Here we extend these methods to develop k-t slice-SPARSE-SENSE and k-t slice-L+S reconstruction models. First-pass perfusion data with MB=3 and rate-2 k-t Poisson-disk undersampling were acquired in 6 patients. The slice-L+S reconstruction showed sharper borders and greater contrast than slice-SPARSE-SENSE and had better image quality scores as assessed by two cardiologists.