Modelling the mechanical properties of the myocardium such as the passive and active responses is an essential component of mathematical cardiac models. In this study, we introduce a new active strain model that is physiologically realistic with parameters that could be calibrated from measured data. By further incorporating a limiting function on the elastic stretch that a living myocyte can experience, the new active strain model can accommodate different levels of active contraction. An existing left ventricular model is further used to simulate left ventricular pump function, which is implemented with an immersed boundary method with finite element extension and further calibrated with clinical data. The comparison with the active stress approach using the same left ventricular model suggests that despite some differences in stresses, the new active strain model can achieve patient-specific modelling.