Cardiac mechanical alternans are believed to be associated with intracellular Ca2 + alternans. However, the mechanisms underlying the genesis of intracellular Ca2 + alternans are unclear. Experimental studies have shown that alternans of systolic Ca2 + under voltage clamp can be produced either by partially inhibiting the Ca2+ release mechanism or by applying small depolarising pulses. In each case the alternans relied on the appearance of propagating waves of Ca2 + release. The aim of the work in this thesis is to develop a theoretical model to understand the mechanisms underlying how the beat-to-beat alternation in the amplitude of the systolic Ca2+ transient is produced. A mathematical model of intracellular Ca2 + handling for a ventricular cardiac cell was developed as a spatially extended object, which is discretized by a group of coupled elements. Each element contains L-type and T-type Ca2 + channels, a subspace into which Ca2 + release takes place, a cytoplasmic space and sarcoplasmic reticulum (SR) release channels (RyR) and uptake sites (SERCA). Inter-element coupling is via Ca2 + diffusion between neighbouring subspaces and cytoplasm spaces. Small depolarising pulses were simulated by a sequence of step changes of cell membrane potential (20 mV) with random block on the L-type channels. Partial inhibition of the release mechanism is mimicked by applying a random reduction of the open probability of the RyR in response to a full stimulation by L type Ca2 + channels. In both cases the simulations are consistent with experimental observations and show that Ca2 + alternans can be generated and sustained through alternation of SR Ca2+ content produced by the propagating wave of Ca2 + release. However, the presented model has certain limitations. In the model, the property of L-type Ca2 + channel is described as an average effect of global current in a cardiac cell, rather than the total current of a large population of unitary L-type Ca2 + current, which is directly associated with the testing potential and which is a very important part of Ca2 + induced Ca2 + release (CICR). To conquer the limitation, an improved version of the Ca2 + handling model was developed by incorporating stochastic unitary L-type Ca2 + channels into the previous model. The improved model has a more biophysically detailed Ca2 + handling model, which can not only simulate proper unitary L-type Ca2 + influx but also produce a good quality of global L-type Ca2 + current. Like the old model the new stochastic model produced the global and regional alternans of Ca2 + transients observed in previous experimental studies as well, and agreed the key role of SR Ca2 + content to genesis of cardiac Ca2 + alternans. The study in this thesis provided novel insights into understanding the mechanisms underlying the genesis of intracellular Ca2 + alternans. In addition, the simulation study of effects of volatile anesthesics on rat ventricle has also been included in this thesis, and is presented in Appendix 1.