Estrogen receptor (ER)-positive breast cancer is responsive to a number of targeted therapies used clinically. Unfortunately, the continuous application of any targeted therapy often results in resistance to the therapy. Our ultimate goal is to use mathematical modelling to optimize alternating therapies that not only decrease proliferation but also stave off resistance. Toward this end, we measured levels of key proteins and proliferation over a 7-day time course in ER-positive MCF7 breast cancer cells. Treatments included endocrine therapy, either estrogen deprivation, which mimics the effects of an aromatase inhibitor, or fulvestrant, an ER degrader. These data were used to calibrate a mathematical model based on key interactions between ER signaling and the cell cycle. We show that the calibrated model is capable of predicting the combination treatment of fulvestrant and estrogen deprivation. Further, we show that we can add a new drug, palbociclib, to the model by measuring only two key proteins, c-Myc and hyperphosphorylated RB1, and adjusting only parameters associated with the drug. The model is then able to predict the combination treatment of estrogen deprivation and palbociclib. Then we added the dynamics of estrogen concentration in the medium into the model and extended the short-term model to a long-term model. The long-term model can simulate various mono- or combination treatments at different doses over 28 days. In addition to palbociclib, we add another Cdk4/6 inhibitor to the model, abemaciclib, which can induce apoptosis at high concentrations. Then the model can match the effects of abemaciclib treatment at two different doses and also capture the apoptosis effects induced by abemaciclib. After calibrating the model to these different treatment conditions, we used the model to explore the synergism among these different treatments. The mathematical model predicts a significant synergism between palbociclib or abemaciclib in combination with fulvestrant. And the predicted synergisms are verified by experiments. This critical synergism between these Cdk4/6 inhibitors and endocrine therapy could reflect the reason that Cdk4/6 inhibitors achieve pronounced success in clinic trails. Lastly, we used protein biomarkers (cyclinD1, cyclinE1, Cdk4, Cdk6 and Cdk2) and palbociclib dose-response proliferation assays to assess the difference between mono- and alternating therapy after 10 weeks of treatments. But neither the protein levels nor palbociclib dose-response show significant differences after 10 weeks of treatment. Therefore, we cannot conclude that alternating therapy delays palbociclib resistance compared with palbociclib mono-treatment after 10 weeks. Longer term experiments or other methods will be needed to uncover any difference. However, in this research we showed that a mechanism-based mathematical model is able to simulate and predict various effects of clinically-used treatments on ER-positive breast cancer cells at different time scales. And this mathematical model has the potential to explore ideas for potential drug treatments, optimize protocols that limit proliferation, and determine the drugs, doses, and alternating schedule for long term experiments. / Doctor of Philosophy / Estrogen receptors are proteins found inside breast cancer cells that are activated by the hormone estrogen. Estrogen-receptor positive breast cancer is the most common type of breast cancer and accounts for about 70% of breast cancer tumors. Endocrine therapy, which inhibits estrogen receptor signaling, and Cyclin-dependent kinase 4 and 6 (Cdk4/6) inhibitors are the preferred first-line therapy for patients with estrogen receptor-positive cancers. We built a mathematical model of MCF7 cells (an estrogen receptor-positive breast cancer cell line) in response to these standard first-line therapies. This mathematical model can capture the experimentally observed protein and cell proliferation changes in response to various treatment conditions, including different drug combinations, different doses, and different treatment durations up to 28 days. The model can then be used to look for more effective treatment possibilities. In particular, our mathematical model predicted a strong synergism between Cdk4/6 inhibitors and endocrine therapy, which could allow significant reductions in drug dosage while producing the same effect. This synergism was verified by experiments. In addition to treatment methods where one drug or combination of several drugs is used continuously, we consider alternating among various therapies in a fixed cycle. The mathematical model can help us determine which drugs and which doses might be most appropriate. Since an alternating therapy doesn't inhibit one particular target non-stop, the hope is that alternating therapies can delay the onset of drug resistance, where the drug becomes less effective or stops working completely. Unfortunately, an initial 10- week experiment to test for differences in resistance to a mono-therapy versus an alternating therapy did not show a significant difference, pointing to the need for longer experiments to see if alternating therapies can actually make a difference in resistance. Mathematical models will be important for determining the drugs, doses, and time intervals to be used in these experiments, as figuring out the best options by trial and error in such long-term experiments is not practical.