## 6 Conclusions
Secondly, for combinations of drugs, the actual dosages and toxicity are not considered. This is perhaps due to the fact that previous studies are dominated by the traditional way of administering hormonal treatment (fixed dosage). However, as adaptive therapy is becoming more prominent, the actual dosages and toxicity of multi-therapy should be accounted for in modeling efforts. Furthermore, in the case of optimal schedule, the best objective is often not clear, which is perhaps due to the lack of a credible set of biomarkers that can track cancer growth. Perhaps the most urgent issue with existing efforts in modeling prostate cancer is the validation of mathematical models. In the early stage of development, models are validated using observations of clinical trends, which can only give qualitative descriptions of the process. As the field progresses, there have been more attempts to validate models against clinical data in a quantitative way. While this is an improvement, efforts are lacking and parameter identification becomes a big problem for the validity of many mathematical models. This issue is further evident when mathematical models attempt to reach clinical application because of the lack of retrospective data sets. The identification of model parameters is coupled with issues of uncertainty, sensitivity, and stochasticity in making any conclusions or forecasts from mathematical models, while useful in providing an indication of likely scenarios, problematic. As many of the existing models are phenomenological, the issue of model identification is further emphasized. Instead, we suggest the use of mechanistic models, which tend to be more complex, but parameters can often be linked to experimental data. The formulation of mechanistic models should be based on well-tested structures from phenomenological models, while acknowledging the biological details. In this way, previous results can aid future studies. As optimal treatments require model predictions far into the future, it severely limits the usage of mathematical models due to increasing uncertainty, especially for long term treatments such as hormonal therapy. To address this issue, we observe that parameters in dynamical systems are often an average of an underlying process over a period of time (e.g., the period of time used in parameter estimation). Thus, the parameters in dynamical models should be updated regularly. There are several means to incorporate parameter evolution in dynamical models, such as applying the Kalman filter [49], incorporating the dynamics of the parameters into the model [47], using weighted fitting to emphasize more recent data [50], or studying the trends of parameter evolution [89]. If the trends of certain parameters can be established for a specific treatment, it can allow for better prediction without higher certainty. An in-depth exploration of parameter evolution could significantly bridge the gap between mathematical models and clinical practices. Furthermore, it could provide a new set of interesting dynamical questions [104]. In summary, mathematical modeling efforts for prostate cancer so far have shown that (1) IAS or adaptive therapy is better than CAS at delaying cancer relapse, if AD cells hold competitive advantages over AI cells, and (2) cancer progression is better associated with the relative velocity of PSA due to variation in PSA production by the tumor and its degradation rate. Moving forward, to close the gap between modeling work and clinical application, the following questions need to be addressed: (1) how to obtain the individualized model parameters without extensive data, (2) what are the most important parameters that affect cancer growth and how to use this knowledge in clinical applications, (3) what is the most appropriate objective for an optimal treatment/schedule study, and (4) how to best quantify the uncertainty in model prediction?