## 2 Elements of Mathematical Models for Clinical Applications
### Applicability in Clinical Settings  
While theoretical work offers insights into the fundamentals of prostate cancer, the complex interactions of various components in a real situation blur the implications from theories. Moreover, in the case of mathematical models, the predictions of the progression of the tumor must be validated against real-world clinical data, which often means fitting and forecasting using incomplete sets of data. Aside from these theoretical issues, clinical applications come with another set of questions that are often case-specific and not explored in depth in theoretical settings. For example, how does one predict whether a certain treatment will be effective for a specific patient? If so, how long will it take for the treatment to be effective? If not completely, then for how long would such treatment be effective? What is the dose level of certain drugs that should be used to achieve optimal results?What is the best combination of drugs to optimize effectiveness? Cancer treatments can be invasive and expensive, which can harm patients physically and financially [1, 6]. Although treating cancer is a nonlinear process, this is not the case in practice; different adjustments to the treatment should be made based on the specific situation. Existing modeling studies attempt to give some hints to possible treatments. However, other features of treatment, such as dosage, combinations of drugs and therapies, can also be optimized to achieve better results. For these reasons, we discuss several important gaps between theoretical work and the clinical application of mathematical models. Real-time estimability:in theoretical studies, researchers often have the luxury of knowing the complete set of a patient's data. However, in the clinical setting, researchers must make do with a smaller but increasing set of a patient's data. This means the parameters for the model cannot be identified with high certainty at the earlier stage, rendering implications from mathematical models unreliable. Uncertainty, identifiability, and sensitivity:another concern that arises with models of high complexity is the sensitivity of the model outputs to variations in the parameters. If the variation is small, it is usually referred to as sensitivity analysis, whereas uncertainty quantification refers to larger variation in the parameters, perhaps due to the large uncertainty in the data. Since measurements and models are imperfect, uncertainty quantification allows physicians to put confidence bounds on a model's predictions. On the other hand, sensitivity analysis can help pinpoint the important parameters in a model, which can aid with decision making during treatment. In close connection with sensitivity and uncertainty, the term identifiability refers to the ability to determine a unique set of model parameters with respect to the given data. This concept is further broken down into structural identifiability and practical identifiability. Structural identifiability refers to the relationship between the inherent structure of the model and perfect data. In essence, structural identifiability asks whether all of the parameters in a model can be uniquely identified given some perfect set of data on the variables in the model. On the contrary, when the data contains errors, the same question refers to the practical identifiability of the model. While these are important topics, there have been few attempts to study these closely related aspects in the case of prostate cancer. Optimal schedule, optimal treatment, and patient classification:while the previous issues deal with the validation of mathematical models in clinical settings, optimal schedule, optimal treatment, and patient classification are important contributions of mathematical models. For instance, patients can be categorized in advance based on whether they would react positively toward a certain type of treatment using mathematical models.