## 4 Mathematical Models in Clinical Settings
### Real-Time Estimability  
Mathematical models need to be validated against data prior to any application. In theoretical studies, the complete set of patient data is often known to the researchers and is used for model validation purposes. This makes sense because existing mathematical models often contain a large number of parameters, so having more data gives a better chance of estimating the parameters. A study of model identification methods by [PERSON] et al. (2019) shows that 1.5 cycles, where a cycle refers to an on- and off-treatment period in IAS, of data is the minimum requirement for model identification in most cases. Yet in clinical settings, a model should be useful even when a limited amount of information is available; however, this is often not the case. If just a few data points are used to parametrize a model, its predictions would be highly unreliable ([PERSON] et al., 2019). On the other hand, if researchers wait for more data as the treatment progresses, it may be too late, perhaps because the cancer becomes resistant and the patient must switch to a different treatment. Furthermore, this issue affects even the classification systems that are introduced along with some mathematical models ([PERSON] et al., 2019; 2020). One might assume that obtaining data more quickly, perhaps by way of a self-measuring device or enticing the patient to visit the hospital more regularly, would resolve this issue; however, this is deceptive. The shape of the patient's data trend (PSA or androgen) is unique. Thus, forcing a collection of data quickly does not expose completely the shape of this trend; instead, this would only oversample certain sections leading to biases. There are two proposed ways to address the issue of data limitation in practice. [PERSON] et al. [61] noticed the issue of the high ratio of the number of parameters to data points, so they assumed that the parameters in the model are the same across patients and the only difference is the initial population of cancer cells. If this assumption holds, the number of parameters needed to be estimated for existing models would decrease dramatically--allowing mathematical models to be useful in clinical studies. However, this is generally not the case. Various studies on personalized medicine, or the idea that each patient should have his own unique set of parameters, show that the parameters vary significantly across patients [88]. This variability between patients makes intuitive sense. The characteristics of a tumor and how the patient reacts to certain drugs should depend on the specificity of the patient's physiology and the tumor's composition, metastatic site, etc. However, it may be possible to show that their assumption holds for patients within a certain group, for example, patients who share certain physical traits. Additionally, some parameters may in fact be relatively constant among patients. Of course, if such categorization is possible, it could potentially be used to resolve this issue. The second possibility is not as clear cut as the first; however, it is inspired by classical physical studies. In this approach, we could consider models that contain mostly (if not only) parameters that can be measured or estimated from laboratory testing. Thus, most of the parameters would not come from data fitting but from actual physical testing of the patients. However, this method has its own set of problems. Perhaps the biggest problem is the construction of the model. One cannot mindlessly add all known mechanisms/factors into the model, There are many limitations by doing so, such as increasing the complexity of the model, existence of unknown biological details, not well tested biological details that may turn out to be incorrect in the future, etc. In order to create such a model, the model formulation rests on the expertise in both mathematical modeling and biological