## 4 Mathematical Models in Clinical Settings
### Real-Time Estimability
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Aspects of Clinical Applications** & **Summary of Findings** & **Future Exploration** \\ \hline Real-time estimability & The estimation of parameters in mathematical models often & Some parameters share similar values across patients, while \\  & require a large quantity of data. & others are more patient-specific. \\  & However, the nature of data & This distinction should be studied in detail. Utilizing multiple data \\  & reliable estimation of parameters & sets is another possibility to allow early estimates of parameters. \\  & for patients may not be possible at & Finally, parameter evolution can \\  & the early stages of treatment. & be accounted for to address the \\  & & limitation of data availability. \\ \hline Uncertainty, sensitivity & Due to the large number of & Local sensitivity analysis and \\ and identifiability & parameters and the heavy reliance & uncertainty quantification should \\ Section 4.2 & on parameter fitting, model & be studied for each patient. Clear \\  & predictions can be unreliable. & links between each parameter and \\  & Furthermore, the issue of & its physical interpretation should \\  & parameter identifiability is often & be established, which potentially \\  & ignored, which can lead to wildly & allows for laboratory \\  & different predictions for a specific & estimates/bounds to resolve \\  & patient. & identifiability. \\ \hline Optimal schedule, & Studies on optimal schedule and & The usefulness of optimal studies \\ optimal treatment, and & treatment yield useful information & and classification hinges on how \\ Section 4.3 & on how intermittent, adaptive, & well the uncertainty in the model \\  & and combination therapies should & can be quantified, which relates to \\  & be carried out. Patient & previous issues. In addition, \\  & classification based on treatment & the objective of optimal studies \\  & effectiveness can be done using & may be extended to include drugs’ \\  & model parameters. However, both & properties, cost, and important \\  & aspects are affected heavily by the & features of each treatment. \\  & estimability of the parameters and & \\  & the uncertainty in the model’s & \\  & forecasts. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Aspects of clinical applications: summary of findings and future exploration. knowledge. One compromise using this approach is to carry out studies that focus on multi-level fitting as done in previous work. This would allow for more accurate estimations of certain parameter values. However, one must carefully isolate the parameters with respect to the data to avoid ambiguity in the biological interpretations of the parameters. An alternative approach to resolve the limitation of data in clinical settings is to account for the evolution of parameters as the treatment progresses, instead of relying extensively on data fitting. The rationale is that the parameters in dynamical models are often stand-ins for more complex processes, which means their values represent the average values over a certain time interval of the underlying processes. Hence, as time goes on, there would be changes accumulated from the underlying processes of the parameters. The effect of the underlying process may be minimal in many cases, but it could also be substantial. This is a complex subject that is akin to a time-scale analysis. Sometimes, fast changing processes can be accounted for using an average as long as the interval of time that the average is taken over is sufficiently long enough for the effect of the fast changing process to be negligible. On the other hand, other processes take a long time for their effects to be noticeable. When it comes to parameter estimations, there should be consideration for the time interval used for data fitting. For example, the extended version of the BK model ([PERSON] et al. [50]) contains a parameter that stands in for the maximal level of serum androgen.