## Implications from Mathematical Models
### Data-Validated Models  
**[PERSON] et al. (2010)**. These early approaches suffer from a common problem: they are observation driven but not data validated, meaning the model is not tested against real-world data quantitatively. This brings us to the work by [PERSON], [PERSON], and [PERSON] (HBA) [40]. In 2010, [PERSON] et al. took a different approach to develop a model for prostate cancer. [PERSON] and colleagues observed that when treatment starts, the PSA level first plummets to a certain level, then decreases more slowly to an undetectable level. [PERSON] et al. proposed that the decay trend of PSA is most naturally modeled by a three-population model with interacting populations via linear rates, which results in a matrix representation. To account for the effect of treatment, [PERSON] et al. use a discrete switching mechanism. When the patient is on treatment, the HBA model takes the form:  
\[\frac{d}{dt} \begin{pmatrix}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\end{pmatrix} = \begin{pmatrix}w_{1,1}^{1}&0&0\\ w_{1,2}^{1}&w_{2,2}^{1}&0\\ w_{1,3}^{1}&w_{2,3}^{1}&w_{3,3}^{1}\end{pmatrix}\begin{pmatrix}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\end{pmatrix} \,, \tag{5}\]  
and during the off-treatment period,  
\[\frac{d}{dt} \begin{pmatrix}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\end{pmatrix} = \begin{pmatrix}w_{1,1}^{0}&w_{2,1}^{0}&0\\ 0&w_{2,2}^{0}&0\\ 0&0&w_{3,3}^{0}\end{pmatrix}\begin{pmatrix}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\end{pmatrix} \,. \tag{6}\]  
The parameters \(w_{ij}\) are the transformation rates from \(x_{i}\) to \(x_{j}\) and for simplicity, the PSA level is taken as the sum of \(x_{1},x_{2},x_{3}\). The model is tested against different sets of clinical data [40; 41]. While the model is phenomenologically based, it has successfully reproduced the highly nonlinear trends in PSA data in many instances [23; 24]; see Figure 4. [PERSON] and colleagues also use the model to explore the classification of cancer patients to determine which treatment (IAS or CAS) may benefit a patient the most. This is done by looking at the numerically estimated parameters for each patient based on their PSA data. Furthermore, the optimal treatment schedule has also been studied using their model [42]. **[PERSON] et al. (2012)**. While purely phenomenological models, such as the HBA model, are useful in establishing the core structure of cancer dynamics, they are perhaps better viewed as a building block to a more comprehensive and biologically tractable model. In this regard, and [PERSON], [PERSON], and [PERSON] (PKN) [43] formulated a mechanistic model of prostate cancer in 2012. The model utilizes the idea of cell quota as previously studied by [PERSON] in a marine ecology context [44] and has gained traction since. The PKN model assumes only two subpopulations of cancer cells that can mutually transform from one to the other. The model assumes that androgen is the limiting nutrient (or driving  
Figure 4: The fitting and forecasting capability of the Hirata, Bruchovsky, and Aihara (HBA) model—figures reproduced from [40] with permission, which is distributed under a Creative Commons Attribution (CC BY) license. (**a**) An example of data fitting using the HBA model. The circles represent patient data. The black curve represents the model result after fitting to the patient’s data during IAS. The grey curve represents the simulated PSA level in the hypothetical situation when continuous androgen suppression therapy (CAS) is carried out. (**b**) An example of forecasting using the HBA model. The vertical dashed line represents the cut-off between fitting and forecasting. force) in the proliferation and death of cancer cells and incorporates it as a cell quota. Specifically, the model presumes that there is a minimum quantity of androgen required for the growth of both AD and AI cells.