## Implications from Mathematical Models
### Data-Validated Models
\[\frac{dx_{i}}{dt} =\underbrace{\mu_{m}\left(1-\frac{q_{i}}{Q_{i}}\right)x_{i}- \underbrace{d_{i}x_{i}}_{\text{death}}-\underbrace{\lambda_{i}(Q_{i})x_{i}}_{ \text{cell transformation}}+\underbrace{\lambda_{j}(Q_{j})x_{j}}_{\text{cell transformation}}, \tag{7}\] \[\frac{dQ_{i}}{dt} =\underbrace{\
u_{m}\frac{q_{m}-Q}{q_{m}-q_{i}}\frac{A}{A+\
u_{h }}-\underbrace{\mu_{m}(Q_{i}-q_{i})}_{\text{uptake}}-\underbrace{bQ_{i}}_{ \text{clarance}}}_{\text{clarance}},\] (8) \[\frac{dP}{dt} =\underbrace{\sum_{i=1,2}\left(\sigma_{0}+\sigma_{i}\frac{Q_{i}^{ m}}{Q_{i}^{m}+\rho_{i}^{m}}\right)x_{i}-\underbrace{\delta P}_{\text{PSA clearance}}}_{\text{PSA production}}. \tag{9}\]  
In this model, \(x_{1}\) and \(x_{2}\) are the AD and AI cancer cells, respectively. \(Q_{i}\) is the intracellular androgen level, in contrast to the serum androgen level \(A\). PSA is represented by the variable \(P\). While the most commonly presented form of the PKN model is in Equation (7), the cell-quota assumption in the model means that the growth term takes a more accurate form of \(\mu_{m}\max\{0,1-\frac{q_{i}}{Q_{i}}\}x_{i}\). In this form, the biological meaning of the model is preserved when \(Q_{i}\) dips below \(q_{i}\). This should be noted in both mathematical analysis and data fitting to avoid negative population simulations [45]. The model is first validated using the data from Akakura [46]. The PKN model requires the input of serum androgen data, so an exponential model is used to interpolate serum androgen levels. Similar to the HBA model, the PKN model also reproduces the time series observed in clinical data (see Figure 5). Additionally, a comparison of the HBA and the PKN models shows that they are equally capable of fitting and forecasting clinical PSA data [23,24]. **[PERSON] and [PERSON] (2016)**. The work by [PERSON] et al. has inspired a series of modifications to address various limitations of the model. First, the PKN model requires interpolation of androgen data, so an assumption of the future androgen level is required, which makes accurate forecasts difficult to obtain. Thus, [PERSON] and [PERSON] (BK) [47] introduced an improved version of the model where the main  
Figure 5: The fitting capability of [PERSON] et al. model—figure reproduced from [PERSON] et al. [43] with permission, which is distributed under a Creative Commons Attribution (CC BY) license. (a) The red circles followed by doted dash lines are patient’s data. The blue curve represents the fitting of the model. The dashed black curve and dotted dashed curves represent the relative level of androgen independent (AI) and androgen dependent (AD) population as predicted by the model. (b) The plot shows the level of intracellular androgen for AD and AI cells (\(Q_{1}\) and \(Q_{2}\), respectively) relative to the minimum cell quotas for AD and AI cells (\(q_{1}\) and \(q_{2}\), respectively). difference is the direct incorporation of androgen as a variable in the model, similar to the [PERSON] et al. model. This allows for the simultaneous fitting of androgen and PSA data. Furthermore, by having androgen as a variable, forecasts can be compared directly to clinical data. Aside from addressing the limitation of forecasting in the PKN model, the BK model also implicitly addresses another limitation, which is pointed out by [PERSON] et al. [23]. Using approximation, [PERSON] et al. showed that the PKN model is not able to reproduce PSA relapse under continuous androgen deprivation therapy. This was missed by the original effort, most likely because their attention was focused on intermittent therapy. [PERSON] et al. [48] show that the main reason for this biological limitation is due to the PKN model having a reversible transformation between the two subpopulations. On the other hand, by having only a transformation from AD to AI cells, which inversely depends on the serum androgen level, the BK model avoids this biological limitation and can produce relapse for CAS.