## Implications from Mathematical Models
### Limitations of Psa as a Proxy for Tumor Size  
**[PERSON] et al. (2001)**. Mathematical models of prostate cancer rely on clinical and experimental data for their validation. For clinical purposes, a model is often validated using the byproduct of prostate cell activity, PSA. However, at the patient level, the correlation between PSA level and cancer volume is questionable. Motivated by this observation, [PERSON] et al. (2003) examined how well PSA represents the prostate (both healthy and cancerous) volume.  
\[\frac{dy}{dt}=\beta_{h}V_{h}+\beta_{c}V_{c}(t)-ky(t) \tag{12}\]  
This simple differential equation assumes linear production rates of PSA based on the proportion of cancerous/healthy cells coupled with a linear degradation rate. \(y(t)\) represents the serum PSA with linear production rates \(\beta_{h}\) and \(\beta_{c}\) from the healthy and cancerous cancer cells, respectively. The volume of healthy cells is assumed to be constant (\(V_{h}\) is constant), while the volume of cancerous cells is governed by exponential growth (\(V_{c}=V_{0}e^{it}\)). The parameter values are estimated using human-derived mouse xenograft LuCaP 23 published in [PERSON] et al. (2003). Although there are limitations in the biological basis of the model, it offers a valuable explanation for the abnormality in the PSA representation of prostate tumor volume. They conclude this abnormality can be explained by the ratio of the PSA degradation rate and tumor growth rate.  
**[PERSON] et al. (2002), [PERSON] and [PERSON] (2003)**. On the other hand, by adjusting the parameter values in [PERSON] et al. to account for differences in mouse and human, [PERSON] et al. (1995) (section 5.3.1) argue that while variation in cancer growth rates can explain some of the poor correlation, the key parameters are the PSA production rates by cancer cells and PSA degradation rates. This conclusion is further supported by the work of [PERSON] and [PERSON] in 2003 ([PERSON] and [PERSON], 2003). Due to the limitation of PSA as a biomarker of prostate cancer, further investigations have been carried out to explore alternative measurements to PSA in predicting cancer progression. For example, the work by [PERSON] et al. (2003) in 2001 on the dynamics of PSA during watchful waiting show that PSA amplitude and relative velocity are better predictors of cancer progression--a conclusion supported by clinical studies ([PERSON] and [PERSON], 2003).  
**[PERSON] (2010), [PERSON] et al. (2012)**. Existing modeling studies sometimes use additional clinical measurement besides PSA as either input or validation for the model. One such example is the work by [PERSON] (2010), where the author constructed a cell kinetics model to track prostate cancer progression from diagnosis to final outcome. In this case, the patient's Gleason score is used to obtain the transition rate of cells. The model was used to study the author's and another patient's data. In a subsequent work, [PERSON] and colleagues ([PERSON] et al., 2012) simplify the model and use it to explain the variability in the recurrence time of prostate cancer patients. They reach a similar conclusion to use PSA doubling time to improve predictive power. There are various potential biomarkers for prostate cancer that are being studied clinically. For a list of potential biomarkers for prostate cancer, the readers are referred to [79]. Modeling work aiming to incorporate multiple biomarkers can potentially result in increasing the accuracy of the model's predictive power.