# The Impact of Intermittent Androgen Suppression Therapy in Prostate Cancer Modeling
## 4 Dynamics of the New Model  
In this section, we carry out mathematical analyses for the newly proposed model, which help to justify its coherence with biological expectations. Previous studies [8,18] have already examined some standard properties of the BK model and its variations, so we only focus on the new model here. We remark that, while PSA serves as an important indicator of how good the model is against clinical data, the dynamics of the system are independent of the dynamics of PSA. In addition, if the dynamics of \(x_{i}\), \(Q\), and \(A\) are known, one can immediately infer the dynamics of \(P\). Thus we will only carry out our analysis with \(x_{i}\), \(Q\), and \(A\). The first proposition establishes a positively invariant region, which is also bounded. This supports the biological validity of the model since we expect \(x_{i}\), \(Q\), and \(A\) to stay non-negative and never become infinite. **Proposition 1**. : _Let \(\overline{A}=\frac{\gamma_{2}+\gamma_{1}A_{0}}{\gamma_{1}}\) and assume \(q_{2}\leq q_{1}<\overline{A}\) and \(\gamma_{2}>\gamma_{1}q_{2}\). Furthermore, we assume that when androgen is abundant, the growth rate is larger than the death rate, meaning \(\mu_{i}(1-q_{i}/\overline{A})>D_{i}(\overline{A})\) for \(i=1,2\). Then, the solutions that start in the region \(\{(x_{1},x_{2},Q,A):0<x_{1}\leq M_{x},0<x_{2}\leq M_{y},Q,A\in[q_{2},\overline {A}]\}\), stay in that region. Here, \(M_{x}=\frac{1}{\delta_{1}}\left(\mu_{1}(1-q_{1}/\overline{A})-D_{1}(\overline {A})\right)\) and_  
\[M_{y}=\frac{1}{2\delta_{2}}\left(\left(\mu_{2}(1-q_{2}/\overline{A})-D_{2}( \overline{A})\right)+\sqrt{4\lambda(q_{2})M_{x}\delta_{2}+\left(\mu_{2}(1-q_ {2}/\overline{A})-D_{2}(\overline{A})\right)^{2}}\right).\]  
Proof. : Since \(A^{\prime}(t)=\gamma_{2}+\gamma_{1}A_{0}(1-\frac{A}{\delta_{0}})-\gamma_{1}A_ {0}(1-u)\), we see that \(A^{\prime}(q_{2})\geq 0\) and \(A^{\prime}(\overline{A})\leq 0\). It follows that \(q_{2}\leq A(t)\leq\overline{A}\) for all \(t>0\) with initial condition in the same region. Similarly, since \(Q^{\prime}(t)=m(A-Q)-\frac{\mu_{1}(Q-q_{1})x_{1}+\mu_{2}(Q-q_{2})x_{2}}{x_{1} +x_{2}}\), we find that \(Q^{\prime}(q_{2})\geq 0\) and \(Q^{\prime}(\overline{A})\leq 0\). Thus, a similar conclusion follows. Consider the rate of change of \(x_{1}\),  
\[x_{1}^{\prime} =\mu_{1}\left(1-\frac{q_{1}}{Q}\right)x_{1}-(D_{1}(Q)+\delta_{1} x_{1})x_{1}-\lambda(Q)x_{1}\] \[\leq\left(\mu_{1}\left(1-\frac{q_{1}}{\overline{A}}\right)-D_{1} (\overline{A})\right)x_{1}-\delta_{1}x_{1}^{2}.\]  
This implies that the limit supremum of \(x_{1}(t)\) is bounded by \(\frac{1}{\delta_{1}}\left(\mu_{1}(1-q_{1}/\overline{A})-D_{1}(\overline{A}) \right)=:M_{x}\). Now consider the rate of change of \(x_{2}\),  
\[x_{2}^{\prime}=\mu_{2}\left(1-\frac{q_{2}}{Q}\right)x_{2}-(D_{2}(Q)+\delta_{2} x_{2})x_{2}+\lambda(Q)x_{1}.\]  
In the limit \(t\to\infty\), \(x_{2}^{\prime}\) is bounded above by  
\[x_{2}^{\prime} \leq\mu_{2}\left(1-\frac{q_{2}}{\overline{A}}\right)x_{2}-(D_{2} (\overline{A})+\delta_{2}x_{2})x_{2}+\lambda(q_{2})M_{x}\] \[=\lambda(q_{2})M_{x}+\left(\mu_{2}\left(1-\frac{q_{2}}{\overline {A}}\right)-D_{2}(\overline{A})\right)x_{2}-\delta_{2}x^{2}=:h(x_{2}).\]  
The function \(h(x_{2})\) is a concave-down parabola. Thus \(\limsup_{t\to\infty}x_{2}(t)\) is bounded above by the larger solution of \(h(x_{2})=0\). Solving for \(h(x)=0\), we obtain the upper bound of the limit supremum of \(x_{2}\),  
\[M_{y}=\frac{1}{2\delta_{2}}\left(\left(\mu_{2}(1-q_{2}/\overline{A})-D_{2}( \overline{A})\right)+\sqrt{4\lambda(q_{2})M_{x}\delta_{2}+\left(\mu_{2}(1-q_{ 2}/\overline{A})-D_{2}(\overline{A})\right)^{2}}\right).\]  
Since \(x_{1}(0)<M_{x}\) and \(x_{2}(0)<M_{y}\), \(x_{1}\) and \(x_{2}\) are bounded above by \(M_{x}\) and \(M_{y}\), respectively. Additionally, if \(x_{i}(0)>0\), then the positivity of \(x_{i}(t)\) follows immediately. The second proposition shows that the system has a boundary equilibrium where AD cells become extinct, during the off-treatment interval.