## 2 Materials and Methods
### Model Development
#### 2.1.2 Lymphoid Compartment  
**Dendritic Cells:**: Mature APCs in the lymphoid compartment (\(A_{M}^{L}\)) arrive via migration from the tumor at a rate \(\mu_{A}\) and undergo natural death at a rate \(\delta_{AM}\), yielding the following equation. \[\frac{dA_{M}^{L}}{dt}=\mu_{A}A_{M}-\delta_{AM}A_{M}^{L}. \tag{11}\]  
**T Cells:**: Naive CTLs (\(T8_{N}^{L}\)) and Th cells (\(T4_{N}^{L}\)) localize to the lymphoid compartment at assumed constant rates \(s_{T8}\) and \(s_{T4}\), respectively. \(T8_{N}^{L}\) and \(T4_{N}^{L}\) cells are activated when they come in contact with mature APCs [22], with maximum rates of activation \(\lambda_{T8}\) and \(\lambda_{T4}\), respectively. This activation is inhibited by activated Tregs in the lymphoid compartment (\(T_{\text{reg}}^{L}\)), most likely through competition for T cell binding sites on APCs [25]. A functional form similar to that in [PERSON] et al. [14] is used to capture this inhibitory effect of Tregs, with half-saturation constant \(TR0\). Both naive T cells are undergo natural death at a rate \(\delta_{TL}\). These assumptions lead to the following equations governing naive T cell dynamics. \[\frac{dT8_{N}^{L}}{dt} = s_{T8}-\frac{\lambda_{T8}T8_{N}^{L}A_{M}^{L}}{1+\frac{T_{\text{ reg}}^{L}}{TR0}}-\delta_{TL}T8_{N}^{L}, \tag{12}\] \[\frac{T4_{N}^{L}}{dt} = s_{T4}-\frac{\lambda_{T4}T4_{N}^{L}A_{M}^{L}}{1+\frac{T_{\text{ reg}}^{L}}{TR0}}-\delta_{TL}T4_{N}^{L}. \tag{13}\]  
All three activated T cell populations--CTLs (\(T8_{A}^{L}\)), Th cells (\(T4_{A}^{L}\)) and Tregs (\(T_{\text{reg}}^{L}\))-- proliferate in the presence of IL-2 [22], secreted by activated Th cells (see Equation (17)). Following [10], the rate of T cell proliferation is taken to be an increasing and saturating function of IL-2 concentration (\(I^{L}\)), with a maximum value of \(\alpha_{i}\), \(i=T8,T4,TR\) and a common half-saturation constant, \(k_{I}\). As in the tumor compartment, Tregs induce contact mediated apoptosis of activated CTLs and Th cells [25], at an assumed rate \(\delta_{R}\). All activated T cell populations migrate to the tumor compartment at rates \(\mu_{i}\), \(i=T8,T4,TR\). In addition, as in the tumor compartment, activated Th cells transform into Tregs in the presence of TGF-\(\beta\)[26], whose lymphoid concentration is represented by \(G^{L}\). This transformation rate is taken to be a Michaelis-Menten-like function with maximum transition rate \(\lambda_{R}\) and half saturation constant \(k_{G}\). Finally, all activated T cells are assumed to have the same average life-span in the lymphoid compartment, and undergo natural death at a rate \(\delta_{TL}\). Together, these assumptions lead to the following equations governing activated T cell dynamics. \[\frac{dT8_{A}^{L}}{dt} = \frac{\lambda_{T8}78_{W}^{L}A_{M}^{L}}{1+\frac{T_{\rm reg}^{L}}{TR0} }+\alpha_{T8}78_{A}^{L}\frac{l^{L}}{k_{I}+l^{L}}-\delta_{R}78_{A}^{L}T_{\rm reg }^{L}-\mu_{T8}78_{A}^{L}-\delta_{TL}T8_{A}^{L}, \tag{14}\] \[\frac{T4_{A}^{L}}{dt} = \frac{\lambda_{T4}74_{W}^{L}A_{M}^{L}}{1+\frac{T_{\rm reg}^{L}}{TR0 }}+\alpha_{T4}74_{A}^{L}\frac{l^{L}}{k_{I}+l^{L}}-\lambda_{R}T4_{A}^{L}\frac{G ^{L}}{k_{G}+G^{L}}-\delta_{R}74_{A}^{L}T_{\rm reg}^{L}-\mu_{T4}T4_{A}^{L}\] (15) \[-\delta_{TL}T4_{A^{\prime}}^{L},\] \[\frac{dT_{\rm reg}^{L}}{dt} = s_{TR}+\lambda_{R}T4_{A}^{L}\frac{G^{L}}{k_{G}+G^{L}}+\alpha_{TR} T_{\rm reg}^{L}\frac{l^{L}}{k_{I}+l^{L}}-\mu_{TR}T_{\rm reg}^{L}-\delta_{ TL}T_{\rm reg}^{L}. \tag{16}\]  
We remark that in our model all Tregs in the lymphoid compartment are assumed to be activated. This is because Tregs have a much lower activation threshold than CTLs and Th cells [27]. We therefore assume a constant localization rate \(s_{TR}\) of activated Tregs. **IL-2 and TGF-\(\beta\):** Lymphoid IL-2 (\(l^{L}\)) is produced by activated Th cells [22], at an assumed rate \(\alpha_{I}\), and undergoes natural degradation at a rate \(\delta_{I}\). Lymphoid TGF-\(\beta\) (\(G^{L}\)) is produced by Tregs [26], at an assumed rate \(\alpha_{G}\), and undergoes natural degradation at a rate \(\delta_{G}\).