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# Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant

• * Corresponding author: Yang Kuang
• Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression across multiple types of cancer. There has recently been interest in combining the ICIs with other forms of treatments, as not all patients benefit from monotherapy. We propose a mathematical model consisting of ordinary differential equations to investigate the combination treatments of the ICI avelumab and the immunostimulant NHS-muIL12. We validated the model using the average tumor volume curves provided in Xu et al. (2017). We initially analyzed a simple generic model without the use of any drug, which provided us with mathematical conditions for local stability for both the tumorous and tumor-free equilibrium. This enabled us to adapt these conditions for special cases of the model. Additionally, we conducted systematic mathematical analysis for the case that both drugs are applied continuously. Numerical simulations suggest that the two drugs act synergistically, such that, compared to monotherapy, only about one-third the dose of both drugs is required in combination for tumor control.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Schematic model representation for Tumor-Immune Interactions. Sharp arrows indicate proliferation/activation. Blocked arrows indicate killing/blocking. Dashed lines indicate proteins on the tumour (V) or T cells (T). IL-12 stimulates the proliferation of activated T-cells. NHS-muIL12's role is to administrate IL-12 to damaged intratumoral regions. PD-L1 is mainly expressed on tumor cells. Anti-PD-L1 binds to PD-L1 to prevent the formation of the PD-1-PD-L1 complex

Figure 2.  Tumor volume data and simulations using model (3)-(6) for each single-agent and combination therapy case. Case (a) administration of no drug. Case (b), (c) treatment with NHS-muIL12 $2\mu g$ and $10 \mu g$ respectively on day $0$. Case (d) administration of avelumab $200 \mu g$ on days 0, 3 and 6. Case (e) treatment with both avelumab ($200 \mu g$) on days 0, 3, 6 and NHS-muIL12 ($2\mu g$) on day 0

Figure 3.  First graph depicts the model simulation for the combination treatment case of avelumab ($200 \mu g$) and NHS-muIL12 ($10\mu g$). The remaining two graphs correspond to each of the drugs administrated with respect to time

Figure 4.  Surface plot depicting the asymptotic behavior of systems (3)-(6) and (18)-(19) for combinations of $\gamma_1$ (avelumab) and $\gamma_2$ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, three regions, namely, 'tumor control', 'tumor escape' and 'intermediate' are identified, which are separated by hyperbolic curves. The hyperbolic curve closer to the origin corresponds to the system (18)-(19) and the other one to (3)-(6). Six representative points indicated by different markers and colors, are chosen and shown on the plot. Their respective tumor volume behavior is plotted in Figure 5 employing the same marker and color notation. Note that the red point falls in the intermediate region, where the behavior is dependent on the system being used

Figure 5.  A subset of tumor volume behaviors of both the full system (3)-(6) and the limiting system (18)-(19) for continuous dosage with $\gamma_1$ (avelumab) and $\gamma_2$ (NHS-muIL12) respectively. The threshold chosen is shown on the plot with a dashed line. We observe that the qualitative tumor volume behaviors between the systems are comparable with the exception of the red point, which lies in the intermediate region as shown in Figure 4

Figure 6.  Surface plot depicting the asymptotic behavior of the full system (3)-(6) for combinations of $\gamma_1$ (avelumab) and $\gamma_2$ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, two regions, namely, 'Tumor Control' and 'Tumor Escape' are identified, which are separated by a hyperbolic curve. Six representative points indicated by different markers and colors, are chosen and shown on the plot

Figure 7.  Surface plot depicting the asymptotic behavior of the limit case system (18)-(19) for combinations of $\gamma_1$ (avelumab) and $\gamma_2$ (NHS-muIL12) drugs, where both are applied continuously. Based on a threshold, two regions, namely, 'Tumor Control' and 'Tumor Escape' are identified, which are separated by a hyperbolic curve. Six representative points indicated by different markers and colors, are chosen and shown on the plot

Table 1.  State Variables of the model system (1)-(2) and (3)-(6).

 Variable Meaning Unit $V$ tumor cell volume mm$^3$ $T$ volume of activated T cells mm$^3$ $L$ free PD-L1 volume mm$^3$ $P$ free PD-1 volume mm$^3$ $A_{1}$ anti-PD-L1 concentration g $A_{2}$ NHS-muIL12 concentration g $Q$ PD-1-PD-L1 volume mm$^3$

Table 2.  Parameters and variables (Var.) of the model system (3)-(6)

 Var. Meaning Value Reference $r$ Tumor cell growth rate $0.213\text{ day}^{-1}$ fitted $\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$ fitted $\delta$ Source of T cell activation 0.02 mm$^{3}$ /day estimated $\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$ [14] $K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter estimated $K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$ estimated $d_{T}$ Death rate of T cells $0-0.5\text{ day}^{-1}$ [14] $d_{A_{1}}$ Degradation rate of Anti-PD-L1 $0.1136\text{ day}^{-1}$ [17] $d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$ [12] $\rho_{p}$ Expression level of PD-1 $3.19\cdot10^{-7}$ - $8.49\cdot 10^{-7}$ [11] $\rho_{L}$ Expression level of PD-L1 $3.56\cdot10^{-7}$ - $1.967\cdot 10^{-6}$ [11] $K_{A_{1}}$ Dissociation constant of free PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter estimated $\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 1-100 [14] $\sigma$ fraction of complex association and dissociation 0.01mm$^{-3}$ estimated $\gamma_1$ continuous infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day estimated $\gamma_2$ continuous infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day estimated $c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}-55^{-1}10^{-6}$ estimated $c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}- 75^{-1}10^{-6}$ estimated

Table 3.  Parameter values used in simulations for equations (3)-(6)

 Variable Meaning Value $r$ Tumor cell growth rate $0.213\text{ day}^{-1}$ $\eta$ Kill rate of tumor cells by T cells $1$ mm$^{-3}$ $\cdot$ day$^{-1}$ $\delta$ Source of activation 0.02 mm$^{3}$ /day $\lambda_{TI_{12}}$ Activation rate of T cells by IL-12 8.81 day$^{-1}$ $K_{A_{2}}$ Dissociation constant of $A_{2}$ $7 \cdot 10^{-14}$ moles/liter $K_{TQ}$ Inhibition of function of T cells by PD-1-PD-L1 $10^{-13}$ mm$^{6}$ $d_{T}$ Death rate of T cells $0.05\text{ day}^{-1}$ $d_{A_{1}}$ Degradation rate of Anti-PD-L1 $0.1136\text{ day}^{-1}$ $d_{A_{2}}$ Degradation rate of NHS-muIL12 0.69 day$^{-1}$ $\rho_{p}$ Expression level of PD-1 $5.84\cdot 10^{-7}$ $\rho_{L}$ Expression level of PD-L1 $2.7635\cdot 10^{-7}$ $K_{A_{1}}$ Dissociation constant of PD-L1 with anti-PD-L1 10$^{-13}$ mol/liter $\epsilon_{v}$ Expression of PD-L1 in tumor cells vs. T cells 50 $\sigma$ fraction of complex association and dissociation 0.001 mm$^{-3}$ $\gamma_1$ prescribed infusion rate of avelumab $10^{-7}-9\cdot 10^{-5}$ g/day $\gamma_2$ prescribed infusion rate of NHS-muIL12 $10^{-9}-2\cdot 10^{-6}$ g/day $c_1$ conversion constant for $A_1$ drug $55^{-1}10^{-7}$ $c_2$ conversion constant for $A_2$ drug $75^{-1}10^{-7}$
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