# A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 and Ionizing Irradiation Treatment

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Assumptions

#### 2.2. Data Derivation from the Study

#### 2.3. Mathematical Modeling

#### 2.3.1. Modeling Therapy with IR (R) and IT (A)

#### 2.3.2. Compartmental Modeling of the Tumor(C), T Cell(T), PD-L1(PL), PD-1(PD), and Complex(S)

#### 2.3.3. Model Reduction: Quasi Steady-State Approximation (QSSA)

## 3. Results

#### 3.1. Simulation using the Cancer-Immune Model with IR and IT Therapy

^{®}, was used for model implementation with ODE45, and parameter estimation was conducted with the nonlinear least square method using MATLAB and Berkeley Madonna. Using the estimated parameters, the model captured the synergistic effect of the combination therapy consisting of IR and IT, as shown in Figure 2 (IR + IT). Additionally, the model failed to capture data around day 22. This is because repeated combination therapies cause a slightly overstated discrepancy in the model. However, the overall data fitting is reliable, and the model captures the efficacy of the combination therapy in mice.

#### 3.2. The Expression Levels of PD-1 and PD-L1

#### 3.3. Global Sensitivity Analysis of the Parameter Space: Latin Hypercube Sampling Partial Rank Correlation Coefficient

#### 3.4. Analysis of Changes in Parameters

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The tumor undergoes logistic growth, and T cells activate the cytokines interleukin 1 (IL1) and interleukin 12 (IL12). IL1 and IL12 are activated by stimulation of the tumor. The tumor expresses PD-L1 on the cell surface, and T cells express both PD-L1 and PD-1 on their surface. The binding process of PD-L1 and PD-1 results in negative feedback in T cells. IT inhibits the growth of both tumor and T cells and inhibits PD-L1 by preventing the formation of the PD-L1-PD-1 complex, which prevents the inhibition of T cell growth.

**Figure 2.**Note that $\mathsf{\alpha}$ PD-L1 is IT (anti-PD-L1). Deng et al. showed the synergistic effect of combination therapy (top left). Simulation results (top right) after data fitting are presented through the mathematical model. This model captures the control, IT, and IR well, and this model also fits combination therapy. The blue curve indicates data fit using the Equations (1)–(6), and the red circle is experimental data. A scatter plot is shown for tumor vs. PD-1+PD-L1 vs. day (bottom). Each figure indicates control, IT, IR, and the combination of IR and IT in turn. Data tips are marked on day 10 (before therapy), 14 (beginning of IR and IT therapy), 20 (during IT therapy), and 35 (final timepoint). The sum of PD-1 and PD-L1 does not always positively correlate with tumor size after therapy. The coordinates $\mathrm{X},\mathrm{Y},\mathrm{and}\mathrm{Z}$, as shown in bottom figure, represent days, PD-1 + PD-L1 concentration and tumor size, respectively. **: p < 0.01; ***: p < 0.001.

**Figure 3.**The $x$ and $y$ axes represent day and tumor size, respectively. Top panel: $\mu $ and $\nu $ are varied from [1 × 10

^{−4}, 0.2] and [1 × 10

^{−4}, 0.2], respectively. Center panel: $\mu $ and $\eta $ are varied from [1 × 10

^{−4}, 0.2] and [0, 1], respectively. Bottom panel:$\eta $ and $\mu $ are varied from [0, 1] and [1 × 10

^{−4}, 0.2], respectively. From these local changes, each parameter positively or negatively influences the change in tumor.

**Figure 4.**The $x$ and $y$ axes represent the sum of PD-1 and PD-L1 and tumor size, respectively. The $x$ axis is plotted on a logarithmic scale. Each graph indicates the two parameters that are varied. $\mu ,\nu ,$ and $\eta $ are varied from [1 × 10

^{−4}, 0.2], [1 × 10

^{−4}, 0.2], and [0, 1], respectively. After beginning combination therapy, the sum of PD-1 and PD-L1 is not positively proportional to tumor size.

**Figure 5.**Tumor growth curve with combination therapy of IR and IT is plotted with the parameter variations using Latin Hypercube Sampling. One thousand cases with four parameters randomly extracted from (7) were conducted. By varying these parameters, some tumors are eliminated, some reach maximal tumor sizes, and the others are in between the two extremes. The amount of combination therapy and other parameter values were unchanged except for $\eta ,\nu ,\mu $, and ${k}_{el}.$

**Figure 6.**PCC and PRCC scatter plots of tumor size versus parameters ${k}_{el},\mu ,\nu ,\mathrm{and}\eta $ on day 10. All four parameters are varied simultaneously. The sample size is 1000. The $x$-axis represents the parameter values in PCC or the residuals of the linear regression between the rank-transformed values of the parameters. The $y$-axis represents the tumor size in PCC or the residuals of the linear regression between the rank-transformed values of tumor versus the rank-transformed values of the parameters. The title of each plot represents the PRCC value with the corresponding p-value. The linear relationship between parameters and tumor variation becomes more apparent with PRCC compared to PCC.

**Figure 7.**Dotted points indicate tumor size on day 14. Combination therapy begins on day 14. An increase in $\nu $ is proportional to tumor growth with PRCC and p-value. Other parameters are less influential.

**Figure 8.**On day 20 (during IT therapy and after IR therapy), PRCC and p-values of four parameters are determined. An increase in $\nu $ is proportional to tumor growth, but other parameters have little influence.

**Figure 9.**In the final phase, the elimination rate of IT, ${k}_{el},$ is proportional to tumor growth. $\nu $ remains proportional to tumor growth.

**Table 1.**Experimental data. Here, the control means without any therapy. IR therapy occurs on day 14 with 12 Gy, and IT therapy begins on day 14 and continues every three days for a total of four doses with $200\mathsf{\mu}\mathrm{g}$. Both indicate a combination therapy of IR and IT and each column represents time and tumor size, respectively.

Control | IR | IT (Anti-PD-L1) | Both | ||||
---|---|---|---|---|---|---|---|

0 | 5 | 0 | 5 | 0 | 5 | 0 | 5 |

13.8587 | 120.487 | 14.0761 | 95.8553 | 13.9674 | 135.2759 | 14.0761 | 94.2126 |

18.0435 | 202.7944 | 18.0435 | 133.8006 | 18.2065 | 206.0865 | 18.0977 | 87.8069 |

21.0326 | 289.9808 | 21.087 | 158.5662 | 21.3043 | 242.3534 | 21.1413 | 53.0777 |

25.0543 | 472.4868 | 24.8913 | 204.7183 | 24.9457 | 316.4249 | 25.1087 | 40.0989 |

28.0977 | 628.6693 | 28.1522 | 303.4149 | 31.1413 | 591.012 | 28.2065 | 27.7271 |

- | - | 30.9783 | 402.0936 | - | - | 31.087 | 24.56 |

- | - | - | - | - | - | 34.9457 | 17.505 |

**Table 2.**Estimated parameters and initial values used in the model are presented together with their descriptions and units.

Initial/Parameters | Description and Units | Estimated Values |
---|---|---|

$C\left(0\right)$ | Tumor initial volume (${\mathrm{mm}}^{3}$) | 5 |

$T\left(0\right)$ | Initial lymphocytic density of CD8+ T cells (${10}^{9}\mathrm{cells}/\mathrm{L}$) | 6 × 10^{−4} (Assumed) |

$PL\left(0\right)$ | Initial concentration of PD-L1 ($\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L})$ | 1 × 10^{−5} (Assumed) |

$PD\left(0\right)$ | Initial concentration of PD-1 $(\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}$) | 1 × 10^{−5} (Assumed) |

${A}_{t}\left(14\right)$ | Initial concentration of Anti-PD-L1 ($\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}$) in tissue | 4 |

$A\left(0\right)$ | Concentration of anti-PD-L1 $\left(\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}\right)$ | 0 |

${R}_{1}\left(14\right)$ | Irradiation (G$\mathrm{y}$) | $12$ |

${k}_{c}$ | Tumor growth rate ($1/{\mathrm{mm}}^{3}/\mathrm{day})$ | 0.29428 |

${C}_{max}$ | Maximum tumor size (${\mathrm{mm}}^{3}$) | 3 × 10^{3} |

${d}_{TC}$ | Maximum tumor death rate by T cells ($1/\mathrm{day})$ | 0.53643 |

${T}_{IC}$ | Half maximum density of T cells $\left({10}^{8}\mathrm{cells}/\mathrm{L}\right)$ | 1 × 10^{3} |

${d}_{RC}$ | Maximum tumor death rate by irradiation ($1/\mathrm{day})$ | 0.5 |

${R}_{IC}$ | Half maximum irradiation $\left(\mathrm{Gy}\right)$ | 8 |

$V$ | Volume in mice ($\mathsf{\mu}\mathrm{L})$ | 50 |

${k}_{I12}$ | T cell activation rate by cytokine $\mathrm{IL}12$ $\left({10}^{8}\mathrm{cells}/\mathrm{L}/\mathrm{day}\right)$ | 10 |

${k}_{I2}$ | T cell proliferation rate by cytokine $\mathrm{IL}12$ $\left(1/\mathrm{day}\right)$ | 1 × 10^{2} |

${k}_{D}$ | Equilibrium constant ($1/\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L})$ | 1 |

$\kappa $ | Inhibition constant of PD-L1 and PD-1 $\left(\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}\right)$ | 10 |

$\eta $ | Expression level ratio of PD-L1 to PD-1 in T cells (unitless) | 0.5 |

${S}_{IC}$ | Half maximum inhibition of PD-L1 and PD-1 $\left(\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}\right)$ | 1 × 10^{3} |

${d}_{T}$ | T cell death rate $\left(1/\mathrm{day}\right)$ | 0.1 |

${d}_{RT}$ | Maximum T cell death rate by irradiation ($1/\mathrm{day})$ | 1 |

$\nu $ | Expression level of PD-L1 on activated T cells (${10}^{3}\mathsf{\mu}\mathrm{g}/\mathrm{cell}/\mathrm{day})$ | 0.1 |

${d}_{PL}$ | Degradation rate of PD-L1 ($1/\mathrm{day})$ | 1 × 10^{−2} |

${d}_{PD}$ | Degradation rate of PD-1 ($1/\mathrm{day})$ | 1 × 10^{−2} |

$\mu $ | Expression level of PD-L1 by tumor vs. T cells $\left(\mathrm{cell}/\mathsf{\mu}\mathrm{L}/{\mathrm{mm}}^{3}\right)$ | 0.1 |

${d}_{APL}$ | Maximum PD-L1 inhibition rate by anti-PD-L1 ($1/\mathrm{day})$ | 20 |

${A}_{IC}$ | Half-maximum inhibition ($\mathsf{\mu}\mathrm{g}/\mathsf{\mu}\mathrm{L}$) | 1 |

${k}_{T}$ | Intercompartment distribution rate $(1/\mathrm{day}$) | 1.5 × 10^{−3} |

${k}_{el}$ | Elimination rate of anti-PD-L1 in tissue $(1/\mathrm{day}$) | 0.1 |

${k}_{A}$ | Elimination rate of anti-PD-L1 ($1/\mathrm{day})$ | 0.05 |

${k}_{1}$ | Delay rate with the mean duration $1/{k}_{1}$(1$/\mathrm{day})$ | 0.15 |

${k}_{R}$ | Elimination rate of ionizing irradiation ($1/\mathrm{day})$ | 0.09 |

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**MDPI and ACS Style**

Byun, J.H.; Yoon, I.-S.; Jeong, Y.D.; Kim, S.; Jung, I.H.
A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 and Ionizing Irradiation Treatment. *Pharmaceutics* **2020**, *12*, 830.
https://doi.org/10.3390/pharmaceutics12090830

**AMA Style**

Byun JH, Yoon I-S, Jeong YD, Kim S, Jung IH.
A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 and Ionizing Irradiation Treatment. *Pharmaceutics*. 2020; 12(9):830.
https://doi.org/10.3390/pharmaceutics12090830

**Chicago/Turabian Style**

Byun, Jong Hyuk, In-Soo Yoon, Yong Dam Jeong, Sungchan Kim, and Il Hyo Jung.
2020. "A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 and Ionizing Irradiation Treatment" *Pharmaceutics* 12, no. 9: 830.
https://doi.org/10.3390/pharmaceutics12090830