# The Optimal Radiation Dose to Induce Robust Systemic Anti-Tumor Immunity

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Data Fitting

^{3}(average relative error of 19.6 ± 16%).

#### 2.2. Predicted Radiation Response

#### 2.3. Optimal Radiation Dose and Dose Fractionation

^{3}, compared to 513 mm

^{3}if the same total dose was delivered in 15 fractions of 2.67 Gy each (Figure 4A).

^{3}for 12.36 Gy × 2 compared to 552 mm

^{3}for 2.38 Gy × 15; Figure 4B). Moreover, simulations suggest that optimal number of fractions and dose per fraction may allow up to a three-fold reduction of the total dose whilst providing a similar volumetric outcome (53.8 mm

^{3}for 7.5 Gy × 8 fractions to a total of 60 Gy total dose vs. 65.3 mm

^{3}for 10 Gy × 2 for a total of 20 Gy; Figure 4A), thereby potentially reducing normal tissue toxicity.

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Experimental Data

^{3}and 21 mm

^{3}on average for primary and secondary tumors, respectively). At the end of the experiment (Day 35), secondary tumors from all groups, except for those that received radiation using fractionations other than 3 × 8 Gy, were excised and analyzed by fluorescence microscopy for the presence of CD8+ T cells. The reported mean value at each experimental time point was extracted from [11] for this study.

#### 4.2. Mathematical Model of Tumors-Immune System Interaction

^{3}); (2) cancer cells dying in a non-immunogenic manner (volume ${D}_{i}\left(t\right)$ mm

^{3}); (3) cancer cells dying in an immunogenic manner (volume ${I}_{i}\left(t\right)$ mm

^{3}); and (4) activated tumor-specific cytotoxic T cells (effector cells; density ${E}_{i}\left(t\right)$ cells/mm

^{3}). Assuming that immune cells do not contribute significantly to the observed tumor volume, we denote the total measurable volume with:

#### 4.3. Parameter Estimation

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RT | Radiotherapy |

APC | Antigen-presenting cell |

CTL | Cytotoxic T lymphocyte |

IR | Irradiation |

BED | Biologically effective dose |

## Appendix A. Supplementary Data

**Figure A1.**Comparison of model simulated growth dynamics with experimental data. Solutions to the proposed model (lines; Equations (3)–(10)) were obtained after performing data fitting (estimated parameters are presented in Table 1). Shown are experimental and simulated tumor volumes for tumors grown without treatment or after monotherapy. Experimental data (circles) derived from Dewan et al. [11].

**Figure A2.**Comparison of model simulated growth dynamics with experimental data. Solutions to the proposed model (lines; Equations (3)–(10)) were obtained after performing data fitting (estimated parameters are presented in Table 1). Shown are experimental and simulated tumor volumes after combination treatment. Experimental data (circles) derived from Dewan et al. [11].

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**Figure 1.**Schematic of the experimental setting and model compartments (circles). Arrows indicate transitions between compartments and T bars indicate killing of cancer cells by immune cells. The proposed model (Equations (3)–(10)) formalizes the relevant mechanisms underlying the in vivo experiment [11] in which carcinoma cells were injected subcutaneously into mice at two spatially-separated sites (left and right flank), and only one site (1, primary tumor) was later irradiated (IR). Some mice received also systemic immunotherapy (9H10), which augments activation of tumor-specific cytotoxic T lymphocytes (CTLs).

**Figure 2.**Comparison of the model simulated growth dynamics with experimental data. Solutions to the proposed model (Equations (3)–(10)) were obtained after performing data fitting (estimated parameters are presented in Table 1). (

**A**) Experimental and simulated tumor volumes for tumors grown without treatment or after monotherapy. (

**B**) Experimental and simulated tumor volumes after combination treatment. Experimental data (red circles) derived from Dewan et al. [11].

**Figure 3.**Comparison of relative T cell density and radiation response curves. (

**A**) Model simulated and experimentally measured changes in immune cell infiltration at the secondary tumor site after the combination of radiotherapy and 9H10 immunotherapy. (

**B**) Proposed interpolation of the model estimated radiation survival fraction ($S{F}_{D}$) and proportion of cells undergoing immunogenic cell death ($A{I}_{D}$) for radiation doses D = 6, 8 and 20 Gy (see Table 1) using Equations (1) and (2).

**Figure 4.**Optimal radiation fractionation and dose per fraction for immune activation. Dependence of the model predicted overall tumor burden at Day 32, i.e., ${V}_{1}\left(32\mathrm{days}\right)+{V}_{2}\left(32\mathrm{days}\right)$, for different total (

**A**) and biologically effective doses (

**B**) for various numbers of fractions. Radiation is delivered daily, and three concurrent doses of 9H10 immunotherapy are applied at different times [11]. (

**C**,

**D**) Optimal number of radiation fractions and corresponding doses per fraction depending on the prescribed total and biologically effective doses. We assume that 9H10 immunotherapy is applied on Days 12,15 and 18.

Parameter | Description | Unit | Value |
---|---|---|---|

r | Viable cancer cells volume doubling time | 1/day | 0.195 |

K | Tumor carrying capacity | mm^{3} | 1423.1 |

a | CTLs’ killing rate | mm^{3}/(cell·day) | 0.0177 |

d | Clearance rate of dying cells | 1/day | 0.264 |

$S{F}_{D}$ | Fraction of viable cancer cells that survive | ||

after radiation dose | |||

$D=20$ Gy | - | 0.265 | |

$D=8$ Gy | - | 0.664 | |

$D=6$ Gy | - | 0.783 | |

$A{I}_{D}$ | Fraction of cells that will undergo | ||

immunogenic cell death after | |||

radiation dose | |||

$D=20$ Gy | - | 0.194 | |

$D=8$ Gy | - | 0.984 | |

$D=6$ Gy | - | 0.367 | |

l | Decay rate of effector cells | 1/day | 0.03 |

w | Baseline T cell recruitment rate | cell/ (mm^{3}·day) | 0.135 |

${w}_{2}$ | Fold change in the baseline T cell recruitment | ||

rate due to immunogenic cell death | - | 15.37 | |

e | Initial fold change in recruitment | ||

of cytotoxic T cells caused by the administered | |||

dose of 9H10 immunotherapy | - | 8.495 | |

$clr$ | 9H10 immunotherapy clearance rate | 1/day | 0.967 |

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**MDPI and ACS Style**

Poleszczuk, J.; Enderling, H.
The Optimal Radiation Dose to Induce Robust Systemic Anti-Tumor Immunity. *Int. J. Mol. Sci.* **2018**, *19*, 3377.
https://doi.org/10.3390/ijms19113377

**AMA Style**

Poleszczuk J, Enderling H.
The Optimal Radiation Dose to Induce Robust Systemic Anti-Tumor Immunity. *International Journal of Molecular Sciences*. 2018; 19(11):3377.
https://doi.org/10.3390/ijms19113377

**Chicago/Turabian Style**

Poleszczuk, Jan, and Heiko Enderling.
2018. "The Optimal Radiation Dose to Induce Robust Systemic Anti-Tumor Immunity" *International Journal of Molecular Sciences* 19, no. 11: 3377.
https://doi.org/10.3390/ijms19113377