# Optimising Hydrogel Release Profiles for Viro-Immunotherapy Using Oncolytic Adenovirus Expressing IL-12 and GM-CSF with Immature Dendritic Cells

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

- In Equation (1), uninfected tumour cells, U, are growing at a rate described by a Gompertz function with proliferation rate r and carrying capacity L. Uninfected cells are infected by virus particles, V, at a frequency-dependent rate with rate constant $\beta $. Killer T cells, K, induce apoptosis in uninfected cells at a frequency-dependent rate with rate constant $\kappa $.
- In Equation (2), tumour cells become infected cells, I, at rate $\beta UV/N$. Infected cells lyse at rate ${d}_{I}I$, and, like uninfected cells, are killed at a frequency-dependent rate.
- In Equation (3), virus particles enter the system at rate ${u}_{V}\left(t\right)$, either by a single injection or release from the hydrogel. Virus particles decay at a rate ${d}_{V}$, and $\alpha $ new viruses are created through lysis.
- In Equation (4), short-term immature DCs, ${D}_{S}$, enter the system at a rate $(1-f){u}_{DC}\left(t\right)$, either by a single injection or release from a hydrogel. They are then stimulated to become mature or activated APCs, ${A}_{M}$, at rate ${s}_{AU}$ or ${s}_{AI}$ due to the interaction with either uninfected or infected tumour cells. They decay at a fast rate ${d}_{S}$, where ${d}_{S}>>{d}_{L}$.
- In Equation (5), long-term immature DCs, ${D}_{L}$, enter the system at rate $f{u}_{DC}\left(t\right)$, where ${u}_{DC}\left(t\right)$ is the rate of injection or release from the hydrogel and f is the fraction of the initially loaded DCs that are long-term DCs, $0<f<1$. Similarly to short-term DCs, they become activated APCs at rate ${s}_{AU}$ and ${s}_{AI}$. These DCs decay slowly at rate ${d}_{L}$.
- In Equation (6), immature APCs, ${A}_{I}$, are recruited to the tumour site by infected cells at rate ${r}_{AI}I$. Immature APCs are then stimulated at the same rate as immature short-term and long-term DCs. The immature APCs decay at rate ${d}_{AI}$.
- In Equation (7), mature APCs are generated through the maturation of introduced long-term and short-term DCs and immature DCs recruited by uninfected tumour cells at rate ${s}_{AU}$ and infected tumour cells at rate ${s}_{AI}$. These cells decay at rate ${d}_{A}$.
- In Equation (8), helper T cels, H, are stimulated by mature APCs at a rate ${s}_{H}$. These cells decay at a rate ${d}_{H}$. We assume the rate APCs stimulate helper cells and helper cells stimulate killer cells is independent of antigen.
- In Equation (9), killer T cells, K, are activated by helper T cells and mature APCs at rates of ${s}_{KH}$ and ${s}_{KA}$, respectively. These cells either die or leave the tumour site at a rate ${d}_{K}$.

## 3. Calibrating Parameters to In Vitro and In Vivo Time-Series Measurements

#### 3.1. DC Release-Profile and Decay Rates

#### 3.2. Tumour, Immune and Viral Parameters

## 4. Optimising the Gel Release Profile

#### 4.1. Constant Release Profile

#### 4.2. Linear Release Profile

#### 4.3. Sigmoidal Release Profile

#### 4.4. Implementing a Genetic Algorithm to Determine the True Optimal Release Curves

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flow diagram for the tumour–virus interaction from co-delivered dendritic cells (DCs) and oncolytic adenovirus expressing IL-12 and GM-CSF. Variables U and I are the uninfected and infected tumour cell populations, respectively; V is the virus population; ${D}_{S}$ and ${D}_{L}$ are the short-lived and long-lived DCs released from the gel, respectively; ${A}_{I}$ is the immature APC population already present at the tumour site; ${A}_{M}$ is the mature APC population; H is the helper T cell population; and K is the killer T cell population. Transition between states (e.g., uninfected to infected) is represented by a solid line, stimulation or activation is represented by a dotted line, death or decay is represented by a double arrow and programmed killing of tumour cells is represented by a dashed line.

**Figure 2.**Fitting the viable dendritic cell (DC) count with and without gel to parameters in Equations (1)–(9) with U = I = V = AI = AM = 0, where DS is the short-term DC population and D

_{L}is the long-term DC population. In panel (

**a**), the decay rate of short-term and long-term DCs, dS and dL, and the fraction of the initial injection that consisted of short-term DCs, f, was fit to the viable DC count after a single injection, where u

_{DC}(t) = 0. In panel (

**b**), the count of DCs released from the gel was fit to the release function u

_{DC}(t) fixing the value of d

_{L}, d

_{S}and f obtained in panel (

**a**). In both figures, circles represent the number of released viable DCs, the black solid lines are the model’s approximations to the data, the blue dashed lines are the long-term DCs in the dish, the yellow dotted lines are the short-term DCs in the dish and the purple dash-dot line is the number of DCs in the gel. The fitted parameter values are in Table 1.

**Figure 3.**Model optimisation of the measurements from Oh et al. [10] following the algorithm in Table 2 for (

**a**) control (PBS and gel) (

**b**) Ad/IL12/GMCSF, (

**c**) DC, (

**d**) DC+Ad/IL12/GMCSF and (

**e**) DC+Ad/IL12/GMCSF+gel. The individual mouse data are plotted as circles with the mean and standard error bar at each time point in blue. The model output for the experiment-optimised parameters is plotted as a solid black line. The parameter values are given in Table 3. In panel (

**f**), the corresponding model populations are plotted for the DC+Ad/IL12/GMCSF+gel experiment in panel (

**e**), note the use of both vertical axes. A

_{I}was not plotted in panel (

**f**) as the magnitude of the population was too small.

**Figure 4.**Effectiveness of a constant gel-release rate f

_{s}(t) (Equation (16)) for DCs, u

_{DC}(t), and virus, u

_{V}(t). In panel (

**a**), the tumour size at day 20 is given as a function of gel-release period t

_{r}, specific to the virus and DCs. The red points in panel (

**a**) correspond to the simulated release profiles in panel (

**b**), t

_{r}= 13 days for DCs and t

_{r}= 1 day for virus, and in panel (

**c**), t

_{r}= 2.6 days for DCs and t

_{r}= 9.8 days for virus. The top row in panels (

**b**,

**c**) correspond to the total number of tumour cells U + I, and the bottom corresponds to the release profile. Note the use of both vertical axes.

**Figure 5.**Effectiveness of a linear gel-release rate f

_{l}(t) (Equation (17)) for DCs, u

_{DC}(t), and virus, u

_{V}(t). In panels (

**a**,

**b**), the tumour size at day 20 is given as a function of the gradient of the release curve a for virus and DCs with planes corresponding to fixed values for the release time t

_{r}which are noted on the plot. In panel (

**c**), the tumour size at day 20 is given as a function of the release time for DCs and virus with planes corresponding to the labelled values of a. In panel (

**d**), the tumour size at day 20 for a decreasing linear release rate (Equation (18)) is plotted for varying initial release rate b for virus and DCs. The green point in panel (

**c**) is the corresponding release profile from the results of Oh et al. [10] in Figure 3e. The red points in panels (

**c**,

**d**) correspond to the simulated release profiles in panel (

**e**) where t

_{r}= 13 for DCs and t

_{r}= 0.5 for virus with a = 3, and in panel (

**f**) where b = 48.7 for DCs and virus. The top row of figures in panels (

**e**,

**f**) corresponds to the total number of tumour cells U + I and the bottom row of figures is the corresponding release profile. Note the use of both vertical axes.

**Figure 6.**Effectiveness of sigmoidal gel-release rate f

_{s}(t) (Equation (14)) for DCs, u

_{DC}(t), and virus, u

_{V}(t). In panel (

**a**), the tumour size at day 20 as a function of k, x

_{0}and t

_{r}is plotted, where the parameters were common for virus and DC release functions. In panel (

**b**), the tumour size at day 20 is plotted as a function of varying k for virus and DCs with t

_{r}= 9 and planes representing x

_{0}= −1, x

_{0}= 1 and x

_{0}= 3. In panel (

**c**), the tumour size at day 20 is plotted as a function of varying x

_{0}for virus and DCs with t

_{r}= 9 and planes representing k = −1, 1 and k = 3. In panel (

**d**), the tumour size at day 20 as a function of t

_{r}for virus and DCs where x

_{0}= 1 and k = 1. The red points in panels (

**c**,

**d**) correspond to the simulated release profiles in panels (

**e**,

**f**). In panel (

**e**), k = 1, t

_{r}= 9 and x

_{0}= 0.5 for u

_{V}and x

_{0}= 3 for u

_{DC}. In panel (

**f**), k = x

_{0}= 1 and t

_{r}= 1.5 for u

_{V}and t

_{r}= 14 for u

_{DC}. The top row of figures in panels (

**e**,

**f**) correspond to the number of tumour cells U + I and the bottom row of figures are the corresponding release profiles. Note the use of both vertical axes.

**Figure 7.**Genetic optimisation of the gel-release profile. Implementing a genetic algorithm we determined the optimal constant, linear and sigmoidal curves for the virus, u

_{V}(t), and DCs, u

_{DC}(t), see Table 5. The corresponding model dynamics are plotted for (

**a**) the total number of immature DCs (DL + DS + AI), (

**b**) the virus (V) and (

**c**) the total number of tumour cells (U + I). In panel (

**c**), the mean and standard error from Oh et al.’s tumour growth under DC+Ad/IL12/GMCSF loaded onto a gel is plotted in blue, and in grey is the model’s predicted tumour growth under DC+Ad/IL12/GMCSF in the absence of the gel.

**Table 1.**Parameter values obtained from fitting the release rate of the DCs and the short-term and long-term decay rates of the DCs to the in vitro viable DC count, see Figure 2.

Parameter | Units | Description | Value |
---|---|---|---|

${d}_{S}$ | day${}^{-1}$ | decay rate of short-term immature DCs | 1.562 |

${d}_{L}$ | day${}^{-1}$ | decay rate of long-term immature DCs | 0.318 |

f | dimensionless | proportion of short-term injected/released DCs | 0.325 |

${a}_{D}$ | DCs/day${}^{2}$ | gradient of release rate | 9.725 $\times {10}^{4}$ |

${b}_{D}$ | DCs/day | constant release rate | 1.463 $\times {10}^{5}$ |

**Table 2.**Experiment-specific optimisation conditions for the measurements of Oh et al. [10]. Equations used to optimise each experiment are listed along with the state variables considered and parameters fitted or fixed. Some parameters were fixed to previous optimisation work in Jenner et al. [36].

Experiment | |||||
---|---|---|---|---|---|

PBS & Gel | Ad/I/G | DC | DC+Ad/I/G | DC+Ad/I/G+Gel | |

Relevant equations | Equation (1) | Equation (1) | Equation (1) | Equation (1) | Equation (1) |

Equation (2) | Equation (2) | Equation (2) | |||

Equation (3) | Equation (3) | Equation (3) | |||

Equation (5) | Equation (5) | Equation (5) | |||

Equation (4) | Equation (4) | Equation (4) | |||

Equation (6) | Equation (6) | ||||

Equation (7) | Equation (7) | Equation (7) | Equation (7) | ||

Equation (8) | Equation (8) | Equation (8) | Equation (8) | ||

Equation (9) | Equation (9) | Equation (9) | Equation (9) | ||

Variables | U | U,I,V | U | U,I,V | U,I,V |

A,H | A_{I},A_{M}, H | A_{I},A_{M}, H | A_{I},A_{M}, H | ||

K | K | K | K | ||

Parameters fit | r,L,U_{0} | β,U_{0} | s_{AU},U_{0} | r_{AI},S_{AI},U_{0} | a_{V},b_{V},U_{0} |

k | k | s_{AU} | |||

Parameters fixed to Table 1 and Table 3 | - | r,L | r,L | r,L,β | r,L,β |

d_{S},d_{L},d_{AI} | d_{S},d_{L},d_{AI},κ | d_{S},d_{L},a_{D},b_{D} | |||

d_{AI},r_{AI},s_{AI},k | |||||

Parameters fixed from [36] | d_{V},α,s_{H},d_{H} | s_{H},d_{H} | d_{V},α,s_{H},d_{H} | d_{V},α,s_{H},d_{H} | |

d_{I},s_{A},d_{A} | d_{A} | d_{I},s_{AU},d_{A} | d_{I},s_{AU},d_{A} | ||

s_{KH},s_{KA},d_{K} | s_{KH},s_{KA},d_{K} | s_{KH},s_{KA},d_{K} | s_{KH},s_{KA},d_{K} |

**Table 3.**Parameter estimates from the sequential optimisation of the model following the algorithm in Table 2 to the experimental measurements in [10]. Certain parameters were fixed to their value in [36], and this is indicated in the table. Parameters that were fit for a particular experiment have their values in

**bold**. Note that Ad/IL12/GMCSF has been shortened to Ad/I/G.

Param | Units | Description | PBS& Gel | Ad/I/G | DC | DC + Ad/I/G | DC + Ad/I/G + gel | |
---|---|---|---|---|---|---|---|---|

Fit | ${d}_{AI}$ | day${}^{-1}$ | Immature DCs decay rate | 1.562 | 1.562 | 1.562 | ||

L | cells $\times {10}^{6}$ | carrying capacity | 13572 | 13572 | 13572 | 13572 | 135720 | |

r | day${}^{-1}$ | growth rate | 0.1045 | 0.1045 | 0.1045 | 0.1045 | 0.1045 | |

${U}_{0}$ | cells $\times {10}^{6}$ | initial tumour size | 20.35 | 85.90 | 85.90 | 85.90 | 47.45 | |

$\beta $ | day${}^{-1}$ | infection rate | - | 0.7295 | - | 0.7295 | 0.7295 | |

$\kappa $ | day${}^{-1}$ | killing rate | - | 0.8225 | 0.218147 | 0.3626 | 0.3626 | |

${s}_{AU}$ | day${}^{-1}$ | APC activation rate by U | - | - | 0.001047 | 0.0022 | 0.05278 | |

${r}_{AI}$ | day${}^{-1}$ | recruitment rate of ${A}_{I}$ | - | - | - | 0.0192 | 0.0192 | |

${s}_{AI}$ | day${}^{-1}$ | APC activation rate by I | - | - | - | 0.0011 | 0.0011 | |

${a}_{V}$ | linear release slope (virus) | - | - | - | - | 0.33791 | ||

${b}_{V}$ | initial linear release (virus) | - | - | - | - | 2.35971 | ||

[36] | $\alpha $ | virus $\times {10}^{10}$ | viral burst size | - | 3500 | - | 3500 | 3500 |

${d}_{I}$ | day${}^{-1}$ | burst rate | - | 1 | - | 1 | 1 | |

${d}_{V}$ | day${}^{-1}$ | viral decay rate | - | 2.3 | - | 2.3 | 2.3 | |

${d}_{A}$ | day${}^{-1}$ | decay of mature APCs | - | 0.23 | 0.23 | 0.23 | 0.23 | |

${d}_{AI}$ | day${}^{-1}$ | decay of immature APCs | - | 1.562 | 1.562 | 1.562 | 1.562 | |

${d}_{H}$ | day${}^{-1}$ | decay of helper T cells | - | 0.23 | 0.23 | 0.23 | 0.23 | |

${d}_{K}$ | day${}^{-1}$ | decay of killer T cells | - | 0.35 | 0.35 | 0.35 | 0.35 | |

${s}_{A}$ | day${}^{-1}$ | APC activation rate | - | 1.2 | - | 7.1 | 7.1 | |

${s}_{KA}$ | day${}^{-1}$ | APC activatet killer T cell | - | 7.1 | 7.1 | |||

${s}_{H}$ | day${}^{-1}$ | helper T cell activation | - | 0.78 | 0.78 | 0.78 | 0.78 | |

${s}_{KH}$ | day${}^{-1}$ | helper T cell activate K | - | 1.6 | 1.6 | 1.6 | 1.6 |

Residual | Coefficient of | Pearson’s Correlation | |
---|---|---|---|

Norm | Determination (${\mathit{R}}^{2}$) | Coefficient | |

PBS and gel | 33,688 | 0.8998 | 0.9487 |

Ad/IL12/GMCSF | 10,905 | 0.6465 | 0.8041 |

DC | 11,931 | 0.8980 | 0.9441 |

DC+Ad/IL12/GMCSF | 7659.4 | 0.7785 | 0.8830 |

DC+Ad/IL12/GMCSF+gel | 5249.7 | 0.5926 | 0.7736 |

**Table 5.**Optimised parameter values for the constant ${f}_{c}$, linear ${f}_{l}$ and sigmoidal ${f}_{s}$ release curves (Equations (12)–(14)) for Ad/IL12/GMCSF and immature DCs from a gel. The virus and DCs were allowed to have individualised curves. Figure 7 is a plot of the tumour volume under the five optimised curves.

Constant | Linear (Increasing) | Sigmoidal | |||||
---|---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{r}}$ | $\mathit{a}$ | ${\mathit{t}}_{\mathit{r}}$ | $\mathit{k}$ | ${\mathit{x}}_{\mathbf{0}}$ | ||

Ad/IL12/GMCSF | ${u}_{V}\left(t\right)$ | 0.1261 | 0.1252 | 14.9893 | 0.05 | 9.2789 | −6.0499 |

DCs | ${u}_{DC}\left(t\right)$ | 13.7732 | 13.1629 | 2.8866 | 13.6311 | 9.1951 | −8.3333 |

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## Share and Cite

**MDPI and ACS Style**

Jenner, A.L.; Frascoli, F.; Yun, C.-O.; Kim, P.S.
Optimising Hydrogel Release Profiles for Viro-Immunotherapy Using Oncolytic Adenovirus Expressing IL-12 and GM-CSF with Immature Dendritic Cells. *Appl. Sci.* **2020**, *10*, 2872.
https://doi.org/10.3390/app10082872

**AMA Style**

Jenner AL, Frascoli F, Yun C-O, Kim PS.
Optimising Hydrogel Release Profiles for Viro-Immunotherapy Using Oncolytic Adenovirus Expressing IL-12 and GM-CSF with Immature Dendritic Cells. *Applied Sciences*. 2020; 10(8):2872.
https://doi.org/10.3390/app10082872

**Chicago/Turabian Style**

Jenner, Adrianne L., Federico Frascoli, Chae-Ok Yun, and Peter S. Kim.
2020. "Optimising Hydrogel Release Profiles for Viro-Immunotherapy Using Oncolytic Adenovirus Expressing IL-12 and GM-CSF with Immature Dendritic Cells" *Applied Sciences* 10, no. 8: 2872.
https://doi.org/10.3390/app10082872