# Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Model

- -
- BCG bacteria within the bladder as B;
- -
- APCs (dendritic cells (DCs) and macrophages) as A;
- -
- activated/matured APC’s after BCG internalization and processing as ${A}_{B}$;
- -
- activated/matured APC’s specific to tumor Ag as ${A}_{T}$;
- -
- effector T lymphocytes consisting mostly of CTLs that react to BCG as ${E}_{B}$;
- -
- effector T lymphocytes consisting mostly of CTLs that react to tumor Ags as ${E}_{T}$;
- -
- IL-2 units injected inside the bladder as ${I}_{2}$;
- -
- tumor cells infected with BCG as ${T}_{i}$;
- -
- tumor cells not infected by BCG as ${T}_{u}$;
- -
- transforming growth factor-beta (TGF-$\beta $) denotes as ${F}_{\beta}$.

## 3. Equilibria

#### 3.1. Equilibrium with $b>0$, ${i}_{2}\ge 0$

- (1)
- From (9’) it follows that one of the possible ${T}_{u}$ is ${T}_{u}=0$.
- (2)
- (10’) it follows ${F}_{\beta}=0$ (via ${T}_{u}=0$).
- (3)
- From (8’) it follows ${E}_{B}{T}_{i}=0$ (via ${T}_{u}=0$).
- (4)
- From (4’) it follows ${A}_{T}=0$ (via ${T}_{u}=0$ and ${E}_{B}{T}_{i}=0$).
- (5)
- From (6’) it follows ${E}_{T}=0$ (via ${T}_{u}=0$ and ${A}_{T}=0$).
- (6)
- From (1’), (2’) the system for A, B it follows (via ${E}_{B}{T}_{i}=0$ and ${T}_{u}=0$)$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & ({p}_{1}A+{\mu}_{B})B=b,\\ \phantom{\rule{1.em}{0ex}}\hfill & [({p}_{1}-\eta )B+{\mu}_{A}]A=\gamma ,\end{array}$$$$\begin{array}{c}{A}^{*}={\displaystyle \frac{\sqrt{{a}_{1}^{2}+4{a}_{0}{a}_{2}}-{a}_{1}}{2{a}_{0}}},\phantom{\rule{1.em}{0ex}}{B}^{*}={\displaystyle \frac{b}{{p}_{1}{A}^{*}+{\mu}_{B}}},\hfill \\ {a}_{0}={p}_{1}{\mu}_{A},\phantom{\rule{1.em}{0ex}}{a}_{1}=b({p}_{1}-\eta )+{\mu}_{A}{\mu}_{B}-\gamma {p}_{1},\phantom{\rule{1.em}{0ex}}{a}_{2}=\gamma {\mu}_{B}.\hfill \end{array}$$
- (7)
- From (3’) it follows ${A}_{B}^{*}={\displaystyle \frac{{p}_{1}}{\beta +{\mu}_{{A}_{1}}}}{A}^{*}{B}^{*}$ (via ${A}^{*}$, ${B}^{*}$).
- (8)
- From (5’) it follows that if ${E}_{B}=0$ then ${I}_{2}=0$ but via (7’) it is impossible. So, from ${E}_{B}{T}_{i}=0$ it follows ${T}_{i}=0$.
- (9)
- From (5’) and (7’) the system for ${E}_{B}^{*}$, ${I}_{2}^{*}$ it follows (via ${A}_{T}={E}_{T}={T}_{i}=0$)$$\begin{array}{c}{\displaystyle \frac{{\beta}_{B}{A}_{B}^{*}{I}_{2}}{{A}_{B}^{*}+g}}={\mu}_{E}{E}_{B},\phantom{\rule{1.em}{0ex}}({A}_{B}^{*}+{E}_{B})\left({q}_{1}-{\displaystyle \frac{{q}_{2}{I}_{2}}{{I}_{2}+{g}_{I}}}\right)+{i}_{2}={\mu}_{{I}_{2}}{I}_{2},\hfill \end{array}$$

#### 3.2. Equilibria with $b=0$, ${i}_{2}\ge 0$

- (1)
- From (2’) it follows $B=0$.
- (2)
- From (3’) it follows ${A}_{B}=0$ (via $B=0$).
- (3)
- From (5’) it follows ${E}_{B}=0$ (via ${A}_{B}=0$).
- (4)
- From (4’) it follows ${A}_{T}=0$ (via ${E}_{B}=0$).
- (5)
- From (6’) it follows ${E}_{T}=0$ (via ${A}_{T}=0$).
- (6)
- From (1’) it follows $A={\displaystyle \frac{\gamma}{{\mu}_{A}}}$ (via $B={E}_{B}=0$).
- (7)
- From (7’) it follows ${I}_{2}={\displaystyle \frac{{i}_{2}}{{\mu}_{{I}_{2}}}}$ (via ${A}_{B}={A}_{T}={E}_{B}={E}_{T}=0$).
- (8)
- From (9’) it follows ${T}_{u}=0$ or ${T}_{u}=K$ (via $B={A}_{T}={E}_{T}=0$).
- (9)
- From (10’) it follows ${F}_{\beta}=0$ or ${F}_{\beta}={\displaystyle \frac{{\alpha}_{\beta ,T}}{{\mu}_{\beta}}}K$ (via ${T}_{u}=0$ or ${T}_{u}=K$).
- (10)
- From (8’) it follows ${T}_{i}=C=const$ (via $B={E}_{B}=0$).

- (1)
- tumor-free (${T}_{u}^{*}=0$)$$\begin{array}{cc}\hfill {E}_{2}=& ({A}^{*},{B}^{*},{A}_{B}^{*},{A}_{T}^{*},{E}_{B}^{*},{E}_{T}^{*},{I}_{2}^{*},{T}_{i}^{*},{T}_{u}^{*},{F}_{\beta}^{*})\hfill \\ \hfill =& \left({\displaystyle \frac{\gamma}{{\mu}_{A}}},0,0,0,0,0,{\displaystyle \frac{{i}_{2}}{{\mu}_{{I}_{2}}}},C,0,0\right),\hfill \end{array}$$
- (2)
- not tumor-free (${T}_{u}^{*}\ne 0$)$$\begin{array}{cc}\hfill {E}_{3}=& ({A}^{*},{B}^{*},{A}_{B}^{*},{A}_{T}^{*},{E}_{B}^{*},{E}_{T}^{*},{I}_{2}^{*},{T}_{i}^{*},{T}_{u}^{*},{F}_{\beta}^{*})\hfill \\ \hfill =& \left({\displaystyle \frac{\gamma}{{\mu}_{A}}},0,0,0,0,0,{\displaystyle \frac{{i}_{2}}{{\mu}_{{I}_{2}}}},C,K,{\displaystyle \frac{{\alpha}_{\beta ,T}}{{\mu}_{\beta}}}K\right).\hfill \end{array}$$

**Remark**

**1.**

## 4. Centralization and Linearization

## 5. Stability

**Lemma**

**1.**

**Corollary**

**1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Remark**

**3.**

**Example**

**5.**

**Remark**

**4.**

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Solution of (3)

#### Appendix A.2. Solution of (4)

#### Appendix A.3. Centralization and Linearization of a Nonlinear Equation

#### Appendix A.4. Schur Complement

**Schur complement**[43]. The symmetric matrix $\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill {B}^{\prime}\hfill & \hfill C\hfill \end{array}\right]$ is negative definite if and only if C and $A-B{C}^{-1}{B}^{\prime}$ are both negative definite.

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**Figure 1.**BCG (denoted by B) invade the APCs (denoted by A) present ed in the urothelium. Bacterial infection stimulates APCs to produce inflammatory cytokines such as IL-2. At the same time, bacteria infect residual cancer cells after surgery designated as ${T}_{u}$. As a result of infection, bacterial Ag appears on the surface of tumor cells, attracting APC (designated as ${A}_{B}$). ${A}_{B}$ engulf tumor infected cells (${T}_{i}$) and transfer the tumor agent to APC (denoted by ${A}_{T}$). Due to the inflammatory environment created by the bacterial infection, ${A}_{B}$ cause CTLs to either mature and track bacterial Ag, denoted as ${E}_{B}$ (Pathway A), or mature and invade tumor cells according to their tumor Ag, starting with a delay $\tau $ (it is indicated by a broken line) due to the time it takes to form the CTL population, designated by ${E}_{B}$ (Pathway B). Two CTL populations can destroy tumor cells ${E}_{B}$ and ${E}_{T}$; and IL-2 will stimulate the maturation of the CTL cell population.

Parameters | Physical Interpretation (Units) | Estimated Value | Reference |
---|---|---|---|

${\mu}_{A}$ | APC half life $\left[day{s}^{-1}\right]$ | 0.038 | [27] |

${\mu}_{{A}_{1}}$ | Activated APC half life $\left[day{s}^{-1}\right]$ | 0.138 | [28] |

${\mu}_{{E}_{1}}$ | Effector cells mortality rate w/o IL-2 $\left[day{s}^{-1}\right]$ | 0.19 | [23] and calculated |

${\mu}_{{E}_{2}}$ | Effector cells mortality rate with IL-2 $\left[day{s}^{-1}\right]$ | 0.034 | [29] |

${\mu}_{B}$ | BCG half life $\left[day{s}^{-1}\right]$ | 0.1 | [30] |

${p}_{1}$ | The rate of BCG binding with APC $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $1.25\times {10}^{-4}$ | [31] adjusted for liters |

${p}_{2}$ | Infection rate of tumor cells by BCG $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $0.028\times {10}^{-6}$ | From model simulation |

${p}_{3}$ | Rate of E deactivation after binding with infected tumor cells $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $1.03\times {10}^{-10}$ | [32] |

${p}_{4}$ | Rate of destruction of infected tumor cells by effector cells $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $1.1\times {10}^{-6}$ | [32] |

$\lambda $ | Production rate of TAA-APC $\left[day{s}^{-1}\right]$ | ${10}^{-8}$ | [33] |

${\beta}_{B}$ | Recruitment rate of effector cells in response to signals released by BCG-infected and activated APC $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]\left[{I}_{2}^{-1}\right]$ | $1.45\times {10}^{8}$ | [34] |

${\beta}_{T}$ | Recruitment rate of effector cells in response to signals released by TAA-infected and activated APC $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]\left[{I}_{2}^{-1}\right]$ | $1.514\times {10}^{6}$ | [35] |

$\gamma $ | Initial APC cell numbers $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | 4700 | [28] |

$\eta $ | Rate of recruited additional resting APCs $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $2.8\times {10}^{-6}$ | [27] |

r | Tumor growth rate $\left[day{s}^{-1}\right]$ | 0.0048–0.0085 | [36] |

b | Bio-effective dose of BCG [c.f.u./week] | $2.2\times {10}^{8}$ | From clinical data provided by Dr. Sarel Halachmi |

$\beta $ | Migration rate of TAA-APC and bacteria activated APC to the lymph node $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | 0.034 | [27] |

$\alpha $ | Efficacy of an effector cell on tumor cell $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $3.7\times {10}^{-6}$ | [37] |

g | Michaelis-Menten constant for BCG activated CTLs and for TAA-CTLs[cells] | ${10}^{13}$ | From model simulation |

${g}_{T}$ | Michaelis-Menten constant for tumor cells[cells] | 5200 | [24] |

K | Maximal tumor cell population [cells] | ${10}^{11}$ | [38] |

${q}_{1}$ | Rate of IL-2 production IU $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | 0.007 | [39] and simulations |

${q}_{2}$ | The proportion of IL-2 used for differentiation of effector cells IU $\left[cell{s}^{-1}\right]\left[day{s}^{-1}\right]$ | $1.2\times {10}^{-3}$ | [35] |

${\mu}_{{I}_{2}}$ | Degradation rate $\left[day{s}^{-1}\right]$ | 11.5 | [35,40] |

$\theta $ | Recruitment rate of Tumor-Ag-activated APC cells in response to signals released after binding effector cells, that react to BCG infection, with infected tumor cells [$1/cel{l}^{-1}$] | 0.01 | From model simulation |

${\alpha}_{\beta ,T}$ | The release term per tumor cell [$pg/cel{l}^{-1}\ast {d}^{-1}$] | $1.38\times {10}^{-4}$ | [23] |

${\alpha}_{T,\beta}$ | Michaelis-Menten saturation dynamics. The dependence on ${F}_{\beta}$ is decreasing from 0 to ${\alpha}_{T,\beta}$ [none] | 0.69 | [23] |

${e}_{T,\beta}$ | Michaelis constant [$pg$] | 10000 | [23] |

${\mu}_{\beta}$ | The constant rate, accounts for degradation of ${F}_{\beta}$ [${d}^{-1}$] | 166.32 | [23] |

${g}_{I}$ | Michaelis-Menten constant for IL-2 [cells] | 10000 | From model calculations |

${i}_{2}$ | Rate of external source [units per treatment] | $8\times {10}^{5}$–$7.7\times {10}^{6}$ | [9] |

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**MDPI and ACS Style**

Bunimovich-Mendrazitsky, S.; Shaikhet, L. Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer. *Processes* **2020**, *8*, 1564.
https://doi.org/10.3390/pr8121564

**AMA Style**

Bunimovich-Mendrazitsky S, Shaikhet L. Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer. *Processes*. 2020; 8(12):1564.
https://doi.org/10.3390/pr8121564

**Chicago/Turabian Style**

Bunimovich-Mendrazitsky, Svetlana, and Leonid Shaikhet. 2020. "Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer" *Processes* 8, no. 12: 1564.
https://doi.org/10.3390/pr8121564