# 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 ${A}_{T}={E}_{T}={T}_{i}=0$)$$\begin{array}{c}{\displaystyle \frac{{\beta}_{B}{A}_{B}^{*}{I}_{2}}{{A}_{B}^{*}+g}}={\mu}_{E}{E}_{B},\phantom{\rule{1.em}{0ex}}({A}_{B}^{*}+{E}_{B})\left(\right)open="("\; close=")">{q}_{1}-{\displaystyle \frac{{q}_{2}{I}_{2}}{{I}_{2}+{g}_{I}}}+{i}_{2}={\mu}_{{I}_{2}}{I}_{2},\hfill \end{array}$$

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

**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.

## References

- Kassouf, W.; Traboulsi, S.L.; Kulkarni, G.S.; Breau, R.H.; Zlotta, A.; Fairey, A.; So, A.; Lacombe, L.; Rendon, R.; Aprikian, A.G.; et al. CUA guidelines on the management of non-muscle invasive bladder cancer. Can. Urol. Assoc. J.
**2015**, 9, 690–704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anastasiadis, A.; de Reijke, T.M. Best practice in the treatment of nonmuscle invasive bladder cancer. Ther. Adv. Urol.
**2012**, 4, 13–32. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bunimovich-Mendrazitsky, S.; Pisarev, V.; Kashdan, E. Modeling and simulation of a low-grade urinary bladder carcinoma. Comput. Biol. Med.
**2015**, 58, 118–129. [Google Scholar] [CrossRef] [PubMed] - Adeloye, D.; Harhay, M.O.; Ayepola, O.O.; Dos Santos, J.P.; David, A.R.; Ogunlana, O.O.; Gadanya, M.A.; Osamor, V.C.; Amuta, O.A.; Iweala, E.E.; et al. Estimate of the incidence of bladder cancer in Africa: A systematic review and Bayesian meta-analysis. Int. J. Urol.
**2019**, 26, 102–112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rodriguez, D.; Sotolongo-Grau, O.; Espinosa, R.; Sotolongo-Costo, R.O.; Miranda, J.A.S.; Antozanz, J.C. Assessment of cancer immunotherapy outcome in terms of the immune response time features. Math. Med. Biol.
**2007**, 24, 287–300. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morales, A.; Eidinger, D.; Bruce, A.W. Intracavity Bacillus Calmette-Guerin in the treatment of superficial bladder tumors. J. Urol.
**1976**, 116, 180–183. [Google Scholar] [CrossRef] - Pettenati, C.; Ingersoll, M.A. Mechanisms of BCG immunotherapy and its outlook for bladder cancer. Nat. Rev. Urol.
**2018**, 15, 615–625. [Google Scholar] [CrossRef] - Lee, C.F.; Chang, S.Y.; Hsieh, D.S.; Yu, D.S. Immunotherapy for bladder cancer using recombinant bacillus Calmette-Guerin DNA vaccines and interleukin-12 DNA vaccine. J. Urol.
**2004**, 171, 1343–1347. [Google Scholar] [CrossRef] - Shapiro, A.; Gofrit, O.; Pode, D. The treatment of superficial bladder tumor with IL-2 and BCG. J. Urol.
**2007**, 177, 81. [Google Scholar] [CrossRef] - Guzev, E.; Halachmi, S.; Bunimovich-Mendrazitsky, S. Additional extension of the mathematical model for BCG immunotherapy of bladder cancer and its validation by auxiliary tool. Int. J. Nonlinear Sci. Numer. Simul.
**2019**, 20, 675–689. [Google Scholar] [CrossRef] - Kitamura, H.; Tsukamoto, T. Immunotherapy for Urothelial Carcinoma. Current Status and Perspectives. Cancers
**2011**, 3, 3055–3072. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malek, T.R.; Bayer, A.L. Tolerance, not immunity, crucially depends on IL-2. Nat. Rev. Immunol.
**2004**, 4, 665–674. [Google Scholar] [CrossRef] [PubMed] - D’onofrio, A.; Gatti, F.; Cerrai, P.; Fresci, L. Delay-induced oscillatory dynamics of tumor-immune system interaction. Math. Comp. Mod.
**2010**, 51, 572–591. [Google Scholar] [CrossRef] - Galach, M. Dynamics of the tumor-immune system competition: The effect of time delay. Int. J. Appl. Math. Comp. Sci.
**2003**, 3, 395–406. [Google Scholar] - Kolmanovskii, V.; Shaikhet, L. Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett.
**2002**, 15, 355–360. [Google Scholar] [CrossRef] [Green Version] - Shaikhet, L. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory. In SIAM Studies in Applied Mathematics; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Xu, S.; Lam, J. A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci.
**2008**, 39, 1095–1113. [Google Scholar] [CrossRef] - Shaikhet, L.; Bunimovich-Mendrazitsky, S. Stability analysis of delayed immune response BCG infection in bladder cancer treatment model by stochastic perturbations. Comput. Math. Methods Med.
**2018**. [Google Scholar] [CrossRef] - Fridman, E.; Shaikhet, L. Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay. Syst. Control. Lett.
**2019**, 124, 83–91. [Google Scholar] [CrossRef] - Berezansky, L.; Bunimovich-Mendrazitsky, S.; Shklyar, B. Stability and controllability issues in mathematical modeling of the intensive treatment of leukemia. J. Optim. Theory Appl.
**2015**, 167, 326–341. [Google Scholar] [CrossRef] - Biot, C.; Rentsch, C.A.; Gsponer, J.R.; Birkhäuser, F.D.; Jusforgues-Saklani, H.; Lemaître, F.; Auriau, C.; Bachmann, A.; Bousso, P.; Demangel, C.; et al. Preexisting BCG-Specific T Cells Improve Intravesical Immunotherapy for Bladder Cancer. Sci. Transl. Med.
**2012**, 4, 137. [Google Scholar] [CrossRef] - Kronik, N.; Kogan, Y.; Schlegel, P.G.; Wölfl, M. Improving T-cell immunotherapy for melanoma through a mathematically motivated strategy: Efficacy in numbers? J. Immunother.
**2012**, 35, 116–124. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bunimovich-Mendrazitsky, S.; Halachmi, S.; Kronik, N. Improving bacillus Calmette-Guérin (BCG) immunotherapy for bladder cancer by adding interleukin 2 (IL-2): A mathematical model. Math. Med. Biol. J. IMA
**2015**, 30, 159–188. [Google Scholar] [CrossRef] - Bunimovich-Mendrazitsky, S.; Shochat, E.; Stone, L. Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer. Bull. Math. Biol.
**2007**, 69, 1847–1870. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bunimovich-Mendrazitsky, S.; Gluckman, J.C.; Chaskalovic, J. A mathematical model of combined bacillus calmette-guerin (BCG) and interleukin (IL)-2 immunotherapy of superficial bladder cancer. J. Theor. Biol.
**2011**, 277, 27–40. [Google Scholar] [CrossRef] [PubMed] - Ludewig, B.; Krebs, P.; Junt, T.; Metters, H.; Ford, N.J.; Anderson, R.M.; Bocharov, G. Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction. Eur. J. Immunol.
**2004**, 34, 2407–2418. [Google Scholar] [CrossRef] [PubMed] - Marino, S.; Kirschner, D.E. The human immune response to Mycobacterium tuberculosis in lung and lymph node. J. Theor. Biol.
**2004**, 227, 463–486. [Google Scholar] [CrossRef] [PubMed] - Yee, C.; Thompson, J.A.; Byrd, D.; Riddell, S.R.; Roche, P.; Celis, E.; Greenberg, P.D. Adoptive T cell therapy using antigen-specific CD8+ T cell clones for the treatment of patients with metastatic melanoma: In vivo persistence, migration, and antitumor effect of transferred T cells. Proc. Natl. Acad. Sci. USA
**2002**, 99, 16168–16173. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Archuleta, R.J.; Mullens, P.; Primm, T.P. The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex. Arch. Microbiol.
**2002**, 178, 311–314. [Google Scholar] [CrossRef] - Wigginton, J.; Kirschner, D. A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. J. Immunol.
**2001**, 166, 1951–1967. [Google Scholar] [CrossRef] [Green Version] - Kuznetsov, V.A.; Makalkin, I.A.; Taylor, M.A.; Perelson, A.S. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol.
**1994**, 56, 295–321. [Google Scholar] [CrossRef] - Vegh, Z.; Mazumder, A. Generation of tumor cell lysate-loaded dendritic cells preprogrammed for IL-12 production and augmented T cell response. Cancer Immunol. Immunother.
**2003**, 52, 67–79. [Google Scholar] [CrossRef] [PubMed] - Fikri, Y.; Pastoret, P.P.; Nyabenda, J. Costimulatory molecule requirement for bovine WC1 + gammadelta T cells proliferative response to bacterial superantigens. Scand. J. Immunol.
**2002**, 55, 373–381. [Google Scholar] [CrossRef] [PubMed] - Kronin, V.; Fitzmaurice, C.J.; Caminschi, I.; Shortman, K.; Jackson, D.C.; Brown, E.L. Differential effect of CD8(+) and CD8(-) dendritic cells in the stimulation of secondary CD4(+) T cells. Int. Immunol.
**2001**, 13, 465–473. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shochat, E.; Hart, D.; Agur, Z. Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols. Math. Model. Methods Appl. Sci.
**1999**, 9, 599–615. [Google Scholar] [CrossRef] - Kogan, Y.; Forys, U.; Shukron, O.; Kronik, N.; Agur, Z. Cellular immunotherapy for high grade Gliomas: Mathematical analysis deriving efficacious infusion rates based on patient requirements. SIAM J. Appl. Math.
**2010**, 70, 1953–1976. [Google Scholar] [CrossRef] [Green Version] - Kronik, N.; Kogan, Y.; Vainstein, V.; Agur, Z. Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol. Immunother.
**2008**, 57, 425–439. [Google Scholar] [CrossRef] - Klinger, M.; Brandl, C.; Zugmaier, G.; Hijazi, Y.; Bargou, R.C.; Topp, M.S.; Gökbuget, N.; Neumann, S.; Goebeler, M.; Viardot, A.; et al. Immunopharmacologic response of patients with B-lineage acute lymphoblastic leukemia to continuous infusion of T cell engaging CD19/CD3-bispecific BiTE antibody blinatumomab. Blood
**2012**, 119, 6226–6233. [Google Scholar] [CrossRef] - Schiphorst, P.P.; Chang, P.C.; Clar, N.; Schoemaker, R.; Osanto, S. Pharmacokinetics of interleukin-2 in two anephric patients with metastatic renal cell cancer. Ann. Oncol.
**1999**, 10, 1381–1383. [Google Scholar] [CrossRef] - Lamm, D.L. Improving Patient Outcomes: Optimal BCG Treatment Regimen to Prevent Progression in Superficial Bladder Cancer. Eur. Urol. Suppl.
**2006**, 5, 654–659. [Google Scholar] [CrossRef] - Nave, O.; Hareli, S.; Elbaz, M.; Iluz, I.H.; Bunimovich-Mendrazitsky, S. BCG and IL—2 model for bladder cancer treatment with fast and slow dynamics based on S PV F method—Stability analysis. Math. Biosci. Eng.
**2019**, 16, 5346–5379. [Google Scholar] [CrossRef] - Haynsworth, E.V. On the Schur Complement. Basel Math. Notes
**1968**, 20, 17. [Google Scholar]

**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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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