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We present a revised mathematical model of the immune response to Bacillus Calmette-Guérin (BCG) treatment of bladder cancer, optimized according to biological and clinical data accumulated during the last decade. The improved model accounts for cytotoxic T lymphocyte differentiation as an integral element of the delayed immune response, as well as the logistic growth terms for cancer cell proliferation. Three equilibria are demonstrated for the proposed model, which is assumed to be influenced by white noise stochastic perturbations that are directly proportional to the system state deviation from an equilibrium. Stability conditions for all equilibria are analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction.

Bladder cancer (BC) is 7th most common cancer (the 4th most common for men) with approximately 356,000 new cases each year and more than 145,000 deaths per year. The highest incidence occurs in industrialized and developed areas such as Europe, North America, and Australia (Jemal et al., [

The treatment of the BC has improved during last 40 years due to development of high definition of cystoscopy, newly technology in the bladder drugs instillation. However, the prognosis of advanced bladder cancer has not improved during the last years (Alexandroff et al. [

BC is most frequently treated with intravesical instillations of an adjuvant immunotherapy with the Bacillus Calmette-Guérin (BCG) bacteria. BCG immunotherapy, originally established by Morales et al. [

In last three decades, it has been commonly accepted that a qualitative understanding of dynamic cancer treatment requires a mathematical framework in which the essential features of this complex process are represented (Byrne, [

One of the main problems encountered in mathematical models described by differential equations is that of their stability. In this work, equilibria stability is analyzed using the Kolmanovskii-Shaikhet general method of Lyapunov functionals construction [

The manuscript is organized as follows: in Section

BCG is thought to encourage tumor elimination by attachment of the BCG to the urothelium and initiation of localized inflammation, which attracts innate immune cells that in turn draw cytotoxic T-lymphocytes (CTLs) and natural killer (NK) cells which attack the tumor cells [

Recent studies have described the BCG-immune system interactions: Biot et al. [

Our model describes the parameters governing the efficacy of BCG treatment for bladder cancer. Based in part upon previous study [

We model the stage in pathogenesis wherein no metastases have yet occurred, such that the entire dynamics of the system take place within the bladder lumen. Therefore, a one-site mathematical model is sufficient.

The BCG treatment model is composed of four nonlinear ODEs to characterize the interactions between the four different biological components, with the local quantity of each noted as follows:

BCG bacteria within the bladder as

Effector T-lymphocytes, principally CTLs that react to BCG and tumor antigens as

Tumor cells infected with BCG as

Tumor cells not infected with BCG as

To summarize briefly, we assume BCG to be introduced into the bladder at a constant rate

the encounter between immune cells and BCG, controlled by parameter

the encounter between immune cells and tumor cells, controlled by parameter

The rate of inactivation of

The equilibria of system (

Putting

Let us assume that system (

Let

For the first equilibrium we obtain the nonlinear system

To obtain sufficient conditions of stability in probability for nonlinear system with the order of nonlinearity higher than one it is enough [

Following Remark

For equations with delay there are two types of stability conditions: delay independent conditions and delay dependent conditions. Following the assumption that in the considered system (

To investigate stability of the linear stochastic delay differential equations (

The trace of the

A

Put

Via Schur complement (see Appendix) instead of the nonlinear Riccati type inequalities (

For conditions (

Conditions (

Using [

If

Let

For positivity of the second equilibrium it is necessary to suppose that

From this it follows that for stability investigation of the second equilibrium it is enough to investigate asymptotic mean square stability of system (

Let

Checking the LMIs (

Note that via (

Put

In this work we present the improved model of BCG immunotherapy in superficial bladder cancer. This study investigates stability of new treatment model with constant instillations of BCG under stochastic perturbations. The model demonstrates several stable states which depend on biologically related parameters and initial conditions.

The innovation of our study is the involvement of CTL cells, specific for the tumor antigen due to BCG infection. Adding BCG to the tumor-immune interaction may increase the immune response, which most benefit from the addition of the antigen. These effector cells capture tumor cells after time delay which has been used for maturation of these effector cells. The entire reaction can take place only with the presence of BCG. We added new terms in two equations (the second and the forth) in system (

It is shown that the considered system has three equilibria describing the different states of the patient. Stability of these states under stochastic perturbations which are directly proportional to the deviation of the patient’s current state from the equilibrium state is investigated. New sufficient conditions for stability in probability of two equilibria and instability of the third equilibrium were obtained using the theoretical method of Lyapunov functionals and the numerical method of linear matrix inequalities (LMIs).

In the first equilibrium we obtain weak immune response because

The third equilibrium is unstable.

The dynamic behavior of the system investigated from the point of view of local stability and a detailed analysis on stability of equilibrium was examined. As proposed in the paper research methods allow continuing and specifying investigation of the considered model and getting its new useful properties.

However, we have to note that this study has following limitations:

This model takes into account 3-biological processes (tumor, immune system, and BCG interactions) only, which are captured by 4 differential equations.

We examine continued BCG treatment with a constant rate

The duration of BCG treatment is not time limited in our model.

By capturing the main parameters of the BCG, maximum tumor size, tumor growth rate, and immune response parameters, we create a silicone model and develop an algorithm for eliminating cancer. We would like to raise awareness in the community of urological-oncological doctors about the possibilities of mathematical modeling and receive quantitative data to improve this model. The ability to plan and predict by calculating a modulated dose of treatment can benefit patients who are unable to take routine treatment because of its serious side effects, as well as to patients who were previously considered not to respond.

The data used to support the findings of this study are available from the corresponding author upon request.

Note that the information about the results described above was presented only in the form of a short abstract on the conference [

The authors declare that they have no conflicts of interest.