## 2 Materials and Methods  
### Model of Cancer and Chemotherapy  
In this section, a mathematical model of tumor growth dynamics, chemotherapy, and the interactions with the immune system will be presented. The model itself and the values of its nominal (mean-population) parameters will be adopted from [23]. The adopted model is not limited to a particular chemotherapeutic drug or cancer subtype. It focuses on the dynamics of a single solid tumor, without accounting for further proliferation or metastatic development. This standard phenomenological model describes the dynamics of four state variables representing the following quantities in a patient with cancer:  
* \(N(t)\) natural killer cell population
* \(L(t)\) cytotoxic T cells population
* \(T(t)\) tumor cell population
* \(u(t)\) amount of anticancer drug in the tumor site  
Natural killer (NK) cells are a key component of the innate immune system, responsible for detecting and eliminating abnormal cells, including tumor cells and virus-infected cells, without prior sensitization. Unlike adaptive immune cells, NK cells do not rely on antigen recognition but instead use a balance of activating and inhibitory receptors to distinguish between healthy and malignant cells. When NK cells recognize a target cell with reduced major histocompatibility complex (MHC) class I expression--a common feature of tumor cells--they release cytotoxic granules containing perforin and granzymes, leading to cell lysis. Cytotoxic T lymphocytes (CTLs), also known as CD8+ T cells, are key players in the adaptive immune system and play a vital role in tumor immunosuppressive. Unlike natural killer (NK) cells, CTLs must be activated through antigen presentation by antigen-presenting cells (APCs) via MHC class I molecules. Once activated, they recognize and attach to tumor cells displaying tumor-specific or tumor-associated antigens. This leads to the induction of apoptosis in target cells, either through the release of perforin and granzymes or by engaging death receptor signaling pathways. The considered mathematical model is defined by differential equations [23]  
\[ \begin{array}{c}\frac{\mathrm{d}N}{\mathrm{d}t}(t)=N(t)(a(1-bN(t))-\alpha_{1}T(t )-k_{N}u(t)),\\ \frac{\mathrm{d}L}{\mathrm{d}t}(t)=rN(t)T(t)-L(t)(\mu+\beta_{1}T(t)+k_{L}u(t)), \\ \frac{\mathrm{d}T}{\mathrm{d}t}(t)=T(t)(c(1-dT(t))-\alpha_{2}N(t)-\beta_{2}L(t )-k_{T}u(t)),\\ \frac{\mathrm{d}u}{\mathrm{d}t}(t)=v(t)-\omega u(t).\end{array}  \tag{1}\]  
Input \(v(t)\in\mathbb{R}\) represents the rate of chemotherapy administration. The state variables form the state vector \(x(t)\in\mathbb{R}^{4}\) defined as  
\[x(t)=\ \Big{(}N(t)\quad L(t)\quad T(t)\quad u(t)\Big{)}^{\mathrm{T}}. \tag{2}\]  
The adopted model is based on the following assumptions [23]:  
1. Tumor cell populations are reduced by both immune effector cells and chemotherapy. 2. Effector cell numbers decline due to natural degradation, their consumption in tumor cell destruction, and the impact of chemotherapy. 3. Chemotherapeutic agents influence both tumor cells and immune effector cells through a mass-action process. 4. Administering the drug at a higher constant rate may lead to a greater reduction in both tumor cells and immune effector cells. The growth of natural killer cells \(N(t)\) follows a logistic model as \(aN(t)(1-bN(t))\), while their suppression occurs through interactions with tumor cells, defined by \(\alpha_{1}T(t)N(t)\). Cytotoxic T cells \(L(t)\) are activated only in response to tumor presence while being generated via a bilinear term \(rN(t)T(t)\). Their natural decline follows a linear process \(-\mu L(t)\), and they are further inactivated upon interaction with tumor cells, represented by \(-\beta_{1}T(t)L(t)\). Tumor cell dynamics \(T(t)\) is modeled using a logistic growth function \(cT(t)(1-dT(t))\). Tumor elimination occurs through the action of both NK cells and cytotoxic T cells, with respective terms \(-\alpha_{2}N(t)T(t)\) and \(-\beta_{2}L(t)T(t)\) accounting for their effects.