In the Steinberg game model, there exists an order of adoption of the strategy by the participants. The doctor, as a "leader", is able to anticipate the response behavior of the tumor and make rational strategies on the basis of the expected tumor behavior; the tumor, as a "follower", is able to respond to the current environment and formulate rational strategies to maximize the target utility in that environment after the doctor has adopted a strategy.

We explore the best treatment strategy by comparing the benefits of doctors and tumor cells in the treatment process. For tumors, we considers three components that affect their growth and reproduction: 1) proliferation under ideal conditions, reflecting the growth of tumors under favorable environments such as abundant resources; 2) environmental constraints, reflecting the hindering effect of resource constraints in the host body and the tumor population's own factors on the value-added of the tumors; and 3) chemotherapy drugs inhibition, reflecting the inhibitory effect of therapeutic drugs on the tumor's proliferation. Accordingly, this paper proposes a gain function for tumors as follows:

$$_=\alpha x-\frac\beta ^-dx$$

Among them, \(_\) refers to the profit of tumor T, \(x\) represents the number of cells in the tumor population, \((>0)\) represents the profit brought by tumor cell proliferation to the tumor population, \(\beta (\beta>0)\) represents the diminishing marginal utility of tumor population proliferation, that is, the inhibitory effect of environmental factors and tumor self factors on its proliferation, \(d\) represents the cost of therapeutic drugs for maintaining the survival of the tumor population. The benefit for the tumor, denoted as \(_\), arises from the increase in tumor count, represented as \(\alpha x\). However, this benefit exhibits diminishing marginal utility as the number of tumors grows, due to the quadratic term \(-\frac\beta ^\), reflecting the scarcity of resources such as nutrients and growth space within the body. Furthermore, during chemotherapy, a larger tumor count results in an increased absorption of chemotherapy doses, leading to a negative impact on the tumors, denoted as \(-dx\).

For the benefit of physicians, we mainly consider these two influencing factors: the target utility consists of two components: 1) inhibition of the value added from the tumor, taking into account the benefit of reducing the number of tumor cells; and 2) limiting the dose of the drug, taking into account the benefit of reducing the toxicity of the drug. That is, the physician must weigh the number of tumor cells and the drug toxicity that comes along with the cost of tumor survival caused by the therapeutic drug. Consequently, this paper proposes the physician's benefit function as follows:

$$_=-\theta ^-(1-\theta )^$$

\(_\) refers to the benefit to physician D, and \(\theta (0<\theta <1)\) refers to the weight of the benefit from reducing the number of cells in the tumour population. On one hand, the benefit for doctors stems from controlling the number of tumors, where a higher tumor count translates to lower benefit, represented as \(-^\). On the other hand, it comes from the control of chemotherapy drugs. The larger the chemotherapy dose, the smaller the benefit \(-^\). It is noteworthy that doctors' priorities between controlling the tumor count and mitigating the side effects of chemotherapy drugs vary based on the situation. When tumors pose a greater threat to life, doctors should prioritize controlling the tumor count, leading to a higher value of \(\theta\). Conversely, in cases where patients exhibit intolerance to the toxic and side effects of chemotherapy drugs, doctors should strictly control the chemotherapy dose, resulting in a higher value of (1 \(-\theta\)).

Based on this understanding, we incorporate it into the model according to the prevalent circumstances in clinical practice, aiming to maximize the doctors' income. Additionally, we delve into two research questions:

Question 1: How can doctors maximize their income by optimizing the chemotherapy effect when solely considering the individual circumstances of patients, where \(\theta\) is a fixed value?

Question 2: Considering both the individual situation of patients and the adjuvant treatment with traditional Chinese medicine, where \(\theta\) is a variable, how can doctors maximize their benefits by optimizing both the chemotherapy and the adjuvant treatment with Chinese medicine?

Hypothetical 1: The physician's medication situation needs to satisfy: 1) the number of tumor cells in the patient's body cannot be less than zero after treatment; 2) the patient is still alive after treatment, the number of tumor cells cannot be greater than the maximum tumor load; and 3) the patient will ultimately die if no treatment is administered.

Maximization of tumor benefit function:

$$\frac_}=\alpha -\beta x-d$$

\(\text\frac_}=0\), we obtain the following result:

According to Hypothetical 1:

$$0<\alpha -\beta q\le d\le \alpha$$

\(q\) refers to the maximum tumor load. Without loss of generality, the order \(q=1\). Thus, there is the following relationship:

$$0<\alpha -\beta \le d\le \alpha$$

From the above conditions, to obtain the optimal treatment, the physician must solve the following optimization problem with constraints:

$$\underset} F\left(d\right)=\theta \right)}^+(1-\theta )^$$

$$\begin\text.\text.& d\le \alpha \\ & \\ & \begind\ge \alpha -\beta \\ \end\end$$

The derivation of the objective function yields the following.

$$\frac=-\frac^}\left(\alpha -d\right)+2\left(1-\theta \right)d$$

\(\text\frac=0\), we come to next equation: \(d=\frac^(1-\theta )}\)

$$d=\frac^(1-\theta )}<\alpha$$

When \(\alpha \ge \frac+\beta\), optimal solution is as follows: \(^=\alpha -\beta\);

When \(\alpha <\frac+\beta\), optimal solution is as follows: \(^=\frac^(1-\theta )}\)

Therefore, if tumor cells have a high proliferative capacity, the physician should use a level of chemotherapeutic agent that is \(\alpha -\beta\). Tumor cells are less able to proliferate, and the level of chemotherapeutic agent that the physician should use is \(\frac^(1-\theta )}\).

Hypothetical 2: Under ideal conditions, physicians can adjust weight by supplementing chemotherapy treatment with other treatment, θ ∈ [0,1].

Based on this, the doctor needs to solve the following optimization problem with constraints:

$$\underset }G\left(d,\theta \right)=\theta \right)}^+(1-\theta )^$$

$$\begin\text.\text.& d\le \alpha \\ & d\ge \alpha -\beta \\ & \begin\theta \le 1\\ \theta \ge 0\end\end$$

First, constructing the Lagrangian function yields the following.

$$L\left(d,\theta ,_,_,_,_\right) =\theta \right)}^+(1-\theta )^+_\left(d-\alpha \right)+_\left(\alpha -\beta -d\right)$$

$$+_\left(\theta -1\right)+_\left(0-\theta \right)$$

Derivation of the objective function separately yields the KKT condition as follows:

$$\frac=-\frac^}\left(\alpha -d\right)+2\left(1-\theta \right)d+_-_=0$$

$$\frac=\right)}^-^+_-_=0$$

$$_\left(d-\alpha \right)=0$$

$$_\left(\alpha -\beta -d\right)=0$$

$$_\left(\theta -1\right)=0$$

$$_\left(0-\theta \right)=0$$

$$\alpha -\beta -d\le 0$$

Solution:

When \(\alpha =1+\beta\), solutions satisfying the KKT condition as \(d=1,\theta <\frac, G\left(d,\theta \right)=1\);

When \(\alpha <1+\beta\), solutions satisfying the KKT condition as \(d=\alpha -\beta ,\theta =1,G\left(d,\theta \right)=1\);\(d=\frac,\theta =\frac,G\left(d,\theta \right)=\frac^}^}\).

Besides, considering whether or not \(^\) takes values on the boundary affects the objective function solution, so the \(\theta^\ast=0\;\text\theta^\ast=1\).

When \(^=0\):

$$\begin\text.\text& d\le \alpha \\ & d\ge \alpha -\beta \end$$

Optimal solution is as follows: \(^=\alpha -\beta\), \(^=0\), \(G\left(^,^\right)=^\).

When \(^=1\):

$$\begin\text.\text& d\le \alpha \\ & d\ge \alpha -\beta \end$$

Optimal solution is as follows: \(^=\alpha\), \(^=1\), \(G\left(^,^\right)=0\).

In summary, the optimal solution is that \(^=\alpha\), \(^=1\). Therefore, with Chinses medicine, physicians should completely eliminate drug toxicity and thus maximize the administration of chemotherapeutic drugs to patients. This conclusion is consistent with common sense, so the model constructed in this paper can better match reality.

In fact, this result is derived under completely ideal assumptions; in practice, conditions for adjuvant therapy are often limited, and physicians can usually only adjust the weights in \(\theta \in [\underset,\overline]\)(\(0<\underset<\overline<1\)). For this reason, according to the physician's benefit function:

$$_\left(d,\theta \right)=-\theta ^-\left(1-\theta \right)^,$$

In this paper, a numerical simulation is utilized for further analysis. The numerical simulation results are shown in Fig. 1.

The numerical simulation results of optimize chemotherapy treatments and adjuvant therapies to maximize benefits. Note: The abscissa represents the maximum value of θ; the ordinate is the dose of chemotherapy drugs

The results reveal that when the capacity for cell proliferation is high (as illustrated in Fig. 1 with α = 2, α = 2.3), as the peak attainable level of Chinese medicine rises, the amount of chemotherapy administered by doctors to the patient remains constant at a particular level. Nonetheless, once the peak attainable level of Chinese medicine crosses a specific threshold, the amount of chemotherapy prescribed by doctors to the patient should be escalated accordingly. On the other hand, when cell proliferation capacity is lower (as depicted in Fig. 1 with α = 1.2, α = 1.5), both the levels of Chinese medicine and chemotherapy remain steady at certain levels until the maximum achievable level of adjuvant therapy reaches a particular threshold. If the maximum achievable level of Chinese medicine surpasses this threshold, then the levels of both Chinese medicine and chemotherapy prescribed by the doctor to the patient should be increased in tandem.

The threshold for changing the level of Chinses medicine and chemotherapy is related to cell proliferation capacity \(\alpha\), proliferation constraints \(\beta\), and the minimum \(\underset\) level of Chinses medicine, with the higher the cell proliferation capacity or the higher the minimum level of adjuvant therapy, the higher the threshold.