5.1 Introduction

The death toll from tumor diseases is on the rise, and the nonlinear dynamics and control of tumor growth have attracted widespread attention [1]. The number of tumor cells is gradually increasing. The most obvious feature is abnormal anti-growth signals. There is a strict control mechanism for normal cells. However, in the continuous process, the static and death signals are turned off to generate cell division signals, which leads to the crazy growth of tumor cells [2, 3]. Tumor cells promote the growth of blood vessels, which are necessary to provide nutrients. This is why the flow of blood in tumor tissues is related to the benign or malignant tumor. Cancer cells are also polarized. They have evolved their camouflage ability in the ongoing battle with immune cells, causing the immune system to mistake them for normal cells, which makes it difficult for chemotherapeutic drugs to distinguish the volume of biological targets [4, 5]. When the differentiation process of normal cells is not controlled, they will evolve into tumor cells. This is the nature of tumor cells, tumor cells continue to proliferate, deprive their limited body energy supply, and ultimately destroy the body’s function and die [6]. Therefore, in order to inhibit the growth of tumor cells, it is urgent to find a treatment that will minimize the damage to oneself.

In the fight against cancer, before the advent of chemotherapy and radiotherapy, there have been no effective measures for the small differences between cancer cells and normal cells [7, 8]. When the side effects of radiotherapy and chemotherapy increased and the targeted therapy was highly targeted and inflexible, scientific research projects began to turn to humans themselves [9]. The complex and unique communities of cell life are called microenvironments by scientists. The microenvironment has many characteristics that affect cell growth, behavior, and how to communicate with other cells nearby [10]. In the oncology world, researchers are committed to understanding the tumor microenvironment and trying to find feasible treatment opportunities. Under normal circumstances, the immune system can recognize and eliminate tumor cells in the tumor microenvironment. However, in order to survive and grow, tumor cells can adopt different strategies to suppress the body’s immune system and fail to kill tumor cells normally, thereby surviving the various stages of the anti-tumor immune response. The above-mentioned characteristics of tumor cells are called immune escape. Tumor cells escape the immune system, not because the immune system cannot recognize them, nor because it is not activated, but cancer cells have evolved a way to prevent T cell activation through specific binding [11,12,13]. Therefore, the medical community has been trying to find many special methods to treat cancer cells to block the activation of T cells and release the immune system.

Chemotherapy not only kills rapidly differentiated tumor cells, but also involves conventional cells. Its side effects are the most obvious, but they can be alleviated by immunotherapy. The closure of immune checkpoints and the success of adoptive cell therapy have made immunotherapy a mature means of treating cancer [14, 15]. Compared with traditional therapies such as surgery, radiotherapy, and chemotherapy, immunotherapy has fewer side effects and better effects, but immunotherapy is difficult to overcome its transient nature. With the rapid increase in tumor patients, immunotherapy is rapidly emerging for the treatment of specific types of cancer, especially tumors with poor immunogenicity [2]. The original intention of immunotherapy is to fight cancer cells through the lethality of immune cells themselves. As a typical immune deficiency syndrome, AIDS is caused by the failure of the immune response and is often attributed to the weakening of the immune level. However, once the activated immune system cannot be stopped, cytokines are produced, which is considered to be an overreaction of the immune system like COVID-19 [16, 17]. Therefore, the combined treatment of chemotherapy and immunotherapy is more reasonable. Immunotherapy refers to a treatment method that artificially enhances or suppresses the body’s immune function to achieve the purpose of curing diseases by referring to the body’s low or hyperimmune state. There are many immunotherapy methods, which are suitable for the treatment of many diseases. Tumor immunotherapy aims to activate the human immune system, relying on its own immune function to kill cancer cells and tumor tissues [18]. Unlike previous surgery, chemotherapy, radiotherapy, and targeted therapy, the target of immunotherapy is not tumor cells and tissues, but the body’s own immune system [19]. Different types of tumor cells interact with different types of immune cells, and these immune cells have the function of helping or attacking tumors [20].

The mechanism of immune regulation varies from person to person, but in the case of special calls, the optimal regulation based on immunotherapy will play a role in reducing tumor cells regardless of specific circumstances. Enhancing tumor antigen presentation can effectively stimulate dendritic cells and improve immunotherapeutic efficacy [21, 22]. The known “predation-prey” between immune cells and tumor cells will cause periodic growth and reduction of cells. This growth and reduction can continue indefinitely or reach a balanced saddle point determined by system parameters [23]. And all of the above is composed of a complex non-linear structure, and it is difficult to achieve the global optimum with conventional optimization methods. Especially for the treatment of the human body, how to rationally use drugs to achieve the minimum harm to the human body is particularly important. So this article proposes a novel evolutionary calculation method, N-Level Hierarchy Optimization (NLHO) algorithm. It is bionic from the hierarchical system of biological populations in the natural world. The hierarchical system refers to the hierarchical phenomenon in which the status of each animal in the animal group has a certain order. The basis of the formation of the hierarchy is the dominance behavior, that is, the “domination-submission” relationship [24]. When the formed hierarchical system stabilizes, lower-ranking people generally show compromise and obedience, but sometimes they also re-struggle to change the hierarchical order, and so on. A stable population will develop for a long time. This is an explanation for the rationality of the hierarchy preserved in evolutionary selection [25]. So for the entire species population, this is conducive to the preservation and continuation of the species. A variety of biological interactions constitute a complex nonlinear growth process of tumor cells, and the main influencing factors of tumor cell populations are the focus of research. Hunting cells refer to immune cells that participate in the removal of foreign objects and strengthen the immune response [26, 27].

In the NLHO algorithm, an N levels optimization structure is designed, which includes the leader level, guider level, executant level and follower level. In the entire population, the individual with the best search position is selected as the leader, who has the grasp of the entire search direction of the team it leads. The second level is the guider level, which executes the tasks issued by the leader and follows the direction of the leader to find the best. Of course, in the whole process, the guider will also refer to the task allocation of the global optimal leader to guide the executants to find the best, so as to prevent the leader of the team from falling into a local optimum. The third level is the executant level, which follows the guider to complete the task, in order to achieve a wider area of coverage search. At the same time, it will also refer to the tasks assigned by the leaders of the ethnic group to make the task goals clearer and speed up the convergence. The last level is the follower level. At this level, followers can be divided into any level to solve different optimization problems. Of course, in the later stage of searching, there may be excessive overlap between population individuals [28].

5.2 Ecological Evolutionary Dynamics Systems Model

This part mainly introduces the mathematical growth model of tumor cells, which takes into account the influence of external factors such as chemotherapy drugs and immunotherapy on tumor cells, as well as the interaction between the two cells. In the following model, T(t) represents the number of tumor cells, I(t) represents the number of immune cells, \(\textit{Con}_{che}(t)\) and \(\textit{Con}_{im}(t)\) represent the blood concentration of chemotherapy drugs and immunotherapy drugs, respectively.

Taking into account the interaction between immune cells and tumor cells, the direct killing of chemotherapeutic drugs and the growth model of tumor cells can be written as

$$\begin{aligned} \begin{aligned} T(t+1) =&\, T(t)+\vartheta _{1} \times T(t) \times \big (1-\vartheta _{2} \times T(t)\big ) \\&- \gamma \times T(t) \times I(t)-\varepsilon \times T(t) \times \textit{Con}_{che}(t) \end{aligned} \end{aligned}$$
(5.1)

where, \(\vartheta _{1}\) stands for inherent growth rate unrelated to immune cells and chemotherapy drugs, \(\vartheta _{2}\) stands for the maximum interaction ability between immune cells and tumor cells, ignoring chemotherapy drugs, \(\gamma \) stands for the growth rate when tumor cells are inactivated and attacked by immune cells, \(\varepsilon \) stands for the stress response coefficient of tumor cells to chemotherapeutics.

Considering the natural growth law of immune cells, we assume that a fixed number of immune cells are produced in a unit time, and these cells have an inevitable life cycle. Tumor cells in the body can stimulate the growth of immune cells, which is a positive non-linear change. In immunotherapy, the addition of immune drugs can produce an immune response, leading to non-linear growth of immune cells. At the same time, in the struggle between immune cells and tumor cells, the immune cells themselves will also cause losses. In chemotherapy, chemotherapy drugs can also cause damage to immune cells.

$$\begin{aligned} \begin{aligned} I(t+1)=&\, I(t)+\vartheta _{3}-\lambda \times I(t) \\ {}&+\frac{\alpha _{1} \times T^{2}(t) \times I(t)}{\beta _{1}+T^{2}(t)}+\frac{\alpha _{2} \times T(t) \times {\text {Con}}_{\text{ im } }(t)}{\beta _{2}+{\text {Con}}_{\textrm{im}}(t)} \\ {}&-\xi _{1} \times T(t) \times I(t)-\xi _{2} \times {\text {Con}}_{\text{ che } }(t) \times I(t) \end{aligned} \end{aligned}$$
(5.2)

where, \(\vartheta _{3}\) stands for rate of continuous inflow, \(\lambda \) stands for natural decay rate without any external effects, \(\alpha _{1}\) stands for maximum recruitment rate caused by tumor cells, \(\alpha _{2}\) stands for the largest proportion of tumor cells caused by immunotherapeutic drugs, \(\beta _{1}\) stands for steepness factor caused by tumor cells, \(\beta _{1}\) stands for steepness coefficient caused by immunotherapeutic drugs, \(\xi _{1}\) stands for stress response coefficient to chemotherapy drugs, \(\xi _{2}\) stands for response rate of tumor cells to immune cells.

At a certain point in time after the injection of chemotherapy drugs, the concentration of the drugs in the body will decrease exponentially. We are adding immune drugs at the same time. We can get the attenuation model of chemotherapy drugs and immune drugs in vivo.

$$\begin{aligned} \begin{aligned} Con_{che}(t+1)=\chi _{che}(t)-e^{-\theta _{1}}Con_{che}(t) \end{aligned} \end{aligned}$$
(5.3)
$$\begin{aligned} \begin{aligned} Con_{im}(t+1)=\chi _{im}(t)-e^{-\theta _{2}}Con_{im}(t) \end{aligned} \end{aligned}$$
(5.4)

where, \(\chi _{che}(t)\) and \(\chi _{im}(t)\) represent the concentration of chemotherapeutic drugs and immune drugs, respectively. \(\theta _{1}\) and \(\theta _{2}\) are the attenuation rates of chemotherapy drugs and immune drugs.

When we qualitatively analyze how to minimize the number of tumor cells remaining in the bloodstream under the premise of using as few drugs as possible, including chemotherapy drugs and immune drugs, this process can be described by quantitative mathematical expressions. From formulas (5.1)–(5.4), we can get:

$$\begin{aligned} \begin{aligned} F_{\min } = \sum _{t=t_0}^{t} \delta ^{t}&\left\{ \omega T^{2}(t) + \int _{0}^{\chi _{\text {che}}(t)} \tan ^{-1}(\bar{U}_{1}^{-1} s) \bar{U}_{1} R_{1} ds \right. \\&\left. + \int _{0}^{\chi _{\text {im}}(t)} \tan ^{-1}(\bar{U}_{2}^{-1} s) \bar{U}_{2} R_{2} ds \right\} \end{aligned} \end{aligned}$$
(5.5)

where, \(\bar{\textit{U}}_{1}\) and \(\bar{\textit{U}}_{2}\) respectively represent the maximum allowable dose of chemotherapy drugs and the dose of a single injection of immunizing agent, \(\delta \) is the discount factor, \(\omega \) is a constant coefficient.

5.3 N-Level Hierarchy Optimization Algorithm

5.3.1 Leader Level of the Hierarchy

First of all, as individuals with high fitness values, leaders have strong self-learning capabilities. Therefore, the iterative formula of design leaders is as follows:

$$\begin{aligned} \begin{aligned} x_{l,j}^{t+1}=x_{l,j}^{t}\big ( 1+randn(\mu _{l},\sigma _{l}) \big ) \end{aligned} \end{aligned}$$
(5.6)

where, \(\textit{i}\) denotes the \(\textit{i}\)th leader in the population, and \(\textit{j}\) is the dimension. \(\textit{t}\) is the number of iterations. Randn is a Gaussian distribution, where the mean \(\mu _{l}\) = 0 and the standard deviation \(\sigma _{l}\) is shown below:

$$\begin{aligned} \begin{aligned} \sigma _{l}={\left\{ \begin{array}{ll} 1&{},f_{l}^{t}\le f_{i}^{t}\\ exp(f_{l}^{t}- f_{i}^{t})&{},f_{l}^{t}\,\,> f_{i}^{t} \end{array}\right. } ,\textit{i}\in [1,2,\cdots N],\textit{i}\ne \textit{l} \end{aligned} \end{aligned}$$
(5.7)

where, \(f_{l}^{t}\) is the fitness value of the \(\textit{l}\)th leader at the \(\textit{t}\)th iteration, and \(f_{i}^{t} \)is the fitness value of any individual in the population that is different from the \(\textit{l}\)th leader.

5.3.2 Guider Level of the Hierarchy

Secondly, as the individuals who guide the general direction of the evolution of the entire population for the leader, the guider must not only learn from the best overall, but also obey the leader’s command.

$$\begin{aligned} \begin{aligned} x_{g,j}^{t+1}=x_{g,j}^{t}&+randn(\mu _{g},\sigma _{g}^{2})\times (x_{l,j}^{t}-x_{g,j}^{t})\\ {}&+s_{1}\times (x_{best,j}^{t}-x_{g,j}^{t}) \end{aligned} \end{aligned}$$
(5.8)

where, \(\textit{g}\) denotes the \(\textit{g}\)th guider in the population, best is the best individual in the current iteration, \(s_{1}\) is the acceptance factor of guider, \(\mu _{g}\) = 0.5.

$$\begin{aligned} \begin{aligned} \sigma _{g}=exp(f_{l}^{t}- f_{g}^{t}) \end{aligned} \end{aligned}$$
(5.9)
$$\begin{aligned} \begin{aligned} s_{1}=exp\left( \dfrac{f_{best}^{t}-f_{g}^{t}}{|f_{g}^{t}|+\varepsilon }\right) \end{aligned} \end{aligned}$$
(5.10)

where, \(\varepsilon \) is an infinitesimal value to prevent a guider from having a fitness value of 0.

5.3.3 Executant Level of the Hierarchy

The executants seek the best as the main body of the entire population. On the one hand, follow the guider’s arrangements, and on the other hand, follow the leader’s direction.

$$\begin{aligned} \begin{aligned} x_{e,j}^{t+1}=x_{e,j}^{t}&+randn(\mu _{e},\sigma _{e}^{2})\times (x_{g,j}^{t}-x_{e,j}^{t})\\ {}&+s_{2}\times (x_{l,j}^{t}-x_{e,j}^{t}) \end{aligned} \end{aligned}$$
(5.11)

where, e is each executant in the population, \(s_{2}\) is the acceptance factor of the executor, \(\mu _{e}\) = 0.8.

$$\begin{aligned} \begin{aligned} \sigma _{e}=exp\left( \dfrac{f_{g}^{t}-f_{e}^{t}}{|f_{e}^{t}|+\varepsilon }\right) \end{aligned} \end{aligned}$$
(5.12)
$$\begin{aligned} \begin{aligned} s_{2}= exp(f_{l}^{t}- f_{e}^{t}) \end{aligned} \end{aligned}$$
(5.13)

5.3.4 Follower Level of the Hierarchy

Finally, there are followers, who themselves will be divided into multiple levels. Learn from each other at different levels, and notify the follow-up executant to check for deficiencies.

$$\begin{aligned} \begin{aligned} x_{f_{n},j}^{t+1}=x_{f_{n},j}^{t}&+randn(\mu _{f_{n}},\sigma _{f_{n}}^{2})\times (x_{f_{n-1},j}^{t}-x_{f_{n},j}^{t})\\ {}&+c_{n}\times rand\times (x_{e,j}^{t}-x_{f_{n},j}^{t}) \end{aligned} \end{aligned}$$
(5.14)

where, \(f_{n}\) is the \(\textit{n}\)-level follower, \( c_{n}\) is the absorption factor of the \(\textit{n}\)-level follower, \(\mu _{f_{n}}=0.8-0.6\times (t/t_{max})\),\( x_{f_{0},j}^{t}= x_{e,j}^{t}\),\(f_{f_{0}}^{t}= f_{e}^{t}\) , n is a natural number greater than 0.

$$\begin{aligned} \begin{aligned} \sigma _{f_{n}}= exp(f_{f_{n-1}}^{t}- f_{f_{n}}^{t}) \end{aligned} \end{aligned}$$
(5.15)
$$\begin{aligned} \begin{aligned} c_{n}= exp(f_{f_{n-1}}^{t}- f_{f_{n}}^{t}) \end{aligned} \end{aligned}$$
(5.16)

5.4 Simulation and Analysis for NLHO

In this experiment, the population size is set to 100, and the maximum number of iterations is set to be 100. Each algorithm is run independently for 50 times, and the spatial dimension is selected according to different test functions. The distribution rates of each level system are LPercent \(=\) 10%, GPercent \(=\) 20%, EPercent \(=\) 40%, and FPercent \(=\) 30%. The value of the updated algebra G \(=\) 10.

For independent tests of 20 test functions, we separately count their mean, minimum and standard deviation to evaluate the performance of NLHO in various aspects by setting the difficulty in different aspects. At the same time, we select some typical algorithms for comparison, such as Taboo Search (TS), Chicken Swarm Optimization (CSO), Genetic Algorithm (GA), Ant Colony Optimization (ACO), and Simulated Annealing (SA), so as to compare the performance of the NLHO algorithm horizontally. The test results are shown in Table 5.1.

Table 5.1 Experimental simulation results
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For the independent tests of the benchmark functions, we calculated 5 parameter indexes respectively, which were their best, worst, median, average and std. deviation. At the same time, in order to verify whether the results are statistically significant, we use the Wilcoxon rank-sum test between NLHO and the other algorithms. “\(+\)”, “−”, and “\(\approx \)” mean that the proposed NLHO is significantly better, significantly worse, and no significantly statistically different in the comparison, respectively.

It can be seen from the experimental results that, compared with the other 5 algorithms, under 20 test functions 60 indicators, NLHO wins 58, 29, 59, 53 and 56 indicators, and they belong to different types of test functions, which reflects the better robustness of NLHO. Moreover, it can be seen from the Wilcoxon rank-sum test that NLHO is not only different from other population algorithms, but also has obvious advantages. The experimental results for each function are discussed in more detail below.

The Ackley function \(f_{1}\) is widely used to test optimization algorithms. It is a continuous experimental function obtained by superimposing an exponential function with a moderately amplified cosine. It is characterized by an almost flat outer area, which is modulated by a cosine wave to form holes or peaks, making the curved surface undulating, but there is a large hole in its center. For the Ackley function, both NLHO and CSO have found the global minimum, and the average value is also equal to the global minimum, so that the standard deviation is also 0. The optimization process diagram of the NLHO algorithm is shown in this article, including the initial iteration diagram, the final result diagram and the intermediate process convergence curve, as shown in Fig. 5.1. The two algorithms are comparable, SA performance is average, while TS, GA and ACO perform poorly.

The Cross-in-Tray function \(f_{2}\) has multiple global minimums. On this function, the mean, minimum and standard deviation of NLHO have reached the best, and the experimental results are shown in Fig. 5.2. At the same time, ACO also found the global optimum, but the mean and standard deviation are slightly inferior to NLHO. TS, CSO and SA performed well, while GA performed average.

The Drop-Wave function \(f_{3}\) is multi-modal and very complicated. In each smaller input domain, its features have multiple ring-shaped peaks and valleys, and the depth of the valleys gradually decreases as the center of the circle shrinks. For the optimization process, it is very easy to fall into the local optimum. For this kind of complex optimization function, NLHO has shown strong optimization ability, and the global minimum, mean and standard deviation have all reached the best. The experimental results are shown in Fig. 5.3. CSO, ACO and SA performed well, and TS and GA performed poorly.

Fig. 5.1
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Ackley

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Fig. 5.2
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Cross-in-tray

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Fig. 5.3
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Drop-wave

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5.5 Develop Therapeutic Strategies for Ecological Evolutionary Dynamics Systems Using NLHO

In this section, we apply the NLHO algorithm to the EEDS model as an experimental verification. According to clinical treatment needs, chemotherapeutic drugs and immune drugs are used as input, and the cost of treatment loss is used as the objective function. Through the iteration of the NLHO algorithm, the optimal therapeutic strategies for patients with a certain basic condition are worked out. For some of the remainder of this sample we will use dummy text to fill out paragraphs rather than use live text that may violate a copyright.

According to clinical medical statistics borrowed from the literature [29], the specific parameters of the dynamic models are presented as Table 5.2. Based on the above, we have completed the establishment of the EEDS model, and determined the specific value of the cost function according to clinical needs. At the same time, the feasibility and effectiveness of the NLHO algorithm are also verified on benchmarks. Apply NLHO to the model of EEDS to develop therapeutic strategies. The best processing strategy is obtained through experiments, which proves the effectiveness and feasibility of the algorithm. The cost function is designed to minimize the number of tumor cells, and also to use the smallest dose of chemotherapeutic drugs and immune drugs to achieve the least harm to the human body.

Table 5.2 Experimental parameter
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When we give the patient the initial number of tumor cells and immune cells, according to the EEDS model and follow certain chemotherapy and immunotherapy plans, we can get the following set of curves of tumor cells and immune cells. As shown in Figs. 5.4 and 5.5, it shows the quantity curve of tumor cells and immune cells. Within a one-year treatment period, the number of tumor cells was successfully reduced to 254. Although the number of immune cells was reduced to 1.52\(\times \)106, significant effects were obtained for the treatment of tumors. Moreover, as shown in Figs. 5.7, 5.8 and 5.9, due to the decline of the body’s immune cells, the immune drug dropped to 0.022 and then increased to about 0.03. This caused the concentration of immune drugs in human blood to rise from the trough of 0.0096 to 0.03, reaching a sufficient level.

Fig. 5.4
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Quantity curve of tumor cells

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Fig. 5.5
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Quantity curve of immune cells

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Fig. 5.6
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Dosage of chemotherapeutic drugs

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Fig. 5.7
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Quantity curve of immune drug

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Fig. 5.8
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Concentration of chemotherapy drugs

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Fig. 5.9
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Concentrations of immunereagents

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The dosage of chemotherapeutic drugs is adaptively and dynamically changed, as shown in Fig. 5.6. Then the concentration of chemotherapeutic drugs in the blood also has the same changing trend, as shown in Fig. 5.8. The reason is to minimize the damage of drugs, and chemotherapy drugs are also harmful, not only killing tumor cells in the body but also destroying immune cells. If chemotherapeutic drugs are put in according to normal treatment methods, normal cells will suffer a lot of erosion, and the impact on the body will be even more significant. However, the drug dose optimized by NLHO will dynamically change adaptively, and the impact on normal cells will be appropriately reduced without affecting the elimination of tumor cells.

5.6 Conclusion

It is a difficult problem to solve optimal therapeutic strategy for EEDS. Theoretically, it can achieve the desired therapeutic effect through reducing the tumor cells through the combined of chemotherapeutic drugs and immune drugs, and minimize the harm to the human body. Benefiting from the concept of heuristic algorithm in evolutionary computing, this chapter has designed the NLHO algorithm via 20 benchmark functions to test NLHO, including unimodal and multimodal, single-mode and multi-mode, single-extreme and multi-extreme, etc. It is compared with the five algorithms of TS, CSO, GA, ACO and SA, and runs independently 50 times to calculate the mean, minimum and standard deviation. It proves that NLHO has good optimization ability and can solve various problems well. At the same time, the development therapeutic strategies of EEDS have achieved very good results. The experimental results have shown that the NLHO algorithm develops therapeutic strategies well, and provides valuable prior knowledge and scientific basis for clinical medicine. Future work will further improve the EEDS, and integrate the optimal control strategy and the evolutionary calculation method for the optimal treatment method.