1 Introduction

1.1 Problem Background

Prior to conducting wet experiments, teammates from the dry experiment project had already constructed models to simulate real-world situations and make accurate predictions. Each model was based on the experimental design and the physical re-quirements of our project. While the dry lab can guide the experimental plan to a certain extent, it also receives feedback from the results and performs further validation.

Basically, the dry experiment has achieved success in two parts. According to pre-vious experiments, attenuated Salmonella can inhibit the expression of tumor cells by reproducing massively in them, thus achieving a therapeutic effect. This discovery fur-ther provides guidance for continuous experiments on attenuated Salmonella. In addi-tion, the feedback from these results is also within expectations. Apart from the above, a dynamic evaluation of the therapeutic effect after using attenuated Salmonella target-ed drug to treat tumors was also conducted. We selected four tumor markers, namely Muc2, Cox-2 CDH-17, and VEGF, and performed simulations to the greatest extent possible using genetic algorithms, etc. The dynamic changes in efficacy were deter-mined through analytic hierarchy process. During model construction, we also accumu-lated valuable experience in this research field.

We hope that this model can provide accurate simulations for the attenuated Sal-monella system and serve as a toolkit that can be disseminated for people to optimize according to their actual needs.

1.2 Our Work

The value judgment of the Attenuated Salmonella

2 Assumptions and Justifications

2.1 We assume that all substances involved in the reaction process are in complete contact.

Based on past experiments, it has been found that the contact between substanc-es significantly affects the progress of the reaction. Therefore, to ensure accurate theo-retical results, it is assumed that the reactants are fully contacted.

2.2 We assume that bacterial growth often follows a logistic curve.

To ensure the efficiency of the simulation, phenomena such as abnormal death of bacteria are not considered in the establishment of this model. Therefore, the growth of the attenuated Salmonella should strictly follow the Logistic curve.

2.3 We assume that the relationship between bacterial growth rate and substrate concentration is described by the Monod equation.

The rate at which bacteria utilize substrates is influenced by various factors, with the most important ones being the maximum substrate utilization rate and half saturation and inhibition constants. To ensure the accuracy of the simulation, it is assumed that the bacterial growth rate is described by the Monod equation, which relates the growth rate to the substrate concentration.

3 Notations

The key mathematical notations used in this paper are listed in Table 1.

Table 1: Notations used in this paper

4 Model of the effect of attenuated Salmonella

4.1 Establishment of a growth model for attenuated Salmonella

Here, we use the method of separation of variables to establish a biological model and discuss the bacterial growth under two conditions: fixed bacterial protein synthesis rate and fixed nutrient content.

And two relationships can be obtained

It can be derived from the following equation

Thereby obtaining the relationship between bacterial growth rate, protein syn-thesis rate, and nutrient utilization

Among them, λ can be regarded as the maximum protein synthesis rate, and it is concluded that there is a Michaelis-Menten relationship in the process of bacterial growth.

4.1.2 Mathematical models of bacterial growth

The growth of Attenuated Salmonella in tumor cells is generally consistent with the SIR model. Normal cells in cancer cells are regarded as receptors to be activated, and the probability constant of actual contact and infection that occurs per unit time is introduced, which is called the infection rate (β). The total number of possible mutual cell contacts is IS, so the expected number of actual infectious events in the △t interval is ISxβ△t, and the number of normal cells decreases accordingly:

It can be expected that a certain number of infected normal cells will recover due to the intervention of the immune system or other medical treatments over time. The probability constant of each infected patient recovering per unit time is introduced, which is called the recovery rate (γ). In the same Δt interval, patients have a certain probability of γ recovery, and the expected number of recovered patients during this time interval is γ△t The total number of cells that should be added accordingly to reflect the recovery is:

Taking into account the above factors, the change in the total number of malignant cells in the tumor can be expressed as:

Simplified to the differential equation is

4.2 The Solution of Attenuated Salmonella's effect

According to the distribution of experimental data, the ordinary differential model of bacterial proliferation was solved by the front Euler method, and the tangent slope method was used to calculate and solve.

Finally, when the infection rate is 0.25 and the cell recovery rate is 0.1, the nor-mal cells in the cell can be effectively treated with treatment.

5 Dynamic evaluation model of efficacy of Attenuated Salmonella

5.1 Establishment of dynamic efficacy model

In order to determine the effect before and after treatment, the dry experi-mental group selected four colorectal cancer tumor markers for data processing and used analytic hierarchy.

5.1.1 Biological models of therapeutic effects

MUC2 is a member of the mucin family and is expressed in goblet cells of the mucosa of the small and large intestines (Betge et al 2016). MUC2 is associated with epithelial cells that cover the intestine, airways, and other mucosa-containing organs. Decreased expression of MUC2 has been reported to be a predictor of poor prognosis, and some prospective studies suggest that the evaluation of adjuvant chemotherapy for stage II and III colon cancer should include MUC2 expression testing to grade patients. In the model, it is represented as C1

COX-2 is an inducible enzyme that regulates the synthesis of prostaglandins and is highly expressed in various epithelial cancers. It is involved in the regulation of apoptosis, angiogenesis, and tumor cell invasion, thereby affecting tumorigenesis. Multiple studies have shown that selective COX-2 inhibitors are an important targeted approach for CRC chemoprevention. In the model, it is represented as C2

CDH17, also known as LI Cadherin, is a cadherin that functions as a calcium-dependent cell adhesion protein, connecting cells through preferential binding to homophilic antigens. High expression of CDH17 is associated with liver metastasis, which is the main cause of death and low survival rate in colorectal cancer patients. In the model, it is represented as C3.

VEGF is a heparin-binding glycoprotein that has strong angiogenic activity in endothelial cells. Angiogenesis is a key factor in the development and progression of colorectal cancer. VEGF is expressed in approximately 50% of colorectal cancers but is lowly expressed in normal colon mucosal cells. Therefore, VEGF is a good diagnostic marker for CRC. In the model, it is represented as C4

5.1.2 Mathematical model to evaluate the dynamic efficacy of attenuated Sal-monella

To more accurately evaluate the dynamic therapeutic effect model, we will use the attenuated Salmonella therapeutic effect model as a guide to evolve the therapeutic effect into a comprehensive consideration of the specific content of the four tumor markers, and use the analytic hierarchy process to evaluate the decision-making of colorectal cancer treatment plans.

We analyzed the relationship between the influencing factors of the model (C1/C2/C3/C4), and then conducted one-to-one comparisons of different influencing factors to construct an evaluation matrix. The relative weights of the elements to be compared in the criteria were calculated through the evaluation matrix. Consistency checks were performed, and the evaluation of the model was given based on the results of the matrix calculation. In the project, we used fuzzy mathematics to transform the fuzzy comprehensive evaluation method into a qualitative evaluation method. The following is our specific modeling process.

5.2 Solution of the effect model

We compared the influencing factors (C1/C2/C3/C4) in pairs, analyzed the importance of each influencing factor, and obtained the evaluation matrix.

Data is checked by consistency test, set coincidence indicator(CI) Consistence Ratio(CR) average random consistency index(RI)

Rank the weights of each influencing factor

Relative importance of a factor at a higher level:The maximum eigenvalue of theevaluation matrix and the corresponding eigenvector

Each element in W represents the proportion of importance under the corresponding judgment criterion. Based on the data and the conclusions of the wet experiment group, we scored the influence of each factor on the treatment of colorectal cancer on a scale of 100. The specific process is as follows:

AHP flowchat

For example, when considering the MUC2 indicator alone, the first element of the score vector is 0.5. Correspondingly, all score vectors (V) are obtained.

When considering the intracellular environmental factors of tumor cells compre-hensively, a specific cellular environment is selected as the research object to obtain the dynamic evaluation performance function image.

6 Sensitivity Analysis

For the dynamic evaluation model of the therapeutic effect of attenuated Sal-monella, due to the specificity of the target organism and significant differences in the bacterial environment, sensitivity analysis is conducted to prove the accuracy of the model.

Through sensitivity analysis by changing evaluation parameters, it is found that the reliability of the therapeutic effect has little difference from that under optimal conditions, indicating the validity of the model.

7 Model Evaluation and Further Discussion

7.1 Strengths

7.1.1 The proliferation curve of Salmonella is quantitatively provided, and the mechanism of its proliferation and expansion is understood, laying a founda-tion for further research.

7.1.2 In the process of solving multivariate problems in Model 2, connections are made to real-life situations and bacterial activity ranges, reducing the pro-cessing time of the algorithm and improving the efficiency of the solution.

7.2 Further Discussion

In future research, the combined effect of attenuated Salmonella and other benefi-cial bacterial strains can be considered. Based on the second model mentioned ear-lier, the analytic hierarchy process can be further used to determine the proportion of two or even multiple bacterial strains under different weights, thereby making a greater contribution to the treatment of relevant cancers.


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