Mathematical Krause-Hegselmann (KH) Model Predicts Chemotherapeutic Resistance to Cisplatin in Human Cervical Cancer Cell Lines

Review Article

Austin J Pathol Lab Med. 2019; 6(1): 1025.

Mathematical Krause-Hegselmann (KH) Model Predicts Chemotherapeutic Resistance to Cisplatin in Human Cervical Cancer Cell Lines

Chen JZ1, Wu R2, Simpkins H2, Sun Z3 and Chen J2*

1Jericho High Senior School, Jericho, New York

2R.W, H.S., J.C., Feinstein Institute for Medical Research, Northwell Health, Pathology and Laboratory Medicine, Staten Island University Hospital, Staten Island, New York 10305

3Z.S., Academy of Math. & Systems Science, Chinese Academy of Sciences, New York

*Corresponding author: Chen J*, Department of Pathology and Laboratory Medicine Staten Island University Hospital, Northwell Health, Staten Island, New York 10305

Received: August 24, 2019; Accepted: September 17, 2019; Published: September 24, 2019


Objective: Solid tumor usually has a complex cellular components and intercellular interaction in a complex tumor microenvironment. We use a mathematical KH model to predict the change of drug resistance to cisplatin in a mixed sensitive and resistant cancer cell population and then use cytosensitivity assay to evaluate if our model can accurately predict this change predicted by KH model.

Methods and Materials: We used the KH model to predict the final increasing resistance in mixed cell population. To confirm the prediction, we focused on a mixed cell population to investigate the change of resistance to cisplatin. We mixed the sensitive and resistant cancer cells and studied their resistance to cisplatin. At the same time, separate sensitive and resistant cells are set up as controls. Our hypothesis is that after mixing, due to intercellular connection and communication (“talking” &“opinion change”), the resistant cells will instigate the sensitive cells to become more resistant, and therefore, the mixed population will show a different property with more resistance. Two human cervical cancer cells 2008 (sensitive) and 2008/C13*(resistant) were grown in RPMI 1640 medium supplemented with 10% FBS. The cytosensitivity was evaluated with MTT assay.

Results: We mixed same amounts of sensitive and resistant cancer cells and treated them with cisplatin and compared with their IC50s of parental cells. In groups of 1000, 2000, 4000, and 6000 cells, the ratios of the mixed cell population IC50s over the average IC50s of sensitive and resistant cells, showed the same tendency, namely, with the longer incubation period from day 2 to day 7, the ratios were higher, which meant the mixed cell population got relatively higher resistance to cisplatin. In other words, after mixing the cells, the sensitivity of the mixed cells were not as simple as the average of sensitive and resistant cells, but showing more resistance, which is interestingly compatible with our model prediction.

Conclusion: For the first time, with a KH model we correctly predicted the increased drug resistance in a mixed cancer cell population, which was demonstrated by our in vitro experiment. Based on the finding above, we see that the mathematical KH model can be employed to supply with novel idea in the research of chemoresistance field. The underlying mechanism is open for further exploration.

Keywords: Krause-Hegselmann (KH) model; Chemotherapy; Cisplatin; Drug resistance; Human cervical carcinoma 


Chemotherapeutic resistance, no matter whether intrinsic or acquired, is a leading obstacle to successful management of patients with cancer. The majority of the current research in chemoresistance field are mainly focusing on comparing resistant cells with sensitive ones, finding out the difference, focusing on the difference to elucidate the resistance mechanisms and then try to figure out a way to reverse them. An irregularly shaped solid tumor, actually is not composed of pure cell population but a cellular mixture from single or multiple clones of stem cells [1], which are characterized with different biological properties including different resistance to chemotherapeutic drugs. Plus the complex tumor microenvironment due to different degree of hypoxia, glucose level, and blood supply [2], the fact that a solid tumor is composed of a mixture of tumor cells with different resistant degree is easily understandable. Thus, how do these different cell populations affect each other in a solid tumor? Are the more resistant cells making the sensitive ones more resistant? If the answer is yes, the current management for patients with solid cancer may need to be retuned. For example, during the current chemotherapy cycles of ovarian cancer, the regimen may not be able to keep same from cycle one to cycle six, but should be retuned according to the profile of biological changes, such as increasing resistance, to further improve patient prognosis. Based on the changing biological properties of a solid tumor with cellular heterogeneity, this kind of adjustment will act in concert with the initial intention of promoting wellness in current precision medicine.

KH model is a simple model often utilized in opinion dynamics to predict the opinion evolution on any given issue of a population. We speculate the tumor cells in a solid tumor would behave like a community and these cells will interact as do people. Based on KH model, our hypothesis is that after mixing the sensitive and resistant cells, due to intercellular connection and communication, the resistant cells will instigate the sensitive cells to become more resistant and not the reverse on drug exposure pressure, and therefore, the mixed population will show a different property from the original status – namely, more resistance, which we call it “1+1>2 effect”. This study is to focus on a mixed cell population to investigate the change of resistance to cisplatin, which is the most commonly used therapeutic drug in chemotherapy. To our knowledge, this is the first paper to use a mathematical KH model to predict and investigate the biologic change.

Methods and Materials

Krause-Hegselmann (KH) model and Prediction for drug resistance

As the picture showed in Figure 1a, each dot in the KH model stands for an individual with different opinion on any type of different topics like entertainment, faith, sport, etc. The KH model is most often used to predict evolution in opinions of persons. Opinions are quantified as numbers between 0 and 1, inclusive, based on extremeness of stance, although the range has no significance and can be changed. Opinions  of individuals  respectively, on any given issue, as a function of time, will change as a result of interaction and influence. A confidence bound ? is given so that agent  will interact/influence each other only if . Each agent’s opinion is updated as follows:

Simply speaking, as the KH model in the figure showed Figure 1, in a community with many individuals with different opinions (a), when the confidence bound (?) from b to f increases enough, the opinion of this same individual group (population) will finalize into one, meaning reaching the same opinion on a single topic finally. This model can be taylored to predict changes in cell resistance, where each cell takes the place of an individual, with “opinions” (sensitive or resistant) represented by their sensitivity to cisplatin.