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Model
Model
1. Introduction
1.1 Background
In our project, we designed a new way to diagnose prostate cancer, which uses PCA3 as a biomarker. Through the usage of toehold switch, our device can display visible readout if a patient has prostate cancer.
After we had finished the design of primer and toehold switch, we planned to use mathematical modeling to prove the sensitivity and the specificity of our product. At the same time, we hoped that we can plot a standard curve for the detection of PCA3 in synthetic urine to determine the effective interval of copies of PCA3 in urine for the detection.
1.2 Objectives
Using the urine samples with different copies of PCA3 to test the fluorescence intensity of each sample, we aim to draw a standard curve that fits for these data.
Another group of statistics will be used to plot the ROC curve (receiver operating characteristic curve). With the curve and the AUC (Area under the Curve of ROC) of it to ensure the reliability of our product.
2. Assumptions and Justifications
Assumption 1: Synthetic urine can chemically represent real urine. Justification: Due to ethical reasons, we were not able to use urine from actual patients to establish our model. Thus, we have to use synthetic urine with a different number of PCA3 mRNA copies to mimic the urine of patients.
Assumption 2: An ROC curve with an AUC value of >80% is effective. Justification: According to ‘glassboxmedicine.com’, ‘a [AUC] greater than 0.8 is excellent performance’.
3. Modelling
3.1 Data Collection
In multiple 2 μl solutions, we mixed different amounts of copies of PCA3 mRNA. We prepared solutions with 0, 50, 100, 200, 300, 400, 500, 800, 1000, 1500, and 2000 copies of PCA3 mRNA. Each PCA3 solution with different copies above were added into 5ml synthetic urine for urine samples. The fluorescent intensity for each individual sample is eventually determined using BioTek Multi-Mode Microplate Reader after finishing the whole procedure of our detection method. The values of the fluorescent intensities are recorded in Figure 1.
Figure 1: Fluorescent Intensities for Solutions with Different mRNA Copies
A scattered plot can be used to record the fluorescent intensities under different amounts of PCA3 mRNA copies, as demonstrated in Figure 2.
Figure 2: Scatter Plot of Fluorescent Intensities
3.2 4PL Regression Model
To establish a model to describe the specific relationship between the fluorescent intensities and PCA3 mRNA copies, we decided to use Four Parameter Logistic (4PL) Regression for data fitting, a model that is commonly used in bioassays. The basic equation for 4PL regression is:
where u and l are the upper and lower asymptote, respectively, i is the point of inflection (the halfway point between u and l ), and H is the Hill’s slope of the curve (related to the steepness of the curve at point c). In our model, y is represented as int , the fluorescent intensity, and x as CmRNA, the number of copies of PCA3 mRNA.
We obtained the values of u,l,i,H using R language (commands presented in Appendix), and our results are as followed:
and after simplification:
A line can be drawn using the above expression, which describes the relationship between the number of PCA3 mRNA copies and the fluorescent intensities, as shown in Figure 3.
Figure 3: Fitted Line Using 4PL Regression
4. ROC Curve and AUC Value
For evaluating the reliability of our products, we developed a receiver operating characteristic curve (ROC curve). ROC curves possess an y-axis of the true positive rate of diagnosis or sensitivity, and the x-axis represents the false positive rate or specificity. To plot a ROC curve, we randomly synthesize urine samples by selecting 300 copies of PCA3 located in the linear elevation of the standard curve. After reverse transcription, PCR, and coupled transcription/translation procedure, the samples go through fluorescence detection. Plotting data on the graph and linking them can obtain an empirical ROC curve to analyze the accuracy of clinical trials. Sensitivity and specificity have an inverse correlation, and a trade-off must take place for each of these points on the graph. The values of points indicate the percentage of true and false positivity, and therefore evaluates the reliability of our method. The AUC (area under the curve) value can be used to describe the effectiveness of our diagnostic method. It refers to the area under the ROC curve and is a measurement of the reliability of the method used. AUC can be interpreted as the process of classifying samples, picking one sample respectively from n1, and one from n0. As the ROC curve is usually above the line of y=x, the AUC would be a value between 0.5 and 1. The closer to 1, the higher the authenticity and reliability, and vice versa. Using the R language, the AUC value is calculated to be 82.4%, which means that our diagnosis method can be deemed effective.
Figure 4.1: ROC Curve and AUC Value.
5. Summary
In our project, the results for diagnosis are interpreted visually after using the toehold switch to translate fluorescent proteins. Thus, it is necessary to establish a model that describes the relationship between the amount of PCA3 mRNA in urine samples and the final fluorescent intensities. In this case, we used the Four Parameter Logistic (4PL) Regression for data fitting and established a model to describe this relationship.
For evaluating the reliability of our products, we developed a receiver operating characteristic curve (ROC curve). The AUC (area under the curve) value is calculated to be 82.4%, which proves the effectiveness of our diagnosis.
