# Development of a Calibration Strip for Immunohromatographic Assay Detection Systems
## 2 Quantitative Detection System  
### Principle of Quantitative Detection  
The Beer-Lambert Law describes the relation between the attenuation of light and the properties of the material through which the light is traveling [18]. It is the basis for the principle of quantitative detection. By definition, it describes the relationship of \(A\) (the absorbance of solution), \(b\) (thickness of medium that absorbs the incident light), and \(c\) (concentration of solution). According to quantum theory, when a monochromatic parallel light irradiates a uniform medium of solution, the total absorbance of the medium is the sum of absorbance of every individual object. i.e.,:  
\[A=\sum_{i=1}^{m}\varepsilon_{i}b_{\text{ci}} \tag{1}\]where \(\epsilon_{\rm i}\) and \(\epsilon_{\rm i}\) are molarity and constant, respectively. When there is only one kind of absorbent medium, Equation (1) can be simplified as:  
\[A=\epsilon bc \tag{2}\]  
In photoelectric detection systems for colloidal gold-based ICG assay, the test line on the detection strip can be regarded as a thin layer of solution with a certain thickness \(b\). Therefore, Equation (2) is simplified as:  
\[A=\mathsf{k^{\prime}}c \tag{3}\]  
where \(k^{\prime}=\epsilon b\) is constant. This indicates that \(A\) (the absorbance of test line) is proportional to \(c\) (concentration of solution). Thus, when stable parallel light irradiates the test line of the strip, the darker the color of test line is, the larger \(A\) is, and the smaller the intensity of reflective light, whereas the lighter the color the test line is, the smaller \(A\) is, and the larger the intensity of reflective light. Collected by optical fiber, reflective light is focused on a photodiode which transfers optical signals into electric signals. Finally, the test strip can be quantitatively detected by analyzing the electric signals.  
Conversely, in image processing detection system for colloidal gold-based ICG assay, when stable parallel light irradiates the surface of the medium, there is no light reflecting or penetrating in the ideal case, which means that all light is absorbed by the medium. Therefore, \(A\) is approximately equal to the integral optical density (_IOD_). _IOD_ is given by:  
\[IOD=\sum_{i=1}^{N}OD^{(i)}=\sum_{i=1}^{N}\lg\frac{\phi_{0}}{\phi^{(i)}} \tag{4}\]  
where \(OD^{(i)}\) is the _IOD_ of pixel \(i\), \(\phi_{0}\) is the reflective optical density of zero concentration of the solution, \(\phi^{(i)}\) is the reflective optical density of pixel \(i\), and \(N\) is the total number of image pixels. In the ideal case, both the background of the strip image and regions outside the test line are white, which indicates that all incident light is reflected. Thus, the density of incident light is equal to the density of reflected light. Additionally, for CCD or CMOS image sensors with linear photoelectric characteristics, the output current of sensors is proportional to the optical density of incident light. Hence:  
\[IOD=\sum_{i=1}^{N}\lg\frac{\phi_{0}}{\phi^{(i)}}=\sum_{i=1}^{N}\lg\frac{I_{0} }{I^{(i)}}=\sum_{i=1}^{N}\lg\frac{G_{0}}{G^{(i)}} \tag{5}\]  
where \(I^{(i)}\) and \(G^{(i)}\) are the output current and gray value of pixel \(i\), respectively, \(I_{0}\) and \(G_{0}\) are the output current and gray value of the strip background, respectively. Thus, the concentration of solution can be calculated by measuring the gray value of test line and background of the image, which indicates that the test strip can be quantitatively detected by performing specific image processing algorithms.