# Advancing Phantom Fabrication: Exploring 3D-Printed Solutions for Abdominal Imaging Research
## 2 Materials and Methods  
### Characterization of Appendix Simulation in an Anthropomorphic Phantom  
The quality of diagnostic images in radiology is generally assessed by three key parameters: noise, contrast, and resolution. In computed tomography (CT), noise, often referred to as image graininess, represents unwanted variations in pixel values within a uniform image region. It is commonly quantified using the standard deviation (SD) of CT numbers within a selected homogenous region of interest (ROI). Noise arises from various sources, including quantum noise, electronic noise, structural noise, and artifacts. In addition to the standard deviation, the signal-to-noise ratio (SNR) is another metric used to quantify noise. Contrast (C) is most often defined as the difference between the mean value of the CT number of the object (\(N\)) and the background (\(N_{0}\)). Also, the contrast-to-noise ratio (CNR) is used as a parameter to describe the contrast. Resolution in CT, also called high-contrast spatial resolution, is the system's ability to distinguish structures from structures that are in proximity. The parameter that describes resolution in CT is the most visible frequency, that is, the maximum number of line pairs that the system can resolve per unit length.  
To evaluate the visibility of the printed object that simulated the appendix, the values of CT numbers were evaluated along the \(x\)- and \(y\)-axes. Figure 1 gives a schematic representation of the evaluation of the circular object corresponding to the appendix. In the idealized case (upper part of the figure), the values of the CT numbers (\(N\)) change discretely, so the boundary between the background and the object is infinitely sharp. This transition describes a rectangular \(\Pi\) function. However, the border between the background and the object is not infinitely sharp, but there is a transitional gradient. The function that describes this transition can be expressed as the convolution of \(\Pi\) and the Gaussian function (G), that is:  
\[f(x)=\Pi(x)\times G(x;\sigma) \tag{1}\]  
where, for our analysis, the significant parameter \(\sigma\) is a measure of the sharpness of the object display, that is, the resolution of the system.  
The result of the convolution of the functions \(\Pi\) and \(G\) can be expressed through the error function erf in the following form:  
\[f(x)=N_{0}-\ \frac{C}{2}\Bigg{\{}\Bigg{[}1-\text{erf}\Bigg{(}\frac{x-\frac{D} {2}-x_{0}}{\sigma}\Bigg{)}\Bigg{]}-\Bigg{[}1-\text{erf}\Bigg{(}\frac{x+\frac{D }{2}-x_{0}}{\sigma}\Bigg{)}\Bigg{]}\Bigg{\}} \tag{2}\]  
Figure 1: Evaluation of the edge sharpness of a circular object.  
The parameters \(x_{0}\) and \(N_{0}\) represent the offset for the \(x\)- and \(y\)-axis, and \(D\) is the diameter of the object.  
Using the non-linear approach method, it is possible to determine the parameters of the function \(f(x)\) so that it best describes the actual measurement results (CT number values). For this purpose, this study used Gnuplot software 6.0 (Release 6.0.1), whose algorithm changes the values of the parameters through an iterative process until the sum of the squared deviations of the actual values from the curve is less than the given limit, which, in this work, had a value of \(10^{-6}\). The results of the nonlinear iterative fitting of the curve are the experimentally determined values of the parameters \(\sigma\), \(C\), \(D\), and others, as shown in Figure 2.