## 2 Materials and Methods
The tooth was then removed from the silicone impressions and prepared for root canal instrumentation using WaveOne Gold (Dentsply Maillefer, Baillagues, Switzerland) of the Primary instrument with the WaveOne motor (Dentsply Maillefer, Baillagues, Switzerland), until the working length was reached. Every WaveOne Gold Primary instrument was used for only one canal. The root canal was irrigated using 3% sodium hypochlorite (CanalPro, Coltene/Whaledent GmbH, Langenau, Germany) before and after instrumentation, and then dried using a paper cone. The teeth were then inserted back into the silicone impressions and were scanned for the second time using CBCT with the same parameters as the first time. Each tooth, after that, was inserted into its own impression as created previously, and was exposed to the periapical digital radiograph, in both directions, for images of after instrumentation. The dataset from the CBCT was processed using the formulation in the previous study, with the data being recorded from the point of 3 mm from the apex foramen, which was 2 mm from the working length [1]. For each corresponding pair of CBCT slices, eight parameters were measured and values were calculated from these parameters. \(M_{b}^{c}\) represented the shortest distance from the mesial root surface to the periphery of the before instrumentation canal; \(M_{a}^{c}\) represented the shortest distance from the mesial root surface to the periphery of the after instrumentation canal; \(D_{b}^{c}\) represented the shortest distance from the distal root surface to the periphery of the before instrumentation canal; and \(D_{a}^{c}\) represented the shortest distance from the distal root surface to the periphery of the after instrumentation canal. The difference between each pair of measurements \(\left|\left(M_{b}^{c}-M_{a}^{c}\right)-\left(D_{b}^{c}-D_{a}^{c}\right)\right|\) value and the ratio between these two differences \(\left|\left(M_{b}^{c}-M_{a}^{c}\right)/\left(D_{b}^{c}-D_{a}^{c}\right)\right|\) value (when the numerator was the smaller of these pairs of numbers) were the transportation and centering ratio in the mesio-distal direction, respectively. \(B_{b}^{c}\) represented the shortest distance from the buccal root surface to the periphery of the after instrumentation canal; \(L_{b}^{c}\) represented the shortest distance from the lingual root surface to the periphery of the before instrumentation canal; and \(L_{a}^{c}\) represented the shortest distance from the lingual root surface to the periphery of the after-instrumentation canal. The difference between each pair of measurements \(\left|\left(B_{b}^{c}-B_{a}^{c}\right)-\left(L_{b}^{c}-L_{a}^{c}\right)\right|\) value and the ratio between these two differences \(\left|\left(B_{b}^{c}-B_{a}^{c}\right)/\left(L_{b}^{c}-L_{a}^{c}\right)\right|\) value (when the numerator was the smaller of these pairs of numbers) were the transportation and centering ratio in the bucco-lingual direction, respectively (Figure 2). The periapical image was processed using the modified method described by a previous study [2]. The trace of the root canal outline was performed using the Sketch software for mobile phones (SamSung Galaxy Note 10, Samsung, Suwon, Korea). Data were transferred to the computer for further measurements and calculations. The Adobe Photoshop CC 2019 software (Adobe, San Jose, CA, USA) was used for overlapping the before and after instrumentation images, creating the composite image for further analysis (Figure 3). After that, the ImageJ 1.52 (National Institutes of Health, Bethesda, Rockville, MD, USA) was used for tracing and measuring the data on the images acquired from the above step. Measurements were performed on each millimeter from the apex foramen to the level of 7 mm from the apex with an accuracy of 0.001 mm and a magnification of 6 (Figure 4).