## 2 Materials and Methods
### Standardized Reorientation of CBCT Images, 3D-Coordinate System, and 3D Measurement  
All measurements were observed by an orthodontic resident (H.J.L.). CBCT images were re-oriented, and measurements were made using ON3D (3D ONS, INC., Seoul, South Korea) software. The 3D coordinates (x,y,z) were constructed using nasion (N) as the reference point (0,0,0). They were assigned according to the N point and the FH plane, which was defined as a plane passing through the bilateral orbitale (Or) points and the midpoint of the bilateral porion (Po) points (Figure 2) [21,22]. Three-dimensional landmarks were traced, and each landmark was assigned its 3D coordinates. The three-dimensional landmarks and values are described in Figures 3 and 4 and Tables 1 and 2. Figure 1: Maximly skeletal expander. (**A**) Pre-expansion, (**B**) Post-expansion. Figure 2: The 3-dimensional coordinate system and standardized reorientation of CBCT Images. (**A**) Frontal view. (**B**) Oblique view. The coordinate system consists of three axes (x, y, and z) with their origin (0,0,0) registered at nasion (N). Frankfort horizontal plane (FH plane) was constructed with bilateral orbitale (Or) points and the midpoint of bilateral porion (Po) points. The x-axis is a transverse axis passing through the N point and also parallel to the FH plane in the frontal view. The y-axis is the anteroposterior axis passing through the N point and perpendicular to the x-axis. The y-axis is parallel to the FH plane in the sagittal plane. The z-axis is the vertical axis perpendicular to both x- and y-axes. Positive values are to the left, posterior, and superior (LPS) to the N point of the subject. Negative values are to the right, anterior, and inferior (RAI) to the N point. PoMP, midpoint of bilateral porion (Po) points. Rt, right. Lt, left. Figure 4: The 3-dimensional values. (**A**) Nasal septal angles (US, MS_C, and LS) in the coronal plane. (**B**) Nasal septal angles (AS, MS_A, and PS) in the axial plane. (**C**) Lateral nasal wall (LNW) angles and widths, and nasal floor (NF) angle. (**D**) Nasal septal length (NSL) in the coronal plane (between US and LS). (**E**) NSL in the axial plane (between AS and PS). Rt, right. Lt, left. \begin{table}
\begin{tabular}{c c} \hline \hline
**Landmarks** & **Definitions** \\ \hline \multicolumn{2}{c}{**Cranial Base landmarks**} \\ N (Nasion) & The most medial and superior point of the frontonasal suture \\ Or (Orbitale) & The most inferior point of the orbital rim \\ Po (Portion) & The most superior point of the external auditory meatus \\ \multicolumn{2}{c}{**Nasal landmarks**} \\ US (upper septum) & The point where the nasal septum and cribriform plate met in the coronal \\ \multicolumn{2}{c}{measurement plane} \\ MS\_C (middle septum\_coronal) & The outermost point of the nasal septum in the coronal measurement plane \\ LS (lower septum) & The point where the nasal septum and palate met in the coronal measurement plane \\ AS (anterior septum) & The most anterior point of the nasal septum in the axial measurement plane \\ MS\_A (middle septum\_axial) & The outermost point of the nasal septum in the axial measurement plane \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of 3-dimensional landmarks. Figure 3: The 3-dimensional landmarks. (**A**) Nasal landmarks on the coronal section passing through the most deviated point in the nasal septum. (**B**) Nasal landmarks on the axial section passing through the most deviated point in the nasal septum. (**C**) Maximillary landmarks on the axial section passing through ANS. (**D**) Mandibular landmarks on the frontal view. Rt, right; Lt, left; US, upper septum; MS_C, middle septum_coronal; LS, lower septum; LNW, lateral nasal wall; NF, nasal floor; AS, anterior septum; MS_A, middle septum_axial; PS, posterior septum; ANS, anterior nasal spine; LF_ANS, lateral fossa_ANS; N, nasion; Pog, pogonion; Me, menton; Go, gonión. The three-dimensional values and coordinates were measured at T1, T1, and T3.