Table 1 Key measurement parameters for alveolar modeling and remodeling

Outer labial cortex bending height (R1)

Distance from the labial bending point to CEJ

Labial cortex bending angle (θ1)

Angle of T1 and T2 labial surface contour line

Labial cortex surface modeling (A1)

The area between T1 and T2 labial surface

Virtual labial cortex surface modeling (VA1)

π (R1)² × θ1/360

Outer palatal cortex bending height (R2)

Distance from the palatal bending point to CEJ

Palatal cortex bending angle (θ2)

Angle of T1 and T2 palatal surface contour line

Palatal cortex surface modeling (A2)

The area between T1 and T2 palatal surface

Virtual palatal cortex surface modeling (VA2)

π (R2)² × θ2/360

Labial inner cortex remodeling (A3)

The area between T1 and T2 labial inner cortical surface

Palatal inner cortex remodeling (A4)

The area between T1 and T2 palatal inner cortical surface

Inner alveolar bending angle (θ3)

Axial angle of T1 and T2 central incisors

Inner labial cortex bending height (R3)

Distance from axial intersection point of T1 and T2 central incisors to T1 labial CEJ

Virtual inner labial cortex modeling (VA3)

π (R3)² × θ3/360

Inner palatal cortex bending height (R4)

Distance from axial intersection point of T1 and T2 central incisors to T1 palatal CEJ

Virtual inner labial cortex modeling (VA4)

π (R4)² × θ3/360