Patent Publication Number: US-2022211470-A1

Title: Removable dental appliances including bendable flaps and arcuate members

Description:
TECHNICAL FIELD 
     This disclosure relates to polymer-based removable dental appliances such as alignment trays. 
     BACKGROUND 
     The field of orthodontics relates to repositioning teeth of a patient for improved function and aesthetic appearance. Orthodontic devices and treatment methods generally involve the application of forces to move teeth into a proper bite configuration, or occlusion. As one example, orthodontic treatment involves the use of slotted appliances, known as brackets, which are fixed to the patient&#39;s anterior, cuspid, and bicuspid teeth. An archwire is typically placed in the slot of each bracket and serves as a track to guide movement of the teeth to desired orientations. The ends of the archwire are usually received in appliances known as buccal tubes that are secured to the patient&#39;s molar teeth. Such dental appliances remain in the mouth of the patient and are periodically adjusted by an orthodontist to check the process and maintain the proper force levels on the teeth until proper alignment is achieved. 
     Orthodontic treatment may also involve the use of polymer-based tooth alignment trays, such as clear tray aligners (CTAs). For example, orthodontic treatment with CTAs includes forming a tray having shells that couple one or more teeth. Each shell is constructed in a position that is deformed from an initial position of a tooth, e.g., a maloccluded position. The deformed position of a respective shell of the CTA applies a force to a respective tooth toward a desired position for the tooth that is an intermediate position between the initial position and a final position resulting from the orthodontic treatment. 
     SUMMARY 
     This disclosure describes removable dental appliances, such as aligner trays, that include at least one bendable flap integrally formed with an appliance body, and methods for making the same. The at least one bendable flap is configured to apply a force to a tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. 
     In some examples, the disclosure describes a removable dental appliance that includes an appliance body configured to at least partially surround a plurality of teeth of a patient. The appliance body defines a shell configured to receive a tooth of the plurality of teeth in an initial position; and a bendable flap integrally formed with the appliance body to extend from a hinge axis of the shell. The bendable flap defines a flap boundary region extending from a first end of the hinge axis around the bendable flap to a second end of the hinge axis. The flap boundary region includes an arcuate member. The bendable flap and the arcuate member are configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. 
     In some examples, the disclosure describes a system including an ordered set of removable dental appliances configured to reposition one or more teeth of a patient. Each removable dental appliance in the set of removable dental appliances includes an appliance body configured to at least partially surround a plurality of teeth of a patient. The appliance body defines a shell configured to receive a tooth of the plurality of teeth in an initial position; and a bendable flap integrally formed with the appliance body to extend from a hinge axis of the shell. The bendable flap defines a flap boundary region extending from a first end of the hinge axis around the bendable flap to a second end of the hinge axis. The flap boundary region includes an arcuate member. The bendable flap and the arcuate member are configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. 
     In some examples, the disclosure describes a method that includes forming a model of dental anatomy of a patient; and forming, based on the model, a removable dental appliance. The removable dental appliance includes an appliance body configured to at least partially surround a plurality of teeth of a patient. The appliance body defines a shell configured to receive a tooth of the plurality of teeth in an initial position; and a bendable flap integrally formed with the appliance body to extend from a hinge axis of the shell. The bendable flap defines a flap boundary region extending from a first end of the hinge axis around the bendable flap to a second end of the hinge axis. The flap boundary region includes an arcuate member. The bendable flap and the arcuate member are configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. 
     In some examples, the disclosure describes a method that includes receiving, by a computing device, a digital representation of a three-dimensional (3D) dental anatomy of a patient, the dental anatomy providing initial positions of a plurality of teeth of the patient. The method also includes determining, by the computing device, dimensions and shapes of a removable dental appliance. The removable dental appliance includes an appliance body configured to at least partially surround a plurality of teeth of a patient. The appliance body defines a shell configured to receive a tooth of the plurality of teeth in an initial position; and a bendable flap integrally formed with the appliance body to extend from a hinge axis of the shell. The bendable flap defines a flap boundary region extending from a first end of the hinge axis around the bendable flap to a second end of the hinge axis. The flap boundary region includes an arcuate member. The bendable flap and the arcuate member are configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. The dimensions and shapes are configured to reposition the one or more teeth of the patient from an initial position to a desired position when the removable dental appliance is worn by the patient. The dimensions and shapes include a position, dimension, and shape of the shell; and a position, dimension, and shape of the bendable flap. The method also includes transmitting, by the computing device, a representation of the removable dental appliance to a computer-aided manufacturing system. 
     In some examples, the disclosure describes a non-transitory computer-readable storage medium that stores computer system-executable instructions that, when executed, configure a processor to receive, by a computing device, a digital representation of a three-dimensional (3D) dental anatomy of a patient, the dental anatomy providing initial positions of a plurality of teeth of the patient. The non-transitory computer-readable storage medium also stores computer system-executable instructions that, when executed, configure a processor to determine, by the computing device, dimensions and shapes of a removable dental appliance. The removable dental appliance includes an appliance body configured to at least partially surround a plurality of teeth of a patient. The appliance body defines a shell configured to receive a tooth of the plurality of teeth in an initial position; and a bendable flap integrally formed with the appliance body to extend from a hinge axis of the shell. The bendable flap defines a flap boundary region extending from a first end of the hinge axis around the bendable flap to a second end of the hinge axis. The flap boundary region includes an arcuate member. The bendable flap and the arcuate member are configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. The dimensions and shapes are configured to reposition the one or more teeth of the patient from an initial position to a desired position when the removable dental appliance is worn by the patient. The dimensions and shapes include a position, dimension, and shape of the shell; and a position, dimension, and shape of the bendable flap. The non-transitory computer-readable storage medium also stores computer system-executable instructions that, when executed, configure a processor to transmit, by the computing device, a representation of the removable dental appliance to a computer-aided manufacturing system. 
     In some examples, the disclosure describes a removable dental appliance including an appliance body configured to at least partially surround a plurality of teeth of a patient, the appliance body defining a shell configured to receive a tooth of the plurality of teeth in an initial position; and a force actuator integrally formed with the appliance body on the shell, where the force actuator defines a spiral configuration, and where the force actuator is configured to apply a force to the tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. 
     The details of one or more examples of this disclosure are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of this disclosure will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A-1E  illustrate facial, oblique, and mesial cross-sectional views of an example removable dental appliance that includes a shell and a bendable flap including an arcuate member configured to apply a force to a tooth of a patient. 
         FIGS. 2A and 2B  are conceptual diagrams illustrating an example removable dental appliance that includes bendable flaps having a spiral configuration. 
         FIGS. 3A-3C  are conceptual diagrams illustrating an example removable dental appliance that includes a bendable flap extending from a hinge axis and a spring bellows opposite the hinge axis. 
         FIGS. 4A-4C  are conceptual diagrams illustrating an example removable dental appliance that includes a bendable flap extending from a slotted hinge axis and jumpers in a plane tangential to a surface of appliance body opposite the hinge axis. 
         FIGS. 5A and 5B  are conceptual diagrams illustrating an example removable dental appliance that includes a bendable flap extending from a slotted hinge axis and a plurality of jumpers bridging flap boundary region in a plane tangential to a surface of appliance body opposite the hinge axis. 
         FIGS. 6A and 6B  are conceptual diagrams illustrating an example removable dental appliance that includes a bendable flap extending from a hinge axis and a spring bellows extending around an entire flap boundary region. 
         FIGS. 7A and 7B  are conceptual diagrams illustrating an example removable dental appliance that includes a bendable flap extending from a hinge axis and a plurality of jumpers bridging flap boundary region. 
         FIG. 8  illustrates a lingual view of an example removable dental appliance that includes a plurality of shells and a bendable flap configured to apply a force to a tooth of a patient. 
         FIGS. 9A-9C  illustrate labial, lingual, and occlusal views of an example removable dental appliance that includes a shell and a plurality of bendable flaps configured to apply a force to a tooth of a patient. 
         FIG. 10  illustrates a lingual view of an example removable dental appliance that includes a plurality of shells and a plurality of bendable flaps configured to apply a force to a tooth of a patient. 
         FIG. 11  illustrates a lingual view of an example removable dental appliance that includes a shell and two bendable flaps configured to apply a force to a tooth of a patient. 
         FIG. 12  illustrates a lingual view of an example removable dental appliance that includes a shell and four bendable flaps configured to apply a force to a tooth of a patient. 
         FIGS. 13A-13F  illustrate occlusal, lingual, and distal cross-sectional views of an example removable dental appliance that includes a shell and four bendable flaps configured to apply a force to a tooth of a patient. 
         FIG. 14  illustrates a buccal view of an example removable dental appliance that includes a shell and three bendable flaps configured to apply a force to a tooth of a patient. 
         FIGS. 15A and 15B  illustrate buccal and occlusal views of an example removable dental appliance that includes a shell and two bendable flaps configured to apply a force to a tooth of a patient. 
         FIG. 16  illustrates a lingual view of an example removable dental appliance that includes a shell and bendable flap with a reinforcing structure configured to apply a force to a tooth of a patient. 
         FIG. 17  illustrates a lingual view of an example removable dental appliance that includes a shell and bendable flap with a reinforcing structure configured to apply a force to a tooth of a patient. 
         FIG. 18  is a block diagram illustrating an example computer environment in which a clinic and manufacturing facility communicate information throughout a dental appliance manufacturing process. 
         FIG. 19  is a flow diagram illustrating an example process of generating digital dental anatomy data. 
         FIG. 20  is a block diagram illustrating an example of a client computing device connected to a manufacturing facility via a network to generate digital dental anatomy data. 
         FIG. 21  is a block diagram illustrating an example computer-aided manufacturing system for construction of a removable dental appliance. 
         FIG. 22  is a flow diagram illustrating a process conducted at a manufacturing facility for construction of a set of removable dental appliances. 
         FIG. 23  is a flow diagram illustrating successive iterations of treatment using an ordered set of removable dental appliances. 
     
    
    
     DETAILED DESCRIPTION 
     This disclosure describes removable dental appliances that include at least one bendable flap integrally formed with an appliance body to extend from a hinge axis of a shell and at least one arcuate member disposed in a respective flap boundary region between the shell and a respective bendable flap. Orthodontic treatment with the removable dental appliances includes the use of at least one bendable flap and at least one arcuate member in the flap boundary region to enable greater control of force vectors applied to the teeth of the patient. The bendable flap and arcuate member apply a force to a tooth to cause movement of the tooth toward a desired position when the removable dental appliance is worn by the patient. For example, a rest position of the bendable flap may intrude into a space defined by the desired position of the tooth. The shell may include a surface that defines a void internal to the shell and shaped to receive the tooth in the desired position. During use of the removable dental appliance, the bendable flap and arcuate member are displaced by the tooth into a deformed position to cause the force, while the surrounding shell remains substantially undeformed. The deformed bendable flap and arcuate member apply the force to a side of the tooth opposite from the void to cause movement of the tooth toward the void. In this way, the removable dental appliance including a bendable flap and arcuate member may be configured to concentrate deformation in at least one of the bendable flap, the hinge axis, or arcuate member. 
     By concentrating the deformation in at least one of the bendable flap, the hinge axis, or the arcuate member, the shell may remain more highly engaged with the tooth. For example, when the removable dental appliance is in a deformed state, e.g., worn by the patient, the shells may have more points of contact with a respective tooth, a greater surface area of contact on a respective tooth, or the like, compared to a removable dental appliance without a bendable flap. In this way, the removable dental appliance may improve engagement of the teeth in the shells, concentrate deformation in the bendable flap and arcuate member, or both. By separating the force generating member (e.g., the bendable flap and arcuate member) and the engagement member (e.g., the shell), the removable dental appliance enables greater control of forces applied to the teeth of a patient. In contrast, removable dental appliances that do not include at least one bendable flap and arcuate member, or other similar features, the appliance body both engages a respective tooth and creates the force required to move the tooth during the course of orthodontic treatment. The degree of tooth engagement (e.g., the amount and positions of shell/tooth contact) affects control of the force applied to the tooth. 
     The bendable flap and arcuate member are configured to control the magnitude, direction, and length of expression of the force applied to a respective tooth. For example, at least one of the position, shape, and dimensions of the bendable flap and/or the arcuate member may result in a desired force vector on the respective tooth. The force vector may be applied to the tooth in a direction or a magnitude that may not be possible to apply to the tooth without the bendable flap and arcuate member. The bendable flap and the arcuate member may also enable expression of a force over a greater distance than a removable dental appliance that relies on deformation of shells of the appliance to express force. For example, the rest position of the bendable flap may extend into a space defined by the tooth in the desired position of the tooth such that, as the tooth moves into a void shaped to receive the tooth in the desired position, the bendable flap continues to express the force of a sufficient magnitude to cause alveolar bone remodeling. Movement of the tooth results in partial relaxation of the bending moments of the bendable flap and/or arcuate member. Some residual stress may remain in the bendable flap and/or arcuate member to ensure positive force levels over the complete range of expression. In this way, the removable dental appliance may improve control of at least one of force vector direction, magnitude, or expression length, to achieve at least one of a desired tooth movement that may not be possible without the bendable flap, a desired tooth movement over a shortened treatment time, a desired tooth with fewer progressions of removable dental appliances in a set of removable dental appliances, or the like, compared to other orthodontic treatments. 
     In some examples, each removable dental appliance of an ordered set of removable dental appliances may result in a greater expression, e.g., compared to a removable dental appliance without bendable flaps and arcuate members, due to the net shape of the appliance remaining relatively constant over the course of each treatment stage. Because the bendable flaps and arcuate member may create force actuators that are isolated to individual teeth, only these focused portions of the appliance (the bendable flaps) need to deform in order to apply directed forces to the teeth. As such, all other portions of the appliance can be made fairly rigid, thereby providing guided channels for tooth movement as well as bracing to resist deformation where movement is not desired. The amount of expression achievable by a single removable dental appliance may be limited by the depth of each tooth-receiving void and the elastic limit of bending possible by each bendable flap. For example, the amount of expression may be greater than 0.25 millimeter (mm) of crown movement, such as greater than 0.5 mm of crown movement or greater than 1 mm of crown movement. When coupled with robust materials that maintain their aesthetics and mechanical properties over a longer in vivo period compared to commonly used thermoplastics, greater expression and control may reduce the number of removable dental appliances in an ordered set of removable dental appliances required to achieve a selected tooth movement, e.g., due to more expression per removable dental appliance; reduce the number of office visits, e.g., due to increased doctor confidence in treatment progress; reduce treatment duration, e.g., due to more continuous forces and decreased round-tripping; and enable more accurate finishes, e.g., due to higher appliance rigidity and positive force application over the complete range of motion. 
       FIGS. 1A-1E  illustrate facial, oblique facial, and mesial cross-sectional views of a portion of an example removable dental appliance  100  that includes a plurality of shells  104 A- 104 D (collectively, “shells  104 ”), shell  104 C including a bendable flap  108 C and arcuate member  109 C configured to apply a force  107 C to a tooth  103 C of a patient. Removable dental appliance  100  includes appliance body  102  configured to at least partially surround plurality of teeth  103 A- 103 D (collectively, “teeth  103 ”) of the mandibular arch  101  of a patient. Appliance body  102  includes shells  104 . Shells  104  may be configured to receive teeth  103 . Bendable flap  108 C and arcuate member  109 C may be configured to apply force  107 C to tooth  103 C to cause a movement of tooth  103 C toward a desired position of tooth  103 C when removable dental appliance  100  is worn by the patient. The desired position may include an intermediate position between the initial position and the final position after orthodontic treatment. 
     In some examples, bendable flap  108 C and arcuate member  109 C may be configured to apply force  107 C to an attachment on tooth  103 C to cause a movement of tooth  103 C toward the desired position. The attachment may include a natural undercut, such as, for example, a cusp tip, a cervical contour, or the like, an artificial undercut, a protrusion, a knob, a handle, or the like. By applying a force  107 C to tooth  103 C via bendable flap  108 C and arcuate member  109 C, removable dental appliance  100  may improve control of at least one of a force vector direction, magnitude, or expression length, to achieve at least one of a desired tooth movement that may not be possible without bendable flap  108 C and arcuate member  109 C, a desired tooth movement over a shortened treatment time, a desired tooth movement with fewer progressions of removable dental appliances in a set of removable dental appliances, or the like, compared to other orthodontic treatments. 
     For purposes of illustration, only teeth  103 , shells  104 , and bendable flap  108 C are shown in  FIGS. 1A-1E , although appliance body  102  may include any number of shells  104  configured to at least partially surround any number of teeth  103 , any number of bendable flaps  108 , and/or any number of arcuate members  109 . For example, the number of teeth  103  on dental arch  101  may be fourteen, less than fourteen (e.g., a patient having one or more extracted teeth), or more than fourteen (e.g., a patient having wisdom teeth or hyperdontia). The number of shells  104  may be fourteen, less than fourteen (e.g., at least one shell configured to surround more than one tooth), or more than fourteen teeth (e.g., more than one shell portions configured to surround one tooth). Additionally, or alternatively, appliance body  102  may include a plurality of bendable flaps  108  on the same or difference shells  104 . Additionally, or alternatively, one or more of each of bendable flaps  108  may include one or more arcuate members  109 . 
     Appliance body  102  is configured to at least partially surround teeth  103  of either the maxillary dental arch or, as shown in  FIGS. 1A-1E , the mandibular dental arch  101  of a patient. For example, appliance body  102  may surround at least one of the facial, lingual, and occlusal surfaces of teeth  103 , overlap a portion of the gingiva of the patient, or the like. In some examples, appliance body  102  may surround different portions of different teeth  103 . 
     Appliance body  102  includes shells  104 . In some examples, appliance body  102  may include a respective shell of shells  104  for each respective tooth of teeth  103 . In other examples, appliance body  102  may include fewer shells than teeth  103 , e.g., shells may receive more than one tooth or a number of teeth  103  may not be surrounded by appliance body  102 . In other examples, appliance body  102  may include more shells  104  than teeth  103 , e.g., two or more shells  104  may surround at least a portion of at least one tooth of teeth  103 . Each respective shell of shells  104  may be shaped to receive at least one respective tooth of teeth  103 . In some examples, shells  104  may surround the facial, lingual, and occlusal portions of teeth  103 . In other examples, shells  104  may surround fewer portions of teeth  103 , such as, only the facial and lingual portions, or only one of the facial or lingual portions of teeth  103 . For example, shells  104 A,  104 B,  104 C, and  104 D may be shaped to surround the lingual, occlusal, and facial portions of tooth  103 A,  103 B,  103 C, and  103 D, respectively. In some examples, shells  104  may define a plurality of voids. For example, appliance body  102  may define a framework configured to contact teeth  103  in selected locations. The selected location may include, for example, portions of interproximal regions between adjacent teeth, portions of occlusal surfaces of teeth, or portions of gingival margins of teeth. The framework may include material concentrated in areas or along lines as needed to resist deformation caused by internal stresses. These internal stresses come as a result of both applied forces and reaction forces acting on appliance body  102 , where the applied forces are generally the result of elastic deformation in bendable flaps  108  as they contact the teeth, and the reaction forces are generally the result of other parts of appliance body  102  (e.g., shells  104 ) contacting tooth surfaces opposite the applied forces and their respective contact points. Key benefits to using a framework may include, for example, reduced material, reduced material cost, reduced fabrication time, increased aesthetics, and increased salivary flow. Such a framework also has the potential to be more rigid than an appliance having constant thickness, provided that increased thickness is used in areas where lines of force are concentrated to cause increased stress in the appliance material. As such, areas of material experiencing lower or minimal stress are removed. This is essentially the process of Generative Design, although it is typically iterative to evaluate stresses and refine the design after successive iterations until diminishing returns are achieved (a threshold is reached) in terms of optimization toward a particular goal, such as maximum rigidity, minimum volume, or a combination thereof. In this way, shells  104  may define a plurality of voids to define a framework contacting at least a portion of at least one of a first interproximal region mesial a respective tooth, a second interproximal region distal the respective tooth, an occlusal surface of the respective tooth, or a gingival margin of the respective tooth. 
     In some examples, a respective shell may not include a bendable flap (e.g., shells  104 A,  104 B, and  104 D). In some examples, a respective shell may apply forces to respective received tooth by deformation of the respective shell. For example, when worn by the patient, shells  104 A,  104 B, and  104 D may deform. The deformation may result in a restorative force as the respective shell moves toward an undeformed configuration. The restorative force may be transferred to the respective tooth via one or more points of contact between the respective shell and the respective tooth. In this way, removable dental appliance  100  may combine some shells  104  that include bendable flaps with some shells  104  that deform to move teeth  103  to desired positions of teeth  103 . In other examples, a respective shell may be configured to be sufficiently stiff so as not to deform. A respective shell that does not deform may provide anchorage for neighboring shells, such as, for example, shells that include a bendable flap  108 . The selection of which shells  104  include bendable flaps  108  may depend on the forces to be exerted on respective teeth  103 , the movements of respective teeth  103 , or both. For example, when deformation of a respective shell  104  does not interfere with the forces to be exerted on neighboring teeth  103  or the movements of neighboring teeth  103 , the respective shell  104  may not include a bendable flap  108 . Conversely, when deformation of a respective shell  104  does interfere with the forces to be exerted on neighboring teeth  103  or the movements of neighboring teeth  103 , the respective shell  104  may include a bendable flap  108  to reduce deformation of the respective shell  104 . 
     In some examples, appliance body  102  may include one or more anchor shells configured to receive one or more anchor teeth. In some examples, anchor teeth may include one or more molar teeth, premolar teeth, or both. In other examples, anchor teeth may include one or more anterior teeth, or a combination of one or more anterior and posterior teeth. Anchor shells may be configured to allow appliance body  102  to deform to result in a force sufficient to move (e.g., force sufficient to cause alveolar bone remodeling) selected teeth without resulting in sufficient force to move the respective anchor teeth. 
     Shell  104 C may be shaped to engage tooth  103 C in an initial position of tooth  103 C. To engage an initial position of tooth  103 C, an internal surface of shell  104 C may contact an at least one selected location, a selected surface area, or both of tooth  103 C. For example, as shown in  FIG. 1C , surface  111 C of shell  104 C may contact at least a portion of an occlusal surface and a lingual surface of tooth  103 C in an initial position. The locations of contact, surface area of contact, or both may affect force  107 C applied by bendable flap  108 C to tooth  103 C, the resulting movement of tooth  103 C, or both. 
     Shell  104 C may also be shaped to receive tooth  103 C in a desired position of tooth  103 C. The desired position of tooth  103 C may be the position after force  107 C has been exerted on tooth  103 C to move tooth  103 C to the extent possible in shell  104 C. For example, surface  111 C may define void  110 C internal to shell  104 C. As shown in  FIG. 1C , void  110 C includes a wedge-shaped void with a maximum depth near the gingival margin of tooth  103 C that tapers to a minimum near axis of rotation  116 C at the incisal edge of tooth  103 C. The wedge shape of void  110 C may be congruent with the path of tooth  103 C as tooth  103 C moves toward the desired position defined by surface  111 C. Tooth  103 C may move through void  110 C toward the desired position until tooth  103 C contacts surface  111 C. In this way, surface  111 C may prevent tooth  103 C from moving beyond the desired position. 
     Removable dental appliance  100  includes at least one bendable flap  108 C. In general, any number of bendable flaps may be positioned on any number of shells  104 . Bendable flap  108 C may be integrally formed with shell  104 C of appliance body  102  to extend from hinge axis  110 C. Hinge axis  110 C extends along an incisal edge of shell  104 C in the mesio-distal direction. Generally, a respective bendable flap  108  may extend from a respective hinge  110  axis extending along any portion of a respective shell, in any direction. By selecting a length and an orientation of a respective hinge axis  110 , removable dental appliance  100  may be configured to apply a respective force via a respective bendable flap  108  to any portion of a respective tooth. 
     As shown in  FIGS. 1A-1E , bendable flap  108 C extends from hinge axis  110 C on a facial surface of appliance body  102  and is positioned on a facial side of removable dental appliance  100 . Appliance body  102  defines a flap boundary region  113 C extending from first end  114 C around bendable flap  108 C to second end  112 C. Flap boundary region  113 C may include an area of reduced shear and tensile stress compared to surrounding portions of appliance body  102 . For example, at least a portion of flap boundary region  113 C includes arcuate member  109 C. 
     Arcuate member  109 C may increase the flexibility of appliance body  102  at flap boundary region  113 C compared to the surrounding appliance body  102 . As illustrated in  FIGS. 1A-1E , arcuate member  109 C may include a spring bellows (e.g., a ribbon of material) extending around at least a portion of flap boundary region  113 C and coupled to shell  104 C and bendable flap  108 C. In some examples, arcuate member  109 C may include a plurality of spring bellows. In other examples, arcuate member  109 C may include one or more jumpers (e.g., a rod of material) defining an arc in a plane tangential to the surface of shell  104 C or extending outside of the plane tangential to the surface of shell  104 C and coupled to shell  104 C and bendable flap  108 C. In some examples, arcuate member  109 C may include any suitable combination of one or more spring bellows, one or more jumpers, or one or more shear reduction regions. 
     Arcuate member  109 C may have an arcuate, sinusoidal, zig-zag, pulsing wave, spiral, helix, serpentine, or folded cross-section in a plane tangential to the surface of shell  104 C and/or a plane perpendicular to the plane tangential to the surface of shell  104 C. The position (e.g., relative to shell  104 C and bendable flap  108 C) and shape of arcuate member  109 C may be selected to allow for cantilever motion of bendable flap  108 C and application of a selected force  107 C to tooth  103 C via bendable flap  108 C when removable dental appliance  100  is worn by the patient. 
     In some examples, arcuate member  109 C may be made of the same material as shell  104 C. For example, arcuate member  109 C may be formed integrally with shell  104 C. In some examples, arcuate member  109 C may be formed by laser cutting portions of appliance body  102  to define arcuate members  109 C. Additionally, or alternatively, arcuate member  109 C may be formed by remolding (e.g., heating and applying a force) portions of appliance body  102 , or coupling additional material to a surface of appliance body  102  (e.g., by adhesion, thermo welding, ultrasonic welding, or the like) to define arcuate members  109 C. In some examples, at least a portion of arcuate member  109 C may be thinner than shell  104 C to allow for greater flexibility, for example, of the spring bellows or the jumper. In some examples, at least a portion of arcuate member  109 C may be thicker than shell  104 C to allow for greater rigidity or toughness of the spring bellows or the jumper. In some examples, arcuate member  109 C may include different or additional material, such as materials having a higher modulus relative to the material of appliance body  102 , metals (wires, ribbons, or sheets), or the like. The materials and manufacture of arcuate member  109 C may be selected to allow for cantilever motion of bendable flap  108 C and application of a selected force  107 C to tooth  103 C via bendable flap  108 C when removable dental appliance  100  is worn by the patient. 
     In examples in which arcuate member  109 C includes a spring bellows, the spring bellows may include a continuous or discontinuous curvilinear portion of appliance body  102 , e.g., an arc, a half wave, a full wave shape, zig-zag, sinusoid, a pulsed wave, or serpentine shape. In some examples, the arcuate displacement may include at least one fold to increase the length and/or flexibility of the spring bellows. The length of the spring bellows may be selected to provide a selected force resulting from the deformation of the spring bellows when removable dental appliance  100  is worn by the patient. 
     In examples in which the spring bellows includes a continuous curve, the arcuate displacement may define an outer radius of curvature, e.g., an outermost surface of the spring bellows. In some examples, the outer radius of curvature may be between about 0.5 millimeters and about 3 millimeters, or about 0.75 millimeters and about 1.5 millimeters, or about 1.0 millimeters. The radius of curvature may be substantially constant or may vary along an interproximal boundary curve. The spring bellows also may define a displacement distance extending between a midline of flap boundary region  113 C and a midline of the spring bellows. In some examples, the displacement distance may be less than about 3 millimeters, or less than about 1 millimeter, or less than about 0.75 millimeters, or about 0.5 millimeters. The displacement distance may be substantially constant or may vary along flap boundary region  113 C. 
     A thickness of the spring bellows may be less than a thickness of shell  104 C and bendable flap  108 C such that the spring bellows deforms more than shell  104 C and bendable flap  108 C to concentrate compression, tension, shear, bending, or torsion in the spring bellows. The thickness of the spring bellows may be between about 0.025 millimeters and about 1.0 millimeter, or between about 0.1 millimeters and about 0.75 millimeters, or between about 0.15 and about 0.6 millimeters, or about 0.3 millimeters. The thickness of the spring bellows may be substantially constant or vary along flap boundary region  113 C. 
     In some examples, the spring bellows may define at least one shear reduction region, e.g., at least one void or cutout in the spring bellows. The at least one shear reduction region may concentrate deformation of spring bellows  108  in selected portions of spring bellows  108 . The location of the terminations of the spring bellows on the shell  104 C and bendable flap  108 C may be selected to provide a selected direction and magnitude of force when removable dental appliance  100  is worn by the patient. In some examples, arcuate member  109 C may include a plurality of spring bellows, each respective spring bellows of the plurality of spring bellows disposed along a respective portion of the flap boundary region. 
     By selecting the shape, length, radius of curvature, and displacement distance of spring bellows, removable dental appliance  100  may control at least one of a direction, a magnitude, and a length of expression of a force on bendable flap  108  resulting from deformation of appliance body  102  when removable dental appliance  100  is worn by the patient. Other spring bellows configurations are described in U.S. Patent Application No. 62/568,982, by Raby, et al., which is incorporated herein by reference in its entirety. 
     In examples in which arcuate member  109 C includes a jumper, the jumpers include an elongate structure extending along a longitudinal axis between a first end coupled to any suitable portion of shell  104 C or a different shell  104  (e.g., not directly adjacent to bendable flap  108 C) and a second end coupled to any suitable portion of bendable flap  108 C. At least a portion of force  107 C results from a deformation of the jumper when removable dental appliance  100  is worn by the patient. For example, when removable dental appliance  100  is worn by the patient, the jumpers may deform to exert at least one of a bending force, a twisting force, a compressive force, a tensile force, or a shear force on the first end and the second end such that selection of the location of the first and second ends may control the direction and magnitude of force  107 C. 
     The jumper may include any suitable shape along the longitudinal axis of the elongate structure, such as, for example, at least one of an arc, fold, zig-zag, sinusoid, spiral, helix, or serpentine shape extending between the first end and the second end of the jumper. In some examples, elongated structure may include at least one fold. In some examples, a medial portion of the jumper (e.g., between the first end and second end) may extend away from a plane tangential to a surface of shell  104 C. In other examples, the medial portion of the jumper may be substantially in the plane tangential to a surface of shell  104 C (e.g., deviate from the plane by less than about 0.5 mm). 
     In some examples, the jumper may include an arcuate shape having an outer radius of curvature (e.g., outermost surface of the jumper) between about 0.5 millimeters and about 5 millimeters. In some examples, the jumper may include a displacement distance (e.g., distance between a plane tangential to a surface of shell  104 C and a midline of the inner radius of the jumper) less than about 2 millimeters, or less than about 1 millimeter, or less than about 0.5 millimeters, or about 0.5 millimeters. The jumper may define a cross-section in a plane perpendicular to a longitudinal axis of the elongated structure having any suitable shape, area, or aspect ratio selected to provide a selected force bendable flap  108 C. The shape, area, or aspect ratio of the cross-section may be constant or vary along the longitudinal axis. 
     The jumper may include any suitable thickness selected to control the magnitude and direction, or the location of concentration, of force  107 C resulting from deformation of appliance body  102  when removable dental appliance  100  is worn by the patient. In some examples, the jumper may be more flexible than the shell to at least one of reduce deformation of the shell or concentrate stress in the jumper when the removable dental appliance is worn by the patient. In some examples, a thickness of appliance body  102  increases near at least one of the first end or the second end of the jumper, for example, to improve toughness of the intersection of the first and second ends and appliance body  102 . A thickness of the jumper may be substantially constant or may vary along elongated structure in a tapered or stepwise manner. In some examples, the thickness of the jumpers may be between about 0.1 millimeters and about 3.0 millimeters, or about 0.3 millimeters and about 1.0 millimeter. 
     In some examples, the appliance body may include a gingival portion coupled to the second end of the jumper (the first end being coupled to bendable flap  108 C) to at least partially anchor appliance body  102  to the alveolar process via the gingiva. In some examples, arcuate member  109 C may include a plurality of jumpers, each respective jumper of the plurality of jumpers including a respective elongated structure extending between a respective first end coupled to a respective position on the shell and a respective second end coupled to a respective position on the bendable flap. 
     By selecting the shape, length, radius of curvature, and displacement distance of jumpers, removable dental appliance  100  may control at least one of a direction, a magnitude, and a length of expression of a force on bendable flap  108  resulting from deformation of appliance body  102  when removable dental appliance  100  is worn by the patient. Other jumper configurations are described in U.S. Patent Application No. 62/569,144, by Raby, et al., which is incorporated herein by reference in its entirety. 
     In some examples, arcuate member  109 C may result in at least a portion of force  107 C, bendable flap  108 C may remain relatively unbent in the deformed portion, or both. Arcuate member  109 C may at least one of enable increased surface contact of bendable flap  108 C with tooth  103 C, reduce build-up of food particles or plaque in flap boundary region  113 C or other portions of the appliance body  102 , and reduce conflict between the bendable flap  108 C and the dental anatomy of the patient, when removable dental appliance  100  is worn by the patient or fitted to the teeth. 
     In some examples, shell  104 C may be thinner or include one or more voids along hinge axis  110 C. The thinner material or voids along hinge axis  110 C may relieve bending stresses in bendable flap  108 C. In some examples, at least a portion of flap boundary region  113 C also may define one or more cutouts or slits in appliance body  102 . Removal of the material from flap boundary region  113 C may effectively nullify shear and tensile stress in flap boundary region  113 C. Additionally or alternatively, at least a portion of flap boundary region  113 C may include an elastomeric polymer or material with a lower elastic modulus than appliance body  102 , an area of reduced thickness of appliance body  102 , or the like, to increase the flexibility of flap boundary region  113 C compared to the surrounding appliance body  102 . In this way, flap boundary region  113 C may allow bendable flap  108 C to deflect in the lingual-facial direction, reduce the amount of deformation in bendable flap  108 C to increase contact area of the number of contacts between bendable flap  108 C and tooth  103 C, or both to improve control over tooth movement. In examples in which flap boundary region  113 C includes an elastomeric material, the elastomeric material may be selected to allow bendable flap  108 C to deflect in the facial-lingual direction, cover at least a portion of flap boundary region  113 C to reduce build-up of food particles or plaque in flap boundary region  113 C or other portions of the appliance body  102 , or both. 
     Bendable flap  108 C and arcuate member  109 C may be configured to apply force  107 C to a facial surface of tooth  103 C. For example, a rest position of bendable flap  108 C may intrude into a space defined by tooth  103 C in a desired position of tooth  103 C such that when removable dental appliance  100  is worn by the patient, an initial position of tooth  103 C may cause a deformation of bendable flap  108 C and arcuate member  109 C. The deformation of bendable flap  108 C and arcuate member  109 C may result in force  107 C, e.g., a restorative force as bendable flap  108 C and arcuate member  109 C move toward an undeformed configuration. The rest position of bendable flap  108 C and arcuate member  109 C may be selected to reduce conflict with the incisal edge of tooth  103 C when removable dental appliance  100  is fitted to the teeth. Additionally or alternatively, bendable flap  108 C may include a ramped surface near a gingival portion of bendable flap  108 C such that the ramped surface deflects bendable flap  108 C and arcuate member  109 C or otherwise reduces conflict with the incisal edge of tooth  103 C when removable dental appliance  100  is fitted to the teeth. 
     In response to force  107 C, tooth  103 C may move through void  110 C toward the desired position until tooth  103 C contacts surface  111 C. In some examples, if only a portion of tooth  103 C contacts surface  111 C, while gaps remain elsewhere, a couple may be formed between the contact point and force  107 C. The resulting couple may cause tooth  103 C to move, e.g., to “walk,” into a position of greater alignment with surface  111 C. For example, tooth  103 C may move in stages of alternating translation and rotation, until tooth  103 C is received in a position of substantial conformity with surface  111 C. In some examples, surface  111 C may be positioned beyond the desired position of tooth  103 C to compensate for relapse of tooth  103 C back toward an intermediate position or the initial position of tooth  103 C. In this way, selecting the shape of internal surface of shell  104 C may enable control of the locations of a force and resulting movement of tooth  103 C. Similar effects are also possible for shells  104 A,  104 B, and  104 D. 
     Force  107 C may be transferred from bendable flap  108 C and arcuate member  109 C to tooth  103 C by one or more contact points of bendable flap  108 C with tooth  103 C. For example, an interior surface of bendable flap  108 C may contact at least a portion tooth  103 C. In some examples, the interior surface of bendable flap  108 C may be shaped to conform to a shape of tooth  103 C in a desired position of tooth  103 C such that contact between bendable flap  108 C and tooth  103 C is increased as tooth  103 C moves toward the desired position. In some examples, the thickness of bendable flap  108 C may be selected to control the number or location of contact points. In some examples, bendable flap  108 C may be divided (e.g., by laser cutting) into a plurality of flaps control the number or location of contact points. In other examples, bendable flap  108 C may include at least one protrusion on an interior surface of bendable flap  108 C. The protrusion may be positioned or shaped to transfer force  107 C to at least one selected portion of tooth  103 C. For example, bendable flap  108 C may include at least one protrusion near the gingival portion of bendable flap  108 C such that the transfer of force  107 C to tooth  103 C is concentrated near the gingival margin. By concentrating the transfer of force near the gingival margin, bendable flap  108 C may more effectively cause a torqueing or root tipping of tooth  103 C. In this way, protrusions on respective bendable flaps may be used to control the transfer of a respective force to achieve or increase the effectiveness of tooth movements, such as, for example, translation, rotation, tipping, torqueing, extrusion, intrusion, or combinations thereof. 
     In some examples, as shown in  FIG. 1C , when removable dental appliance  100  is worn by the patient, axis of rotation  116 C may be substantially fixed or anchored through appliance body  102  to other parts of the dental anatomy, such as, for example, teeth  103 A,  103 B, and  103 D. Application of force  107 C to a portion of tooth  103 C near the gingival margin by bendable flap  108 C and arcuate member  109 C may form a couple with axis of rotation  116 C. A couple may include two opposing forces offset by some distance. For example, as force  107 C moves tooth  103 C with a center of resistance located near the center of the root of tooth  103 C, the fixed axis of rotation  116 C of shell  104 C may apply a second opposing force to the incisal edge of tooth  103 C. By forming a couple with axis of rotation  116 C, force  107 C may result in rotation  118 C of tooth  103 C toward void  110 C, e.g., a root tipping or torqueing movement. In this way, the locations of contact, surface area of contact, or both of surface  111 C of shell  104 C may affect force  107 C applied to tooth  103 C, the resulting movement of tooth  103 C, or both. 
     When removable dental appliance  100  is fitted to, or removed from, teeth  103 , bendable flap  108 C and arcuate member  109 C may deflect in the lingual-facial direction as bendable flap  108 C and arcuate member  109 C deforms to accommodate tooth  103 C. The deflection may cause stress near first and second ends  114 C and  112 C of hinge axis  110 C, and/or where arcuate member  109 C is coupled to shell  104 C and bendable flap  108 C. To reduce stress caused by deflection of bendable flap  108 C and/or arcuate member  109 C, appliance body  102  may define stress concentration reduction regions. The circular stress concentration reduction regions may include a diameter that is at least greater than a width of flap boundary region  113 C. As bendable flap  108 C and arcuate member  109 C deflect, stress may be distributed around the circular stress concentration reduction region to reduce localized concentration of stress that may otherwise tear appliance body  102  or cause wearing of appliance body  102 . Reducing localized concentration of stress may reduce wear on appliance body  102  and increase the useable life of removable dental appliance  100 . 
     By allowing deflection of bendable flap  108 C in the lingual-facial direction, bendable flap  108 C and arcuate member  109 C may be configured to apply force  107 C to a side of tooth  103 C opposite from void  110 C to cause movement of tooth  103 C toward void  110 C. For example, bendable flap  108 C may be configured to intrude into a space defined by the desired position of tooth  103 C when bendable flap  108 C is in a rest position. In some examples, the desired position of tooth  103 C is a position after tooth  103 C contacts at least a portion of the surface of appliance body  102  defining void  110 C internal to shell  104 C. As shown in  FIG. 1E , bendable flap  108 C intrudes into the space defined by tooth  103 C. By intruding into the space defined by tooth  103 C in the desired position, bendable flap  108 C and arcuate member  109 C may apply force  107 C to tooth  103 C through the movement of tooth  103 C into void  110 C. For example, as illustrated in  FIG. 1C , bendable flap  108 C and arcuate member  109 C may apply force  107 C to tooth  103 C when tooth  103 C is in an initial position. As seen in  FIG. 1D , bendable flap  108 C and arcuate member  109 C applies force  107 C to tooth  103 C when tooth  103 C is in the desired position. When tooth  103 C is in the desired position, force  107 C may be greater than a minimum force to cause alveolar bone remodeling. In this way, removable dental appliance  100  may achieve complete expression of tooth  103 C through void  110 C to a position of substantial conformity with surface  111 . 
     In some examples, appliance body  102  may include gingival regions  106 A,  106 B,  106 C, and  106 D (collectively, “gingival regions  106 ”) that overlap at least a portion of the gingiva (e.g., gingival margins) of the patient. For example, gingival regions may extend around the gingival portion of shells  104 , where teeth  103  meets the gingiva. Gingival regions  106  may be configured to use at least a portion of the gingiva, the alveolar process, or both for anchorage. For example, when worn by the patient, gingival regions  106  may at least partially contact the gingiva to access additional bracing provided by gingival regions  106  indirectly engaging with the alveolar process without impeding mobility of teeth  103 . Additionally, or alternatively, by increasing an extent of shells  104  with gingival regions  106 , greater force may be applied to a selected tooth of teeth  103  while using the more rigid alveolar process as an anchor instead of neighboring teeth. As such, gingival regions  106  may allow better control of tooth movements relative to a fixed reference (the alveolar process), without causing unwanted reactionary movements of neighboring teeth. In some examples, appliance body  102  may exclude gingival regions  106 . 
     In some examples, appliance body  102  may include a unitary material, e.g., a single, uniform material. The unitary material may include a single polymer, or homogeneous mixture of one or more polymers. For example, removable dental appliance  100  may consist of a single, continuous 3D printed or thermoformed component. In other examples, appliance body  102  may include a multi-layer material. Multi-layer materials may enable one or more portions of appliance body  102  to be formed with a plurality of layers having different elastic modulus to enable selection of force characteristics, displacement characteristics, or both of bendable flap  108 C. The multi-layer material may include multiple layers of a single material, e.g., a single polymer, or multiple layers of a plurality of materials, e.g., two or more polymers, a polymer and another material. For example, removable dental appliance  100  may consist of a multilayer 3D printed or thermoformed component. Suitable polymers may include, but are not limited to, (meth)acrylate polymer; epoxy; silicones; polyesters; polyurethanes; polycarbonate; thiol-ene polymers; acrylate polymers such as urethane (meth)acrylate polymers, polyalkylene oxide di(meth)acrylate, alkane diol di(meth)acrylate, aliphatic (meth)acrylates, silicone (meth)acrylate; polyethylene terephthalate based polymers such as polyethylene terephthalate glycol (PETG); polypropylene; ethylene-vinyl acetate; or the like. The thickness of appliance body  102  may range between about 0.10 millimeters and about 2.0 millimeters, such as between about 0.2 and about 1.0 millimeters, or between about 0.3 millimeters and about 0.75 millimeters. In the same or different examples, removable dental appliance  100  may include chamfers or fillets on edges of appliance body  102  and other spaces. Such chamfers or fillets may improve patient comfort and reduce the visibility of removable dental appliance  100 . In the same or different examples, removable dental appliance  100  may include at least one reinforcement structure to increase the stiffness of an area of appliance body  102  (e.g., bendable flap  108 C or arcuate member  109 C) to increase the strength of an area of appliance body  102  (e.g., hinge axis  110 C). 
     In some examples, removable dental appliance  100  may include metallic components configured to enhance forces applied by removable dental appliance  100  to one or more of the surrounded teeth. For example, the metallic component may comprise a metal wire having any suitable cross sectional shape (e.g., circular, rectilinear, or a ribbon) extending through at least a portion of appliance body  102 , such as bendable flap  108 C or arcuate member  109 C. In some examples, removable dental appliance  100  may include one or more other metal components, such as metal occlusal components, where greater durability is needed to overcome the stress of high-pressure occlusal contact, such as caused by bruxing, or mastication. In some examples, removable dental appliance  100  may include catches to connect to an anchorage device implanted within the patient, e.g., a temporary anchorage device or mini-screw. For example, catches may be positioned on anchor shells to connect to an anchorage device on anchor teeth. In this manner, such removable dental appliances  100  may provide a hybrid construction of metal and plastic. While plastic components may be generally clear for reduced visibility, metal components may include plating or other coloring to reduce visibility of removable dental appliance  100  when worn by the patient. For example, metal components positioned near teeth  103  of a patient when worn may include white colored coating or plating, such as, for example, rhodium, silver, white anodized titanium, Teflon, PTFE, and the like, or be formed of a white colored metal, such as, for example, rhodium, silver, white anodized titanium, and the like. Metal components positioned elsewhere may be colored to generally match tissue color within the mouth of the patient. 
     In some examples, a respective bendable flap or plurality of bendable flaps may define a spiral configuration.  FIGS. 2A-2C  are conceptual diagrams illustrating an example removable dental appliance  2000  that includes bendable flaps  2008  having a spiral configuration. Removable dental appliance  2000  may be the same as or substantially similar to removable dental appliance  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     The spiral configuration of bendable flaps  2008  may enable a bendable flap (or a plurality of bendable flaps) to apply a force near a center of the spiral and distribute a corresponding deformation around a perimeter of the spiral. For example, as illustrated in  FIG. 2A , appliance body  2002  may include two bendable flaps  2008 A defining a single flap boundary region  2013 A defining a double spiral configuration. When in a rest position, centers  2015 A of bendable flaps  2008 A may intrude into a space defined by a desired position of a tooth. In a deformed position, when worn by a patient, bendable flaps  2008 A may deform to concentrate a deformation force on the tooth near centers  2015 A. The deformation force of bendable flaps  2008 A may be transferred to appliance body  2002  around a perimeter  2010  of the spiral configuration. In some examples, as illustrated in  FIG. 2B , the spiral configuration may include a quadruple spiral configuration having a plurality of bendable flaps  2008 C,  2008 D,  2008 E,  2008 F. In some examples, removable dental appliance  2000  may include arcuate members as discussed above in reference to  FIGS. 1A-1E . 
     In some examples, a respective bendable flap may define slotted edges, have thinner material along a hinge axis, and thinner material defining an arcuate member opposite the hinge axis.  FIGS. 3A-3C  are conceptual diagrams illustrating an example removable dental appliance  2100  that includes a bendable flap  2108  extending from a hinge axis  2110  and a spring bellows  2109  opposite hinge axis  2110 . Removable dental appliance  2100  may be the same as or substantially similar to removable dental appliance  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     As illustrated in  FIGS. 3A-3C , bendable flap  2108  extends from appliance body  2102  at hinge axis  2110 . Hinge axis  2110  may include voids  2117  defining a thinner region of appliance body  2102 . In some examples, voids  2117  may include a series of notches to form a living hinge that may act more a compliant mechanism than an energy storage member. Bendable flap  2108  defines slotted sides  2113 . Slotted sides  2113  may include apertures extending through appliance body  2102 . In other examples, slotted sides  2113  may include any suitable type of area of reduced shear resistance compared to adjacent portions of appliance body  2102 . Appliance body  2102  includes arcuate member  2109  opposite hinge axis  2110 . As illustrated in  FIG. 3B , arcuate member  2109  may define a displacement of appliance body  2102  away from a plane tangent to the surface of bendable flap  2108 . As illustrated in the cross-sectional view of  FIG. 3C , a thickness of arcuate member  2109  and hinge axis  2110  may be substantially less than a thickness of other portions of appliance body  2102 , including bendable flap  2108  and shell  2104 . In some examples, arcuate member  2109  may act as a spring and store potential energy in addition to the energy stored at the hinge axis  2110 . By combining the combining the Modulus of Elasticity of bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109 , a composite modulus of elasticity may closely approximates a horizontal line at about 10N than any of bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  could achieve independently. In some examples, bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  may be simpler and easier to engineer and manufacture compared to, for example, a continuous spring bellows surrounding a U-shaped bendable flap. For example, it may be easier to model and compute the forces in bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  compared to, for example, a continuous spring bellows surrounding a U-shaped bendable flap. Such simplified modeling may reduce computational intensity or time when determining positions, dimensions, and shapes of bendable flaps and arcuate member to provide selected forces to teeth to achieve a selected treatment plan. Additionally, or alternatively, bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  may be simpler to machine because only linear cut paths may be needed. In some examples, bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  may be mass produced as premanufactured parts and attached later to a formed appliance body  2102 . In such example, bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  may be formed using continuous linear extrusion of material and cutting bendable flap  2108 , hinge axis  2110 , and/or arcuate member  2109  into individual parts of any given width. In some examples, at least a portion of appliance body  2102  adjacent bendable flap  2108  may define voids (e.g., slots  2013  may be enlarged). In some examples, by not transferring force directly to appliance body  2108  adjacent bendable flap  2108  deformation of appliance body  2108  may be reduced when such a concern is realized or in example where the close proximity of neighboring structures might be such that appliance body  2108  fit to the teeth may be compromised due to such deformation. 
     In some examples, an appliance body may include a bendable flap extending from a hinge axis including slots and defining slotted edges, and arcuate member opposite the hinge axis that includes zigzag springs in a plane tangential to a surface of the appliance body.  FIGS. 4A-4C  are conceptual diagrams illustrating an example removable dental appliance  2200  that includes a bendable flap  2208  extending from a slotted hinge axis  2210  and jumpers  2209  in a plane tangential to a surface of appliance body  2202  opposite hinge axis  2210 . Removable dental appliance  2200  may be the same as or substantially similar to removable dental appliances  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     As illustrated in  FIGS. 4A-4C , bendable flap  2208  extends from appliance body  2202  at slotted hinge axis  2210 . As discussed above, slotted hinge axis  2210  may have an increased flexibility compared to a hinge axis that is not slotted. Bendable flap  2208  defines slotted sides  2213 . Slotted sides  2213  may include apertures extending through appliance body  2202 . In other examples, slotted sides  2213  may include any suitable type of area of reduced shear resistance compared to adjacent portions of appliance body  2202 . Appliance body  2202  includes arcuate members  2209  opposite hinge axis  2210 . Arcuate members  2209  include zigzag shaped springs in a plane tangential to a surface of appliance body  2202 . In some examples, arcuate members  2209  may enable bendable flap  2208  to move in a direction perpendicular to slotted hinge axis  2210  and in a lingual-labial direction. The movement of bendable flap  2208  may, in some examples, improve expression during movement of a respective tooth. In some examples, the configuration illustrated in  FIGS. 4A-4C  may be effective at isolating the shell  2204  from reaction forces. In some examples, bendable flap  2208 , hinge axis  2210 , and/or arcuate members  2209  may be simpler to machine because an end mill or laser cutter may be used to cut the features into appliance body  2202  after thermoforming appliance body  2202 . As such, the configuration illustrated in  FIGS. 4A-4C  may be suited to design constraints and methods of manufacture that call for an appliance of substantially constant thickness. In some examples, by removing material along hinge axis  2210 , deformation of bendable flap  2208  may be reduced to increase contact area with a tooth or allow for more predictable contact points. In some examples, the configuration illustrated in  FIGS. 4A-4C  may be more comfortable for the patient by protruding less into the direction of the tongue, lips, or cheeks. Many variations of the arcuate members  2209  are possible, such as, for example, one or more jumpers or variation in the amplitude, width, length, attachment points, or the like of the one or more jumpers. 
     In some examples, an appliance body may include a bendable flap extending from a hinge axis including slots and a plurality of arcuate members bridging a flap boundary region.  FIGS. 5A and 5B  are conceptual diagrams illustrating an example removable dental appliance  2300  that includes a bendable flap  2308  extending from a slotted hinge axis  2310  and a plurality of jumpers  2309  bridging flap boundary region  2313  in a plane tangential to a surface of appliance body  2302  opposite hinge axis  2310 . Removable dental appliance  2300  may be the same as or substantially similar to removable dental appliances  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     As illustrated in  FIGS. 5A and 5B , bendable flap  2308  extends from appliance body  2302  at slotted hinge axis  2310 . Bendable flap  2308  defines flap boundary region  2213 . Appliance body  2302  includes a plurality of arcuate members  2309  extending from shell  2304  to bendable flap  2308 . Although six arcuate members  2309  are illustrated, in other examples, appliance body  2303  may include fewer or more arcuate members  2309 . Arcuate members  2309  include zigzag shaped springs in a plane tangential to a surface of appliance body  2302 . In some examples, arcuate members  2309  may enable bendable flap  2308  to move in the plane tangential to a surface of appliance body  2302  and in a lingual-labial direction. The movement of bendable flap  2208  may, in some examples, improve expression during movement of a respective tooth. Additionally, or alternately, the plurality of arcuate member  2309  may improve control of the direction or magnitude of a force applied by bendable flap  2308  to a surface of a tooth. The configuration illustrated in  FIGS. 5A and 5B  may enable increased force compared to other configuration by placing additional arcuate members  2309  along the sides of bendable flap  2308 . In some examples, arcuate members  2309  may be omitted from the distal end of bendable flap  2308  (e.g., the end most distant from the hinge axis) to allow bendable flap  2308  to be positioned more closely to neighboring dental structures or force actuators. In some examples, the configuration illustrated in  FIGS. 5A and 5B  may allow for more flexible bendable flap  2308  by decreasing the number of arcuate members  2309  and increasing the length of arcuate members  2309 . This is made possible by the increased length available when utilizing flap boundary region  2313  on the lateral sides of bendable flap  2308  in addition to the region at the distal end of bendable flap  2308 . 
     In some examples, an appliance body may include a bendable flap extending from a hinge axis and an arcuate member defining a spring bellows bridging at least a portion of a flap boundary region.  FIGS. 6A and 6B  are conceptual diagrams illustrating an example removable dental appliance  2400  that includes a bendable flap  2408  extending from a hinge axis  2410  and a spring bellows  2409  extending around an entire flap boundary region  2413 . Removable dental appliance  2400  may be the same as or substantially similar to removable dental appliance  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     As illustrated in  FIGS. 6A and 6B , bendable flap  2408  extends from appliance body  2402  at hinge axis  2410 . Hinge axis  2410  may include one or more voids  2417 . Appliance body  2402  includes arcuate member  2409  extending around an entire flap boundary region  2413 . As illustrated in  FIGS. 6A and 6B , arcuate member  2409  may define a displacement of appliance body  2402  away from a plane tangential to the surface of shell  2404 . As illustrated in the cross-sectional view of  FIG. 6A , a thickness of arcuate member  2409  and hinge axis  2410  may be substantially less than a thickness of other portions of appliance body  2402 , including bendable flap  2408  and shell  2404 . The relatively thinner arcuate member  2409  and hinge axis  2410  may be more flexible than the surrounding appliance body  2402  or bendable flap  2408 . Additionally, or alternatively, in some examples, one or more portions of arcuate member  2409  and hinge axis  2410  may include slots to reduce shear stress in selection regions. As illustrated in  FIG. 6B , arcuate member  2409  may include a tapered spring bellows. The tapered spring bellows may be disposed in a plane tangential to a surface of appliance body  2402  near hinge axis  2410  and protrude further from the plane at other positions around flap boundary region  2413 . In this way, portions of arcuate member  2409  nearer hinge axis  2410  may be less flexible than portions of arcuate member  2409  farther from hinge axis  2410 . In some examples, arcuate member  2409  may include a plurality of undulations toward and away from the plane tangential to a surface of appliance body  2402 . Such undulation may improve control of a direction and/or magnitude of force applied by arcuate member  2409  to bendable flap  2408 . The configuration illustrated in  FIGS. 6A and 6B  may increase patient comfort by eliminating exposed edges of appliance material, provide substantially greater forces than other examples by increasing the effective length of the spring bellows to include the lateral sides of the flap, and/or reduce buildup of food and plaque, compared to appliance bodies having voids defining arcuate members. In some examples, appliance body  2402  may include fillets or chamfers to improve patient comfort and/or reduce buildup of food or plaque in corners or inside edges of appliance body  2402 . In some examples, appliance body  2402  may be thermoformed without the need for post-processing, such as machining or cutting. In some examples, appliance body  2402  may be 3D printable without the need for support structures on or near bendable flap  2408 , depending on appliance orientation in the printer (because of the elimination of exposed edges that might lack localized support). 
     In some examples, an appliance body may include a bendable flap extending from a hinge axis and an arcuate members defining jumpers bridging at least a portion of a flap boundary region.  FIGS. 7A and 7B  are conceptual diagrams illustrating an example removable dental appliance  2500  that includes a bendable flap  2508  extending from a hinge axis  2510  and a plurality of jumpers  2509  bridging flap boundary region  2513 . Removable dental appliance  2500  may be the same as or substantially similar to removable dental appliance  100  discussed above in reference to  FIGS. 1A-1E , except for the differences describe herein. 
     As illustrated in  FIGS. 7A and 7B , bendable flap  2508  extends from appliance body  2502  at hinge axis  2510 . Hinge axis  2510  may include one or more voids  2517 . Appliance body  2502  includes jumpers  2509  bridging flap boundary region  2513 . As illustrated in  FIGS. 7A and 7B , jumpers  2509  may define a displacement of appliance body  2502  away from a plane tangential to the surface of shell  2504 . As illustrated in the cross-sectional view of  FIG. 7B , a thickness of jumpers  2509  and hinge axis  2510  may be substantially less than a thickness of other portions of appliance body  2502 , including bendable flap  2508  and shell  2504 . The relatively thinner jumpers  2509  and hinge axis  2510  may be more flexible than the surrounding appliance body  2502  or bendable flap  2508 . Additionally, or alternatively, in some examples, one or more portions of jumpers  2509  and hinge axis  2510  may include slots to reduce shear stress in selection regions. In some examples, the force may be lessened from by interrupting the continuity of jumpers  2509  with discrete through holes or shear reduction area, thus reducing the total area of jumpers  2509  without reducing thickness to the point of compromised durability, formability, printability, or the like. Jumpers  2509  can also be placed on the lateral sides only of bendable flap  2508  to reduce aspect ratio or length overall of bendable flap  2508 . In some examples, the voids defined by appliance body  2502  (e.g., jumpers  2509 ) may increase salivary flow around the teeth and through the appliance, which can be beneficial toward the flushing out of acid which can, given prolonged contact with the teeth, lead to demineralization of tooth enamel, white spot lesions, dental caries, gingivitis, or the like. 
     In general, a respective bendable flap and arcuate members may be integrally formed with a respective shell on any one of a lingual, facial, or occlusal surface of a respective appliance body.  FIG. 8  illustrates a lingual view of a portion of an example removable dental appliance  200  that includes a plurality of shells, one of which is labeled in  FIG. 8  as shell  204 , a bendable flap  208  and an arcuate member  209  configured to apply a force  207  to a lingual surface of a tooth  203  of a patient. Removable dental appliance  200  may be the same as or substantially similar to removable dental appliance  100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-7B , except for the differences described herein. Like removable dental appliance  100 , removable dental appliance  200  may include an appliance body  202  configured to at least partially surround a plurality of teeth of mandibular arch  201  of a patient. Appliance body  202  defines a shell  204  shaped to engage tooth  203  in an initial position of tooth  203  and also receive tooth  203  in a desired position. Bendable flap  208  is integrally formed with appliance body  202  to extend from a hinge axis  210  of shell  204 . Appliance body  202  also includes arcuate member  209  bridging flap boundary region  213 , which includes a first end  212  and a second end  214 . Appliance body  202  may or may not include gingival regions, similar to appliance body  102 . 
     In the example of  FIG. 8 , bendable flap  208  and arcuate member  209  are positioned on a lingual side of appliance body  202  and is configured to apply force  207  to a lingual surface of tooth  203 . For example, a rest position of bendable flap  208  intrudes into a space defined by tooth  203  in a desired position of tooth  203 . When removable dental appliance  200  is worn by the patient, an initial position of tooth  203  causes a deformation of bendable flap  208  and arcuate member  209 . The deformation of bendable flap  208  results in force  207 . Force  207  is transferred from bendable flap  208  to tooth  203  by one or more contact points of bendable flap  208  with tooth  203 , e.g., a surface of bendable flap  208  or protrusions from bendable flap  208 . In some examples, bendable flap  208  and arcuate member  209  are positioned and shaped to concentrate force  207  near the incisal edge of tooth  203 . By concentrating force  207  near the incisal edge of tooth  203 , bendable flap  208  and arcuate member  209  may result in a torqueing of tooth  203  about an axis of rotation  216  in direction of rotation  218 . Although not shown in  FIG. 8 , a surface of shell  204  may define a void internal to shell  204 . As tooth  203  is torqued about axis of rotation  216 , tooth  203  moves into the void and the facial surface of tooth  203  may contact the surface of shell  204 . 
     In some examples, a plurality of bendable flaps and a plurality of arcuate members may be on opposing sides of an appliance body.  FIGS. 9A-9C  illustrate labial, lingual, and occlusal views of an example removable dental appliance  300  that includes a plurality of shells, one of which is labeled in  FIGS. 9A-9C  as shell  304 , bendable flaps  308 A and  308 B (collectively, “bendable flaps  308 ”) and arcuate members  309 A and  309 B (collectively, “arcuate members  309 ”) configured to apply respective forces  307 A and  307 B (collectively, “forces  307 ”) to tooth  303  of a patient. Removable dental appliance  300  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-8 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  300  may include appliance body  302  configured to at least partially surround a plurality of teeth, one of which is labeled in  FIGS. 9A-9C  as tooth  303 , of the mandibular arch  301  of a patient. Appliance body  302  may or may not include gingival regions, similar to appliance body  102 . Shell  304  may be shaped to engage tooth  303  in an initial position of tooth  303  and receive tooth  303  in a desired position of tooth  303 . Appliance body  302  may define flap boundary region  313 A having first end  312 A and second end  314 A. Bendable flap  308 A may be integrally formed with appliance body  302  to extend from hinge axis  310 A of shell  304 . 
     In contrast to removable dental appliances  100  and  200 , removable dental appliance  300  additionally includes second bendable flap  308 B and second arcuate member  309 B. Like bendable flap  308 A, bendable flap  308 B may be integrally formed with appliance body  302  to extend from hinge axis  310 B of shell  304 , where appliance body  302  defines flap boundary region  313 B having first end  312 B and second end  314 B. As seen in  FIG. 9C , bendable flap  308 A is positioned on the facial side of removable dental appliance  300 , whereas bendable flap  308 B is positioned on the lingual side of removable dental appliance  300 . Bendable flap  308 A and arcuate member  309 A may be configured to apply force  307 A to a facial surface of tooth  303 . Bendable flap  308 B and arcuate member  309 B may be configured to apply force  307 B to a lingual surface of tooth  303 . As seen in  FIGS. 3A and 3B , forces  307 A and  307 B may be centered at about the same height of tooth  303  relative to a horizontal plane. In other examples, forces  307 A and  307 B may be centered at different heights of tooth  303 . 
     Bendable flaps  308 A and  308 B may be positioned to form a couple. For example, force  307 A may be substantially opposed to, and separated by a distance from, force  307 B. The couple of forces  307 A and  307 B may result in a rotation of tooth  303  about an axis  316  approximately centered in tooth  303  and extending in the occlusal-gingival direction. Although not shown in  FIGS. 9A-9C , a surface of shell  304  may define a void internal to shell  304  and shaped to receive tooth  303  in a desired position of tooth  303 . For example, bendable flap  308 A may be configured to apply force  307 A to a facial-distal surface of tooth  303  opposite from the void to cause movement of tooth  303  toward the void. Similarly, the surface of shell  304  may define a second void internal to shell  304  and shaped to receive tooth  303  in a desired position of tooth  303 . For example, bendable flap  308 B may be configured to apply force  307 B to a lingual-mesial surface of tooth  303  opposite from the second void to cause movement of tooth  303  toward the second void. In this way, bendable flaps  308  and shell  304  may be integrally formed to cause a movement of tooth  303  toward a desired position of tooth  303 . 
     In some examples, a plurality of bendable flaps and a plurality of arcuate members may be on the same side of an appliance body.  FIG. 10  illustrates a lingual view of an example removable dental appliance  400  that includes a plurality of shells, one of which is labeled in  FIG. 10  as shell  404 , a plurality of bendable flaps  408 A and  408 B (collectively, “bendable flaps  408 ”), and a plurality of arcuate members  409 A and  409 B (collectively, “arcuate members  409 ”) configured to apply respective forces  407 A and  407 B (collectively, “forces  407 ”) to tooth  403  of a patient. Removable dental appliance  400  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-9 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  400  may include appliance body  402  configured to at least partially surround plurality of teeth, one of which is labeled in  FIG. 10  as tooth  403 , of the mandibular arch  401  of a patient. Appliance body  402  may or may not include gingival regions, similar to appliance body  102 . Shell  404  may be shaped to engage tooth  403  in an initial position of tooth  403  and receive tooth  403  in a desired position of tooth  403 . Appliance body  402  may include bendable flap  408 A extending from hinge axis  410 A of shell  404  and arcuate member  409 A extending around flap boundary region  413 A from first end  412 A to second end  414 A. Additionally, like removable dental appliance  300 , removable dental appliance  400  may include second bendable flap  408 B extending from second hinge axis  410 B and second arcuate member  409 B extending around flap boundary region  413 B from first end  412 B to second end  414 B. 
     As illustrated in  FIG. 10 , bendable flaps  408 A and  408 B are positioned on the lingual side of removable dental appliance  400 . Bendable flap  408 A and arcuate member  409 A may be configured to apply force  407 A to a lingual surface near the incisal edge of tooth  403 . Whereas bendable flap  408 B and arcuate member  409 B may be configured to apply force  407 B to a lingual surface near the gingival margin of tooth  403 . Forces  407 A and  407 B may be concentrated at about the center of tooth  403  (e.g., the center of an axis extending in the mesial-distal direction across tooth  403 ). In other examples, forces  407 A and  407 B may be concentrated at different locations on tooth  403 . 
     Forces  407 A and  407 B may have similar or dissimilar magnitudes. In examples in which the magnitude of forces  407 A and  407 B are similar, tooth  403  may be translated in the facial direction. In examples in which the magnitude of forces  407 A and  407 B are dissimilar, tooth  403  may be both translated in the facial direction and tipped. For example, if force  407 A has a greater magnitude than force  407 B, tooth  403  may translate in the facial direction with an occlusal tipping in the facial direction. In some examples, dissimilar forces  407 A and  407 B may be used to reduce the moment of the resulting force on tooth  403 , e.g., by considering the center of resistance of tooth  403 . In this way, bendable flaps  408 A and  408 B may be configured to result in linear translation of tooth  403 . 
     In some examples, a plurality of bendable flaps and a plurality of arcuate members on the same side of an appliance body may be configured to concentrate a respective plurality of forces. For example,  FIG. 11  illustrates a lingual view of an example removable dental appliance  500  that includes a plurality of shells, one of which is labeled in  FIG. 11  as shell  504 , and a plurality of bendable flaps  508 A and  508 B and respective arcuate members  509 A and  509 B configured to apply respective forces  507 A and  507 B to tooth  503  of a patient. 
     Removable dental appliance  500  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-10 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  500  may include appliance body  502  configured to at least partially surround a plurality of teeth, including tooth  503 , of the mandibular arch  501  of a patient. Appliance body  502  may or may not include gingival regions. Shell  504  may be shaped to engage tooth  503  in an initial position of tooth  503  and receive tooth  503  in a desired position of tooth  503 . Appliance body  502  may define flap boundary region  513 A having first end  512 A and second end  514 A. Bendable flap  508 A may be integrally formed with appliance body  502  to extend from hinge axis  510 A of shell  504 . Additionally, like removable dental appliance  400 , removable dental appliance  500  may include second bendable flap  508 B integrally formed with appliance body  502  to extend from hinge axis  510 B of shell  504 , where appliance body  502  defines flap boundary region  513 B having first end  512 B and second end  514 B. Bendable flaps  508 A and  508 B may be positioned on the lingual side of removable dental appliance  500 . 
     As seen in  FIG. 11 , bendable flaps  508 A and  508 B and arcuate members  509 A and  509 B may be configured to apply forces  507 A and  507 B, respectively, to a lingual surface near the center of tooth  503 . Forces  507 A and  507 B may be concentrated near the center of tooth  503  (e.g., center of a mesial-distal axis). In other examples, forces  507 A and  507 B may be concentrated at different locations on tooth  503  along a mesial-distal axis. In some examples, arcuate members  509 A and  509 B may be connected along a common axis  515 . In other examples, at least one of arcuate members  509 A and  509 B may be coupled to both bendable flaps  508 A and  508 B along axis  515 . By concentrating both forces  507 A and  507 B on the same portion of tooth  503 , removable dental appliance  500  may transfer a greater magnitude of force to the portion of tooth  503  than may be possible with a single bendable flap. In this way, removable dental appliance  500  may achieve movement of tooth  503  in a shorter duration compared to a dental appliance without bendable flaps  508 A and  508 B. 
     In some examples, a plurality of bendable flaps may include four or more bendable flaps and a corresponding plurality of arcuate members on the same side of an appliance body. The plurality of bendable flaps and plurality arcuate members may be configured to concentrate a respective plurality of forces in one or more locations on a respective tooth. For example,  FIG. 12  illustrates a lingual view of an example removable dental appliance  600  that includes a plurality of shells, one of which is labeled in  FIG. 12  as shell  604 , a plurality of bendable flaps  608 A,  608 B,  608 C, and  608 D (collectively, “bendable flaps  608 ”) and a plurality of arcuate members  609 A,  609 B,  609 C, and  609 D (collectively, “arcuate members  609 ”) configured to apply forces  607 A,  607 B,  607 C, and  607 D to tooth  603  of a patient. 
     Removable dental appliance  600  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-11 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  600  may include appliance body  602  configured to at least partially surround a plurality of teeth, including tooth  603 , of the mandibular arch  601  of a patient. Appliance body  602  may or may not include gingival regions. Shell  604  may be shaped to engage tooth  603  in an initial position of tooth  603  and receive tooth  603  in a desired position of tooth  603 . Appliance body  602  may define flap boundary region  613 A having first end  612 A and second end  614 A. Bendable flap  608 A may be integrally formed with appliance body  602  to extend from hinge axis  610 A of shell  604 . Additionally, like removable dental appliance  500 , removable dental appliance  600  may include second bendable flap  608 B integrally formed with appliance body  602  to extend from hinge axis  610 B of shell  604 , where appliance body  602  defines flap boundary region  613 B having first end  612 B and having second end  612 A in common with bendable flap  608 A. 
     In contrast to removable dental appliance  500 , removable dental appliance  600  may also include third bendable flap  608 C, a third arcuate member  609 C, fourth bendable flap  608 D, and a fourth arcuate member  609 D. Bendable flaps  608  may be positioned on the lingual side of removable dental appliance  600 . As illustrated in  FIG. 12 , bendable flaps  608  and arcuate members  609  may be configured to apply forces  607 A,  607 B,  607 C, and  607 D (collectively, “force  607 ”) to a lingual surface near the center of tooth  603 . Force  607  may be centered near the center of tooth  603  (e.g., a center of tooth  603  along a mesial-distal axis across  603 ). In other examples, force  607  may be concentrated at different locations on tooth  603 . For example, respective bendable flaps  608  and arcuate members  609  may be configured to apply a respective force of forces  607  at any position on the respective bendable flap of bendable flaps  608 . By concentrating forces  607  on the same portion of tooth  603 , removable dental appliance  600  may transfer a greater magnitude of force to the portion of tooth  603  than may be possible with one or two bendable flaps. 
     By transferring a greater magnitude of force to tooth  603 , bendable flaps  608  and arcuate members  609  may cause a movement of tooth  603  in a shorter duration of time compared to fewer bendable flaps. For example, bendable flaps  608  may cause a translation of tooth  603  in a facial direction in a reduced amount of time compared to other dental appliances without bendable flaps  608  and/or and arcuate members  609 . In other examples, bendable flaps  608  and arcuate members  609  may be configured to produce other tooth movements, or combinations of tooth movements, such as, for example, at least one of rotation, translation, tipping, torqueing, extrusion, and intrusion. Alternatively, or additionally, the greater magnitude of force to tooth  603  may enable removable dental appliance  600  to cause a movement of tooth  603  that may not be possible with fewer bendable flaps, e.g., translation of a premolar. In this way, removable dental appliance  600  may achieve a movement of tooth  603  that requires a relatively higher magnitude of force applied to tooth  603 , a movement of tooth  603  in a shorter duration, or both compared to a dental appliance without bendable flaps  608 . 
     In some examples, one or more bendable flaps and arcuate members are configured to cause an intrusion of a respective tooth. For example,  FIGS. 13A-13F  illustrate occlusal, lingual, and distal cross-sectional views of an example removable dental appliance  700  that includes a plurality of shells, one of which is labeled in  FIG. 7  as shell  704 , a plurality of bendable flaps  708 A,  708 B,  708 C, and  708 D (collectively, “bendable flaps  708 ”) and a plurality of arcuate members  709 A,  709 B,  709 C, and  709 D (collectively, “arcuate members  709 ”) configured to apply respective forces  707 A,  707 B,  707 C, and  707 D (collectively, “forces  707 ”) to tooth  703  of a patent.  FIG. 7A  illustrates an occlusal view of a portion of removable dental appliance  700  surrounding tooth  703 , in which tooth  703  is in an initial position, e.g., a maloccluded position.  FIG. 7B  illustrates an occlusal view of a portion of removable dental appliance  700  surrounding tooth  703 , in which tooth  703  is in a desired position, e.g., a final position after orthodontic treatment or an intermediate position achieved by use of removable dental appliance  700 .  FIG. 7C  illustrates a lingual view of removable dental appliance  700  surrounding tooth  703 , in which tooth  703  is in an initial position.  FIG. 7D  illustrates a lingual view of a portion of removable dental appliance  700  surrounding tooth  703 , in which tooth  703  is in a desired position.  FIG. 7E  illustrates a cross-sectional view of removable dental appliance  700  surrounding tooth  703  in an initial position.  FIG. 7F  illustrates a cross-sectional view of a portion of removable dental appliance  700  surrounding tooth  703  in a desired position. 
     Removable dental appliance  700  is the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-12 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  700  includes appliance body  702  configured to at least partially surround a plurality of teeth, including tooth  703 , of the mandibular arch  701  of a patient. Appliance body  702  may or may not include gingival regions, similar to appliance body  102 . Shell  704  may be shaped to engage tooth  703  in an initial position of tooth  703  and receive tooth  703  in a desired position of tooth  703 . Additionally, like removable dental appliance  600 , the example of removable dental appliance  700  includes four bendable flaps  708  of arcuate members  709 . For example, appliance body  702  defines flap boundary regions  713 A,  713 B,  713 C, and  713 D (collectively, “flap boundary regions  713 ”) having first end  712 A, second end  714 A, third intersection  712 B, and fourth intersection  714 B. Each respective bendable flap of bendable flap  708  is integrally formed with appliance body  702  to extend from a respective hinge axis  710 A,  710 B,  710 C, and  710 D (collectively, “hinge axes  710 ”) of shell  704 . 
     As seen in  FIGS. 13A-13F , bendable flaps  708  are positioned on or near an occlusal plane of removable dental appliance  700 . Each respective bendable flap of bendable flaps  708  and respective arcuate members  709  are configured to apply a respective force of forces  707  to an occlusal surface of tooth  703 . Forces  707  may be distributed substantially evenly across the occlusal surface of tooth  703  or concentrated in one or more portions of the occlusal surface of tooth  703 . In examples in which forces  707  are evenly distributed, forces  707  may cause an intrusion of tooth  703 . In other examples, forces  707  may be concentrated in one or more areas of tooth  703  to cause other movements in addition to intrusion, such as, tipping. 
     As shown in  FIG. 7E , bendable flaps  708  are shaped to engage tooth  703  in an initial position of tooth  703 . For example, when the removable dental appliance is worn by the patient, bendable flaps  708  and arcuate members  709  deform to an initial deformed position. Forces  707  are the greatest when bendable flaps  708  and arcuate members  709  are in the initial deformed position. In some examples, surface  711  of shell  704  corresponds to the shape of tooth  703  when bendable flaps  708  are in the deformed position. By corresponding to the shape of tooth  703  in the deformed position, bendable flaps  708  engage a greater portion of tooth  703 . In this way, when the magnitude of force  707  is the greatest, control of the direction of forces  707  is the greatest. As tooth  703  moves in response to forces  707 , bendable flaps  708  may become progressively less engaged with tooth  703  and forces  707  progressively diminish. For example, as illustrated in  FIG. 7F , when tooth  703  reaches a desired position, void  710  is formed between surface  711  and tooth  703 . Movement of tooth  703  continues until force  707  is insufficient to cause alveolar bone remodeling. For example, forces  707  may be insufficient to cause alveolar bone remodeling when bendable flaps  708  move to a position where one or more of flap boundary regions  709  converge to contact one another. 
     In other examples, surface  711  may correspond to the shape of tooth  703  in the desired position of tooth  703 . For example, bendable flaps  708  may be in a state of lesser engagement with tooth  703  when tooth  703  is in an initial position. As tooth  703  moves in response to forces  707 , the engagement of bendable flaps  708  may increase. In some examples, couples may be formed between tooth  703  and surface  711  such that tooth  703  moves in one or more translations or rotations as bendable flaps  708  progressively engage with tooth  703 . 
     In some examples, one or more bendable flaps and one or more arcuate members are configured to cause an extrusion of a respective tooth. For example,  FIG. 14  illustrates a buccal view of a portion of an example removable dental appliance  800  that includes a plurality of shells, one of which is labeled in  FIG. 14  as shell  804 , a plurality of bendable flaps  808 A,  808 B, and  808 C (collectively, “bendable flaps  808 ”) and a plurality of arcuate members  809 A,  809 B, and  809 C (collectively, “arcuate members  809 ”) configured to apply respective forces  807 A,  807 B, and  807 C (collectively, “forces  807 ”) to tooth  803  of a patent. 
     Removable dental appliance  800  is the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-13 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  800  includes an appliance body  802  configured to at least partially surround a plurality of teeth, including tooth  803 , of the mandibular arch  801  of a patient. Appliance body  802  may or may not include gingival regions. Shell  804  may be shaped to engage tooth  803  in an initial position of tooth  803  and receive tooth  803  in a desired position of tooth  803 . Additionally, like removable dental appliance  400 , the example of removable dental appliance  800  includes a plurality of bendable flaps  808  and a plurality of arcuate member  809 . Appliance body  802  defines flap boundary region  813 A having first end  812 A and second end  814 A, flap boundary region  813 B having first end  812 B and second end  814 B, and flap boundary region  813 C having first end  812 C and second end  814 C. Each respective bendable flap of bendable flaps  808  integrally formed with appliance body  802  to extend from a respective hinge axis  810 A,  810 B, and  810 C (collectively, “hinge axes  810 ”) of shell  804 . 
     As seen in  FIG. 14 , bendable flaps  808  are positioned near a gingival margin of tooth  803 . Each respective bendable flap of bendable flaps  808  and arcuate members  809  are configured to apply a respective force of forces  807  to surface of tooth  803  below the height of contour of tooth  803 . The height of contour of tooth  803  is the greatest amount of convexity or bulge of the crown. Bendable flaps  808  may be distributed substantially evenly across the facial and lingual portions of shell  804  such that forces  807  are distributed substantially evenly across both the lingual and facial sides of tooth  803 . For example, although not shown in  FIG. 14 , a respective bendable flap positioned on a facial side of a respective shell may have a corresponding bendable flap positioned on a lingual side of the respective shell. In this way, forces  807  may be configured to extrude tooth  803 , rather than merely translating or rotating tooth  803 . In other examples, bendable flaps  808  and arcuate members  809  may be configured to cause other movements in addition to extrusion, such as, tipping, torqueing, translating, or rotating. 
     In some examples, one or more bendable flaps and arcuate members may be configured to cause a translation of a respective tooth. For example,  FIGS. 15A and 15B  illustrate buccal and occlusal views of an example removable dental appliance  900  that includes a plurality of shells, one of which is labeled in  FIGS. 15A and 15B  as shell  904 , a plurality of arcuate members  909 A and  909 B (collectively, “arcuate members  909 ”) and a plurality of bendable flaps  908 A and  908 B (collectively, “bendable flaps  908 ”) configured to apply respective forces  907 A and  907 B (collectively, “forces  907 ”) to tooth  903  of a patient.  FIG. 15A  illustrates a buccal view of a portion of removable dental appliance  900  surrounding tooth  903 , in which tooth  903  is in an initial position, e.g., a maloccluded position.  FIG. 15B  illustrates an occlusal view of a portion of removable dental appliance  900  surrounding tooth  903 , in which tooth  903  is in an initial position. 
     Removable dental appliance  900  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-14 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  900  includes appliance body  902  configured to at least partially surround a plurality of teeth, including tooth  903 , of the mandibular arch  901  of a patient. Appliance body  902  may or may not include gingival regions. Shell  904  may be shaped to engage tooth  903  in an initial position of tooth  903  and receive tooth  903  in a desired position of tooth  903 . Additionally, like removable dental appliance  300 , the example of removable dental appliance  900  includes two bendable flaps  908 . For example, appliance body  902  defines flap boundary region  913 A having first end  912 A and second end  914 A, and flap boundary region  913 B having first end  912 B and second end  914 B. Each respective bendable flap of bendable flap  908  integrally formed with appliance body  902  extends from a respective hinge axis  910 A and  910 B (collectively, “hinge axes  910 ”) of shell  904 B. 
     As seen in  FIGS. 15A and 15B , bendable flaps  908  are positioned on opposing facial and lingual sides of removable dental appliance  900  near an interproximal region between shell  904  and the adjacent shell. Each respective bendable flap of bendable flaps  908  and arcuate members  909  are configured to apply a respective force of forces  907  to a surface of tooth  903  near the interproximal region of tooth  903  and the adjacent tooth. The magnitude of force  907 A may be substantially similar to the magnitude of force  907 B to cause a distal translation of tooth  903 . In other examples, forces  907  may have dissimilar magnitudes or facially or lingually biased positions to cause a distal translation and a translation in either a facial or lingual direction, i.e., a translation along an angled axis relative to one of the ordinal tooth axes. 
     In some examples, at least one portion of an appliance body includes a reinforcing structure. The reinforcing structure may be configured to increase the stiffness of at least one bendable flap and/or arcuate member. As one example,  FIG. 16  illustrates a lingual view of a portion of an example removable dental appliance  1000  that includes a plurality of shells, one of which is labeled in  FIG. 16  as shell  1004 , and at least one bendable flap  1008  configured to apply force  1007  to tooth  1003  of a patient. 
     Removable dental appliance  1000  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-15 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  1000  includes an appliance body  1002  configured to at least partially surround a plurality of teeth, including tooth  1003 , of the mandibular arch  1001  of a patient. Shell  1004  is shaped to engage tooth  1003  in an initial position of tooth  1003  and receive tooth  1003  in a desired position of tooth  1003 . Additionally, like removable dental appliance  200 , removable dental appliance  1000  includes bendable flap  1008  positioned on a lingual side of appliance body  1002 . Appliance body  1002  defines flap boundary region  1009  having first end  1012  and second end  1014 . Bendable flap  1008  is integrally formed with appliance body  1002  extends from a hinge axis  1010  of shell  1004 . Appliance body  1002  may or may not include a gingival region. 
     As seen in  FIG. 16 , appliance body  1002  may include a reinforcing structure  1005 . Reinforcing structure  1005  is positioned on bendable flap  1008 . In other examples, the reinforcing structure may be positioned adjacent bendable flap  1008 , such as on arcuate member  1009 . Reinforcing structure  1005  is configured to increase the stiffness of at least a portion of appliance body  1002 , such as bendable flap  1008  in  FIG. 16 . For example, reinforcing structure  1005  includes a strip of additional material to increase the stiffness of bendable flap  1008  when, as discussed above, bendable flap  1008  is deformed when removable dental appliance  1000  is worn by the patient. In some examples, reinforcing structure  1005  may include a jumper, as discussed above. In the example of  FIG. 16 , the additional material forming reinforcing structure  1005  is the same material as appliance body  1002 . In other examples, the reinforcing structure may include at least one material having a higher elastic modulus than the appliance body material, such as, for example, a different polymer, a biocompatible metal, or the like. In some examples, one or more edges of reinforcing structure  1005  may be chamfered or filleted. Chamfering or filleting at least a portion of reinforcing structure  1005  may improve patient comfort. In this way, reinforcing structure  1005  may increase a stiffness of bendable flaps  1008  to increase a magnitude of the force provided by bendable flap  1008 , improve control of the force provided by bendable flap  1008  and the resulting movement of tooth  1003 , or both. 
     In some examples, rather than being on a bendable flap, a reinforcing structure may be positioned adjacent one or more bendable flaps. For example,  FIG. 17  illustrates a lingual view of a portion of an example removable dental appliance  1100  that includes a plurality of shells, one of which is labeled in  FIG. 17  as  1104 , bendable flaps  1108 A and  1108 B and arcuate members  1109 A and  1109 B configured to apply respective forces  1107 A and  1107 B to tooth  1103  of a patient. 
     Removable dental appliance  1100  may be the same as or substantially similar to removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  described above in reference to  FIGS. 1A-16 , except for the differences described herein. For example, like removable dental appliance  100 , removable dental appliance  1100  includes an appliance body  1102  configured to at least partially surround a plurality of teeth, such as tooth  1103 , of the mandibular arch  1101  of a patient. Shell  1104  is shaped to engage tooth  1103  in an initial position of tooth  1103  and receive tooth  1103  in a desired position of tooth  1103 . Additionally, like removable dental appliance  400 , removable dental appliance  1100  includes bendable flaps  1108 A and  1108 B (collectively, “bendable flaps  1108 ”) positioned on a lingual side of appliance body  1102 . Appliance body  1102  defines flap boundary region  1109 A having first end  1112 A and second end  1114 A, and bendable flap boundary region  1109 B having first end  1112 B and second end  1114 B. Bendable flaps  1108  are integrally formed with appliance body  1102  to extend from respective hinge axes  1110 A and  1110 B of shell  1104 . Appliance body  1102  may or may not include gingival regions, similar to appliance body  102 . Also, like removable dental appliance  1000 , appliance body  1102  includes a reinforcing structure  1105 . 
     Reinforcing structure  1105  is positioned adjacent bendable flaps  1108 . Reinforcing structure  1105  is configured to increase the stiffness of at least a portion of appliance body  1102 . For example, reinforcing structure  1105  includes a thickened region of shell  1104  or a strip of additional material on shell  1104  to increase the stiffness of a region of appliance body  1102  between first and second hinge axes  1110 A and  1110 B. When removable dental appliance  1100  is worn by the patient, bendable flaps  1108  deform. The deformation of bendable flaps  1108  may cause a deformation or stress in the region of appliance body  1102  between first and second hinge axes  1110 A and  1110 B. The additional material forming reinforcing structure  1105  may reduce the flexibility of the region of appliance body  1102  between first and second hinge axes  1110 A and  1110 B to resist this deformation and reduce the stress in the region of appliance body  1102  between first and second hinge axes  1110 A and  1110 B. In this way, reinforcing structure  1105  may improve concentration of deformation in bendable flaps  1108  to improve control of forces  1107  and the resulting movement of tooth  1103 . Additionally, or alternatively, the additional material forming reinforcing structure  1105  may improve the durability of at least a portion of appliance body  1102 . In this way, reinforcing structure  1105  may reduce the likelihood of permanent deformation or breakage of the at least a portion of appliance body  1102  when removable dental appliance  1100  is worn by the patient or fitted to the teeth. 
       FIG. 18  is a block diagram illustrating an example computer environment  10  in which clinic  14  and manufacturing facility  20  communicate information throughout a manufacturing process of a set of removable dental appliances  22  for patient  12 . The set of removable dental appliances  22  may include at least one of removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 . As discussed above, removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500  include a plurality of shells, at least one bendable flap, and at least one arcuate member. Initially, an orthodontic practitioner of clinic  14  generates one or more images of a dental anatomy of patient  12  using any suitable imaging technique and generates digital dental anatomy data  16  (e.g., a digital representation of patient&#39;s 12 tooth structure). For example, the practitioner may generate X-ray images that can be digitally scanned. Alternatively, the practitioner may capture digital images of the patient tooth structure using, for example, conventional computed tomography (CT), laser scanning, intra-oral scanning, CT scans of dental impressions, scans of dental casts poured from impressions, ultrasound instrumentation, magnetic resonance imaging (MRI), or any other suitable method of three-dimensional (3D) data acquisition. In other embodiments, the digital images may be provided using a hand-held intra-oral scanner such as the intra-oral scanner using active wavefront sampling developed by Brontes Technologies, Inc. (Lexington, Mass.) and described in PCT Publication No. WO 2007/084727 (Boerjes, et al.), which is incorporated herein by reference in its entirety. Alternatively, other intra-oral scanners or intra-oral contact probes may be used. As another option, the digital dental anatomy data  16  may be provided by scanning a negative impression of patient&#39;s 12 teeth. As still another option, the digital dental anatomy data  16  may be provided by imaging a positive physical model of patient&#39;s 12 teeth or by using a contact probe on a model of patient&#39;s 12 teeth. The model used for scanning may be made, for example, by casting an impression of patient&#39;s 12 dentition from a suitable impression material such as alginate or polyvinylsiloxane (PVS), pouring a casting material (such as orthodontic stone or epoxy resin) into the impression, and allowing the casting material to cure. Any suitable scanning technique may be used for scanning the model, including those described above. Other possible scanning methods are described in U.S. Patent Publication No. 2007/0031791 (Cinader et al.), which is incorporated herein by reference in its entirety. 
     In addition to providing digital images by scanning the exposed surfaces of the teeth, it is possible to image non-visible features of the dentition, such as the roots of patient&#39;s 12 teeth and patient&#39;s 12 jaw bones. In some embodiments, the digital dental anatomy data  16  is formed by providing several 3D images of these features and subsequently “stitching” them together. These different images need not be provided using the same imaging technique. For example, a digital image of teeth roots provided with a CT scan may be integrated with a digital image of the teeth crowns provided with an intraoral visible light scanner, for example, as describe in U.S. Patent Application No. 62/787,025, by Raby et al., which in incorporated herein by reference in its entirety. Scaling and registering of two-dimensional (2D) dental images with 3D dental images is described in U.S. Pat. No. 6,845,175 (Kopelman, et al.), which is incorporated herein by reference in its entirety, and U.S. Patent Publication No. 2004/0029068 (Badura, et al.), which is incorporated herein by reference in its entirety. Issued U.S. Pat. No. 7,027,642 (Imgrund, et al.), which is incorporated herein by reference in its entirety, and U.S. Pat. No. 7,234,937 (Sachdeva, et al.), which is incorporated herein by reference in its entirety, describe using techniques of integrating digital images provided from various 3D sources. Accordingly, the term “imaging” as it is used herein is not limited to normal photographic imaging of visually apparent structures, but includes imaging of dental anatomies that are hidden from view. The dental anatomy may include, but is not limited to, any portion of crowns or roots of one or more teeth of a dental arch, gingiva, periodontal ligaments, alveolar bone, cortical bone, implants, artificial crowns, bridges, veneers, dentures, orthodontic appliances, or any structure that could be considered part of the dentition before, during, or after treatment. 
     To generate digital dental anatomy data  16 , a computer must transform raw data from the imaging systems into usable digital models. For example, for raw data representing the shapes of teeth received by a computer, the raw data is often little more than a point cloud in 3D space. Typically, this point cloud is surfaced to create 3D object models of the patient&#39;s dentition, including one or more teeth, gingival tissue, and other surrounding oral structure. For this data to be useful in orthodontic diagnosis and treatment, the computer may “segment” dentition surfaces to produce one or more discrete, movable 3D tooth object models representing individual teeth. The computer may further separate these tooth models from the gingiva into separate objects. 
     Segmentation allows a user to characterize and manipulate the teeth arrangement as a set of individual objects. Advantageously, the computer may derive diagnostic information such as arch length, bite setting, interstitial spacing between adjacent teeth, and even American Board of Orthodontics (ABO) objective grading from these models. As a further benefit, the digital orthodontic setups may provide flexibility in the manufacturing process. By replacing physical processes with digital processes, the data acquisition step and data manipulation steps can be executed at separate locations without the need to transport stone models or impressions from one location to another. Reducing or eliminating the need for shipping physical objects back and forth can result in significant cost savings to both customers and manufacturers of customized appliances. 
     After generating digital dental anatomy data  16 , clinic  14  may store digital dental anatomy data  16  within a patient record in a database. Clinic  14  may, for example, update a local database having a plurality of patient records. Alternatively, clinic  14  may remotely update a central database (optionally within manufacturing facility  20 ) via network  24 . After digital dental anatomy data  16  is stored, clinic  14  electronically communicates digital dental anatomy data  16  to manufacturing facility  20 . Alternatively, manufacturing facility  20  may retrieve digital dental anatomy data  16  from the central database. Alternatively, manufacturing facility  20  may retrieve preexisting digital dental anatomy data  16  from a data source unassociated with clinic  14 . 
     Clinic  14  may also forward prescription data  18  conveying general information regarding a practitioner&#39;s diagnosis and treatment plan for patient  12  to manufacturing facility  20 . In some examples, prescription data  18  may be more specific. For example, digital dental anatomy data  16  may be a digital representation of the dental anatomy of patient  12 . The practitioner of clinic  14  may review the digital representation and indicate at least one of desired movements, spacing, or final positions of individual teeth of patient  12 . For example, the desired movements, spacing, and final positions of individual teeth of patient  12  may affect the forces to be applied to the teeth of patient  12  at each stage of treatment by each removable dental appliance of the set of removable dental appliances  22 . As discussed above, the forces applied by each removable dental appliance (e.g., removable dental appliance  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 ) of the set of removable dental appliances  22  may be determined by selecting the dimensions, shapes, and positions of at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), at least one bendable flap (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), at least one reinforcing structures (e.g., reinforcing structures  1005  or  1105 ), and the like. The at least one of desired movements, spacing, or final positions of individual teeth of patient  12  may enable the practitioner, a technician at manufacturing facility  20 , and a computer at manufacturing facility  20  to determine at least one of selected dimensions, shapes, and positions of at least one of the shells, bendable flaps, arcuate members, and reinforcing structures. In this way, digital dental anatomy data  16  may include at least one of practitioner, technician, or computer selected dimensions, shapes, and positions of at least one of shells, bendable flaps, arcuate members, and reinforcing structures of each of removable dental appliance of the set of removable dental appliances  22  to result in the desired movement of the teeth of patient  12 . Following review of the digital representation, the digital dental anatomy data  16  that includes the selected dimensions, shapes, and positions of shells, bendable flaps, arcuate members, and reinforcing structures of each removable dental appliance of the set of removable dental appliances  22 , may be forwarded to manufacturing facility  20 . Manufacturing facility  20  may be located off-site, or located with clinic  14 . 
     For example, each clinic  14  may include on-site equipment for manufacturing facility  20  such that a treatment plan and digital design may be performed entirely by a clinical practitioner, or an assistant, in the clinical setting, using software installed locally. The manufacturing may be performed in the clinic, as well, by using a 3D printer (or by other methods of additive manufacturing). A 3D printer allows manufacturing of intricate features of a dental appliance or a physical representation of the dental anatomy of patient  12  through additive printing. The 3D printer may use iterative digital designs of original dental anatomy of patient  12  as well as a desired dental anatomy of patient  12  to produce multiple digital appliances, digital appliance patterns customized to produce the desired dental anatomy of patient  12 , or both. Manufacturing may include post-processing to remove uncured resin and remove support structures, or to assemble various components, which may also be necessary and could also be performed in a clinical setting. 
     Manufacturing facility  20  utilizes digital dental anatomy data  16  of patient  12  to construct the set of removable dental appliances  22  to reposition teeth of patient  12 . Sometime thereafter, manufacturing facility  20  forwards the set of removable dental appliances  22  to clinic  14  or, alternatively, directly to patient  12 . For example, the set of removable dental appliances  22  may be an ordered set of removable dental appliances. Patient  12  then wears the removable dental appliances  22  in the set of removable dental appliances  22  sequentially over time according to a prescribed schedule to reposition the teeth of patient  12 . For example, patient  12  may wear each removable dental appliance in the set of removable dental appliances  22  for a period of between about 1 week and about 6 weeks, such as between about 2 weeks and about 4 weeks, or about 3 weeks. Optionally, patient  12  may return to clinic  14  for periodic monitoring of the progress of the treatment with removable dental appliances  22 . 
     During such periodic monitoring, a clinician may adjust the prescribed schedule of patient  12  for wearing the removable dental appliances in the set of removable dental appliances  22  sequentially over time. Monitoring generally includes visual inspection of the teeth of patient  12  and may also include imaging to generate digital dental anatomy data. In some examples, the clinician may decide to interrupt the treatment of patient  12  with the set of removable dental appliances  22 , for example, by sending the newly generated digital dental anatomy data  16  to manufacturing facility  20  in order to produce a new set of removable dental appliances  22 . In some examples, the clinician may send newly generated digital dental anatomy data  16  to manufacturing facility  20  following the completion of the prescribed schedule of the treatment with removable dental appliances  22 . In some examples, following the completion of the prescribed schedule of the treatment with removable dental appliances  22 , the clinician may request a new set of removable dental appliances from manufacturing facility  20  to continue treatment of patient  12 . 
       FIG. 19  is a flow diagram illustrating process  30  conducted at clinic  14  in accordance with one example of this disclosure. Initially, a practitioner at clinic  14  collects patient identity and other information from patient  12  and creates a patient record ( 32 ). As described above, the patient record may be located within clinic  14  and optionally configured to share data with a database within manufacturing facility  20 . Alternatively, or additionally, the patient record may be located within a database at manufacturing facility  20  that is remotely accessible to clinic  14  via network  24  or within a database that is remotely accessible by both manufacturing facility  20  and clinic  14 . 
     Next, digital dental anatomy data  16  of patient  12  may be generated using any suitable technique ( 34 ), to thereby create a virtual dental anatomy. Digital dental anatomy data  16  may be comprised of a two-dimensional (2D) image, a three-dimensional (3D) representation of the dental anatomy, or both. 
     In one example, 3D representations of a dental anatomy are generated using a cone beam computerized tomography (CBCT) scanner, such as an i-CAT 3D dental imaging device (available from Imaging Sciences International, LLC; 1910 N Penn Road, Hatfield, Pa.). Clinic  14  stores the 3D digital dental anatomy data  16  (in the form of radiological images) generated from the CBCT scanner in the database located within clinic  14 , or alternatively, within manufacturing facility  20 . The computing system processes the digital dental anatomy data  16  from the CBCT scanner, which may be in the form of a plurality of slices, to compute a digital representation of the tooth structure that may be manipulated within the 3D modeling environment. 
     If 2D radiological images are used ( 36 ), then the practitioner may further generate 3D digital data ( 38 ). The 3D digital dental anatomy data  16  may be produced by, for example, forming and subsequently digitally scanning a physical impression or casting of the tooth structure of patient  12 . For example, a physical impression or casting of a dental arch of patient  12  may be scanned using a visible light scanner, such as an OM- 3 R scanner (available from Laser Design, Inc. of Minneapolis, Minn.) or an ATOS scanner (available from GOM GmbH of Braunschweig, Germany). Alternatively, the practitioner may generate the 3D digital dental anatomy data  16  of the occlusal service by use of an intra-oral scan of the dental arch of patient  12 , or existing 3D tooth data. In one example, the method of forming a digital scan from a casting or an impression described in U.S. Pat. No. 8,491,306, by Raby et al., which is incorporated herein by reference in its entirety, may be used. In the same or different examples, techniques for defining a virtual tooth surface and virtual tooth coordinate system as described in U.S. Patent Application Publication No. 2013/0325431, by See et al., which is incorporated herein by reference in its entirety, may be used. In any case, the digital data are digitally registered within the 3D modeling environment to form a composite digital representation of a tooth structure, which may include the tooth roots as well as the occlusal surfaces. 
     In one example, 2D radiological images and the 3D digital data for the occlusal surface of the dental arch are registered by first attaching registration markers (e.g., fiducial markers or a pedestal having known geometry) to the tooth structure of patient  12  prior to generating both the radiological images and the 3D digital scan. Thereafter, the digital representation of the registration markers within the 2D radiological image and the 3D digital data may be aligned within a 3D modeling environment using registration techniques described in U.S. Pat. No. 8,491,306. 
     In another example, 3D digital data of the tooth structure is generated by combining two 3D digital representations of the tooth structure. For example, a first 3D digital representation may be a relatively low-resolution image of the roots obtained from a CBCT scanner (e.g., an i-CAT 3D dental imaging device) and the second 3D digital representation may be a relatively high-resolution image of the crowns of the teeth obtained from an industrial CT scan of an impression or a visible light (e.g., laser) scan of a casting of the dental arch of the patient. The 3D digital representations may be registered using a software program that enables the 3D representations to be manipulated within a computer environment (e.g., Geomagic Studio software (available from 3D Systems, Inc.; 333 Three D Systems Circle, Rock Hill, S.C.), or alternatively, registration techniques described in U.S. Pat. No. 8,491,306 may be used. 
     Next, a computer system executing 3D modeling software renders a resultant digital representation of the tooth structure, including the occlusal surface as well as the root structure of the patient&#39;s dental arch. Modeling software provides a user interface that allows the practitioner to manipulate digital representations of the teeth in 3D space relative to the digital representation of the patient&#39;s dental arch. By interacting with the computer system, the practitioner generates treatment information, such as by selecting indications of the desired positions, final positions, or both of individual teeth of patient  12 , duration of a respective stage of treatment, or number of treatment stages, the direction or magnitude of forces on the teeth of patient  12  during a stage of treatment, or the like ( 40 ). In some examples, bendable flaps may be used during at least one, but fewer than all stages of treatment. For example, the desired positions of individual teeth of patient  12 , duration of a respective stage of treatment, or number of treatment stages may affect the direction or magnitude of forces on the teeth of patient  12  at each stage of treatment by each removable dental appliance of the set of removable dental appliances  22 . As discussed above, the forces applied by each removable dental appliance (e.g., removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 ) of the set of removable dental appliances  22  may be determined by selecting the dimensions, shapes, and positions of at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), bendable flaps (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), reinforcing structures (e.g., reinforcing structures  1005  or  1105 ), and the like. In this way, updating the database with diagnostic and treatment information ( 40 ) may include determining or selecting by the practitioner, a technician, or automatically by a computer the dimensions, shapes, and positions of the plurality of shells, at least one bendable flap, at least one reinforcing structure, and the like of each of removable dental appliance of the set of removable dental appliances  22  to result in the desired movement of the teeth of patient  12 . 
     Once the practitioner has finished conveying general information regarding a diagnosis and treatment plan within the 3D environment, the computer system updates the database associated with the patient record to record the prescription data  18  conveying general information regarding a diagnosis and treatment plan as specified by the practitioner ( 42 ). Thereafter, the prescription data  18  is relayed to manufacturing facility  20  for manufacturing facility  20  to construct one or more removable dental appliances including at least one bendable flap, such as removable dental appliances  22  ( 44 ). 
     Although described with respect to an orthodontic practitioner located at an orthodontic clinic, one or more of the steps discussed with respect to  FIG. 19  may be performed by a remote user, such as a user located at manufacturing facility  20 . For example, the orthodontic practitioner may only send radiological image data and an impression or casting of the patient to manufacturing facility  20 , where a user interacts with a computer system to develop a treatment plan within a 3D modeling environment. Optionally, a digital representation of the treatment plan within the 3D modeling environment may then be transmitted to the orthodontic practitioner of clinic  14 , who may review the treatment plan and either send back his or her approval, or indicate desired changes. 
       FIG. 20  is a block diagram illustrating an example of a client computer  50  connected to manufacturing facility  20  via network  24 . In the illustrated example, client computer  50  provides an operating environment for modeling software  52 . Modeling software  52  presents a modeling environment for modeling and depicting the 3D representation of the teeth of patient  12 . In the illustrated example, modeling software  52  includes user interface  54 , alignment module  56 , and rendering engine  58 . 
     User interface  54  provides a graphical user interface (GUI) that visually displays the 3D representation of patient&#39;s 12 teeth. In addition, user interface  54  provides an interface for receiving input from practitioner  60  of clinic  14 , e.g., via a keyboard and a pointing device, a touchscreen, or the like, for manipulating patient&#39;s 12 teeth within the modeled dental arch. 
     Modeling software  52  may be accessible to manufacturing facility  20  via network interface  70 . Modeling software  52  interacts with database  62  to access a variety of data, such as treatment data  64 , 3D data  66  relating to the tooth structure of patient  12 , and patient data  68 . Database  62  may be represented in a variety of forms including data storage files, lookup tables, or a database management system (DBMS) executing on one or more database servers. The database management system may be a relational (RDBMS), hierarchical (HDBMS), multi-dimensional (MDBMS), object oriented (ODBMS or OODBMS) or object relational (ORDBMS) database management system. The data may, for example, be stored within a single relational database, such as SQL Server from Microsoft Corporation. Although illustrated as local to client computer  50 , database  62  may be located remote from the client computer  50  and coupled to the client computer  50  via a public or private network, e.g., network  24 . 
     Treatment data  64  describes diagnosis or repositioning information for the teeth of patient  12  selected by practitioner  60  and positioned within the 3D modeling environment. For example, treatment data  64  may include the dimensions, shapes, and positions of at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), at least one bendable flap (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), at least one reinforcing structure (e.g., reinforcing structures  1005  or  1105 ), and the like that may result in a selected magnitude and direction of force vectors to be applied to teeth of a patient (e.g., teeth  103 ) throughout the treatment plans. 
     Patient data  68  describes a set of one or more patients, e.g., patient  12 , associated with practitioner  60 . For example, patient data  68  specifies general information, such as a name, birth date, and a dental history, for each patient  12 . 
     Rendering engine  58  accesses and renders 3D data  66  to generate the 3D view presented to practitioner  60  by user interface  54 . More specifically, 3D data  66  includes information defining the 3D objects that represent each tooth (optionally including roots), and jaw bone within the 3D environment. Rendering engine  58  processes each object to render a 3D triangular mesh based on viewing perspective of practitioner  60  within the 3D environment. User interface  54  displays the rendered 3D triangular mesh to practitioner  60 , and allows practitioner  60  to change viewing perspectives and manipulate objects within the 3D environment. 
     U.S. Pat. No. 8,194,067, to Raby et al., which is incorporated by reference in its entirety, and U.S. Pat. No. 7,731,495, by Eisenberg, which is incorporated by reference in its entirety, describe other examples for computer systems and 3D modeling software having user interfaces that may be used with the techniques described herein. 
     Client computer  50  includes processor  72  and memory  74  to store and execute modeling software  52 . Memory  74  may represent any volatile or non-volatile storage elements. Examples include random access memory (RAM) such as synchronous dynamic random access memory (SDRAM), read-only memory (ROM), non-volatile random access memory (NVRAM), electrically erasable programmable read-only memory (EEPROM), and FLASH memory. Examples may also include non-volatile storage, such as a hard-disk, magnetic tape, a magnetic or optical data storage media, a compact disk (CD), a digital versatile disk (DVD), a Blu-ray disk, and a holographic data storage media. 
     Processor  72  represents one or more processors such as a general-purpose microprocessor, a specially designed processor, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a collection of discrete logic, or any type of processing device capable of executing the techniques described herein. In one example, memory  74  may store program instructions (e.g., software instructions) that are executed by processor  72  to carry out the techniques described herein. In other examples, the techniques may be executed by specifically programmed circuitry of processor  72 . In these or other ways, processor  72  may be configured to execute the techniques described herein. 
     Client computer  50  is configured to send a digital representation of a 3D tooth structure of a patient, and optionally, treatment data  64  and/or patient data  68  to computer  80  of manufacturing facility  20  via network  24 . Computer  80  includes user interface  82 . User interface  82  provides a GUI that visually displays the 3D representation of the digital model of teeth. In addition, user interface  82  provides an interface for receiving input from a user, e.g., via a keyboard and a pointing device, for manipulating teeth of a patient within the digital representation of the 3D tooth structure of the patient. 
     Computer  80  may further be configured to automatically determine dimensions and shapes of each removable dental appliance of a set of removable dental appliances  22 . The dimensions and shapes of removable dental appliance  22  may include a position, dimension, and shape (e.g., at least one of at least one position, at least one dimension, and at least one shape) of at least one of the plurality of shells, at least one bendable flap, at least one arcuate member, at least one reinforcing structure, and the like, such that removable dental appliance  22  is configured to reposition the one or more teeth from their initial positions to final positions when the removable dental appliance is worn by the patient. As discussed above with respect to  FIGS. 1-17 , the position, dimension, and shape of at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), at least one bendable flap (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), at least one reinforcing structure (e.g., reinforcing structures  1005  or  1105 ), and the like may affect the magnitude, direction, and length of expression of a force applied to the teeth when the removable dental appliance is worn by the patient. For example, the position, dimensions, and shape of a respective bendable flap and/or arcuate member may determine, at least in part, the magnitude, direction, and length of expression of the force resulting from a deformation of the bendable flap and/or arcuate member when the removable dental appliance is worn by the patient. The position, dimensions, and shape of the arcuate member and/or an optional reinforcing structure may concentrate deformation in selected regions of a respective bendable flap to control the direction of force applied to the teeth. Also, the position, dimensions, and shape of a respective shell of the plurality of shells may affect the location(s) of engagement of a respective shell with a respective tooth. The location(s) of engagement may affect the direction of the force applied to the respective tooth. Computer  80  may analyze at least one of the magnitude, direction, and length of expression of at least one force resulting from a deformation of the respective bendable flap and/or arcuate member when the removable dental appliance is worn by the patient to determine at least one of position, dimension, and shape of a respective shell, a respective bendable flap, a respective arcuate member, a respective reinforcing structure, or the like that will result in a desired movement of a respective tooth of a patient when the removable dental appliance is worn by the patient. 
     Computer  80  may present a representation of the removable dental appliance  22  for user to review, including review of dimensions and shapes. Alternatively, or additionally, computer  80  may accept input from a user to determine dimensions and shapes of a set of removable dental appliances  22  for patient  12 . For example, the user input may influence at least one of an automatically determined dimensions or shapes. Computer  80  may transmit, or otherwise send, a digital model of the set of removable dental appliance  22 , the dimensions and shapes of the set of removable dental appliances  22 , or both, to computer-aided manufacturing system  84  for production of the set of removable dental appliances  22 . 
     Client computer  50  and computer  80  are merely conceptual representations of an example computer system. In some examples, the functionalities described with respect to client computer  50 , computer  80 , or both may be combined into a single computing device or distributed among multiple computing devices within a computer system. For example, cloud computing may be used for digital design of dental appliances described herein. In one example, the digital representations of tooth structures are received at one computer at the clinic, while a different computer, such as computer  80 , is used to determine the shapes and dimensions of a removable dental appliance. In addition, it may not be necessary for that different computer, such as computer  80 , to receive all of the same data in order for it determine shapes and dimensions. Shapes and dimensions may be determined, at least in part, based on knowledge derived through analysis of historical cases or virtual models of exemplary cases, without receiving a complete 3D representation of the case in question. In such an example, data transmitted between client computer  50  and computer  80 , or otherwise utilized to design a custom dental appliance may be significantly less than the complete data set representing a complete digital dental model of a patient. 
       FIG. 21  is a block diagram illustrating an example computer-aided manufacturing system  1500  for construction of removable dental appliance  1522 . The example of computer-aided manufacturing system  1500  includes an additive manufacturing system  1502  in communication with computer  1504  and coupled to build material source  1510 . In some examples, computer-aided manufacturing system  1500  may include computer-aided manufacturing system  84  of  FIG. 20 . For example, computer  1504  may be the same as or substantially similar to computer  80 . Build material source  1510  includes a source of at least one polymeric material, such as, for example, at least one of the polymeric materials of appliance body  102  discussed above. Dental appliance  1522  may be the same as or substantially similar to at least one of removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 . In some examples, dental appliance  1522  includes one dental appliance of a set of dental appliances  22 . 
     Additive manufacturing system  1502  includes a moveable platform  1508  and an extrusion head  1506 . Movable platform  1508  and extrusion head  1506  are configured to manufacture dental appliance  1522 . For example, computer  1504  controls extrusion head  1506  and moveable platform  1508  to manufacture removable dental appliance  1522 . Controlling, by computer  1504 , extrusion head  1506  may include at least one of controlling a material feed rate from build material source  1510  to extrusion head  1506 , controlling a deposition rate of build material on dental appliance  1522 , controlling a temperature of extrusion head  1506 , and controlling a position of extrusion head  1506 . By controlling at least one of a material feed rate, a material deposition rate, a temperature of extrusion head  1506 , and a position of extrusion head  1510 , computer  1504  may control manufacture of a position, dimension, and shape of at least a portion of dental appliance  1522 . Controlling, by computer  1504 , movable platform  1508  may include at least one of controlling a translation of moveable platform in a plane normal to the direction of material deposition from extrusion head  1506  and controlling an elevation of moveable platform along an axis substantially parallel to the direction of material deposition from extrusion head  1506 . By controlling at least one of a translation and elevation of moveable platform  1508 , computer  1504  may control manufacture of a position, dimension, and shape of at least a portion of dental appliance  1522 . 
     Although  FIG. 21  illustrates a computer-aided manufacturing system  1500  configured for Fused Deposition Modeling (FDM), computer-aided manufacturing system  1500  may also be configured for stereolithography (SLA), inverse vat polymerization additive manufacturing, inkjet/polyjet additive manufacturing, or other methods of additive manufacturing. In examples in which computer-aided manufacturing system  1500  is configured for polyjet printing, computer-aided manufacturing system  1500  may be configured to print multiple materials in a single print, thereby allowing a high modulus material for the rigid components of dental appliance  1522  (e.g., shells) and a low modulus or elastomeric material for the less rigid components of dental appliance  1522  (e.g., bendable flaps and/or arcuate members). Further, with polyjet additive manufacturing, the modulus may be varied selectively across the dental appliance  1522 , and a different modulus may be used for the bendable flaps and/or arcuate members than is used for the shells, for different parts of a bendable flap and/or arcuate members, or for different parts of a shell, for example. Similarly, a different modulus may be used for the anchoring shells than is used for the shell used to reposition individual teeth. 
     Additionally, or alternatively, manufacturing a dental appliance may include thermoforming and using a femtosecond laser controlled by a multi-axis robot or CNC machine to cut away material, such as to form slots, hinges, and spring features. In some instances, cut depth may be controlled to selectively ablate material and reduce the thickness of the appliance in certain areas, such as to form a more flexible hinge axis or to increase the flexibility of a spring element. 
     Additionally, or alternatively, manufacturing a dental appliance may include forming at least a portion of the appliance, if not the entire appliance, by milling or otherwise machining the appliance from a solid block of material. 
     Additionally, or alternatively, manufacturing a dental appliance may include, especially where varying thickness or reinforcements are needed, thermoforming the appliance body, and using a multi-axis robot to dispense, via a heated extrusion nozzle, hot thermoplastic material onto an appliance of otherwise uniform thickness. This can serve to create structures of greater thickness in areas. In similar fashion, a photocurable resin may be dispensed onto the surface and light cured, either immediate after dispensing or after all features have been laid down. 
     Additionally, or alternatively, manufacturing a dental appliance may include, the prefabrication of bendable flaps and/or arcuate members. The prefabricated bendable flaps and/or arcuate members may include material such as stainless steel, titanium, or nickel titanium (NiTi), and bonded or fastened to the appliance body which is formed by other means, such as by thermoforming or 3D printing. The advantage in this approach is to allow for smaller structures having greater force delivery. In such cases, the computing device would be used to select from among a discrete set of premanufactured flaps the meet the required force and deflection criteria to achieve the prescribed movement, and place to determine the best position(s) for placement on each tooth. 
       FIG. 22  is a flow diagram illustrating process  1600  conducted at manufacturing facility  20  for construction of set of removable dental appliances  22 . In some examples, set of removable dental appliances  22  may include at least one of removable dental appliance  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 . Computer  80  at manufacturing facility  20  receives digital dental anatomy data  16  including initial positions of one or more teeth of the patient and prescription data  18  ( 1602 ) from clinic  14 . Alternatively, computer  80  may retrieve the information from a database located within or otherwise accessible by computer  80 . A trained user associated with computer  80  may interact with a computerized modeling environment running on computer  80  to develop a treatment plan relative to the digital representation of the patient&#39;s tooth structure and generate prescription data  18 , if clinic  14  has not already done so. In other examples, computer  80  may automatically develop a treatment plan based solely on the patient&#39;s tooth structure and predefined design constraints. 
     Once computer  80  receives patient&#39;s tooth structure, computer  80  determines dimensions and shapes of a removable dental appliance for the patient ( 1604 ). The dimensions and shapes of the removable dental appliance are configured to reposition the one or more teeth of the patient from their initial positions to desired positions when the removable dental appliance is worn by the patient. In the same or additional examples, computer  80  determines dimensions and shapes of set of removable dental appliances  22  for the patient configured to be worn in series. 
     In some examples, determining dimensions and shapes of the removable dental appliance includes selecting, with computer  80 , the dimensions and shapes of the removable dental appliance according to a set of predefined design constraints. The set of predesigned design constraints may include one or more factors, including, but not limited to, at least one of a minimum and a maximum localized force applied to one or more of the surrounded teeth, at least one of a minimum and a maximum rotational force applied to one or more of the surrounded teeth, at least one of a minimum and a maximum translational force applied to one or more of the surrounded teeth, at least one of a minimum and a maximum total force applied to one or more of the surrounded teeth, and at least one of a minimum and a maximum stress or strain applied to the removable dental appliance, when the removable dental appliance is worn by the patient and the surrounded teeth are in their initial positions. 
     Computer  80  may use finite element analysis (FEA) techniques to analyze forces on the teeth of a patient as well as the removable dental appliance during the determination of the dimensions and shapes of the removable dental appliance. For example, computer  80  may apply FEA to a solid model of the teeth of a patient as the modeled teeth move from their initial positions to their final positions representing a treatment including an ordered set of removable dental appliances. Computer  80  may use FEA to select the appropriate removable dental appliance to apply the desired forces on the teeth. In addition, computer  80  may use a virtual articulator to determine contact points between the teeth throughout the movement of the modeled teeth during the treatment. Computer  80  may further include occlusal contact forces, such as interdigitation forces, in the FEA forces analysis in combination with forces from the removable dental appliance during the design of dental appliances in an ordered set of removable dental appliances. Computer  80  may further determine an order in which teeth are to be moved to optimize the application of forces, reduce treatment time, improve patient comfort, or the like. 
     In some examples, determining dimensions and shapes of a removable dental appliance (e.g., removable dental appliance  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 ) includes selecting, with computer  80  thicknesses of the appliance body (e.g., appliance body  102 ,  202 ,  302 ,  402 ,  502 ,  602 ,  702 ,  802 ,  902 ,  1002 , and  1102 ), at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), at least one bendable flap (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), at least one reinforcing structures (e.g., reinforcing structures  1005  or  1105 ), or the like, to provide a stiffness suitable to reposition the one or more teeth of the patient from their initial positions to final positions when the removable dental appliance is worn by the patient. In some examples, the selected thickness may range between about 0.10 millimeters and about 2.0 millimeters, such as between about 0.2 and about 1.0 millimeters, or between about 0.3 millimeters and about 0.75 millimeters. In some examples, computer  80  may further select a material of the removable dental appliance according to the predefined design constraints. 
     The dimensions and shapes of a removable dental appliance for the patient may be presented to a user via user interface of  82  of computer  80  ( 1606 ). In examples in which dimensions and shapes of the removable dental appliance are presented to a user via user interface of  82 , the user may have the opportunity to adjust the design constraints or directly adjust the dimensions and shapes of removable dental appliance before the design data is sent to computer-aided manufacturing system  84 . In some examples, the dimensions and shapes of the removable dental appliance may be presented to a user by computer  80  directly as the removable dental appliance is manufactured by computer-aided manufacturing system  84 . For example, computer  80  may send a digital model of the removable dental appliance to computer-aided manufacturing system  84 , and computer-aided manufacturing system  84  manufactures removable dental appliance according to the digital model from computer  80 . 
     However, even in examples where the dimensions and shapes of a removable dental appliance for the patient may be presented to a user via user interface of  82  of computer  80 , following user approval, computer  80  sends a digital model of the removable dental appliance to computer-aided manufacturing system  84  ( 1608 ), and computer-aided manufacturing system  84  manufactures the removable dental appliance according to the digital model from computer  80  ( 1610 ). 
     In some examples, computer-aided manufacturing system  84  may include a 3D printer. Forming appliance body (e.g., appliance body  102 ,  202 ,  302 ,  402 ,  502 ,  602 ,  702 ,  802 ,  902 ,  1002 , and  1102 ) may include printing the surfaces of at least one of the plurality of shells (e.g., shells  104 ,  204 ,  304 ,  404 ,  504 ,  604 ,  704 ,  804 ,  904 ,  1004 ,  1104 ,  2004 ,  2104 ,  2204 ,  2304 ,  2404 , or  2504 ), at least one bendable flap (e.g., bendable flaps  108 C,  208 ,  308 A,  308 B,  408 A,  408 B,  508 A,  508 B,  608 A,  608 B,  608 C,  608 D,  708 A,  708 B,  708 C,  708 D,  808 A,  808 B,  808 C,  908 A,  908 B,  1008 ,  1008 A,  1108 B,  2008 ,  2108 ,  2208 ,  2408 , or  2508 ), at least one arcuate member (e.g., arcuate members  109 C,  209 ,  309 A,  309 B,  409 A,  409 B,  509 A,  509 B,  609 A,  609 B,  609 C,  609 D,  709 A,  709 B,  709 C,  709 D,  809 A,  809 B,  809 C,  909 A,  909 B,  1009 ,  1009 A,  1109 B,  2009 ,  2109 ,  2209 ,  2409 , or  2509 ), at least one reinforcing structures (e.g., reinforcing structures  1005  or  1105 ), or the like with the 3D printer. In other examples, forming appliance body may include printing representations of the teeth of a patient (e.g., teeth  103 ) with the 3D printer, thermoforming appliance body over the representations of the teeth of a patient, and trimming excess material (optionally automated by CNC or robotic machinery such as, e.g., end mill or LASER cutter) to form the plurality of shells, at least one bendable flap, at least one arcuate member, at least one reinforcing structure, and like. The representations of the teeth of a patient may include raised surfaces to facilitate forming at least one of the plurality of shells, the at least one bendable flap, at least one arcuate member, at least one reinforcing structure, and the like, in the thermoformed and trimmed appliance body. 
     The techniques of  FIG. 22  may be applied to design and manufacture of each of an ordered set of removable dental appliances  22 . For example, each removable dental appliance in the ordered set of removable dental appliances  22  may be configured to incrementally reposition the teeth of the patient. In this manner, the ordered set of removable dental appliances  22  may be configured to reposition the teeth of the patient to a greater degree than any one of the removable dental appliances within the set of the removable dental appliances  22 . Such an ordered set of removable dental appliances  22  may specifically be configured to incrementally reposition the one or more teeth of the patient from their initial positions to desired positions as the removable dental appliances of the ordered set of removable dental appliances  22  for the patient are worn sequentially by the patient. 
     In some examples, the techniques described with respect to  FIG. 22  may be embodied within a computer-readable storage medium, such as a computer-readable storage medium of computer  50 , computer  80 , or both. The computer-readable storage medium may store computer-executable instructions that, when executed, configure a processor to perform the techniques described with respect to  FIG. 22 . 
     Following the design of set of removable dental appliances  22 , manufacturing facility  20  fabricates set of removable dental appliances  22  in accordance with the digital dental anatomy data  16  and prescription data  18  ( 1610 ). Construction of removable dental appliances  22  may include 3D printing, thermoforming, injection molding, lost wax casting, 5-axis milling, laser cutting, hybrid plastic and metal manufacturing techniques, such as snap-fitting and overmolding, as well as other manufacturing techniques. 
       FIG. 23  is a flow diagram  1700  illustrating successive iterations of treatment using an ordered set of removable dental appliances. The ordered set of removable dental appliances is configured to reposition one or more teeth of a patient. In some examples, the ordered set of removable dental appliances may include at least one of removable dental appliances  100 ,  200 ,  300 ,  400 ,  500 ,  600 ,  700 ,  800 ,  900 ,  1000 ,  1100 ,  2000 ,  2100 ,  2200 ,  2300 ,  2400 , or  2500 . 
     Treatment begins with the first iteration of treatment ( 1702 ). At the beginning of the first iteration of treatment, the teeth of a patient are at their initial positions as represented by detention state X ( 1704 ). A scan of the teeth of a patient, for example, as described above with respect to  FIG. 18 , are taken to facilitate the design of the ordered set of removable dental appliances ( 1706 ). From the scan of teeth of a patient, a computer, e.g., computer  50 , determines at least one, such as two, different shapes and dimensions for removable dental appliances in the ordered set: first setup X a    1708 A and second setup X b    1708 B. Example techniques for creating a digital model of the teeth of a patient are described in U.S. Pat. No. 8,738,165 to Cinader, et al., which is incorporated herein by reference in its entirety. The computer may determine first setup X a    1708 A and second setup X b    1708 B by first adjusting the digital model of the teeth of a patient to create a model of the desired position of the teeth of a patient following the therapy. Then, the computer may create the shape and dimensions for removable dental appliances in the ordered set based on the time and forces required to move the teeth of a patient from the initial positions to their desired positions. For example, the computer model may adjust the thicknesses, positions, shapes, and dimensions of at least one of the plurality of shells, at least one bendable flap, at least one reinforcing structure, and the like of the removable dental appliances in the ordered set to produce the forces required to move the teeth of a patient from the initial positions to the desired positions. The modeled forces applied by removable dental appliances in the ordered set may further be based on the incremental positional movements of the teeth of a patient during the treatment. In this manner, the computer may design each of the removable dental appliances in the ordered set according to expected forces applied on the teeth in the predicted positions of the teeth at the time during the treatment the removable dental appliances in the ordered set is to be worn by the patient. 
     In some examples, at least one, such as three, different removable dental appliances in the set of removable dental appliances can be manufactured using each of first setup X a    1708 A and second setup X b    1708 B to produce at least two, such as six, removable dental appliances in the set of removable dental appliances. For example, first setup X a    1708 A may be used to manufacture first removable dental appliance (RDA) X a, SOFT    1710 A, second RDA X a, MEDIUM    1710 B, and third RDA X a, HARD    1710 C; and second setup X b    1708 B may be used to manufacture fourth RDA X b, SOFT    1710 D, fifth RDA X b, MEDIUM    1710 E, and sixth RDA X b, HARD    1710 F. First, second, and third RDAs  1710 A to  1710 C may be substantially the same shape and dimensions, but may comprise materials with different stiffness characteristics. For example, the second and third RDAs  1710 B and  1710 C may have higher stiffness characteristics than first RDA  1710 A, and third RDA  1710 C may have higher stiffness characteristics than second RDA  1710 B. Similarly, the fourth, fifth, and sixth RDAs  1710 D to  1710 F may be substantially the same shape and dimensions, but comprise materials with different stiffness characteristics. In some examples, first RDA  1710 A may have the same stiffness characteristics as the fourth RDA  1710 D, such as a relatively soft polymeric material. Similarly, second RDA  1710 B may have the same stiffness characteristics as the fifth RDA  1710 E, such as a relatively stiffer polymeric material than first RDA  1710 A. Likewise, third RDA  1710 C may have the same stiffness characteristics as the sixth RDA  1710 F, such as a relatively stiffer polymeric material than second RDA  1710 B. 
     RDAs  1710 A to  1710 F in the ordered set of removable dental appliances may be worn in sequence over time by the patient. For example, each of RDAs  1710 A to  1710 F in the ordered set of removable dental appliances may be worn between about 1 week and about 6 weeks, such as between about 2 weeks and about 4 weeks, or about 3 weeks. Following the treatment plan using RDAs  1710 A to  1710 F, the teeth of a patient may be at their final positions for the first iteration of treatment as represented by detention state X+1 ( 1712 ). 
     Once teeth of a patient are at or near dentition state X+1, the patient may return to the clinician who may evaluate the result of the first iteration of treatment ( 1714 ). If the first iteration of treatment has resulted in acceptable final positions of the teeth of a patient, then the treatment may be ended ( 1716 ). However, if the first iteration of treatment did not result in acceptable final positions of the teeth of a patient, one or more additional iterations of treatment may be performed. To begin the next iteration of treatment, the clinician may take another scan of the teeth of a patient to facilitate the design of a subsequent ordered set of removable dental appliances ( 1706 ). In some examples, evaluation of the result of the first iteration of treatment may include taking another scan of the teeth of a patient, in which case beginning the next iteration of treatment may simply involve forwarding the digital model of the teeth of a patient to a manufacturing facility so that another ordered set of removable dental appliances may be manufactured for the patient based on the new positions of the teeth of a patient. In yet other examples, the newly acquired scan may be used to create one or more iterations of removable dental appliances in the clinician&#39;s facility. 
     The techniques of  FIG. 23  represent one specific example, and a variety of modifications may be made to the techniques of  FIG. 23  within the spirit of this disclosure. For example, an ordered set of removable dental appliances may include more or less than six removable dental appliances. As another example, each removable dental appliance in the ordered set of removable dental appliances may have unique shapes and dimensions, and each removable dental appliance in the ordered set of removable dental appliances may be made of material having substantially the same or similar stiffness characteristics. 
     Various examples have been described. These and other examples are within the scope of the following claims.