Objective: The aim of this case series study is to present the rationality and scientific evidence of a new design for a Double and Triple abutment with their specific new concept of Biodynamic Optimized Peri-implant Tissue (BOPiT).

Methods: The innovative design of these abutments with a paraboloid geometry was based on BOPiT, simultaneously involving the principles of mechanobiology, biotensegrity, and mechanotransduction. Thus, individuals rehabilitated with different extents of edentulous spaces using the innovative double and triple abutments on osseointegrated dental implants are included in this case series. The double and triple abutments support 2 or 3 dental crowns on a single implant, respectively. Clinic and radiographic examinations are presented at T1 (loading after 4 months of the implant insertion surgeries with the appropriate metalloceramic prostheses) and T2 [final examination with a follow-up time ≥ 3 to 12 years (average of 7.2 years)].

Conception and design of the Double Abutment (DA) and Triple Abutment (TA): Scientific Rationality Traditional abutments are generally cylindrical nowadays, having two curved edges at the base and an edge that connects the two vertices on the side face (total of 3 edges) considering its strength as a unit, while the new DA and TA features a paraboloid geometric design along its entire length.

The paraboloid is a mathematical equation that generates quadric surfaces of two types: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval-shaped cup and can have a maximum or minimum point. In an appropriate coordinate system, with the three axes x, y and z, it can be represented by the equation: z/c=x2 /a2 + y2 /b2, where a and b are constants that determine the degree of curvature in the x-z and y-z planes, respectively (an elliptic paraboloid opens upwards). The hyperbolic paraboloid, or hypar, is a doubly ruled surface (composed of multiple lines, whose union forms the surface itself), usually saddle-shaped. Its coordinated system is also based on three axes, manifested in the equation: z/c = y2/b2 - x2/a2. For c>0, i.e., a hyperbolic paraboloid that opens downward along the x-axis and along the y-axis (i.e., the parabola in the x=0 plane is open upward and the parabola in the y=0 plane opens downward).

The design of DA (elliptical paraboloid) and TA (hyperbolic paraboloid) allows high stiffness with reduced bending stresses and equalization of forces. Additionally, TA hyperbolic paraboloid structures are the most complex and possess a unique combination of structural and architectural properties, resultant from its double curvature, concave convex and suited to carry the in-plane shear forces and transmit uniform eccentric axial forces to the other members of the system. Vectorial loads are minimized, preventing the traditional axis of transverse coordinates that imply early bone loss in the implant abutment connection area. These designs represent the interaction that exists between all biological phenomena and mathematics.

The new abutments are multiple mechanical load organizers, and support 2 (DA) or 3 (TA) crowns on each abutment, with an innovative geometric design that is totally passive (no welds), resulting from mathematical and physical conformations coupled in geometric vector models.

DA and TA are protected by the following patents: "Pilar geminado sobre implante osseointegrado e corretivo extensor compensatório de inclinações" [Double abutment on osseointegrated implant and compensating inclination extensor corrector (PI BR 0505827-9)]; "Splint abutment over osseointegrated implant and compensatory slanted coping" (United States Patent 9039416 and Canadian Patent No. 2,630,592); "System of triple abutments with connect (United States Patent 11,701,206); "Sistema de pilares trigeminados e múltiplos sobre implantes osseointegrados e guias direcionadores" [Triple and multiple abutments on osseointegrated implants and guide directors (PI BR 0800298-3, Brazil)]. All patents belong to the Institute of Technology and Research (Instituto de Tecnologia e Pesquisa -, Belo Horizonte, Minas Gerais, Brazil).

Patients This clinical investigation was designed as a pre-post case series study and is reported in accordance with the PROCESS (Preferred Reporting of Case Series in Surgery) and Consensus-based Clinical Case Reporting Guideline Development. All procedures performed in this study were in accordance with the ethical standards of the 1975 Declaration of Helsinki, revised in 2013, and received approval from the Ethics Research Committee of the Federal University of Minas Gerais, Brazil (CAEE: 60410222.2.0000.5149).

The selected group of participants were informed about the procedures and signed an informed consent form before participating in the study. Patients were recruited from a private clinic in the city of Belo Horizonte - Brazil (BR), between September 2009 and January 2019.

Participants were in good systemic health, non-diabetic and non-smokers. Prior to dental implant insertion surgeries, pre-surgical evaluation, detailed oral examination including complete periodontal examination, dental biofilm control and coronary polishing, as well as scaling and root planning and treatment of carious lesions (when necessary) were performed.

Since these new abutments break important paradigms in implant dentistry, such as passive multiple prostheses supported on a single implant with biomechanical benefits, the investigators have cautiously chosen to perform only ~5 cases/year, aiming at reaching a minimum follow-up time of 5 years.

This consecutive case series comprises a sample of 37 individuals/43 cases (16 men and 21 women) with different extensions of edentulous spaces rehabilitated using DA (n=28) and TA (n=15) in 43 osseointegrated implants. All cases using DA and TA had occlusal function with their respective antagonistic pairs (natural, restored or implanted teeth). Radiographic exams (analogue or digital) are presented at T1 (loading after 4 months of implant insertion surgeries with the appropriate metal-ceramic prostheses) and at T2 (final exam) with an average follow-up time of 7.2 years (>3 to 12 years).

The implants were cylindrical with internal hexagon and without surface treatment, presenting 4.0 mm in diameter and length varying between 10 and 13 mm (Neodent® - Grupo Straumann, Curitiba, Paraná, Brazil).

Clinical and laboratorial procedures

(a) Selection of components for molding: analogues, transfers and Universal Cast to Long Abutment (UCLA) with metallic base in cobalt chromium compatible with the diameters and type of hexagon of the implant system platforms installed (all cases in this study being with internal hexagon); (b) molding of the osseointegrated implant using a compatible transfer and antagonists; (c) Confection of the manual work model; (d) use of UCLA with metallic base to adapt the design of DA or TA in acrylic, with individualized design according to the measurements and paraboloid surfaces of the DA (elliptical) and TA (hyperbolic). The use of angulation in DA and TA is individualized, versatile and dependent on the position of the implant, when necessary; (d) casting the pillars in cobalt chromium alloy and polishing.; (e) Confection of crowns according to the chosen material (porcelain or resins) and (f) the DA or TA abutments are screwed into the mouth and the crowns cemented. All laboratorial procedures were performed in one single dental laboratory.

Inclusion Criteria:

• good systemic health
• non-diabetic
• non-smokers
• implant replacement need

Exclusion Criteria:

• former and current smokers
• diabetes
• periodontitis

This consecutive case series comprises a sample of 37 individuals/43 cases (16 men and 21 women) with different extensions of edentulous spaces rehabilitated using DA (n=28) and TA (n=15) in 43 osseointegrated implants.