## 1 Introduction  
Primary stability (PS) plays a crucial role in attaining and maintaining the mineralized bone at the implant interface level for a long time [1, 2, 3, 4]. PS was reported to represent the resultant force of friction generated by the implant surface and the host bone contact after implant insertion [5; 6; 7]. A high implant insertion torque (IT) probably constitutes an important factor in reaching a high PS with the reduction in implant micromotion [2; 3]. It has been shown that implant micromotion during the early phases after implant insertion impair the osseointegration process, resulting in a fibrous capsule formation and implant loosening. In the early 1990s, a limit of 150 \(\upmu\)m of micromotion was considered as the cut-off for a successful osseointegration [8]. Despite this, with the evolution of implant systems and surface modifications, this variable has been reconsidered [9; 10; 11]. A recent review of [PERSON] et al. [12] showed a 32% higher micromotion in failed implants compared to the osseointegrated ones, but it also concluded that considering a universal tolerable limit for micromotion in osseointegrated implants could be misleading. Traditionally, dental implants take from 3 to 6 months to fully heal after their insertion, avoiding micromovements. However, with the emergence of immediate loading protocols, by which the implant loading can be performed immediately after surgery or within 48 h (ILP), the importance of PS increased dramatically. The rate of failures in ILP protocols has been reported as higher than conventional loading (risk ratio = 2.09, 95% confidence interval (CI) [1.18; 3.69], \(p\) = 0.01) [13]. Consequently, specific IT values have been suggested for ILP. For example, a minimum value of 60 Ncm was indicated by [PERSON] et al. [14] for single teeth and an IT of 45 and 32 Ncm for implants that have to support partial and full arch restorations, respectively. However, with a 2-year retrospective study, [PERSON] et al. [15] asserted that also an IT \(<\) 32 Ncm can be compatible with ILP success. Among the important factors influencing PS, there can be cited the bone quality, the bone density, the bone volume, the length and diameter of implants, the surgeon's experience and skills, the thread shape, the implant macrogeometry, the surgical site morphology, the drilling technique, the preparation of osteotomies, and the surgical technique [3; 4; 5; 9; 16; 17; 18; 19; 20]. The presence of poor bone quality and quantity has been reported to decrease the implant success [1; 5; 21]. PS is known to be related to the mechanical stability achieved with different surgical techniques and procedures: under-preparation, bone condensation, and bicortical fixation [5; 18; 22]. In fact, an under-preparation of the implant site in cancellous bone has been already proved to be helpful in improving implant PS [5; 6; 18; 22; 23; 24; 25; 26]. In recent years, polyurethane has been reported to have biomechanical properties similar to human bone and also a more uniform average cell size in respect to natural bone [16; 18; 20]. It is a standardized and homogenous material that can be provided in different thicknesses and densities, from D1 to D4 [20; 21]. These characteristics make the polyurethane foam a suitable material for biomechanical tests of dental implants, for example, the IT, removal torque (RT), and resonance frequency analysis (RFA) assessment [17; 21; 27; 28; 29]. In particular, the latter is known to be a valuable parameter to measure the implant stability quotient (ISQ) after its positioning, becoming a noninvasive technique to identify implant failure risk and add information on the predictability of dental implant procedures [30]. Thus, the main objective of this in vitro study has been the evaluation of IT, RT, and RFA values of different diameter conical implants (3.3 and 4 \(\times\) 13 mm) placed in under-prepared sites on 10 and 20 pounds per cubic foot (PCF) density polyurethane blocks, with and without the addition of a 30 PCF density cortical sheet.