Besides retention, it is also of vital importance not to cause excessive load on implants。18 In natural teeth, the periodontal ligament acts as an intermediate cushion to buffer the occlusal loads。19 However, in the osseointegrated dental implant, occlusal loads are transmitted directly to the surrounding bones。 When overloading happens, high deformations (above 2000–3000 microstrain) occur in the bone around the implants。 When pathological overloading occurs (over 4000 microstrain), stress and strain gradients exceed the physiological limits of the bone, which may cause micro-fractures at the bone–implant interface, fracture of the implant, loosening of components of the implant system, and unwanted bone resorption。20,21

Recognizing the damage done by overloading, clinicians pay close attention to the stress and strain developed in peri- implant bone when using different prosthetic designs。 Three- dimensional finite element analysis (3D FEA) has been considered a precise and appropriate approach for investigat- ing stress and strain distribution in bone and offers many advantages over other methods in simulating the complexity of clinical situations。22  To date, there has been little   previous

2。 Materials and methods

2。1。 Model design

To obtain the geometry of a totally edentulous patient’s mandible, a computed tomography (CT) examination was carried out on a volunteer, with approval from the ethnical committee of Peking University School of Stomatology (IRB00001052-07051)。 Her mandible and mandibular over- denture were scanned。 The CT examination files were then imported into Mimics8。0 (Materialise, Leuven, Belgium)。 Straumann implants (Straumann, Basel, Switzerland; diame- ter: 4。1 mm, length: 10 mm, screw-shaped) and Locator attachment systems (Zest Anchors, Escondido, CA, USA; diameter: 3。85 mm, length: 3。85 mm) were chosen as over- denture retainers for this biomechanical analysis。 The three- dimensional geometries of the edentulous mandible and prosthetic components were modelled in SolidWorks 2008 (SolidWorks Corporation, Ve´ lizy-Villacoublay, France)。

The geometries of the mandible, overdenture, implant and attachment systems were then meshed using Abaqus 6。8 (Simulia Corporation, Ve´ lizy-Villacoublay, France)。 Four 3D FE models of an edentulous mandible supporting an implant overdenture were designed (Fig。 1), each with different numbers of implants in the anterior area of mandible between the mental foramina。 All implants were vertically positioned and well  distributed  in  the  interforaminal  region,  at  least 6 mm mesial to the mental foramen, as   follows:

• Model A, a single implant was located in the midline of the jaw。

• Model B, the overdenture was retained by two implants 20 mm apart。

• Model C, the overdenture was retained by three implants with the central one in the midline of the jaw and other two a distance of 18 mm to either side。

• Model D, the overdenture was retained by  four implants 12 mm apart。

The models were meshed with 3D four-node tetrahedron elements。 The total numbers of elements and nodes are listed in Table 1。 A refined mesh was generated in the interforaminal region to faithfully reproduce the complex strain distribution observed in peri-implant bone。

2。2。 Material properties

The edentulous jaw was composed of a 2-mm constant cortical bone layer around a cancellous bone core, covered by a 2-mm thick mucosa。 The Locator attachment system was composed of three parts: abutment, nylon replacement   male