**1. Introduction**

216 Simulated Annealing – Single and Multiple Objective Problems

[43] Kennedy, J., Eberhart, R. C. Particle Swarm Optimization. In Proceeding of IEEE International Conference Neural Networks, Perth, Australia; 1995, 1942-1948.

> Biomaterials should simultaneously satisfy many requirements and possess properties such as non-toxicity, corrosion resistance, thermal conductivity, strength, fatigue durability, biocompatibility and sometimes aesthetics. A single composition with a uniform structure may not satisfy all such requirements. Natural biomaterials often possess the structure of Functionally Graded Materials (FGMs) which enables them to satisfy these requirements. FGMs provide the structure with which synthetic biomaterials should essentially be formed. The size of biomaterial components is relatively small. In the case of dental applications, the components are generally smaller than 20 mm. This substantially reduces the difficulty of fabricating such materials due to a mismatch in thermal expansion which causes micro crack formation during the cooling cycle.

> Biomaterials are essential for life and health in certain cases. They have a generally high added value for their size. Thus, biomaterials form one of the most important areas for the application of FGMs. It is an area for which FGMs, at the present time, are sufficiently developed for practical use. The dental implant is used for restoring the function of chewing and biting, and therefore eating, which is the most fundamental activity of human beings required for living. We are living in an era of longer life expectancy and thus, dental care becomes especially important for better quality of life in old age.

> Implant may be classified to "implant'' as an artificial bone for medical use and ''dental implant'' as an artificial tooth for dental use. The specified properties are slightly different depending on their use. The implants in orthopaedics are used mostly as structurally enforced artificial bone which is inserted inside the corpus. Medical implants lay more weight on strength, toughness, torque in mechanical properties and the specific problem of tribology and abrasion resistance in artificial joint. Dental implant is usually much smaller and is used to reconstruct the masticatory function when tooth root is completely lost or extracted.

© 2012 Sadollah and Bahreininejad, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Sadollah and Bahreininejad, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Implant is placed in the jawbone in the manner to penetrate from the inside to the outside of the bone. The function is quite different at the inside of bone, outside and at the boundary. In the inside of jaw bone, bone affinity and stress relaxation are important and in the outside of bone, that is, in oral cavity, the sufficient strength is necessary. In the application of human body implant, FGM is usually composed of Collagen Hydroxyapatite (HAP) and titanium (Watari et al., 2004; Hedia, 2005).

Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 219

(1)

(1 ) *V V m C* = − (2)

with parameter *m*, as in Equation (1). The volume fractions of the two-phase composite FGM dental implant can be calculated from the following equations (Hedia, 2005; Wang et al.,

where *Vc* denotes the volumetric fraction of HAP/Col (ceramic), *Vm* denotes the volumetric fraction of titanium (metal), *m* is a constant to define the variation in material composition, *y* is the vertical position within the implant region and *h* is the total length of the implant. Fig. 1 shows the schematic view of FGM dental implant with graded material composition used

*C <sup>y</sup> <sup>V</sup> <sup>h</sup>* <sup>=</sup>

**Figure 1.** Schematic view of FGM dental implant with graded material composition.

*c*

Accordingly, the Young modulus and Poisson ratio can be calculated as (Hedia, 2005):

0 2

*E E Ev E E*

2/3

*mm cc v v V vV* = + (4)

(3)

3

( ) ( )( ) *c m cm*

 + − <sup>=</sup> +− −

*c m cm m*

*E E Ev v*

where *E0* is the equivalent Young modulus at different regions of the implant, *Ec* is the Young modulus of HAP/Col, *Em* is the Young modulus of titanium. *vc* and *vm* are the Poisson ratios for HAP/Col and titanium, respectively. The HAP/Col and titanium compositions vary according to the relative length of *y/h*, with respect to the material gradient *m*, meaning that *m* governs the variation in the volumetric fraction of the titanium to HAP/Col compositions. Referring to the properties of FGM implant, the values of *Ec* and *Em* are kept

*m*

2007; Yang & Xiang, 2007):

in dentistry.

HAP is indeed a principal component in human bones and related tissues. HAP inclusion in forming the dental implant material can bring about an enhanced biocompatibility with the native hosting tissues. The main advantages of using FGM dental implant are: 1) reduction of stress shielding effect on the surrounding bones that usually arises in the presence of fully metallic implants (Hedia, 2005), 2) improvement of biocompatibility with bone tissues (Watari et al., 2004), 3) preventing the thermal-mechanical failure at the interface of HAP coated metallic implants (Wang et al., 2007) and 4) meeting the biomechanical requirements at each region of the bone while enhance the bone remodeling, hereby maintaining the bone's health status (Yang & Xiang, 2007). The latter is more related to volume fraction of FGM. The first three aspects of using FGM implant have been investigated in the previous studies (Watari et al., 2004; Hedia, 2005; Wang et al., 2007).

However, limited knowledge has been available in the effect on bone remodeling due to the use of FGM dental implants. The other issue needs to be systematically studied is how to devise an optimal FGM pattern for dental implant application. It has been widely accepted that a mating mechanical property to the host bone should be made in order to avoid stress shielding (Hedia & Mahmoud, 2004; Hedia et al., 2006) and promote osseointegration and bone remodeling (Chu et al., 2006; Yang & Xiang, 2007). However, there are few reports available which examine whether or not a mating property could result in the best remodeling consequence and ensure a long-term success.

Recently, optimization of FGM dental implant was studied by Lin et al. (2009) using the Response Surface Methodology (RSM) and the results show the incompatibility of properties with each other and the need for using multi-objective algorithms to overcome the problem. Another issue concerns the existing material engineering technology which may not allow us to make such mating pattern for individuals in a cost efficient way. As a result, how to optimally tailor FGM pattern for remodeling is of noteworthy implication in developing FGM implantation.

This chapter aims at extending a more realistic FGM design for dental implantation. Using Simulated Annealing (SA), the multi-objective optimization model was developed to optimize FGM gradient pattern for desirable on-going bone turnover outcome and mechanical responses. SA algorithm has shown great potential for solving optimization problems as they conduct global stochastic search. The multi-objective optimization problem was solved using SA and the results were compared with the RSM.
