A dental implant is a biocompatible titanium device that is surgically placed into the mandibular or maxillary bone to support a prosthetic tooth crown in order to replace missing teeth. Dental implants are an effective, safe and predictable solution for those who have lost a tooth or teeth due to dental caries, periodontal disease, injuries or other reasons. Despite the high success rate of implantation when the implants are correctly designed, manufactured and inserted in bone of good quality and quantity, implant failures occur, especially in compromised sites and patients .

Bone, as a living tissue, is capable of optimizing its internal structure by redistributing its apparent density under the influence of external loads to fulfil its maximum function. This paradigm is known as the Wolff’s law . The process of bone redistributing its apparent density is termed ‘bone remodelling’. Since the initial publication by W olff , many theories describing the bone remodelling process have been proposed. F rost suggested that internal and external remodelling should be distinguished, where internal remodelling refers to the alteration of density of bone tissue and external remodelling is the apposition and removal of bone tissue on the bone surface. P auwels concluded that changes in the magnitude of the habitual mechanical stimulation lead to an ‘atrophy’ or ‘hypertrophy’ of the osseous structure. F rost also suggested a trigger criterion for remodelling based on the strain magnitude. This concept was formally developed later by C owin & H egedus and C owin & N achlinger using the concept of adaptive elasticity under small strains, which was the first mathematically rigorous theory for adaptive growth and remodelling of bone. C owin & F iroozbakhsh , F iroozbakhsh & C owin , and C owin et al. used the strain tensor as the stimuli for bone remodelling and assumed a quadratic relationship between the strain and the rate of adaptation. C arter proposed the concept of ‘lazy zone’, a threshold value, indicating that bone remodelling takes place only when the external stimuli exceed the ‘lazy zone’. This concept was incorporated by H uiskes et al. who used the strain energy density as the stimulus signal to control bone remodelling. M ullender et al. and M ullender & H uiskes developed a physiological approach to simulate bone remodelling, which assumed that osteocytes are sensitive to mechanical loading and are active in controlling the adaptation of bone mass in their environment.

When an implant or implants are surgically placed into the jawbone, the mechanical environment is altered. According to Wolff’s law, bone rearranges its internal structure by remodelling in response to implant placement and loading. Although many applications of bone remodelling theories are documented on the prediction of density distribution in peri-implant bone tissues , the effect of percentage of bone–implant contact on bone remodelling outcomes has not been considered. Full contact between the bone and the implant is often assumed in existing biomechanical research, which assumes that 100% bone–implant contact is achieved. This is inconsistent with clinical observations, which show that bone–implant contact depends on a variety of factors such as implant surface topography and bone quality. The present study, based on a newly proposed bone remodelling algorithm, aims to evaluate the effect of percentage of bone to implant contact on density distribution of jawbone through a two-dimensional (2D) finite element model.

Bone remodelling theories

Some popular bone remodelling theories are summarized below.

A popular bone remodelling theory suggested by H uiskes et al. is expressed as

dρdt=(B(Uρ−(1+s)K)ifUρ>(1+s)K0if(1−s)K≤Uρ≤(1+s)KB(Uρ−(1−s)K)ifUρ<(1−s)K)dρdt=⎛⎝⎜⎜⎜⎜⎜B(Uρ−(1+s)K)0B(Uρ−(1−s)K)ifUρ>(1+s)Kif(1−s)K≤Uρ≤(1+s)KifUρ<(1−s)K⎞⎠⎟⎟⎟⎟⎟
d ρ d t = B U ρ − ( 1 + s ) K i f U ρ > ( 1 + s ) K 0 i f ( 1 − s ) K ≤ U ρ ≤ ( 1 + s ) K B U ρ − ( 1 − s ) K i f U ρ < ( 1 − s ) K

where ρ is the apparent density of bone tissue; dρ / dt is the rate of change in density; B is a constant regulating the rate of the remodelling process; U is the strain energy density (SED); s is in percentage denoting the region of the ‘lazy zone’ around the threshold value K , which is a reference value of SED ( Fig. 1 ); and the ‘lazy zone’ describes the bone not having a net change in apparent density. In Eq. (1) , U can be expressed as:

U=12σ⋅εU=12σ⋅ε
U = 1 2 σ ⋅ ε

where σ and ε are the stress and strain tensors of the bone tissue, respectively.

The assumed local adaptation as a function of the SED with the effect of ‘lazy zone’.

For multiple-load cases, the SED can be expressed as:

U=1n∑ni=1UiU=1n∑ni=1Ui
U = 1 n ∑ i = 1 n U i

where n is the number of load cases and U _i is the SED under load case i .

Similar to the theory of H uiskes et al. , W einans et al. adopted the strain energy per unit of bone mass (i.e. U / ρ as the bone remodelling stimuli) and proposed a time-dependent remodelling equation:

dρdt=B(Uaρ−k),0<ρ≤ρcbdρdt=B(Uaρ−k),0<ρ≤ρcb
d ρ d t = B U a ρ − k , 0 < ρ ≤ ρ c b

where ρ = ρ ( x,y,z ) is the apparent density of bone tissue at position ( x,y,z ); U _a is in the same form as Eq. (3) and ρ _cb is the maximum apparent density which is equivalent to that of a cortical bone.

The algorithm by W einans et al. was applied to predict density distribution in a normal proximal femur with results showing a discontinuous pattern . The algorithm showed an unstable condition in its mathematical operation. Owing to the nature of the differential equations used in the algorithm to describe the adaptive remodelling process, the simulation produces discontinuous configurations, a phenomenon called ‘checker-board’. As discussed by R uimerman & H uiskes , the resulting density distributions correlate well with those in a real proximal femur only when the density distributions were locally averaged from discontinuous density patterns in the femoral head where the trabecular bone is located.

M ullender et al. and M ullender & H uiskes suggested a new algorithm which simulated bone remodelling as a self-organizational control process. In their theory, bone is assumed to have N sensor cells distributed uniformly over its volume. An arbitrary sensor i measures a signal S _i , which is given as:

Si=UiρiSi=Uiρi
S i = U i ρ i

where U _i is the SED and ρ _i is the apparent density at the location of the sensor. The density ρ ( x , t ) at location x is regulated by the stimulus value Ф( x , t ), to which all sensor cells contribute, relative to their distance from x . Hence,

Φ(x,t)=∑Ni=1fi(x)(Si−K)Φ(x,t)=∑Ni=1fi(x)(Si−K)
Φ ( x , t ) = ∑ i = 1 N f i ( x ) ( S i − K )

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