The temporomandibular joint (TMJ) is essential to numerous key human functions, in particular, expression of emotion, chewing, swallowing and speech. Generally considered a load bearing joint, the TMJ is prone to a range of disorders due to structural and/or positional abnormalities \cite{Tanaka2008}. Research conducted in the 1980s showed that 16\% to 59\% of those tested, suffered symptoms and 33\% to 86\% showed clinical signs \cite{Carl}. The most common joint pathology for the TMJ is osteoarthritis, which can lead to critical deformities both functional and morphological \cite{Zarb}.

For severe cases, a total joint replacement is considered to be a functional solution, by which the joint is replaced with an implant which connects to the upper mandible via a series of screws. The success of such implants has reached an acceptable level of reliability as reinforced by a 10 year study which reported a cumulative survival rate of 97. 9\% for ITI implants \cite{Blanes2007}. It has nonetheless been shown that the success of such implants is influenced by adaptation in the host bone \cite{Young1998}, caused by a change in the stress state in certain key areas.

As such it is critical to consider the morphological changes in bone quality following joint replacement when designing implants, in order to minimise the effect of stress shielding and ensure high success rates for implants in the long term. \subsection{Project Statement} The mandibular bone’s masticatory function is intrinsically linked to it’s biomechanical function based on Wolff’s law which determines bone density, bone microarchitecture and bone turnover. Reconstruction of large bone defects of the mandible is commonly performed using titanium implants.

This project addresses the development of an algorithm to determine wether stress shielding in adjacent bone which would ultimately lead to bone resorption and potential implant loosening. Using bone imaging together with three dimensional FE analysis will determine the stress/strain states around mandibular implants under physiological loading conditions. A suitable bone remodelling algorithm must be developed and tested with the goal of providing a tool which can be of use in future implant design research. \subsection{Project Aims}

The encompassing objective for this project is to develop a suitable method of FE analysis to assess the effects of stress shielding and consequent bone adaptation following jaw surgery. The key aim however, is to build and test a robust algorithm to simulate the remodelling process. The various rates and constants in the adopted adaptation model must be tested and verified. Sensitivity to mesh density and type must be considered in order to determine the sensitivity of the algorithm. The following goals are expected to be delivered by the end of the project: begin{itemize} \item Develop an algorithm to simulate remodelling in the jaw bone \item Investigate the effect of the density change, gradient and lazy zone \item Analyse and discuss the sensitivity of the model to element type and size \item Compare continuous and discontinuous solutions \item Apply model to existing jaw model, and discuss the validity of such analysis \end{itemize} \subsection{Report format} This report will follow the process taken to design and establish the validity of an algorithm to account for bone remodelling effects in the bone.

A literature review will outline key findings from previous bone remodelling studies. From this review, a method is established to be used as the basis for all following analysis. A preliminary algorithm will first be applied to a homogeneous model, followed by models of increasing complexity and finally the jaw bone. \section{Literature Review} % \item JAW PROBLEMS % \item JAW SURGERY % \item SURGERY ISSUES – bone Remodellling is critical to success % \item BONE RemodellING % Wolff’s Law % Rate equations for Remodelling – Strain adaptive Remodelling Huiskes et al. \item Relationship between E rho % \item CLINICAL TRIALS VS. SIMULATION vs usd vs python vs umat % \item Model already exists % “Minimisation of stress shielding is an important factor for the long term success of implants as it can cause excessive bone resorption and compromise the outcome of an implant. ” – Schmidutz2014} Jaw joint replacement surgery is well established, however failures can arise due to bone resorption, cancellous bone growth around the implant and choice of implant material leading to screws becoming loose.

Reduction of stress shielding is therefore a key factor to ensure the long term success of an implant \cite{Schmidutz2014}. Bone remodelling due to implants has been studied using radiology on animals since 1991 \cite{Pilliar1991}. Along with continued development of dental implant applications, it has become critical to the design of implants, to assess the effects of bone remodelling \cite{Linreview}. Finite element (FE) modelling has been shown to have some success in doing so, although problems are often encountered related to discontinuities in the density continuum and convergence of the algorithm \cite{Jacobs1995}. subsection{Bone Remodelling} Bone Remodelling is the process by which new bone tissue is formed and old bone tissue removed from the skeleton. The remodelling process is dependent on the mechanical loading of the bone. According to Wolff’s Law, developed in 1892 by the German surgeon, Julius Wolff, `every change in the function of a bone is followed by certain definite changes in its internal architecture and its external conformation’ \cite{wolff1892}. Increased loading will change the trabecular architecture of the bone by increasing density and hence stiffness.

This in turn reduces the effect of the increased load on the bone, by reducing the strain. The inverse is also true of reduced loads, which cause the bone to reduce in density and ultimately resorb \cite{Beau}. \begin{sloppypar} There are in fact two types of bone Remodelling which have been found to occur, `surface’ and `internal’ remodelling \cite{frost1964dynamics}. The former refers to resorption or deposition of bone, whereas the latter refers to the resorption or reinforcement of lamellar bone within the existing osteons \cite{Cowin1976}. \end{sloppypar}

Bone Remodelling involves three types of cells, osteoblasts, osteocytes and osteoclasts. Following the osteoblastic process, the osteoblasts develop into lacunae, otherwise known as osteocytes. These are responsible for the provision of calcium and other ions between the blood plasma and bone minerals \cite{mclean1968fundamentals}. If aged, the osteons can be resorbed to maintain the physiological function of the bone \cite{Cowin1976}. From experimental studies conducted in the past 30 years, Turner notes the following three rules to consider for bone adaptation: begin{itemize} \item Bone adaptation is driven by dynamic and not static loading \item Only a short duration necessary to initiate adaptation, which consequently diminishes with time \item Bone cells become accustomed to routine loading, and are more responsive to changes in loading \end{itemize} Currently, no bone remodelling theories are based on jaw bones, but are instead based on long bones.

These theories are considered to be analogous to the jaw bone given the modification of relevant constants and parameters \cite{Eser2013}. subsection{Remodelling Algorithms} Bone remodelling studies carried out on animals have suggested that the bone adaptation process is heavily influenced by mechanical loading conditions \cite{Eymard2003}. A number of theories exist in the literature for predicting the change in bone properties. Continuum damage mechanics is one such theory, which uses the damage status of the bone to predict the bone growth \cite{Idhammad2013}. This theory states that a damaged bone will cause bone apposition, reducing porosity and increasing the density.

Another commonly adopted method is that of \cite{Weinans1992}, which attributes the local bone adaptation to a local mechanical stimulus. The general remodelling rate equation described by the rate of change of density is \begin{equation} \frac{\partial \rho}{\partial t}= K(S – S_{ref}) \end{equation} where $S – S_{ref}$ is the difference between the mechanical stimulus at a point, and the reference stimulus at that point, whilst $K$ is a rate constant. The Mechanostat theory suggests that the bone remodelling process is controlled by the difference in minimum effective strain (MES) \cite{Frost1983}.

Frost proposed that bone health was maintained by a biomechanical feedback system in which the peak mechanical strain was kept within an acceptable range. A drawback in the use of the MES is that the strain term in the equation is a tensor which requires directional properties to be defined. The concept of a daily stress stimulus incorporated the time dependency characteristic stated by \cite{Turner1997} to the Mechanostat theory of Frost. Beaupre et al. (1990) defined the daily stress stimulus to be \begin{equation} psi_{daily} = \big(\sum_i n_i \sigma^m \big)^{\frac{1}{m}} \end{equation} with $n_i$ the number of loading cycles and $m$ a constant to be determined empirically. The problem with the use of this stimulus arise due to the fact that loading cycles and chewing patterns can vary across population groups. This would require copious statistical data to be gathered across age, cultural and gender groups. The most commonly used stimulus indicator is the strain energy density (SED).

To its advantage, the strain energy density is a scalar quantity, hence eliminating the need for directional properties in the remodelling process. The SED is calculated \begin{equation}U = \frac{1}{2} \sigma \cdot \epsilon \end{equation} where $\sigma$ is the mechanical stress and $\epsilon$ is the strain. The use of the SED can be specified to the SED of the trabeculae which is approximated by $U/\rho$ \cite{Carter1989a}. This value represents the strain energy per unit of bone mass and assumes the bone to be continuous.

As such the stimulus value commonly adopted is \begin{equation} S(x,y,z,t) = \frac{U(x,y,z,t)}{\rho (x,y,z,t)} \end{equation} Most bone remodelling algorithms adopt a `lazy zone’, which defines a region of normal activity surrounding the reference stimulus, as opposed to a linear remodelling rate relation. In general, the remodelling process occurs more slowly with the use of a lazy zone than with the use of a linear rate equation \cite{Beau}. The use of bone remodelling rate equations causes discontinuities in the resultant density distribution \cite{Weinans1992}.

Those equations were based on the assumption of a continuus material, which is far from the reality of the bone at the microscopic tissue level. One proposed solution to this problem is to model the bone at the substructural level \cite{Prendergast1996}, however the size and complexity of this model would make it inappropriate for practical use. \subsection{Relationship between density and Young’s Modulus} The relationship presented by Carter and Hayes for the stiffness of bone is \cite{Carter1977} \begin{equation} E = C\epsilon^{0. 06} \rho^{\gamma} \end{equation} where $C$ is an empirical constant to be determined, $\epsilon$ the strain rate during loading and $\rho$ the apparent density. Often the effect of the strain rate is ignored and so for trabecular bone \cite{Weinans1992} \begin{equation} E = 3790 \rho^3 \end{equation} A mathematical analysis of the stability of different values of $\gamma$ shows that the algorithm is unstable for values of $\gamma$ greater than one and as such the bone adaptation process is a chaotic.

One such theory is the Stanford theory \cite{Beau}, which has been used in literature to study bone adaptation around dental implants \cite{Mellal2004}. %\subsection{Choice of Subroutine} %The bone Remodelling algorithm must run either for a set amount of time, or until convergence is reached. There are two known ways to implement the algorithm. The first is to run Abaqus using a python script, which loops until the strain energy densities stabilise. The second method is to use a sub-routine such as UMAT or USDFLD.