# Structural Simulation of a Bone-Prosthesis System of the Knee Joint

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

- The final model is represented as a triangle mesh, but the interior density of the bone must be known for stress estimation
- The algorithm to extract and classify bone structures from medical data works on a slice-by-slice basis, rather than using the dataset as a whole; this reduces the chance of detecting long structures spanning over different slices; moreover, it can detect only cortical structures
- The segmentation requires too much human intervention, since the user must set a number of seed points to initialise the algorithm and clean the resulting images from incorrectly assigned pixels
- No error control/estimation is provided by the various sub-units of the tool

## 2. System Overview

## 3. Bone Modelling

#### 3.1. CT Data

#### 3.2. Classification and Volume Segmentation

_{k}the k-th class. Let us define a number of feature functions F

_{1}(.), …, F

_{M}(.) on the pixel domain. We use a simple generative approach [3] to learn the model for the posterior probability P (c

_{k}∣F

_{1}(z), …, F

_{M}(z))) for each pixel z. We model the likelihoods P (F

_{1}(z), …, F

_{M}(z)∣c

_{k}) using GMMs (one for each class). Manually-annotated training datasets are used to learn the likelihoods, using the EM algorithm [3]. We assume that the priors P(c

_{k}) are uniformly distributed. One might argue that these probabilities are different for each class and that we can compute them from our training sets by counting the number of pixels in each class. However, too many uncontrolled elements can affect the percentage of bone pixels over the total volume, such as biological factors (e.g., age, sex, osteoporosis), and machine setup parameters (e.g., percentage of patient tissues contained into the imaged volume). Hence, equiprobability of the priors P(c

_{k}) is a reasonable choice. Assuming equiprobable priors, by the Bayes' theorem we get

_{k}(z) = P (c

_{k}∣F

_{1}(z), …, F

_{M}(z))) are used to classify unseen data. A MAP rule is used to get crisp classifications. For each pixel z the most probable labelling is

_{C}(z) = max [p̂

_{k}(z)]. Being conservative, a strict rejection rule is employed to retain only highly probable classifications (low uncertainty). Thus, we accept this classification if p̂

_{C}≥ ε where ε is a user-defined threshold which bounds the classification uncertainty. If we restrict to two classes, c

_{1}and c

_{2}, and to a single feature F

_{1}(z) for the sake of visualisation, our simple rejection option can be depicted using the diagram in Figure 3. Pixels classified as cortical bone with high uncertainty are used for FEM simulation as well (see Section 6.), accounting for this classification uncertainty.

_{s}= (F

_{s,N}(.), F

_{s,E}(.), F

_{s,S}(.), F

_{s,W}(.)) as the mean of the HU values for four regions in the surrounding window: North (N), East (E), South (S), and West (W) (see Figure 4). In our implementation, we use the HU value of the interest pixel together with the feature vectors υ

_{s}. We choose these features to selectively evaluate the surrounding context of a pixel in four directions. We do not use circular features since they would bring less useful information, being independent from orientation. We also use simple mean instead of more complex distance weighting since distance is accounted for using multiple scales. Due to the dimension of feature windows, especially in larger scales, a special treatment should be reserved to border pixels. In our implementation, we do not bother about image borders since in our application interest pixels always lie in the central area of CT scans.

#### 3.3. Bone Mechanical Parameters

## 4. Prosthesis Modelling

## 5. Mathematical Description and Multi-Scale Modelling of the Bone-Prosthesis Contact

_{n}= σ

_{n}/u

_{n}, and friction coefficient, µ = |σ

_{τ}|/σ

_{n}, on the interface between bone and prosthesis.

**u**

^{ε}be the interface jump in the displacement vector, ε be the small parameter, denoting the ratio between the coatings roughness and the macro-size of the bone-prosthesis system;

**n**

^{micro}denotes the unit outward normal vector of the bone and g + εg̅ is the initial interface gap (see Figure 10). Then, the local non-penetration condition for each coating point x ∈ S

^{ε}(see [16]) is

**Proposition.**For the local non-penetration contact condition (3), the anisotropic macroscopic non-penetration condition will be given in the form

_{nn}, k

_{nτ}and k

_{ττ}can be found for each known surface micro-profile from the following formulas

_{nτ}and k

_{ττ}, will imply some tangential drag forces, which can be interpreted as friction forces. The drag forces caused by k

_{nτ}, k

_{ττ}can be represented in the form of the Coulomb's friction with the homogenised friction coefficients $\mu =\frac{\mid {k}_{n\tau \mid}}{{k}_{nn}}+\frac{{k}_{\tau \tau}}{{k}_{nn}}\left|\frac{{u}_{\tau}}{{u}_{n}}\right|$. This macroscopic contact condition will be used for numerical computations of the macro-problem on. The main result here is that even starting with the frictionless contact micro-problem with a rough interface, we end up with a macro-problem containing friction.

_{nn}, and the friction coefficient, µ, for four coating layers with different porosity and roughness are calculated for the case of the full micro-contact between the bone and the prosthesis. The coatings' surfaces were given as volumetric voxel grid, containing key-points. We interpolated the surfaces by orthogonal parabolic splines (see Figure 11) to obtain coordinates of the micro-normal vector in each point of the micro-contact surface by the standard formula $n(x,y)=\frac{(-{z}_{x}(x,y),-{z}_{y}(x,y),1)}{\sqrt{1+{z}_{x}^{2}(x,y)+{z}_{y}^{2}(x,y)}}$. It can be seen that for given materials the magnitude of the friction coefficient can differ in several orders. The friction coefficient strongly depends on the roughness and the porosity of the coating layer.

_{α}, b

_{α}, α = n, τ, are material constants and the average penetration depth h is introduced as a displacement of the mean line of the rough surface. According to [18], c

_{n}, c

_{τ}are proportional to $\frac{1}{\frac{1-{v}_{\mathit{\text{Prosth}}}^{2}}{{E}_{\mathit{\text{Prosth}}}}+\frac{1-{v}_{\mathit{\text{coat}}}^{2}}{{E}_{\mathit{\text{coat}}}}+\frac{1-{v}_{\mathit{\text{Bone}}}^{2}}{{E}_{\mathit{\text{Bone}}}}}$, where E

_{coat}, ν

_{coat}are effective Young's modulus and Poisson's ratio of the coating calculated from the simple analytic Hashin composite sphere model (Christensen [19]) on the basis of its porosity and elastic properties of the coating alloy, E

_{alloy}, ν

_{alloy}. Furthermore, according to our terminology, h =< k

_{nn}u

_{n}+ k

_{nτ}u

_{τ}‒ g

_{n}> and < g

_{n}>= Rt/2. Here < · > denotes the averaging in the cross-section orthogonal to the macro-normal i.e. $<\cdot >:=\frac{1}{\left|{S}_{0}\right|}{\int}_{{S}_{0}}\cdot d\widehat{x}$. We obtain then the following macroscopic contact law:

_{n}are presented in Table 2. The proposed method for estimation of macroscopic contact conditions is implemented in the software KneeMech [20]. The steps for performing simulations can be seen in Figure 12. The macroscopic contact stiffness k

_{n}and friction coefficient µ are calculated by using the mechanical parameters for the prosthesis and the coating as well as porosity and roughness of the coating. After that, the prosthesis is manually positioned using a GUI. Then, the full mathematical macroscopic contact problem for the bone-prosthesis system can be constructed. It consists of equilibrium equations (8) with constitutive elastic relations (9) for the bone and prosthesis, contact (3), (11) and boundary (12) conditions:

_{n}(x) = (σ(x) ·

**n**(x)) ·

**n**(x) is the normal stress, σ

_{τ}(x) = σ(x) ·

**n**(x) − σ

_{n}

**n**(x) denotes the tangential stress vector, ${[u]}_{n}\left(x\right):=(\mathbf{\text{u}}\left(x\right)\mid {S}_{0}^{\mathit{\text{Prosth}}}-\mathbf{\text{u}}(x)\mid {S}_{0}^{\mathit{\text{Bone}}})\cdot \mathbf{n}(x)$, is the jump in the normal displacement, [

**u**]

_{τ}= [

**u**] − [u]

_{n}

**n**(x) is the vector of jumps in the tangential displacements, go and t are components of given vectors of the boundary displacements and traction (see Figure 13 for g

_{0}= 0).

## 6. Simulation

#### 6.1. Finite Element Discretisation

_{Bone}∪ Ω

_{Prosth}, = { υ

_{i}∈ H

^{1}(Ω), i = 1,…, N | υ

_{i}(x) = g

_{0i}(x), x ∈ Γ

_{u}}. The elasticity problem without contact can be rewritten in a weak formulation as follows: find u

_{i}∈ such that for all υ

_{i}∈

_{i}∈ .

_{h}consisting of tetrahedral elements. Each component of displacements over each element is approximated by linear polynomials. On the basis of these tetrahedra it is possible to generate a system of piecewise linear global basis functions {Φ

_{ξ}}. Using this basis we can span a space ${\mathcal{V}}_{h}=\left\{{\upsilon}_{i}\in C\left({\overline{\Omega}}_{h}\right),i=1,\dots ,N|{\upsilon}_{i}\left(x\right)={g}_{0i}\left(x\right),x\in {\Gamma}_{u}^{h}\right\},$ where ${\Gamma}_{u}^{h}$ is an approximation of the boundary Γ

_{u}by the triangles correspondingly to Ω

_{h}.

**v**

_{h})

_{i}is an approximation of the i-th component of the displacement field defined over Ω̅

_{h}then

_{p}is the total number of grid points. The term (15) in (14) can be approximated as follows:

_{N}by a finite element mesh.

**u**

_{h})

_{i}∈

_{h}, i = 1, ..,N which minimises the functional

_{n}, δ

_{τ}> 0 are small penalty parameters, chosen with respect to (6) as follows:

_{Ch}is the discretised contact surface and [.]

_{+}≔ max {0,.}. It is important to remark that all nodal points in Γ

_{Ch}are interface points. This means that in these points the displacements can have different values in different materials. The corresponding linear system is solved using a preconditioned conjugate gradient solver [26].

#### 6.2. Hierarchical mesh coarsening

#### 6.3. Dealing with Uncertainties

_{c}+ 1, where N

_{c}is the number of uncertain mechanical parameters.

_{0}, E

_{i}, η

_{i}(i = 1,2) (see also section 3.3.) solved with timestep Δt, uncertainty modelling should consider expressions such as

^{(i)}we can write its Young's modulus E

^{(i)}in the form:

_{j}is a noise symbol, an unknown such that |ε

_{j}| < 1. ${E}_{0}^{(i)}$ is the central value of the Young's modulus, while ${E}_{j}^{(i)}$ is the radius of its uncertainty.

^{(i)}, we can write it in the form

^{(i)}and ${K}_{j}^{(i)}$ comes from the Young's modulus uncertainty ${E}_{j}^{(i)}$.

_{c}, however, the global stiffness matrix is obtained combining local matrices in the form of Equation 24, and it will therefore have the form:

_{0}is the same matrix that would be obtained in the deterministic problem (ignoring uncertainties), and the K

_{i}are N

_{c}matrices obtained from the uncertainties.

_{0}+ Σ M

_{i}ε

_{i}where

_{i}, we can write u as the sum of a linear part u

_{a}and a linearisation of the nonlinear part u

_{l}. The linear part will be in the form

_{i}, i = 1,…, N

_{c}in [0,1]. This can only be satisfied (polynomial identity rule) if the coefficients of ε

_{i}on each side of the equation are equal, and therefore it must be

_{c}+ 1 sharp problems, all with the same coefficient matrix (M

_{0}).

_{0}is found the u

_{i}can all be computed independently And since the coefficient matrix is the same in all equations, most global objects, preconditioners, and decompositions can be computed only once and then reused across equations, reducing computational time.

_{0}+ Σ |u

_{i}| where the absolute value is considered component by component. However, for subsequent computations, such as stress and strain evaluation, it is better to keep the displacements in their affine form, as the information on the correlation between components of the stress tensor greatly improves the bounds of the von Mises equivalent stress.

_{i}. Adding up the relative uncertainty for each noise symbol gives the total relative uncertainty.

## 7. Discussion

**5.5**MPa, while in the second case it's only

**1.8**MPa. As expected, the contact of the prosthesis with spongy bone results in much higher stresses. It should be remarked that both examples are artificial and were chosen to pick up extreme cases: they are irrelevant for clinical practice, but show the importance of choosing an appropriate cutting plane and prosthesis size.

## Acknowledgments

^{∗}Data is usually acquired using spiral scanning. Pixel values are interpolated from spiral data^{†}For this reason, hereafter we will use the terms pixel and voxel interchangeably.

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**Figure 2.**A CT scan of a 80-years-old patient, affected by osteoporosis and arthrosis. A manual labelling is also shown together with the relative distribution of Hounsfield Units (HU) values of two tissue classes: soft tissue and trabecular bone.

**Figure 3.**Rejection rule. Probability distributions for two classes, conditioned by the HU value (our simplest feature). The threshold ε is used for rejection of low-probability classifications (see text). The shadowed region depicts the rejection interval.

**Figure 5.**An example of a slice classified using our classification method. The colour codes are as follows: White for cortical bone, blue for trabecular bone, red for soft tissue, and black for background.

**Figure 6.**Surface model showing of the cortical tissue of a knee joint. Notice that the the bones of the articulation are neatly separated.

**Figure 11.**Fragment of the coatings surface interpolated by parabolic splines using given data points.

**Figure 12.**Simulation sequence: set the properties of the prosthesis, obtain contact conditions, position the prosthesis, and solve the macroscopic contact problem.

**Figure 13.**Resulting bone-prosthesis system and scheme of boundary conditions. Arrows represents directions of the loading forces. The bottom part of the bone is fixed.

**Figure 14.**Tetrahedral mesh of the tibia with four levels of coarsening corresponding to the values given in Table 3.

**Figure 16.**Sample screenshot from the KneeMech® program, showing the result of the simulation for a tibia.

**Table 1.**Parameter settings used for the acquisition of the knee CT data employed in our experiments.

Parameter | Value |
---|---|

Exposure | 200 Sv |

kVp | 140 kiloVolt |

Slice thickness | 1.3 mm |

Slice spacing | 0.6 mm |

Slice resolution | 512 × 512 pixels |

Number of slices | 70–140 |

**Table 2.**Normal contact stiffness and friction coefficients for effective non-penetration condition in case of full microscopic contact. Parameters for k

_{nn}(u) using the expression (7) in case of solution-dependent micro-contact surface.

Coating | |||

Roughness, Rt, µm | 150 ± 50 | 200 ± 100 | 200 ± 100 |

Porosity, % | 15 ± 10 | 30 ± 10 | 15 ± 10 |

Normal contact stiffness k_{nn} | 0.99 | 0.96 | 0.87 |

Friction coefficient | 9.78e-04 | 5.45e-02 | 2.21e-01 |

Contact param. for k_{nn}(u) from (7) | |||

a_{n} | 4.933230 | 2.337140 | 1.766300 |

b_{n} | 0.497877 | 0.382857 | 0.427986 |

Coarsening Levels | Number of Nodes | Number of Elements | Node reduction | Time (sec.) |
---|---|---|---|---|

0 | 346 783 | 1 636 175 | 1.0 | 135 |

1 | 81 816 | 341 152 | 4.2 | 65 |

2 | 58 640 | 230 642 | 6.2 | 56 |

3 | 57 186 | 223 763 | 6.4 | 56 |

Position of the prosthesis | |||

Equivalent stress (MPa) | 1.25 | 1.1 | 1.18 |

E (GPa) | Equivalent stress (MPa) | Vertical displacement (mm) |
---|---|---|

5 | 1.18 | 0.0316 |

10 | 1.19 | 0.0233 |

14.7 | 1.20 | 0.0152 |

Angle, ° | Equivalent stress (MPa) | Horizontal displacement (mm) |
---|---|---|

0 | 1.1 | 0.0152 |

45 | 1.2 | 0.00722 |

© 2008 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Andrä, H.; Battiato, S.; Bilotta, G.; Farinella, G.q.M.; Impoco, G.; Orlik, J.; Russo, G.; Zemitis, A.
Structural Simulation of a Bone-Prosthesis System of the Knee Joint. *Sensors* **2008**, *8*, 5897-5926.
https://doi.org/10.3390/s8095897

**AMA Style**

Andrä H, Battiato S, Bilotta G, Farinella GqM, Impoco G, Orlik J, Russo G, Zemitis A.
Structural Simulation of a Bone-Prosthesis System of the Knee Joint. *Sensors*. 2008; 8(9):5897-5926.
https://doi.org/10.3390/s8095897

**Chicago/Turabian Style**

Andrä, Heiko, Sebastiano Battiato, Giuseppe Bilotta, Giovanni q M. Farinella, Gaetano Impoco, Julia Orlik, Giovanni Russo, and Aivars Zemitis.
2008. "Structural Simulation of a Bone-Prosthesis System of the Knee Joint" *Sensors* 8, no. 9: 5897-5926.
https://doi.org/10.3390/s8095897