# Multicriteria Decision Analysis in Improving Quality of Design in Femoral Component of Knee Prostheses: Influence of Interface Geometry and Material

Authors: Ali Jahan; Marjan Bahraminasab
Journal: Advances in Materials Science and Engineering
 (2015)

---

## Abstract

Knee prostheses as medical products require careful application of quality and design tool to ensure the best performance. Therefore, quality function deployment (QFD) was proposed as a quality tool to systematically integrate consumer’s expectation to perceived needs by medical and design team and to explicitly address the translation of customer needs into engineering characteristics. In this study, full factorial design of experiment (DOE) method was accompanied by finite element analysis (FEA) to evaluate the effect of inner contours of femoral component on mechanical stability of the implant and biomechanical stresses within the implant components and adjacent bone areas with preservation of the outer contours for standard Co-Cr alloy and a promising functionally graded material (FGM). The ANOVA revealed that the inner shape of femoral component influenced the performance measures in which the angle between the distal and anterior cuts and the angle between the distal and posterior cuts were greatly influential.  In the final ranking of alternatives, using multicriteria decision analysis (MCDA), the designs with FGM was ranked first over the Co-Cr femoral component, but the original design with Co-Cr material was not the best choice femoral component, among the top ranked design with the same material.


---

## Body

## 1. Introduction

Poor estimation of the main product parameters may lead to high cost due to subsequent redesign or even to product failure. Development of regulatory and functional needs in a product is tremendously complex as it involves a number of compromises linking with the consumer. Furthermore, many compromises required during product design often lead to a suboptimal final design and to a reduction in the design process effectiveness. The process of selecting for best design parameters to fulfill the desired requirements deals with a lot of trade-offs. Functional specifications which can be related to the material and geometrical design need to be identified properly and to be mapped for the connections between them. Therefore, a systematic approach, which enables the designers to determine and map the correlation between the different functional specifications and design parameters, is highly demanded. It seems that there is no attempt for applying the combined quality function deployment-finite element analysis-multiattribute decision making-design of experiment (QFD-FEA–MADM–DOE) approach for design improvement.One of the medical products that still lack sufficient design solutions is total knee replacements (TKRs), which suffer from aseptic loosening and eventual revision surgery [1]. This problem deals with several mechanical factors (Figure 1) which are associated with many material and shape aspects [2–5]. Currently, various TKR designs are available in which the femoral components are similar in basic design, but the angles and lengths related to inner contours may differ [6]. These variations in the inner shape of prosthetic femur might adversely affect either the component or bone performances. To the best of the authors’ knowledge, the mechanical stability of the implant and the biomechanical stresses within the implant components and adjacent bone areas of different inner contours of femoral components has not been investigated yet. In the present study, therefore, the stress behavior of different inner contours of femoral components with preservation of the outer contours is evaluated and compared.Figure 1
Aseptic loosening problem, the related leading causes, and the engineering design solutions.


## 2. Materials and Methods

The methodology in this study contains four main steps: (1) determining the design variables and the design of computational experiments, (2) quality function deployment to translate the voice of customer to technical terms and performance outputs, and to determine their relative importance, (3) finite element analysis to predict the performance outputs for the design options generated using DOE, and (4) analysis of variance (ANOVA) to find significant factors and to evaluate the main and interaction effects of design variables, and then multiattribute decision making to rank the candidate design options. These steps are shown in Figure2 and explained in the next section in detail.Figure 2
Steps used in the methodology.


### 2.1. Design Variables and Design of Experiments

Various knee prosthesis designs are currently available in which the basic designs of femoral components are analogous; however the interface geometry of this component including the location pegs, the angles associated with the inner contour, and their respective lengths vary from a particular design to another [6]. The influence of location pegs on stress shielding effect was studied by Wang et al. [7] in a two-dimensional finite element analysis and by Bahraminasab et al. [8] in three-dimensional FEA and with respect to the materials used. In the present study, in addition to the effect of materials, the influences of inner cuts (anterior, posterior, and oblique cuts) on the biomechanical performance were investigated, whilst the outer contours of prosthetic femur were preserved as the original design, since the articulating surfaces are complex curves (single curve or multicurves) with precise transitions which identify the contact behavior with the polyethylene component [9] and are design-specific. Here, four geometrical design variables were taken into account including the angle between the distal and oblique anterior cuts (β), the angle between the distal and anterior cuts (γ), the angle between the distal and oblique posterior cuts (δ), and the angle between the distal and posterior cuts (ε). While these variables were changed between the ranges identified in Table 1, the peg position, total length (L
T), length of distal cut (L
D), total height of anterior and anterior oblique cuts (H
A), and total height of posterior and posterior oblique cuts (H
P) were kept constant (points 1–4 were fixed). Figure 3 presents two-dimensional (2D) profile of femoral component with four geometrical design variables and constraints used.Table 1
Geometrical variables and their respective levels.
Run number
Factor 1
Factor 2
Factor 3
Factor 4
A: beta (deg)
B: gamma (deg)
C: delta (deg)
D: epsilon (deg)
1
125
98
140
88
2
125
98
125
88
3
140
90
140
94
4
125
98
140
94
5
125
90
125
94
6
140
98
140
94
7
140
98
140
88
8
125
90
140
88
9
125
98
125
94
10
140
90
125
94
11
140
90
140
88
12
140
98
125
94
13
125
90
125
88
14
125
90
140
94
15
140
90
125
88
16
140
98
125
88Figure 3
Design variables of the inner contour.Femoral component as a key feature in current knee prostheses has been the focus of many studies related to materials. Scholars have been working on tailor and development of new biomaterials for this component to avoid mismatch of engineered materials properties with those of the biological ones and to provide acceptable performance without short-term and long-term failure. These include the generation of new metallic alloys and engineering ceramics [10]; additionally, recent trend has concentrated on the biomedical uses of functionally graded materials [11–13] as the biological materials often have such a structure in human body and are multifunctional. In this regard, a porous ceramic-metal FGM for being used in the main body of the femoral component is under study [14, 15]. Furthermore, the use of porous materials with advantage of multifunctionality, where bone ingrowth to the implant concurrently can couple with appropriate stiffness, strength, and density, is of particular interest [16]. The application of porous material was previously suggested for the femoral component and the location pegs in total knee prosthesis as a means to help reduce the stress shielding and micromotion as leading causes of aseptic loosening [8]. Therefore, in the current study, in addition to the geometry variables, two material options were also considered for the main body of femoral component including standard Co-Cr alloy and new functionally graded material. Furthermore, porous Ti material properties were chosen for the femoral peg when the base material was FGM.To plan the computational experiments, a full factorial design was used. Full factorial design covers all possible combinations of a set of factors including both continuous and categorical ones. This design is the most conservative of all design types, as there is little ambiguity when all combinations of the factor settings are examined. The number of experiments (runs) in full factorial design depends on the number of factors levels; for example, designs with only two-level factors contain2
f experimental runs where f is the number of factors. Unfortunately, the runs number grows exponentially in the number of factors; thus full factorial designs become unpractical when there are a large number of factors. However, for the purpose of the current study, it is useful since there are only 4 factors with two levels, which results in 16 computational experiments for each material. Table 1 shows the variables (factors) and their respective levels used, and Figure 4 represents the two-dimensional profiles of generated designs.Figure 4
Two-dimensional profiles of generated designs.


### 2.2. Quality Function Deployment

Extraction of all the relevant material/design selection criteria/indices needs a broad engineering knowledge and this sometimes makes it difficult for practitioners to take a decision related to design. In fact, even mistakenly missing a criterion, the results may be adversely affected [17]. Quality functional deployment is a systematic method for incorporating voices of the customers into the appropriate technical requirements, for each stage of product development and production. The structure of QFD can be thought of as a framework of a house, as shown in Figure 5. The house of quality translates the customer needs into material and design requirements that usually meet specific target values (Figure 5).Figure 5
The simplified house of quality.The external walls of the house are the customer requirements. On the left side is a listing of the voice of the customer or what the customer expects in the product. The relative importance of the customers’ requirements can be judged based on priority scale developed as 1: not important, 2: important, 3: much more important, 4: very important, and 5: most important. The ceiling or second floor of the house contains the technical descriptors. Consistency of the product is provided through materials and design characteristics. The interior walls of the house are the relationships between customer requirements and technical descriptors. To quantify the relations between the customers’ needs and technical requirements (“Whats” and “Hows”), an appropriate scale is set for assigning the relative importance as 5: moderate, 7: strong, and 9: very strong. The technical characteristics may be beneficial (higher values are desired) or nonbeneficial (lower values are preferred) or target based [18]. The roof of the house is the interrelationship between technical descriptors. Reich and Levy [19] improved the analysis related to roof of quality house when technical/engineering characteristics influence each other in asymmetric ways and their mutual influence varies in relation to different customer requirements. Given that in this research there is no significant subjective correlation between engineering characteristics, the roof of quality house was detached. Once the house of quality matrix is developed with the necessary data, the weight for each technical requirement is calculated using the following equation:(1)
W
j
=
∑
i
=
1
n
R
i
j
×
C
i
j
=
1,2
,
…
,
m
,where W
j is the absolute weight for jth technical requirement, n is the number of customers’ requirements, C
i is the priority assigned to ith customer requirement, m is the number of engineering characteristics, and R
i
j is weight assigned to the relationship between jth technical requirement and ith customer requirement. The last two rows of the prioritized technical descriptors are the absolute and relative weights. Figure 6 shows the process of translating customer needs to material and design selection criteria.Figure 6
Process of translating customer needs to material and design selection criteria.House of quality (HoQ) can be used as an initial aid to deal with the trade-offs and can be coupled with multicriteria decision analysis to deal with the design selection process strategically.


### 2.3. Finite Element Analysis

The 3D models of femoral component, polyethylene insert (related to commercial implant), and femur, from the previous studies [14, 20], were used to model the two-dimensional components (redrawn from the lateral sagittal section through the femoral peg center). Furthermore, the outer contour of femoral component was redrawn based on the original dimensions of the design using spline and was fixed in all design options. The finite element model was then modified to simulate various inner contours according to Table 1 and assessed using “ABAQUS V 6.9” software. Two types of materials were used for femoral components of TKR including Co-Cr alloy (as a standard material) and FGM (as a promising material) where 60% porous material properties were considered for the FGM peg according to the previous peg design optimization study [8]. For FGM, the gradation occurs in the direction from titanium with 70% porosity at uppermost surface of femoral component (bone interface) to alumina ceramic with no porosity at the lowermost surface based on the previous research on FGM optimization [15]. The material properties of a two-phase FGM were calculated based on the rule of mixture and volume fractions as follows [21]:(2)
V
c
=
y
h
k
V
m
=
1
-
V
c
p
=
A
y
h
n
1
-
y
h
z
,where(3)
n
+
z
/
n
n
1
-
n
/
n
+
z
z
≥
A
≥
0
.Here, h is the thickness of the component made of FGM, and k is the material gradient (controlling the variation in volume fraction of Ti and Al2O3). The porosity, p, is governed within the FGM structure by three parameters: A, n, and z. Young’s modulus and Poisson’s ratio are given by the following relationships:(4)
E
=
E
0
1
-
p
1
+
p
5
+
8
ν
37
-
8
ν
/
8
1
+
ν
23
+
8
ν
E
0
=
E
c
E
c
+
E
m
-
E
c
V
m
2
/
3
E
c
+
E
m
-
E
c
V
m
2
/
3
-
V
m
ν
=
ν
m
V
m
+
ν
c
V
c
.In the above equations, E and ν signify Young’s modulus and Poisson’s ratio at various regions of the FGM implant, correspondingly, and E
0 is the equivalent elastic modulus at different zones of the implant when the porosity is zero, and lastly E
c, E
m, ν
c, and ν
m denote Young’s moduli and Poisson’s ratios for Ti and Al2O3, respectively. For porous Ti, Gibson and Ashby’s [22] equations were used to calculate the material properties as follows: (5)
E
f
=
C
1
E
a
p
∗
2
p
∗
=
p
f
p
a
=
1
-
P
t
,where p
∗, p
f, p
a, and P
t are, in turn, the relative density of the foam, density of the foam, density of the solid Ti, and porosity. The elastic moduli of the foam alloy and the solid alloy are represented by E
f and E
a, respectively. For constant parameter, C
1, a value of 1.0 was suggested as the porosity does not highly affect Poisson’s ratio [23]. All material properties in the current paper were presumed to be homogeneous isotropic, linearly elastic, except for the properties of polyethylene, which was assumed to be elastic-plastic (see Table 2 and Figure 7).Table 2
Materials properties used in FEA.
Materials
E (MPa)
ν
Co-Cr alloy
240000
0.3
Alumina ceramic
350000
0.21
Ti
100000
0.3
Cortical bone
17000
0.3
Cancellous bone
400
0.3Figure 7
Elastic-plastic behavior of ultra high molecular weight polyethylene.Contact conditions were defined to apply finite sliding for pairs of surfaces including master and slave surfaces. The contact was applied both between the distal surface of the femoral component and proximal surface of tibial insert (articulating surfaces) and between the distal surface of the femur and proximal surface of femoral component (bone/implant interface surfaces). The friction coefficients between the femoral and tibial components were presumed to be 0.04 for Co-Cr alloy and 0.03 for alumina (last layer of FGM) [24, 25]. The friction coefficients between the femur and the prosthetic femur were assigned as 0.54 to simulate the porous-coated Co-Cr alloy and 0.62 and 0.65 for 60% and 70% porous Ti, respectively, which were associated with the porosity of femoral peg and the uppermost layer of FGM correspondingly. The values of porosity and the respective friction coefficients have been measured and reported in several studies [16, 26]. Increase in porosity was assumed to cause an increase in coefficient of friction as it has been identified that highly porous coatings have higher surface frictional characteristics [16]. Meanwhile, all components were meshed using 4-node bilinear plane stress quadrilateral reduced-integration (CPS4R) and 3-node linear plane stress triangular (CPS3) elements. Using reduced-integration causes less time to run the analysis but sometimes it may influence the accuracy. However, fine mesh of these elements can yield the acceptable results. Thus the global mesh size was chosen to be 1 mm (see Figure 8) based on a convergence test on the solution of von-Mises stresses on the femur and the peak contact stress of the polyethylene insert. Applying this mesh size resulted in 598 elements for polyethylene insert in all models and 1083 and 2535 elements for femoral component and femur in commercial implant model, respectively. The number of elements was 980–1380 and 2347–2587, respectively, for femoral component and femur in different design models.Figure 8
Finite element mesh including CPS4R and CPS3.The FEA of models with different inner contours for both materials were run under a load of 600 N (pressure) applied to the top surface of femur. For boundary conditions, the distal surface of the polyethylene insert was fully constrained from rotation and translation, and the femur was constrained from rotating in two directions whereas it was allowed to translate in the inferior-superior direction. To acquire the stress values in the femur, 10 regions of interest (ROI) were defined behind the interfaces (inner contours) as given in Figure9. Totally, the stresses were obtained at 250 random points, and after checking the normal distribution of data in each region, the means and standard deviations of stresses were calculated within each region. The minimum of mean stresses and the maximum of stresses STDV of ROIs were taken into account as performance outputs to indicate the stress shielding. Maximum tangential displacement (CSLIP) of femoral component relative to the femur was predicted as a measure of micromotion and implant stability.Figure 9
Regions of interest in the femoral bone.


### 2.4. Multiattribute Decision for Design Selection

TOPSIS has superior characteristics over other multiattribute decision making (MADM) methods and have been used widely for real world selection problems. Therefore in this paper, the extended version [27] of this technique will be used for ranking of material and design options. TOPSIS technique is based on the principle that the optimal point should have the shortest distance from the positive ideal solution and the farthest from the negative ideal solution (Figure 10).Figure 10
Ideal and anti-ideal solutions with two benefit criteria.The steps of extended TOPSIS can be expressed as follows.(1) Convert the raw measuresx
i
j into the standardized measures r
i
j:(6)
r
i
j
=
1
-
x
i
j
-
T
j
Max
⁡
x
i
j
m
a
x
,
T
j
-
Min
⁡
x
i
j
m
i
n
,
T
j
.(2) Develop a set of importance weights (w
j) for criteria.(3) Multiply the columns of the normalized decision matrix by the associated weights:(7)
V
i
j
=
r
i
j
w
j
;
j
=
1,2
,
3
,
…
,
n
;
i
=
1,2
,
3
,
…
,
m
.(4) Identify the PIS:(8)
V
1
+
,
V
2
+
,
V
3
+
,
…
,
V
n
+
=
M
a
x
i
V
i
j
∣
i
=
1
,
…
,
m
.(5) Identify the NIS:(9)
V
1
-
,
V
2
-
,
V
3
-
,
…
,
V
n
-
=
M
i
n
i
V
i
j
∣
i
=
1
,
…
,
m
.(6) Develop a distance measure for each alternative to both ideal (D
+) and nadir (D
-) (10)
D
i
+
=
∑
j
=
1
n
V
i
j
-
V
j
+
2
0.5
i
=
1,2
,
3
,
…
,
m
D
i
-
=
∑
j
=
1
n
V
i
j
-
V
j
-
2
0.5
i
=
1,2
,
3
,
…
,
m
.(7) Calculate the relative closeness to the ideal solution according to(11)
C
i
=
D
i
-
D
i
-
+
D
i
+
;
i
=
1,2
,
3
,
…
,
m
;
0
<
C
i
<
1
.(8) Rank alternatives by maximizing the ratio in Step (7). The larger the index value, the better the performance of the alternative.


## 2.1. Design Variables and Design of Experiments

Various knee prosthesis designs are currently available in which the basic designs of femoral components are analogous; however the interface geometry of this component including the location pegs, the angles associated with the inner contour, and their respective lengths vary from a particular design to another [6]. The influence of location pegs on stress shielding effect was studied by Wang et al. [7] in a two-dimensional finite element analysis and by Bahraminasab et al. [8] in three-dimensional FEA and with respect to the materials used. In the present study, in addition to the effect of materials, the influences of inner cuts (anterior, posterior, and oblique cuts) on the biomechanical performance were investigated, whilst the outer contours of prosthetic femur were preserved as the original design, since the articulating surfaces are complex curves (single curve or multicurves) with precise transitions which identify the contact behavior with the polyethylene component [9] and are design-specific. Here, four geometrical design variables were taken into account including the angle between the distal and oblique anterior cuts (β), the angle between the distal and anterior cuts (γ), the angle between the distal and oblique posterior cuts (δ), and the angle between the distal and posterior cuts (ε). While these variables were changed between the ranges identified in Table 1, the peg position, total length (L
T), length of distal cut (L
D), total height of anterior and anterior oblique cuts (H
A), and total height of posterior and posterior oblique cuts (H
P) were kept constant (points 1–4 were fixed). Figure 3 presents two-dimensional (2D) profile of femoral component with four geometrical design variables and constraints used.Table 1
Geometrical variables and their respective levels.
Run number
Factor 1
Factor 2
Factor 3
Factor 4
A: beta (deg)
B: gamma (deg)
C: delta (deg)
D: epsilon (deg)
1
125
98
140
88
2
125
98
125
88
3
140
90
140
94
4
125
98
140
94
5
125
90
125
94
6
140
98
140
94
7
140
98
140
88
8
125
90
140
88
9
125
98
125
94
10
140
90
125
94
11
140
90
140
88
12
140
98
125
94
13
125
90
125
88
14
125
90
140
94
15
140
90
125
88
16
140
98
125
88Figure 3
Design variables of the inner contour.Femoral component as a key feature in current knee prostheses has been the focus of many studies related to materials. Scholars have been working on tailor and development of new biomaterials for this component to avoid mismatch of engineered materials properties with those of the biological ones and to provide acceptable performance without short-term and long-term failure. These include the generation of new metallic alloys and engineering ceramics [10]; additionally, recent trend has concentrated on the biomedical uses of functionally graded materials [11–13] as the biological materials often have such a structure in human body and are multifunctional. In this regard, a porous ceramic-metal FGM for being used in the main body of the femoral component is under study [14, 15]. Furthermore, the use of porous materials with advantage of multifunctionality, where bone ingrowth to the implant concurrently can couple with appropriate stiffness, strength, and density, is of particular interest [16]. The application of porous material was previously suggested for the femoral component and the location pegs in total knee prosthesis as a means to help reduce the stress shielding and micromotion as leading causes of aseptic loosening [8]. Therefore, in the current study, in addition to the geometry variables, two material options were also considered for the main body of femoral component including standard Co-Cr alloy and new functionally graded material. Furthermore, porous Ti material properties were chosen for the femoral peg when the base material was FGM.To plan the computational experiments, a full factorial design was used. Full factorial design covers all possible combinations of a set of factors including both continuous and categorical ones. This design is the most conservative of all design types, as there is little ambiguity when all combinations of the factor settings are examined. The number of experiments (runs) in full factorial design depends on the number of factors levels; for example, designs with only two-level factors contain2
f experimental runs where f is the number of factors. Unfortunately, the runs number grows exponentially in the number of factors; thus full factorial designs become unpractical when there are a large number of factors. However, for the purpose of the current study, it is useful since there are only 4 factors with two levels, which results in 16 computational experiments for each material. Table 1 shows the variables (factors) and their respective levels used, and Figure 4 represents the two-dimensional profiles of generated designs.Figure 4
Two-dimensional profiles of generated designs.


## 2.2. Quality Function Deployment

Extraction of all the relevant material/design selection criteria/indices needs a broad engineering knowledge and this sometimes makes it difficult for practitioners to take a decision related to design. In fact, even mistakenly missing a criterion, the results may be adversely affected [17]. Quality functional deployment is a systematic method for incorporating voices of the customers into the appropriate technical requirements, for each stage of product development and production. The structure of QFD can be thought of as a framework of a house, as shown in Figure 5. The house of quality translates the customer needs into material and design requirements that usually meet specific target values (Figure 5).Figure 5
The simplified house of quality.The external walls of the house are the customer requirements. On the left side is a listing of the voice of the customer or what the customer expects in the product. The relative importance of the customers’ requirements can be judged based on priority scale developed as 1: not important, 2: important, 3: much more important, 4: very important, and 5: most important. The ceiling or second floor of the house contains the technical descriptors. Consistency of the product is provided through materials and design characteristics. The interior walls of the house are the relationships between customer requirements and technical descriptors. To quantify the relations between the customers’ needs and technical requirements (“Whats” and “Hows”), an appropriate scale is set for assigning the relative importance as 5: moderate, 7: strong, and 9: very strong. The technical characteristics may be beneficial (higher values are desired) or nonbeneficial (lower values are preferred) or target based [18]. The roof of the house is the interrelationship between technical descriptors. Reich and Levy [19] improved the analysis related to roof of quality house when technical/engineering characteristics influence each other in asymmetric ways and their mutual influence varies in relation to different customer requirements. Given that in this research there is no significant subjective correlation between engineering characteristics, the roof of quality house was detached. Once the house of quality matrix is developed with the necessary data, the weight for each technical requirement is calculated using the following equation:(1)
W
j
=
∑
i
=
1
n
R
i
j
×
C
i
j
=
1,2
,
…
,
m
,where W
j is the absolute weight for jth technical requirement, n is the number of customers’ requirements, C
i is the priority assigned to ith customer requirement, m is the number of engineering characteristics, and R
i
j is weight assigned to the relationship between jth technical requirement and ith customer requirement. The last two rows of the prioritized technical descriptors are the absolute and relative weights. Figure 6 shows the process of translating customer needs to material and design selection criteria.Figure 6
Process of translating customer needs to material and design selection criteria.House of quality (HoQ) can be used as an initial aid to deal with the trade-offs and can be coupled with multicriteria decision analysis to deal with the design selection process strategically.


## 2.3. Finite Element Analysis

The 3D models of femoral component, polyethylene insert (related to commercial implant), and femur, from the previous studies [14, 20], were used to model the two-dimensional components (redrawn from the lateral sagittal section through the femoral peg center). Furthermore, the outer contour of femoral component was redrawn based on the original dimensions of the design using spline and was fixed in all design options. The finite element model was then modified to simulate various inner contours according to Table 1 and assessed using “ABAQUS V 6.9” software. Two types of materials were used for femoral components of TKR including Co-Cr alloy (as a standard material) and FGM (as a promising material) where 60% porous material properties were considered for the FGM peg according to the previous peg design optimization study [8]. For FGM, the gradation occurs in the direction from titanium with 70% porosity at uppermost surface of femoral component (bone interface) to alumina ceramic with no porosity at the lowermost surface based on the previous research on FGM optimization [15]. The material properties of a two-phase FGM were calculated based on the rule of mixture and volume fractions as follows [21]:(2)
V
c
=
y
h
k
V
m
=
1
-
V
c
p
=
A
y
h
n
1
-
y
h
z
,where(3)
n
+
z
/
n
n
1
-
n
/
n
+
z
z
≥
A
≥
0
.Here, h is the thickness of the component made of FGM, and k is the material gradient (controlling the variation in volume fraction of Ti and Al2O3). The porosity, p, is governed within the FGM structure by three parameters: A, n, and z. Young’s modulus and Poisson’s ratio are given by the following relationships:(4)
E
=
E
0
1
-
p
1
+
p
5
+
8
ν
37
-
8
ν
/
8
1
+
ν
23
+
8
ν
E
0
=
E
c
E
c
+
E
m
-
E
c
V
m
2
/
3
E
c
+
E
m
-
E
c
V
m
2
/
3
-
V
m
ν
=
ν
m
V
m
+
ν
c
V
c
.In the above equations, E and ν signify Young’s modulus and Poisson’s ratio at various regions of the FGM implant, correspondingly, and E
0 is the equivalent elastic modulus at different zones of the implant when the porosity is zero, and lastly E
c, E
m, ν
c, and ν
m denote Young’s moduli and Poisson’s ratios for Ti and Al2O3, respectively. For porous Ti, Gibson and Ashby’s [22] equations were used to calculate the material properties as follows: (5)
E
f
=
C
1
E
a
p
∗
2
p
∗
=
p
f
p
a
=
1
-
P
t
,where p
∗, p
f, p
a, and P
t are, in turn, the relative density of the foam, density of the foam, density of the solid Ti, and porosity. The elastic moduli of the foam alloy and the solid alloy are represented by E
f and E
a, respectively. For constant parameter, C
1, a value of 1.0 was suggested as the porosity does not highly affect Poisson’s ratio [23]. All material properties in the current paper were presumed to be homogeneous isotropic, linearly elastic, except for the properties of polyethylene, which was assumed to be elastic-plastic (see Table 2 and Figure 7).Table 2
Materials properties used in FEA.
Materials
E (MPa)
ν
Co-Cr alloy
240000
0.3
Alumina ceramic
350000
0.21
Ti
100000
0.3
Cortical bone
17000
0.3
Cancellous bone
400
0.3Figure 7
Elastic-plastic behavior of ultra high molecular weight polyethylene.Contact conditions were defined to apply finite sliding for pairs of surfaces including master and slave surfaces. The contact was applied both between the distal surface of the femoral component and proximal surface of tibial insert (articulating surfaces) and between the distal surface of the femur and proximal surface of femoral component (bone/implant interface surfaces). The friction coefficients between the femoral and tibial components were presumed to be 0.04 for Co-Cr alloy and 0.03 for alumina (last layer of FGM) [24, 25]. The friction coefficients between the femur and the prosthetic femur were assigned as 0.54 to simulate the porous-coated Co-Cr alloy and 0.62 and 0.65 for 60% and 70% porous Ti, respectively, which were associated with the porosity of femoral peg and the uppermost layer of FGM correspondingly. The values of porosity and the respective friction coefficients have been measured and reported in several studies [16, 26]. Increase in porosity was assumed to cause an increase in coefficient of friction as it has been identified that highly porous coatings have higher surface frictional characteristics [16]. Meanwhile, all components were meshed using 4-node bilinear plane stress quadrilateral reduced-integration (CPS4R) and 3-node linear plane stress triangular (CPS3) elements. Using reduced-integration causes less time to run the analysis but sometimes it may influence the accuracy. However, fine mesh of these elements can yield the acceptable results. Thus the global mesh size was chosen to be 1 mm (see Figure 8) based on a convergence test on the solution of von-Mises stresses on the femur and the peak contact stress of the polyethylene insert. Applying this mesh size resulted in 598 elements for polyethylene insert in all models and 1083 and 2535 elements for femoral component and femur in commercial implant model, respectively. The number of elements was 980–1380 and 2347–2587, respectively, for femoral component and femur in different design models.Figure 8
Finite element mesh including CPS4R and CPS3.The FEA of models with different inner contours for both materials were run under a load of 600 N (pressure) applied to the top surface of femur. For boundary conditions, the distal surface of the polyethylene insert was fully constrained from rotation and translation, and the femur was constrained from rotating in two directions whereas it was allowed to translate in the inferior-superior direction. To acquire the stress values in the femur, 10 regions of interest (ROI) were defined behind the interfaces (inner contours) as given in Figure9. Totally, the stresses were obtained at 250 random points, and after checking the normal distribution of data in each region, the means and standard deviations of stresses were calculated within each region. The minimum of mean stresses and the maximum of stresses STDV of ROIs were taken into account as performance outputs to indicate the stress shielding. Maximum tangential displacement (CSLIP) of femoral component relative to the femur was predicted as a measure of micromotion and implant stability.Figure 9
Regions of interest in the femoral bone.


## 2.4. Multiattribute Decision for Design Selection

TOPSIS has superior characteristics over other multiattribute decision making (MADM) methods and have been used widely for real world selection problems. Therefore in this paper, the extended version [27] of this technique will be used for ranking of material and design options. TOPSIS technique is based on the principle that the optimal point should have the shortest distance from the positive ideal solution and the farthest from the negative ideal solution (Figure 10).Figure 10
Ideal and anti-ideal solutions with two benefit criteria.The steps of extended TOPSIS can be expressed as follows.(1) Convert the raw measuresx
i
j into the standardized measures r
i
j:(6)
r
i
j
=
1
-
x
i
j
-
T
j
Max
⁡
x
i
j
m
a
x
,
T
j
-
Min
⁡
x
i
j
m
i
n
,
T
j
.(2) Develop a set of importance weights (w
j) for criteria.(3) Multiply the columns of the normalized decision matrix by the associated weights:(7)
V
i
j
=
r
i
j
w
j
;
j
=
1,2
,
3
,
…
,
n
;
i
=
1,2
,
3
,
…
,
m
.(4) Identify the PIS:(8)
V
1
+
,
V
2
+
,
V
3
+
,
…
,
V
n
+
=
M
a
x
i
V
i
j
∣
i
=
1
,
…
,
m
.(5) Identify the NIS:(9)
V
1
-
,
V
2
-
,
V
3
-
,
…
,
V
n
-
=
M
i
n
i
V
i
j
∣
i
=
1
,
…
,
m
.(6) Develop a distance measure for each alternative to both ideal (D
+) and nadir (D
-) (10)
D
i
+
=
∑
j
=
1
n
V
i
j
-
V
j
+
2
0.5
i
=
1,2
,
3
,
…
,
m
D
i
-
=
∑
j
=
1
n
V
i
j
-
V
j
-
2
0.5
i
=
1,2
,
3
,
…
,
m
.(7) Calculate the relative closeness to the ideal solution according to(11)
C
i
=
D
i
-
D
i
-
+
D
i
+
;
i
=
1,2
,
3
,
…
,
m
;
0
<
C
i
<
1
.(8) Rank alternatives by maximizing the ratio in Step (7). The larger the index value, the better the performance of the alternative.


## 3. Results and Discussion

### 3.1. Influence of Design Variables

The effects of the design variables on the responses were determined, after checking the normal probability plot of residuals using Design Expert software version 7.0.0. Prestatistical analysis demonstrated less contribution of material to the design objectives comparing to geometrical variables. For instance, the effect of material on the minimum of mean stresses at different region was less than all geometrical variables except forδ (the angle between the distal and oblique posterior cuts). Furthermore, material was identified not to be statistically significant factor on the maximum standard deviation of ROI stresses. For maximum micromotion at bone-implant interface, the influence of material was higher than δ and β (the angle between the distal and oblique anterior cuts) but much lower than γ (the angle between the distal and anterior cuts) and ε (the angle between the distal and posterior cuts). However, the obtained values showed the superiority of FGM over Co-Cr alloy for all above mentioned objectives. Hence, for more precise evaluation of geometrical variables, ANOVA was conducted separately based on the collected data for each material. Table 3 shows the ANOVA table of the model for the first response, minimum of mean stresses at ROI in femur, when the material used was Co-Cr alloy. The F value of the model (F
=
41.81188) is high and the p value is less than 0.05, meaning that the model is statistically significant. Consequently, the main significant terms are β (the angle between the distal and oblique anterior cuts), γ (the angle between the distal and anterior cuts), and ε (the angle between the distal and posterior cuts) but it seems delta is not significant. Furthermore, the interaction effects (effect of one factor on the response changes depends on the level of another factor) of βε and γε are significant.Table 3
ANOVA results for minimum of mean stresses in femur for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
16.17977
10
1.617977
41.81188
0.0003
A-beta
0.426201
1
0.426201
11.01392
0.021∗
B-gamma
4.833423
1
4.833423
124.9056
0.0001∗
C-delta
0.055955
1
0.055955
1.445983
0.283
D-epsilon
8.626668
1
8.626668
222.9309
<0.0001∗
AB
0.209491
1
0.209491
5.413679
0.0675
AC
0.018615
1
0.018615
0.481052
0.5188
AD
0.284554
1
0.284554
7.353474
0.0422∗
BC
0.029604
1
0.029604
0.765017
0.4218
BD
1.676965
1
1.676965
43.33623
0.0012∗
CD
0.018299
1
0.018299
0.472881
0.5223
Residual
0.193483
5
0.038697
Total
16.37326
15
∗Denotes significant factors.The interactions ofβε and γε on minimum of mean stresses are given in Figures 11(a) and 11(b). It can be indicated that changes in β does not significantly influence the response when the angle between the distal and posterior cuts is at the high level of this factor (ε = 94°, dash line in the diagram), while for the low level of this factor (ε = 88°, continuous line in the diagram), the changes in β cause higher variations in the minimum stress. Furthermore, the interaction of factors γε implies that the effect of factor γ (the angle between the distal and anterior cuts) on minimum of mean stresses is greater for low level of factor ε. Based on both interactions low level of factor ε positively affects the minimum stress of ROI. However, higher values of β and lower values of γ are desirable within the studied ranges of variables, for this response as the higher stress levels in the bone result in the reduction of stress shielding effect and the successive bone resorption.Figure 11
Interaction effects of design variables on minimum of mean stresses of ROI: (a) interaction effect ofβ and ε, (b) interaction effect of γ and ε.
(a)
(b)The ANOVA table of the model for the maximum standard deviation of ROI stresses (Table4) shows that the main influential factors are γ, δ, and ε; additionally, the interaction effect of δε is also significant. As it can be seen in Figure 12(a), when the angle between the distal and anterior cuts increases, STDV of stresses in the bone as a measure of nonuniformity in the stress distribution of the femur will decrease. Furthermore, the interaction of δε shows that changes in STDV by changing the angle between the distal and oblique posterior cuts are dramatic when the angle between the distal and posterior cuts is at the low level of this factor (ε = 88°). However, at the high level of this factor (ε = 94°) changes in the response by varying the angle between the distal and oblique posterior cuts are very slight. The results within the studied ranges of these two factors indicated that higher value of ε and lower value of δ can reduce the STDV of stresses.Table 4
ANOVA results for maximum STDV of stresses in femur for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
7.708196
10
0.77082
12.39438
0.0062
A-beta
0.092434
1
0.092434
1.486293
0.2772
B-gamma
3.893795
1
3.893795
62.6102
0.0005∗
C-delta
0.787533
1
0.787533
12.66312
0.0162∗
D-epsilon
1.484719
1
1.484719
23.87351
0.0045∗
AB
0.349913
1
0.349913
5.626419
0.0638
AC
0.106938
1
0.106938
1.71951
0.2467
AD
0.008447
1
0.008447
0.135817
0.7276
BC
0.01
1
0.01
0.160796
0.705
BD
0.003122
1
0.003122
0.050203
0.8316
CD
0.971295
1
0.971295
15.61791
0.0108∗
Residual
0.310955
5
0.062191
Total
8.019152
15
∗Denotes significant factors.Figure 12
Effects of design variables on maximum STDV of stresses in ROIs: (a) main effect ofγ, (b) interaction effect of δ and ε.
(a)
(b)The ANOVA table of the model for the maximum micromotion at bone-implant interface is presented in Table5, which identifies that the main significant factors are γ, β, and ε, and the influential interactions are γε and βγ. From Figure 13 it can be interpreted that lower values for the angle between the distal and posterior cuts and higher values for the angle between the distal and anterior cuts (ε = 88° and γ = 98°) will reduce the micromotion at the interface (Figure 13(a)). Meanwhile, the interaction of βγ also shows that higher value of γ causes less micromotion, and at high level of factor γ (γ = 98°), changes of β have negligible effect on the response, while at the low level of factor γ (γ = 90°), variations of β more influence the response (Figure 13(b)).Table 5
ANOVA results for maximum micromotion at femur-implant interface for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
11975.86
10
1197.586
1918.347
<0.0001
A-beta
90.0601
1
90.0601
144.2624
<0.0001∗
B-gamma
2613.254
1
2613.254
4186.029
<0.0001∗
C-delta
2.0449
1
2.0449
3.275614
0.1301
D-epsilon
9015.503
1
9015.503
14441.44
<0.0001∗
AB
89.6809
1
89.6809
143.6549
<0.0001∗
AC
1.1025
1
1.1025
1.766034
0.2413
AD
2.9929
1
2.9929
4.794163
0.0801
BC
2.5281
1
2.5281
4.049625
0.1003
BD
158.5081
1
158.5081
253.9055
<0.0001∗
CD
0.1849
1
0.1849
0.296181
0.6097
Residual
3.1214
5
0.62428
Total
11978.98
15
∗Denotes significant factors.Figure 13
Effects of design variables on maximum micromotion at femur-implant interface: (a) interaction effect ofγ and ε, (b) interaction effect of γ and β.
(a)
(b)The influential parameters for FGM femoral component were also identified. Same factors are significant for minimum of mean stresses and maximum of STDV of stresses (see Table6). However, for micromotion the influential factors are different. This is probably due to different interface characteristics of the two materials.Table 6
Comparison of significant design factors for Co-Cr and FGM in all responses.
Influential factors
Min of mean stress
Max of STDV stress
Max of micromotion
Co-Cr
β
(
+
)
γ
(
-
)
ε
(
-
)
β
ε
(
-
)
γ
ε
(
+
)
δ
(
-
)
γ
(
-
)
ε
(
-
)
δ
ε
(
+
)
β
(
+
)
γ
(
-
)
ε
(
+
)
β
γ
(
-
)
γ
ε
(
-
)
FGM
β
(
+
)
γ
(
-
)
ε
(
-
)
β
ε
(
-
)
γ
ε
(
+
)
γ
(
-
)
ε
(
-
)
δ
ε
(
+
)
β
(
+
)
γ
(
-
)
ε
(
+
)
δ
(
+
)
β
γ
(
-
)
γ
ε
(
-
)
β
ε
(
-
)
γ
δ
(
-
)
δ
ε
(
+
)
Positive (+) or negative (−) effect of the main and interactions effects (coefficient of regression models).


### 3.2. House of Quality for Shaping and Weighting Design Selection Criteria

The conceptual design begins with gathering of the voices of the customer as functional requirements. Expectations of customers (patients), medical, and design teams from implant (femoral component) in a knee replacement surgery are shown in Figure14. Needs start the redesign/design process and a high-quality design will strike the best combination of wants and needs in the specifications.Figure 14
Customers performance needs in design of femoral component and complementary requirements detected by the medical and design team.Figure15 shows how the technical requirements are divided to design selection criteria (for ranking of design alternatives for each material) and criteria of material and design selection (ranking of designs for both materials). Given that it is important to maximize the minimum observed mean stress in 10 ROIs (Figure 9), the noncompensatory model of Maximin was used to consider the mean stress of the most vulnerable region. In the same way, for criterion of stresses STDV, maximum observed standard deviation in different regions was reported. For design selection criteria, in addition to minimum of stress mean, maximum of stress STDV, and maximum contact slip at femoral component/bone interface, area of cross section as an index of mass was considered. However, in final selection, multiplication of cross section area and density of material was used to be representative of mass index. Meanwhile, the maximum peg stress and maximum stress at corner points of inner contour were taken into account. Maximum peg stress which was on the peg root is very important due to the transition from the main body to the peg. The high stress in this region is even worse when the cross section is porous. This is critical because of the adverse effect on the component which might cause premature peg failure through root fracture. Maximum stress at corner points of inner contour is also important from the stress concentration point of view, especially for the FGM with the porous layer interfacing the bone. For final selection these two criteria were replaced by safety index of peg and safety index of main body in which the differences between yield of material and maximum observed stress were used. This is to avoid misleading results of traditional ratio based performance indices [28]. Furthermore, in final selection, biocompatibility and hardness were also considered. Table 7 demonstrates house of quality for translation of customer needs to engineering characteristics. Absolute weights (importance) for material and design selection criteria are also shown in this table. For instance, in criterion of “stress level in femur,” absolute weight (36) is multiplication of importance degree (4) for related customer requirement (Req 5 as indicated in Figure 14) and type of relationship between customer and technical requirement (strong: 9). The last two rows in Table 7, respectively, show relative importance of criteria (weight), for “design selection” (Tables 8 and 9) and “material and design selection” (Table 10), that will be used for ranking via TOPSIS method.Table 7
House of quality including requirements, engineering characteristics, and obtained weights.
Degree of importance
Engineering characteristics (How)








Stress level in femur (A)
Nonuniformity of stress distribution in femur (B)
Displacement at bone-implant interface (C)
Stress level at femoral corner points (F)
Stress level in peg (D)
Mass (E)
Biocompatibility (G)
Hardness (H)
Prioritized customerRequirements (what)
Req1
2
●
Req2
3
⊙
●
Req3
5
◯
●
Req4
4
●
Req5
4
●
⊙
Req6
4
●
Absolute weight
36
28
36
21
27
43
45
36
Relative weight for design selection criteria (input for Tables8 and 9)
0.19
0.15
0.19
0.11
0.14
0.23
—
—
Relative weight for material and design selection criteria (input for Table10)
0.13
0.10
0.13
0.08
0.10
0.16
0.17
0.13
Prioritized technical descriptors
+9 ● strong.
+7 ⊙ moderate.
+5 ◯weak.Table 8
Performance of Co-Cr alloy designs in different criteria and ranking orders scenarios by TOPSIS method.
Objectives
Max
Min
Min
Min
Min
Min
Relative closeness to the ideal solution
Ranking
Relative weights of design criteria
0.19
0.15
0.19
0.11
0.14
0.23
A
B
C
D
E
F
Design 1
2.919
3.116
34.70
24.67
900.79
115.4
0.4080
12
Design 2
3.605
3.428
44.60
24.74
918.17
114.7
0.4075
13
Design 3
1.570
2.189
20.13
26.22
804.59
72.9
0.6180
1
Design 4
1.771
2.221
21.35
26.26
925.80
108.2
0.4315
10
Design 5
2.962
2.682
35.46
27.05
901.99
73.9
0.5865
4
Design 6
4.244
1.884
45.55
27.35
874.13
109.8
0.5212
6
Design 7
1.600
1.306
20.27
27.79
849.11
122.3
0.4487
9
Design 8
1.804
1.336
21.50
27.86
867.89
86.9
0.5750
5
Design 9
1.213
2.464
87.99
17.00
934.89
101.5
0.2863
16
Design 10
1.352
1.815
97.54
25.52
813.68
79.0
0.4637
8
Design 11
0.845
0.958
61.94
10.05
779.58
69.0
0.6107
2
Design 12
0.851
1.309
62.35
10.12
883.21
109.8
0.4086
11
Design 13
1.293
2.470
90.74
17.13
885.27
85.9
0.3960
15
Design 14
1.408
1.813
99.12
25.55
892.91
68.3
0.4690
7
Design 15
0.892
1.148
63.25
10.11
796.96
68.3
0.6033
3
Design 16
0.871
1.311
60.43
10.10
866.49
116.5
0.4000
14
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: maximum peg stress (MPa).
E: area of cross section (mm2).
F: maximum stress at corner points of inner contour (MPa).Table 9
Performance of FGM designs in different criteria and ranking orders scenarios by TOPSIS method.
Objectives
Max
Min
Min
Min
Min
Min
Relative closeness to the ideal solution
Ranking
Relative weights of design criteria
0.19
0.15
0.19
0.11
0.14
0.23
A
B
C
D
E
F
Design 1
3.100
2.655
30.61
20.16
900.79
14.4
0.6029
4
Design 2
3.818
3.449
39.58
23.54
918.17
14.3
0.5540
8
Design 3
1.707
2.208
17.93
14.09
804.59
13.1
0.6566
3
Design 4
1.925
2.245
19.16
14.34
925.8
12.2
0.6011
5
Design 5
3.150
2.668
31.24
21.26
901.99
29.0
0.4385
14
Design 6
4.466
1.866
40.36
24.17
874.13
13.8
0.6730
1
Design 7
1.741
1.249
18.03
14.71
849.11
14.7
0.6581
2
Design 8
1.961
1.349
19.25
14.97
867.89
33
0.4678
11
Design 9
1.327
2.456
79.44
12.76
934.89
12.8
0.4582
12
Design 10
1.480
1.814
87.88
15.83
813.68
18.6
0.4582
13
Design 11
0.932
1.017
57.53
7.55
779.58
22.2
0.5300
9
Design 12
0.941
1.326
57.93
7.86
883.21
12.0
0.5603
6
Design 13
1.353
2.460
83.24
12.88
885.27
17.5
0.4180
15
Design 14
1.462
1.835
90.14
15.81
892.91
18.4
0.4120
16
Design 15
0.940
1.206
58.59
7.59
796.96
22.6
0.5103
10
Design 16
0.934
1.328
58.56
7.88
866.49
13.6
0.5552
7
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: maximum peg stress (MPa).
E: area of cross section (mm2).
F: maximum stress at corner points of inner contour (MPa).Table 10
Ranking of top materials and designs scenarios with regard to all technical criteria obtained from QFD.
Objectives
Max
Min
Min
Max
Min
Max
Max
Max
Relative closeness to the ideal solution
Ranking
Relative weights
0.13
0.10
0.13
0.08
0.10
0.16
0.17
0.13
Design options
A
B
C
D
E
F
G
H
Co-Cr alloy
Original
1.86
1.69
21.93
421.4
7.04
339.5
1
1
0.4531
6
Design 3
1.570
2.189
20.13
423.78
6.68
377.1
1
1
0.4554
5
Design 11
0.845
0.958
61.94
439.95
6.47
381
1
1
0.4142
8
Design 15
0.892
1.148
63.25
439.89
6.61
381.7
1
1
0.4064
9
Design 5
2.962
2.682
35.46
422.95
7.49
376.1
1
1
0.4367
7
FGM
Design 6
4.466
1.866
40.36
24.83
3.03
18.2
3
3
0.5868
1
Design 7
1.741
1.249
18.03
34.29
2.95
17.3
3
3
0.5809
2
Design 3
1.707
2.208
17.93
34.91
2.79
18.9
3
3
0.5584
3
Design 1
3.100
2.655
30.61
28.84
3.13
17.6
3
3
0.5537
4
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: safety index of peg (difference between yield of material and maximum peg stress which is 450 and 49 MPa for Co-Cr alloy and FGM resp.).
E: weight index (area of cross section∗ density (g/cm)).
F: safety index of main body (difference between yield of material and maximum stress at corner points of inner contour in which yields of 450 and 32 MPa for Co-Cr alloy and FGM were considered, resp.).
G: biocompatibility of material.
H: hardness of interface material with PE insert.Figure 15
One by one matching of technical requirements with selection criteria.


### 3.3. Ranking of Design Alternatives

Tables8 and 9, respectively, show an overview of designs performance for Co-Cr alloy and FGM. Among the criteria for design selection, only criterion of “minimum of mean stress in different regions” has maximization objective and for the other criteria, including “maximum of STDV of stresses in different regions,” “maximum contact slip at femoral component/bone interface,” “maximum peg stress,” “area of cross section,” and “maximum stress at corner points of inner contour” less is better. In both tables, relative weights are obtained from QFD’s absolute weight, with regard to considered criteria for design selection stage.It can be seen from Tables8 and 9 that the designs 3, 11, 15, and 5 are the four top designs for Co-Cr alloy, and designs 6, 7, 3, and 1 are the top ranked designs for FGM. Table 10 shows these eight high rank designs and the original design with Co-Cr alloy. To select the best material and design, criteria of “minimum of stress mean in different regions,” “safety index of peg,” “safety index of main body,” “biocompatibility of material,” and “hardness of interface material with PE insert” should be maximized. For other criteria, including “maximum of stress STDV in different regions,” “maximum contact slip at femoral component/bone interface,” and “Weight index” lower values are desirable. The most striking result to emerge from Table 10 is that design 6 with FGM is the top rank design. It can also be seen from ranking orders that the designs with FGM are better than Co-Cr alloy. Design 3 is the only common option in the list of four top rank designs of each material. It clarifies the high contribution of selected material in the performance of femoral component. What is interesting here is that the original design has the second rank in the designs made from Co-Cr alloy. It is due to advantages of Co-Cr alloy-design 3 over original one in criteria “maximum contact slip at femoral component/bone interface,” “Safety index of peg,” “weight index,” and “safety index of main body.” Although it seems that by considering only a single performance output (stress mean in different regions), the original design is better than Co-Cr alloy-design 3, it is clear that design 5 is better than the original one in this condition.


## 3.1. Influence of Design Variables

The effects of the design variables on the responses were determined, after checking the normal probability plot of residuals using Design Expert software version 7.0.0. Prestatistical analysis demonstrated less contribution of material to the design objectives comparing to geometrical variables. For instance, the effect of material on the minimum of mean stresses at different region was less than all geometrical variables except forδ (the angle between the distal and oblique posterior cuts). Furthermore, material was identified not to be statistically significant factor on the maximum standard deviation of ROI stresses. For maximum micromotion at bone-implant interface, the influence of material was higher than δ and β (the angle between the distal and oblique anterior cuts) but much lower than γ (the angle between the distal and anterior cuts) and ε (the angle between the distal and posterior cuts). However, the obtained values showed the superiority of FGM over Co-Cr alloy for all above mentioned objectives. Hence, for more precise evaluation of geometrical variables, ANOVA was conducted separately based on the collected data for each material. Table 3 shows the ANOVA table of the model for the first response, minimum of mean stresses at ROI in femur, when the material used was Co-Cr alloy. The F value of the model (F
=
41.81188) is high and the p value is less than 0.05, meaning that the model is statistically significant. Consequently, the main significant terms are β (the angle between the distal and oblique anterior cuts), γ (the angle between the distal and anterior cuts), and ε (the angle between the distal and posterior cuts) but it seems delta is not significant. Furthermore, the interaction effects (effect of one factor on the response changes depends on the level of another factor) of βε and γε are significant.Table 3
ANOVA results for minimum of mean stresses in femur for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
16.17977
10
1.617977
41.81188
0.0003
A-beta
0.426201
1
0.426201
11.01392
0.021∗
B-gamma
4.833423
1
4.833423
124.9056
0.0001∗
C-delta
0.055955
1
0.055955
1.445983
0.283
D-epsilon
8.626668
1
8.626668
222.9309
<0.0001∗
AB
0.209491
1
0.209491
5.413679
0.0675
AC
0.018615
1
0.018615
0.481052
0.5188
AD
0.284554
1
0.284554
7.353474
0.0422∗
BC
0.029604
1
0.029604
0.765017
0.4218
BD
1.676965
1
1.676965
43.33623
0.0012∗
CD
0.018299
1
0.018299
0.472881
0.5223
Residual
0.193483
5
0.038697
Total
16.37326
15
∗Denotes significant factors.The interactions ofβε and γε on minimum of mean stresses are given in Figures 11(a) and 11(b). It can be indicated that changes in β does not significantly influence the response when the angle between the distal and posterior cuts is at the high level of this factor (ε = 94°, dash line in the diagram), while for the low level of this factor (ε = 88°, continuous line in the diagram), the changes in β cause higher variations in the minimum stress. Furthermore, the interaction of factors γε implies that the effect of factor γ (the angle between the distal and anterior cuts) on minimum of mean stresses is greater for low level of factor ε. Based on both interactions low level of factor ε positively affects the minimum stress of ROI. However, higher values of β and lower values of γ are desirable within the studied ranges of variables, for this response as the higher stress levels in the bone result in the reduction of stress shielding effect and the successive bone resorption.Figure 11
Interaction effects of design variables on minimum of mean stresses of ROI: (a) interaction effect ofβ and ε, (b) interaction effect of γ and ε.
(a)
(b)The ANOVA table of the model for the maximum standard deviation of ROI stresses (Table4) shows that the main influential factors are γ, δ, and ε; additionally, the interaction effect of δε is also significant. As it can be seen in Figure 12(a), when the angle between the distal and anterior cuts increases, STDV of stresses in the bone as a measure of nonuniformity in the stress distribution of the femur will decrease. Furthermore, the interaction of δε shows that changes in STDV by changing the angle between the distal and oblique posterior cuts are dramatic when the angle between the distal and posterior cuts is at the low level of this factor (ε = 88°). However, at the high level of this factor (ε = 94°) changes in the response by varying the angle between the distal and oblique posterior cuts are very slight. The results within the studied ranges of these two factors indicated that higher value of ε and lower value of δ can reduce the STDV of stresses.Table 4
ANOVA results for maximum STDV of stresses in femur for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
7.708196
10
0.77082
12.39438
0.0062
A-beta
0.092434
1
0.092434
1.486293
0.2772
B-gamma
3.893795
1
3.893795
62.6102
0.0005∗
C-delta
0.787533
1
0.787533
12.66312
0.0162∗
D-epsilon
1.484719
1
1.484719
23.87351
0.0045∗
AB
0.349913
1
0.349913
5.626419
0.0638
AC
0.106938
1
0.106938
1.71951
0.2467
AD
0.008447
1
0.008447
0.135817
0.7276
BC
0.01
1
0.01
0.160796
0.705
BD
0.003122
1
0.003122
0.050203
0.8316
CD
0.971295
1
0.971295
15.61791
0.0108∗
Residual
0.310955
5
0.062191
Total
8.019152
15
∗Denotes significant factors.Figure 12
Effects of design variables on maximum STDV of stresses in ROIs: (a) main effect ofγ, (b) interaction effect of δ and ε.
(a)
(b)The ANOVA table of the model for the maximum micromotion at bone-implant interface is presented in Table5, which identifies that the main significant factors are γ, β, and ε, and the influential interactions are γε and βγ. From Figure 13 it can be interpreted that lower values for the angle between the distal and posterior cuts and higher values for the angle between the distal and anterior cuts (ε = 88° and γ = 98°) will reduce the micromotion at the interface (Figure 13(a)). Meanwhile, the interaction of βγ also shows that higher value of γ causes less micromotion, and at high level of factor γ (γ = 98°), changes of β have negligible effect on the response, while at the low level of factor γ (γ = 90°), variations of β more influence the response (Figure 13(b)).Table 5
ANOVA results for maximum micromotion at femur-implant interface for Co-Cr alloy.
Source
Sum of squares
df
Mean square
F value
p valueProb > F
Model
11975.86
10
1197.586
1918.347
<0.0001
A-beta
90.0601
1
90.0601
144.2624
<0.0001∗
B-gamma
2613.254
1
2613.254
4186.029
<0.0001∗
C-delta
2.0449
1
2.0449
3.275614
0.1301
D-epsilon
9015.503
1
9015.503
14441.44
<0.0001∗
AB
89.6809
1
89.6809
143.6549
<0.0001∗
AC
1.1025
1
1.1025
1.766034
0.2413
AD
2.9929
1
2.9929
4.794163
0.0801
BC
2.5281
1
2.5281
4.049625
0.1003
BD
158.5081
1
158.5081
253.9055
<0.0001∗
CD
0.1849
1
0.1849
0.296181
0.6097
Residual
3.1214
5
0.62428
Total
11978.98
15
∗Denotes significant factors.Figure 13
Effects of design variables on maximum micromotion at femur-implant interface: (a) interaction effect ofγ and ε, (b) interaction effect of γ and β.
(a)
(b)The influential parameters for FGM femoral component were also identified. Same factors are significant for minimum of mean stresses and maximum of STDV of stresses (see Table6). However, for micromotion the influential factors are different. This is probably due to different interface characteristics of the two materials.Table 6
Comparison of significant design factors for Co-Cr and FGM in all responses.
Influential factors
Min of mean stress
Max of STDV stress
Max of micromotion
Co-Cr
β
(
+
)
γ
(
-
)
ε
(
-
)
β
ε
(
-
)
γ
ε
(
+
)
δ
(
-
)
γ
(
-
)
ε
(
-
)
δ
ε
(
+
)
β
(
+
)
γ
(
-
)
ε
(
+
)
β
γ
(
-
)
γ
ε
(
-
)
FGM
β
(
+
)
γ
(
-
)
ε
(
-
)
β
ε
(
-
)
γ
ε
(
+
)
γ
(
-
)
ε
(
-
)
δ
ε
(
+
)
β
(
+
)
γ
(
-
)
ε
(
+
)
δ
(
+
)
β
γ
(
-
)
γ
ε
(
-
)
β
ε
(
-
)
γ
δ
(
-
)
δ
ε
(
+
)
Positive (+) or negative (−) effect of the main and interactions effects (coefficient of regression models).


## 3.2. House of Quality for Shaping and Weighting Design Selection Criteria

The conceptual design begins with gathering of the voices of the customer as functional requirements. Expectations of customers (patients), medical, and design teams from implant (femoral component) in a knee replacement surgery are shown in Figure14. Needs start the redesign/design process and a high-quality design will strike the best combination of wants and needs in the specifications.Figure 14
Customers performance needs in design of femoral component and complementary requirements detected by the medical and design team.Figure15 shows how the technical requirements are divided to design selection criteria (for ranking of design alternatives for each material) and criteria of material and design selection (ranking of designs for both materials). Given that it is important to maximize the minimum observed mean stress in 10 ROIs (Figure 9), the noncompensatory model of Maximin was used to consider the mean stress of the most vulnerable region. In the same way, for criterion of stresses STDV, maximum observed standard deviation in different regions was reported. For design selection criteria, in addition to minimum of stress mean, maximum of stress STDV, and maximum contact slip at femoral component/bone interface, area of cross section as an index of mass was considered. However, in final selection, multiplication of cross section area and density of material was used to be representative of mass index. Meanwhile, the maximum peg stress and maximum stress at corner points of inner contour were taken into account. Maximum peg stress which was on the peg root is very important due to the transition from the main body to the peg. The high stress in this region is even worse when the cross section is porous. This is critical because of the adverse effect on the component which might cause premature peg failure through root fracture. Maximum stress at corner points of inner contour is also important from the stress concentration point of view, especially for the FGM with the porous layer interfacing the bone. For final selection these two criteria were replaced by safety index of peg and safety index of main body in which the differences between yield of material and maximum observed stress were used. This is to avoid misleading results of traditional ratio based performance indices [28]. Furthermore, in final selection, biocompatibility and hardness were also considered. Table 7 demonstrates house of quality for translation of customer needs to engineering characteristics. Absolute weights (importance) for material and design selection criteria are also shown in this table. For instance, in criterion of “stress level in femur,” absolute weight (36) is multiplication of importance degree (4) for related customer requirement (Req 5 as indicated in Figure 14) and type of relationship between customer and technical requirement (strong: 9). The last two rows in Table 7, respectively, show relative importance of criteria (weight), for “design selection” (Tables 8 and 9) and “material and design selection” (Table 10), that will be used for ranking via TOPSIS method.Table 7
House of quality including requirements, engineering characteristics, and obtained weights.
Degree of importance
Engineering characteristics (How)








Stress level in femur (A)
Nonuniformity of stress distribution in femur (B)
Displacement at bone-implant interface (C)
Stress level at femoral corner points (F)
Stress level in peg (D)
Mass (E)
Biocompatibility (G)
Hardness (H)
Prioritized customerRequirements (what)
Req1
2
●
Req2
3
⊙
●
Req3
5
◯
●
Req4
4
●
Req5
4
●
⊙
Req6
4
●
Absolute weight
36
28
36
21
27
43
45
36
Relative weight for design selection criteria (input for Tables8 and 9)
0.19
0.15
0.19
0.11
0.14
0.23
—
—
Relative weight for material and design selection criteria (input for Table10)
0.13
0.10
0.13
0.08
0.10
0.16
0.17
0.13
Prioritized technical descriptors
+9 ● strong.
+7 ⊙ moderate.
+5 ◯weak.Table 8
Performance of Co-Cr alloy designs in different criteria and ranking orders scenarios by TOPSIS method.
Objectives
Max
Min
Min
Min
Min
Min
Relative closeness to the ideal solution
Ranking
Relative weights of design criteria
0.19
0.15
0.19
0.11
0.14
0.23
A
B
C
D
E
F
Design 1
2.919
3.116
34.70
24.67
900.79
115.4
0.4080
12
Design 2
3.605
3.428
44.60
24.74
918.17
114.7
0.4075
13
Design 3
1.570
2.189
20.13
26.22
804.59
72.9
0.6180
1
Design 4
1.771
2.221
21.35
26.26
925.80
108.2
0.4315
10
Design 5
2.962
2.682
35.46
27.05
901.99
73.9
0.5865
4
Design 6
4.244
1.884
45.55
27.35
874.13
109.8
0.5212
6
Design 7
1.600
1.306
20.27
27.79
849.11
122.3
0.4487
9
Design 8
1.804
1.336
21.50
27.86
867.89
86.9
0.5750
5
Design 9
1.213
2.464
87.99
17.00
934.89
101.5
0.2863
16
Design 10
1.352
1.815
97.54
25.52
813.68
79.0
0.4637
8
Design 11
0.845
0.958
61.94
10.05
779.58
69.0
0.6107
2
Design 12
0.851
1.309
62.35
10.12
883.21
109.8
0.4086
11
Design 13
1.293
2.470
90.74
17.13
885.27
85.9
0.3960
15
Design 14
1.408
1.813
99.12
25.55
892.91
68.3
0.4690
7
Design 15
0.892
1.148
63.25
10.11
796.96
68.3
0.6033
3
Design 16
0.871
1.311
60.43
10.10
866.49
116.5
0.4000
14
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: maximum peg stress (MPa).
E: area of cross section (mm2).
F: maximum stress at corner points of inner contour (MPa).Table 9
Performance of FGM designs in different criteria and ranking orders scenarios by TOPSIS method.
Objectives
Max
Min
Min
Min
Min
Min
Relative closeness to the ideal solution
Ranking
Relative weights of design criteria
0.19
0.15
0.19
0.11
0.14
0.23
A
B
C
D
E
F
Design 1
3.100
2.655
30.61
20.16
900.79
14.4
0.6029
4
Design 2
3.818
3.449
39.58
23.54
918.17
14.3
0.5540
8
Design 3
1.707
2.208
17.93
14.09
804.59
13.1
0.6566
3
Design 4
1.925
2.245
19.16
14.34
925.8
12.2
0.6011
5
Design 5
3.150
2.668
31.24
21.26
901.99
29.0
0.4385
14
Design 6
4.466
1.866
40.36
24.17
874.13
13.8
0.6730
1
Design 7
1.741
1.249
18.03
14.71
849.11
14.7
0.6581
2
Design 8
1.961
1.349
19.25
14.97
867.89
33
0.4678
11
Design 9
1.327
2.456
79.44
12.76
934.89
12.8
0.4582
12
Design 10
1.480
1.814
87.88
15.83
813.68
18.6
0.4582
13
Design 11
0.932
1.017
57.53
7.55
779.58
22.2
0.5300
9
Design 12
0.941
1.326
57.93
7.86
883.21
12.0
0.5603
6
Design 13
1.353
2.460
83.24
12.88
885.27
17.5
0.4180
15
Design 14
1.462
1.835
90.14
15.81
892.91
18.4
0.4120
16
Design 15
0.940
1.206
58.59
7.59
796.96
22.6
0.5103
10
Design 16
0.934
1.328
58.56
7.88
866.49
13.6
0.5552
7
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: maximum peg stress (MPa).
E: area of cross section (mm2).
F: maximum stress at corner points of inner contour (MPa).Table 10
Ranking of top materials and designs scenarios with regard to all technical criteria obtained from QFD.
Objectives
Max
Min
Min
Max
Min
Max
Max
Max
Relative closeness to the ideal solution
Ranking
Relative weights
0.13
0.10
0.13
0.08
0.10
0.16
0.17
0.13
Design options
A
B
C
D
E
F
G
H
Co-Cr alloy
Original
1.86
1.69
21.93
421.4
7.04
339.5
1
1
0.4531
6
Design 3
1.570
2.189
20.13
423.78
6.68
377.1
1
1
0.4554
5
Design 11
0.845
0.958
61.94
439.95
6.47
381
1
1
0.4142
8
Design 15
0.892
1.148
63.25
439.89
6.61
381.7
1
1
0.4064
9
Design 5
2.962
2.682
35.46
422.95
7.49
376.1
1
1
0.4367
7
FGM
Design 6
4.466
1.866
40.36
24.83
3.03
18.2
3
3
0.5868
1
Design 7
1.741
1.249
18.03
34.29
2.95
17.3
3
3
0.5809
2
Design 3
1.707
2.208
17.93
34.91
2.79
18.9
3
3
0.5584
3
Design 1
3.100
2.655
30.61
28.84
3.13
17.6
3
3
0.5537
4
A: minimum of stress mean in different regions explained in Figure9 (MPa).
B: maximum of stress STDV in different regions (MPa).
C: maximum contact slip at femoral component/bone interface (μm).
D: safety index of peg (difference between yield of material and maximum peg stress which is 450 and 49 MPa for Co-Cr alloy and FGM resp.).
E: weight index (area of cross section∗ density (g/cm)).
F: safety index of main body (difference between yield of material and maximum stress at corner points of inner contour in which yields of 450 and 32 MPa for Co-Cr alloy and FGM were considered, resp.).
G: biocompatibility of material.
H: hardness of interface material with PE insert.Figure 15
One by one matching of technical requirements with selection criteria.


## 3.3. Ranking of Design Alternatives

Tables8 and 9, respectively, show an overview of designs performance for Co-Cr alloy and FGM. Among the criteria for design selection, only criterion of “minimum of mean stress in different regions” has maximization objective and for the other criteria, including “maximum of STDV of stresses in different regions,” “maximum contact slip at femoral component/bone interface,” “maximum peg stress,” “area of cross section,” and “maximum stress at corner points of inner contour” less is better. In both tables, relative weights are obtained from QFD’s absolute weight, with regard to considered criteria for design selection stage.It can be seen from Tables8 and 9 that the designs 3, 11, 15, and 5 are the four top designs for Co-Cr alloy, and designs 6, 7, 3, and 1 are the top ranked designs for FGM. Table 10 shows these eight high rank designs and the original design with Co-Cr alloy. To select the best material and design, criteria of “minimum of stress mean in different regions,” “safety index of peg,” “safety index of main body,” “biocompatibility of material,” and “hardness of interface material with PE insert” should be maximized. For other criteria, including “maximum of stress STDV in different regions,” “maximum contact slip at femoral component/bone interface,” and “Weight index” lower values are desirable. The most striking result to emerge from Table 10 is that design 6 with FGM is the top rank design. It can also be seen from ranking orders that the designs with FGM are better than Co-Cr alloy. Design 3 is the only common option in the list of four top rank designs of each material. It clarifies the high contribution of selected material in the performance of femoral component. What is interesting here is that the original design has the second rank in the designs made from Co-Cr alloy. It is due to advantages of Co-Cr alloy-design 3 over original one in criteria “maximum contact slip at femoral component/bone interface,” “Safety index of peg,” “weight index,” and “safety index of main body.” Although it seems that by considering only a single performance output (stress mean in different regions), the original design is better than Co-Cr alloy-design 3, it is clear that design 5 is better than the original one in this condition.


## 4. Conclusions

The present study was designed to determine the effect of inner shape geometries and material on performance of femoral component using FEA, DOE, QFD, and MADM techniques. The findings complement earlier studies on advantages of FGM over Co-Cr alloy, which is the most prevalent material for this component. The results also showed that the angle between the distal and anterior cuts and the angle between the distal and posterior cuts are the most influential factors in all responses; however the interactions were also observed. An important outcome of the present study is the fact that the geometry of femoral component using standard Co-Cr alloy is not the optimum one; the performance of component can be improved either by geometry modification for Co-Cr alloy or optimized geometry for FGM. The present study confirms previous findings that the current design of femoral component still is not the optimum and contributes additional evidence that suggests more broadly research on dimension of interface geometry in addition to angles. However, it would be more fruitful to focus on improving geometry of component with functionally graded material. Since the study was conducted using 2D FEA, the generalizability of these results may subject to some limitations. Nevertheless, these findings enhance our understanding on the most important factors and will serve as a base for future 3D studies. It was shown that how applying quality and design tools in the product-planning phase can help in translating the customer’s requirements. MADM methods in combination of DOE can improve design process in which DOE determines important design factors and MADM helps in ranking of design scenario with diverse qualitative and quantitative data. Design decision making problems need different information in which it is not possible to find the best design only based on design of experiments methods and the related optimization software. Together these results provide important insights into designer on how to improve performance of current knee implants.



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*Source: 693469-2015-06-21.xml*