Damage, significantly impairing the quality of life, coxarthrosissal can meet a young age. Therefore, nowadays the prosthesis implantation is performed in the most active life period of the patient. Besides, it is known that the replacement of the prosthesis will be necessary later [1]. When acetabular cups were measured, the results were analysed and the differences were determined, I took the coordinate systems needed to describe the movement of the prosthesis

Dudás, Illés

Monostori, Renáta

Library of National Technical University "Kharkiv Polytechnic Institute"

ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 
133 
UDC 621.9 
 
Renáta Monostori, Illés Dudás 
University of Miskolc, Hungary 
 
MATHEMATICAL ANALYSIS OF HIP JOINT PROSTHESIS 
 
Damage, significantly impairing the quality of life, coxarthrosissal can meet a young age. 
Therefore, nowadays the prosthesis implantation is performed in the most active life period of the 
patient. Besides, it is known that the replacement of the prosthesis will be necessary later [1].  
When acetabular cups were measured, the results were analysed and the differences were 
determined, I took the coordinate systems needed to describe the movement of the prosthesis. 
 
INTRODUCTION 
Arthrosis is one of the most common musculoskeletal disorders in our 
lifetime, almost everyone has heard about it or has met someone who was operated 
due to the disease. 
 
 
 
Figure 1 – Location of prosthesis – with the help of medical imaging [2] 
 
The wear of the hip joint involves joint pain, limping and reduced mobility. 
First, in the morning, while getting up people feels as if hips became rusty, 
then it disappears later, and hips start "operating". After prolonged sitting or lying 
down there are similar complaints. The movement of the joint progressively 
decreases, due to the pain and the restraint of movement, the person is limping. 
Later the pain becomes persistent, walking ability decreases, the person has to stop 
more and more often, and it is difficult to put on socks and shoes [3]. 
Type of the hip joint prosthesis 
The anatomic hip prosthesis may be cemented and cementless depending on 
the type of fixation used to hold the implant in place. 
The so called classical cement fixation has a history of about 6 decades. 
 
© Renáta Monostori, Illés Dudás, 2017 
ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 
134 
The main point of this implant family is that the plastic sleeve made of plastic with 
high abrasion resistance and the femur stem made of rapid hardening polyacrylate. 
This is the most commonly used type of prosthesis, which makes almost any 
prosthesis implant technically feasible [3]. 
Construction (Figure 2): 
- socket; 
- metal head - ball; 
- stem of different types. 
The method without cement is recommended especially for younger patients. 
In their case the organism itself provides the fixture of the prosthesis [3], as the 
bone “grows in” the surface of the prosthesis from the bonebed matching the size 
of the prosthesis. 
 
 
Figure2 – Parts of the prosthesis [3] 
 
1. DEFINITION OF A COORDINATE SYSTEM FOR MODELLING 
I was created a mathematical model for the hip prosthesis, it is necessary to 
determine the coordinate systems that make the movement of the acetabular cup 
and the femur head more transparent. [5, 6] We need the coordinate system of the 
acetabular cup as a standing coordinate system, since the socket is getting fixed in 
the basin bone [7]. However, the femur head can move both in a standing 
coordinate system and in a rotating coordinate system. 
The former is the resting position, the latter is the case when the patient 
moves. 
These coordinate systems were applied on the basis of the general 
mathematical model of Illés Dudás (Figure 3) [4, 8]. 
ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 
135 
 
Figure 3 – Coordinate systems 
 
where: 
K0(x0, y0, z0) – standing coordinate system of the acetabular cup 
K1(x1, y1, z1) – standing coordinate system of the femur head 
K1F(x1F, y1F, z1F) – moving (rotating) coordinate system of the femur head 
d(b, a, c) – split distance between the acetabular cup and 
femur head (Figure 4) 
φ1 – angle value 
 
 
Figure 4 – Status of the acetabular cup and femur head 
ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 
136 
2. DETERMINATION OF THE TRANSFORMATION MATRICES 
FOR THE ANALYSIS OF THE MOVING 
The two-parameter equation of the femur head: 
 
1
1F 1
1
R sin u cos
r R cos u
R sin u sin
   
  
 
    
 (1) 
 
The matrix below shows how to get from K0  to K1 (Figure 3): 
 
0,1
1 0 0 b
0 1 0 a
M
0 0 1 c
0 0 0 1
 
 
 
 
 
 
 (2) 
 
The following matrix shows how to get from K1 to K1F (Figure 5):  
 
1 1
1 1
1,1F
1 1
cos sin 0 0
0 cos sin 0
M
sin 0 cos 0
0 0 0 1
    
    
 
    
 
 
, (3) 
 
 
 
Figure 5 – Figures to the M1,1F matrix 
 
The following total transformation matrix shows how to get from the standing 
coordinate system of the acetabular cup to the rotating coordinate system of the 
femur head: 
ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 
137 
1 1
1 1
0,1F
1 1
cos sin 0 b
0 cos sin a
M
sin 0 cos c
0 0 0 1
   
  
 
  
 
 
 (4) 
Based on these we can determinate the two parameter-equation of the 
acetabular cup: 
     
     
     
1 11 1
1 11 1
0,cup 0,1F 1,F
1 11 1
cos d sin u cos sin d cos u b
cos d cos u sin d sin u sin a
sin d sin u cos cos d sin u sin c
1
R R
R R
r M r
R R
            
 
            
   
 
              
 
 
 (5) 
 
3. SUMMARY 
We have introduced the applications and types of hip prosthesis (acetabular 
cup, femur head). It can be seen that these prostheses are expensive components 
with very complex geometry. It is very important to choose the material to be used. 
Relying on the general mathematical model of Illés Dudás, I used a mathematical 
model to examine the function and geometry of the prosthesis. Using this model, 
additional wear, geometry and manufacturing tests can be carried out as a 
continuation of the research. The mathematical analysis I have made can be applied 
to any hip prosthesis geometry. 
 
ACKNOWLEDGEMENT 
The described article/presentation/study was carried out as part of the EFOP-
3.6.1-16-00011 “Younger and Renewing University – Innovative Knowledge City 
– institutional development of the University of Miskolc aiming at intelligent 
specialisation” project implemented in the framework of the Szechenyi 2020 
program. The realization of this project is supported by the European Union, co-
financed by the European Social Fund. 
 
References: 1 Szendrői M., Sólyom, L.: Modern approach to surgical treatment of osteoarthritis of 
the hip, Lege Artis Medicinea, 2001; 2 http://www.videoklinika.hu/csipoprotezis_nemcsak_az_ortoped-
sebesz_szemszogebol/; 3 http://www.ortopedszakorvos.eu/betegsegekrol; 4 Dudás, I.: The Theory and 
Practice of Worm Gear Drives Kogan Page US., USA, 2004; 5 Dudás, L.: Modelling and simulation 
of a new worm gear drive having point-like contact, Engineering with Computers 29.3 (2013): 251-
272.; 6 Balajti, Zs.: New Modelling of Computer Aided Design of worms in the Same Axis, Journal 
Manufacturing and Industrial Engeneering, No. 2 Volume XL, ISSN 1338-6549, Slovakia, Kosice, 
2012, pp.: 26-29; 7 Balajti, Zs., Abel, J.: Applying projective geometry in design of worm 
manufacturing KEY ENGINEERING MATERIALS 581: pp.: 77-81. (2014) 7th International 
Congress of Precision Machining, ICPM 2013. Miskolc, Magyarország: 2013.10.03 -2013.10.05; 8 
Litvin, F. L.: Gear geometry and applied theory Englewood Cliffs, Prentice Hall, NJ., 1994


Mathematical analysis of hip joint prosthesis

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 133 UDC 621.9  Renáta Monostori, Illés Dudás University of Miskolc, Hungary  MATHEMATICAL ANALYSIS OF HIP JOINT PROSTHESIS  Damage, significantly impairing the quality of life, coxarthrosissal can meet a young age. Therefore, nowadays the prosthesis implantation is performed in the most active life period of the patient. Besides, it is known that the replacement of the prosthesis will be necessary later [1].  When acetabular cups were measured, the results were analysed and the differences were determined, I took the coordinate systems needed to describe the movement of the prosthesis.  INTRODUCTION Arthrosis is one of the most common musculoskeletal disorders in our lifetime, almost everyone has heard about it or has met someone who was operated due to the disease.    Figure 1 – Location of prosthesis – with the help of medical imaging [2]  The wear of the hip joint involves joint pain, limping and reduced mobility. First, in the morning, while getting up people feels as if hips became rusty, then it disappears later, and hips start "operating". After prolonged sitting or lying down there are similar complaints. The movement of the joint progressively decreases, due to the pain and the restraint of movement, the person is limping. Later the pain becomes persistent, walking ability decreases, the person has to stop more and more often, and it is difficult to put on socks and shoes [3]. Type of the hip joint prosthesis The anatomic hip prosthesis may be cemented and cementless depending on the type of fixation used to hold the implant in place. The so called classical cement fixation has a history of about 6 decades.  © Renáta Monostori, Illés Dudás, 2017 ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 134 The main point of this implant family is that the plastic sleeve made of plastic with high abrasion resistance and the femur stem made of rapid hardening polyacrylate. This is the most commonly used type of prosthesis, which makes almost any prosthesis implant technically feasible [3]. Construction (Figure 2): - socket; - metal head - ball; - stem of different types. The method without cement is recommended especially for younger patients. In their case the organism itself provides the fixture of the prosthesis [3], as the bone “grows in” the surface of the prosthesis from the bonebed matching the size of the prosthesis.   Figure2 – Parts of the prosthesis [3]  1. DEFINITION OF A COORDINATE SYSTEM FOR MODELLING I was created a mathematical model for the hip prosthesis, it is necessary to determine the coordinate systems that make the movement of the acetabular cup and the femur head more transparent. [5, 6] We need the coordinate system of the acetabular cup as a standing coordinate system, since the socket is getting fixed in the basin bone [7]. However, the femur head can move both in a standing coordinate system and in a rotating coordinate system. The former is the resting position, the latter is the case when the patient moves. These coordinate systems were applied on the basis of the general mathematical model of Illés Dudás (Figure 3) [4, 8]. ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 135  Figure 3 – Coordinate systems  where: K0(x0, y0, z0) – standing coordinate system of the acetabular cup K1(x1, y1, z1) – standing coordinate system of the femur head K1F(x1F, y1F, z1F) – moving (rotating) coordinate system of the femur head d(b, a, c) – split distance between the acetabular cup and femur head (Figure 4) φ1 – angle value   Figure 4 – Status of the acetabular cup and femur head ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 136 2. DETERMINATION OF THE TRANSFORMATION MATRICES FOR THE ANALYSIS OF THE MOVING The two-parameter equation of the femur head:  11F 11R sin u cosr R cos uR sin u sin           (1)  The matrix below shows how to get from K0  to K1 (Figure 3):  0,11 0 0 b0 1 0 aM0 0 1 c0 0 0 1       (2)  The following matrix shows how to get from K1 to K1F (Figure 5):   1 11 11,1F1 1cos sin 0 00 cos sin 0Msin 0 cos 00 0 0 1               , (3)    Figure 5 – Figures to the M1,1F matrix  The following total transformation matrix shows how to get from the standing coordinate system of the acetabular cup to the rotating coordinate system of the femur head: ISSN 2078-7405. Резание и инструменты в технологических системах, 2017,  выпуск 87 137 1 11 10,1F1 1cos sin 0 b0 cos sin aMsin 0 cos c0 0 0 1           (4) Based on these we can determinate the two parameter-equation of the acetabular cup:                1 11 11 11 10,cup 0,1F 1,F1 11 1cos d sin u cos sin d cos u bcos d cos u sin d sin u sin asin d sin u cos cos d sin u sin c1R RR Rr M rR R                                              (5)  3. SUMMARY We have introduced the applications and types of hip prosthesis (acetabular cup, femur head). It can be seen that these prostheses are expensive components with very complex geometry. It is very important to choose the material to be used. Relying on the general mathematical model of Illés Dudás, I used a mathematical model to examine the function and geometry of the prosthesis. Using this model, additional wear, geometry and manufacturing tests can be carried out as a continuation of the research. The mathematical analysis I have made can be applied to any hip prosthesis geometry.  ACKNOWLEDGEMENT The described article/presentation/study was carried out as part of the EFOP-3.6.1-16-00011 “Younger and Renewing University – Innovative Knowledge City – institutional development of the University of Miskolc aiming at intelligent specialisation” project implemented in the framework of the Szechenyi 2020 program. The realization of this project is supported by the European Union, co-financed by the European Social Fund.  References: 1 Szendrői M., Sólyom, L.: Modern approach to surgical treatment of osteoarthritis of the hip, Lege Artis Medicinea, 2001; 2 http://www.videoklinika.hu/csipoprotezis_nemcsak_az_ortoped-sebesz_szemszogebol/; 3 http://www.ortopedszakorvos.eu/betegsegekrol; 4 Dudás, I.: The Theory and Practice of Worm Gear Drives Kogan Page US., USA, 2004; 5 Dudás, L.: Modelling and simulation of a new worm gear drive having point-like contact, Engineering with Computers 29.3 (2013): 251-272.; 6 Balajti, Zs.: New Modelling of Computer Aided Design of worms in the Same Axis, Journal Manufacturing and Industrial Engeneering, No. 2 Volume XL, ISSN 1338-6549, Slovakia, Kosice, 2012, pp.: 26-29; 7 Balajti, Zs., Abel, J.: Applying projective geometry in design of worm manufacturing KEY ENGINEERING MATERIALS 581: pp.: 77-81. (2014) 7th International Congress of Precision Machining, ICPM 2013. Miskolc, Magyarország: 2013.10.03 -2013.10.05; 8 Litvin, F. L.: Gear geometry and applied theory Englewood Cliffs, Prentice Hall, NJ., 1994

http://repository.kpi.kharkov.ua/bitstream/KhPI-Press/31318/1/RITS_2017_%2087_Monostori_Mathematical.pdf

Mathematical analysis of hip joint prosthesis

Abstract

Similar works

Full text

Available Versions

Library of National Technical University "Kharkiv Polytechnic Institute"

Electronic National Technical University "Kharkiv Polytechnic Institute" Institutional Repository (eNTUKhPIIR)

Mathematical analysis of hip joint prosthesis

Abstract

Similar works

Full text

Available Versions

Library of National Technical University "Kharkiv Polytechnic Institute"

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

Electronic National Technical University "Kharkiv Polytechnic Institute" Institutional Repository (eNTUKhPIIR)