- Email: [email protected]
A porous implant model for a knee joint
45
Int J Biomed Comput, 29 (1991) 45-59
Elsevier Scientific Publishers Ireland Ltd.
A POROUS IMPLANT MODEL FOR A KNEE JOINT
P.N. TANDON and AMITA CHAURASIA Department
of Mathematics,
Harcouri
Butler Technological
Institute.
Kanpur 208002
(India)
(Received February 21st. 1991) (Accepted May 23rd. 1991)
A study of the lubrication mechanism occurring in knee joint replacement is presented. The ideal&d model has been shown to produce results consistent with those in normal situations. In the present problem viscoelastic fluid has been considered to represent the synovial fluid in the fluid-film region and purely viscous Newtonian fluid in the porous layer due to filtration. Because of exact solution not being possible for the governing non-linear partial differential equations, the perturbation method has been used to obtain approximate solutions. The effect of an increase in the viscoelastic parameter of the lubricant is similar to that of an increase in concentration of hyaluronic acid molecules in synovial fluid. Important deductions are made for load carrying capacity and coefficient of friction and it has been shown that the slip velocity plays an important role in maintaining the self-adjusting nature of human joints. Keywords:
Knee joint replacement; Lubrication mechanism; Porous implant model
Introduction
The synovial joints provided by nature in the human body to carry out trouble free motion of one bone past another have long been identified as a bearings system. In fact, these joints function as excellent bearings in biological conditions. They support the considerable loads involved, and provide low friction coefficient service over a long span of one’s lifetime. However, in certain cases the joints develop certain abnormalities, e.g. osteoarthrosis. The synovial joint constitutes a tribological system similar to any other bearing so that the principal failure and their remedies are the same. In engineering systems such situations are tackled by improving or changing the lubricant. A total replacement is also required at times if the disorder becomes very severe. Both these remedial procedures can also be adapted for biological systems and therefore attempts have been made in both the directions for human joints. Joint replacement is widely used for treatment when joints fail to support loads or normal functions due to some accidental force or diseases. A number of different designs have emerged ranging from hinged to unconstrained configurations. Attempts have also been made to produce data in the laboratory for extrapolating the Correspondence to: P.N. Tandon, Department of Mathematics, Harcourt Butler Technological Institute, Kanpur 208002. India.
0020-7101/91/$03.50 0 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
46
P.N. Tandon and A. Chaurasia
results for the human body. At present, most joint replacements are carried out at the hip, knee and fingers. It is advantageous to predict the lifetime of the joint prosthesis and the ability to produce the effect of load or patient activity on the prosthesis, e.g. ball and socket for hip and hinge configuration for knee, elbow and thumb [I]. During recent years, arthroplasty of the human knee joint with endoprosthesis is a common mode of treatment for degenerated knee joints. Advancing the state of art in design and utilization of these endoprosthetic devices requires thorough understanding of the stress behaviour of healthy joints and reconstructed combination of the bone-bone cement prosthesis device. The reaction of living bone depends among other things on the mechanical stresses to which it is subjected. The replacement of the articular surfaces of a human joint for the treatment of arthritis has come into wide practice in recent years [2]. Recently a comparison has been made of the lubrication mechanism believed to occur in normal and artificial joints [3]. After considering both metal on metal and metal on plastic specimens of artificial joints removed from the patients after various periods of implantation, important deductions were made about modes of lubrication and wear [4,5]. Walker [6] confirmed that the coefficient of friction for metal to metal is much more than that for metal and plastic. Further, there may be interaction between metal and plastic causing adhesion but the lubricant in between can therefore be expected to reduce frictional forces to about one-fifth or a quarter of the dry value [7]. Scales et al. [8] have also reported that the frictional effects experienced in various models of prosthesis subjected to a normal loading cycle show that metal and plastic arrangements prove to be much superior to lower the frictional resistances. Porous ceramics [9], metallic [lo] and polymeric [ 1l] materials are being considered for use in joint prosthetic stabilization via tissue in-growth into the surface undulations and pores of the replacement. However, there are a number of factors which suggest that ceramics, being inert, high strength abrasion resistant, will be the material of choice for use in highly corrosive environments [12]. There have been two important comparative studies of models of replacement via tissue in-growth and cementation for impaction and fixation of prosthesis in animals [13,14]. The studies confirmed that tissue in-growth provided better implant performance than using cementation. A number of investigators have been examining the concept of bone in-growth into microporous materials as a means of prosthetic fixation [15]. Tandon et al. [ 16,171 have also proposed suitable models for artificial human knee and hip joint replacements introducing porous plastic so that the bone in-growth into the pores may provide better impaction than using cement as compared to those employing steel or rigid plastic frames. It has also been observed that porous implants also help in maintaining an inbuilt cooling mechanism and nutritional transport in almost the same way as in normal joints. This investigation is an attempt to derive some conclusions for a simplified model of knee joint replacement. The proposed model assumes a two region flow model: flow of viscous fluid in the porous matrix and squeeze film lubrication in between the two approaching arm’s of the joint with viscoelastic fluid as lubricant to represent the synovial fluid. The arms of the hinge are impacted or connected into the shafts of the femur and tibia, the upper arm is fixed and its end is covered with a
47
Fig. 1. Model
for an artificial
knee joint
replacement
and its geometrical
counterpart.
porous layer whereas the lower one moves through an angle from CYto 0 as shown in Fig. 1. As the two arms approach each other, viscous fluid enters into the porous matrix which in turn increases the concentration of suspended particles in the fluid-film region. The pore size of the porous material of the lower arm has been considered to be so small that only the suspending medium enters into the porous matrix. Thus, the normal process of imbibition and exudation is maintained in this model. The model presents improvement over earlier models and it is much closer to reality. Formulation of the Problem Referring to Fig. I, the proposed model may be considered as a two-dimensional squeeze film lubrication in between two approaching surfaces. The upper one is fixed and porous and the other one is rigid and movable about the hinge. Under the assumption of small 0, the expression for film thickness is given by h’ = x’ tan 6 = de
(1)
The basic governing equation in the fluid-film fluid can be written as
a*d
apt ax,=&
[
__ayt2
a*d
- 3rl0, - X,) ___
af*
region applicable for viscoelastic
ad
__
2
( ay*>I
(2)
48
P. N. Tendon und A. Chuurusiu
and the equation of continuity is au’ _+?C=()
ax
(3)
af
where u‘ and v’ are velocities along the coordinated axes and p’ is the hydrodynamic pressure. The flow of viscous fluid in the porous matrix is governed by Darcy’s law; we have the following equations:
cco
-aP ax*
4
-
4
-,=__ U
-t=__
(4)
aF#
V
po
w
are the velocity components in the porous matrix. Thus, the pressure P satisfies the Laplace equations,
a*P + axl*
§*F’
.
=(-j
(5)
af2
Boundary conditions
u’
ad at y’= 0 af ’
=
-
(J’ __
U’ =
-
xc&
4
y’=--
at y’ = h’
aP
-
.
aty’=O
cl0 W v’ = -
x)w,
at y’ = h’
P’ = 0,
at x’ = 0 and L’
-1 P = 0,
at x’ = 0 and L’
aFf
ay'
P’(x’)
= 0, at y' = - H’
= F(x),
0)
(6)
Model for knee
49
is the permeability, H’ is the cartilage thickness and u’ is the where u’ = q/A, C#J parameter associated with the slip parameter. The following are non-dimensional variables: X’
x--,
y=--, Y'
24’
u=-
,
ho
ho
V'
p=p’
UO
NO2
“Z-,
UO
(7)
F
-1 P
=-
L
PUo2
=-, L'
’
h
ho
=---, h’ ho
The governing equations are transformed
Re
ap TT
a2t4 _-33_ ay2
a2u - au 2 a? (>ay
2!K+2?=, ax
u’=
Re=-
PUohO Pil
into the non-dimensional
form:
(8)
(9)
ay
aF
-Re
tfq---
r_= -Re
$I---
(10)
ax
aF
(11)
ay
and
a2F _+E=()
(12)
ay2
ax2
where, Re is the Reynolds number, 4, = qVho2. The boundary conditions in nondimensional form are written as u=-u-,
au
ay
u=xue, v=
aty=O
aty=h
- Re 4, -
aF
aty=O
ay ’ v=
- xw,
aty=h
(13)
50
P.N.
P = 0,
at x = 0 and L
P = 0,
at x = 0 and L
-
aF
=
o
aY
P(x) =
.
Tandon and A. Chaurusia
at x = 0 and L
P(x,O)
For the present analysis the non-dimensional
film thickness is given by
h = x9
(14)
Solution of the Problem
Under the assumption of very thin cartilage, it is assumed that aP/ay is linear across the matrix and is zero at the outer surface of the porous bearing shell. This assumption leads to
a2F
= constant = K, aY2
(15)
Using Eqns. (12) and (15) and boundary condition (13), we obtain
iq, = 0 =
KReH ho*
a2P
ax2
(1’5)
In order to obtain the solution for the fluid-film region all the variables are assumed in the sequences of the function in ascending powers of small parameters E, e.g. f=fe
+ ef, + E2f + * * .
(17)
Introducing u, p in this form in Eqn. (8) and collecting the zeroth and first order terms we obtain the following equations:
ap, at% ax ay2
(18)
ap, a2u, au0 2 a2uo 3--ax ay2 ( ay > a9
(19)
Re----=-
Re-=--
Model for
51
knee
Solutions of the zeroth and first order velocity distribution
where 0, - a)h*
2(h - u)
are given below.
-4xw(g 1- cJJ (h - a)
and u1z-p
Re
aP,
2
ax
3h2 (xw@* 00, - h) -- 3 uh2 2(h - u)~
2
1 _
XW8
(h - a)3 ’
3h6
h’
h5
(x00)* (h - a)*
2(h - a) - 8(h - u)~ + 8(h - a)* h3
h’
A2=l+
h4
4(h - a)* - 2(h - a)
Pressure distribution Expressions for the zeroth and first order pressure distribution
x4 e3
x+
X3
+-_----6
x3
e* 3
+
1241 es
5~’e6
1841
1404,
x6
x5 e4
2w --
+
I
x4
e3
8u
are
x7 - e6
x6 e3 369,
1
+
28~4,
(21)
and
~x6 E, 6
~x7 E2 +
7
F, ‘g
x_--- x4 e4 691
x5 e4 20&
(22)
P. N. Tendon und A. Churusiu
52
where
C,=L+-+-
C,=_--_
L4 e3
LS e3
12041
2004 I
L3
L6 e3
6
3Wl L6 es
+
L7 e4 -+28&l
5L’ e6
l&#q +
+
L3 e2 3
L4e3 8a
14041
E, = DI + DI E2 = D2 + Ds
e6 D2=-
5
(
61
0, =
Ds =
9W
--_--~ WJ,
2
2a24,
4u”
es a3 2u
e6 o3 2u2
L6 E, F, = - -6
L’ E2
e4
~~ e3 -+L WI
F2=---
2iw2
Lo
20&q
I
Load carrying capacity
The load carrying capacity of the joint in this case is defined as
w=
s L
P dx
-L.
=.
w, + E w,
(23)
Model for
53
knee
where P = PO + e PI
WO
L4
=
+24+-
L4 e2 _-_-12
L' es
LS e3 -_
12Wl
40a
L1e3 11241
1
9 L8tIb 2160@,
w, = Closure-time The closure time for a particular angle is given by t, = to + e t1
where to = -
11 =-
1
Of
wO
s Bi
1
B/
wl
s Bi
csd@
c6de
L5 e3 L2 c,= -+-+2
c4 =
6%
12Ou&
L4 e2 L7e3 L4 -++----_-----1124,
24 2
c,
L6 e4
c-2
=
12 c3
+
c 4
Re
CPI
Cl
>
Coefjcent of friction The coefficient of friction can be obtained by dividing frictional drag by load capacity
F
q
=-w
54
P. N. Tandon and A. Chaurasia
where
s L
F=
T
dx
-L
and r is the shear stress on the walls. Torque
The torque about the hinge is
s L
?lL
P dx
-L
Results and Discussion The problem of lubrication and friction has been analysed for a simplified model for a knee joint replacement. Figures 2-6 and Tables I-IV depict the representative computational results for pressure distribution, load capacity of the tibia1 replacement and time of approach for particular angular gap and torque. Figure 2 depicts the pressure distribution within the tibia1 replacement as a function of the angle 8. The pressure distribution increases B varies from a to 0. These results are in good agreement with that of normal situation in the knee joint [18].
Fig. 2. Variation
of pressure distribution
with I
for differentvalues
of slip parameter
and 0.
55
I6 8=30° lb-
1l2-
IO-
a-
O-Fig. 3. Variation
of friction coefficient
Fig. 4. Variation
of friction coefficient
with slip parameter
with slip parameter
for different
for different
values of viscoelastic parameter.
8.
56
P. N. Tandon und A, Chuurusiu
Further, we also observe that as the slip velocity at the porous boundary increases, the pressure decreases owing to the increased filtration of base fluid into the porous cartilage. Figure 3 describes the variation of the coefficient of friction with slip velocity for different values of the viscoelastic parameter. It has been observed that the friction coefficient decreases as the viscoelastic parameter increases. It increases with the slip parameter. In osteoarthritic conditions, hyaluronic acid molecules die out and lose their characteristics. At times, when viscosity decreases, external medicines are added to change the overall viscosity of the synovial fluid [19]. Figure 4 depicts the variation of the coefficient of friction with slip parameter for different values of the femoro-tibia1 angle. We observe from this that as the femorotibia1 angle increases, the friction coefficient increases. Figures 5 and 6 depict the variation of load carrying capacity with slip parameter for different values of femoro-tibia1 angle and viscoelastic parameter. The load carrying capacity increases as the viscoelastic parameter increases. The increasing values of the viscoelastic parameter describe the increase in the concentration of the suspended hyaluronic acid molecules which, in turn, increases the over all viscosity of the lubricant. This helps in sustaining greater loads [20]. It is concluded that at the closing end of the walking cycle, the slip velocity existing at the porous boundary plays an important role in the self-adjusting nature of the artificial joint in a similar manner to normal subjects. Further, we may observe from Fig. 6 that the load capacity increases as 8 varies from CYto 0. Tables I and II describe the variation of closure time for different values of slip parameter and viscoelastic parameter. It is clear that the closure time decreases as the slip parameter increases and it increases as the viscoelastic parameter increases owing to the increased apparent viscosity due to the increase in the viscoelastic parameter 1211. Tables III and IV depict the variation of torque with viscoelastic parameter for different values of the slip parameter and at different angles. The torque increases
TABLE
I
VARIATION
OF CLOSURE
W = 5.03542
x IO’*; E = 0.01.
(I
TIME
WITH
o
w2
60-45”
45-30”
30-15”
15-O”
0.1 0.2
1.319400 0.72135
I .330402 0.741386
I .36288 I 0.75204
1.380456 0.112591
0.3 0.4
0.417899 0.2lOOl8
0.529921 0.421119
0.541937 0.500379
0.553871 0.537506
01
Q4
0.3
a2 Q-4
Fig. 5. Variation of load carrying capacity with slip parameter for different 0. gr
es300
0
0.1
I
I
1
02
a3
04
Q----)
Fig. 6. The variation of load carrying capacity with slip parameter for parameter.
different values of viscoelastic
58
P. N.
Tandon
and
A. Chaurasia
as the viscoelastic parameter increases and it decreases when the slip parameter increases. We observe from Table IV that as the femoro-tibia1 angle decreaes the torque increases. As the viscoelastic parameter increases, the overall apparent viscosity increases and this, in turn, increases the torque as more viscous fluid offers greater resistance. TABLE
II
VARIATION
OF CLOSURE
TIME WITH e
W = 5.03542 x 1012;c = 0.1. e
442
0 0.01 0.02 0.03
TABLE
60-45”
45-30”
30-15”
15-O”
0.98756 1.319400 1.83094 2.43897
1.012374 1.330402 2.09876 2.83777
1.12980 I .362881 2.28112 3.00343
I .29876 I .380256 2.41562 3.19876
0.1
0.2
0.3
0.4
9.202502 x lOI 9.2687 x lOI 9.308856 x lOI 1.038001 x 10”
1.116334 x 2.89982 x 3.12255 x 4.180164 x
III
TORQUE a
e
0 0.01 0.02 0.03
TABLE
lOI IO”’ lOI lOI
3.350519 x 5.626355 x 6.362514 x 8.46929 x
IO” lOIs lOI lOI
I.441051 x 2.062187 x 3.444784 x 4.477259 x
10’5 lOI IO” 1O’5
IV
TORQUE e
0
0.01 0.02 0.03
8
60°
45”
30”
15”
2.03975 x 10” 7.920125 x IO” 9.24701 x IO” 1.02386 x lOI
9.202502 x lOI 9.2687 x lOI
3.570718 x lOI 3.916508 x IO”
8.66425 x lOI 9.52774 x lOI
9.308856 x lOI 1.038001 x 10”
4.612407 x IO” 5.092287 x lOIs
2.235273 x IO’” 3.705689 x IO’-’
Model for knee
59
References Adams JC: Outline of Orrhopaedics, English Language Book Society, Edinburgh, 1967. Swanson SAV, Freeman MAR and Health JC: J Bone Jr Surg, 55(B) (1973) 759. Barnet CH and Cobbaldt AF: Lubrication within living joints, J Bone Jt Surg, 44B(8) (1962) 662. Pilliar RM, Comeron HV and McNal I: Porous surface layered prosthetic devices, Biomed Eng. 4 (1975) 1. Radin EL and Paul IL, A consolidated concept ofjoint lubrication, J Bone JI Surg, 54(A) (1972) 607. Walker PS: The friction of internal artilicial joints. In Proc. Conf on Human Locomotor Engineering, Institution of Mechanical Engineers, 1971, p. 123. Duff-Barelay I and Spillman DT: Total hip joint prosthesis - a laboratory study of friction and wear. In Proc. Symp. Lubrication and Wear in Living and Artijicial Human joims, Institution of Mechanical Engineers. 1966, Vol. 181(3J). p. 90. 8 Scales JT, Kelly P and Goddard D: Friction torque studies of total joint replacements - the use of simulator. In Proc. Symp. on Lubrication and Wear in Join& 1969. 9 Klawitter JJ and Hulbert SF: Application of porous ceramics for attachment of load bearing internal orthopedic appliances, J Biomarer Res, 5 (1971) 161. IO Galante J, Rostoker W, Week R and Ray RD: Sintered fiber metal compositer as a basis for attachment of implants to bone, J Bone Jf Surg, 5(3A) (1971). II Taylor DF and Smith FB: Porous methylmethacrylate as an implant material, J Biomed Mafer Res Symp, 2 (1972) 467.
12 Kriegel WW and Palmour L: Ceramics in Severe Environmems. Plenum Press, New York, 1971. 13 Hulbert SF, Richbourgh HL, Klawitter JJ and Sauer BW: Evaluation of a metal ceramic composite hip prosthesis, J Biomed Murer Res Symp, 6 (1975) 189. 14 Nunamakar DN, Black J and Tornzo RG: A comparison of biological in-growth and cementation for fixation of the total hip prostesis. In Int. Biomoferial Symposium, Clemson University, Clemson, SC, 1975. 15 Klawitter J, Weinstein AM, Hulbert SF and Sauer BW: Tissue in growth and mechanical locking for anchorage of prosthesis in locomotor system. In Advance in Artificial Hip and Knee Joint Technology, New York, 1976, p. 422. 16 Tandon PN and Jaggi S: On wear and lubrication in an artificial knee joint replacement, Int J Mech Sci, (1981). 17 Tandon PN and Jaggi S: A polar model for synovial fluid with reference to human joint. Inr J Mech Sci, 21(3) (1979) 161. 18 Morrison JB: Mechanics of knee joint in relation to normal walking, J Biomech. 3 (1970) 5. 19 Tandon PN, Agarwal R and Chaurasia A, A model for ankle joint articulation. In Proc. oj Biomechanics, Wiley Eastern, New Delhi, 1989, p. 309. 20 Tandon PN and Chaurasia A: Microstructural effects on hip joint articulation, Appl Math. Modelling. 14 (1990) 312-319. 21 Tandon PN and Rakesh L: A model for lubrication mechanism in knee joint replacement, Def Sfi J, 36(l) (1986) 45.
Int J Biomed Comput, 29 (1991) 45-59
Elsevier Scientific Publishers Ireland Ltd.
A POROUS IMPLANT MODEL FOR A KNEE JOINT
P.N. TANDON and AMITA CHAURASIA Department
of Mathematics,
Harcouri
Butler Technological
Institute.
Kanpur 208002
(India)
(Received February 21st. 1991) (Accepted May 23rd. 1991)
A study of the lubrication mechanism occurring in knee joint replacement is presented. The ideal&d model has been shown to produce results consistent with those in normal situations. In the present problem viscoelastic fluid has been considered to represent the synovial fluid in the fluid-film region and purely viscous Newtonian fluid in the porous layer due to filtration. Because of exact solution not being possible for the governing non-linear partial differential equations, the perturbation method has been used to obtain approximate solutions. The effect of an increase in the viscoelastic parameter of the lubricant is similar to that of an increase in concentration of hyaluronic acid molecules in synovial fluid. Important deductions are made for load carrying capacity and coefficient of friction and it has been shown that the slip velocity plays an important role in maintaining the self-adjusting nature of human joints. Keywords:
Knee joint replacement; Lubrication mechanism; Porous implant model
Introduction
The synovial joints provided by nature in the human body to carry out trouble free motion of one bone past another have long been identified as a bearings system. In fact, these joints function as excellent bearings in biological conditions. They support the considerable loads involved, and provide low friction coefficient service over a long span of one’s lifetime. However, in certain cases the joints develop certain abnormalities, e.g. osteoarthrosis. The synovial joint constitutes a tribological system similar to any other bearing so that the principal failure and their remedies are the same. In engineering systems such situations are tackled by improving or changing the lubricant. A total replacement is also required at times if the disorder becomes very severe. Both these remedial procedures can also be adapted for biological systems and therefore attempts have been made in both the directions for human joints. Joint replacement is widely used for treatment when joints fail to support loads or normal functions due to some accidental force or diseases. A number of different designs have emerged ranging from hinged to unconstrained configurations. Attempts have also been made to produce data in the laboratory for extrapolating the Correspondence to: P.N. Tandon, Department of Mathematics, Harcourt Butler Technological Institute, Kanpur 208002. India.
0020-7101/91/$03.50 0 1991 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
46
P.N. Tandon and A. Chaurasia
results for the human body. At present, most joint replacements are carried out at the hip, knee and fingers. It is advantageous to predict the lifetime of the joint prosthesis and the ability to produce the effect of load or patient activity on the prosthesis, e.g. ball and socket for hip and hinge configuration for knee, elbow and thumb [I]. During recent years, arthroplasty of the human knee joint with endoprosthesis is a common mode of treatment for degenerated knee joints. Advancing the state of art in design and utilization of these endoprosthetic devices requires thorough understanding of the stress behaviour of healthy joints and reconstructed combination of the bone-bone cement prosthesis device. The reaction of living bone depends among other things on the mechanical stresses to which it is subjected. The replacement of the articular surfaces of a human joint for the treatment of arthritis has come into wide practice in recent years [2]. Recently a comparison has been made of the lubrication mechanism believed to occur in normal and artificial joints [3]. After considering both metal on metal and metal on plastic specimens of artificial joints removed from the patients after various periods of implantation, important deductions were made about modes of lubrication and wear [4,5]. Walker [6] confirmed that the coefficient of friction for metal to metal is much more than that for metal and plastic. Further, there may be interaction between metal and plastic causing adhesion but the lubricant in between can therefore be expected to reduce frictional forces to about one-fifth or a quarter of the dry value [7]. Scales et al. [8] have also reported that the frictional effects experienced in various models of prosthesis subjected to a normal loading cycle show that metal and plastic arrangements prove to be much superior to lower the frictional resistances. Porous ceramics [9], metallic [lo] and polymeric [ 1l] materials are being considered for use in joint prosthetic stabilization via tissue in-growth into the surface undulations and pores of the replacement. However, there are a number of factors which suggest that ceramics, being inert, high strength abrasion resistant, will be the material of choice for use in highly corrosive environments [12]. There have been two important comparative studies of models of replacement via tissue in-growth and cementation for impaction and fixation of prosthesis in animals [13,14]. The studies confirmed that tissue in-growth provided better implant performance than using cementation. A number of investigators have been examining the concept of bone in-growth into microporous materials as a means of prosthetic fixation [15]. Tandon et al. [ 16,171 have also proposed suitable models for artificial human knee and hip joint replacements introducing porous plastic so that the bone in-growth into the pores may provide better impaction than using cement as compared to those employing steel or rigid plastic frames. It has also been observed that porous implants also help in maintaining an inbuilt cooling mechanism and nutritional transport in almost the same way as in normal joints. This investigation is an attempt to derive some conclusions for a simplified model of knee joint replacement. The proposed model assumes a two region flow model: flow of viscous fluid in the porous matrix and squeeze film lubrication in between the two approaching arm’s of the joint with viscoelastic fluid as lubricant to represent the synovial fluid. The arms of the hinge are impacted or connected into the shafts of the femur and tibia, the upper arm is fixed and its end is covered with a
47
Fig. 1. Model
for an artificial
knee joint
replacement
and its geometrical
counterpart.
porous layer whereas the lower one moves through an angle from CYto 0 as shown in Fig. 1. As the two arms approach each other, viscous fluid enters into the porous matrix which in turn increases the concentration of suspended particles in the fluid-film region. The pore size of the porous material of the lower arm has been considered to be so small that only the suspending medium enters into the porous matrix. Thus, the normal process of imbibition and exudation is maintained in this model. The model presents improvement over earlier models and it is much closer to reality. Formulation of the Problem Referring to Fig. I, the proposed model may be considered as a two-dimensional squeeze film lubrication in between two approaching surfaces. The upper one is fixed and porous and the other one is rigid and movable about the hinge. Under the assumption of small 0, the expression for film thickness is given by h’ = x’ tan 6 = de
(1)
The basic governing equation in the fluid-film fluid can be written as
a*d
apt ax,=&
[
__ayt2
a*d
- 3rl0, - X,) ___
af*
region applicable for viscoelastic
ad
__
2
( ay*>I
(2)
48
P. N. Tendon und A. Chuurusiu
and the equation of continuity is au’ _+?C=()
ax
(3)
af
where u‘ and v’ are velocities along the coordinated axes and p’ is the hydrodynamic pressure. The flow of viscous fluid in the porous matrix is governed by Darcy’s law; we have the following equations:
cco
-aP ax*
4
-
4
-,=__ U
-t=__
(4)
aF#
V
po
w
are the velocity components in the porous matrix. Thus, the pressure P satisfies the Laplace equations,
a*P + axl*
§*F’
.
=(-j
(5)
af2
Boundary conditions
u’
ad at y’= 0 af ’
=
-
(J’ __
U’ =
-
xc&
4
y’=--
at y’ = h’
aP
-
.
aty’=O
cl0 W v’ = -
x)w,
at y’ = h’
P’ = 0,
at x’ = 0 and L’
-1 P = 0,
at x’ = 0 and L’
aFf
ay'
P’(x’)
= 0, at y' = - H’
= F(x),
0)
(6)
Model for knee
49
is the permeability, H’ is the cartilage thickness and u’ is the where u’ = q/A, C#J parameter associated with the slip parameter. The following are non-dimensional variables: X’
x--,
y=--, Y'
24’
u=-
,
ho
ho
V'
p=p’
UO
NO2
“Z-,
UO
(7)
F
-1 P
=-
L
PUo2
=-, L'
’
h
ho
=---, h’ ho
The governing equations are transformed
Re
ap TT
a2t4 _-33_ ay2
a2u - au 2 a? (>ay
2!K+2?=, ax
u’=
Re=-
PUohO Pil
into the non-dimensional
form:
(8)
(9)
ay
aF
-Re
tfq---
r_= -Re
$I---
(10)
ax
aF
(11)
ay
and
a2F _+E=()
(12)
ay2
ax2
where, Re is the Reynolds number, 4, = qVho2. The boundary conditions in nondimensional form are written as u=-u-,
au
ay
u=xue, v=
aty=O
aty=h
- Re 4, -
aF
aty=O
ay ’ v=
- xw,
aty=h
(13)
50
P.N.
P = 0,
at x = 0 and L
P = 0,
at x = 0 and L
-
aF
=
o
aY
P(x) =
.
Tandon and A. Chaurusia
at x = 0 and L
P(x,O)
For the present analysis the non-dimensional
film thickness is given by
h = x9
(14)
Solution of the Problem
Under the assumption of very thin cartilage, it is assumed that aP/ay is linear across the matrix and is zero at the outer surface of the porous bearing shell. This assumption leads to
a2F
= constant = K, aY2
(15)
Using Eqns. (12) and (15) and boundary condition (13), we obtain
iq, = 0 =
KReH ho*
a2P
ax2
(1’5)
In order to obtain the solution for the fluid-film region all the variables are assumed in the sequences of the function in ascending powers of small parameters E, e.g. f=fe
+ ef, + E2f + * * .
(17)
Introducing u, p in this form in Eqn. (8) and collecting the zeroth and first order terms we obtain the following equations:
ap, at% ax ay2
(18)
ap, a2u, au0 2 a2uo 3--ax ay2 ( ay > a9
(19)
Re----=-
Re-=--
Model for
51
knee
Solutions of the zeroth and first order velocity distribution
where 0, - a)h*
2(h - u)
are given below.
-4xw(g 1- cJJ (h - a)
and u1z-p
Re
aP,
2
ax
3h2 (xw@* 00, - h) -- 3 uh2 2(h - u)~
2
1 _
XW8
(h - a)3 ’
3h6
h’
h5
(x00)* (h - a)*
2(h - a) - 8(h - u)~ + 8(h - a)* h3
h’
A2=l+
h4
4(h - a)* - 2(h - a)
Pressure distribution Expressions for the zeroth and first order pressure distribution
x4 e3
x+
X3
+-_----6
x3
e* 3
+
1241 es
5~’e6
1841
1404,
x6
x5 e4
2w --
+
I
x4
e3
8u
are
x7 - e6
x6 e3 369,
1
+
28~4,
(21)
and
~x6 E, 6
~x7 E2 +
7
F, ‘g
x_--- x4 e4 691
x5 e4 20&
(22)
P. N. Tendon und A. Churusiu
52
where
C,=L+-+-
C,=_--_
L4 e3
LS e3
12041
2004 I
L3
L6 e3
6
3Wl L6 es
+
L7 e4 -+28&l
5L’ e6
l&#q +
+
L3 e2 3
L4e3 8a
14041
E, = DI + DI E2 = D2 + Ds
e6 D2=-
5
(
61
0, =
Ds =
9W
--_--~ WJ,
2
2a24,
4u”
es a3 2u
e6 o3 2u2
L6 E, F, = - -6
L’ E2
e4
~~ e3 -+L WI
F2=---
2iw2
Lo
20&q
I
Load carrying capacity
The load carrying capacity of the joint in this case is defined as
w=
s L
P dx
-L.
=.
w, + E w,
(23)
Model for
53
knee
where P = PO + e PI
WO
L4
=
+24+-
L4 e2 _-_-12
L' es
LS e3 -_
12Wl
40a
L1e3 11241
1
9 L8tIb 2160@,
w, = Closure-time The closure time for a particular angle is given by t, = to + e t1
where to = -
11 =-
1
Of
wO
s Bi
1
B/
wl
s Bi
csd@
c6de
L5 e3 L2 c,= -+-+2
c4 =
6%
12Ou&
L4 e2 L7e3 L4 -++----_-----1124,
24 2
c,
L6 e4
c-2
=
12 c3
+
c 4
Re
CPI
Cl
>
Coefjcent of friction The coefficient of friction can be obtained by dividing frictional drag by load capacity
F
q
=-w
54
P. N. Tandon and A. Chaurasia
where
s L
F=
T
dx
-L
and r is the shear stress on the walls. Torque
The torque about the hinge is
s L
?lL
P dx
-L
Results and Discussion The problem of lubrication and friction has been analysed for a simplified model for a knee joint replacement. Figures 2-6 and Tables I-IV depict the representative computational results for pressure distribution, load capacity of the tibia1 replacement and time of approach for particular angular gap and torque. Figure 2 depicts the pressure distribution within the tibia1 replacement as a function of the angle 8. The pressure distribution increases B varies from a to 0. These results are in good agreement with that of normal situation in the knee joint [18].
Fig. 2. Variation
of pressure distribution
with I
for differentvalues
of slip parameter
and 0.
55
I6 8=30° lb-
1l2-
IO-
a-
O-Fig. 3. Variation
of friction coefficient
Fig. 4. Variation
of friction coefficient
with slip parameter
with slip parameter
for different
for different
values of viscoelastic parameter.
8.
56
P. N. Tandon und A, Chuurusiu
Further, we also observe that as the slip velocity at the porous boundary increases, the pressure decreases owing to the increased filtration of base fluid into the porous cartilage. Figure 3 describes the variation of the coefficient of friction with slip velocity for different values of the viscoelastic parameter. It has been observed that the friction coefficient decreases as the viscoelastic parameter increases. It increases with the slip parameter. In osteoarthritic conditions, hyaluronic acid molecules die out and lose their characteristics. At times, when viscosity decreases, external medicines are added to change the overall viscosity of the synovial fluid [19]. Figure 4 depicts the variation of the coefficient of friction with slip parameter for different values of the femoro-tibia1 angle. We observe from this that as the femorotibia1 angle increases, the friction coefficient increases. Figures 5 and 6 depict the variation of load carrying capacity with slip parameter for different values of femoro-tibia1 angle and viscoelastic parameter. The load carrying capacity increases as the viscoelastic parameter increases. The increasing values of the viscoelastic parameter describe the increase in the concentration of the suspended hyaluronic acid molecules which, in turn, increases the over all viscosity of the lubricant. This helps in sustaining greater loads [20]. It is concluded that at the closing end of the walking cycle, the slip velocity existing at the porous boundary plays an important role in the self-adjusting nature of the artificial joint in a similar manner to normal subjects. Further, we may observe from Fig. 6 that the load capacity increases as 8 varies from CYto 0. Tables I and II describe the variation of closure time for different values of slip parameter and viscoelastic parameter. It is clear that the closure time decreases as the slip parameter increases and it increases as the viscoelastic parameter increases owing to the increased apparent viscosity due to the increase in the viscoelastic parameter 1211. Tables III and IV depict the variation of torque with viscoelastic parameter for different values of the slip parameter and at different angles. The torque increases
TABLE
I
VARIATION
OF CLOSURE
W = 5.03542
x IO’*; E = 0.01.
(I
TIME
WITH
o
w2
60-45”
45-30”
30-15”
15-O”
0.1 0.2
1.319400 0.72135
I .330402 0.741386
I .36288 I 0.75204
1.380456 0.112591
0.3 0.4
0.417899 0.2lOOl8
0.529921 0.421119
0.541937 0.500379
0.553871 0.537506
01
Q4
0.3
a2 Q-4
Fig. 5. Variation of load carrying capacity with slip parameter for different 0. gr
es300
0
0.1
I
I
1
02
a3
04
Q----)
Fig. 6. The variation of load carrying capacity with slip parameter for parameter.
different values of viscoelastic
58
P. N.
Tandon
and
A. Chaurasia
as the viscoelastic parameter increases and it decreases when the slip parameter increases. We observe from Table IV that as the femoro-tibia1 angle decreaes the torque increases. As the viscoelastic parameter increases, the overall apparent viscosity increases and this, in turn, increases the torque as more viscous fluid offers greater resistance. TABLE
II
VARIATION
OF CLOSURE
TIME WITH e
W = 5.03542 x 1012;c = 0.1. e
442
0 0.01 0.02 0.03
TABLE
60-45”
45-30”
30-15”
15-O”
0.98756 1.319400 1.83094 2.43897
1.012374 1.330402 2.09876 2.83777
1.12980 I .362881 2.28112 3.00343
I .29876 I .380256 2.41562 3.19876
0.1
0.2
0.3
0.4
9.202502 x lOI 9.2687 x lOI 9.308856 x lOI 1.038001 x 10”
1.116334 x 2.89982 x 3.12255 x 4.180164 x
III
TORQUE a
e
0 0.01 0.02 0.03
TABLE
lOI IO”’ lOI lOI
3.350519 x 5.626355 x 6.362514 x 8.46929 x
IO” lOIs lOI lOI
I.441051 x 2.062187 x 3.444784 x 4.477259 x
10’5 lOI IO” 1O’5
IV
TORQUE e
0
0.01 0.02 0.03
8
60°
45”
30”
15”
2.03975 x 10” 7.920125 x IO” 9.24701 x IO” 1.02386 x lOI
9.202502 x lOI 9.2687 x lOI
3.570718 x lOI 3.916508 x IO”
8.66425 x lOI 9.52774 x lOI
9.308856 x lOI 1.038001 x 10”
4.612407 x IO” 5.092287 x lOIs
2.235273 x IO’” 3.705689 x IO’-’
Model for knee
59
References Adams JC: Outline of Orrhopaedics, English Language Book Society, Edinburgh, 1967. Swanson SAV, Freeman MAR and Health JC: J Bone Jr Surg, 55(B) (1973) 759. Barnet CH and Cobbaldt AF: Lubrication within living joints, J Bone Jt Surg, 44B(8) (1962) 662. Pilliar RM, Comeron HV and McNal I: Porous surface layered prosthetic devices, Biomed Eng. 4 (1975) 1. Radin EL and Paul IL, A consolidated concept ofjoint lubrication, J Bone JI Surg, 54(A) (1972) 607. Walker PS: The friction of internal artilicial joints. In Proc. Conf on Human Locomotor Engineering, Institution of Mechanical Engineers, 1971, p. 123. Duff-Barelay I and Spillman DT: Total hip joint prosthesis - a laboratory study of friction and wear. In Proc. Symp. Lubrication and Wear in Living and Artijicial Human joims, Institution of Mechanical Engineers. 1966, Vol. 181(3J). p. 90. 8 Scales JT, Kelly P and Goddard D: Friction torque studies of total joint replacements - the use of simulator. In Proc. Symp. on Lubrication and Wear in Join& 1969. 9 Klawitter JJ and Hulbert SF: Application of porous ceramics for attachment of load bearing internal orthopedic appliances, J Biomarer Res, 5 (1971) 161. IO Galante J, Rostoker W, Week R and Ray RD: Sintered fiber metal compositer as a basis for attachment of implants to bone, J Bone Jf Surg, 5(3A) (1971). II Taylor DF and Smith FB: Porous methylmethacrylate as an implant material, J Biomed Mafer Res Symp, 2 (1972) 467.
12 Kriegel WW and Palmour L: Ceramics in Severe Environmems. Plenum Press, New York, 1971. 13 Hulbert SF, Richbourgh HL, Klawitter JJ and Sauer BW: Evaluation of a metal ceramic composite hip prosthesis, J Biomed Murer Res Symp, 6 (1975) 189. 14 Nunamakar DN, Black J and Tornzo RG: A comparison of biological in-growth and cementation for fixation of the total hip prostesis. In Int. Biomoferial Symposium, Clemson University, Clemson, SC, 1975. 15 Klawitter J, Weinstein AM, Hulbert SF and Sauer BW: Tissue in growth and mechanical locking for anchorage of prosthesis in locomotor system. In Advance in Artificial Hip and Knee Joint Technology, New York, 1976, p. 422. 16 Tandon PN and Jaggi S: On wear and lubrication in an artificial knee joint replacement, Int J Mech Sci, (1981). 17 Tandon PN and Jaggi S: A polar model for synovial fluid with reference to human joint. Inr J Mech Sci, 21(3) (1979) 161. 18 Morrison JB: Mechanics of knee joint in relation to normal walking, J Biomech. 3 (1970) 5. 19 Tandon PN, Agarwal R and Chaurasia A, A model for ankle joint articulation. In Proc. oj Biomechanics, Wiley Eastern, New Delhi, 1989, p. 309. 20 Tandon PN and Chaurasia A: Microstructural effects on hip joint articulation, Appl Math. Modelling. 14 (1990) 312-319. 21 Tandon PN and Rakesh L: A model for lubrication mechanism in knee joint replacement, Def Sfi J, 36(l) (1986) 45.