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Quantitative analysis of joint lubrication
Wear, 61(1980) 203 - 218 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
QUANTITAT~E
203
ANALYSIS OF JOINT LUBRICATION
PAUL MARNELL
Department of ChemicalEngineering, Manhattan College, New York, N.Y. 10451 (U.S.A.) RICHARD K. WHITE
1702 Walnut St., Allentown, Pa. 18104 (U.S.A.) (Received March 19,1979)
Summary A comprehensive theoretical analysis of the extent of elastohydrodynamic lubrication in human joints is presented. The analytical model is developed from existing experimental data on the geometry, loading, kinetics and elastic properties of the hip joint and the viscous properties of synovial fluid. Results of a compu~r-generated numerical solution of the lubrication equations are given which demonstrate that elastohydrodynamic lubrication does not persist within human joints. An alternative lubrication mechanism based on the information obtained from the analysis is discussed.
1, Introduction Though it has been 45 years since MacConail [ l] first proposed that hydrodynamic lubrication may be the controlling mechanism for human joints and numerous other hypotheses [2, 31 have since been proposed opposing this view, there have been no quantitative lubrication analyses undertaken to explore his proposition. Today hydrodynamic lubrication is dismissed on the basis of qualitative arguments and other mech~isms are now debated f4]. The purpose of this paper is to present a detailed comprehensive mathematical analysis undertaken to ascertain whether hydrodynamic lub~cation is dominant in human joints and to propose an alternative mechanism for study which is based on the information, analysis and results generated during the course of this investigation.
2. Selection of a theoretical model In this section we qu~ti~ the primary factors in an elastohydrodynamic* lubrication analysis, i.e. bearing (joint) geometry, load history, *The prefix “elasto” indicates that the deformation (cartilage layers) is considered in the analysis.
of the bearing surfaces
204
journal (femur) angular velocity, lubricant (synovial fluid) viscosity and bearing surface elasticity. Three secondary phenomena, i.e. the flow of synovial fluid through the cartilage layers, the generation of normal stresses in synovial fluid and the viscoelastic properties of synovial fluid, have been shown by Marnell [ 51 to be negligible for this analysis. The adjective “secondary” in the preceding sentence is used only with reference to the lubrication analysis under discussion. These phenomena may be of primary importance in other situations. 2.1. Joint geometry The hip joint is assumed to be a ball (femur) and socket (acetabulum) bearing with a femur radius rf of about 2.5 cm. The nominal joint clearance C defined as the difference between the acetabulum and the femur radii is about 0.05 cm [6] . The deviation from sphericity has been shown to be about 1% by Rydell [ 71 and also by White and Elrod [ 81 and its effect in this analysis is negligible. The coordinate system used in this paper is shown in Fig. 1; it is fixed in the acetabulum.
Fig. 1. The spherical coordinate system used in the analysis.
Even though the surfaces of the femur and the acetabulum are spherical, their layers of articular cartilage vary in thickness. The maximum thicknesses are in the range 0.1 - 0.2 cm [ 91. Generally published cross sections of the hip joint cartilage layers indicate that the variations in thickness can be approximated by h,f(O, 4) = 0.1 - (O.lE+r)e cm
h,,(8, qb)= 0.1 + (0.3/n)8 cm
0 -c e -c 142 0-c 8 < A/3
(1) (2a)
205
h,,(B, 4) = 0.58 - (l.l4/n)O
cm
n/3 < 8 < n/2
(2b)
where h,, and hCf are the corresponding acetabulum and femoral cartilage layer thicknesses. The geometric characteristics of the other components of the hip joint such as the fat pads and the teres ligament are assumed to be negligible perturbations on the ball and socket geometry. 2.2. Load history and femur angular velocity history The normalized load history of Fig. 2 is obtained by using the in vivo experimental data of Rydell [ 71 on the history of the load on an artificial femoral head during walking and transforming them to the coordinate system fixed in the acetabulum. The present analysis is quite general though and an arbitrary load history can be used as an input parameter to accommodate any difference in load history between artificial and natural hip joints.
Fig. 2. A representative femur loading history during walking. The load components refer to a coordinate system fiied in the acetabulum; they are normalized relative to the total body weight B.
206
The history of the femur angular velocity vector w during walking is developed from the measurements by Eberhart et al. [lo] of the rotation of the femur relative to the pelvis. The time histories of the measured saggital, frontal and transverse plane projection angles are given in Fig. 3. However, since projection effects are not significant it can be shown [ 5] that the components of o can be approximated by w, = da,/dt
Pa)
wY = daJdt
(3b)
wz = da,/dt
(3c)
!
0
I
2
l
sime.t,wc
Fig. 3. The time histories of the projection angles (measured in degrees) G, af and &.
2.3. Synovial fluid viscosity Data obtained by Block and Dittenfass [ 11] on the viscous behavior of normal synovial fluid indicate that its viscosity can be represented by 17= mD”-l where Q is the coefficient of viscosity in poises (dyn s cmm2) and D is the
(4)
207
cone-in-cone viscometer shearing rate in seconds. (Any consistent set of units can be used; the c.g.s. system is used throughout this paper.) For a normal knee joint n =$ (5a) m = 47 dyn s113cm-’ (5b) Based on these data for simple shear flow it is assumed that for the flow conditions present in the hip joint during walking synovial fluid behaves as a power law fluid [12] ; i.e. its viscosity is given by 77= m{(~A:A)1’2}n-1
(6)
where the numerical values of m and n are given by eqns. (5) and A is the rate of deformation tensor. For the simple flow field of the cone-in-cone viscometer, the term within the traces in eqn. (6) reduces to the shearing rate D. Synovial fluid also exhibits thixotropic behavior [ 131. The steady state viscosity is about 10% smaller than the initial viscosity at a shearing rate of 1000 s-l. At D = 10 s-l the difference is about 1%. Since we are concerned with lubrication during walking where the synovial fluid will approach a well sheared state, the small thixotropic effect will be absent after a number of steps have been taken. Hence the time-dependent (thixotropic) viscosity effect is assumed to be negligible. This has also been demonstrated by Meyers et al. [ 141. 2.4. Cartilage elasticity Since cartilage is deformable it can yield under the pressure generated in the synovial fluid when the joint is loaded. The relation between the local radial displacement of the bearing surfaces (acetabulum and femur) and the pressure field of the lubricant (synovial fluid) film is a necessary input to an elastohydrodynamic lubrication analysis. The procedure developed to obtain this relation is quite detailed and is presented elsewhere [ 51. It consists of a series of analysis to obtain a kernel function K defined by a/2 s(f%f#~,
0
=
r”,
2n
K(e,$;d’, #‘)p(d’,4’, JS
0
0
t) sine’ d#’ de’
(7)
where r, = rf + C is the radius of the acetabulum, p(8 ‘, $‘, t) is the fluid film pressure at point (0 ‘, 4’) on the acetabulum at time t, K(8, 4; 8’, 4’) is the algebraic difference between the radial displacements of the acetabulum and the femur cartilage surfaces at point (e,#) due to a unit load acting on each of these surfaces at point (0 ‘, 4’) and S (0 , 9, t) is the algebraic difference between the radial displacements of the acetabulum and the femur cartilage surfaces at point (0 , 4) due to the fluid film pressure field. Here we briefly summarize the analyses required to obtain the kernel function and compute S(0 , 9, t).
208
(1) The displacement equation (often called the Navier equation) for cartilage is developed by specializing Biot’s formulation [ 151 for the stressstrain law of a porous fluid-saturated matrix whose interconnecting pores contain a viscous compressible fluid. Thus assuming that cartilage consists of an incompressible matrix, i.e. any volume change it exhibits is due to a corresponding change in its porosity f and that its pore fluid is incompressible, the displacement equation for the cartilage matrix reduces to G*S+(X+G)Ve-VP,=0
(8)
where the shear modulus G and X are material properties (both are called Lame constants), S is the displacement vector for the matrix, pc is the pore liquid pressure, e is the matrix dilatation (change in volume divided by original volume) defined by e=V.S
(9)
and * is the vector Laplacian operator defined by [16] a=gv.
-VXVX
(10)
The displacement s of the pore liquid is introduced by means of Darcy’s law f(i
4)=-&l, c1
(11)
where h is the permeability of the cartilage, /J is the viscosity of the pore liquid and the dots above s and S indicate differentiation with respect to time, e.g. S is the pore liquid velocity vector. These expressions neglect inertia effects in both media, the effect of viscous dissipation on the cartilage matrix deformation and the effect of cartilage matrix dilatation on the flow of the pore liquid. The first effect is of no concern under the slowly varying conditions present in the hip during walking, and the latter two effects are also of minor consequence when the cartilage layers of the femur and acetabulum are loaded only by the pressure field of the assumed lubricant film, i.e. the surfaces of the cartilage layers do not touch and consolidation of the cartilage matrix (expulsion of its pore liquid) is negligible. Torzilli and Mow [4] have studied the situation where the latter two effects are of primary interest and the corresponding terms are included in eqns. (8) and (11). (2) It is shown that the deformation due to dilatation of the cartilage when loaded by an external fluid pressure field is negligible when compared with that due to shear. Under these conditions eqns. (8) and (11) can be reduced to G*S=Vp,
(12)
v”p,
(13)
= 0
Note that only the value for the shear modulus is required for a numerical solution to these equations.
209
(3) Since the cartilage layer is thin relative to its radius of curvature, radial (normal) component of eqn. (12) reduces to GV2S, =
;I&
the
(14)
Equations (13) and (14) are solved for a planar cartilage layer of uniform thickness and infinite extent subjected to a unit force. The resulting kernel function is shown to be applicable to the real situation of a finite curved cartilage layer of variable thickness. This approximation is valid because the value of the kernel function approaches zero at a small distance from the point of load application while the variation in cartilage thickness is negligible over the same distance. (4) Equation (7) can be integrated using this infinite layer kernel function by expanding the pressure into a two-dimensional Taylor series about (0, 4). From this it follows that (15) i.e. the net local cartilage radial displacement depends solely on the local second derivative of the pressure field and the local thicknesses of the femoral and acetabulum layers. This result considerably simplifies the elastohydrodynamic analysis which is discussed in the next section.
3. Development
of the system of equations
We postulate elastohydrodynamic lubrication and based on the model described in Section 2 outline the development of the system of equations for determining the thickness of the synovial fluid film in the hip joint during walking. If upon solving this system of equations we find that the calculated film thickness is less than the minimum film thickness required for this type of lubrication, it will follow that the hypothesis is not valid and another mechanism is dominant. In this study it is assumed that the lower limit for the film thickness in elastohydrodynamic lubrication is lo-’ cm. This is roughly twice the diameter of the hyaluronic acid molecule [ 171 which is a key component of synovial fluid. We define the local volumetric flow vector per unit length by QZ
J 0
h Vdr
(16)
where h is the local film thickness and V the synovial fluid velocity vector. From the conservation of mass for an incompressible fluid it follows that (17)
210
If there were no deformation of the cartilage layers the local film thickness would be given by h(0,d,0
= h,(~,4%0 = C -fix*
-f2.Y*
(18)
-fG*
where x*(t), y*(t) and z*(t) are the time-dependent X, y and z coordinates respectively of the center of the femur, and fi
= sin 8 cos $I
(19a)
f2 = sin e sin Q
(I=)
f3 = cos
WC)
8
This result follows directly from the geometry of a ball within a circular cavity as shown in Fig. 4. With cartilage deformation we have (20)
h(e,ti,t) = hs(e,@,t) + s(e,+,t) where 6 is given by eqn. (15).
Fig. 4. The geometry for describing the instantaneous femur position. The z axis is directed along the polar axis of the hemispherical “cavity”.
Combining eqns. (17), (18), (20) and (15) yields dy* z+f3-+
da* dt
0.34 -hEt G
;’ -_p(e,~,
t)
(21)
where
There are four unknowns in eqn. (21), namely x* , y*, z* and p. It can be shown [ 51 that 4 is determined solely but in a rather complex manner by p, h and w, and hence the same four unknowns (o is an input parameter). Three additional equations are available from the load equations. The transient load W(t) transferred to the femur (an input parameter) must be equal to the integrated effect of the pressure field acting over the surface of the acetabulum. Thus
211 112
W,(t)
21
ss
= -
0
n/2 W,(t)=
GW
277
(pr,2sin8
J-s
0
n12
The boundary
sin4
t23b)
2n
sin 6 d# de ) cos 8
(prz S$
0
d$de)sine
0
w=(t) = -
p(n/2,9,
@rz sin 19d@ de) sin 13cos $
0
(23~)
0 condition
for the pressure field is
0 = 0
(24)
while the initial conditions
are (25a)
p(e,9,0)=0 X*(o) = x;
(25b)
Y*(O) = $I
(25~)
z”(0) = 2;
(25d)
Equation (24) expresses the fact that the synovial fluid pressure at the hemispherical edge is equal to the ambient joint pressure which is essentially zero. At the start of the walking cycle the joint load is very nearly zero; this condition is specified by eqn. (25a). The initial position of the center of the femur is specified by three arbitrary initial coordinates xi, yg and 2:. Equations (21) and (23) constitute a set of four coupled equations, one of which is a strongly non-linear partial differential equation and three of which are integral equations, for the four variablesp(8, 9, t), x*(t), y*(t) and z*(t). Though an analytical solution is doubtful, an algorithm for a numerical solution is possible. Details of the algorithm and the corresponding computer program developed for its implementation have been presented by Marnell [51.
4. Results, conclusions
and discussion
In order to minimize the required computer time (a single run of four walking cycles requires 30 min on an IBM 360) a case that would allow the use of relatively large time intervals in the numerical calculation and would also be most favorable for elastohydrodynamic lubrication was studied. Thus the input body weight B was taken to be a nominal 25 lbf (10’ dyn), the joint clearance C was assumed to be only 0.001 cm and the shear modulus G was assumed to be only 1 X lo6 dyn cmT2. Values for all other input parameters, i.e.h,,, hcf, w, 17, r, and W/B were as specified in Section 2. However, as will be discussed, even this case of a lightly loaded close-fitting bearing with highly compliant surfaces failed to demonstrate sustained elasto-
212
hydrodynamic less favorable
lubrication. Therefore study of more representative cases was not undertaken.
though
4.1. Results and interpretation The variations of the coordinates of the center of the femur with time for four walking cycles are given in Figs. 5 - 7. Owing to the periodic nature of the joint load and angular velocity vectors they too are periodic. After about four walking cycles the femur has almost settled into its equilibrium orbit, as witnessed by the almost horizontal nature of the maximum and minimum displacement loci. Over a major portion of the walking cycle both the x* and y* displacements exceed 10m3 cm, the maximum that could occur if the surfaces were rigid (they would touch at this value). However, as the load is reduced to zero during the latter half of a walking cycle the deformed cartilage layers expand against the fluid film and reverse the direction of motion of the femur, thereby decreasing the values of X* and y* . At the end of each cycle the X* and y* coordinates are less than 10M3 cm, as required if the surfaces are to be separated by a fluid film. The z* coordinate decreases with time since the squeeze film action of the x and y loads opposes that due to the z load, i.e. the x and y loads
2
0
z
I
Tim,t
L
3
,sBc
Fig. 5. The x* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cms2).
4
213
16
I
unumryul surface d+xement
0
I
2
3
4
Time,t,ssr.
Fig. 6. The y* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cmm2).
squeeze some fluid into the cavity while the z load only squeezes fluid from the cavity. The net result is a slight gain over the first several cycles such that the femur lifts out of the acetabulum from an initial z* displacement of 5.23 X 10m4 cm to an equilibrium value of about -1 X lo-’ cm. The limiting values for the x* , y* and z* coordinates at the end of a walking cycle are about 10 X lo-“, 6 X 10V4 and -1 X 10e4 cm, a situation that leads to minimum film thicknesses which are substantially less than 1 X lo-* cm, the nominal lower bound for hydrodynamic lubrication. This is illustrated in Fig. 8 where the variation of the film thickness with the polar angle 8 using the longitudinal angle 4 as a parameter at a time of 3.5 s is presented. In the region 60” G 8 < 90” the x and y loads tend to force the femur onto this acetabulum along the 54” longitudinal. The effect of the angular velocity vector is not large enough to help drag fluid into this region and maintain an adequate hydrodynamic film thickness. Even though the load in the r direction is greater than that in the y direction, the fact that wY is greater than o, offsets this and leads to a larger film profile at $I = 0. The inflections of the curves occur at 8 = 60” ; this is to be expected as the acetabulum cartilage layer starts to decrease with 0 at this point (see eqn. (21).
Fig. 7. The z* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cm-‘).
The minimum film thickness profile occurs at about 4~= 54” which roughly corresponds to the angle with the x axis of the resultant of the x and y loads. In other words the film profile is smallest at about that value of $I which is directly under the loading. This is also illustrated in Fig. 9 where the corresponding longitudinal variation of the film thickness is presented for 19= 45’. When 6 is about 54” the film thickness is again a minimum. The polar pressure profiles (with 4 as a parameter) in the joint at a time of 3.5 s are presented in Fig. 10. As expected the maximum pressure profile is that for 9 close to 54”. Since the x and y loads tend to force the femur onto the acetabulum along the 54’ longitudinal, the pressure also exhibits its maximum there. 4.2. Conclusions and discussion Within the range of validity of the postulated analytical model, elastohydrodynamic lubrication does not persist within the hip joint during normal walking, i.e. the joint load and the relative motion of the bearing surfaces are such that the joint surfaces are not separated by lubricant film of the order of 10e4 cm. Factors of cartilage surface irregularity [4], flow of synovial fluid and the pore liquid through the cartilage layers, selective filtration of the macro-
215
0
0
10
a
30
40
5u
60
m
\
60
8
PokrAngle,0,dy Fig. 8. The film thickness rofiles at maximum loading for case 1 at 3.5 s (B = 10’ dyn, C = 0.001 cm, G = 1 X 10 P dyn cm-‘).
molecules of synovial fluid at the cartilage surfaces [4] and the specific affinity between synovial fluid and cartilage [ 181 which are not of significance in elastohydrodynamic lubrication no doubt play an important part in the lubrication of synovial joints. These phenomena may be accommodated in a postulated model for lubrication which is essentially a combined boundary and internally induced hydrostatic lubrication model. Thus cartilage asperities are of the same magnitude as the calculated minimum film thickness; hence these surface peaks touch and interact with each other. However, since the cartilage layers are filled with liquid only a small portion of the load is actually carried by the interacting peaks. The majority of the load is carried by the liquid that pervades the cartilage matrix. The load will be transferred to the cartilage matrix in proportion to the amount of pore liquid expelled. Further, the compliance of the liquidfilled cartilage matrix prevents rigid peak shearing, i.e. opposing peaks slide across each other. Since there is a strong affinity between cartilage and synovial fluid, these interacting surfaces are probably coated with the macromolecules of synovial fluid which allows them to pass over each other without significant resistance or wear. It has been experimentally determined that articular cartilage in duo maintains its liquid content with time. The reasons for this are based on the unique properties of cartilage and synovial fluid and the transient behavior of the joint load and angular velocity vectors. Let us assume for the moment a constant load and fixed region of application. The elasticity and surface irregularities of cartilage tend to
216
16
1.4
Ll
,o-
OE
0.6-
40
84
/w
I
/#
I
MO
,
m
,
z&3
,
A?0
I
240
0
longitudinal Angle, z deg.
Fig. 9. A longitudinal film thickness profile at maximum loading for case 1 at 3.5 s (B = 107 dyn, C = 0.001 cm, G = 1 X lo6 dyn cmh2). Fig. 10. The pressure profiles at maximum loading for case 1 at 3.5 s (B = 10’ dyn, C = 0.001 cm, G = 1 X 106 dyn cme2).
confine the synovial fluid within virtually non-connecting valleys so that no clear uninterrupted flow channel exists between the region of loading and the joint extremities. Hence the synovial fluid is forced to flow through the articular cartilage layers and the peaks if it is to be expelled from the valleys between the cartilage surfaces. If the synovial fluid is to flow through cartilage, it must displace an equivalent amount of pore liquid. However, since the permeability of cartilage is extremely small, the escape of liquid from the cartilage is slow. As the synovial fluid tries to flow through the cartilage, its macromolecules tend to filter out on the surface of the finely pored cartilage matrix and form a “gel” which increases the resistance to its flow to the
217
extremities of the joint. Hence we have a situation wherein the synovial fluid is essentially “contained” in the valleys between the cartilage asperities. It is this “contained” synovial fluid which probably transmits the majority of the load through the pore liquid of the cartilage to the underlying bone structure. The remaining fraction of the load is carried by the cartilage matrix and results in simultaneous expulsion of some of its pore liquid. Even though the resistance to flow of the “contained” synovial fluid and the pore liquid is quite large, after a sufficient period of time they would escape to the joint extremities if the direction, magnitude and point of application of the load remained constant. Fortunately, this is not the case and the variation in the direction, magnitude and point of application of the load are such that the synovial fluid and the pore liquid, rather than flow inexorably out of the joint, shuttle around in the region of loading. Hence, though at any one instant the load may tend to drive synovial fluid and pore liquid out of the loading region (at an extremely slow rate), later in the walking cycle the change in the direction of the load vector and the relative motion of the cartilage surfaces are such that they are “pumped” back into the region. The cartilage deformation profile varies and undulates throughout the walking cycle and also helps to maintain the internal “pumping” within the region of loading.
References 1 M. A. MacConaill, The function of intra-articular fibrocartilages, with special reference to the knee and inferior radio-ulnar joints, J. Anat., 66 (1932) 210 - 227. 2 V. C. Mow, The role of lubrication in biomechanical joints, J. Lubr. Technol., 91 (1969) 320 - 328. 3 E. L. Radin and I. L. Paul, A consolidated concept of joint lubrication, J. Bone Jt Surg., 54A (3) (April 1972) 607 - 616. 4 P. A. Torzilli and V. C. Mow, On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function - II. The analysis, solution and conclusions, J. Biomech., 9 (1976) 587 - 606. 5 P. Marnell, A theoretical evaluation of the persistence of hydrodynamic lubrication in the hip joint during walking, Doctoral Thesis, Columbia University, 1973. 6 P. S. Walker, D. Dowson, M. D. Longfield and V. Wright, Boosted lubrication in synovial joints by fluid entrapment and enrichment, Ann. Rheum. Dis., 2 7 (6) (Nov. 1968) 513 - 520. 7 N. W. Rydell, Forces Acting on the Femoral Head Prosthesis, Univ. Gotteborg, Sweden, 1966. 8 R. K. White and H. G. Elrod, Program for Correlation of Hip Contour Measurements, Columbia Univ., New York, 1968. 9 C. H. Barnett, D. V. Davies and M. A. MacConaill, Synouial Joints, Charles C. Thomas, Springfield, Ill., 1961. 10 H. D. Eberhart, V. T. Inman and B. Bresler, in P. E. Klopsig (ed.), Human Limbs and Their Substitutes, McGraw Hill, New York, 1954. 11 B. Block and L. Dittenfass, Rheological study of human synovial fluid, Aust. N.Z. J. Surg., 33 (1963) 108 - 113. 12 R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960.
218 13 D. V. Davies, Properties of synovial fluid, Proc. Inst. Mech. Eng. London, 181 (35) (1966-67) 25 - 29. 14 R. R. Meyers, S. Negami and R. K. White, Dynamic mechanical properties of synovial fluid, Biorheology, 3 (1966) 197 - 209. 15 M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (2) (1955) 182 - 185. 16 P. Moon and D. E. Spencer, Field Theory For Engineers, Van Nostrand, New York, 1961. 17 A. G. Ogston and J. E. Stanier, The dimensions of the particle of hyaluronic acid complex in synovial fluid, Biochem. J., 49 (1951) 585. 18 C. W. McCutchen, The frictional properties of animal joints, Wear, 5 (1962) 1 - 17.
QUANTITAT~E
203
ANALYSIS OF JOINT LUBRICATION
PAUL MARNELL
Department of ChemicalEngineering, Manhattan College, New York, N.Y. 10451 (U.S.A.) RICHARD K. WHITE
1702 Walnut St., Allentown, Pa. 18104 (U.S.A.) (Received March 19,1979)
Summary A comprehensive theoretical analysis of the extent of elastohydrodynamic lubrication in human joints is presented. The analytical model is developed from existing experimental data on the geometry, loading, kinetics and elastic properties of the hip joint and the viscous properties of synovial fluid. Results of a compu~r-generated numerical solution of the lubrication equations are given which demonstrate that elastohydrodynamic lubrication does not persist within human joints. An alternative lubrication mechanism based on the information obtained from the analysis is discussed.
1, Introduction Though it has been 45 years since MacConail [ l] first proposed that hydrodynamic lubrication may be the controlling mechanism for human joints and numerous other hypotheses [2, 31 have since been proposed opposing this view, there have been no quantitative lubrication analyses undertaken to explore his proposition. Today hydrodynamic lubrication is dismissed on the basis of qualitative arguments and other mech~isms are now debated f4]. The purpose of this paper is to present a detailed comprehensive mathematical analysis undertaken to ascertain whether hydrodynamic lub~cation is dominant in human joints and to propose an alternative mechanism for study which is based on the information, analysis and results generated during the course of this investigation.
2. Selection of a theoretical model In this section we qu~ti~ the primary factors in an elastohydrodynamic* lubrication analysis, i.e. bearing (joint) geometry, load history, *The prefix “elasto” indicates that the deformation (cartilage layers) is considered in the analysis.
of the bearing surfaces
204
journal (femur) angular velocity, lubricant (synovial fluid) viscosity and bearing surface elasticity. Three secondary phenomena, i.e. the flow of synovial fluid through the cartilage layers, the generation of normal stresses in synovial fluid and the viscoelastic properties of synovial fluid, have been shown by Marnell [ 51 to be negligible for this analysis. The adjective “secondary” in the preceding sentence is used only with reference to the lubrication analysis under discussion. These phenomena may be of primary importance in other situations. 2.1. Joint geometry The hip joint is assumed to be a ball (femur) and socket (acetabulum) bearing with a femur radius rf of about 2.5 cm. The nominal joint clearance C defined as the difference between the acetabulum and the femur radii is about 0.05 cm [6] . The deviation from sphericity has been shown to be about 1% by Rydell [ 71 and also by White and Elrod [ 81 and its effect in this analysis is negligible. The coordinate system used in this paper is shown in Fig. 1; it is fixed in the acetabulum.
Fig. 1. The spherical coordinate system used in the analysis.
Even though the surfaces of the femur and the acetabulum are spherical, their layers of articular cartilage vary in thickness. The maximum thicknesses are in the range 0.1 - 0.2 cm [ 91. Generally published cross sections of the hip joint cartilage layers indicate that the variations in thickness can be approximated by h,f(O, 4) = 0.1 - (O.lE+r)e cm
h,,(8, qb)= 0.1 + (0.3/n)8 cm
0 -c e -c 142 0-c 8 < A/3
(1) (2a)
205
h,,(B, 4) = 0.58 - (l.l4/n)O
cm
n/3 < 8 < n/2
(2b)
where h,, and hCf are the corresponding acetabulum and femoral cartilage layer thicknesses. The geometric characteristics of the other components of the hip joint such as the fat pads and the teres ligament are assumed to be negligible perturbations on the ball and socket geometry. 2.2. Load history and femur angular velocity history The normalized load history of Fig. 2 is obtained by using the in vivo experimental data of Rydell [ 71 on the history of the load on an artificial femoral head during walking and transforming them to the coordinate system fixed in the acetabulum. The present analysis is quite general though and an arbitrary load history can be used as an input parameter to accommodate any difference in load history between artificial and natural hip joints.
Fig. 2. A representative femur loading history during walking. The load components refer to a coordinate system fiied in the acetabulum; they are normalized relative to the total body weight B.
206
The history of the femur angular velocity vector w during walking is developed from the measurements by Eberhart et al. [lo] of the rotation of the femur relative to the pelvis. The time histories of the measured saggital, frontal and transverse plane projection angles are given in Fig. 3. However, since projection effects are not significant it can be shown [ 5] that the components of o can be approximated by w, = da,/dt
Pa)
wY = daJdt
(3b)
wz = da,/dt
(3c)
!
0
I
2
l
sime.t,wc
Fig. 3. The time histories of the projection angles (measured in degrees) G, af and &.
2.3. Synovial fluid viscosity Data obtained by Block and Dittenfass [ 11] on the viscous behavior of normal synovial fluid indicate that its viscosity can be represented by 17= mD”-l where Q is the coefficient of viscosity in poises (dyn s cmm2) and D is the
(4)
207
cone-in-cone viscometer shearing rate in seconds. (Any consistent set of units can be used; the c.g.s. system is used throughout this paper.) For a normal knee joint n =$ (5a) m = 47 dyn s113cm-’ (5b) Based on these data for simple shear flow it is assumed that for the flow conditions present in the hip joint during walking synovial fluid behaves as a power law fluid [12] ; i.e. its viscosity is given by 77= m{(~A:A)1’2}n-1
(6)
where the numerical values of m and n are given by eqns. (5) and A is the rate of deformation tensor. For the simple flow field of the cone-in-cone viscometer, the term within the traces in eqn. (6) reduces to the shearing rate D. Synovial fluid also exhibits thixotropic behavior [ 131. The steady state viscosity is about 10% smaller than the initial viscosity at a shearing rate of 1000 s-l. At D = 10 s-l the difference is about 1%. Since we are concerned with lubrication during walking where the synovial fluid will approach a well sheared state, the small thixotropic effect will be absent after a number of steps have been taken. Hence the time-dependent (thixotropic) viscosity effect is assumed to be negligible. This has also been demonstrated by Meyers et al. [ 141. 2.4. Cartilage elasticity Since cartilage is deformable it can yield under the pressure generated in the synovial fluid when the joint is loaded. The relation between the local radial displacement of the bearing surfaces (acetabulum and femur) and the pressure field of the lubricant (synovial fluid) film is a necessary input to an elastohydrodynamic lubrication analysis. The procedure developed to obtain this relation is quite detailed and is presented elsewhere [ 51. It consists of a series of analysis to obtain a kernel function K defined by a/2 s(f%f#~,
0
=
r”,
2n
K(e,$;d’, #‘)p(d’,4’, JS
0
0
t) sine’ d#’ de’
(7)
where r, = rf + C is the radius of the acetabulum, p(8 ‘, $‘, t) is the fluid film pressure at point (0 ‘, 4’) on the acetabulum at time t, K(8, 4; 8’, 4’) is the algebraic difference between the radial displacements of the acetabulum and the femur cartilage surfaces at point (e,#) due to a unit load acting on each of these surfaces at point (0 ‘, 4’) and S (0 , 9, t) is the algebraic difference between the radial displacements of the acetabulum and the femur cartilage surfaces at point (0 , 4) due to the fluid film pressure field. Here we briefly summarize the analyses required to obtain the kernel function and compute S(0 , 9, t).
208
(1) The displacement equation (often called the Navier equation) for cartilage is developed by specializing Biot’s formulation [ 151 for the stressstrain law of a porous fluid-saturated matrix whose interconnecting pores contain a viscous compressible fluid. Thus assuming that cartilage consists of an incompressible matrix, i.e. any volume change it exhibits is due to a corresponding change in its porosity f and that its pore fluid is incompressible, the displacement equation for the cartilage matrix reduces to G*S+(X+G)Ve-VP,=0
(8)
where the shear modulus G and X are material properties (both are called Lame constants), S is the displacement vector for the matrix, pc is the pore liquid pressure, e is the matrix dilatation (change in volume divided by original volume) defined by e=V.S
(9)
and * is the vector Laplacian operator defined by [16] a=gv.
-VXVX
(10)
The displacement s of the pore liquid is introduced by means of Darcy’s law f(i
4)=-&l, c1
(11)
where h is the permeability of the cartilage, /J is the viscosity of the pore liquid and the dots above s and S indicate differentiation with respect to time, e.g. S is the pore liquid velocity vector. These expressions neglect inertia effects in both media, the effect of viscous dissipation on the cartilage matrix deformation and the effect of cartilage matrix dilatation on the flow of the pore liquid. The first effect is of no concern under the slowly varying conditions present in the hip during walking, and the latter two effects are also of minor consequence when the cartilage layers of the femur and acetabulum are loaded only by the pressure field of the assumed lubricant film, i.e. the surfaces of the cartilage layers do not touch and consolidation of the cartilage matrix (expulsion of its pore liquid) is negligible. Torzilli and Mow [4] have studied the situation where the latter two effects are of primary interest and the corresponding terms are included in eqns. (8) and (11). (2) It is shown that the deformation due to dilatation of the cartilage when loaded by an external fluid pressure field is negligible when compared with that due to shear. Under these conditions eqns. (8) and (11) can be reduced to G*S=Vp,
(12)
v”p,
(13)
= 0
Note that only the value for the shear modulus is required for a numerical solution to these equations.
209
(3) Since the cartilage layer is thin relative to its radius of curvature, radial (normal) component of eqn. (12) reduces to GV2S, =
;I&
the
(14)
Equations (13) and (14) are solved for a planar cartilage layer of uniform thickness and infinite extent subjected to a unit force. The resulting kernel function is shown to be applicable to the real situation of a finite curved cartilage layer of variable thickness. This approximation is valid because the value of the kernel function approaches zero at a small distance from the point of load application while the variation in cartilage thickness is negligible over the same distance. (4) Equation (7) can be integrated using this infinite layer kernel function by expanding the pressure into a two-dimensional Taylor series about (0, 4). From this it follows that (15) i.e. the net local cartilage radial displacement depends solely on the local second derivative of the pressure field and the local thicknesses of the femoral and acetabulum layers. This result considerably simplifies the elastohydrodynamic analysis which is discussed in the next section.
3. Development
of the system of equations
We postulate elastohydrodynamic lubrication and based on the model described in Section 2 outline the development of the system of equations for determining the thickness of the synovial fluid film in the hip joint during walking. If upon solving this system of equations we find that the calculated film thickness is less than the minimum film thickness required for this type of lubrication, it will follow that the hypothesis is not valid and another mechanism is dominant. In this study it is assumed that the lower limit for the film thickness in elastohydrodynamic lubrication is lo-’ cm. This is roughly twice the diameter of the hyaluronic acid molecule [ 171 which is a key component of synovial fluid. We define the local volumetric flow vector per unit length by QZ
J 0
h Vdr
(16)
where h is the local film thickness and V the synovial fluid velocity vector. From the conservation of mass for an incompressible fluid it follows that (17)
210
If there were no deformation of the cartilage layers the local film thickness would be given by h(0,d,0
= h,(~,4%0 = C -fix*
-f2.Y*
(18)
-fG*
where x*(t), y*(t) and z*(t) are the time-dependent X, y and z coordinates respectively of the center of the femur, and fi
= sin 8 cos $I
(19a)
f2 = sin e sin Q
(I=)
f3 = cos
WC)
8
This result follows directly from the geometry of a ball within a circular cavity as shown in Fig. 4. With cartilage deformation we have (20)
h(e,ti,t) = hs(e,@,t) + s(e,+,t) where 6 is given by eqn. (15).
Fig. 4. The geometry for describing the instantaneous femur position. The z axis is directed along the polar axis of the hemispherical “cavity”.
Combining eqns. (17), (18), (20) and (15) yields dy* z+f3-+
da* dt
0.34 -hEt G
;’ -_p(e,~,
t)
(21)
where
There are four unknowns in eqn. (21), namely x* , y*, z* and p. It can be shown [ 51 that 4 is determined solely but in a rather complex manner by p, h and w, and hence the same four unknowns (o is an input parameter). Three additional equations are available from the load equations. The transient load W(t) transferred to the femur (an input parameter) must be equal to the integrated effect of the pressure field acting over the surface of the acetabulum. Thus
211 112
W,(t)
21
ss
= -
0
n/2 W,(t)=
GW
277
(pr,2sin8
J-s
0
n12
The boundary
sin4
t23b)
2n
sin 6 d# de ) cos 8
(prz S$
0
d$de)sine
0
w=(t) = -
p(n/2,9,
@rz sin 19d@ de) sin 13cos $
0
(23~)
0 condition
for the pressure field is
0 = 0
(24)
while the initial conditions
are (25a)
p(e,9,0)=0 X*(o) = x;
(25b)
Y*(O) = $I
(25~)
z”(0) = 2;
(25d)
Equation (24) expresses the fact that the synovial fluid pressure at the hemispherical edge is equal to the ambient joint pressure which is essentially zero. At the start of the walking cycle the joint load is very nearly zero; this condition is specified by eqn. (25a). The initial position of the center of the femur is specified by three arbitrary initial coordinates xi, yg and 2:. Equations (21) and (23) constitute a set of four coupled equations, one of which is a strongly non-linear partial differential equation and three of which are integral equations, for the four variablesp(8, 9, t), x*(t), y*(t) and z*(t). Though an analytical solution is doubtful, an algorithm for a numerical solution is possible. Details of the algorithm and the corresponding computer program developed for its implementation have been presented by Marnell [51.
4. Results, conclusions
and discussion
In order to minimize the required computer time (a single run of four walking cycles requires 30 min on an IBM 360) a case that would allow the use of relatively large time intervals in the numerical calculation and would also be most favorable for elastohydrodynamic lubrication was studied. Thus the input body weight B was taken to be a nominal 25 lbf (10’ dyn), the joint clearance C was assumed to be only 0.001 cm and the shear modulus G was assumed to be only 1 X lo6 dyn cmT2. Values for all other input parameters, i.e.h,,, hcf, w, 17, r, and W/B were as specified in Section 2. However, as will be discussed, even this case of a lightly loaded close-fitting bearing with highly compliant surfaces failed to demonstrate sustained elasto-
212
hydrodynamic less favorable
lubrication. Therefore study of more representative cases was not undertaken.
though
4.1. Results and interpretation The variations of the coordinates of the center of the femur with time for four walking cycles are given in Figs. 5 - 7. Owing to the periodic nature of the joint load and angular velocity vectors they too are periodic. After about four walking cycles the femur has almost settled into its equilibrium orbit, as witnessed by the almost horizontal nature of the maximum and minimum displacement loci. Over a major portion of the walking cycle both the x* and y* displacements exceed 10m3 cm, the maximum that could occur if the surfaces were rigid (they would touch at this value). However, as the load is reduced to zero during the latter half of a walking cycle the deformed cartilage layers expand against the fluid film and reverse the direction of motion of the femur, thereby decreasing the values of X* and y* . At the end of each cycle the X* and y* coordinates are less than 10M3 cm, as required if the surfaces are to be separated by a fluid film. The z* coordinate decreases with time since the squeeze film action of the x and y loads opposes that due to the z load, i.e. the x and y loads
2
0
z
I
Tim,t
L
3
,sBc
Fig. 5. The x* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cms2).
4
213
16
I
unumryul surface d+xement
0
I
2
3
4
Time,t,ssr.
Fig. 6. The y* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cmm2).
squeeze some fluid into the cavity while the z load only squeezes fluid from the cavity. The net result is a slight gain over the first several cycles such that the femur lifts out of the acetabulum from an initial z* displacement of 5.23 X 10m4 cm to an equilibrium value of about -1 X lo-’ cm. The limiting values for the x* , y* and z* coordinates at the end of a walking cycle are about 10 X lo-“, 6 X 10V4 and -1 X 10e4 cm, a situation that leads to minimum film thicknesses which are substantially less than 1 X lo-* cm, the nominal lower bound for hydrodynamic lubrication. This is illustrated in Fig. 8 where the variation of the film thickness with the polar angle 8 using the longitudinal angle 4 as a parameter at a time of 3.5 s is presented. In the region 60” G 8 < 90” the x and y loads tend to force the femur onto this acetabulum along the 54” longitudinal. The effect of the angular velocity vector is not large enough to help drag fluid into this region and maintain an adequate hydrodynamic film thickness. Even though the load in the r direction is greater than that in the y direction, the fact that wY is greater than o, offsets this and leads to a larger film profile at $I = 0. The inflections of the curves occur at 8 = 60” ; this is to be expected as the acetabulum cartilage layer starts to decrease with 0 at this point (see eqn. (21).
Fig. 7. The z* coordinate of the femur center as a function of time for case 1 (B = 10’ dyn, C = 0.001 cm, G = 1 X lo6 dyn cm-‘).
The minimum film thickness profile occurs at about 4~= 54” which roughly corresponds to the angle with the x axis of the resultant of the x and y loads. In other words the film profile is smallest at about that value of $I which is directly under the loading. This is also illustrated in Fig. 9 where the corresponding longitudinal variation of the film thickness is presented for 19= 45’. When 6 is about 54” the film thickness is again a minimum. The polar pressure profiles (with 4 as a parameter) in the joint at a time of 3.5 s are presented in Fig. 10. As expected the maximum pressure profile is that for 9 close to 54”. Since the x and y loads tend to force the femur onto the acetabulum along the 54’ longitudinal, the pressure also exhibits its maximum there. 4.2. Conclusions and discussion Within the range of validity of the postulated analytical model, elastohydrodynamic lubrication does not persist within the hip joint during normal walking, i.e. the joint load and the relative motion of the bearing surfaces are such that the joint surfaces are not separated by lubricant film of the order of 10e4 cm. Factors of cartilage surface irregularity [4], flow of synovial fluid and the pore liquid through the cartilage layers, selective filtration of the macro-
215
0
0
10
a
30
40
5u
60
m
\
60
8
PokrAngle,0,dy Fig. 8. The film thickness rofiles at maximum loading for case 1 at 3.5 s (B = 10’ dyn, C = 0.001 cm, G = 1 X 10 P dyn cm-‘).
molecules of synovial fluid at the cartilage surfaces [4] and the specific affinity between synovial fluid and cartilage [ 181 which are not of significance in elastohydrodynamic lubrication no doubt play an important part in the lubrication of synovial joints. These phenomena may be accommodated in a postulated model for lubrication which is essentially a combined boundary and internally induced hydrostatic lubrication model. Thus cartilage asperities are of the same magnitude as the calculated minimum film thickness; hence these surface peaks touch and interact with each other. However, since the cartilage layers are filled with liquid only a small portion of the load is actually carried by the interacting peaks. The majority of the load is carried by the liquid that pervades the cartilage matrix. The load will be transferred to the cartilage matrix in proportion to the amount of pore liquid expelled. Further, the compliance of the liquidfilled cartilage matrix prevents rigid peak shearing, i.e. opposing peaks slide across each other. Since there is a strong affinity between cartilage and synovial fluid, these interacting surfaces are probably coated with the macromolecules of synovial fluid which allows them to pass over each other without significant resistance or wear. It has been experimentally determined that articular cartilage in duo maintains its liquid content with time. The reasons for this are based on the unique properties of cartilage and synovial fluid and the transient behavior of the joint load and angular velocity vectors. Let us assume for the moment a constant load and fixed region of application. The elasticity and surface irregularities of cartilage tend to
216
16
1.4
Ll
,o-
OE
0.6-
40
84
/w
I
/#
I
MO
,
m
,
z&3
,
A?0
I
240
0
longitudinal Angle, z deg.
Fig. 9. A longitudinal film thickness profile at maximum loading for case 1 at 3.5 s (B = 107 dyn, C = 0.001 cm, G = 1 X lo6 dyn cmh2). Fig. 10. The pressure profiles at maximum loading for case 1 at 3.5 s (B = 10’ dyn, C = 0.001 cm, G = 1 X 106 dyn cme2).
confine the synovial fluid within virtually non-connecting valleys so that no clear uninterrupted flow channel exists between the region of loading and the joint extremities. Hence the synovial fluid is forced to flow through the articular cartilage layers and the peaks if it is to be expelled from the valleys between the cartilage surfaces. If the synovial fluid is to flow through cartilage, it must displace an equivalent amount of pore liquid. However, since the permeability of cartilage is extremely small, the escape of liquid from the cartilage is slow. As the synovial fluid tries to flow through the cartilage, its macromolecules tend to filter out on the surface of the finely pored cartilage matrix and form a “gel” which increases the resistance to its flow to the
217
extremities of the joint. Hence we have a situation wherein the synovial fluid is essentially “contained” in the valleys between the cartilage asperities. It is this “contained” synovial fluid which probably transmits the majority of the load through the pore liquid of the cartilage to the underlying bone structure. The remaining fraction of the load is carried by the cartilage matrix and results in simultaneous expulsion of some of its pore liquid. Even though the resistance to flow of the “contained” synovial fluid and the pore liquid is quite large, after a sufficient period of time they would escape to the joint extremities if the direction, magnitude and point of application of the load remained constant. Fortunately, this is not the case and the variation in the direction, magnitude and point of application of the load are such that the synovial fluid and the pore liquid, rather than flow inexorably out of the joint, shuttle around in the region of loading. Hence, though at any one instant the load may tend to drive synovial fluid and pore liquid out of the loading region (at an extremely slow rate), later in the walking cycle the change in the direction of the load vector and the relative motion of the cartilage surfaces are such that they are “pumped” back into the region. The cartilage deformation profile varies and undulates throughout the walking cycle and also helps to maintain the internal “pumping” within the region of loading.
References 1 M. A. MacConaill, The function of intra-articular fibrocartilages, with special reference to the knee and inferior radio-ulnar joints, J. Anat., 66 (1932) 210 - 227. 2 V. C. Mow, The role of lubrication in biomechanical joints, J. Lubr. Technol., 91 (1969) 320 - 328. 3 E. L. Radin and I. L. Paul, A consolidated concept of joint lubrication, J. Bone Jt Surg., 54A (3) (April 1972) 607 - 616. 4 P. A. Torzilli and V. C. Mow, On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function - II. The analysis, solution and conclusions, J. Biomech., 9 (1976) 587 - 606. 5 P. Marnell, A theoretical evaluation of the persistence of hydrodynamic lubrication in the hip joint during walking, Doctoral Thesis, Columbia University, 1973. 6 P. S. Walker, D. Dowson, M. D. Longfield and V. Wright, Boosted lubrication in synovial joints by fluid entrapment and enrichment, Ann. Rheum. Dis., 2 7 (6) (Nov. 1968) 513 - 520. 7 N. W. Rydell, Forces Acting on the Femoral Head Prosthesis, Univ. Gotteborg, Sweden, 1966. 8 R. K. White and H. G. Elrod, Program for Correlation of Hip Contour Measurements, Columbia Univ., New York, 1968. 9 C. H. Barnett, D. V. Davies and M. A. MacConaill, Synouial Joints, Charles C. Thomas, Springfield, Ill., 1961. 10 H. D. Eberhart, V. T. Inman and B. Bresler, in P. E. Klopsig (ed.), Human Limbs and Their Substitutes, McGraw Hill, New York, 1954. 11 B. Block and L. Dittenfass, Rheological study of human synovial fluid, Aust. N.Z. J. Surg., 33 (1963) 108 - 113. 12 R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960.
218 13 D. V. Davies, Properties of synovial fluid, Proc. Inst. Mech. Eng. London, 181 (35) (1966-67) 25 - 29. 14 R. R. Meyers, S. Negami and R. K. White, Dynamic mechanical properties of synovial fluid, Biorheology, 3 (1966) 197 - 209. 15 M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (2) (1955) 182 - 185. 16 P. Moon and D. E. Spencer, Field Theory For Engineers, Van Nostrand, New York, 1961. 17 A. G. Ogston and J. E. Stanier, The dimensions of the particle of hyaluronic acid complex in synovial fluid, Biochem. J., 49 (1951) 585. 18 C. W. McCutchen, The frictional properties of animal joints, Wear, 5 (1962) 1 - 17.