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The role of synovial fluid filtration by cartilage in lubrication of synovial joints—II. Squeeze-film lubrication: Homogeneous filtration
J R,omuchunrcr Prmted
I” Grc;,t
Vol 26, No. 10. pp
II51
0021 9290;93 $6.00+000 e: 1993 Pergamon Press Ltd
1160. 1993.
Bnfam
THE ROLE OF SYNOVIAL FLUID FILTRATION BY CARTILAGE IN LUBRICATION OF SYNOVIAL JOINTS--II. SQUEEZE-FILM LUBRICATION: HOMOGENEOUS FILTRATION M. Institute
of Theoretical
HLAVQEK
and Applied Mechanics, Czech Academy 12849 Prague 2, Czech Republic
of Sciences, 49 VySehradski,
AbstractA mathematical model of synovial film filtration and synovial gel formation at normal approach of cartilage surfaces in the human hip joint is presented. The biphasic mixture model presented in Part I of this paper [Hlavii-ek, J. Biomechanics 26, (1993)] for synovial fluid and that of Mow and his collaborators [J. Biomech. Enyng 102, 73-84 (1980)] for cartilage are used. A general analysis of filtration of an axially symmetric synovial squeeze-film between two cartilage layers at normal approach is given. The geometrically simple case much idealising the human hip joint is also considered: two cartilage discs with a synovial film in between are compressed by the steady loading of the half human weight. The homogeneous film filtration process where the synovial gap remains parallel and the macromolecular concentration in the gap is spatial homogeneous (to the thin-film approximation) and time-dependent is numerically analysed. If the hyaluronic acid concentration of the synovial gel at equilibrium is 50 mgiml at least, the resulting stable gel layer thickness for the homogeneous filtration in the human hip joint and for normal synovial fluids is about 0.1 pm. being almost independent of the loading. Inflammatory synovial fluid shows values several times lower.
phase
INTRODUCTION
Newtonian
fluid and
biphasic
model
(Mow
et al. 1980) for the SF and cartilage were used. respecSynovial fluid (SF) forms an interface with the articular cartilage in synovial joints and plays an important role in their lubrication. SF contains a small concentration c of hyaluronic acid (HA)-protein complex that forms a continuous network. According to Maroudas (1967, 1969) and Walker et al. (1968), the concentration of the complex in the gap between the cartilages increases during the squeeze-film action due to the penetration of low molecular weight substances from SF into the cartilage. Increasing the SF viscosity in the gap delays the approach of the cartilages, and the formation of gels (Maroudas, 1969) protects their surfaces, if sliding motion ensues. Assuming a linear variation of SF viscosity with c, an inverse proportionality of the film thickness to c and no flow into the cartilage, Dowson et al. (1970) calculated the squeeze-film thickness between rigid impermeable surfaces as a function of time. An initial film thickness of 10 pm was assumed and the time for this to fall to 1 pm was calculated. However, Maroudas (1967. 1969) predicted the final thickness of the gel to be 0.02-0.03 pm and stated that the flow from the gap into the cartilage should be more significant than the flow sideways in the gap, when the film thickness is less than about 0.5 pm. Hou et al. (1988, 1990, 1992) presented a squeeze-film lubrication model of a rigid impermeable spherical indenter approaching a thin permeable cartilage layer supported by a rigid impermeable subchondral bone. A single-
Received in final form 3 March 1993.
tively. In the model the viscosity of Newtonian SF does not change during the squeeze-film action. Lai and Mow (1978) analysed one-dimensional filtration of SF between two parallel semipermeable rigid boundaries approaching with a constant speed and used the classical diffusion equation. In the present paper the SF is a mixture of two incompressible fluids--see Part I of the paper (HlaviEek, 1993). The viscous phase is formed by the HA-protein complex, the ideal one by all low molecular weight substances. As to the cartilage, the biphasic linear model of Mow et al. (1980) is used, i.e. cartilage is modelled as a diffusing mixture of an elastic porous solid matrix and an ideal fluid. The ideal-fluid phase only can flow across the SF-cartilage interface as the macromolecules of the HA-protein complex are too large to penetrate into cartilage pores. In Part II of this paper the boundary conditions at the synovial-film-cartilage interface are ob.tained using the balances of mass, momentum and energy and following Hou et al. (1989). Moreover, the viscous phase of.the SF is assumed to adhere to the cartilage surface. The adherence of the HA-protein complex to the cartilage surface has been experimentally proved by the method of microelectrophoresis (Maroudas, 1969). In the filtration model by Hou et al. (1988, 1990, 1992) a single-phase viscous SF passes this interface as a whole and the ‘no-slip’ condition for the SF would lead to incorrect boundary value problems. Hou et al. had to formulate a special kinematic condition (‘pseudo-no-slip’) in order to obtain a correct formulation for their model.
1151
1152
M. HLAVA~EK
A general analysis of filtration of an axially symmetric synovial squeeze-film by two cartilage layers is given in the present part of the paper. A geometrically simple case much idealising the human hip joint is numerically analysed: two cartilage discs with a synovial film in between are compressed by the half weight of the human body. For this geometry there exists such a squeeze-film process that the synovial gap remains parallel and the macromolecular concentration c is homogeneous in the gap (to the thin-film approximation) and increasing with time. Some preliminary results for this case have been reported earlier (Hlavacek, 1990). MIXTURE MODEL OF ARTICULAR
(1)
4::’ + (@“’P!“‘),i = 0 incompressible
(2)
phases, and
the constitutive equations $?‘= _@i)fi&. ,,, f&=
I
_j$”
1
*!3’= - @Q)j&.
=rp$qGjlqj5)‘)).
‘I
+
1,2 yields (5)
Equation (4),, i.e. the first of equations (4), yields the volume flux q!r) of the ideal fluid relative to the solid phase in the fbrm of the Darcy law, q!l)=~cl)(~~l)_~!2))= I I I with the cartilage permeability
_&
sly
(7)
After excluding irj1)-3j2) from equations (4), and (5), we obtain with the use of equation (7)
For small cartilage deformation. considered the linear relation
(8) Mow et al. (1980)
d!f”=XEi:2:bij+2~1;:i’.:) II
(9)
where I?/‘) is the matrix displacement vector and 1, fi are the elastic Lame material constants of the porous matrix. The model in this linear formulation corresponds to Biot’s model of consolidation of soils (Biot, 1941). BOUNDARY
CONDITIONS
AT THE INTERFACES
We refer to Hou et al. (1989), where the boundary conditions at the interface of two biphasic mixtures (one phase being fluid, the other solid), with the displacement (and therefore the velocity) of the solid phase continuous at the interface, are given. Using the same procedure for the interface s.,, between the cartilage (taken as a mixture of an ideal fluid and the elastic porous solid matrix) and the SF (taken as a mixture of an ideal and a viscous fluid according to Part I of this paper), we obtain the following boundary conditions on s* written in the Cartesian coordinates Xi: v~2)=~~2),
#f’e
(6)
k^defined by
{+p/~~
(k^fi,i),i=?F\.
the momentum equations d!“’ .+$!“‘=0 u., I ’ the continuity equations
a =
[~(“(tl”--912’)+912’],i=0.
CARTILAGE
Articular cartilage is a tissue composed essentially of a fibre-reinforced solid matrix phase (collagen fibres and proteoglycan macromolecules) and an interstitial fluid phase (water). The compressive viscoelastic behaviour of cartilage is governed by the flow of water within the tissue, its exudation and imbibition across the articular surfaces and by the (visco-)elastic resistance of the cartilage matrix. Although the porosity of the tissue is large (about 80%) the permeability is relatively low due to the small pore size. A mixture approach was first used on cartilage by Mow et al. (1980). The basic equations of cartilage considered as a two-phase mixture (the idea1 fluid and the elastic porous solid) can be written in the following form (Holmes, 1986):
for intrinsically
while summing equations (2) for
p(‘)(v11)-v12))ni=P(l)(E11)-o12))ni,
IJ’
(3)
Here and in what follows, a comma followed by a subscript stands for the respective partial derivative and the summation is made over pairs of identical Latin subscripts in the range l-3. The superscripts (1) and (2) are used to indicate that the quantity is associated with the fluid and the solid phase, respectively. The ‘roofed quantities refer to the cartilage. i$), ey), tiy) and @(a)are the partial stress tensor, the velocity, the diffusive drag force and the volume fraction of the (a)-phase, respectively. 0 is the fluid pressure, 5:;)’ the stress component of the solid phase that is due to matrix deformation and k stands for the diffusive drag coefficient. As the mixture is saturated, G(r) + +(‘) = 1. Inserting equations (3) into equation (1) and summing the result gives
[a~~‘+~$)
-p”‘v~“(v~“-v~‘)]nj -“)91’)(0j’)-Oj2))]nj,
=[8,!i”+csf’-p [u!~)v~‘)+u~~)v~)-t~(l)v~‘)v~)(v!’)-v/~))]n~ =[a;;)c;l)+aij
-(2)-(z) vi -2p1
*‘l’F:l’3:l’(B!l’_812’)]ni. (10)
Here, analogous to equation (I, 1) of Part I of this paper, fi(m)=$a)@(u), where PC’)are the partial mass densities and $l)=p”), st2) are constant true mass densities of incompressible phases in the cartilage. ni is the unit normal to s* and p(“), vf), c$) are the partial mass densities, the velocities and the partial stresses of SF, respectively, referred for a = 1 to the ideal and a=2 to the viscous fluid phase. See equations (I, 1) and (I, 5) for the definitions of p(O)and ai(g).In equations (10) the zero heat flux is assumed and the internal energy density term is absent, as the fluid phase is
Synovial fluid filtration-11
ideal (inviscid). Equations ( 10)2, (lo), and (lo), follow from the balance of mass, momentum and energy, respectively (Hou et al., 1989). Equation (lo), is a kinematic condition of continuity of the viscous- and solid-phase velocities. This condition follows from the fact that the HA-protein complex present in the SF becomes adsorbed onto the cartilage surface, as proved by the method of microelectrophoresis by Maroudas (1969). The boundary conditions on the interface s, between the biphasic cartilage and a non-porous rigid subchondral bone are of the form (Hou et al., 1989) +o,
$(l’($ ‘“-P)ni=O, ‘
(B!f’+d;f’)nj=li, (11)
where ni is the unit normal to s+ and fi is the traction vector applied on the cartilage by the rigid bone. NORMAL APPROACH
OF CARTILAGE
LAYERS
Let us consider a thin axially symmetric synovial film between two cartilage layers of a constant thickness h^(Fig. 1). The SF-cartilage and cartilage-subchondral-bone interfaces are denoted by s*, s* and s+,s +, respectively. In what follows, an asterisk and a cross up or down indicate quantities on s* or s.,,and S+ or s+. respectively. Two systems of cylindrical coordinates, I, 0, z and r, f&i, in the synovial film and in the lower cartilage layer, respectively, are introduced (Figs 1 and 3). The radii of curvature of s*, s* are large compared to the film length, that is large compared to h^and this is large compared to the synovial film thickness h(r, t). Then, without any loss of generality. s* and s* can be assumed to be symmetrical with respect to the plane z=O, the symmetry axis being r=O. The cartilage layers are compressed in the zdirection by the force y’(t) = p(t) H(t),
(12)
where H(t) denotes the Heaviside unit step function and P(t) is a given smooth function of time t. For the axial syrrimetry, equations (4)2 and (8) with the use of equation (9) yield the following equations in the r 10, z-coordinates for ti(‘) r ’ CF’and p in the lower cartilage layer ( - h^
1153
j,?= tiCjs+(X +$)(rfip)),,/r
Let us introduce the non-dimensional ables or constants as follows: r=r’f,
(primed) variz=z’i,
2=Ph,
t=t’t,v=iJf,s=2/*,~=Rjr,riP’=1;:!*’K,aJ2’=t21;(f)h, ;(*,=yj?!ff/t;
pq,2),,
i/t;
b ’ ” v, =v,, ‘(“v , v~‘=y~?‘f~,
$w=;‘;.“q
b=o’*,
p’=$‘;‘f’q&
p=p’d.
f,f,t
In the case of the human hip joint analysed below, the orders of quantities used and given in the SI units are: r- 10-z, z- 10-7-10-5, t- 101, h^- 10-3, ni- 106, k^-lo-‘6,+, ;(C. 103, &“‘, @=)wlO”, tl- 10-2-10’ for qC2)- lo-*, r]- 102-10’ for qC2)- 10-l. Here q(l) and qC2)= cpare the volume fractions of the ideal and the viscous phase of the SF, respectively. The order of SF viscosity q follows from equations (I, 17), and (I, 31). We also have V-1O-3, 6- 10-4-10-3, C?- 10-l. If 5,‘~ I/fi and the primed quantities are of the order 1, then writing equation (14) with the use of the primed quantities and keeping the terms of order IO6 only, i.e. dropping the lower-order terms, equation (14) with the help of equation (9) takes on the simple form --p= -6 + 8rie. Similarly, equation (lo), written for the r-direction yields @‘=O. We can easily find that also in equation (lo), the terms containing v,!~)-v!~) and C!1)-t!2) are small and may be dropped. Then: taking ;he scalar product between equation (IO), and vi2)=iri2), subtracting the result from (lo), and using equations (1.Q (I, 13a), (3), we obtain p=fi for cp+l. Thus, for thin synovial films and cartilage layers and physiological loading of human hip joints the boundary conditions [equations (10) and (I l)] become I’
@‘=O,
v(2) = 9’2, 7s
#a)e=O
I
’
4,
u)=(iw
*
7
p=B,
on s&=0)
(15)
and ti@)=O, 1.2?)=0, p,;=O on s+ (2= -h^). I
Fig. I. Squeeze-film geometry.
are
characteristic values of the variables r, z, t (for example, their average values in the film and time span). As the cartilage is attached to the rigid bone, both tir’ and fir) are of the order of h^ and this explains the definition of 9;!2),t2’::‘. This idea is due to Armstrong (1986), who used this perturbation analysis for a thin single-phase elastic layer with a rigidly fixed surface. p is the total pressure in the SF [see equations (I, 7) and (I, 13a)]. Let us insert equations (I, l), (I, 5). (3) and (9) into equations (10) and use the r, 8, z (or i)-coordinates with n, = ni = 1, n, = ne = 0. For illustration we give here the result for equation (lo), written on s* in the z-direction only:
,
bone
(13)
~(~B,r),,lr+~~,.;i=(r~0!2)),,/r+3~~~,
v(2) = 9’2’
subchondral!
+fl(rt$f,‘).,,h-,
(16)
Approximating s + , s+ quadratically in r and denoting
1154
M. HLAVACEK
by hb(r, t) the vertical distance of their points, we have hb(r, t)=h,(O, t)+r*/%
where W is the radius of curvature of s+ , s+ at r =O. Figure 1 shows that h(r, t)=hb(O, t)+r*/C%--2(h^+iii:)). Using equations load
(17)
(3) and (15), we have for the total
9’(t) = -2?&:‘6
+ &i;))r dr = 2xTpr dr.
0
0
(18)
First, let us consider the case of the SF with a Newtonian viscous phase. Setting n = 1 in equations (I, 35 and 36) and using equations (I, 8), , (I, 13a)l, (6) and (15),, as v_(*)= -h 12, we get the equations of a squeezed synovial’tfilm in the form
ditions on s.,, [equations (15),-,] and on s+ [equations (16)], the ambient conditions for r-co [equations (22)] and the initial conditions [equations (20) and (21)]. Fluid motion in the cartilage is governed by the permeability k^and, therefore, by the left-hand side of equation (13),: its first and second terms correspond to the flow in the radial and vertical directions, respectively. For k^=O (no fluid diffusion) equation (13),, together with the initial conditions given by equations (20), yields the incompressibility condition for the porous matrix for any t. In this case 1 can be excluded from equation (13),, 2 using equation (13),, which yields B., =fi [(r$*‘). Jr1 ., +fitijfjpp
B.0= ji(rt;~$/r These are the momentum
+ fiti:fbp.
equations
(23) for an incom-
(~3r~,,ll2tt~).llr+~Cr(vl~‘-~p,,)l,,lr=~.,-2q~~), pressible single-phase solid with shear modulus F, ((PHAh3rp.,/12~N),r/r=((PHAh).l+h[r(PHAv~~’l,r/r, ~N((PHA)=~((PHA/(PHAO)~I
1.62 x 10-l’ r~;;d.~~.
I&&=
(19)
Here vHA is the HA volume concentration. The Newtonian viscosity tfN and the permeability k of the synovial film were introduced in Part I, equations (I, 17)* and (I, 25). If P(O)=O,then Y(t) is continuous at t=O [see Fig. 2(a)] and equation (12) yields the quiescent initial conditions fi=p=o,
i+o,
lif’=o
@GO).
(20)
Further, we set %A@, ~HAO.
hbo
boundary r-00 as
O)=%IA~~
MO,
O)=hbo,
(21)
with a scalar parameter 6, commonly called the hydrostatic pressure. The boundary conditions given by equations (15),_ 6 and (16) must be replaced for this material by -p=
-j+2fi#,
fi(*)=O, r
$i*)=O
$*~+ti~*~=O . . at f= -6.
at z*=O, (24)
Let us note that equations (24),, 2 guarantee the continuity of the stress vector, but not that of the pressure on se, while equations (24)~~~do not guarantee zero pressure gradient on s+ . We come back to the general case k^#O (with diffusion). To get equations for j’, fi’j?, ii’$‘, we use the primed quantities in equations (13), (15),_ 6, (16) and (20). For example, equation (13), takes the form a[~2(r’8’,,.),r./r’+8’,4,P.]=~(rts’jf)),,.t,/r’+;‘:tb,.
being given constants. Also, the ambient conditions for p and (P,.,~are assumed at
, (25)
where the parameter E is defined by P(m,
~)=O,
(PHA(a,
~)=(PHAO.
(22)
Thus, if P(t), (PHAO and hbOare given then, for seven unknown functions p(r, t), (PHA(r,t), h(r, t), MO, t), fi(r, f, t), lij2)(r, 2, t) and ir:*)(r, 2, t) there are seven equations, equations (19),,,, (17), (18) and (13),_,, together with the boundary conditions on s* [equations (15),, 2,4-6] and on s+ [equations (16)], equations (22) and the initial conditions [equations (20) and (21)]. PERTURBATION ANALYSIS OF THE CARTILAGE LAYER
Approximate solutions in the cartilage will be found due to the fact that the cartilage forms a thin layer and that substantial fluid motion in the cartilage occurs only in a thin boundary layer at the articular surface for physiological time scales. For the time being let us assume that p(r, t) is known. I?*) fir) and fi in the lower cartilage layer (Fig. 1) rn;s; satisfy equations (13), the boundary con-
2 = k^tii/lh^*.
(26)
In the perturbation analysis it is assumed that Ii’!,*), I?:?, j’ can be represented by the asymptotic series in terms of the small parameter 8=&h lo-‘, a’;,*‘= lY;,; + C&l’,‘,+ 6^2A’(2)+. 0 I, * . ., etc.
(27)
Then, on substituting these series into equations (13), (15),_,, (16) and (20) written in the primed quactities and collecting terms of the same power of 6, the original problem is replaced by a series of simpler problems for the coefficients of the asymptotic series. For E- 1 the terms in the series given by equations (27) descend with increasing n and the series can converge. In this case the diffusion terms in equation (25) apply throughout the yhole ca$lage thickness. Inserting 1-1, k- 10-16, M- 106, h- 10m3 into equation (26), we find that this is the case as late as about lo4 s after the beginning of load application. For the time scale I- 10’ s equation (26) yields L- lo-‘. It can be found, however, that for 0 & 1 the terms in the series given by
Synovial
fluid filtration-II
equations (27) increase with n and the series diverge. Thus, the expansion given by equations (27) cannot be used throughout the whole cartilage thickness under physiological conditions, i.e. before 10’s since the load initiation. As 8e82 in our case, equation (25) shows that the solution up to order 8’ corresponds to the l=O case (without diffusion) if fi’,;.3 - 1. However, at places where $‘,pC2 + 1, the diffusion term 2b’.1.4. in equation (25) can matter even if 16 1. This is the case, for example, in a thin boundary layer of s*. Across this boundary layer the pressure jump at the interface obtained for the case l=O must be smoothed out by diffusion. The thickness &, of this boundary layer where diffusion in the f-direction applies is obtained from equation (26) with O- 1 in the form h,, -(E tit)?
(28)
We see that h^,, increases with time as t”‘. For example, for t = 10 s we get fi,, - 10m4- lo-’ m. There exists another boundary layer in the cartilage at s+ due to the fact that $‘,@#O at s+ for the solution without diffusion, but this boundary layer is of no importance near s*. As the system of equations (13), (15),_,, (16) and (20) is linear, its solution is taken as a superposition of two special solutions, ;‘“=;l;f’+;;;;,
;(:“=;$‘+$;J,
r
(29)
6=61+h,.
The first solution (denoted by subscript I) is that without diffusion (R =O, single-phase incompressible material [see equations (23)] with the boundary conditions given by equations (24). The second solution (subscript II) is that with diffusion (f #O) and is different from zero in the boundary layer of s* of the thickness A,, only. The boundary conditions on s., for the latter are chosen such that a superposition of both the solutions satisfies the original boundary conditions [equations (15),_,]. Using the non-dimensional r’, i’, li$‘, 1;;‘:’ defined above and 6;‘. pi’ defined by fi;‘=fi&i, p;‘=p/fi in equations (13), with k=O, (23) and (24), substituting the asymptotic series for ti$), 1;$’ and 0; in a form similar to equation (27) and collecting terms of the same power of 8, we found ti;i$ l;$L and & for n=O, 1, 2, which mation
resulted
in the 8’ order
1155
Now, equations (15), (29) and (31) yield the boundary conditions of the II-solution to order 8’ in the form @,I= -i’
v,
I(rli~fj),,/r+Iiiiil(fj It
at z*=O.
(32)
from equation (28) that &,4h if lo4 s. For such t we can consider the IIsolution to be defined in the half-plane 2~0, rather than in the strip -k < i < 0, and we put $2, uni, A(2) fir,-+0 Ilr ’ Let us introduce
for i-
- x
a small parameter
&&/r
(33)
sd by
b,=(Gt)l’2
and write equations (13), (32), (33) and (20) using the non-dimensional r’, t’, p’ defined above and ,Y, A!!(Z) G:i)=ti$)$, %I,, ’ G;$f), _ & defined by f=Y&, A(2’=$;:?hdr fi,,=&A?. Let us assume that ti;;$), %I2 G;$‘, and &, can be represented by asymptotic series in terms of 8,. G;;y)= t$fi
+ a,ti$~
, etc.
+
and insert these series into equations (13), (32), (33) and (20) written in non-dimensional forms. The order c?: system of equations yields G;;i@i=0 and the following problem for ;;;E,b: fi”‘2’
112”0,2’,2’,
=
;+2, 112”O. f’
for 1”<0.
-g2v
at
-tt(2) = U IIP”O.2,’
f”=O,
ti;;:t,), -+O
at i”+
I’;;f)o=O
at t’=O+,
-
(34)
CXI ,
and for &,: d;,O,i.l..=&,O,l. A, -pv’ PIlO’
for f”
j;,o+O
at 2”-+ --co,
1,,,o=O
at t’=O+,
P
(35)
where V’=(r’p’,,,),,,/r’ = VF2/fi. The order 8: solutjon to equations (34) and (35) takes for t$h’/i& the form [see Sneddon (1972), equations (2-16-35) and (2- 16-33)]
1)/2/I, ti$(r,&
1;;:‘=&” V(2+3i/h-i3/f3)/6fl,
2= -h2V
follows
approxi-
j?,=p+PV(2-i2/P)/2,
,=o,
t +h2/i%-
d,,(r, 2, 4 =
lP’=Pp,,(P/PI,
t?;;;,*+ti;;;
ii22 q&G)‘/2 ^
t)= -h2
-$ (>.I
s
’ W, 7)
Ed7
’
o (t-zp
l/2
f
W,
___
4
o(‘-T)112Ed~~
(30)
V=(r~.~),.lr.
(36)
To order A2 it follows that JJ=p+P
v,
I(rtijf’).,/r+ tiif’=O.
C;2:+IP I?., =o, I, ~Cii~~‘,=~~ V
“$=O,
fi,,*=hV
at f=O, at f= -it.
(31)
Now, within the approximation used, GE in equation (17) can be expressed with the aid of equations (29)2, A (30), and (36),, while P,~* and vi:) in equations (19) can be obtained through equations (29),, (30), and
M. HLAVACEK
1156
(36), and through equations (29),, (30), and (36),, respectively. At last, h(r, t) may he excluded from equations (19) by substituting equation (17) into equations (19). Thus, the problem for p, (PHA, h, hi,, fi, 1;:’ and ;P’ defined above is simplified to that for p(r, t), (PHA(r,t) and &(O, t) that satisfy equations (19)i, 2 and (18), together with the ambient boundary conditions [equations (22)] and the initial conditions [equations (20) and (21)], if 9(‘(t), hbO and &.,A0 are known. This general problem will be analysed later numerically. In what follows, only the case of homogeneous filtration will be solved. SQUEEZE-FILM
OF SYNOVIAL
GEL
So far, we analysed a fluid-like film (qN< a). As mentioned in Part I of this paper, with increasing cnA the macromolecular network becomes more permanent and the SF changes gradually into the synovial gel (SG) with a solid-like matrix. Let us denote the quantities referring to the SG by a circle above them. Similarly to cartilage, this gel can be modelled as a biphasic mixture of an ideal fluid and a solid porous matrix and described by equations (l)-(8) with the ‘roofs’ replaced by circles. The boundary conditions on s*, the SG-cartilage interface, have also the form of equations (lo), the terms on their left-hand sides being provided with circles. Again, the terms with the velocity differences can be dropped for thin films and cartilage layers and the same method used above for the SF films gives for the axial symmetry and n, = 1, ng=nr=O the boundary conditions on s* in the form i(Z)= 9’2’ +(2’+2’, ;“‘=Q~‘, b=fi, &;‘e+‘e, 6i 2’e --%‘. Here &!?’ ii the stress tensor component orthe s%d phase &re to matrix deformation [compare with d$” in equation (3)2 for cartilage] referred to in Part I as the swelling stress. According to the equilibrium filtration experiments by Maroudas (1969) in the human hip joint loaded by the body weight the normal component &St)’ of the swelling stress is at least one order lower than the total normal stress in the SG &i’,’+&i’,’ = -i + &y=‘:“, i.e. ]?r~Z’e) Q b, up to cnA equal about 50 mg/ml for the normal SF. Only above this value GE” rises quickly with increasing cuA at a constant total normal stress. Assuming that the shear stress 6:” is of the same or lower order than iit’e, the above boundary conditions have the form of equations (15), with i written instead of p. We see that the boundary conditions on s, are the same for both the SF and SG. Following the deduction of equations (I, 35,36) for the SF film, we find for a thin SG film successively $,Z=O, &jf,‘i=O, i{F,‘J’Z=O(even for a non-linear constitutive law of the gel matrix), $‘f’F (l,J'Z=O. AS ]“~~~]~);~“~I, if “~‘,“‘=it~~ on s*,s*, it holds that ?2’e=F(2’ ..A m a thin gel film. This last equation is the counterpart of equation (I, 33) and is also obtained from there for qn-+co. Thus, in a thinfilm approximation it follows that equations (19) describe both the SF (with a finite t,Q and the SG film (for qn-+co). We conclude that in the case of normal
SF for cuA up to about 50 mg/ml the equations of the synovial squeeze-film are the same for both the SF and the SG. Only above this value up to the maximum value of CHA (according to Part I, about 56 mg/ml) the swelling stress may be comparable to the pressure and the boundary conditions given by equations (15),_, are no more valid. HOMOGENEOUS
FILTRATION
Rather than solving numerically the general axially symmetric problem defined above (a system of integro-partial differential equations), in what follows we shall deal with the following simpler case much idealising the human hip joint. Two discs (cartilages) of thickness h and fixed radius R (hg R) are compressed (Fig. 2) by the force s(t) given by equation (12), with P(t)=P”[t-(t-t,)H(t-t,)]/t,,
(37)
where fl and t, > 0 are constant [Fig. 2(a)]. Between the porous elastic discs saturated with the ideal fluid there is a synovial film of constant thickness he and constant HA-concentration @raO at t =0 (ho
(PHA.r=O.
As (PHA ,z =0 for a thin film, it follows then that (mu*is a function of t only. For homogeneous filtration [equation (38)], if it exists, qua, uI A(2’, u2 A(2)and fi cannot be given at r= R in advance and we set only p=O
Besides equation assume
atr=R.
(39)
(38), for the time being, we also h,,=O.
(40)
First, let us consider the case of Newtonian SF and deformable cartilage. We shall show that for 9(t) given by equation (12) there exist h(t), qHA(t) and
Fig. 2. Homogeneous
squeeze-film
model.
Synovial fluid filtration--~11 p(r, t) for 0 0 that meet equations (19) and (18)2 with the upper integral bound R and equation (39). With ths use of equations (38), (40), (29)i and (3O)i ,4 we find that equation (19)2 takes the form ~1~(t)~:~+a~(t)1,‘+u,(t)=O, which yields V,V=O. Then. as p,, = 0 at r = 0, equations (39) and ( 18)2 yield for the loading given by equation (12) p(r, r)=28(t)(l
-r2/R2)/nR2.
(41)
Equations (30) and (36) were deduced for cartilage layer loaded by a general loading function p(r, t) as asymptotic approximations to orders 8’ and @, respectively. However, for p(r, t) in the form of equation (41) it holds that V,,=O. A closer examination reveals that for such p(r, t) we get t;;~f,‘=ti$~=$,=O for n > 3 and t;,, = li;;‘$ = 0. &,. = li;‘r’,t/= G$?j” =0 for n> 1. Thus, equations (36) and (30) are exact for the pressure distribution given by equation (41). As V,,=O. using equations (29),, (30), and (36)r. we have j,?,=O, while equations (29),, (3O)a and (36), yield 6:‘: = 0 in the cartilage. It follows that 4::’ and 1;::) are functions oft only. Therefore, for the Newtonian SF there exists a homogeneous filtration process defined by equations (38) and (39) during which the synovial gap remains parallel [i.e. equation (40) is valid], with a homogeneous (i.e. independent of r) flow into the cartilage. Inserting equations (12), (37) and (41) into equations (36), after integration we obtain with the use of equations ( 10j2. (6)2. (29), and (30)2,4 for the loading given by equation (37) q;:‘(t)=
- 16F@ [t’~2-(t-t,)“2
H(t-t,)]
X ,$1’2~R4rCX3:2,$/2 for teh
-,?-^ /kM (hdtcch).
(42)
In fact, it is in this case only that the II-solution in the cartilage layer can be replaced by that in the halfplane. Inserting equation (42) into equations (19) and using equations (38) (40) and (36), gives a system of ordinary differential equations for h(t), q”*(t). Here we give the result for step loading [Fig. 2(b)] Y(t)=
&Y(f),
(43)
that is obtained from the loading given by equation (37) for t,+O. Equation (42) takes the form q::‘(t)=
-81%2(&&t)1/2/zR4.
1157
during this rapid transient response (R = 13 mm. E=8.8 MPa, ~=0.02 Pas, P’=445 N). A pocket-type film configuration was formed that flattened rapidly in time and a parallel film was rebuilt within some milliseconds after the load initiation. As shown above. the parallel fluid film corresponds to film pressure profile given by equation (41). Thus, leaving aside the rapid transient response due to the SF viscosity. as the initial conditions at t=O+ we take the film pressure profile from equations (41) and (43) and the instantaneous cartilage deformation and pressure given by the I-solution in equations (30). It follows from equations (30),, (36), and (43) that tij:‘=liifi does not depend on time, so that C!:)=O for t>O and small deformations. In this case the equations for h(t). q”*(t) are obtained from equations (19) in the form h,,= -8&/~~/12r],~+N)/rrR~. y+,~,,=8&mNInR4h, h=h,.
cpHA=~HAO at t=O
(45)
with
N(h, (P”A, t)=2K2(l;/n~t)“‘+k((PHA)h.
(46)
Equations (45) can also be obtained in another way in this special case instead of by superposing the Iand II-solutions. In fact, we can easily find for V,, = 0 that inserting equations (27) into equations (13) gives q”=;;~f’= &, = 0 for n > 2, while only &, lij$‘, 6; and Gp2) are non-zero. The equation for 6; can b.e solved through the Laplace transform, which leads for t’& 1 to 4::) given by equation (44) and for t-, x to $:‘=
-8PIKilnR4.
(47)
Equation (30), gives the instantaneous P-J&* on s* at t=O+: p-&*=2(p-j,+)=
pressure
- VP;
drop 148)
where (41) and (43) yield equations (30),, V= -~&CR’. In the case of a rigid porous cartilage we have for @(r, i) in the lower disc for t>O, using equations (8) (16),, (15), and C:fi=O,
(44)
The cartilage alone (as a mixture of an elastic matrix and an ideal fluid) would respond at t=O+ to step loading [equation (43)] as an elastic body [see Armstrong (1986)], and the viscous SF alone would respond as a rigid body. This would break continuity conditions [equations (15), .2] on s.+ at t =O+. Thus, if conditions given by equations (15),,, are to be also satisfied at t=O’, the viscous SF must retard the instantaneous cartilage deformation. The time scale of this transient response t,=rlN/&?m 10-a s in our case. Rybicki et nl. (1979) analysed numerically the film profile of a Newtonian fluid between two elastic cartilage discs compressed by step loading [equation (43)]
(rd.,).,lr+~.~n=O. $=p(r) fi,?=O The solution for loading (43) is of the form
at i=O, at i= -&. given by equations
(41) and
j(r, .?)=p(r)+4%(2+Z/~)/rrR” and the rigid porous cartilage tions (44) and (46), are
counterparts
of equa-
q::‘= -8i%&R4, N(h, q&=2ifl+k(&dh.
i39)
1158
M.
HLAVAEEK
initial deformafion
R
pressure
$ (2) air-cc&.
Fig. 3. Schematic picture of initial cartilage deformation and cartilage pressure at loading and deloading. Deformation and pressure in deformable cartilage at t =O+ and for t ~0 are shown schematically in Fig. 3. Due to the instantaneous pressure drop (48) on s, at t = O+ the ideal phase starts to flow rapidly from the gap into the cartilage. Equation (44) shows that for small t the flux 4:: into the cartilage is very high for deformable cartilage, it decreases with time and for t -+ co it tends to that for the rigid cartilage [compare equations (47) and (49),]. SQUEEZE-FILM
LENGTH
ESTIMATION
In the human hip joint loaded by the body weight it holds for most part of the film that h+(ti:zl, jii$] -+I$$], where Gary given to order 2’ in equation (31)s corresponds to the k =0 case. Thus, within this approximation the film radius R(t) of an axially symmetric synovial film compressed by a step loading does not depend on t during the gel-forming process. The film radius can be found on using the vertical displacement of s* for dry frictionless contact (without the fluid film in between) of rigid spheres covered with incompressible layers. Following thin elastic Armstrong (1986) who analysed plane strain, we easily obtain for the radius R(t)%:R(O) of axially symmetric squeeze-film during the gel-forming process R(0)=2(FL3W,f/Xp)“6
(50)
where gRer= a/2, with B being the curvature radius of s*, s* at r = 0, t = O- and W,, is the effective curvature radius (sphere against plane). Equation (50) is valid for &R(O) 4 1 and this R(0) will be used for the disc radius R in equations (45). NON-NEWTONIAN
SYNOVIAL
FLUID
So far, we have assumed the SF to be Newtonian. According to Fig. 1 of Part I of this paper, the normal SF shows a strong shear-thinning effect, while the inflammatory SF is approximately Newtonian. The experimental points shown there for both the normal and the inflammatory SFs have been approximated by the power laws [equation (I, 3111 of the form
in three regions of the shear rate Y= vale. Here m, n, w are constant (m, n taking different values in different intervals of Y). See Fig. 1 in Part I for these intervals and the values of m, n, w. n= 1 corresponds to the Newtonian behaviour. The shear-thinning effect will be taken into account approximately in the following way. Instead of a shear-thinning SF we consider a fictitious Newtonian one with a variable effective viscosity qer (h, (PHA). qsr is defined as the viscosity of the shear-thinning SF given by equations (51), with Y replaced by the volume-averaged shear rate ( Iyl).= 1 of the fictitious Newtonian SF. Setting n= 1 in equation (I, 33) and integrating over the synovial gap of height h and radius R, we obtain with the use of equations (41) and (43)
Replacing in equation (51) q and Yby qer and (IY))n=l from the above equation, respectively, we obtain for rler &f(h,
(52)
(P~~)=[tl~(2~~/3nR’)“-‘]““.
Note that here r~ef=~N for n= 1. Now, within the above approximation, h(t) and vu,+(t) for the shearthinning SF can be obtained by solving the problem defined by equations (45) with &r, N and YINgiven by equations (52), (46)2 and (51),, respectively. RESULTS AND CONCLUSIONS HOMOGENEOUS
FOR THE
CASE
For step loading [equation (43)] and p= 500 N equations (45) with the use of equations (52), (51),, (46), and (19), are solved by the Runge-Kutta method, starting with ho = 20 pm, to = 1 x 10F3 s. The value t =0 cannot be used, as N -+ co for t-+0+. Higher ho and lower to > 0 give very small corrections to h(t) and qHA(t). For the normal human hip joint we choose the following representative values for cartilage: R= 10e3 m, l,f=0.5 m (Dowson, 1970) fi =0.5 MPa (Armstrong, $?=1+2b= 1.5 MPa, 1986), ,&=6x lo-i6 m4N-2s-1 (McCutchen, 1962). We take R = R(0) and equation (50) yields R = 1.5 cm. As to the SF, three types of fluids are considered. According to Fig. 1 in Part I, we take in equations SF: n= 1, (52) for the normal (51) and n=3.26 x lo-‘, m=4x 10m3 Pas for Y’Yl> m=l.99Pas” for Y2
Synovial
fluid filtration
loadings and all SFs. This value was obtained in Part I using the results of equilibrium filtration experiments with the normal SF (Maroudas, 1969). Figure 4 shows h(t) and cuA(r)=#~uA(t), where the HA true mass density pgl= 1.5 g/ml (see Part I), for the above three SFs. Denoting h, t at cHAmal by pm, hminr r,,,, we get for the normal SF h,i,=O.ll t max=4.2 s. for the inflammatory SF h,,,=0.04 pm, t max= 3.7 s, and for the Newtonian SF with the normal CHAO? h,i, =0.09 pm, t,,, =3.4 s. h,i, for the normal and Newtonian SFs differ only slightly, showing that the shear-thinning effect of the normal SF on hmin iS lOW. cuA,, matters more. hmin for the inflammatory SF is about one-third of that for the normal one at the same cnAmax. cuA,,ar of the inflammatory SF is not known, but it may be higher than 56 mg/ml, because of a lower HA molecular weight of the inflammatory fluid. This would make h,,, a bit lower in the inflammatory case, but this change should not be high, as cuA,r is very high near h,i, (Fig. 4). If P= 100 N and 2500 N and the other parameters take the values given above, we find for the normal SF h,,,=O.l2pm, t,,, = 12.5 s, R= 1.1 cm and h Ill,”=O.lO pm. t,,,= 1.4s, R = 1.9 cm, respectively. Surprisingly, h,i, (in contrast to t,,,) practically does not depend on l? hmin does not also change much if A, i( and 1, fi vary in the physiological range. For k=O we get the curves h(t), cnA(t) indistinguishable from those for k#O. This means that diffusion of the ideal phase through the macromolecular network along the synovial film is small and the term kh in equation (46), can be neglected with respect to the first term. However, setting l=O, k#O [i.e. using the term kh in equation (46), only] is equivalent to cartilage absence for parallel films, as subchondral bone is impermeable. Now, cnA increases due to diffusion of the ideal phase along the film only and with the same R it takes hours to reach cnAmax. while h,,, is about one order lower than that with the cartilage present. For the rigid porous cartilage and normal SF, using equations (45) with the aid of equations (52). (51),, (49)* and (19),. we get h,,,=O.O4pm, t,,,=23 s. Thus, the squeeze-film time t,,, is much shorter and h In!”several times higher for deformable than for rigid cartilage. h. ,‘06[m]
Fig. 4. Variation of h and cHA with time for normal. matory
and
BM26:10-B
Newtonian
synovial fluids [Fig. 2(b)].
for step
inflamloading
II
113
The value of hmin obtained in this paper for the normal SF (i.e. 0.1 hum) is about 5 times higher than that estimated by Maroudas (1969) who assumed that h at which the total flux of the SF along the gap becomes equal to that of the ideal fluid into the cartilage was about 0.5 pm. However, our calculation shows that this occurs at about h=2 pm. For h=0.5 pm cuA should be already 4 times higher than cnAO. This explains the difference and indicates that the estimation hmin =0.02 pm by Maroudas (1969) for the normal SF is too low. For step loading [equation (43)] the instantaneous behaviour of cartilage is the same as that of an incompressible single-phase elastic solid immediately after the application of load at t =O, with a pressure jump on s, [the I-solution in equations (30)]. This jump is proportional to (rp,,),,/r in the axially symmetric case. However, for t>O both mixtures require the pressure continuity p=i on s* and, therefore, a boundary layer of thickness h,,-(ifit)1’2 in the cartilage near s, with a very high pressure gradient (perpendicularly to s*) is formed to smooth out the pressure jump (see the II-solution). The flow direction on s* is given by the sign of(rp,,),,. In our homogeneous case the flux of the ideal phase into the cartilage, high shortly after t=O, diminishes with time [see equation (44)] and, theoretically. tends for t-t zc to the constant flux for the rigid porous cartilage. The flux of the SF sideways is also high shortly after t q =O. but drops more quickly with time than that into the cartilage. For h below 2pm the total flux into the cartilage exceeds that along the gap, the increase of cnA becomes visible (Fig. 4) and cnA shoots up for values above 30 mg/ml. At the beginning of the filtration process the viscosity n is important. At that time the shear rate is high, but n’s of the normal and the inflammatory SFs do not differ much. Later, at lower shear rates, the flux sideways is low and a larger difference in rf of both the fluids can hardly already apply. Here the difference in cHAOmatters more than that in rl. The protein portion of the HA--protein complex plays no role in this filtration model (with a possible exception for the value of cnAmar). Now. if the load drops to zero at t = f in a stepwise fashion (Fig. 3) both the pressure in the SF and the part of stress in the cartilage due to the instantaneous response of a single-phase cartilage at t =0 are removed and only the pressure in the boundary layer (IIsolution) remains. Thus, another pressure jump is formed on s*, but this time the ideal fluid flows through s, in the opposite direction, in our case out of the cartilage. This pumping effect works obviously also for a pulsating continuous load. A step loading has been analysed here because of its simplicity. The present study seems to corroborate the idea of effective filtration by cartilage of a physiologically squeezed film of normal SF in human hip joints loaded by the steady body weight. Further experimental investigations are needed. In the first place, experiments similar to those by Maroudas ( 1969)
1160
should
M. HLAV~~EEK
be
made
cHAmaxdepending
with various SFs to find on cHAo, the HA molecular weight,
the ionic strength of the SF, etc.
REFERENCES Armstrong, C. G. (1986) An analysis of the stresses in a thin layer of articular cartilage in a synovial joint. Engng. Med. 15, 55-61. Biot, M. A. (1941) General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164. Dowson, D., Unsworth, A. and Wright, V. (1970) Analysis of boosted lubrication in human joints. J. Mech. Engng. Sci. 12, 364-369.
HlavZek, M. (1990) Synovial fluid filtration by cartilage in synovial joints. In Theoretical and Applied Mechanics, Proc. 6th Nat1 Cong. Theoretical and Applied Mechanics, Vol. 4, 84-87, Publ. House Bulg. Acad. Sci. Sofia. HlavBEek, M. (1993) The role of synovial fluid filtration by cartilage in lubrication of synovial joints-I. Mixture model of synovial fluid. J. Biomechnnics 26, 1145-1150. Holmes, M. H. (1986) Finite deformation of soft tissue: Analysis of a mixture model in uni-axial compression. J. biomech. Engng. 108,372-381.
Hou J. S., Lai, W. M., Holmes, M. H. and Mow, V. C. (1988) Boundary conditions and fluid flow through cartilage under squeeze-film action. Advances in Bioengineering, Winter Annual Meeting of ASME, Chicago. Hou, J. S., Holmes, M. H., Lai, W. M. and Mow, V. C. (1989) Boundary conditions at the cartilage-synovial fluid inter-
face for joint lubrication
and theoretical
verifications.
J. biomech. Engng. 111, 78-87. Hou, J. S., Holmes, M. H., Lai, W. M. and Mow, V. C. (1990) Squeeze-film lubrication for articular cartilage with synovial fluid. In Biomechanics of Diarthrodial Joints (Edited bv Mow. V. C.. Ratcliffe. A. and Woo. S. L. Y.L ,. iol. II, 3i7-367.’ Springer, New York. Hou, J. S., Mow, V. C., Lai, W. M. and Holmes, M. H. (1992) An analysis of the squeeze-film lubrication mechanism for articular cartilage. J. Biomechanics 25, 247-259. Lai, W. M. and Mow, V. C. (1978) Ultrafiltration of synovial fluid by cartilage. J. Engng. Mech. Div. Am. Sot. Ciu. Engrs 104, 79-96. Marobdas, A. (1967) Hyaluronic acid films. Proc. Instn Mech. Engrs 181, 35, 122-124. Maroudas, A. (1969) Studies on the formation of hyaluronic acid films. In Lubrication and Wear in Joints (Edited by Wright, V.). Sector, London, 124-133. McCutchen, C. W. (1962) The frictional nronerties of animal . . joints. Wear 5, l-17. Mow, V. C., Kuei, S. C., Lai, W. M. and Armstrong, C. G. (1980) Biphasic creep and stress relaxation of articular cartilage: Theory and experiment. J. biomech. Engng. 102, 73-84. Rybicki, E. F., Glaeser, W. A., Strenkowski, J. S. and Tamm, M. A. (1979) Effects of cartilage stiffness and viscosity on a non-porous complaint bearing lubrication model for living joints. J. Biomechanics 12, 403-409. Sneddon, I. H. (1972) The Use oflntegral Transfbrms, p. 103. McGraw-Hill, New York. Walker, P. S., Dowson, D., Longfield, M. D. and Wright, V. (1968) Boosted lubrication in synovial joints by fluid entrapment and enrichment. Ann. rheum. Dis. 27,512-520.
I” Grc;,t
Vol 26, No. 10. pp
II51
0021 9290;93 $6.00+000 e: 1993 Pergamon Press Ltd
1160. 1993.
Bnfam
THE ROLE OF SYNOVIAL FLUID FILTRATION BY CARTILAGE IN LUBRICATION OF SYNOVIAL JOINTS--II. SQUEEZE-FILM LUBRICATION: HOMOGENEOUS FILTRATION M. Institute
of Theoretical
HLAVQEK
and Applied Mechanics, Czech Academy 12849 Prague 2, Czech Republic
of Sciences, 49 VySehradski,
AbstractA mathematical model of synovial film filtration and synovial gel formation at normal approach of cartilage surfaces in the human hip joint is presented. The biphasic mixture model presented in Part I of this paper [Hlavii-ek, J. Biomechanics 26, (1993)] for synovial fluid and that of Mow and his collaborators [J. Biomech. Enyng 102, 73-84 (1980)] for cartilage are used. A general analysis of filtration of an axially symmetric synovial squeeze-film between two cartilage layers at normal approach is given. The geometrically simple case much idealising the human hip joint is also considered: two cartilage discs with a synovial film in between are compressed by the steady loading of the half human weight. The homogeneous film filtration process where the synovial gap remains parallel and the macromolecular concentration in the gap is spatial homogeneous (to the thin-film approximation) and time-dependent is numerically analysed. If the hyaluronic acid concentration of the synovial gel at equilibrium is 50 mgiml at least, the resulting stable gel layer thickness for the homogeneous filtration in the human hip joint and for normal synovial fluids is about 0.1 pm. being almost independent of the loading. Inflammatory synovial fluid shows values several times lower.
phase
INTRODUCTION
Newtonian
fluid and
biphasic
model
(Mow
et al. 1980) for the SF and cartilage were used. respecSynovial fluid (SF) forms an interface with the articular cartilage in synovial joints and plays an important role in their lubrication. SF contains a small concentration c of hyaluronic acid (HA)-protein complex that forms a continuous network. According to Maroudas (1967, 1969) and Walker et al. (1968), the concentration of the complex in the gap between the cartilages increases during the squeeze-film action due to the penetration of low molecular weight substances from SF into the cartilage. Increasing the SF viscosity in the gap delays the approach of the cartilages, and the formation of gels (Maroudas, 1969) protects their surfaces, if sliding motion ensues. Assuming a linear variation of SF viscosity with c, an inverse proportionality of the film thickness to c and no flow into the cartilage, Dowson et al. (1970) calculated the squeeze-film thickness between rigid impermeable surfaces as a function of time. An initial film thickness of 10 pm was assumed and the time for this to fall to 1 pm was calculated. However, Maroudas (1967. 1969) predicted the final thickness of the gel to be 0.02-0.03 pm and stated that the flow from the gap into the cartilage should be more significant than the flow sideways in the gap, when the film thickness is less than about 0.5 pm. Hou et al. (1988, 1990, 1992) presented a squeeze-film lubrication model of a rigid impermeable spherical indenter approaching a thin permeable cartilage layer supported by a rigid impermeable subchondral bone. A single-
Received in final form 3 March 1993.
tively. In the model the viscosity of Newtonian SF does not change during the squeeze-film action. Lai and Mow (1978) analysed one-dimensional filtration of SF between two parallel semipermeable rigid boundaries approaching with a constant speed and used the classical diffusion equation. In the present paper the SF is a mixture of two incompressible fluids--see Part I of the paper (HlaviEek, 1993). The viscous phase is formed by the HA-protein complex, the ideal one by all low molecular weight substances. As to the cartilage, the biphasic linear model of Mow et al. (1980) is used, i.e. cartilage is modelled as a diffusing mixture of an elastic porous solid matrix and an ideal fluid. The ideal-fluid phase only can flow across the SF-cartilage interface as the macromolecules of the HA-protein complex are too large to penetrate into cartilage pores. In Part II of this paper the boundary conditions at the synovial-film-cartilage interface are ob.tained using the balances of mass, momentum and energy and following Hou et al. (1989). Moreover, the viscous phase of.the SF is assumed to adhere to the cartilage surface. The adherence of the HA-protein complex to the cartilage surface has been experimentally proved by the method of microelectrophoresis (Maroudas, 1969). In the filtration model by Hou et al. (1988, 1990, 1992) a single-phase viscous SF passes this interface as a whole and the ‘no-slip’ condition for the SF would lead to incorrect boundary value problems. Hou et al. had to formulate a special kinematic condition (‘pseudo-no-slip’) in order to obtain a correct formulation for their model.
1151
1152
M. HLAVA~EK
A general analysis of filtration of an axially symmetric synovial squeeze-film by two cartilage layers is given in the present part of the paper. A geometrically simple case much idealising the human hip joint is numerically analysed: two cartilage discs with a synovial film in between are compressed by the half weight of the human body. For this geometry there exists such a squeeze-film process that the synovial gap remains parallel and the macromolecular concentration c is homogeneous in the gap (to the thin-film approximation) and increasing with time. Some preliminary results for this case have been reported earlier (Hlavacek, 1990). MIXTURE MODEL OF ARTICULAR
(1)
4::’ + (@“’P!“‘),i = 0 incompressible
(2)
phases, and
the constitutive equations $?‘= _@i)fi&. ,,, f&=
I
_j$”
1
*!3’= - @Q)j&.
=rp$qGjlqj5)‘)).
‘I
+
1,2 yields (5)
Equation (4),, i.e. the first of equations (4), yields the volume flux q!r) of the ideal fluid relative to the solid phase in the fbrm of the Darcy law, q!l)=~cl)(~~l)_~!2))= I I I with the cartilage permeability
_&
sly
(7)
After excluding irj1)-3j2) from equations (4), and (5), we obtain with the use of equation (7)
For small cartilage deformation. considered the linear relation
(8) Mow et al. (1980)
d!f”=XEi:2:bij+2~1;:i’.:) II
(9)
where I?/‘) is the matrix displacement vector and 1, fi are the elastic Lame material constants of the porous matrix. The model in this linear formulation corresponds to Biot’s model of consolidation of soils (Biot, 1941). BOUNDARY
CONDITIONS
AT THE INTERFACES
We refer to Hou et al. (1989), where the boundary conditions at the interface of two biphasic mixtures (one phase being fluid, the other solid), with the displacement (and therefore the velocity) of the solid phase continuous at the interface, are given. Using the same procedure for the interface s.,, between the cartilage (taken as a mixture of an ideal fluid and the elastic porous solid matrix) and the SF (taken as a mixture of an ideal and a viscous fluid according to Part I of this paper), we obtain the following boundary conditions on s* written in the Cartesian coordinates Xi: v~2)=~~2),
#f’e
(6)
k^defined by
{+p/~~
(k^fi,i),i=?F\.
the momentum equations d!“’ .+$!“‘=0 u., I ’ the continuity equations
a =
[~(“(tl”--912’)+912’],i=0.
CARTILAGE
Articular cartilage is a tissue composed essentially of a fibre-reinforced solid matrix phase (collagen fibres and proteoglycan macromolecules) and an interstitial fluid phase (water). The compressive viscoelastic behaviour of cartilage is governed by the flow of water within the tissue, its exudation and imbibition across the articular surfaces and by the (visco-)elastic resistance of the cartilage matrix. Although the porosity of the tissue is large (about 80%) the permeability is relatively low due to the small pore size. A mixture approach was first used on cartilage by Mow et al. (1980). The basic equations of cartilage considered as a two-phase mixture (the idea1 fluid and the elastic porous solid) can be written in the following form (Holmes, 1986):
for intrinsically
while summing equations (2) for
p(‘)(v11)-v12))ni=P(l)(E11)-o12))ni,
IJ’
(3)
Here and in what follows, a comma followed by a subscript stands for the respective partial derivative and the summation is made over pairs of identical Latin subscripts in the range l-3. The superscripts (1) and (2) are used to indicate that the quantity is associated with the fluid and the solid phase, respectively. The ‘roofed quantities refer to the cartilage. i$), ey), tiy) and @(a)are the partial stress tensor, the velocity, the diffusive drag force and the volume fraction of the (a)-phase, respectively. 0 is the fluid pressure, 5:;)’ the stress component of the solid phase that is due to matrix deformation and k stands for the diffusive drag coefficient. As the mixture is saturated, G(r) + +(‘) = 1. Inserting equations (3) into equation (1) and summing the result gives
[a~~‘+~$)
-p”‘v~“(v~“-v~‘)]nj -“)91’)(0j’)-Oj2))]nj,
=[8,!i”+csf’-p [u!~)v~‘)+u~~)v~)-t~(l)v~‘)v~)(v!’)-v/~))]n~ =[a;;)c;l)+aij
-(2)-(z) vi -2p1
*‘l’F:l’3:l’(B!l’_812’)]ni. (10)
Here, analogous to equation (I, 1) of Part I of this paper, fi(m)=$a)@(u), where PC’)are the partial mass densities and $l)=p”), st2) are constant true mass densities of incompressible phases in the cartilage. ni is the unit normal to s* and p(“), vf), c$) are the partial mass densities, the velocities and the partial stresses of SF, respectively, referred for a = 1 to the ideal and a=2 to the viscous fluid phase. See equations (I, 1) and (I, 5) for the definitions of p(O)and ai(g).In equations (10) the zero heat flux is assumed and the internal energy density term is absent, as the fluid phase is
Synovial fluid filtration-11
ideal (inviscid). Equations ( 10)2, (lo), and (lo), follow from the balance of mass, momentum and energy, respectively (Hou et al., 1989). Equation (lo), is a kinematic condition of continuity of the viscous- and solid-phase velocities. This condition follows from the fact that the HA-protein complex present in the SF becomes adsorbed onto the cartilage surface, as proved by the method of microelectrophoresis by Maroudas (1969). The boundary conditions on the interface s, between the biphasic cartilage and a non-porous rigid subchondral bone are of the form (Hou et al., 1989) +o,
$(l’($ ‘“-P)ni=O, ‘
(B!f’+d;f’)nj=li, (11)
where ni is the unit normal to s+ and fi is the traction vector applied on the cartilage by the rigid bone. NORMAL APPROACH
OF CARTILAGE
LAYERS
Let us consider a thin axially symmetric synovial film between two cartilage layers of a constant thickness h^(Fig. 1). The SF-cartilage and cartilage-subchondral-bone interfaces are denoted by s*, s* and s+,s +, respectively. In what follows, an asterisk and a cross up or down indicate quantities on s* or s.,,and S+ or s+. respectively. Two systems of cylindrical coordinates, I, 0, z and r, f&i, in the synovial film and in the lower cartilage layer, respectively, are introduced (Figs 1 and 3). The radii of curvature of s*, s* are large compared to the film length, that is large compared to h^and this is large compared to the synovial film thickness h(r, t). Then, without any loss of generality. s* and s* can be assumed to be symmetrical with respect to the plane z=O, the symmetry axis being r=O. The cartilage layers are compressed in the zdirection by the force y’(t) = p(t) H(t),
(12)
where H(t) denotes the Heaviside unit step function and P(t) is a given smooth function of time t. For the axial syrrimetry, equations (4)2 and (8) with the use of equation (9) yield the following equations in the r 10, z-coordinates for ti(‘) r ’ CF’and p in the lower cartilage layer ( - h^
1153
j,?= tiCjs+(X +$)(rfip)),,/r
Let us introduce the non-dimensional ables or constants as follows: r=r’f,
(primed) variz=z’i,
2=Ph,
t=t’t,v=iJf,s=2/*,~=Rjr,riP’=1;:!*’K,aJ2’=t21;(f)h, ;(*,=yj?!ff/t;
pq,2),,
i/t;
b ’ ” v, =v,, ‘(“v , v~‘=y~?‘f~,
$w=;‘;.“q
b=o’*,
p’=$‘;‘f’q&
p=p’d.
f,f,t
In the case of the human hip joint analysed below, the orders of quantities used and given in the SI units are: r- 10-z, z- 10-7-10-5, t- 101, h^- 10-3, ni- 106, k^-lo-‘6,+, ;(C. 103, &“‘, @=)wlO”, tl- 10-2-10’ for qC2)- lo-*, r]- 102-10’ for qC2)- 10-l. Here q(l) and qC2)= cpare the volume fractions of the ideal and the viscous phase of the SF, respectively. The order of SF viscosity q follows from equations (I, 17), and (I, 31). We also have V-1O-3, 6- 10-4-10-3, C?- 10-l. If 5,‘~ I/fi and the primed quantities are of the order 1, then writing equation (14) with the use of the primed quantities and keeping the terms of order IO6 only, i.e. dropping the lower-order terms, equation (14) with the help of equation (9) takes on the simple form --p= -6 + 8rie. Similarly, equation (lo), written for the r-direction yields @‘=O. We can easily find that also in equation (lo), the terms containing v,!~)-v!~) and C!1)-t!2) are small and may be dropped. Then: taking ;he scalar product between equation (IO), and vi2)=iri2), subtracting the result from (lo), and using equations (1.Q (I, 13a), (3), we obtain p=fi for cp+l. Thus, for thin synovial films and cartilage layers and physiological loading of human hip joints the boundary conditions [equations (10) and (I l)] become I’
@‘=O,
v(2) = 9’2, 7s
#a)e=O
I
’
4,
u)=(iw
*
7
p=B,
on s&=0)
(15)
and ti@)=O, 1.2?)=0, p,;=O on s+ (2= -h^). I
Fig. I. Squeeze-film geometry.
are
characteristic values of the variables r, z, t (for example, their average values in the film and time span). As the cartilage is attached to the rigid bone, both tir’ and fir) are of the order of h^ and this explains the definition of 9;!2),t2’::‘. This idea is due to Armstrong (1986), who used this perturbation analysis for a thin single-phase elastic layer with a rigidly fixed surface. p is the total pressure in the SF [see equations (I, 7) and (I, 13a)]. Let us insert equations (I, l), (I, 5). (3) and (9) into equations (10) and use the r, 8, z (or i)-coordinates with n, = ni = 1, n, = ne = 0. For illustration we give here the result for equation (lo), written on s* in the z-direction only:
,
bone
(13)
~(~B,r),,lr+~~,.;i=(r~0!2)),,/r+3~~~,
v(2) = 9’2’
subchondral!
+fl(rt$f,‘).,,h-,
(16)
Approximating s + , s+ quadratically in r and denoting
1154
M. HLAVACEK
by hb(r, t) the vertical distance of their points, we have hb(r, t)=h,(O, t)+r*/%
where W is the radius of curvature of s+ , s+ at r =O. Figure 1 shows that h(r, t)=hb(O, t)+r*/C%--2(h^+iii:)). Using equations load
(17)
(3) and (15), we have for the total
9’(t) = -2?&:‘6
+ &i;))r dr = 2xTpr dr.
0
0
(18)
First, let us consider the case of the SF with a Newtonian viscous phase. Setting n = 1 in equations (I, 35 and 36) and using equations (I, 8), , (I, 13a)l, (6) and (15),, as v_(*)= -h 12, we get the equations of a squeezed synovial’tfilm in the form
ditions on s.,, [equations (15),-,] and on s+ [equations (16)], the ambient conditions for r-co [equations (22)] and the initial conditions [equations (20) and (21)]. Fluid motion in the cartilage is governed by the permeability k^and, therefore, by the left-hand side of equation (13),: its first and second terms correspond to the flow in the radial and vertical directions, respectively. For k^=O (no fluid diffusion) equation (13),, together with the initial conditions given by equations (20), yields the incompressibility condition for the porous matrix for any t. In this case 1 can be excluded from equation (13),, 2 using equation (13),, which yields B., =fi [(r$*‘). Jr1 ., +fitijfjpp
B.0= ji(rt;~$/r These are the momentum
+ fiti:fbp.
equations
(23) for an incom-
(~3r~,,ll2tt~).llr+~Cr(vl~‘-~p,,)l,,lr=~.,-2q~~), pressible single-phase solid with shear modulus F, ((PHAh3rp.,/12~N),r/r=((PHAh).l+h[r(PHAv~~’l,r/r, ~N((PHA)=~((PHA/(PHAO)~I
1.62 x 10-l’ r~;;d.~~.
I&&=
(19)
Here vHA is the HA volume concentration. The Newtonian viscosity tfN and the permeability k of the synovial film were introduced in Part I, equations (I, 17)* and (I, 25). If P(O)=O,then Y(t) is continuous at t=O [see Fig. 2(a)] and equation (12) yields the quiescent initial conditions fi=p=o,
i+o,
lif’=o
@GO).
(20)
Further, we set %A@, ~HAO.
hbo
boundary r-00 as
O)=%IA~~
MO,
O)=hbo,
(21)
with a scalar parameter 6, commonly called the hydrostatic pressure. The boundary conditions given by equations (15),_ 6 and (16) must be replaced for this material by -p=
-j+2fi#,
fi(*)=O, r
$i*)=O
$*~+ti~*~=O . . at f= -6.
at z*=O, (24)
Let us note that equations (24),, 2 guarantee the continuity of the stress vector, but not that of the pressure on se, while equations (24)~~~do not guarantee zero pressure gradient on s+ . We come back to the general case k^#O (with diffusion). To get equations for j’, fi’j?, ii’$‘, we use the primed quantities in equations (13), (15),_ 6, (16) and (20). For example, equation (13), takes the form a[~2(r’8’,,.),r./r’+8’,4,P.]=~(rts’jf)),,.t,/r’+;‘:tb,.
being given constants. Also, the ambient conditions for p and (P,.,~are assumed at
, (25)
where the parameter E is defined by P(m,
~)=O,
(PHA(a,
~)=(PHAO.
(22)
Thus, if P(t), (PHAO and hbOare given then, for seven unknown functions p(r, t), (PHA(r,t), h(r, t), MO, t), fi(r, f, t), lij2)(r, 2, t) and ir:*)(r, 2, t) there are seven equations, equations (19),,,, (17), (18) and (13),_,, together with the boundary conditions on s* [equations (15),, 2,4-6] and on s+ [equations (16)], equations (22) and the initial conditions [equations (20) and (21)]. PERTURBATION ANALYSIS OF THE CARTILAGE LAYER
Approximate solutions in the cartilage will be found due to the fact that the cartilage forms a thin layer and that substantial fluid motion in the cartilage occurs only in a thin boundary layer at the articular surface for physiological time scales. For the time being let us assume that p(r, t) is known. I?*) fir) and fi in the lower cartilage layer (Fig. 1) rn;s; satisfy equations (13), the boundary con-
2 = k^tii/lh^*.
(26)
In the perturbation analysis it is assumed that Ii’!,*), I?:?, j’ can be represented by the asymptotic series in terms of the small parameter 8=&h lo-‘, a’;,*‘= lY;,; + C&l’,‘,+ 6^2A’(2)+. 0 I, * . ., etc.
(27)
Then, on substituting these series into equations (13), (15),_,, (16) and (20) written in the primed quactities and collecting terms of the same power of 6, the original problem is replaced by a series of simpler problems for the coefficients of the asymptotic series. For E- 1 the terms in the series given by equations (27) descend with increasing n and the series can converge. In this case the diffusion terms in equation (25) apply throughout the yhole ca$lage thickness. Inserting 1-1, k- 10-16, M- 106, h- 10m3 into equation (26), we find that this is the case as late as about lo4 s after the beginning of load application. For the time scale I- 10’ s equation (26) yields L- lo-‘. It can be found, however, that for 0 & 1 the terms in the series given by
Synovial
fluid filtration-II
equations (27) increase with n and the series diverge. Thus, the expansion given by equations (27) cannot be used throughout the whole cartilage thickness under physiological conditions, i.e. before 10’s since the load initiation. As 8e82 in our case, equation (25) shows that the solution up to order 8’ corresponds to the l=O case (without diffusion) if fi’,;.3 - 1. However, at places where $‘,pC2 + 1, the diffusion term 2b’.1.4. in equation (25) can matter even if 16 1. This is the case, for example, in a thin boundary layer of s*. Across this boundary layer the pressure jump at the interface obtained for the case l=O must be smoothed out by diffusion. The thickness &, of this boundary layer where diffusion in the f-direction applies is obtained from equation (26) with O- 1 in the form h,, -(E tit)?
(28)
We see that h^,, increases with time as t”‘. For example, for t = 10 s we get fi,, - 10m4- lo-’ m. There exists another boundary layer in the cartilage at s+ due to the fact that $‘,@#O at s+ for the solution without diffusion, but this boundary layer is of no importance near s*. As the system of equations (13), (15),_,, (16) and (20) is linear, its solution is taken as a superposition of two special solutions, ;‘“=;l;f’+;;;;,
;(:“=;$‘+$;J,
r
(29)
6=61+h,.
The first solution (denoted by subscript I) is that without diffusion (R =O, single-phase incompressible material [see equations (23)] with the boundary conditions given by equations (24). The second solution (subscript II) is that with diffusion (f #O) and is different from zero in the boundary layer of s* of the thickness A,, only. The boundary conditions on s., for the latter are chosen such that a superposition of both the solutions satisfies the original boundary conditions [equations (15),_,]. Using the non-dimensional r’, i’, li$‘, 1;;‘:’ defined above and 6;‘. pi’ defined by fi;‘=fi&i, p;‘=p/fi in equations (13), with k=O, (23) and (24), substituting the asymptotic series for ti$), 1;$’ and 0; in a form similar to equation (27) and collecting terms of the same power of 8, we found ti;i$ l;$L and & for n=O, 1, 2, which mation
resulted
in the 8’ order
1155
Now, equations (15), (29) and (31) yield the boundary conditions of the II-solution to order 8’ in the form @,I= -i’
v,
I(rli~fj),,/r+Iiiiil(fj It
at z*=O.
(32)
from equation (28) that &,4h if lo4 s. For such t we can consider the IIsolution to be defined in the half-plane 2~0, rather than in the strip -k < i < 0, and we put $2, uni, A(2) fir,-+0 Ilr ’ Let us introduce
for i-
- x
a small parameter
&&/r
(33)
sd by
b,=(Gt)l’2
and write equations (13), (32), (33) and (20) using the non-dimensional r’, t’, p’ defined above and ,Y, A!!(Z) G:i)=ti$)$, %I,, ’ G;$f), _ & defined by f=Y&, A(2’=$;:?hdr fi,,=&A?. Let us assume that ti;;$), %I2 G;$‘, and &, can be represented by asymptotic series in terms of 8,. G;;y)= t$fi
+ a,ti$~
, etc.
+
and insert these series into equations (13), (32), (33) and (20) written in non-dimensional forms. The order c?: system of equations yields G;;i@i=0 and the following problem for ;;;E,b: fi”‘2’
112”0,2’,2’,
=
;+2, 112”O. f’
for 1”<0.
-g2v
at
-tt(2) = U IIP”O.2,’
f”=O,
ti;;:t,), -+O
at i”+
I’;;f)o=O
at t’=O+,
-
(34)
CXI ,
and for &,: d;,O,i.l..=&,O,l. A, -pv’ PIlO’
for f”
j;,o+O
at 2”-+ --co,
1,,,o=O
at t’=O+,
P
(35)
where V’=(r’p’,,,),,,/r’ = VF2/fi. The order 8: solutjon to equations (34) and (35) takes for t$h’/i& the form [see Sneddon (1972), equations (2-16-35) and (2- 16-33)]
1)/2/I, ti$(r,&
1;;:‘=&” V(2+3i/h-i3/f3)/6fl,
2= -h2V
follows
approxi-
j?,=p+PV(2-i2/P)/2,
,=o,
t +h2/i%-
d,,(r, 2, 4 =
lP’=Pp,,(P/PI,
t?;;;,*+ti;;;
ii22 q&G)‘/2 ^
t)= -h2
-$ (>.I
s
’ W, 7)
Ed7
’
o (t-zp
l/2
f
W,
___
4
o(‘-T)112Ed~~
(30)
V=(r~.~),.lr.
(36)
To order A2 it follows that JJ=p+P
v,
I(rtijf’).,/r+ tiif’=O.
C;2:+IP I?., =o, I, ~Cii~~‘,=~~ V
“$=O,
fi,,*=hV
at f=O, at f= -it.
(31)
Now, within the approximation used, GE in equation (17) can be expressed with the aid of equations (29)2, A (30), and (36),, while P,~* and vi:) in equations (19) can be obtained through equations (29),, (30), and
M. HLAVACEK
1156
(36), and through equations (29),, (30), and (36),, respectively. At last, h(r, t) may he excluded from equations (19) by substituting equation (17) into equations (19). Thus, the problem for p, (PHA, h, hi,, fi, 1;:’ and ;P’ defined above is simplified to that for p(r, t), (PHA(r,t) and &(O, t) that satisfy equations (19)i, 2 and (18), together with the ambient boundary conditions [equations (22)] and the initial conditions [equations (20) and (21)], if 9(‘(t), hbO and &.,A0 are known. This general problem will be analysed later numerically. In what follows, only the case of homogeneous filtration will be solved. SQUEEZE-FILM
OF SYNOVIAL
GEL
So far, we analysed a fluid-like film (qN< a). As mentioned in Part I of this paper, with increasing cnA the macromolecular network becomes more permanent and the SF changes gradually into the synovial gel (SG) with a solid-like matrix. Let us denote the quantities referring to the SG by a circle above them. Similarly to cartilage, this gel can be modelled as a biphasic mixture of an ideal fluid and a solid porous matrix and described by equations (l)-(8) with the ‘roofs’ replaced by circles. The boundary conditions on s*, the SG-cartilage interface, have also the form of equations (lo), the terms on their left-hand sides being provided with circles. Again, the terms with the velocity differences can be dropped for thin films and cartilage layers and the same method used above for the SF films gives for the axial symmetry and n, = 1, ng=nr=O the boundary conditions on s* in the form i(Z)= 9’2’ +(2’+2’, ;“‘=Q~‘, b=fi, &;‘e+‘e, 6i 2’e --%‘. Here &!?’ ii the stress tensor component orthe s%d phase &re to matrix deformation [compare with d$” in equation (3)2 for cartilage] referred to in Part I as the swelling stress. According to the equilibrium filtration experiments by Maroudas (1969) in the human hip joint loaded by the body weight the normal component &St)’ of the swelling stress is at least one order lower than the total normal stress in the SG &i’,’+&i’,’ = -i + &y=‘:“, i.e. ]?r~Z’e) Q b, up to cnA equal about 50 mg/ml for the normal SF. Only above this value GE” rises quickly with increasing cuA at a constant total normal stress. Assuming that the shear stress 6:” is of the same or lower order than iit’e, the above boundary conditions have the form of equations (15), with i written instead of p. We see that the boundary conditions on s, are the same for both the SF and SG. Following the deduction of equations (I, 35,36) for the SF film, we find for a thin SG film successively $,Z=O, &jf,‘i=O, i{F,‘J’Z=O(even for a non-linear constitutive law of the gel matrix), $‘f’F (l,J'Z=O. AS ]“~~~]~);~“~I, if “~‘,“‘=it~~ on s*,s*, it holds that ?2’e=F(2’ ..A m a thin gel film. This last equation is the counterpart of equation (I, 33) and is also obtained from there for qn-+co. Thus, in a thinfilm approximation it follows that equations (19) describe both the SF (with a finite t,Q and the SG film (for qn-+co). We conclude that in the case of normal
SF for cuA up to about 50 mg/ml the equations of the synovial squeeze-film are the same for both the SF and the SG. Only above this value up to the maximum value of CHA (according to Part I, about 56 mg/ml) the swelling stress may be comparable to the pressure and the boundary conditions given by equations (15),_, are no more valid. HOMOGENEOUS
FILTRATION
Rather than solving numerically the general axially symmetric problem defined above (a system of integro-partial differential equations), in what follows we shall deal with the following simpler case much idealising the human hip joint. Two discs (cartilages) of thickness h and fixed radius R (hg R) are compressed (Fig. 2) by the force s(t) given by equation (12), with P(t)=P”[t-(t-t,)H(t-t,)]/t,,
(37)
where fl and t, > 0 are constant [Fig. 2(a)]. Between the porous elastic discs saturated with the ideal fluid there is a synovial film of constant thickness he and constant HA-concentration @raO at t =0 (ho
(PHA.r=O.
As (PHA ,z =0 for a thin film, it follows then that (mu*is a function of t only. For homogeneous filtration [equation (38)], if it exists, qua, uI A(2’, u2 A(2)and fi cannot be given at r= R in advance and we set only p=O
Besides equation assume
atr=R.
(39)
(38), for the time being, we also h,,=O.
(40)
First, let us consider the case of Newtonian SF and deformable cartilage. We shall show that for 9(t) given by equation (12) there exist h(t), qHA(t) and
Fig. 2. Homogeneous
squeeze-film
model.
Synovial fluid filtration--~11 p(r, t) for 0
-r2/R2)/nR2.
(41)
Equations (30) and (36) were deduced for cartilage layer loaded by a general loading function p(r, t) as asymptotic approximations to orders 8’ and @, respectively. However, for p(r, t) in the form of equation (41) it holds that V,,=O. A closer examination reveals that for such p(r, t) we get t;;~f,‘=ti$~=$,=O for n > 3 and t;,, = li;;‘$ = 0. &,. = li;‘r’,t/= G$?j” =0 for n> 1. Thus, equations (36) and (30) are exact for the pressure distribution given by equation (41). As V,,=O. using equations (29),, (30), and (36)r. we have j,?,=O, while equations (29),, (3O)a and (36), yield 6:‘: = 0 in the cartilage. It follows that 4::’ and 1;::) are functions oft only. Therefore, for the Newtonian SF there exists a homogeneous filtration process defined by equations (38) and (39) during which the synovial gap remains parallel [i.e. equation (40) is valid], with a homogeneous (i.e. independent of r) flow into the cartilage. Inserting equations (12), (37) and (41) into equations (36), after integration we obtain with the use of equations ( 10j2. (6)2. (29), and (30)2,4 for the loading given by equation (37) q;:‘(t)=
- 16F@ [t’~2-(t-t,)“2
H(t-t,)]
X ,$1’2~R4rCX3:2,$/2 for teh
-,?-^ /kM (hdtcch).
(42)
In fact, it is in this case only that the II-solution in the cartilage layer can be replaced by that in the halfplane. Inserting equation (42) into equations (19) and using equations (38) (40) and (36), gives a system of ordinary differential equations for h(t), q”*(t). Here we give the result for step loading [Fig. 2(b)] Y(t)=
&Y(f),
(43)
that is obtained from the loading given by equation (37) for t,+O. Equation (42) takes the form q::‘(t)=
-81%2(&&t)1/2/zR4.
1157
during this rapid transient response (R = 13 mm. E=8.8 MPa, ~=0.02 Pas, P’=445 N). A pocket-type film configuration was formed that flattened rapidly in time and a parallel film was rebuilt within some milliseconds after the load initiation. As shown above. the parallel fluid film corresponds to film pressure profile given by equation (41). Thus, leaving aside the rapid transient response due to the SF viscosity. as the initial conditions at t=O+ we take the film pressure profile from equations (41) and (43) and the instantaneous cartilage deformation and pressure given by the I-solution in equations (30). It follows from equations (30),, (36), and (43) that tij:‘=liifi does not depend on time, so that C!:)=O for t>O and small deformations. In this case the equations for h(t). q”*(t) are obtained from equations (19) in the form h,,= -8&/~~/12r],~+N)/rrR~. y+,~,,=8&mNInR4h, h=h,.
cpHA=~HAO at t=O
(45)
with
N(h, (P”A, t)=2K2(l;/n~t)“‘+k((PHA)h.
(46)
Equations (45) can also be obtained in another way in this special case instead of by superposing the Iand II-solutions. In fact, we can easily find for V,, = 0 that inserting equations (27) into equations (13) gives q”=;;~f’= &, = 0 for n > 2, while only &, lij$‘, 6; and Gp2) are non-zero. The equation for 6; can b.e solved through the Laplace transform, which leads for t’& 1 to 4::) given by equation (44) and for t-, x to $:‘=
-8PIKilnR4.
(47)
Equation (30), gives the instantaneous P-J&* on s* at t=O+: p-&*=2(p-j,+)=
pressure
- VP;
drop 148)
where (41) and (43) yield equations (30),, V= -~&CR’. In the case of a rigid porous cartilage we have for @(r, i) in the lower disc for t>O, using equations (8) (16),, (15), and C:fi=O,
(44)
The cartilage alone (as a mixture of an elastic matrix and an ideal fluid) would respond at t=O+ to step loading [equation (43)] as an elastic body [see Armstrong (1986)], and the viscous SF alone would respond as a rigid body. This would break continuity conditions [equations (15), .2] on s.+ at t =O+. Thus, if conditions given by equations (15),,, are to be also satisfied at t=O’, the viscous SF must retard the instantaneous cartilage deformation. The time scale of this transient response t,=rlN/&?m 10-a s in our case. Rybicki et nl. (1979) analysed numerically the film profile of a Newtonian fluid between two elastic cartilage discs compressed by step loading [equation (43)]
(rd.,).,lr+~.~n=O. $=p(r) fi,?=O The solution for loading (43) is of the form
at i=O, at i= -&. given by equations
(41) and
j(r, .?)=p(r)+4%(2+Z/~)/rrR” and the rigid porous cartilage tions (44) and (46), are
counterparts
of equa-
q::‘= -8i%&R4, N(h, q&=2ifl+k(&dh.
i39)
1158
M.
HLAVAEEK
initial deformafion
R
pressure
$ (2) air-cc&.
Fig. 3. Schematic picture of initial cartilage deformation and cartilage pressure at loading and deloading. Deformation and pressure in deformable cartilage at t =O+ and for t ~0 are shown schematically in Fig. 3. Due to the instantaneous pressure drop (48) on s, at t = O+ the ideal phase starts to flow rapidly from the gap into the cartilage. Equation (44) shows that for small t the flux 4:: into the cartilage is very high for deformable cartilage, it decreases with time and for t -+ co it tends to that for the rigid cartilage [compare equations (47) and (49),]. SQUEEZE-FILM
LENGTH
ESTIMATION
In the human hip joint loaded by the body weight it holds for most part of the film that h+(ti:zl, jii$] -+I$$], where Gary given to order 2’ in equation (31)s corresponds to the k =0 case. Thus, within this approximation the film radius R(t) of an axially symmetric synovial film compressed by a step loading does not depend on t during the gel-forming process. The film radius can be found on using the vertical displacement of s* for dry frictionless contact (without the fluid film in between) of rigid spheres covered with incompressible layers. Following thin elastic Armstrong (1986) who analysed plane strain, we easily obtain for the radius R(t)%:R(O) of axially symmetric squeeze-film during the gel-forming process R(0)=2(FL3W,f/Xp)“6
(50)
where gRer= a/2, with B being the curvature radius of s*, s* at r = 0, t = O- and W,, is the effective curvature radius (sphere against plane). Equation (50) is valid for &R(O) 4 1 and this R(0) will be used for the disc radius R in equations (45). NON-NEWTONIAN
SYNOVIAL
FLUID
So far, we have assumed the SF to be Newtonian. According to Fig. 1 of Part I of this paper, the normal SF shows a strong shear-thinning effect, while the inflammatory SF is approximately Newtonian. The experimental points shown there for both the normal and the inflammatory SFs have been approximated by the power laws [equation (I, 3111 of the form
in three regions of the shear rate Y= vale. Here m, n, w are constant (m, n taking different values in different intervals of Y). See Fig. 1 in Part I for these intervals and the values of m, n, w. n= 1 corresponds to the Newtonian behaviour. The shear-thinning effect will be taken into account approximately in the following way. Instead of a shear-thinning SF we consider a fictitious Newtonian one with a variable effective viscosity qer (h, (PHA). qsr is defined as the viscosity of the shear-thinning SF given by equations (51), with Y replaced by the volume-averaged shear rate ( Iyl).= 1 of the fictitious Newtonian SF. Setting n= 1 in equation (I, 33) and integrating over the synovial gap of height h and radius R, we obtain with the use of equations (41) and (43)
Replacing in equation (51) q and Yby qer and (IY))n=l from the above equation, respectively, we obtain for rler &f(h,
(52)
(P~~)=[tl~(2~~/3nR’)“-‘]““.
Note that here r~ef=~N for n= 1. Now, within the above approximation, h(t) and vu,+(t) for the shearthinning SF can be obtained by solving the problem defined by equations (45) with &r, N and YINgiven by equations (52), (46)2 and (51),, respectively. RESULTS AND CONCLUSIONS HOMOGENEOUS
FOR THE
CASE
For step loading [equation (43)] and p= 500 N equations (45) with the use of equations (52), (51),, (46), and (19), are solved by the Runge-Kutta method, starting with ho = 20 pm, to = 1 x 10F3 s. The value t =0 cannot be used, as N -+ co for t-+0+. Higher ho and lower to > 0 give very small corrections to h(t) and qHA(t). For the normal human hip joint we choose the following representative values for cartilage: R= 10e3 m, l,f=0.5 m (Dowson, 1970) fi =0.5 MPa (Armstrong, $?=1+2b= 1.5 MPa, 1986), ,&=6x lo-i6 m4N-2s-1 (McCutchen, 1962). We take R = R(0) and equation (50) yields R = 1.5 cm. As to the SF, three types of fluids are considered. According to Fig. 1 in Part I, we take in equations SF: n= 1, (52) for the normal (51) and n=3.26 x lo-‘, m=4x 10m3 Pas for Y’Yl> m=l.99Pas” for Y2
Synovial
fluid filtration
loadings and all SFs. This value was obtained in Part I using the results of equilibrium filtration experiments with the normal SF (Maroudas, 1969). Figure 4 shows h(t) and cuA(r)=#~uA(t), where the HA true mass density pgl= 1.5 g/ml (see Part I), for the above three SFs. Denoting h, t at cHAmal by pm, hminr r,,,, we get for the normal SF h,i,=O.ll t max=4.2 s. for the inflammatory SF h,,,=0.04 pm, t max= 3.7 s, and for the Newtonian SF with the normal CHAO? h,i, =0.09 pm, t,,, =3.4 s. h,i, for the normal and Newtonian SFs differ only slightly, showing that the shear-thinning effect of the normal SF on hmin iS lOW. cuA,, matters more. hmin for the inflammatory SF is about one-third of that for the normal one at the same cnAmax. cuA,,ar of the inflammatory SF is not known, but it may be higher than 56 mg/ml, because of a lower HA molecular weight of the inflammatory fluid. This would make h,,, a bit lower in the inflammatory case, but this change should not be high, as cuA,r is very high near h,i, (Fig. 4). If P= 100 N and 2500 N and the other parameters take the values given above, we find for the normal SF h,,,=O.l2pm, t,,, = 12.5 s, R= 1.1 cm and h Ill,”=O.lO pm. t,,,= 1.4s, R = 1.9 cm, respectively. Surprisingly, h,i, (in contrast to t,,,) practically does not depend on l? hmin does not also change much if A, i( and 1, fi vary in the physiological range. For k=O we get the curves h(t), cnA(t) indistinguishable from those for k#O. This means that diffusion of the ideal phase through the macromolecular network along the synovial film is small and the term kh in equation (46), can be neglected with respect to the first term. However, setting l=O, k#O [i.e. using the term kh in equation (46), only] is equivalent to cartilage absence for parallel films, as subchondral bone is impermeable. Now, cnA increases due to diffusion of the ideal phase along the film only and with the same R it takes hours to reach cnAmax. while h,,, is about one order lower than that with the cartilage present. For the rigid porous cartilage and normal SF, using equations (45) with the aid of equations (52). (51),, (49)* and (19),. we get h,,,=O.O4pm, t,,,=23 s. Thus, the squeeze-film time t,,, is much shorter and h In!”several times higher for deformable than for rigid cartilage. h. ,‘06[m]
Fig. 4. Variation of h and cHA with time for normal. matory
and
BM26:10-B
Newtonian
synovial fluids [Fig. 2(b)].
for step
inflamloading
II
113
The value of hmin obtained in this paper for the normal SF (i.e. 0.1 hum) is about 5 times higher than that estimated by Maroudas (1969) who assumed that h at which the total flux of the SF along the gap becomes equal to that of the ideal fluid into the cartilage was about 0.5 pm. However, our calculation shows that this occurs at about h=2 pm. For h=0.5 pm cuA should be already 4 times higher than cnAO. This explains the difference and indicates that the estimation hmin =0.02 pm by Maroudas (1969) for the normal SF is too low. For step loading [equation (43)] the instantaneous behaviour of cartilage is the same as that of an incompressible single-phase elastic solid immediately after the application of load at t =O, with a pressure jump on s, [the I-solution in equations (30)]. This jump is proportional to (rp,,),,/r in the axially symmetric case. However, for t>O both mixtures require the pressure continuity p=i on s* and, therefore, a boundary layer of thickness h,,-(ifit)1’2 in the cartilage near s, with a very high pressure gradient (perpendicularly to s*) is formed to smooth out the pressure jump (see the II-solution). The flow direction on s* is given by the sign of(rp,,),,. In our homogeneous case the flux of the ideal phase into the cartilage, high shortly after t=O, diminishes with time [see equation (44)] and, theoretically. tends for t-t zc to the constant flux for the rigid porous cartilage. The flux of the SF sideways is also high shortly after t q =O. but drops more quickly with time than that into the cartilage. For h below 2pm the total flux into the cartilage exceeds that along the gap, the increase of cnA becomes visible (Fig. 4) and cnA shoots up for values above 30 mg/ml. At the beginning of the filtration process the viscosity n is important. At that time the shear rate is high, but n’s of the normal and the inflammatory SFs do not differ much. Later, at lower shear rates, the flux sideways is low and a larger difference in rf of both the fluids can hardly already apply. Here the difference in cHAOmatters more than that in rl. The protein portion of the HA--protein complex plays no role in this filtration model (with a possible exception for the value of cnAmar). Now. if the load drops to zero at t = f in a stepwise fashion (Fig. 3) both the pressure in the SF and the part of stress in the cartilage due to the instantaneous response of a single-phase cartilage at t =0 are removed and only the pressure in the boundary layer (IIsolution) remains. Thus, another pressure jump is formed on s*, but this time the ideal fluid flows through s, in the opposite direction, in our case out of the cartilage. This pumping effect works obviously also for a pulsating continuous load. A step loading has been analysed here because of its simplicity. The present study seems to corroborate the idea of effective filtration by cartilage of a physiologically squeezed film of normal SF in human hip joints loaded by the steady body weight. Further experimental investigations are needed. In the first place, experiments similar to those by Maroudas ( 1969)
1160
should
M. HLAV~~EEK
be
made
cHAmaxdepending
with various SFs to find on cHAo, the HA molecular weight,
the ionic strength of the SF, etc.
REFERENCES Armstrong, C. G. (1986) An analysis of the stresses in a thin layer of articular cartilage in a synovial joint. Engng. Med. 15, 55-61. Biot, M. A. (1941) General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164. Dowson, D., Unsworth, A. and Wright, V. (1970) Analysis of boosted lubrication in human joints. J. Mech. Engng. Sci. 12, 364-369.
HlavZek, M. (1990) Synovial fluid filtration by cartilage in synovial joints. In Theoretical and Applied Mechanics, Proc. 6th Nat1 Cong. Theoretical and Applied Mechanics, Vol. 4, 84-87, Publ. House Bulg. Acad. Sci. Sofia. HlavBEek, M. (1993) The role of synovial fluid filtration by cartilage in lubrication of synovial joints-I. Mixture model of synovial fluid. J. Biomechnnics 26, 1145-1150. Holmes, M. H. (1986) Finite deformation of soft tissue: Analysis of a mixture model in uni-axial compression. J. biomech. Engng. 108,372-381.
Hou J. S., Lai, W. M., Holmes, M. H. and Mow, V. C. (1988) Boundary conditions and fluid flow through cartilage under squeeze-film action. Advances in Bioengineering, Winter Annual Meeting of ASME, Chicago. Hou, J. S., Holmes, M. H., Lai, W. M. and Mow, V. C. (1989) Boundary conditions at the cartilage-synovial fluid inter-
face for joint lubrication
and theoretical
verifications.
J. biomech. Engng. 111, 78-87. Hou, J. S., Holmes, M. H., Lai, W. M. and Mow, V. C. (1990) Squeeze-film lubrication for articular cartilage with synovial fluid. In Biomechanics of Diarthrodial Joints (Edited bv Mow. V. C.. Ratcliffe. A. and Woo. S. L. Y.L ,. iol. II, 3i7-367.’ Springer, New York. Hou, J. S., Mow, V. C., Lai, W. M. and Holmes, M. H. (1992) An analysis of the squeeze-film lubrication mechanism for articular cartilage. J. Biomechanics 25, 247-259. Lai, W. M. and Mow, V. C. (1978) Ultrafiltration of synovial fluid by cartilage. J. Engng. Mech. Div. Am. Sot. Ciu. Engrs 104, 79-96. Marobdas, A. (1967) Hyaluronic acid films. Proc. Instn Mech. Engrs 181, 35, 122-124. Maroudas, A. (1969) Studies on the formation of hyaluronic acid films. In Lubrication and Wear in Joints (Edited by Wright, V.). Sector, London, 124-133. McCutchen, C. W. (1962) The frictional nronerties of animal . . joints. Wear 5, l-17. Mow, V. C., Kuei, S. C., Lai, W. M. and Armstrong, C. G. (1980) Biphasic creep and stress relaxation of articular cartilage: Theory and experiment. J. biomech. Engng. 102, 73-84. Rybicki, E. F., Glaeser, W. A., Strenkowski, J. S. and Tamm, M. A. (1979) Effects of cartilage stiffness and viscosity on a non-porous complaint bearing lubrication model for living joints. J. Biomechanics 12, 403-409. Sneddon, I. H. (1972) The Use oflntegral Transfbrms, p. 103. McGraw-Hill, New York. Walker, P. S., Dowson, D., Longfield, M. D. and Wright, V. (1968) Boosted lubrication in synovial joints by fluid entrapment and enrichment. Ann. rheum. Dis. 27,512-520.