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A model of force transmission in the tibio-femoral contact

incorporating fluid and mixtures

Citation for published version (APA):

Schreppers, G. J. M. A., Sauren, A. A. H. J., & Huson, A. (1991). A model of force transmission in the tibio-femoral contact incorporating fluid and mixtures. Proceedings of the Institution of Mechanical Engineers. Part H: Journal of Engineering in Medicine, 205(4), 233-241. https://doi.org/10.1243/PIME_PROC_1991_205_299_02, https://doi.org/10.1243/PIME_PROC_1991_205_075_02

DOI:

10.1243/PIME_PROC_1991_205_299_02 10.1243/PIME_PROC_1991_205_075_02 Document status and date:

Published: 01/01/1991 Document Version:

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(2)

233

A

model of

force transmission in the tibio-femoral

contact incorporating fluid and mixtures

GJ M A Schreppers, PhD,A A H J Sauren,PhD andA 8uson,PhD

Faculty of Mechanica! Engineering, Eindhoven University of Technology, The Netherlands

An ~xisymmetricfinite ~lement mo~el is formu!ated ~hich c~mprise~ a rigid spherical indentor, a meniscal ring and an articular cartllage layer, both consldered as mixture materlals wh/ch are Interactmg with an idealfluid sub-system.

F.ro~parameter studies it is concluded that the application of the mixture theory incomparison with solid modelling only leads to

slgnific~ntefJects when the~u.tersurfaces of the co'!'ponents aren~tsealed. The load distribution appears to change enormously during relaxatlOn of.the modeis. InltlQl~y thelarges~fractl?nofthe load IS borne by thefluidin the cavity, while at the end, when the system has reached ltS final configuratl.on, the menlscal ring bears the major part of the load. Further, the length of the relaxation period

app~ars to depend on thema~n1tudeof the step change of the load. Fmally, the curvature of the spherical indentor appears to have sl!1nificant efJects on the lo.admg of the meniscal ring, only immediately after the step changes of load are applied, and these efJects dlsappear as soon as the fluld starts to exude from the modeis.

1 INTRODUCfION

The present study is intended to provide a model of the tibio-femoral contact complex transmitting loads via the contact between the femur and tibia, both by direct contact of the cartilage-covered articular surfaces and indirectly via the menisci and the synovial fluid. Because of the complex character of the mechanical behaviour of this joint, a stepwise modelling approach is adopted. During every step, parameter studies are perfonned to investigate the contribution of the relevant components to this behaviour. The first step of the modelJing process was reported by Schrepperset al. (I) and consisted of an axisymmetric model comprising the ends of the femur and tibia and a toroid ring in between, representing the meniscus. These models were based on the assumption that load is fully transmitted by direct contact between the cartilage layers and indir-ect contact via the meniscal ring. All joint components were considered as solids.

The important conclusion of this study was the soft layers at the ends of the femur and tibia exert an important influence on the force transmission in the models studied. For models with soft layers the geometry of the tibial plateau seems to be of negligible importance, and the load borne by the meniscus increases, as compared to corresponding models without soft layers.

In reality both the articular cartilage layers and menisci are not solids but hydrated tissues and the con-tacts between these components are lubricated by syn-ovial fluid which also has the structure of a hydrated mixture. These tissues comprise free fluid that can move through the solid matrix of the cartilage and the hyal· uronic acid-protein complex of the synovia. In the liter-ature the cartilage is often described as a mixture (2-7), while the synovia is considered as a viscous fluid (8). A complex interaction between the cartilage layers and the synovial !luid occurs when the joint is loaded. To

Th. MS was receiv.don26 November1991and was accepted for publicationOn

2JMarch1992.

account for these effects cartilage and menisci have to be modelled as mixtures.

The present paper considers, with respect to Schrep-' perset al. (I), the next step in the modelling process of the force transmission in the tibio-femoral contact complex, in which mixtures and fluid are incorporated in an axisymmetric model. A model is defined using finite defonnation mixture elements which are coupled to a sub-system consisting of an ideal fluid.

2 DFSCRIPTION OF THE MODEL

The model (Fig. 1) is axisymmetric and contains a planar disc, representing thc articular cartilage layer, a spherical indentor and, in between, a toroid with a wedge-shaped cross-section, representing the meniscus (9). In the unloaded situation the upper end plane of the articular cartilage layer is in contact with the indentor only at tbe axis of symmetry. The lower end plane of the cartilage layer is fixed to a rigid foundation. The lower end plane of the meniscal ring rests fully on this layer while the upper surface of the meniscal ring matches the spherical indentor.

Frictionless sliding of the meniscal ring along the articular cartilage layer and the spherical indentor as weil as sliding of the articular cartilage layer along the spherical indentor is allowed. Both the articular cartil-age layer and the meniscal ring are defonnable mixtures of asolid and a fluid. The values for the stiffnesses of the soft layers and the bony components differ enor-mously and therefore the question arises as to whether it is necessary for the load distribution to take the deformability of the bony components into account. From studies performed by Schreppers (9) the assump-tion of rigidity of the bony parts in these models does not appear to affect the load distribution. For efficiency reasons this deformability is left out of the model in the present studies: the spherical indentor and the founda-tion are rigid and impervious. Tbe cavity enclosed by the cartilage layer, the meniscal ring and the rigid sphere is filled with an ideal fluid. For the time being, in the model this cavity is considerably larger than in the

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234 G J M A SCHREPPERS. A A H J SAUREN AND A HUSON I I I I

>1

~

Mi lure components

m

Fluid componenl 20mm 18 8

IE

I tE I I I 1Axisof symmelry

(

I

I I I

Fig. 1 Half of the cross-section of the axisymmetric model

where E and y represent tbe Young modulus and

Poisson ratio respectively. Furtbermore, the interaction between the solid and the fluid phase is described by Darcy's law:

wherekis the constant permeability coefficient and V,v.

and À represent the gradient operator, tbe solid phase velocity and the hydrostatic pressure of the fluid in the mixture respectively.

The numerical values for the material parameters are listed in Table1. Using tbe equilibrium condition

V .(t - À/)= 0 (3)

together witb equations (1) and (2), the deformations in the model and the field of hydrostatic pressureÀcan be calculated as functions of time. An axial load of 250 N real knee joint, in order to keep the model initially easier to describe. Fluid flow across the interfaces of the fluid-mixture and mixture-mixture contacts is allowed. The outflow at the outer radiu of tbe meniscal ring and the articular cartilage layer is free while the fluid pres-sure is assumed to be zero at these places. Itis as umed that no fluid layer is present between contacting sur-faces. Tbe material behaviour is characterized by a linear coupling between the components tij of the second Piola-Kirchhoff stress tensor and the com-ponentsei)of the Green-Lagrange strain tensor

accord-ing to Hooke's law:

is applied at time t

=

0 and tbe response of tbe model is

calculated for tbe next 25000 seconds using non-linear finite element techniques. Tbe articular cartilage layer and tbe meniscaJ ring are divided into four-node iso-parametric mixture element with lin ar interpolation functions for the displacements and hydrostatic pre

-ures.

From the performed analysis the load distribution in this model appears to depend strongly on time. In the reference model, presented by Schreppers et al. (I), the load was borne by both direct contact between the femoral and tibial components and contact through the meniscal ring. In the model considered here, a third sub-connection is consituted by the fluid enclosed in the cavity. Figure 2 shows the load distribution over these three sub-connections as a function of time. Initially just after the load has been applied, the flow in the

mixture components has not yet started and about 75 per cent of the total load is borne by the fluid sub-system, while the direct and meniscal contact bear 7 and 18 per cent respectively. As time proceeds fluid can flow out of the model at the circumferential outer surface of the meniscal ring and the articular cartilage layer and the pressure in the fluid sub-system decreases. Because of the outfiow, the compression of the total model increases and the fiuid sub-system carries a continuall_ decreasing part of the totalload. Finally after 10 000 seconds the hydrostatic pressure is approximatel zero all over the model and the load is borne only b direct contact (16 per cent) and the meniscal ring (84 per cent). Flow across the upper surface of the eartilage la er anI oecurs in the regions where it is contacting tbe Huid sub-system or the meniscal ring. The outflow c a es in the zone where it is contacting the irnpervious spherical indentor. (2) (1)

o

- y 2(1

+

Y) 0

o

I

o

- Y

v .

V. - V . (kVÀ) = 0

r

e,,]

r

1 er:

=.!.

0 e:: E - y elf - y

(4)

FOR E TRANSMISSION IN THE TIBIO·FEMORAL CONTA INCORPORATING FLUID AND MIXTURES 235

4 IERFACE CONDITIONS 3 PARAMETER STUDIES Table I Values for material parameters

Firstly, the interface conditions with respect to the fluid flow are considered here. Higginson and Norman (6) questioned the oecessity of taking ioto account the car-tilage components in the tibio-femoral joint as mixtures because of their very low permeability. Both the menis-cus and cartilage layer compri a dense network of

fibres at their outer surfaces and it might be speculated that this layer hampers the fluid flow across these sur-faces.

Five modeIs are coosidered which differ from each other with respect to the boundary conditions for the hydrostatic pressure. Two basic parameters are con-sidered. Firstly, there is the presence or absence of fluid in the joint cavity. The other concerns the fluid flow. A weil as models in which fluid flow across the outer sur-faces of components i allowed, model with sealed mixture components and models with solid componeots are also considered. In model A (Fig. 3) the meniscal ring and articular cartilage layers consist of solid material while the cavity is empty. Model Bis similar to model A, with the cavity being filled with an ideal f1uid. Models C aod 0 are similar to models A and B respect-ively, except th at the meniscal ring and articular cartil-age layers are mixtures which are sealed at their outer surfaces. Finally, model E is the model presented in the previous section, which is identical to model

o

apart from the fact that the outer surfaces are not sealed. The version of model C without sealed surfaces is Dot taken into account here. The reason for this is that it is con-sidered to be a non-realist ic model as DO pressure is built up in the cavity although the space i enclosed. The constitutive behaviour of all models is the same as that of the model described in the previou section on the understanding that models A and B have zero per-meability.

In Fig. 4a aDd b the undefonned and defonned element meshes for models A and Bare shown. Tbe axial compr ssion of both models is clearly visible and larger for model A than for model B. The cootact area between the rigid sphere and the tibial compoo ot is larger ~ r model A than for model B. Th se effect can

0.001 0.001 0.3 0.4 20.0 10.0 E MPa v k mm2fNs Meniscal ring Articular carlilage

In comparison with the models presented in the first step of the modelling process (1), in this work an addi-tional function for the meniscal ring with respect to the building up of pressure in the cavity is found. To gain more insight into this function, parameter studies have been performed with respect to the interface conditions and the load that is applied to the model respectively. In the following the consequences of the transition from the solid material models to the mixture material models are studied systematically.

The meniscal ring has a dual function in this model. Initially, just after the load is applied the enclosure of the fluid cavity by the meniscal ring resuIts in pressure being built up in the cavity, while finally, when fluid pressure approximates zero, the meniscal ring carries the larger part of the totalload.

~

Dir Icuntact

~ Meniscal ring

m

Fluid sub· ySlem 0.80

,9

0.60 '5 ~;;; :<; -g

.3

0.40 0.20 10 100 1000 T S

Fig. 2 Load distribution versus time

(5)

236 G J M A S HREPPERS. A A H J SA REN AND A H SON

A B

c

0

• Fluid 'ub-y(ems

~

Mixture component

D

Solid components

E

Fig. 3 Half of the cross-sections of models A to E. Impervious outer surfaces are indicated by bold lines

be deduced from the larger axial stiffness of model B as a result of the load-bearing capacity of the fluid in the cavity. The volume of the cavity remains unchanged in model B while it is reduced with increasing laad for model A.

Figure 5a and b represents the principal stresses related to the Cauchy stress tensor for models A and B in the integration points. Compressive stresses in the rz plane are indicated by dashed squares while the tensiIe

stresses are represented by squares built up from solid hnes. Compressive stresses in the circumferential direc-tion are shown by crosses made up from dashed lines whereas tensile stresses in the circumferential direction are shown by crosses made up from solid lines. In bath models the stresses in the meniscal ring are mainly directed circumferentially. In model B (cavity filled) at the inner side of the meniscal ring the larger campres-sive stresses in the rz plane are mainly radially directed, while for model A they are more axially oriented at this place.

Figures 6 and 7 show the axial compression and the fraction of the total load that is transmitted by the meniscal ring respectively versus time for all five modeis. Because fluid flow is absent in models A and B their behaviour is constant with time. From Fig. 6 the Part H: Journalof Engineering in Medicine

axial tiffness for the models with a fluid-filled cavity (B,

o

and E) appears to be larger than for the models with an empty cavity (A and C), because the f1uid bears part of the laad. For the farmer models the loading of the meniscal ring is smaller. Initially, the curves of model E are very close to the curves of model 0, but as time proceeds more and more fluid is squeezed out and at the end the hydrostatic pressure equals zero everywhere, yielding the same conditions as in model A.

5 LOADINGS

In the parameter studies described by Schreppers et al.

(1), the joint load ranged from 0 to 1000 .The laad distribution appeared to depend on this laad. In tb is work all analyses were done for a step change of the laad from 0 to 250 N. The influence of the magnitude of the step is still unknown. Two possible effe ts ar pr -posed beforehand. The first effect concern an incr a -ing total compression of the model for larger loads. Thus more fluid has to he squeezed ut of the model, resulting in a larger time to elapse until na further changes occur a a re uit of the tep change of tb I ad. In the following this period is called tbe rel ation periad. The other effi ct implies an increase of tbe initial

(6)

FORCE TRANS MISS ION IN THE TIBIO-FEMORAL ONTA INCORPORATING FLUID AND MIXTURES ... ... _..••••••••.•••• 1:;::::::::::...•... (a) ModelA Undeformed geometry - - - - Deformed geometry 237 . . . n. . . : : : : : : : : : : : : : : : : : · (b)Model B

Fig. 4 Finite element meshe

pressure in the cavity for larger loads. The resulting fluid velocities will increase so that the relaxation period is expected to be shorter.

Taking these contradictory effects into account, it cannot be predicted whether the relaxation period will increase or decrease for larger loads. Therefore, the effect of the magnitude of the load is investigated by performing an additional analysis with model E for a loading step of 500 N.

In Fig. 8 the pressure in the cavity is given as a func-tion of time. Figure 9 shows the fracfunc-tion of the load transmitted by the meniscal ring versus time for both loadings. From both figures it can be seen that the relaxation period for the loading of 500 N is approx-imately half of the relaxation period for the loading of 250 N. After relaxation the fraction q is smaller for the

higher loadings than for the lower loadings which is

consistent with the results of the olid models d scribed by Schrepperset al. (I).

6 SURFACE GEOMETRY

Both for the reference model and for the reduced model, presented by Schreppers et al. (I) and Schreppers (9) respectively, the curvature of the contact surface(s) appeared to have only minor effects on the load dis-tribution in the models when the contact surfaces ar covered by soft layers. This is an important character-istic for the mechanical behaviour of the tibio-femoral contact complex. Therefore we want to know whether it also applies when mixture materials are used and the cavity is filled with fluid. Starting from m del E tw models with different curvatures of the rigid ph rical indentor are defined. These model are indi ated by F

(7)

238 G J M A SCHREPPERS, A A H J A REN AND A HUSON 0 [] [] 0

m

[J [] [] [] lIJ [] [J [] lIJ (J []

lIJ

(J El []

lIJ

(J [] El []

lIJ

[) El [] :1

t

[] ~

·

!

.; 0'

.,

,

"

"

o

o

[J

0

o

0

0

o

0

o []

[] 0

[]

o

0

·

··

·

'I "

,

" "

o

G

DO

GO

DO

DO

GO

Ii " lf

·

= ,

Principal stresses I'n .mtegrat'IOn points S. ee text for furth er explanation Fig. 5 A

/ '

,

..2...- - - -

. / /

.

...

...:.:.

...

,

-

-:."""-

-

_

...

E

--.;~--~

•••••• '" - - - -

-

-

-

--c

---0.10 la 0.00

~

...

~

...

~

•.., 0.1 Fig.6 Axial c o m '

S-

T ParI H J 250 N pressIon of th

: oumal arE

,._",g ;,

0 • att = 0 e models A

M

'"

.

,,;~

' 10E ,

"'0'

hm,. f

or a load step of

(8)

FORCE TRANSMISSION IN THE TIBIO·FEMORAl CONTA IN ORPORATING Fl 10 AND MIXTURES

UlO,---,

239 A

...

0.60

"

/

I

/

I

/

,

O.~

/ B / ' •.••.••. ·---7--~-~---

..,.

.' 0"0 D ...•...•~~..•.... .- I'--~''';-' -E o.w~--

~~---c

0.00l -...L...L...L...~'!-:-...~~... 0.1 JO 100 1000 10000 T

Fig. 7 Part of the load borne by the meniscal ring for lhe models A to E versus time for a load step of 250 N atT = 0S

and G and their radius is 20 and 60 mm respectively. Thus, in model G the wedge of the meniscal ring and the cavity are smaller while for model F they are larger. To both models a step change of the load from 0 to 250

N is applied. After the load is applied, some time will elapse until only the solid materials bear the load. Then the fraction of the load borne by the meniscal ring can be expected to be approximately the same for the models E,F and G. Whether this will be the case for the initial response when the Huid bear part of the load, i not clear beforehand.

In Fig. 10the parts of the load borne by the meni,cal ring versus time for the models E, F and Gare shown as results of the performed numerical analyses. From this figure the initial loadings of the meniscal ring appear to differ for the three models up to 10per cent.

The relaxation period for the model G, witb tbe larger radius, is smaller, while tbe relaxation period for tbe model F witb tbe smaller spherical radius is larger in

comparison witb modelE.

7 CONCLUSIONS AND D1SCUSSIO

From these analyses the following conclusions can he drawn:

I. Th application of the mixture theory on thc model of the libio-femoral contact complex only lead to significant effect if the outer surfaccs of the com-ponents are not sealed.

2. Irthese surfaccs are not sealed the load distribution depends on the load history in such a way that the

2 . 0 0 , - - - , 250 N 1.60 1.20 0.80 0.40 500

-"

"

"

"

\ \ \ \ \ \ \ \ \ \ \ \ \

,

,

,

1000 10000 ..!... s

Fig.8 Pressureinthe cavity versus time for model E under load steps of 250 N and 500 N alT= 0S

(9)

240 G J M A SCHREPPERS. A A H J SAURE AND A HUSON

I.()(),.---.

0.80

.-"

0.60 /

"

I

...

I I 0.40 II .-" , " , " , SooN 0.20

---,

2S0N 0.00 0.1 10 100 1000 10000 T s

Fig. 9 Part of the load borne by the meniscal ring versus time for model E under load steps of 250 N and 500 N atT= 0S

time that elapses until no further changes resulting from the applied load occur is rather large for the chosen parameter values.

3. The fluid-filled cavity carries a large part (up to 75 per cent) of the total load applied on the model and this fraction decrea es to zero when the fluid is being squeezed out of the model.

4. The length of the relaxation period appears to depend on the magnitude f the step change of the load in such a way that it decreases for incr a ing load steps.

5. Finally in the presented modeis, variation of the cur-vature of the spherical indentor appears to have sig-nificant elfects on the loading of the meniscal ring, just after the step changes of load are applied. Thes effects disappear when the fluid is leaving the modeis.

Because of the st rong abstraction of the actual modeis, care must be taken when translating these char-acteristics to the real knee joint. The simplifications in the models pertain to the two-dimensional nature of the geometry, the chosen values of material properties, the fluid-solid interaction and the absence of meniscal attachments to tbe tibial plateau. Tbe load alues and loading rates were primarily chosen to demonstrate tbe capabilities of the present modeis. Any attempt to relate them10 me characteristic of identified human move-ment, for example walking or running, would be highly speculative. However. it is supposed that the performed analyses properly describe the effects of basic para-meters in the tibio-femoral contact complex.

To achieve a more realistic description, two main aspects should be considered in fulure steps of the

mod-\ . 0 0 , - - - . . . , 0.60 0.80

"

.-/

"

I I I I I I I I I I

"

"

"

"

--

-G 0.40 0. 20F- E - " . F __

-

----O. 00 ' - - ' -...""--...---'-..._'-"---'-...'---'-...~... 0.1 10 100 1000 10000 T s

Fig.10 Part of the load borne by the meniscal ring versus time for the models E F and Gunder a load step of 250Nat t = 0

(10)

FOR E TRANSMISSION I THE TIBIO·FEMOR L INCORPOR TlNG FL ID A MI TRES 241

elling process of the force transmission in the tibio-femoral contact complex. The first aspect concerns the interaction of ftuid and matrix in the hydrated tissues. Because electrolytes are dissolved in the ftuid phase of the articular cartilage and the proteoglycan aggregates of the solid matrix are ionized, local concentrations of e1ectrical loadings are created by forcing the ftuid to move out of the model. Therefore, the time that elapses until the load distribution is stationary will probably be smaller in reality.

The other aspect concerns the possibilities of ftuid outftow from the cavity when load is applied. In reality, ftuid may escape more easily because the meniscus will not fully enclose the cavity, as is the case in the model. Although quantitative data about the resistance against the ftuid outftow are not yet available, these possibilities should be the subject of research in further steps of the modelling process. The authors of the present paper are convineed that this can only be done in an appropriate way if these ftows are considered as unknown quantities in the model and are implicitly calculated for the applied loadings. Therefore, the viscosity of the ftuid has to be taken into account.

Finally, a possible way to arrive at a complete model of the tibio-femoral contact complex needs to be described. Firstly, attention has to be focused on the interaction of mixture components with a Newtonian ftuid. Then squeeze film effects in both the direct and indirect contact area have to be investigated. This aspect could cause the relaxation period of the model to decrease considerably. Next, the application of more complex loading patterns, such as harmonie axial loads,

©IMechE 1991

can be performed. When a thr e-dimensional formula-tion is applied, bending of the knee mayalso be simu-lated. Finally, more detailed descriptions of the material and geometry of the real joint can be implemented. [n thi stage, for example physical non-linear material behaviour and effect re uIting from moving electrical loadings in the articular cartilage can be taken into account.

REFERE CES

I Scbreppers, G. J. M. A., Sauren, A. A. H. J. and HUSOD, A. A

numerical model of the load transmission in the tibio-femoraJ

contact area.J.Engng Medieine,1990,204,53-59.

2 Mow, V.

c.,

Holmes, M. H. and ui, W.M fJuid transport and

mechanical properties of articular cartilage: a review. J.

Bio-meehanies,1984, 17, 337-394.

3 Spilker, R. L., Sub, J.-K. and Mow, V.C.Effects of friction on the

unconfined compressive response of articular cartilage: a finite

element analysis.J.Biomeeh. Engng,1990, 112, 138-146.

4 Spilker, R. L. and Sub, J.-K. Formulation and evaluation of a finite

element model for the biph ic model of hydrated soft tis ues.

Compul. Slruets, 1990,35,425-439.

5 Dowson, D., Unswortb, A. and Wrigbt, V. Analysis of boosted

lub-rication in humanjoin .J.Meeh. Engng ei., 1970, 12,3 .

6 HigginsOD, G. R. and NormaD, R. The lubrication of porous elastic

solids with reference to the functioning of human joints.J. Meeh.

Engng Sci.,1984,16,250-257.

7 Walker, P. S., Dowson, D., Longfield, M. D. and Wrigb V.

Boosted lubrication in synovial joints by Duid entrapment and

enrichmenl.Ann. Rheum. Dis.,1968,27,512-520.

8 Droogeodijk, L. On the lubrication of synoviaJ joints. PhD thesis,

Twente niversity, Th Netherlands, 1984.

9 Schreppers, G. J. M. A. For transmi ion in the tibio-femoral

contact complex. PhD thesis, Eindhoven niv rsity of Technology,

Tbe etherlands, 1991.

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