J. Biomechanics Vol. 25, No. 1, pp. 81-90, 1992 Printedin Great Britain

0

ml-9290/92 ss.oo+.Cm 1991 PergamonPress plc

A THEORETICAL MODEL OF THE KNEE AND ACL: THEORY AND EXPERIMENTAL VERIFICATION DEB

A. LOCH, ZONGPING Luo, JACK L. LEWIS* and NATHANIEL J. STEWART

Biomechanics Laboratory, Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 55455, U.S.A. three-dimensional mathematical model of the human knee joint was developed to examine the role of single ligaments, such as an anterior cruciate ligament (ACL) graft in ACL reconstruction, on joint motion and tissue forces. The model is linear and valid for small motions about an equilibrium position. The knee joint is modeled as two rigid bodies (the femur and the tibia) interconnected by deformable structures, including the ACL or ACL graft, the cartilage layer, and the remainder of the knee tissues (modeled as a single element). The model was demonstrated for the equilibrium condition of the knee in extension with an anterior tibia1 force, causing anterior drawer and hyperextension. The knee stiffness matrix for this condition was measured for a human right knee in vitro. predicted model response was compared with experimental observations. Qualitative agreement was found between model and experiment, validating the model and its assumptions. T’hemodel was then used to predict the change in graft and cartilage forces and joint motion of the knee due to an increment of load in the normal joint both after ACL removal and with various altered states simulating ACL reconstrucfions. Results illustrate the interdependence between loads in the ACL graft, other knee structures, and contact force. Stiffer grafts and smaller maximum unloaded length of the ligament lead to higher graft and contact forces. Changes in cartilage stiffness alter load sharing between ACL graft and other joint tissues. Abstract-A

condition because of the inherent biological and experimental variability and the interdependence of The anterior cruciate ligament (ACL) is one of the variables. Experimentally measurements of such most frequently injured ligaments in the human knee quantities as cartilage contact stresses and other soft joint. The consequence of injury is often knee instabiltissue loads are difficult or impossible to perform. An ity and limitation of function, and surgical reconstrucalternative approach to experimental testing is to tion is sometimes necessary. Reconstructions have stimulate the reconstructive process with a mathematbeen perfozmed using a wide variety of autogenous ical model and then vary the parameters in the model graft materials, both with and without artificial aug- to assess their effect. mentation (Butler et al., 1981). These procedures, Various theoretical models of the knee have been however, sometimes lack long-term stability and func- developed. Grood and Hefzy (1983) have reviewed tion (Sidles et al., 1988). For example, Sachs et al. these models; only the immediately relevant models (1990) reported 12% effusion and 20% flexion con- will be mentioned here. Piziali and Rastegar (1977) tracture in 180 knees one year after knee ligament formed a single general nonlinear stiffness matrix for surgery. It is usually not clear what causes poor the entire knee joint. The amount of physical property functional outcome, but the process may involve data needed to describe the knee was minimized by an initially overtight graft (Yoshiya et al., 1987; using only one stiffness element. However, the model O’Donoghue et al., 1966), too early graft loading was limited in its ability to isolate individual struc(Noyes et al., 1983), poor graft quality, or an improtures and allow their physical parameters to be maperly placed graft (Engebretsen et al., 1989; Hefty and nipulated. The nonlinearity of the stiffness matrix also Grood, 1986; Sidles et al., 1988). The state of the joint limited its utility. Models reported by Wismans et al. in an unsuccessful reconstruction is also often unclear. (1980), Grood and Hefzy (1982) and Andriacchi et al. There is usually pain and crepitus with or without (1983) identify and model specific structural elements instability; whether this is due to cartilage and menisand allow the physical parameters of each structure to cus degeneration, stretching of other soft tissues, or be manipulated individually. However, this requires bony changes is unknown. The causative factors in the individual structures to be defined and anatomical poor outcome have been studied experimentally in and mechanical characteristics measured, an often human cadaver joints and animals, but the tests are difficult or impossible task. often difficult to interpret and relate to the human The approach adopted in this paper is a combination of these previous models, in that the knee is modeled with a generalized stiffness matrix to account Receivedin pnalfonn 24 May 1991. for structures other than the ACL or the contact *Address correspondence to: Jack L. Lewis, Biomechanics surfaces. Individual elements are used to represent the Laboratory, Department of Orthopaedic Surgery, University of Minnesota, Box 289 UMHC, 420 Delaware St. S.E., ACL and cartilage layer. This avoids the necessity of measuring anatomical and material property data for Minneapolis, MN 55455, U.S.A. INTRODUCTION

81

82

D. A. LOCH et al.

individual structures not of particular interest, while allowing manipulation of ACL and cartilage parameters. This also allows direct measurement of model parameters via the generalized stiffness matrix. The model is intended to be as simple as possible while still including three-dimensional effects and demonstrating the realistic features of knee mechanics. The significant features of the model are: (1) A single stiffness element, measured experimentally, is used to model the entire knee joint with the exception of the ACL graft and articular cartilage. (2) A linear spring element is used to model the ACL graft. (3) The bone surfaces are approximated by regular surfaces (spheres for the femoral condyles and a plane for the tibia1 plateau). (4) A deformable cartilage spring is used to represent the articular cartilage layer between each femoral condyle and the tibia1 surface. (5) The model predicts incremental changes in ligament and cartilage contact forces and the resulting incremental rigid body motion about a (loaded) equilibrium position.

MODEL

FORMULATION

In this model, the mechanics of the human knee joint is reduced to a system of two rigid bodies representing the femur and the tibia, and deformable structural elements to simulate the ACL, the cartilage, and the rest of the knee tissues connecting the rigid bodies. For large displacements the human knee joint shows a nonlinear relationship between the load and the displacement (Piziali and Rastegar, 1977), which is difficult to model mathematically. However, if the displacements are small, the knee joint structure has linear characteristics. In this investigation we assume small displacements about an equilibrium position so that the linear elastic theory can be applied. Thus, the model presented here applies to an increment of load applied to an equilibrium position. Figure 1 shows the simplified model of the knee joint. The linear knee stiffness element includes the effects due to the menisci, joint capsule, soft tissues, and all the ligaments except the ACL. The ACL is modeled as a linear spring. The cartilage layer is modeled as a lumped linear spring. The femoral condyles are approximated by two fixed rigid spheres, each maintaining a contact point on the cartilage. The tibia1 plateau is represented by a plane rigidly attached to the tibia. The six-degree-of-freedom position of the tibia is described relative to that of the femur. Two Cartesian coordinate systems are selected as follows: one is fixed to the moving rigid body (X, Y, 2) and the other is attached to the fixed rigid body (x, y, z). Both coordinate systems coincide at the initial equilibrium position, that is, both coordinate systerns have the same origin and their axes are collinear

7 I i

: ACL

Graft

: t !

Femur

:

\

i

‘_ Knee Stiffness Element

Cal

Fig. 1. Diagram of the idealized knee joint used to form the theoretical model, an anterior view of a right knee. The idealized surfaces are represented by solid black lines, while the bones are drawn in with dashed lines to give anatomical perspective.

at the equilibrium position. From this definition, points on the tibia are fixed with respect to the XYZ coordinate system and the position of the tibia is described relative to the xyz coordinate system. For a right knee, the x, y, z axes point to the anterior, medial and superior directions, respectively. The initial common origin of these coordinate systems is inside the tibia (the actual origin for this model was determined experimentally as described in a later section), The six degrees of freedom of the moving rigid body are expressed by the origin displacements u,, uy, u, in the xyz coordinate system and the rotations of the XYZ coordinate system, &, O,, 8,, relative to the xyz coordinate system. The equilibrium equations for the tibia are: {Cf}={f}-I~ncc}-{f*cL}-{1;.ycr}=O,

(I)

where {f } are the applied forces, {_I&} are the forces generated by the joint structures other than ACL and cartilage, {fAn ) are the forces generated by the ACL, and {f;,,yer} are the contact forces acting through the cartilage spring layer. It is assumed that

where

1x.0 >= CXtnce1 {u>,

(2)

uk!LI= CGZLIb>,

(3)

I&c, I= C&W,1 W,

(4)

A

is the displacement

83

theoretical model of the knee and ACL

vector. Equation (1) can thus be

written as (5) [Klmct] is the stiffness matrix of the knee joint without the ACL and cartilage stiffness, and is determined experimentally (see details in the next section). [KacL] is the stiffness matrix of the ACL, and [K1,yer] is the stiffness matrix of the cartilage layer. The formulation of [KacL] and [K1.ycr] is shown in the Appendix. (f} contains the components of the applied force:

ness was tested first to serve as a check on model predictions, and the ACL was then cut and stiffness tests repeated, to compute [KLn.. J. From the definition in equation (5), the relationship between the external loads {f }and displacements {u} of the tibia1 origin is

(f )=CK,eintl (IO,

where [Kjoint], the total knee stiffness matrix, can be formulated with or without the contribution of the ACL. When the ACL is not present, [Kjoint] is written as [K$$cL],where CK%'"l

= C&me I+ CK.,erl.

When the ACL is present

ffl=

(8)

[Kjoint] is written as

CK$%l,where ’

(6)

wheref,,f,, fz are the components of an applied force, m,, my, m, are the components of the applied moment and X,, Y,,2, are the position components of the applied force in the XYZ coordinate system. It is noted that the position components xI, y/,z, are replaced by Xr, Y/, Z, after neglecting the higherorder terms (u,, u,,, uz, 6,., B,, 6,) in equation (A3). DETERMINATION

(7)

OF THE STIFFNESS MATRICES

CK$%l= C&J + C&ye,1 + CKCLI.

(9)

If the force is applied as six components individually, the six independent 6 x 1 force columns can be written as a force matrix: fi

0

0

0

0

0

0 f* 0 Cfl= r 0 0 f;

0 0

0 0

0 0

0 0 Om4 0 0 0 0 0 0 0 0

0 0 m5 0 0 m6

From equations (7) and (lo), the stiffness matrix [Kjoint] is obtained as [Kjoint] = [f] [u] - I, or

To apply the model, specific values of the three stiffness matrices must be provided. [K,& depends ‘fl 0 0 0 0 0 on the attachment sites, and length and stiffness of the Of*0 0 0 0 ACL as shown in equation (AS). The length and 0 0 f3 0 0 0 attachment sites of the ACL on the femur and the CKjoint 3= tibia were estimated from the work of Odensten and 0 0 Om, 0 0 Gillquist (1985). The stiffness value of the ACL was 0 0 0 0 m, 0 assumed to be 200 N/mm, taken from the linear re10 0 0 0 0 m6 gion of the load-elongation curve determined by other investigators (Claes and Schmid, 1985). [Kuyer] was determined from equation (AlO), with the stiffness 11x2 Kc1 ux3 %4 %5 %6 of the cartilage layer, k,,assumed to be 25 N/mm UY4 UY5 %l UY2 UY3 %6 based on the elastic modulus of 5 N/mm’ (Hori and f&l %2 uz3 %4 Kz5 %6 X Mockros, 1976) with an approximated cartilage thick0 xl &2 &3 0x4 0x5 8x6 ’ ness of 2 mm and a contact area of 10 mm*. e eY2 %3 eY4 eYS eY6 Yl The stiffness matrix for the remainder of the knee i e 21 812 ez3 structures, [KLncc], was determined by experimenter4 ezS ez6 ally measuring the total knee stiffness matrix for the (11) case with no ACL, and then subtracting the cartilage where each column of the 6 x 6 matrix [u] contains stiffness. This model is only valid for small motion around the equilibrium position for which [Kknsa] is the six displacements of the tibial coordinate frame measured, and different knee positions will result in that resulted from the application of force shown in different stiffness matrices. In this paper, the focus is the corresponding column of matrix [f]. Since we on the establishment of the stiffness matrix from the are assuming linearity between load and displacement parameters, the stiffness matrix [KLno.] must equilibrium state of 89 N anterior drawer applied on be symmetric. As shown in the Appendix, [Kkn] and the tibia 11 mm below the tibia1 surface, with the knee are symmetric also; thus, [&,t,,J is symmetin extension. This particular position was chosen for [f&] the study of ACL function in extension and the role of ric. Because of symmetry, there are at most 21 lndeAn experiment is used the graft in flexion contractures. The intact joint stiff- pendent constants in [K&j.

r

84

D. A. LOCHet al.

to measure these 21 constants. In principle, six independent displacements produced by each of the six independent loads provide sufficient information to solve for the 21 independent constants in the stiffness matrix. In order to estimate [KLocc], an experiment was performed using a knee-loading system built previously in the Biomechanics Laboratory, University of Minnesota (Lewis et al., 1989). One human cadaver right knee was used. The tendons, capsular structures, quadriceps muscle, and patella were all left intact. Both the femur and tibia were set in cylinders of methacrylate bone cement, which were then fixed within aluminum holders in the machine. The origin of the xyz coordinate system, or the origin of the X YZ coordinate system at initial equilibrium position, was defined as the point along the centroid of the methacrylate cement cylinder on the moving tibia 11 mm below the knee joint surface. A 420 mm long aluminum rod along the centroid of the tibia1 cylinder, and an aluminum ring 95 mm in diameter with four paired screws, orthogonal to the rod, were also used to assist in loading the tibia to achieve six independent forces and moments. The loads were applied to the knee at terminal extension. Extension was chosen because the ACL is more highly loaded here than at other flexion angles and is a position of clinical interest. However, because there is little motion at this flexion angle when the knee is loaded, accurate experimental measurement of incremental motions is difficult. A schematic diagram of the experiment is shown in Fig. 2. The motion of the origin in the tibia relative to the fixed femur was measured by a six-degree-of-freedom instrumented spatial linkage (ISL), also designed in

I

Methacrylate

k \ Loading Ring

Fig. 2. Diagram of the experimental apparatus used to obtain the mechanical properties for the theoretical model. The same apparatus used to verify several of the predictions made with the theoretical model. An instrumented spatial linkage (ISL) was used to measure the motion of the tibia relative to the femur.

the Biomechanics Laboratory (Kirstukas, 1989). The ISL had a nominal sensitivity of 0.5 mm or 0.5”. One end of the ISL was fixed to the femur holder and the other end was connected to the holder moving with the tibia. The voltage outputs from the ISL were recorded and stored in a computer data acquisition system, and the voltage data transformed into three origin translations and three rotations. The knee machine and data reduction procedures are described by Lewis et al. (1989). The initial equilibrium state was produced by hanging an 89 N weight on a cable-pulley assembly, which produced a force in the +X direction at the tibia1 origin. At this equilibrium position, six loads, fx&.h, m,, my and m, [relate tofirf2,f3, m4, ms and m6 in equation (lo), respectively], were superposed one at a time. Moments were created by applying a force couple about the appropriate axes. Forces and moments were applied either on the ring forfx,f,,fz and m, or on the bar for m, and m,. The loads were sensed by a load cell and outputs recorded by the computer data acquisition system. The test was first performed on the intact knee at extension. After the necessary data were obtained, the ACL was cut at its tibia1 insertion through a parapatellar incision and the test repeated. In addition to the small probable nonlinearities in the knee specimen, there were inevitable experimental errors in the force and motion measurements; in particular: (1) The exact orientation and position of Cartesian coordinate systems were difficult to locate on the cadaver knee. This led to some error in applying six loads along corresponding directions. To reduce this error, each load state was repeated three times and the average position of the tibia was taken from the ISL. (2) The viscoelastic properties of the soft tissues have a significant effect on the measurement. To reduce this effect, the loads were applied 20-30 s before the position of the tibia was determined from the ISL. (3) The ISL has a sensitivity of 0.5 mm or 0.5”. Thus, small displacements are inaccurate. The offdiagonal components of [u] were assumed to be zero if they were less than 0.5 mm or 0.5”. Because of the symmetry of the stiffness matrix, for a set of prescribed applied forces, the resulting displacements are constrained. Due to small nonlinearities and experimental error, this was not strictly the case; therefore, a stiffness matrix computed from equation (11) and experimental data would not be exactly symmetric. Rather than average the nonsymmetric stiffness matrix to make it symmetric, the measured displacements were adjusted to result in a symmetric stiffness matrix. To ensure that the symmetry of the stiffness matrix is satisfied, the off-diagonal components of [u] were adjusted as follows. The inverse of the stiffness matrix was formulated to be symmetric by

85

A theoretical model of the knee and ACL

The units used are Newtons, mm and degrees. Moments were higher with the ACL present to generate measurable rotations. With the experimentally determined loads and displacements shown in equations (15) and (16), applying the symmetry requirement to [uJ from equation (14), and subtracting [KIPycr] from [Kzzn], the stiffness matrix [KLnee] was determined to be

where [Kj,,,,,] is the stiffness matrix obtained from the direct experimental data and equation (7), and is the symmetric stiffness matrix obtained C&&m after manipulating [u]. By suitable manipulation of equation (7), and use of [f] = [f]‘,

T.Kjointl-l=CUICfl-l~ (CKjolntl-l)T=Cfl-lCUIT~ Substitution

(13)

of these into equation (12) gives

CGnec

l~lndjustedCfl~l~lfCUICfl~l +Cfl-’ CUlTI, =t{C~lCfl-1C~lTCf3~~ (14) [Uladjustcd where is the displacement matrix after manipulation. The components of and [u] differed by a small amount. For the loads applied, the measured vj3 was less than 0.5 mm, the resolution of the ISL. If this value was assumed to be zero, the cartilage effect would be eliminated and the [u] matrix would be singular. Thus, a nonzero u33 had to be assumed. It was found that altering u33 from 0.3 mm to 0.8 mm in the theoretical model caused little change in the load sharing between the ACL and [KLnes]; the final u33 was arbitrarily chosen to be 0.5 mm. The applied loads and the corresponding measured displacements for the two test series are listed below: For the knee without the ACL

23.6

0

0

446

460

0

33.2

0

0

0

0

0

0

101

0

0

0

446

0

0

391,000

46,900 37,500

460

0

0

46,900

48,400 -1840

0

0

37,500

-1840

[u]mdjusted

[U]adjustcd

CFl=

55.0

0

0

0

0

0

0

52.8

0

0

0

0

0

0

50.3

0

0

0

0 0

0 0

0 0

3090 0

0 2760

0 0

0

0

0

0

0

2800

2.860

CUJ=

0

0

0

0

1.588

0

0

0

0

0.500

0

0

0

0 0

0 0.573 -0.573

0

0

-0.539

-1.558 0

I=

-17.5

0 0

0 -0.516

0

-0.487

4.526

(16)

0.630 4.291

Fot the knee with the ACL

IF]=i‘j

[V]=

‘;

5;6

3;lo

3;70

41,900

The units are Newtons, mm and radians. [KIayer] was obtained from equation (AIO). The approximate contact point locations measured from the cadaver knee were (0,26, 11) and (0, -26,ll) in the XYZ coordinate system. It is noted that the component kss between 0, and my, the flexion-extension stiffness, is not significantly lower than the other rotation diagonal

0

0.630

’

(19)

(15)

- 1.364

- 17.5

(17)

..‘v

i -1.891 2.387 0

0.630 1.537 00

0

0

-2.097 1.418 0

0.934 0

;

;

““0”

O 0.510

:

;

3.994

(18)

86

D. A. LOCH et al.

components, contrary to intuition, since this is approximately parallel to the flexion-extension axis. The reason for this is that in this particular equilibrium position, 89 N anterior tibia1 load, the knee was resisting extension and was much stiffer in extension than in its unloaded state. The theoretical estimation of the knee stiffness matrix with the ACL is the summation of the stiffness matrices, CKACLI, CKayerl, and C&nccl.[KACL] is derived from equations (A5) and (A6). The approximate attachment sites (mm) of the ACL in the initial XYZ coordinate system measured from the cadaver knee was (- 10, -6,41) on the femur and (16, 11.5, 11) on the tibia. By using equation (9), the resulting joint stiffness with ACL present, [Kg;], is

[X$:$1 =

95.4

48.4

- 82.9

- 1040

2580

- 70.0

48.4

65.8

- 55.8

-999

1420

-35.3

- 55.8

196

1710

- 2440

60.6

1710

422,000

3160

38,600

- 2440

3160

111,000

-3380

60.6

38,600

-3380

41,900 1

--p;2

i

2580 - 70.0

-999 1420 -35.3

Equation (20) is the proposed stiffness matrix of the experimental knee joint. For a given set of applied loads, {f), the model will predict motion of the tibia relative to the femur, plus incremental change in forces in the ACL and at the contact points. Two contact points are assumed. Since load changes are relative to an initial loaded state, positive, or tensile, contact forces can occur since they represent reduction of the contact force. At some point, contact would be lost, in which case a solution for a single contact could be used. However, this would require some knowledge or assumption about the magnitude of the initial contact force, which was not done here. Therefore, two contact forces are assumed to exist for all the cases in this paper. To assess the accuracy of model predictions, comparisons of predictions and experimental data were performed. A first check was a comparison of displacements measured experimentally with the ACL intact, equation (18), with displacements predicted, using [K$$i] = [K&e] + [I&r], from equation (20). The predicted displacements using [K$%], equation (20), due to loads in equation (17), which should be the same as those in equation (18), are 2.028 -0.444

-0.435 1.379

Cul~%icted = i 0.251 0.229

0.133 0.115 - 0.286 - 1.089

- 2.063 - 0.229 Note that, differences differences ical model

In general, displacements predicted using [K$y,Q agreed well with the experimentally measured displacements. Using the criterion of twice the nominal sensitivity of the ISL, 1 mm or l”, there were four out of a possible 36 significant differences between components in equations (18) and (21), uZB., I+, ugyr user. The theoretical model was less stiff in flexion-extension and axial torsion than the actual joint, although the direction of changes in use,, and use, were correct. Since the theoretical model is simplified from the real knee joint and experimental errors exist as pointed out before, the difference between the predicted displacements and the experimental displacements is reasonable.

0.232 0.125 1 0.43 -0.057 0.172 0.057

0.251 0.136 - 0.077 0.688 -0.573 - 0.630

other than experimental error, the only between equations (18) and (21) are due to between the actual ACL and the theoretof the ACL.

’

(20)

A second check of model validity was via the anterior displacement and axial rotation. For applied incremental anterior tibia1 force, it would be expected that u, would increase and would increase further with cutting the ACL. Comparing model predictions with the ACL to those without the ACL (Table 1) shows u, increasing to 2.038 mm with the ACL and to 2.862 mm without the ACL, consistent with our expectations. For an axial tibia1 rotation moment, it would be expected that 13.would be positive and tlz would increase with cutting the ACL. Model predictions show 6, to be 4.0” with the ACL and 6.9” after ACL cutting (after scaling for the higher m,), supporting the validity of model predictions. Although the stiffness matrices were determined from testing a single knee and should be confirmed by more systematic testing, the responses predicted are intuitively reasonable. To demonstrate the potential usefulness of the model, the stiffness values obtained were used to examine the influence of some surgical variables in ACL reconstruction.

- 2.289 -0.311 0.176 -0.516 5.329 0.630

-0.293 -0.159 0.090 i ’ -0.745 0.859 6.933

(21)

Assessment of the ACL and ACL graf?

Several additional cases were examined with the model. Case 1 was the normal joint with intact ACL,

87

A theoreticai model of the knee and ACL Table 1. Model predictions of change in model variables for five hypothetical knee states. Case 1 has an additional 55 N of anterior drawer placed on the intact knee at its equilibrium position. Case 2 differs from Case 1 in that the ACL is missing. 0.se 3 differsfrom Case 1 in that the ACL (or anatomically placed graft) is 4 times as stiff. Case 4 difiii from Case 1 in that the cartilaw is 4 times as stiff. Case 5 differs from Case 1 in that I!,,,,the resting length of the ACL graft, is lengthened such that the ACL does not carry load until the anterior drawer is increased by 27.5 N 1

Cases

u, (mm) Tibia motion

& (“) 6, (0)

2.038 0.201 -2.201

Point 1 Point 2

-8.6 -4.0

Graft force (N) Cartilage force (N)

36.7

3

2 2.862 0 0 0 0 0

1.935 0.226 - 2.226 41.3 -9.7 -4.5

4 1.941 0.172 -0.172 41.0 -19.1 -3.6

5 2.450 0.101 -0.101 18.4 -4.3 - 2.0

Graft locations (mm): (16.0,11.5,11.0) on tibia; (-10.0, -6.0,Ill.O) on femur. Cartilage contact locations on tibia (mm)-Point I: (0,26,11); Point 2: (0, - 2611).

to an incremental anterior tibial force of 55 N, in addition to the equilibrium force of 89 N with the knee in extension. Case 2 was the same load state with the ACL eliminated. Case 3 was the same as Case 1, except that the ACL (or ACL graft) was four times as stiff, k=800 N/mm. This could represent a heavier biological graft such as a large segment of patellar tendon, or a prosthesis such as a GorTex ligament (Chen, 1980). Case 4 was the same as Case 1 except that the cartilage layer was four times as stiff. This could occur with pathological changes in, or thinning of, cartilage. Finally, Case 5 represents an initially slack ACL graft. Due to the initial slackness, the load-displacement relation is bilinear: the ACL does not carry load until the anterior drawer force reached 27.5 N above the equilibrium loading. As mentioned, comparing the motions of Cases 1 and 2 demonstrates that the model predicts what is intuitively reasonable. Displacements are generally in the expected direction; displacements increase with removal of the ACL. This comparison also demonstrates the interaction between contact force and ACL force. Higher ACL forces correspond to higher contact forces. A comparison of Cases 1 and 3 shows that increasing the graft stiffness alters joint motion and increases graft and contact forces. A comparison of Cases 1 and 4 shows that increasing cartilage layer stiffness increases contact force and graft force magnitudes. This illustrates the high interdependence of the joint structural elements. Finally, a comparison of Cases 1 and 5 shows that when the maximum unloaded length of the ACL Lo is increased by 0.7 mm (that required to avoid load below 27.5 N additional anterior drawer) a change in graft force of 50% results. subjected

DIXXJSSION The purpose of this work was to develop a threedimensional structural model of the knee joint that was simple but would allow study of particular structural elements, such as the ACL, by varying element

parameters. The model described here appears to achieve this purpose. The model is useful for demonstrating how changes in geometrical or material properties of one joint structural element intluence joint motion at an equilibrium point, and influence forces and motions of other joint elements. There are limitations and assumptions in the model, however, so it is limited in its accuracy and in the questions it can address. The ACL model is highly simplified since the multibanded ACL was replaced with a single band. This was done for the convenience of model development. An improved ACL model should include several bands and a stiffness directly measured for the test knee. The knee stiffness matrix (whole joint less the ACL and cartilage layer) is rather imprecise; more accurate motion measurement should be used to establish it. Reproducibility and accuracy of stiffness components should be established. Additional checks on accuracy of the resulting stiffness matrices should be provided to further validate this model. A major limitation of the model is the basic concept of incremental loading about a preloaded equilibrium position. This approach was chosen to allow for simple model formulation with the use of linearization, and for simpler experimental measurement of model parameters. Because of this approach, if this modeling method is to be applied to a wide range of motion, stiffness matrices and joint surface parameters must be determined at each equilibrium configuration, With only one equilibrium position measured, total joint laxity and the change of ACL mechanics with flexion angle cannot be simulated. Some of these questions may be addressable with some relatively simple extensions of this modeling approach; these possibilities are being explored. In spite of these limitations, however, the model does achieve its goals of allowing assessment of the influence of surgical variables on some features of joint mechanics. The model emphasizes the load sharing that occurs between the ACL and other joint structures, and how joint contact stiffness alters this load sharing. This demonstrates the interdependence

88

D. A.

LOCH

of the joint structures and emphasizes the point that ACL reconstruction is a ‘joint’ reconstruction, not just a ‘ligament’ reconstruction. The same point has relevance in the study of osteoarthritis (OA). If OA is mechanical failure due to wear of cartilage, the process may involve joint structures other than the cartilage since many other structures can influence cartilage stresses. OA may be the end result of failure of the joint as a functional unit or organ, not just a cartilage disease (Freeman, 1972; Radin et al., 1990). The specific cases evaluated also give an insight into ACL-joint interaction. In general, higher ACL forces lead to higher joint contact forces. Stiffer grafts or smaller values for Lo lead to higher ACL forces. This may have relevance to ligament reconstruction and osteoarthritis since a tighter graft could lead to greater wear. The model also demonstrates that although the cartilage layer does not exert a shear force that can restrain anterior tibia1 force, changes in cartilage or contact stiffness can change the load sharing between the ACL and other structures. In particular, stiffening of the joint contact leads to an increase in ACL or graft force. For example, this may have relevance to menisectomy since it has been shown that loss of the meniscus increases contact stiffness (Kurosawa et al., 1980; Krause et al., 1976). According to the model predictions, this could lead to an increased ACL or graft force, which would assumedly increase the chance of ACL or graft rupture. Note that this scenario would only be directly valid for load cases where there are minimal shear forces resisted by the meniscus. Finally, the model predicts a 50% change of graft force with a change of Lo of only 0.7 mm. The accuracy of the model parameters must still be established, but the high sensitivity of graft force to L,-,agrees with the experimental measurements (Lewis et al., 1990), which have shown that decreasing Lo by 3 mm resulted in a 50% increase in graft force. Both model and experiment support the hypothesis that Lo is the variable causing large variations in graft force using current surgical methods of ACL reconstruction. The limitation of the model to small motions about an equilibrium point, i.e. a linearized model, is considerable and there are other models available without this limitation. Therefore, it is useful to point out where the proposed model may find application. Any state in which the joint can be in static equilibrium under load is potentially a state that can be analyzed. This would be at the extremes of motion, where the ligaments would be loaded; so the model could be used for many of the states where the ligaments are loaded. The model could be used to assess how loads are redistributed, and how contact location and forces change with ligament reconstructions or stretching. The authors are using it to analyze the mechanics of flexion contracture of the knee, in which the ACL or other tissue blocks full extension of the knee. Most tests of ligament integrity are performed in states

er ai.

which load the ligament and put the joint in a state of static equilibrium. The model would be useful in analyzing these states and assessing effects of surgical variables on these states. As discussed in the Introduction, the model is a simplified version of more complete, complex models; it uses elements of several of the previous approaches. It follows the spirit of the model by Pizialli and Rastegar (1977), in lumping all structures in a single stiffness matrix, but avoids the difficulties of the nonlinearities of the stiffness matrix in that model. It follows the linearization approach of Grood and Hefzy (1982) but avoids the large number of individual structural elements and properties of that model, while still allowing altering features of a single ligament. The models by Andriacchi et al. (1983) and Wismans et al. (1980) allow for large motions and deformations, but they also require a large number of property values. These two models could be used to compute a stiffness matrix for the joint at an equilibrium position of the joint, but we suspect that such a matrix measured directly, such as we have done, would be more accurate and reliable. They could compute such matrices at new points, however, which may be easier than repeated measurements as is required in our model. An advantage of our model over the other models is that the parameters of the joint stiffness, other than the ACL, can be measured directly. This is both easier and more accurate than measuring parameters for each individual element in the discrete structures models. We suspect that our model would be advantageous over the other models in those instances where one or a few equilibrium states are of interest, and the effects of a few structural elements are to be analyzed. Acknowledgements-This work was supported by the National Institute of Health grants AR38398 and AR39255, and OREF/Bristol-Meyers-Squibb/Zimmer Institutional Grant.

REFERENCES Andriacchi, T. P., Milosz, R. P., Hampton, S. J. and Galante, J. 0. (1983) Model studies of the stiffness characteristics of the human knee joint. J. Biomechanics 16,23-29. Butler, D. L., Grood, E. S., Noyes, F. R., Olmstead, M. L., Hohn, R. B., Arnoczky, S. P. and Siegel, M. G. (1981) Mechanical properties of primate vascularized vs nonvascularized patellar tendon grafts; changes over time. J. orthop. Res. 7, 68-79.

Chen, E. H. (1980) Material design analysis of the prosthetic anterior cruciate ligament. .I. biomed. Muter. Res. 14, 567-586. Claes, L. E. and Schmid, R. K. (1985) Visco-elastic properties of human knee ligaments. In Proc. 9th Annual Meeting of the Societyfor Riomaterials, p. 55, Birmingham, Alabama. Engebretsen, L., Lew, W. D., Lewis, J. L. and Hunter, R. E. (1989) Knee mechanics after repair of the anterior cruciate ligament. A cadaver study of ligament augmentation. Acta oFthop. stand. 60, l-7.

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Freeman. M. A. R. 11972) Pathonenesis of osteoarthritis: a hypdthesis. In Mddern ‘Trends in Orthopaedics (Edited by Apley, A. G.). Butterworths, London. Grood, E. S. and Hefty, M. S. (1982) An analytical technique

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A theoretical model of the knee and ACL for modeling knee joint stiffness-Part 1: ligamentous forces. J. biomech. Engng 104, 330-337. Grood, E. S. and Hefty, M. S. (1983) Sensitivity of insertion location on deformation of the anterior cruciate ligament. Proc. 1983 Biomechanics Symp., pp. 105-108. ASME, New York. Hefty, M. S. and Grood, E. S. (1986) Sensitivity of insertion locations on length patterns of anterior cruciate ligament fibers. J. biomech. Engng 103, 73-82. Hori, R. Y. and Mockros, L. F. (1976) Indentation tests of human articular cartilage. J. Biomechanics 9, 259-268. Kirstukas, S. J. (1989) Design and calibration of 6R instrumented spatial linkages for anatomical joint motion measurement. MS thesis, University of Minnesota. Krause, W. R., Pope, M. H., Johnson, R. J. and Wilder D. G. (1976) Mechanical changes in the knee after meniscectomy. J. Bone Jt Surg. !%A, 599-604. Kurosawa, H., Fukubageshi, T. and Nakajima, H. (1980) Load-bearing model of the knee joint: physical behavior of the knee joint with or without menisci. Cfin. Orthop. 149, 283-290. Lewis, J. L., Lew, W. D., Engebretsen, L., Hunter, R. E. and Kowalczyk, C. (1990) Factors effecting graft force in surgical reconstruction of the anterior cruciate ligament. J. orthop. Res. 8, 514-521. Lewis, J. L., Lew, W. D., Hill, J. A., Hanley, P., Ohland, K., Kirstukas, S. and Hunter, R. E. (1989) Knee joint motion and ligament forces before and after ACL reconstruction. J. biomech. Engng 111, 97-106. Noyes, F. R., Butler, D. L., Paulos, L. E. and Grood, E. S. (1983) Intra-articular cruciate reconstruction I: perspectives on graft strength, vascularization, and immediate motion after replacement. Clin. Orthop. Rel. Res. 172, 71-77. Odensten, M. and Gillquist, J. (1985) Functional anatomy of the anterior cruciate ligament and a rationale for reconstruction. J. Bone Jt Surg. 67A, 257-262. O’Donoghue, D. H., Rockwood, C. A. Jr, Frank, G. R., Jack, S. C. and Kenyon, R. (1966) Repair of the anterior cruciate ligament in dogs. J. Bone Jt Surg. 48A, 503-519. Piziali, R. L. and Rastegar, J. C. (1977) Measurement of the nonlinear coupled stiffness characteristics of the human knee. J. Biomechanics 10, 45-51. Radin, E. L., Burr, D. B., Fyhrie, D., Brown, T. D. and Boyd, R. D. (1990) Characteristics ofjoint loading as it applies to osteoarthritis. In Biomechanics of Diarthrodial Joints (Edited by Mow, V. C., Ratcliffe, A. and Woo, S. L.-Y.), pp. 437451. Springer, New York. Sachs, R. A., Reznik, A., Daniel, D. M. and Stone, M. L. (1990) Complications of knee ligament surgery. In Knee Ligaments: Structure, Function, Injury, and Repair (Edited by Daniel, D., Akeson, W. and O’Connor, J.), pp. 505-520. Raven Press, New York. Sidles, J. A., Larson, R. V., Garbini, J. L., Downey, D. J. and Matsen, F. A. III (1988) Ligament length relationships in the moving knee. J. orthop. Res. 6, 593-610. Sokolnikoff, I. S. (1956) Mathematical Theory of Elasticity, 2nd Edn. McGraw Hill, New York. Wismans, J., Veldpaus, F., Janssen, J., Huson, A. and Struben, P. (1980) A three-dimensional mathematical model of the knee-ioint. J. Biomechanics 13. 677-685. Yoshiya, S., Andrish,~J. T., Manley, M. T. and’Bauer, T. W. (1987) Graft tension in anterior cruciate ligament reconstruction. An in uivo study in dogs. Am. J. Sports Med. 15, 464470. APPENDIX

Transformation matrices between fixed and moving coordinates

The two-dimensional transformation between two Cartesian coordinate systems (X, Y) and (x, y) can be represented as

where 6 is the rotation angle between the two coordinate systems. a, and uy are the translation components of the origin relative to the xy coordinate system. For small motions, cos 6 = 1 and sin 6 = 0. Therefore, equation (Al) can be simplified as

(f)=(:, -:>(“y>+(:;). CA2) In three dimensions, a small-displacement transformation involving rotations about, and translations along, the fixed x-, y- and z-axes can be calculated by

or

(A3)

where u,, t+, u, are the displacement components of the origin relative to the xyz coordinate system and t?,, 19,,,6, are the rotation angle components between the two coordinate systems relative to the xyz coordinate system. In the current problem, the X YZ coordinate system is fixed on the moving tibia, while the xyz coordinate system is fixed on the static femur. Equation (A3) transforms the position of points expressed in the tibia1 coordinate frame to positions described relative to the fixed femoral frame. Formation of stt#iaessmatrix for the ACL element [Ran] is the stiffness matrix of the ACL and is formulated by the force-displacement relationship of a single spring in three dimensions, connected to points on the tibia and the femur f,=k(L-Li)X,

(A4)

where f, is the spring force vector, k is the stiffness, x is the unit vector along the spring, L, is the ACL length at the initial equilibrium position, and L is the final length, with

x=i[(x,--A)i+(y,-B)j+k,-C)k],

where X,, Y,, Z, are the initial position components in the xyz coordinate system of the attachment site on the tibia and have the same values as given in the X YZ coordinate system; A, B, C are the position components in the xyz coordinate system of the attachment site on the femur; and x,, yI, z, are the final position components in the xyz coordinate system of the attachment site on the tibia. For small motion, using equation (A3) to replace the x,, y,, z, with u,, uyr u,, f?,, 0,, 6, and neglecting the second-order terms, the component equa-

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et al.

tions of equation (A4) can be obtained as

X,-A X,-A X,-A X,-A Y,-B Y.-B Y.-B Y.-B 2,-c z,-c z,-c z,-c Z.-C z,-c

X,-A

X,-A

Y,-B

Y*-B

X,C-Z,A 0

0

Y,A-X,B

I\09I 0.

The moment of the ACL force about the origin is

where X., Y,, 2, replace x., y,, z, after the higher-order terms in equation (A3) are neglected. The stiffness matrix [I&J is then formed from equations (AS) and (A6). Formation ofstiflness matrix due to the cartilage layer [KImrcr]is formulated by six equilibrium equations of the spring layer force. As shown in Fig. 1, the two contact points are approximately symmetrical at the initial equilibrium position along the X axis, and are Xz, Yz, 2z and Xz, - Yz, Zz on the tibia1 plateau. At each contact point, the equilibrium equations are

where I0 is the initial thickness of the layer, Z1 and Z2 are the z-components of the contact point on the femur and the tibia, respectively, when the knee is in a displaced configuration. The coordinate Z, can be replaced by the z-coordinate of the sphere center relative to the displaced X YZ frame minus the sphere radius R. The constant coordinate Z2 can be similarly substituted for; using z,, the z-coordinate of the sphere center relative to the xyz frame, equation (A8) becomes L,+d,=(Z,-R)-(Z.-R-l,). Coordinate Z, can be replaced using the inverse form of equation (A3) to obtain d,=(x,8,-y.8,+z,-u,)-z, or

647) Because displacements are small, x,=X2, y0= Y2; thus, d,=X2tIl-

whereh,,A,,h.andmLr, mlyy mu are the components of the spring layer force and moment about the origin, respectively. X2, Y, in the XYZ coordinate system replace x2, y, in the xyz coordinate system after the higher-order terms are neglected in equations (A3). k, is the stiffness of the (layer) spring, and d, is the change in thickness of the (layer) spring under load (positive d, indicates an increase in thickness). From the frictionless assumption on the cartilage surface, the normal directions of both the tibia1 and femoral surfaces at contact points have to be parallel to the spring direction. This then gives the constraint equations as 10+d,=Z,-Z2,

648)

Y2&-u,.

(A9)

Using equation (A9) to replace d, in equation (A7), repeating this procedure on another contact point, and then adding the two results, the cartilage layer stiffness matrix for the two contact points is obtained as

~KtD&[

;

_3jx2

jy;

-i;;

;I.

(AlO) . ,