Abstract A new methodology for modeling articular tibio-femoral contact based on therecently developed asymptotic model of frictionless elliptical contact interaction betweenthin biphasic cartilage layers is presented. The developed mathematical model of articu-lar contact is extended to the case of contact between arbitrary viscoelastic incompressiblecoating layers. The approach requires use of the smooth contact surface geometry and ef-ficient contact points detection methods. A generalization of the influence surface theorybased method for representing articular surfaces from the unstructured noisy surface data isproposed. The normal contact forces are determined analytically based on the exact solutionfor elliptical contact between thin cartilage layers modeled as viscoelastic incompressiblelayers. The effective geometrical characteristics of articular surfaces are introduced for usein the developed asymptotic models of elliptical contact between articular surfaces.
Multibody dynamic simulations of physical exercise of a human skeleton based on the rigidmultibody approach [1] as well as on the flexible multibody approach [2] require modelingof the distributed internal forces generated by articular contact in joints. It is believed thatdynamic and impact patterns of the contact pressures play an important role in the devel-opment and progression of knee joint osteoarthritis [3]. Thus, multibody dynamic modelsof the knee joint capable of predicting contact stresses would be useful for studying themechanical aspects of this joint degenerative disease.
In several multibody dynamic models for the tibio-femoral joint [4–6], the articular con-tact problem is resolved under the assumption of a rigid contact formulation, when the con-tact between the surface of each femoral condyle and the surface of the tibia takes place at
The financial support from the European Union Seventh Framework Programme under contract numberPIIF-GA-2009-253055 is gratefully acknowledged.
I. Argatov (�)Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth SY23 3BZ Wales, UKe-mail: [email protected]
a single point and no deformation is considered in the articular cartilage layers due to thecontact loading. In contrast to the rigid contact model, the deformable contact model, whichtakes into account deformation of the articular cartilage layers, requires not only a descrip-tion of the articular surface geometry, but also additional information about the deformationbehavior of articular cartilage. As it was observed in [7], the advantage of deformable artic-ular contact model over the rigid contact model is two-fold: (1) It is not restricted to con-traform contact, and conforming surfaces can also be considered; (2) The knee multibodydynamic model with deformable contact has a higher numerical stability.
A multibody knee contact modeling methodology [8, 9] based on the deformable contactmodel should include the implementation of an efficient mathematical model for calculatingcontact pressures and the resulting contact forces. A number of musculoskeletal modelsof the knee joint employ different forms of the elastic Winkler foundation model [7]. It isknown [10] that this model is appropriate for describing the stress-deformation behaviorof thin compressible elastic coating layers, and it fails to represent contact interaction ofincompressible layers. At the same time, it was shown [11] that the instantaneous responseof a biphasic cartilage layer under distributed normal forces is in perfect agreement with thecorresponding solution for a bonded thin incompressible elastic layer.
In recent years, finite-element (FE) models have been increasingly used to simulate ar-ticular contact [12, 13]. In particular, a mathematical model of distributed contact using anumber of contacting patches was employed in [14]. At that, a uniform stress distributionover each contact patch was assumed. The advantage of FE models over the elastic founda-tion model consists in their ability to evaluate the subsurface stresses. Moreover, FE modelsare not confined to simple geometrical configurations, which are necessary for deriving an-alytical solutions like those used in the rigid contact model. However, as it was observedin [15], in comparison with simple deformable contact models, FE models are too time con-suming for the simulation of the knee joint dynamics in real activities such as the gait cycle.Very recently, a novel surrogate modeling approach for performing computationally effi-cient three-dimensional elastic contact with general surface geometry was proposed in [16]in order to lower the high computational cost of repeated contact analysis within multibodydynamic simulations. The method [16] fits a computationally cheap surrogate contact modelto data points sampled from a computationally expensive FE elastic contact model.
A new methodology for modeling tibio-femoral contact presented in this study is basedon the recently developed asymptotic model of frictionless elliptical contact interactionbetween thin biphasic [17] and viscoelastic [18] layers. The approach requires use of thesmooth contact surface geometry and efficient contact points detection methods. While thesubchondral bone is assumed to be rigid, we study different models for the articular carti-lage which is considered to be a thin layer of isotropic linear-elastic or viscoelastic (com-pressible or incompressible) material. The normal contact forces are determined analyticallybased on the exact solution for elliptical contact between thin viscoelastic compressible orincompressible cartilage layers.
As it was observed in [14], an anatomical based multibody dynamics model requires anaccurate description of the articular surfaces in order to solve the articular contact problem.In this study, we present a generalization of the method [19] for representing articular sur-faces from the unstructured experimental surface data, which can be used for regularizationof noisy surface data. Finally, we introduce the effective geometrical characteristics of ar-ticular surfaces for using in the developed asymptotic models of elliptical contact betweenarticular surfaces.
Development of an asymptotic modeling methodology for tibio-femoral 5
Fig. 1 Knee joint coordinatesystems
2 Articular surface geometry
The geometrical data of the tibia and femur are assumed to be given in Cartesian coordinatesystems (x1, x2, x3) and (x1, x2, x3), respectively. Following [7], the positive x1-axis is di-rected anteriorly, the positive x2-axis is pointed medially, and the positive x3-axis is directedproximally (Fig. 1). To describe the relative position of the femur with respect to the tibia,let us assume that the tibia is considered to be rigidly fixed. In such a case, the coordinatesx and x can be referred to as the “space-fixed” and “body-fixed” [20]. We assume that in thefully extended position of the joint, the directions of the corresponding coordinate axes ofboth coordinate systems coincide.
Let the position of an arbitrary point P on the femoral surface is represented by the vectorr(P ) in the body-fixed coordinate system. To describe the position vector r(P ) of the samepoint in the space-fixed system, one needs the transition vector r(O) from the origin of thetibial coordinate system (point O) to the origin of the femoral coordinate system (point O)and the rotation transformation matrix R (for its description, see [7]). According to thesedefinitions, the following relation holds [7, 20]:
r(P ) = r(O) + Rr(P ). (1)
Consider now an arbitrary point P on the tibial surface and a distance vector between thepoints P and P (see Fig. 1), i. e.,
d(P, P ) = r(P ) − r(P ). (2)
Following [9, 21], we introduce the normal contact distance vector, d0, between the articularsurfaces in such a way that it is parallel to each of the surface normals. The correspondingpoints P 0 and P 0 are called the potential contact points. At that, d0 = d(P 0, P 0).
The length of vector d0 with the proper sign taken into account will be called the pseudo-penetration and will be denoted as follows:
δ0 = −d0 · n0. (3)
Here, n0 is the outer normal to the tibial surface at the point P 0, and the dot denotes scalarproduct. We will assume that for any admissible position of the femur relative to the tibia,there is only a pair of the potential contact points P 0 and P 0 for each pair of femoral and
6 I. Argatov
Fig. 2 Pseudo-penetration of the contacting bodies
tibial condyles. Note that this assumption is in agreement with the geometric compatibilityof rigid bodies condition used in [4–6]. Therefore, if δ0 = 0, then the articular surfacescontact each other at a single point. In this case, a single tangent plane exists to both femoraland tibial surfaces. If d0 · n0 > 0 (and d0 · n0 < 0, where n0 is the outer normal to thefemoral surface at the point P 0), then there is no contact between the surfaces and δ0 < 0(see Fig. 2a). And finally, the penetration condition states that if d0 ·n0 < 0 (and d0 · n0 > 0),then the contact between the articular surfaces exists and δ0 > 0 (see Fig. 2b).
Furthermore, let us introduce a local Cartesian coordinate system (ξ1, ξ2, ζ ) with the cen-ter at the point P 0 in such a way that the positive ζ -axis points along the normal vector n0.Locally, that is in the vicinity of points P 0 and P 0, the equations of both articular surfacescan be written as follows:
ζ = −φ0(ξ), ζ = −δ0 + φ0(ξ). (4)
It is assumed that locally the tibia and femur occupy the domains ζ ≤ −φ0(ξ) and ζ ≥−δ0 + φ0(ξ), respectively.
In view of (4), we define the local gap function as
φ(ξ) = φ0(ξ) + φ0(ξ). (5)
In the next section, following [4], we may assume that the functions φ0(ξ) and φ0(ξ) can beapproximated by polynomials in ξ1 and ξ2 of degrees n and n as follows:
φ0(ξ) =n∑
p=2
p∑
q=0
apqξp−q
1 ξq
2 , φ0(ξ) =n∑
p=2
p∑
q=0
apqξp−q
1 ξq
2 . (6)
The coefficients apq and apq are calculated by minimizing the functions
N∑
j=1
(ζ j −
n∑
p=2
p∑
q=0
apq
(ξ
j
1
)p−q(ξ
j
2
)q
)2
,
N∑
j=1
(ζ j −
n∑
p=2
p∑
q=0
apq
(ξ
j
1
)p−q(ξ
j
2
)q
)2
, (7)
where N and N are the numbers of measured surface points, and (ξj
1 , ξj
2 , ζ j ) and (ξj
1 , ξj
2 , ζ j )
are the measured coordinates of the j th point on the tibial surface (j = 1, . . . ,N ) and on thefemoral surface (j = 1, . . . , N ), respectively.
Development of an asymptotic modeling methodology for tibio-femoral 7
3 Contact constitutive relations
In this section, we consider analytical models of deformable tibio-femoral contact underthe assumption that the local gap function can be approximated by an elliptical paraboloid.The articular cartilages are modeled as thin compressible and incompressible elastic or vis-coelastic layers.
3.1 Elastic foundation model. Elliptical contact of thin compressible elastic layers
Consider a frictionless contact between two thin linear elastic layers of constant thicknesses,h1 and h2, firmly attached to rigid substrates with continuously varying curvatures. Let usassume that in the undeformed state, the surfaces of the layers touch at a single point denotedby P 0. Introducing a Cartesian coordinate system (η1, η2, ζ ) with the center at the point P 0
such that the coordinate plane ζ = 0 coincides with the common tangent plane to the layersurfaces, without loss of generality, we may assume that with the accuracy up to terms oforder |η|3 the gap function, ϕ(η), defined as the distance between the layer surfaces alongthe ζ -axis, is represented by an elliptic paraboloid
ϕ(η) = (2R1)−1η2
1 + (2R2)−1η2
2. (8)
Let w(η) and w(η) be the vertical displacement functions for the surface points of thelayers representing the tibial and femoral articular cartilages, respectively, due to the actionof the surface pressures p(η). Given that the materials of the layers are elastic with Young’smoduli E1 and E2, and Poisson’s ratios ν1 and ν2, which are assumed to be not too close to0.5, we will have
w(η) = −E−11 A1h1p(η), w(η) = E−1
2 A2h2p(η). (9)
Here, the index 1 refers to the tibia and 2 refers to the femur, and the following notation isused:
An = (1 + νn)(1 − 2νn)
1 − νn
. (10)
Let also δ0 be the vertical approach of the rigid substrates. Then the following equationshould hold in the contact region, ω, where the contact pressure is positive:
w(η) − w(η) = δ0 − ϕ(η), η ∈ ω. (11)
Substituting the expressions (9) into (11), we obtain the contact condition in the form
(E−1
1 A1h1 + E−12 A2h2
)p(η) = δ0 − ϕ(η). (12)
From (12), it immediately follows that
p(η) = k(δ0 − ϕ(η)
), (13)
where k is the Winkler foundation modulus given by
k =(
(1 + ν1)(1 − 2ν1)h1
(1 − ν1)E1+ (1 + ν2)(1 − 2ν2)h2
(1 − ν2)E2
)−1
. (14)
8 I. Argatov
It is obvious that if the layers’ materials are similar (i.e., E1 = E2 = E and ν1 = ν2 = ν),formula (14) simplifies to the following one:
k = (1 − ν)E
(1 + ν)(1 − 2ν)h. (15)
Here, h = h1 + h2 is the joint thickness.Integrating the contact pressure distribution (13) over the contact region ω, we obtain the
contact force
P =∫∫
ω
p(η) dη. (16)
In the case of a paraboloidal gap function (8), in view of (13), we find that the contactarea ω is an elliptical domain with the semi-axes a = √
2R1δ0 and b = √2R2δ0. According
to (8), (13), and (16), we will have
P = πk√
R1R2δ20 . (17)
Equation (17) represents the force-displacement relationship for the case of ellipticalcontact of thin compressible coatings. The fact that a thin elastic layer with its Poisson’sratio not too close to 0.5 behaves like a Winkler elastic foundation was first rigorouslyestablished in [22]. The case of elliptical contact in the framework of elastic foundationmodel was considered in detail in [23]. The elastic foundation model based on (15) wasused for multibody dynamic simulations of knee contact mechanics in a number of papers[7, 24]. Formula (14) was recently considered in [15]. A discussion of analytical modelsemployed for describing articular contact is presented in [15] along with a comparativestudy of four different models: the classical Hertz contact model, Elastic foundation model,a new modified elastic foundation model, which takes into account the Hertzian type contactfor relatively low conforming surfaces and small contact areas, and the finite-element model.
3.2 Asymptotic model for elliptical contact of thin incompressible elastic layers
It is readily seen that the coefficient An defined by formula (10) vanishes as νn → 0.5, andconsequently the Winkler foundation modulus k tends to infinity (see (14), (15)).
Based on the asymptotic analysis of the frictionless contact problem for a thin elasticlayer bonded to a rigid substrate in the thin-layer limit [25], the following contact constitu-tive relations for thin incompressible layers can be established instead of (9):
w(η) = −E−11 B1h
31Δp(η), w(η) = E−1
2 B2h32Δp(η). (18)
Here, Δ = ∂2/∂η21 + ∂2/∂η2
2 is the Laplace differential operator, and
Bn = νn(1 + νn)(4νn − 1)
3(1 − νn)2. (19)
Substituting the expressions (18) into the contact condition (11), we obtain
−(E−1
1 B1h31 + E−1
2 B2h32
)Δp(η) = δ0 − ϕ(η). (20)
Equation (20) should hold over the whole contact region ω. According to [10, 26], we im-pose the following boundary conditions on the contour Γ of the contact region ω:
p(η) = 0,∂p
∂n(η) = 0, η ∈ Γ. (21)
Development of an asymptotic modeling methodology for tibio-femoral 9
Here, ∂/∂n is the normal derivative.In the case of a paraboloidal gap function (8), the exact solution to the problem (20), (21)
was obtained in [10] in the form
p(η) = p0
(1 − η2
1
a2− η2
2
b2
)2
. (22)
The maximum contact pressure p0 and the semi-axes a and b of the elliptical contact regionω satisfy the following system of algebraic equations [10]:
δ0 = 4p0
m
(1
a2+ 1
b2
), (23)
1
2R1= 4p0
ma2
(3
a2+ 1
b2
),
1
2R2= 4p0
mb2
(1
a2+ 3
b2
). (24)
Here, we introduced the notation
m = (E−1
1 B1h31 + E−1
2 B2h32
)−1. (25)
If the layers’ materials are similar, formula (25) simplifies to the following one:
m = 3(1 − ν2)2E
ν(4ν − 1)(1 + ν)(h31 + h3
2). (26)
Integrating the contact pressure distribution (22) over the contact region ω and takinginto account (16), (23), and (24), we obtain the force-displacement relationship:
P = πm
3MP (s)R1R2δ
30 . (27)
Here, s = b/a is the aspect ratio of the contact area, and the factor MP (s) is given by
MP (s) = s(3s2 + 1)(s2 + 3)
(s2 + 1)3. (28)
According to (24), the following relation holds true [17]:
s2 =√(
R1 − R2
6R1
)2
+ R2
R1− (R1 − R2)
6R1. (29)
Equation (27) represents the force-displacement relationship for the case of ellipticalcontact of thin incompressible coatings.
3.3 Asymptotic model for elliptical contact of thin compressible viscoelastic layers
For the sake of simplicity, we assume that Poisson’s ratios ν1 and ν2 of the viscoelasticlayers are time independent. Then, applying the viscoelastic correspondence principle to theassociated elastic equation (12), one arrives at the following governing integral equation:
h
E∞
∫ t
0−Φα(t − τ)
∂p
∂τ(η, τ ) dτ = δ0(t) − ϕ(η)H(t). (30)
10 I. Argatov
Here, δ0(t) is the variable approach of the rigid substrates, t is a time variable, H(t) isthe Heaviside step function such that H(t) = 1 for t > 0 and H(t) = 0 for t ≤ 0, E∞ is aharmonic mean of the relaxed elastic moduli E∞
1 and E∞2 , Φα(t) is the compound creep
function, i. e.,
Φα(t) = α1Φ1(t) + α2Φ2(t), E∞ = 2E∞1 E∞
2
E∞1 + E∞
2
,
α1 = 2E∞2 h1 A1
(E∞1 + E∞
2 )(h1 + h2), α2 = 2E∞
1 h2 A2
(E∞1 + E∞
2 )(h1 + h2).
Recall that the constants A1 and A2 are defined by formula (10).The force-displacement relationship is given be the following equation [18]:
P (t) = π
h
√R1R2E∞
∫ t
0−Ψα(t − τ)
d
dτ
(δ2
0(τ ))dτ. (31)
Here, Ψα(t) is the compound relaxation function determined by the relaxation functionsΨ1(t) and Ψ2(t) of the viscoelastic compressible coatings as follows:
Ψα(t) = 1
α1Ψ1(t) + 1
α2Ψ2(t).
Note that if the layer’s materials follow a standard linear viscoelastic solid model, wehave
Φn(t) = 1 − (1 − ρn) exp(−t/Tn),
Ψn(t) = 1 − (1 − 1/ρn) exp(−t/(ρnTn)
),
where Tn is the characteristic relaxation time of strain under applied step of stress, ρn =E∞
n /E0n is the ratio of E∞
n to the unrelaxed elastic modulus E0n .
3.4 Asymptotic model for elliptical contact of thin incompressible viscoelastic layers
Applying the viscoelastic correspondence principle to the associated elastic equation (20),we arrive at the following governing integro-differential equation:
− h3
E∞
∫ t
0−Φβ(t − τ)Δ
∂p
∂τ(η, τ ) dτ = δ0(t) − ϕ(η)H(t). (32)
Here, Δ = ∂2/∂η21 + ∂2/∂η2
2 is the Laplace operator, Φβ(t) is the compound creep functiondetermined by the formulas
Φβ(t) = β1Φ1(t) + β2Φ2(t),
β1 = 2E∞2 h3
1 B1(ν1)
(E∞1 + E∞
2 )(h1 + h2)3, β2 = 2E∞
1 h32 B2(ν2)
(E∞1 + E∞
2 )(h1 + h2)3. (33)
At that, the constants B1 and B2 are defined by formula (19). As before, we require that thecontact pressure distribution p(η, t) should satisfy the boundary conditions (21).
Development of an asymptotic modeling methodology for tibio-femoral 11
The general solution of (32) in the case (8) was derived in [18]. Integrating the obtainedcontact pressures over the elliptical contact region, we arrive at the following equation:
P (t) = πm
3MP (s)R1R2
∫ t
0−Ψα(t − τ)
d
dτ
(δ3
0(τ ))dτ. (34)
Here, MP (s) is the factor defined by formula (28), and Ψβ(t) is the compound relaxationfunction for the viscoelastic incompressible coatings given by
Ψβ(t) = 1
β1Ψ1(t) + 1
β2Ψ2(t).
It should be noted that (31) and (34) are valid for the case of contact interaction consistingof a monotonic loading phase and a monotonic unloading phase. Thus, they can be appliedfor modelling contact forces in impact situations. We underline that the case of repetitiveloading requires a special treatment.
Example 1 Let us consider the application of (34) for the tibio-femoral contact in theweight-bearing region [32] with the gap function given by (37) and (38). In this case, wehave RL
1 = 6.9 mm and RL2 = 29.9 mm, RM
1 = 26.5 mm and RL2 = 30.0 mm. According to
(29), we obtain sL = 1.65 and sM = 0.51 for the lateral and medial compartment that meansthat the elliptical contact areas are oriented in the ML and AP directions, respectively. Sub-stituting this value into (28), we readily get MP (sL) = 1.68 and MP (sM) = 1.48.
In order to determine the parameter m and the compound relaxation function Ψβ(t), wemake use of the correspondence established in [18] between the short-time biphasic model[11, 27] and the Maxwell viscoelastic model
Ψn(t) = E0ne
−t/τn , n = 1,2,
where E0n is the instantaneous elastic modulus, τn is a relaxation time. According to [18],
for incompressible layers (ν1 = ν2 = 0.5) we have E0n = 3μsn and τn = h3
n/(3μsnkn), whereμsn is the shear modulus of the solid phase of the cartilage tissue (n = 1,2), k1 and k2
are the cartilage permeabilities. Finally, note that in the case of the Maxwell model, theelastic moduli E∞
1 , E∞2 , and E∞ in (32), (33) should be replaced with E0
1 , E02 , and E0 =
2E01E
02/(E
01 + E0
2), respectively.
4 Effective geometrical characteristics of articular contact
4.1 Approximation of the articulating femur and tibia geometries by elliptic paraboloids
Because of the complexity of human knee joint geometry, it is difficult to obtain analyticalsolutions for the contact pressure distribution in the knee joint under physiological load-ing conditions even under simplifying assumptions about the articular cartilage mechanicalbehavior. In [28], different analytical solutions were compared with finite element methodsolution in the case of axisymmetric articular joint idealized as a system of a rigid ball witha cartilage layer in frictionless contact with a hemispherical layer of cartilage attached toa rigid base. Spherical surfaces were used for representing the medial and lateral femoralcondyles in a number of papers [5, 29]. Also, in two-dimensional models [29, 30], the tibia
12 I. Argatov
surface is generally represented as a parabolic profile. In [15], a toroidal geometry was se-lected to represent the geometry of the medial (or the lateral) femoral and tibial componentsin a total knee replacement.
Elliptical paraboloids are widely used in contact mechanics for approximating the inter-acting non-axisymmetric surfaces. Based on high-resolution MRI, it was shown [31] that allarticular surfaces of the knee joints of healthy volunteers displayed predominantly convexovoid shapes, except for the central aspect of the medial tibia (with the highest degree of con-cavity). At that, none of the articulating surfaces displayed saddle-like properties. In [32],the principal curvature radii of the femoral and tibial cartilage surfaces were measured in theweight-bearing regions of the medial and lateral compartments of three-dimensional mod-els from the MP images obtained from 11 young healthy male individuals (age 30.5 ± 5.1years). In particular, in the lateral condyle of the femur, the average radii were 22 ± 4 mmand 25 ± 4 mm in AP and ML directions, respectively. In the medial condyle of the femur,the average radii were 34 ± 5 mm and 21 ± 3 mm in AP and ML directions, respectively. Inthe lateral (medial) plateau of the tibia, the average radii were 37±10 mm and −43±11 mm(−95 ± 38 mm and −29 ± 7 mm), respectively. Here, positive and negative values representconvex and concave surfaces, respectively.
According to [32], the shape functions φ0(ξ) and φ0(ξ) entering (4) are specified asfollows:
φL0 (ξ) = −
(x2
1
37− x2
2
43
), φM
0 (ξ) = −(−x2
1
95− x2
2
29
), (35)
φL0 (ξ) = x2
1
22+ x2
2
25, φM
0 (ξ) = x21
34+ x2
2
21. (36)
Here, the coordinates are assumed to be measured in millimeters, whereas the indexes “L”and “M” denote the quantities referring to the lateral and medial compartments, respectively.
Substituting (35) and (36) into (5), we obtain
φL(ξ) =(
1
22+ 1
37
)x2
1 +(
1
25− 1
43
)x2
2 , (37)
φM(ξ) =(
1
34− 1
95
)x2
1 +(
1
21− 1
29
)x2
2 . (38)
Thus, as a first approximation, actual articulating surfaces in tibio-femoral contact canbe represented by elliptic paraboloids. The parameters of such an approximation (i.e., theprincipal radii of curvature) generally depend on the position of the central contact point(point P 0 in Fig. 2).
Of course, differences in actual articulating surfaces and idealized elliptic paraboloidsurfaces influence contact mechanics and especially the distribution of contact pressures.This question is considered in [33], where a perturbation solution is obtained under theassumption that the subchondral bones are rigid and shaped close to elliptic paraboloids.
4.2 Analytical approximations for articular contact surfaces
To model the articular contact, one needs to describe the articular surface geometry in theframework of a certain mathematical model. A number of surface-fitting methods for rep-resenting the three-dimensional topography of articular surfaces, and in particular, B-splinemethod [34, 35], use a structured data set and provide a limited continuity of the fitted
Development of an asymptotic modeling methodology for tibio-femoral 13
articular surface. Methods to represent articular surfaces from the unstructured data weresuggested in [36, 37] and are based on a parametric polynomial representation.
In order to effectively deal with non-ordered data points, a new method for the represen-tation of articular surfaces, which is based on the influence surface theory of elastic plates,was introduced in [19]. A set of N ≥ 3 points (ξ j , ζ j ) (j = 1, . . . ,N ) is approximated withthe function
w(ξ) = b0 + b1ξ1 + b2ξ2 +N∑
k=1
fk
∣∣ξ − ξ k∣∣2
ln∣∣ξ − ξ k
∣∣, (39)
where |ξ | =√
ξ 21 + ξ 2
2 . For N + 3 coefficients f1, . . . , fN , b0, b1, and b2, we have the fol-lowing system of N + 3 linear algebraic equations:
b0 + b1ξj
1 + b2ξj
2 +N∑
k=1
fk
∣∣ξ j − ξ k∣∣2
ln∣∣ξ j − ξ k
∣∣ = ζ j , j = 1, . . . ,N, (40)
N∑
k=1
fk = 0,
N∑
k=1
ξk1 fk = 0,
N∑
k=1
ξk2 fk = 0. (41)
It is clear that the method [19] provides unlimited continuity of the fitted articular surface(with exception of the data points, where the function (39) is only continuously differen-tiable).
Since experimental measurements of surface data always contain a degree of measure-ment uncertainty, it was observed in [34] that a surface-fitting method which consists ofinterpolating the measured surface data may result in some degree of surface roughness. In[19], it was also noted that one limitation of the fitting method (39)–(41) is that it requiresthe fitting surface to pass through all measured surface points. This means that the fittingaccuracy of the method [19] is partially controlled by the accuracy of the measurement in-strument. At that, due to the noisy nature of measured data, forcing the fitting surface to passthrough all measured surface data points may not produce an optimal fitting surface [19].
In view of the observation made above, we propose the following regularization of themethod [19]. Given the measured surface data points (ξ j , ζ j ) (j = 1,2, . . . ,N ), the fittingsurface must satisfy the optimization criterion
minf1,...,fN ,b0,b1,b2
1
2
N∑
j=1
fj
(b0 + b1ξ
j
1 + b2ξj
2 +N∑
k=1
fk
∣∣ξ j − ξ k∣∣2
ln∣∣ξ j − ξ k
∣∣)
(42)
subject to (41) and the unilateral constraint
1
N
N∑
j=1
(b0 + b1ξ
j
1 + b2ξj
2 +N∑
k=1
fk
∣∣ξ j − ξ k∣∣2
ln∣∣ξ j − ξ k
∣∣ − ζ j
)2
≤ ε2. (43)
Here, ε represents a given tolerance level. Note that the physical meaning of the optimiza-tion criterion (42) is to find the point forces f1, . . . , fN and the rigid body displacementparameters b0, b1, and b2 such that the plate potential energy reaches its minimum under therelaxed unilateral geometrical constraint (42).
Now, the approximation (39) will not necessarily pass through the set of surface datapoints (ξ j , ζ j ) (j = 1,2, . . . ,N ), but it will minimize the plate potential energy (42), while
14 I. Argatov
simultaneously keeping the sum of squired distances (w(ξ j ) − ζ j )2 under the given toler-ance level.
Note that spline smoothing methods for regularization of noisy data were consideredin [34].
4.3 Effective geometrical characteristics in the case of thin compressible layers
In order to apply the force displacement relationship (17) or (31), one needs to evaluate thegeometric parameters R1 and R2 appearing in the paraboloid approximation (8) of the localgap function (5). In other words, the local gap function (5) must be approximated as follows:
φ(ξ) = ϕ(ξ) + ϕ(ξ), (44)
where ϕ(ξ) is in a sense a small discrepancy, while ϕ(ξ), according to (8), should be takenin the form
ϕ(ξ) = ξ 21
(cos2 θ
2R1+ sin2 θ
2R2
)+ ξ 2
2
(sin2 θ
2R1+ cos2 θ
2R2
)+ ξ1ξ2 sin 2θ
(1
2R1− 1
2R2
). (45)
Observe that the angle θ introduced in (45) has the meaning of the angle between the positiveξ1-axis and the positive η1-axis.
Considering the contact problem for the gap function (44) in the framework of the elasticfoundation model, we will have the following contact pressure and contact force (see (13)and (16)):
p(ξ) = k(δ0 − ϕ(ξ) − ϕ(ξ)
), (46)
P = k
∫∫
ω
(δ0 − ϕ(ξ) − ϕ(ξ)
)dξ . (47)
Here, ω is the new contact region that somehow slightly differs from the elliptical contactregion ω corresponding to the gap function ϕ(ξ).
Under the assumption that the gap variation ϕ(ξ) introduces a small variation into thecontact region ω and the contact force P , we may use the approximate relationship
P = k
∫∫
ω
(δ0 − ϕ(ξ)
)dξ . (48)
Consequently, according to (44), (47), and (48), the following approximate equation holds:∫∫
ω
(φ(ξ) − ϕ(ξ)
)dξ = 0. (49)
Thus, in view of (49), we suggest the following optimization criterion for choosing theparameters R1 and R2:
minR1,R2,θ
∫∫
ω∗|φ(ξ) − ϕ(ξ)|dξ . (50)
Here, ω∗ is a characteristic area. In particular, the domain ω∗ should contain a maximumcontact area for a class of admissible contact loadings. Note that an estimate for ω∗ canbe obtained based on a polynomial approximation for φ(ξ) (see (5) and (6)). Furthermore,because of the fact that the gap function depends on the orientation of the femur with respect
Development of an asymptotic modeling methodology for tibio-femoral 15
to the tibia, and the aspect ratio of the contact zone changes with this orientation, in the placeof ω∗ one can substitute a circular domain, which surrounds the maximum elliptical contactzone for all admissible orientations and contact loadings. Finally, observe that the shape ofthe local gap function φ(ξ) outside of the contact area does not play any role in evaluatingthe contact pressure and contact force.
4.4 Effective geometrical characteristics in the case of thin incompressible layers
Considering now the contact problem for the gap function (44) in the framework of theasymptotic model for incompressible elastic layers, we will have the following problem forthe contact pressure (see (20), (21), and (25)):
−m−1Δp(ξ) = δ0 − ϕ(ξ) − ϕ(ξ), ξ ∈ ω, (51)
p(ξ) = 0,∂p
∂n(ξ) = 0, ξ ∈ Γ . (52)
Here, Γ is the contour of the contact domain ω.Under the assumption that the gap variation ϕ(ξ) introduces a small variation into the
elliptical contact region ω corresponding to the gap function ϕ(ξ), we derive from (51) and(52) the following limit problem for the variation of the contact pressure:
m−1Δp(ξ) = ϕ(ξ), ξ ∈ ω, (53)
p(ξ) = 0,∂p
∂n(ξ) = 0, ξ ∈ Γ. (54)
Moreover, the gap variation ϕ(ξ) will not greatly influence the resulting contact force, if∫∫
ω
p(ξ) dξ = 0. (55)
Applying the second Green’s formula and taking into account (53) and (54), we reduce(55) to the following one:
∫∫
ω
(ξ 2
1 + ξ 22
)ϕ(ξ) dξ = 0. (56)
Thus, in view of (56), we suggest the following optimization criterion for determiningthe parameters R1 and R2 (compare with (50)):
minR1,R2,θ
∫∫
ω∗
(φ(ξ) − ϕ(ξ)
)2dξ . (57)
Indeed, two of the three necessary optimality conditions for (57) have the form∫∫
ω∗
(φ(ξ) − ϕ(ξ)
)(ξ 2
1 cos θ + ξ 22 sin θ + ξ1ξ2 sin 2θ
)dξ = 0,
∫∫
ω∗
(φ(ξ) − ϕ(ξ)
)(ξ 2
1 sin θ + ξ 22 cos θ − ξ1ξ2 sin 2θ
)dξ = 0.
Now, adding the equations above, we readily obtain∫∫
ω∗
(φ(ξ) − ϕ(ξ)
)(ξ 2
1 + ξ 22
)dξ = 0. (58)
16 I. Argatov
In other words, (58) is a necessary optimality condition for (57).Comparing (56) and (58), in view of (44), we come to the conclusion that (56) and (58)
coincide if their integration domains coincide. This motivates the choice of the optimizationcriterion (57).
4.5 Determining the effective geometrical characteristics from experimental surface data
The optimization criteria (50) and (57) can be written as
minR1,R2,θ
∫∫
ω∗|φ(ξ) − ϕ(ξ)|σ dξ , (59)
where σ = 1 in the case (50) and σ = 2 in the case (57).The optimization criterion (59) requires a continuous representation of the gap function
φ(ξ), while originally only the coordinates of experimental surface data points (ξj
1 , ξj
2 , ζ j )
(j = 1, . . . ,N ) and (ξj
1 , ξj
2 , ζ j ) (j = 1, . . . , N ) have been provided from the measurementexperiment (see (7)). For instance, the analytical approximation (39) can be sued. However,it would be useful to have a possibility to determine the parameters R1 and R2 directly fromthe experimental surface data. That is why, the following discrete variant of the optimizationcriterion (59) makes sense.
Given the measured surface data points (ξj
1 , ξj
2 , ζ j ) (j = 1, . . . ,N ) and (ξj
1 , ξj
2 , ζ j ) (j =1, . . . , N ), the two sets of effective geometrical parameters R0
1 , R02 , θ0 and R0
1 , R02 , θ0 must
satisfy the criteria
minR0
1 ,R02 ,θ0
∑
ξ j ∈ω∗
∣∣ζ j − ϕ0(ξ j
)∣∣σ , (60)
minR0
1 ,R02 ,θ0
∑
ξj ∈ω∗
∣∣ζ j − ϕ0(ξ
j )∣∣σ , (61)
where ϕ0(ξ) and ϕ0(ξ) are given by
ϕ0(ξ) = ξ 21
(cos2 θ0
2R01
+ sin2 θ0
2R02
)+ ξ 2
2
(sin2 θ0
2R01
+ cos2 θ0
2R02
)
+ ξ1ξ2 sin 2θ0
(1
2R01
− 1
2R02
), (62)
ϕ0(ξ) = ξ 21
(cos2 θ0
2R01
+ sin2 θ0
2R02
)+ ξ 2
2
(sin2 θ0
2R01
+ cos2 θ0
2R02
)
+ ξ1ξ2 sin 2θ0
(1
2R01
− 1
2R02
). (63)
According to (60) and (61), the tibial and femoral surfaces are represented locally by theeffective elliptic paraboloids (62) and (63), whose orientations with respect to the positiveξ1-axis is determined by the angles θ0 and θ0, respectively. Now, the effective geometricalparameters R1 and R2 appearing in the elliptic paraboloidal approximation (8) can be deter-mined from R0
1 , R02 , θ0 and R0
1 , R02 , θ0 following a standard procedure used in the Hertzian
contact mechanics [23].
Development of an asymptotic modeling methodology for tibio-femoral 17
5 Discussion and conclusions
The asymptotic methodology for tibio-femoral articular contact developed in this paper isbased on an asymptotic theory for a thin compressible or incompressible viscoelastic layerattached to a rigid substrate. As it was shown in [18], the viscoelastic contact model for in-compressible layers incorporates the asymptotic model [11, 17, 27] for short-time responseof biphasic layers as a special case, corresponding to the Maxwell model of viscoelasticmaterial.
In the case of elastic layers, the contact constitutive relations can be represented as fol-lows (see (17) and (27)):
P = EMRlδn0 . (64)
Here, E is Young’s elastic modulus, R = √R1R2 is a geometric mean of the curvature radii
R1 and R2, the factor M is a function of the thicknesses of the layers h1 and h2, Poisson’sratio ν, and the aspect ratio s of the elliptical contact region. At that, for compressible layers,n = 2 and l = 1, while for incompressible layers, n = 3 and l = 2. Note that the dimensionof M is L2−n−l , where L is the dimension of length. Equation (64) was used in a numberof publications on multibody simulations [38]. In particular, it incorporates the Hertzianforce-displacement relationship with n = 3/2 and l = 1/2.
From (64) (see also (26) and (27)), it is readily seen that in the case of incompressiblelayers, the contact force, P , is inversely proportional to the joint thickness cubed, h3, whilein the case of thin compressible coatings, the contact force is simply inversely proportional tothe joint thickness (see (15) and (17)). This implication is very important from the viewpointof applications of the articular contact modelling to osteoarthritic joints. Indeed, the changeof the articular cartilage thickness has been widely used as an indicator of its degenerativestatus.
In order to take into account the effect of energy dissipation during the elasto-plasticcontact interaction, the following Hunt–Crossley equation [39] has been widely employedfor modelling impact situations:
P (t) = bδp
0 (t)δq
0 (t) + kδn0 (t). (65)
The stiffness coefficient k and the damping coefficient b depend on material and geometricproperties of colliding bodies. As it was observed in [38], an important aspect of (65) isthat damping depends on the indentation, which is physically sound since the contact areaincreases with deformation and a plastic region is more likely to develop for larger contactdisplacements. For biomechanical applications, (65) was used in [9, 43, 44].
Note that (65) also incorporates the so-called Lankarani–Nikravesh contact-impact forcemodel (with p = n and the coefficient b inversely proportional to the initial impact velocity).A contact detection methodology for the automatic detection of precise instant of contactin contact-impact analysis in multibody dynamics was developed in [40] with a specialemphasis put on the Lankarani–Nikravesh force model, which was recently discussed indetail in [41]. For biomechanical applications, the Lankarani–Nikravesh model was recentlyused in [42] for describing articular contact between intervertebral discs to study the cervicalspine dynamics.
In the case of viscoelastic layers, according to (64), the contact constitutive relations canbe represented as follows (see also, (31) and (34)):
P (t) = E∞MRl
∫ t
0−Ψ (t − τ)
d
dτ
(δn
0 (τ ))dτ. (66)
18 I. Argatov
Here, E∞ is the relaxation modulus, Ψ (t) is the compound relaxation function.It should be noted that (64) and (66) do not exactly describe the initial short time interval
of contact interaction, while the contact zone does not exceed the joint thickness of thelayers. However, if the maximum characteristic size of the contact zone achieved during theloading phase is much grater than each thickness of the layers, the overall error introducedby this initial interval will be relatively small, just like it was shown in [45] with respect tothe influence of the superseismic stage of contact on the Hertzian impact theory.
Observe that in contrast to (65), the force-displacement relationship (66) introduces theviscous mechanism of energy dissipation and is likely to be more physically sound in view ofthe biphasic nature of articular cartilage. Note also that the consideration of viscous effectsin quasistatic or dynamic simulations could be important, in particular, in the simulation oftotal knee replacement [15].
We underline that (64) and (66) depend on the geometrical parameters of the articularsurfaces in contact, and their accurate determination is an important step in applicationsof these equations. In this study, we introduced the effective geometrical characteristics ofarticular surfaces for use in the developed asymptotic models of elliptical contact betweenarticular surfaces.
It is interesting to observe following [7], where the effects of different mathematical de-scriptions of articular contact and articular surface geometry on the kinematic characteristicsof the knee model were investigated, that close approximations of the articular surfaces bypolynomials are not necessary, since the motion characteristics were not influenced greatlyby the degree of the polynomial approximations for the curved tibial surfaces. This wascaused by the size of the contact area, which covered small surface irregularities and madethe contribution of the contact pressure distribution to the net contact force less dependenton the irregularities. Thus, this observation supports the necessity to operate with the effec-tive geometrical characteristics of articular surfaces. It should be also emphasized that theanalytical models for the contact force using the local geometrical characteristics (princi-pal radii of curvature of the articular surfaces at the potential contact points P 0 and P 0) incontrast to the effective geometrical characteristics are restricted to simple geometries and,therefore, their applicability to real articular contact geometries is limited.
The objective of this study is to describe analytically the articular tibio-femoral contactfor applications in multibody dynamic simulations of the human knee joint. As the mainresult of the present paper, simple asymptotic models of elliptical contact between the ar-ticular cartilage layers have been established based on the recently developed asymptoticmodel of frictionless elliptical contact interaction between thin biphasic or viscoelastic car-tilage layers. The asymptotic models use the effective geometrical characteristics of articularsurfaces, which can be determined from the introduced optimization criteria.
References
1. Eberhard, P., Spägele, T., Gollhofer, A.: Investigations for the dynamical analysis of human motion.Multibody Syst. Dyn. 3, 1–20 (1999)
2. Kłodowski, A., Rantalainen, T., Mikkola, A., Heinonen, A., Sievänen, H.: Flexible multibody approachin forward dynamic simulation of locomotive strains in human skeleton with flexible lower body bones.Multibody Syst. Dyn. 25, 395–409 (2011)
3. Herzog, W., Federico, S.: Considerations on joint and articular cartilage mechanics. Biomech. Model.Mechanobiol. 5, 64–81 (2006)
4. Wismans, J., Veldpaus, F., Janssen, J., Huson, A., Struben, P.: A three-dimensional mathematical modelof the knee-joint. J. Biomech. 13, 677–679 (1980) 681–685
5. Abdel-Rahman, E.M., Hefzy, M.S.: Three-dimensional dynamic behaviour of the human knee joint un-der impact loading. Med. Eng. Phys. 20, 276–290 (1998)
Development of an asymptotic modeling methodology for tibio-femoral 19
6. Ling, Z.-K., Guo, H.-Q., Boersma, S.: Analytical study on the kinematic and dynamic behaviors of aknee joint. Med. Eng. Phys. 19, 29–36 (1997)
7. Blankevoort, L., Kuiper, J.H., Huiskes, R., Grootenboer, H.J.: Articular contact in a three-dimensionalmodel of the knee. J. Biomech. 24, 1019–1031 (1991)
9. Machado, M., Flores, P., Claro, J.C.P., Ambrósio, J., Silva, M., Completo, A., Lankarani, H.M.: Devel-opment of a planar multibody model of the human knee joint. Nonlinear Dyn. 60, 459–478 (2010)
10. Barber, J.R.: Contact problems for the thin elastic layer. Int. J. Mech. Sci. 32, 129–132 (1990)11. Ateshian, G.A., Lai, W.M., Zhu, W.B., Mow, V.C.: An asymptotic solution for the contact of two biphasic
cartilage layers. J. Biomech. 27, 1347–1360 (1994)12. Wilson, W., van Donkelaar, C.C., van Rietberger, R., Huiskes, R.: The role of computational models in
the search for the mechanical behaviour and damage mechanisms of articular cartilage. Med. Eng. Phys.27, 810–826 (2005)
13. Wu, J.Z., Herzog, W., Epstein, M.: Evaluation of the finite element software ABAQUS for biomechanicalmodelling of biphasic tissues. J. Biomech. 31, 165–169 (1997)
14. Caruntu, D.I., Hefzy, M.S.: 3-D anatomically based dynamic modeling of the human knee to includetibio-femoral and patello-femoral joints. J. Biomech. Eng. 126, 44–53 (2004)
15. Pérez-González, A., Fenollosa-Esteve, C., Sancho-Bru, J.L., Sánchez-Marín, F.T., Vergara, M.,Rodríguez-Cervantes, P.J.: A modified elastic foundation contact model for application in 3D modelsof the prosthetic knee. Med. Eng. Phys. 30, 387–398 (2008)
17. Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonicloading. J. Biomech. 44, 759–761 (2011)
18. Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid sub-strates. Appl. Math. Model. 35, 3201–3212 (2011)
19. Wang, J.H.-C., Ryu, J., Han, J.-S., Rowen, B.: A new method for the representation of articular surfacesusing the influence surface theory of plates. J. Biomech. 33, 629–633 (2000)
20. Huiskes, R., Van Dijk, R., de Lange, A., Woltring, H.J., Van Rens, Th.J.G.: Kinematics of the humanknee joint. In: Berme, N., Engin, A.E., Correia da Silva, K.M. (eds.) Biomechanics of Normal andPathological Human Articulating Joints, pp. 165–187. Martinus Nijhoff, Dordrecht (1985)
21. Glocker, Ch.: Formulation of spatial contact situations in rigid multibody systems. Comput. MethodsAppl. Math. 177, 199–214 (1999)
22. Aleksandrov, V.M., Vorovich, I.I.: Contact problems for the elastic layer of small thickness. J. Appl.Math. Mech. 28, 425–427 (1964)
23. Johnson, K.L.: Contact Mechanics. Cambridge Univ. Press, Cambridge (1985)24. Pandy, M.G., Sasaki, K., Kim, S.: A three-dimensional musculoskeletal model of the human knee joint.
Part 1: theoretical construction. Comput. Methods Biomech. Biomed. Eng. 1, 87–108 (1997)25. Argatov, I.I.: The pressure of a punch in the form of an elliptic paraboloid on a thin elastic layer. Acta
Mech. 180, 221–232 (2005)26. Chadwick, R.S.: Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math.
62, 1520–1530 (2002)27. Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers.
J. Biomech. 30, 371–375 (1997)28. Li, G., Sakamoto, M., Chao, E.Y.S.: A comparison of different methods in predicting static pressure
distribution in articulating joints. J. Biomech. 30, 635–638 (1997)29. Kücük, H.: The effect of modeling cartilage on predicted ligament and contact forces at the knee. Com-
put. Biol. Med. 36, 363–375 (2006)30. Abdel-Rahman, E.M., Hefzy, M.S.: A two-dimensional dynamic anatomical model of the human knee
joint. J. Biomech. Eng. 115, 357–365 (1993)31. Hohe, J., Ateshian, G., Reiser, M., Englmeier, K.-H., Eckstein, F.: Surface size, curvature analysis, and
assessment of knee joint incongruity with MRI in vivo. Magn. Reson. Med. 47, 554–561 (2002)32. Koo, S., Andriacchi, T.P.: A comparison of the influence of global functional loads vs. local contact
anatomy on articular cartilage thickness at the knee. J. Biomech. 40, 2961–2966 (2007)33. Argatov, I., Mishuris, G.: Contact problem for thin biphasic cartilage layers: perturbation solution. Q. J.
Mech. Appl. Math. 64, 297–318 (2011)34. Ateshian, G.A.: A B-spline least-squares surface fitting method for articular surfaces of diarthrodial
joints. J. Biomech. Eng. 115, 366–373 (1993)35. Dhaher, Y.Y., Delp, S.L., Rymer, W.Z.: The use of basis functions in modelling joint articular surfaces:
application to the knee joint. J. Biomech. 33, 901–907 (2000)
20 I. Argatov
36. van Ruijven, L.J., Beek, M., van Eijden, T.M.G.J.: Fitting parametrized polynomials with scattered sur-face data. J. Biomech. 32, 715–720 (1999)
37. Hirokawa, S., Ueki, T., Ohtsuki, A.: A new approach for surface fitting method of articular joint surfaces.J. Biomech. 37, 1551–1559 (2004)
38. Gilardi, G., Sharf, I.: Literature survey of contact dynamics modelling. Mech. Mach. Theory 37, 1213–1239 (2002)
39. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl.Mech. 42, 440–445 (1975)
40. Flores, P., Ambrósio, J.: On the contact detection for contact-impact analysis in multibody systems.Multibody Syst. Dyn. 24, 103–122 (2010)
41. Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for softmaterials in multibody dynamics. Multibody Syst. Dyn. 25, 357–375 (2011)
42. Monteiro, N.M.B., da Silva, M.P.T., Folgado, J.O.M.G., Melancia, J.P.L.: Structural analysis of the in-tervertebral discs adjacent to an interbody fusion using multibody dynamics and finite element cosimu-lation. Multibody Syst. Dyn. 25, 245–270 (2011)
43. Silva, M.P.T., Ambrósio, J.A.C., Pereira, M.S.: Biomechanical model with joint resistance for impactsimulation. Multibody Syst. Dyn. 1, 65–84 (1997)
44. Guess, T.M., Thiagarajan, G., Kia, M., Mishra, M.: A subject specific multibody model of the knee withmenisci. Med. Eng. Phys. 32, 505–515 (2010)
45. Argatov, I.I.: Asymptotic modeling of the impact of a spherical indenter on an elastic half-space. Int. J.Solids Struct. 45, 5035–5048 (2008)
