Initial Material Properties

This section describes the material properties that are applied to the geometry in the finite element knee model.

Ligaments and Tendons (spring representation)

The model includes 15 ligament bundles. Stiffness and reference strain of each of these bundles are summarized in Table 5. For each bundle, the stiffness was evenly divided between the springs included in that bundle. The number of springs will be determined with a convergence study.

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Fig. 47 The nonlinear force-length curve that is used to define ligament behavior.

Note that every ligament is modeled as having a toe region that ends at 6% strain.

Prestrain

Each ligament is modeled as a bundle of fibers and prestrain (Table 5) is defined with respect to each fiber’s length in the reference state, therefore the prestrain is defined with respect to each fiber’s length in the joint’s position during imaging. The prestrain is used to define the slack length (Fig. 47).

Stiffness

Stiffness values (\(k_{eq}\)) (Table 5) define the ligament bundle’s equivalent stiffness, where each fiber has the same stiffness. Each fiber in a ligament bundle with \(n\) fibers (Fig. 48) has a stiffness of \(k_{fib} = k_{eq}/n\). Similarly, there are \(m\) springs (Fig. 48) that compose a fiber. Each spring in a fiber has the same stiffness value \(k_{spr}=m/k_{fib}\).

Adjacent fibers in the ligament mesh are connected with cross springs (Fig. 48). There are \(p=(n-1)(m+1)\) cross spring in a ligament mesh. each cross spring only carries tension forces, and has a constant stiffness of 10 \(N/mm\).

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Fig. 48 An example of a ligament mesh with \(n=6\) fibers in parallel, and are oriented in the superior-inferior direction. Each fiber contains \(m=19\) springs in series. There are also \(p=100\) cross springs that are oriented in the anterior-posterior direction. The cross springs connect adjacent ligament fibers. Note that this is used as an example, and not used to describe the insertion locations or size of any specific ligament.

The literature normally reportes stiffness values in units of \(N/mm\) or \(N/\epsilon\). To avoid limiting potential sources, values for both unit types are used. Every spring in the finite element model has a force-displacement curve that is similar to Fig. 47. Ligaments with a stiffness with units of \(N/\epsilon\) have the spring’s stiffness converted to units of \(N/mm\) based on the spring’s length in the reference state and the assigned prestrain.

Table 5 Material properties of the ligament and tendons that are modeled as bundles of springs.
Tissue Name Stiffness Reference Strain Comments
amAcl 2120 (N/strain) [AW12] -0.01 [HCA+16] ACL stiffness magnitude from [AW12] and amACL to plACL ratio from [KSL+16]
plAcl 2880 (N/strain) [AW12] -0.04 [HCA+16] ACL stiffness magnitude from [AW12] and amACL to plACL ratio from [KSL+16]
alPcl 5625 (N/strain) [AW12] 0.06 [HCA+16] PCL stiffness magnitude from [AW12] and alPCL to pmPCL ratio from [KSL+16]
pmPcl 3375 (N/strain) [AW12] -0.03 [HCA+16] PCL stiffness magnitude from [AW12] and alPCL to pmPCL ratio from [KSL+16]
sMclProx 1375 (N/strain) [AW12] 0.07 [HCA+16] The average of the anterior middle and posterior MCL springs reported in [HCA+16]
sMclDist 1375 (N/strain) [AW12] 0.07 [HCA+16] The average of the anterior middle and posterior MCL springs reported in [HCA+16]
dMcl 1000 (N/strain) [AW12] 0.03 [AW12]  
lcl 2000 (N/strain) [AW12] 0.05 [AW12]  
all 750 (N/strain) [EKH+15] 0.05 The prestrain value is arbitrarily defined
pfl 1000 (N/strain) [EKH+15] -0.02 [EKH+15]  
oplMid 1000 [EKH+15] -0.03 [EKH+15]  
Mpfl 29.4 (N/mm) [HHD+14] -0.05 [SVL+16]  
Lpfl 16 (N/mm) [MSIA09] 0.01 [SVL+16]  
PT 4334 (N/mm) [HBMA+06] 0.02 [SVL+16]  
pMFL 49 (N/mm) [KHC+94] 0.0 The prestrain value is arbitrarily defined
aMFL 49 (N/mm) [KHC+94] 0.0 The prestrain value is arbitrarily defined
amHorn 125 (N/mm) [ELS+14] 0.0 The prestrain value is arbitrarily defined
pmHorn 125 (N/mm) [ELS+14] 0.0 The prestrain value is arbitrarily defined
alHorn 150 (N/mm) [ELS+14] 0.0 The prestrain value is arbitrarily defined
plHorn 130 (N/mm) [ELS+14] 0.0 The prestrain value is arbitrarily defined

Ligaments and Tendons (continuum representation)

If in the future, a ligament and/or tendon is being modeled as a continuum, the tissue will be modeled as a transversely isotropic material.

Fiber Direction

The fibers are assumed to be uniformly distributed in, and perfectly bonded to the matrix. The fiber directions are defined in the image state, and run along the length of the ligament. Similar to [NBJvdG+17], the ACL and PCL are split into two fiber orientations each. These different orientations represent the amACL, plACL, alPCL, and pmPCL bundles of the ligament.

Material Model

Ligament/tensons will be modeled with a transversely isotropic hyperelastic material model. A Holzapfel-Gesser-Ogden [NBJvdG+17] hyperelastic model will be used. The properties for the ligament and tendons are shown in Table 6. Tissues that were not modeled in [NBJvdG+17] were assigned the same values as the MCL.

Table 6 Material properties of the ligament and tendons if they are modeled as a continuum. Note that all of the \(K_1\) and \(K_2\) come from [NBJvdG+17].
Tissue Name \(K_1\) \(K_2\) prestrain Comments
amAcl 52.27 5.789 0.03 [NBJvdG+17]  
plAcl 52.27 5.789 0.03 [NBJvdG+17]  
PCL 46.18 2.758 0.06 [HCA+16]  
MCL 41.21 5.351 0.07 [HCA+16]  
dMcl 41.21 5.351 0.03 [AW12] The \(K_1\) \(K_2\) values are taken from the MCL
lcl 41.21 5.351 0.05 [AW12]  
all 41.21 5.351 0.05 The prestrain value is arbitrarily defined and the \(K_1\) \(K_2\) values are taken from the MCL
pfl 41.21 5.351 -0.02 [EKH+15] The \(K_1\) \(K_2\) values are taken from the MCL
oplMid 41.21 5.351 -0.03 [EKH+15] The \(K_1\) \(K_2\) values are taken from the MCL
Mpfl 41.21 5.351 -0.05 [SVL+16] The \(K_1\) \(K_2\) values are taken from the MCL
Lpfl 41.21 5.351 0.01 [SVL+16] The \(K_1\) \(K_2\) values are taken from the MCL
PT 41.21 5.351 0.02 [SVL+16] The \(K_1\) \(K_2\) values are taken from the MCL
pMFL 41.21 5.351 0.0 The prestrain value is arbitrarily defined and the \(K_1\) \(K_2\) values are taken from the MCL
aMFL 41.21 5.351 0.0 The prestrain value is arbitrarily defined and the \(K_1\) \(K_2\) values are taken from the MCL
Meniscal Horns 41.21 5.351 0.0 The prestrain value is arbitrarily defined and the \(K_1\) \(K_2\) values are taken from the MCL

Prestrain

A uniform prestrain will be assigned to the ligament/tendon [MEHW16] with the prestrain values defined in Table 5.

Cartilage

The model includes 4 cartilage geometries. The cartilage is modeled using an elastic foundation model [LKST15]. Contact pressures are calculated using a pressure-overclosure relationship.

\[\frac{Pressure}{Overclosure} = \frac{(1 - \nu)E}{(1+\nu)(1-2\nu)}\frac{1}{t}\]
Table 7 Material properties of the cartilage geometries.
Tissue Name E (MPa) Poisson’s ratio Thickness (mm)
Femoral Cartilage 12 [DKB10] 0.45 [DKB10] 3 [LKST15]
Medial Tibial Cartilage 12 [DKB10] 0.45 [DKB10] 3 [LKST15]
Lateral Tibial Cartilage 12 [DKB10] 0.45 [DKB10] 3 [LKST15]
Patellar Cartilage 12 [DKB10] 0.45 [DKB10] 3 [LKST15]

Menisci

The model includes two meniscus geometries. The menisci are modeled as transversely isotropic linearly elastic [DKB10]. The circumferential elastic modulus is greater than the axial and radial values. Young’s modulus and Poisson’s ration in the axial direction is equal to radial direction.

Table 8 Material properties of the menisci geometries.
Tissue Name \(E_{circumferential}\) (MPa) \(\nu_{circumferential}\) \(E_{axial}=E_{radial}\) (MPa) \(\nu_{axial}=\nu_{radial}\)
Medial Meniscus 120 [DKB10] 0.45 [DKB10] 20 [DKB10] 0.3 [DKB10]
Lateral Meniscus 120 [DKB10] 0.45 [DKB10] 20 [DKB10] 0.3 [DKB10]

Mass Properties

Mass properties are required because we are using an explicit finite element model. Nominal masses are assigned to reduce the amount of intertial effects in the model. Every ligament node is assigned a mass of 0.0001 kg. The cartilage and menisci are assigned a density of 0.000001 kg/mm^2. The bones are rigid bodies, however the connector elements that are rigidly attached to the bones have a rotational inertia of \(I_{11}=I_{22}=I_{33}=0.0001\) kg*mm^2, and \(I_{12}=I_{23}=I_{13}=0.0\).

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