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.
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 \(N/mm\).
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.
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.
\[\frac{Pressure}{Overclosure} = \frac{(1 - \nu)E}{(1+\nu)(1-2\nu)}\frac{1}{t}\]
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 \(I_{12}=I_{23}=I_{13}=0.0\).
[HHD+14] | Mirco Herbort, Christian Hoser, Christoph Domnick, Michael J. Raschke, Simon Lenschow, Andre Weimann, Clemens Kösters, and Christian Fink. MPFL reconstruction using a quadriceps tendon graft: Part 1: Biomechanical properties of quadriceps tendon MPFL reconstruction in comparison to the Intact MPFL. A human cadaveric study. The Knee, 21(6):1169–1174, December 2014. URL: http://www.sciencedirect.com/science/article/pii/S0968016014001719, doi:10.1016/j.knee.2014.07.026. |
[KHC+94] | (1, 2) T. Kusayama, C. D. Harner, G. J. Carlin, J. W. Xerogeanes, and B. A. Smith. Anatomical and biomechanical characteristics of human meniscofemoral ligaments. Knee surgery, sports traumatology, arthroscopy: official journal of the ESSKA, 2(4):234–237, 1994. |
[SVL+16] | Colin R. Smith, Michael F. Vignos, Rachel L. Lenhart, Jarred Kaiser, and Darryl G. Thelen. The Influence of Component Alignment and Ligament Properties on Tibiofemoral Contact Forces in Total Knee Replacement. Journal of Biomechanical Engineering, 138(2):0210171–02101710, February 2016. URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4844247/, doi:10.1115/1.4032464. |
[AW12] | Shahram Amiri, David R. Wilson, Shahram Amiri, and David R. Wilson. A Computational Modeling Approach for Investigating Soft Tissue Balancing in Bicruciate Retaining Knee Arthroplasty, A Computational Modeling Approach for Investigating Soft Tissue Balancing in Bicruciate Retaining Knee Arthroplasty. Computational and Mathematical Methods in Medicine, Computational and Mathematical Methods in Medicine, 2012, 2012:e652865, October 2012. URL: http://www.hindawi.com/journals/cmmm/2012/652865/abs/, http://www.hindawi.com/journals/cmmm/2012/652865/abs/, doi:10.1155/2012/652865, 10.1155/2012/652865. |
[DKB10] | (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17) Yasin Y. Dhaher, Tae-Hyun Kwon, and Megan Barry. The effect of connective tissue material uncertainties on knee joint mechanics under isolated loading conditions. Journal of Biomechanics, 43(16):3118–3125, December 2010. URL: http://www.sciencedirect.com/science/article/pii/S0021929010004380, doi:10.1016/j.jbiomech.2010.08.005. |
[ELS+14] | (1, 2, 3, 4) Michael B. Ellman, Christopher M. LaPrade, Sean D. Smith, Matthew T. Rasmussen, Lars Engebretsen, Coen A. Wijdicks, and Robert F. LaPrade. Structural Properties of the Meniscal Roots. The American Journal of Sports Medicine, 42(8):1881–1887, August 2014. URL: https://doi.org/10.1177/0363546514531730, doi:10.1177/0363546514531730. |
[EKH+15] | (1, 2, 3, 4, 5, 6, 7) Joseph A. Ewing, Michelle K. Kaufman, Erin E. Hutter, Jeffrey F. Granger, Matthew D. Beal, Stephen J. Piazza, and Robert A. Siston. Estimating patient-specific soft-tissue properties in a TKA knee. Journal of Orthopaedic Research, pages n/a–n/a, September 2015. URL: http://onlinelibrary.wiley.com/doi/10.1002/jor.23032/abstract, doi:10.1002/jor.23032. |
[HCA+16] | (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) Michael D. Harris, Adam J. Cyr, Azhar A. Ali, Clare K. Fitzpatrick, Paul J. Rullkoetter, Lorin P. Maletsky, and Kevin B. Shelburne. A Combined Experimental and Computational Approach to Subject-Specific Analysis of Knee Joint Laxity. Journal of Biomechanical Engineering, 138(8):081004–081004, June 2016. URL: http://dx.doi.org/10.1115/1.4033882, doi:10.1115/1.4033882. |
[KSL+16] | (1, 2, 3, 4) Mohammad Kia, Kevin Schafer, Joseph Lipman, Michael Cross, David Mayman, Andrew Pearle, Thomas Wickiewicz, and Carl Imhauser. A Multibody Knee Model Corroborates Subject-Specific Experimental Measurements of Low Ligament Forces and Kinematic Coupling During Passive Flexion. Journal of Biomechanical Engineering, 138(5):051010–051010, March 2016. URL: http://dx.doi.org/10.1115/1.4032850, doi:10.1115/1.4032850. |
[LKST15] | (1, 2, 3, 4, 5) Rachel L. Lenhart, Jarred Kaiser, Colin R. Smith, and Darryl G. Thelen. Prediction and Validation of Load-Dependent Behavior of the Tibiofemoral and Patellofemoral Joints During Movement. Annals of Biomedical Engineering, 43(11):2675–2685, April 2015. URL: http://link.springer.com/article/10.1007/s10439-015-1326-3, doi:10.1007/s10439-015-1326-3. |
[SVL+16] | Colin R. Smith, Michael F. Vignos, Rachel L. Lenhart, Jarred Kaiser, and Darryl G. Thelen. The Influence of Component Alignment and Ligament Properties on Tibiofemoral Contact Forces in Total Knee Replacement. Journal of Biomechanical Engineering, 138(2):021017–021017, January 2016. URL: http://dx.doi.org/10.1115/1.4032464, doi:10.1115/1.4032464. |