Contents
Target Outcome
Coefficients of a desired constitutive model fitting tissue stress-strain response
Prerequisites
Infrastructure
Python. Python is a programming language (GPL compatible license, http://www.python.org/). Python is the default scripting environment for Open Knee(s); for more details, please refer to Infrastructure/ScriptingEnvironment.
SciPy. SciPy is a Python based open-source software for mathematics, science, and engineering (BSD license, see http://www.scipy.org/).
Spyder. Spyder is an interactive development environment for Python with advanced editing, interactive testing, debugging and introspection features (MIT license, see http://code.google.com/p/spyderlib/).
npTDMS. npTDMS is Cross-platform, NumPy based module for reading TDMS files produced by LabVIEW. TDMS files are the binary files used for the robotic testing raw data. (LGPL license, see https://pypi.python.org/pypi/npTDMS/).
Previous Protocols
For more details, see Specifications/ExperimentationTissueMechanics.
Protocols
Input
- Sample tissue geometric properties; width, height, thickness OR diameter, thickness
- Sample tissue time-displacement, time-force profiles for multistep stress-relaxation tests
- Sample tissue image data (with time stamp) (to reconstruct local tissue strains)
Constitutive Models and Parameters
Material properties and the constitutive models are adapted from literature as described below. It should be noted that these parameters and the constitutive representations may me modified in upcoming stages of modeling and simulation through the lifecycle of the model.
Bone
All bones (femur, tibia, fibula, patella) will be assumed to be rigid bodies. Bones have a much higher stiffness than the other knee tissues. Rigid body assumption simplifies computation, therefore decreasing the computational cost and facilitates definition of joint kinematics and/or kinetics as loading and boundary conditions. A density of 1e-9 tonnes/mm^3 (identical to water; consistent with spatial units of mm). It should be noted that density assignment will not have importance on static simulations without the action of gravity. It is provided as a place holder.
Cartilage
Cartilage will be modeled as a nearly incompressible Neo-Hookean material defined by FEBio setting C2 parameter of FEBio material type Mooney-Rivlin (uncoupled) [1]. This is a fairly simplified representation of cartilage’s mechanical behavior [2]. We believe that it would be adequate and computationally less challenging for joint level simulations while providing an opportunity to understand local mechanical environment on and within the cartilage. Cartilage constitutive model and coefficients will be similar to previous modeling study [3], which reported an elastic modulus of 15 MPa and Poisson’s ratio of 0.475. Corresponding Neo-Hookean material coefficients are noted below.
Density* (tonnes/mm^3) |
C1 (MPa) |
C2 (MPa) |
K (MPa) |
1e-9 |
2.54 |
0 |
100 |
All units are consistent with spatial units of mm
*Density assignment is a place holder; it will not have importance on static simulations without the action of gravity.
Ligaments and Tendons
Ligaments and tendons will be modeled as nearly incompressible, transversely isotropic, hyperelastic material with a Mooney-Rivlin ground substance (Neo-Hookean by setting C2 = 0) [1]. This type of representation accommodates tensile dominant behavior of the ligaments dictated by their fiber alignment across their longitudinal axis [4]. The parameters will be identical to a previous modeling study [5], which fitted data from literature. These values are noted below.
Ligament |
Density* (tonnes/mm^3) |
C1 (MPa) |
C2 (MPa) |
K# (MPa) |
C3 (MPa) |
C4 |
C5 (MPa) |
λm |
ACL |
1e-9 |
1.95 |
0 |
146.41 |
0.0139 |
116.22 |
535.039 |
1.046 |
PCL |
1e-9 |
3.25 |
0 |
243.9 |
0.1196 |
87.178 |
431.063 |
1.035 |
MCL |
1e-9 |
1.44 |
0 |
793.65 |
0.57 |
48.0 |
467.1 |
1.063 |
LCL$ |
1e-9 |
1.44 |
0 |
793.65 |
0.57 |
48.0 |
467.1 |
1.063 |
PL |
1e-9 |
2.75 |
0 |
206.61 |
0.065 |
115.89 |
777.56 |
1.042 |
QT& |
1e-9 |
2.75 |
0 |
206.61 |
0.065 |
115.89 |
777.56 |
1.042 |
All units are consistent with spatial units of mm.
ACL: anterior cruciate ligament. PCL: posterior cruciate ligament. LCL: lateral collateral ligament. MCL: medial collateral ligament. PL: patellar ligament. QT: quadriceps tendon.
*Density assignment is a place holder; it will not have importance on static simulations without the action of gravity.
#Bulk modulus (K) is calculated as (K = 1/D); D obtained from source literature.
$LCL properties were assumed to be identical to MCL.
&QT properties were assumed to be identical to PL. This assumption should not have significance as anticipated use of QT is to transfer loads to patella in a distributed manner.
Constitutive modeling of the ligaments requires specification of a fiber direction. In FEBio, under the material definition, fiber type can be specified as “vector” so that the initial direction of all fibers in the material point are along the direction of the vector [1]. For each ligament and tendon this direction will be defined as the direction of the longest edge of the oriented bounding box of the tissue to approximate the longitudinal alignment of the tissue.
Meniscus
Menisci will be modeled as nearly incompressible, transversely isotropic, hyperelastic material with a Mooney-Rivlin ground substance (Neo-Hookean by setting C2 = 0) [1]. This material model is seemingly more complicated than transversely orthothropic linear elastic models in literature. Yet, it will allow a convenient way to represent the behavior of meniscus which is largely dictated by its circumferential stiffness (based on fiber alignment) with the capacity to sustain compressive loading [6]. For convenience, the parameters will be identical to those used in a previous modeling study of meniscus [7] and they are noted below.
Density* (tonnes/mm^3) |
C1 (MPa) |
C2 (MPa) |
K# (MPa) |
C3 (MPa) |
C4 |
C5 (MPa) |
λm |
1e-9 |
4.61 |
0 |
92.16 |
0.1197 |
150.0 |
400.0 |
1.019 |
All units are consistent with spatial units of mm.
*Density assignment is a place holder; it will not have importance on static simulations without the action of gravity.
#Bulk modulus (K) is calculated as (K = 1/D); D obtained from source literature. It should be noted that due to differences in dilatational component of constitutive models (used in here and in source literature), equivalence of K is an approximation.
Constitutive modeling of the menisci requires specification of fiber direction. In FEBio, fiber direction can be specified for each element individually, in the ElementData section [1]. The meniscus fibers will be oriented in the circumferential direction. To accomplish this, a best-fit oval will be calculated to represent the shape of the meniscus in the transverse plane. The fibers in each element will be oriented so that their direction will be in parallel with the tangent of the closest point on the oval.
In Situ Ligament Strain
Prestrain formulation of FEBio PreStrain Plugin is described in detail in literature [8]; from this literature:
“The gradient of the local mapping from the stress-free to the prestressed reference configuration is represented by Fp, which will be referred to as the prestrain gradient. The total elastic deformation gradient Fe is determined by the composited deformation gradient, ”
Prestrain type will be defined as “in-situ stretch”, where the prestrain gradient is calculated based on the initial fiber stretch. The in situ stretch can be defined at the element level in the ElementData section of the FEBio input file, or the initial stretch can be defined for all fibers in one ligament. We will use the isochoric prestrain generator option, as the material will be assumed to be incompressible. The update rule type will be chosen as “prestrain”, meaning that Fp will be updated from the initial prestrain gradient given, to eliminate distortion due to incompatibility with the reference geometry (as opposed to enforcing the given in-situ stretch, resulting in possible distortion of the geometry). We will apply the prestrain using an elimination of distortion approach, where the goal is to eliminate distortion induced by the incompatibility of the initial prestrain gradient [8]. One average initial stretch will be defined for all fibers in each ligament as:
Ligament |
Initial Stretch |
ACL* |
1.016 |
PCL# |
1.0 |
MCL* |
1.034 |
LCL* |
1.027 |
PL# |
1.0 |
QT# |
1.0 |
*Initial strain was set to average of values reported in the modeling study of Dhaher et al.[9], who referred to previous knee models [5] and experimental data [10].
#Initial strain set to zero due to lack of data, similar to Dhaher et al. [9]
Since the prestrain update rule is chosen to eliminate distortion, the above values would only be used as an initial guess for fiber stretch. Due to changes in cross sectional area of the ligament, in order to maintain equilibrium, the solver will update the fiber stretches as needed. Due to this fact, it may be unnecessary for the modeler to define “anterior”, “posterior” etc. regions (as done in previous studies [5],[9]), as the values are likely to change. Thus, one average value will be given as the initial guess for fiber stretch for all regions in the ligament. After the solver determines the updated initial stretch values, they can be compared to the above literature values, and the average initial guess may be calibrated in upcoming modeling stages, if necessary.
Output
- Fit to tissue-specific stress-strain data for a desired constitutive relationship
- Constitutive relationship formulation
- Numerical values of coefficients of the constitutive relationship
- Fit error representing the quality of tissue-specific representation of material properties
References
1. FEBio User’s Manual Version 2.8. Available at: https://help.febio.org/FEBio/FEBio_um_2_8/index.html. (Accessed: 13th August 2018)
2. Henak, C. R., Anderson, A. E. & Weiss, J. A. Subject-specific analysis of joint contact mechanics: application to the study of osteoarthritis and surgical planning. J Biomech Eng 135, 021003 (2013).
3. Donahue, T. L. H., Hull, M. L., Rashid, M. M. & Jacobs, C. R. A finite element model of the human knee joint for the study of tibio-femoral contact. J Biomech Eng 124, 273–280 (2002).
4. Weiss, J. A., Gardiner, J. C., Ellis, B. J., Lujan, T. J. & Phatak, N. S. Three-dimensional finite element modeling of ligaments: technical aspects. Med Eng Phys 27, 845–861 (2005).
5. Peña, E., Calvo, B., Martínez, M. A. & Doblaré, M. A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint. J Biomech 39, 1686–1701 (2006).
6. Fithian, D. C., Kelly, M. A. & Mow, V. C. Material properties and structure-function relationships in the menisci. Clin. Orthop. Relat. Res. 252, 19–31 (1990).
7. Shriram, D., Praveen Kumar, G., Cui, F., Lee, Y. H. D. & Subburaj, K. Evaluating the effects of material properties of artificial meniscal implant in the human knee joint using finite element analysis. Sci Rep 7, 6011 (2017).
8. Maas, S. A., Erdemir, A., Halloran, J. P. & Weiss, J. A. A general framework for application of prestrain to computational models of biological materials. J Mech Behav Biomed Mater 61, 499–510 (2016).
9. Dhaher, Y. Y., Kwon, T.-H. & Barry, M. The effect of connective tissue material uncertainties on knee joint mechanics under isolated loading conditions. J Biomech 43, 3118–3125 (2010).
10. Gardiner, J. C. & Weiss, J. A. Subject-specific finite element analysis of the human medial collateral ligament during valgus knee loading. J. Orthop. Res. 21, 1098–1106 (2003).