Target Outcome

Coefficients of a desired constitutive model fitting tissue stress-strain response

Prerequisites

Infrastructure

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.

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.

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.

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.

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).