1. BME 615 LIGAMENT/TENDON MECHANICS
Ray Vanderby
Biomedical Engineering
Orthopedics and Rehabilitation
University of Wisconsin

2. Ligaments and tendons are similar - but not the same tissues
There are differences in:
Function (levels and frequency of loading)
Attachments
Subtle differences in composition
Cells (primarily FB) are phenotypically different

4. Ligament Hierarchy
Frank et al., in Woo and Buckwalter, 1987

5. Ligament Fibers from SEM

6. Ligament Histology Woo and Buckwalter, Injury and Repair of Musculoskeletal Soft Tissues, AAOS, 1987

7. Why Ligaments? Important anatomical structures Check rein
Proprioceptive feedback
Guide joint motion
Relative simplicity
Biologically
Everybody has them and hurts them
Anatomically
Mechanically
Accessible (surgically)
Easy to test (mechanically)
Display typical soft tissue behavior

8. Knee Ligament Injury and Healing
When these ligaments are injured, they mount a healing response that has been studied extensively. Ligament injuries occur across all age groups and activity levels. MCL and ACL injuries can occur through contact (Lee Evens) or non-contact activity (woman) and combined MCL-ACL injuries are common. These ligaments vary greatly in there healing responses. The ACL does not heal well and as a result is often surgically replaced with a graft. On the other hand, it is clinically it is found that the MCL heals well on its own, even though studies show mechanical and biological deficits for years post injury.

9. Biomechanics versus Necromechanics

10. Ligament Homeostasis

11. Effect of Age on Ligament
S. Woo, from Knee Ligaments, Daniel et al. eds. 1990

12. Testing Methods
S. Woo, from Knee Ligaments, Daniel et al. eds. 1990

13. Mechanical Testing Methods
unloaded
stretched
Freshly harvested tissue
Grip bones in bath
Silicon grease optical markers
Stretch with servo-electric load frame
Video analysis for displacement from L0
Record force transducer on video tape

14. Pseudo-Elastic Behavior

15. Typical Behavior of Ligaments
S. Woo, from Knee Ligaments, Daniel et al. eds. 1990

16. Typical Behavior of Rat Ligaments

17. Elastic Phenomenological Models
Plot  versus  and curve fit
Algebraic curve fit
Exponential curve fit
Power law curve fit
Logarithmic curve fit

18. Nonlinear Elastic Models (1D Pseudoelastic)
Wertheim (1847) 2 = a 2 + b 
Morgan (1960)  = a n
Kenedi et al. (1964)  = k d
 = A [eB - 1]
Ridge and Wright (1964)  = C + k b
 = x + y ln 

19. Nonlinear Elastic Models (3D Pseudoelastic)
Apparent Poisson’s ratios in ligament & tendon
Screen and Chang (2007) in rat tail tendon 0.08
Hewitt et al., (2001) in capsular ligament
Lynch et al., (2003) in bovine flexor tendon 2.98
Quapp and Weiss (1998) suggest combining soft 3D material model for ground substance (Mooney Rivlin model – we will study later) with phenomenological stiffening in one direction to represent the fibers:
Mooney Rivlin model for ground substance
Fibers in 1 direction
SP Reese et al., J Biomech, 2010

20. Nonlinear Elastic Models (3D Pseudoelastic)
SP Reese et al., J Biomech, 2010

21. Nonlinear Elastic Models (3D Pseudoelastic)

22. Nonlinear Elastic Models (3D Pseudoelastic)
JA Weiss, Med Eng & Phys, 2005
Weiss reports that measured in situ strains are in the 3-10% range!
Experimental testing with preload suggests that much of this must be unfolding of crimped fibers and straightening of a compressed structure.
If this was a real load (above preload), it would be incredibly inefficient to have to load tissues this much with each knee flex and it would damage the tissue

23. Nonlinear Elastic Models SLY Woo, et al. Chapt
Nonlinear Elastic Models SLY Woo, et al. Chapt. 6, Basic Orthopaedic Biomechanics, Mow and Hayes, ed. 1997
F = ci(x-i)

24. Microstructural Models
Each spring represents the recruitment of tensile elements (collagen or elastin fibers)
Superimposed upon ground substance behavior
In the limiting case, a probability density function is used represent fiber recruitment

25. Elastic Microstructural Model (Hurschler, et al.)

26. where: is the tissue stretch ratio
is the straightening stretch ratio, which is the stretch after which fibers begin to bear load
is the axial stress-stretch law of the fiber
is the stretch at which the first fibers begin to bear load
is a Weibel PDF to describe recruitment of fibers as a function of straightening stretch
are constants to describe the PDF
is the percent of fibers recruited

27. Note, Weibull PDF is scaled to data (area under each PDF = 1)

28. Viscoelastic Behavior

29. Viscoelastic Behavior

31. Viscoelasticity
Time dependent behavior in soft hydrated cells & tissues
Linear and nonlinear viscoelastic models frequently used to phenomenologically describe mechanical behavior of ligaments and other biological tissues
Usually model based on curve fitting creep or relaxation experiment at one level of loading
Most commonly used model is quasi-linear viscoelasticity (QLV)

32. Linear Viscoelasticity 1
Constitutive equations are Boltzmann integrals,
J(t) = creep compliance
E(t) = relaxation response (axial Young's relaxation modulus)
Derivative and convolution theorems for Laplace transforms
convert to
s(s) = s E(s)e(s) and e(s) = s J(s)s(s).

33. Summary
Linear viscoelasticity requires “elastic” stress strain to be linear
Linear models including “spectral damping” are frequency dependent on their level of damping. Above is NOT true in ligaments (as well as most other soft tissues)
Simple linear models (e.g. 3 parameter solid) are not robust and limited for biomechanical applications!

34. Quasi-Linear Viscoelasticity 1
Assumes that time dependent response J(t-) is separable from stress dependent response e()
This means that while the magnitude of the compliance depends on stress, the overall shape of the creep curve is independent of stress. QLV admits non-linearity in the elastic portion of the stress strain curve e().

35. Quasi-Linear Viscoelasticity 2
where G(t-) is called the “reduced relaxation function”
Assumes that time dependent response G(t-) is separable from strain dependent response e()
This means that while the magnitude of the stiffness depends on strain, the overall shape of the relaxation curve is independent of strain!

36. Quasi-Linear Viscoelasticity 3
where e() is called the “elastic response” and is
obtained by curve fitting an “elastic” stress-strain curve
A simple derviative with respect to strain give the stiffness term for the QLV Boltzmann integral

37. Quasi-Linear Viscoelasticity 4
In Woo chapter in course notes and in Abramowitch & Woo paper in supplemental materials, a QLV model is fit a ramp function under displacement control up to a constant strain and then held.
With these stress strain data, material parameters C, 1 and 2 are obtained for reduced relaxation function.
Elastic material constants A and B can be fit by this process or from “quasi-static” behavior.
Then the model is used to predict a more complex load history and compared to experimental results.

38. Quasi-Linear Viscoelasticity 5

39. Mechanics of Failure

40. Mechanics of Failure – Insertions & mid-substance
Woo & Buckwalter, eds., Injury and Repair of the Musculoskeletal Soft Tissues, AAOS, 1987

41. Mechanics of Failure – Age & Rate
Woo & Buckwalter, eds., Injury and Repair of the Musculoskeletal Soft Tissues, AAOS, 1987

42. Mechanics of Healing Tissue – Repair & Mobilization (extra-capsular ligaments heal, intra-capsular ligaments do not)
Woo & Buckwalter, eds., Injury and Repair of the Musculoskeletal Soft Tissues, AAOS, 1987

43. Mechanics of Ligament and Tendon Failure
Bear in mind that most tests are done with joints positioned so that ligament fibers are stressed as uniformly as possible!!!!
Hence, ligament loads and “stresses” can be thought of as upper bound on failure loads.
In most cases of trauma, the joint is not at an optimal angle, so ligament fibers will fail one at a time (like tearing a phonebook) and at lower loads.
Tendon testing requires gripping soft tissue –
hence stress concentrations reduce peak load or stress

44. Mechanics of Ligament and Tendon Failure
By testing to failure through a range of joint positions, a more robust but still phenomenological failure theory can be formulated.
Alternatively, a micromechanical failure theory can be formulated by setting a stretch limit on fibers after they are recruited so they fail when stretched too far. Then, the PDF should not be continuous in either direction, and it must be formulated as a function of joint position (since fiber recruitment will occur in different rates at different joint positions).

45. Ligament Path &Tendinopathy
Tendonitis
Tendonosis
Partial Tears
Ligament Sprains (Degrees)
Tendon Strains (stretch or partial tear)
Muscle pulls
Insertion issues (enthesis)