Page 192 - Advances in Biomechanics and Tissue Regeneration

188 9. COMPUTATIONAL MUSCULOSKELETAL BIOMECHANICS OF THE KNEE JOINT

9.4 JOINT STABILITY ANALYSES

Stability of a mechanical or biological structure is defined as its ability to withstand small perturbations without
hypermobility or excessive motions causing damage to the system. As a musculoskeletal system, knee joint stability
is maintained by an intricate interplay between active musculature and passive tissues. Knee instability usually man-
ifests itself by giving way, excessive laxity, and pain. So, knee joint stability assessment and evaluation of parameters
affecting it are crucial in injury prevention and treatment managements. In clinical context, knee joint stability is usu-
ally evaluated by its laxity under external physiological loads and disturbances. There are a few tests to evaluate the
knee stability for different injuries; for example, Lachman and pivot shift tests are performed to detect anterior cruciate
ligament injuries [137, 138].
The first stability analysis of the human spine by the minimization of its potential energy was performed by
Bergmark [139].Wehaveimplementedthe same approachtoevaluatefor thefirst timethestability of theknee
joint in gait [140]. To do so, muscle forces are initially calculated in the equilibrium phase of the study as described
in preceding paragraphs. At the final deformed configuration, all muscles are then replaced by uniaxial elements
F
with a force-dependent axial stiffness [132]. Muscle stiffness k ¼ q is taken linearly proportional to the computed
L
total (passive and active) muscle force (F) and inversely proportional to its current length (L). q is a constant dimen-
sionless coefficient taken the same for all muscles [106, 139, 141]. After replacing muscles with springs, perturba-
tion and buckling analyses, under a unit load at and along GRF and at the deformed loaded configuration, are
performed. Stability analyses are iteratively performed for different values of q to evaluate the minimum (critical)
q below which the system ceases to be stable [140].Alowercritical q suggests a more stable system so that, at
the extreme, the critical q¼0 indicates that the passive system is stable under given loads and does not need
any additional stiffness from musculature for stability although muscle forces are very likely needed to maintain
equilibrium.
Short of a nonlinear postbuckling analysis, linear buckling and perturbation analyses are two common stability
tools to evaluate the stability (divergence type) of a structure under given static-dynamic loading conditions. Buckling
analysis at a given q predicts the reserve additional load (buckling load) that the system can support before exhibiting
instability; at the neutral stable condition, this buckling load approaches zero. The perturbation analysis, on the other
hand, quantifies the system response under a small perturbation that tends to infinity as the system becomes unstable.
The critical q in the intact knee joint at heel strike was found at about 14. In this case, as q approaches the critical q,
displacement in the perturbation analysis substantially increases, whereas the lowest buckling (or reserve) load drops
to nil (Fig. 9.5).

FIG. 9.5 Displacement at unit perturbation load along the ground reaction force (on the left side) and lowest buckling force (Fcr) (on the right side) at
different values of q (muscle stiffness coefficient) in linear perturbation and buckling analyses performed at the deformed loaded configuration of the
intact knee joint at the initiation of contact (heel strike) in gait.

I. BIOMECHANICS