Modeling of failure mode in knee ligaments depending on the strain rate

The knee is one of the most commonly injured joints. The stability of the knee is dependent on several muscles, the meniscus and four major ligaments. These knee ligaments are the medial collateral ligament (MCL), the lateral collateral ligament (LCL), the anterior cruciate ligament (ACL), and the posterior cruciate ligament (PCL).

The effect of strain rate on the failure modes of the bone-ligament-bone complex has been investigated. There are three failure mechanisms [1, 2]. The first is midsubstance tearing in which the fibers of the ligament pull apart in the tissue, generally during a fast loading rate. The second mode is ligament-bone interface failure in which the fibers fail at the ligament-bone insertion without fracture of the underlying bone. The final mode is tibia or femoral avulsion fractures, usually occurring during relatively slow loading rates [1, 2].

Ligaments and bone have nonlinear time and history-dependent viscoelastic properties. These properties can be expressed in terms of stress-strain relationships with rate as an additional parameter. Figure 1 and 2 show typical stress-strain relationships of the ligament and bone respectively. For the ligament the slope of the stress-strain curve becomes nearly constant in the prefailure phase. The curve is composed of four regions; toe, linear, failure, and complete failure [1‐4]. The ligament failure mode varies with different types of ligament (ACL, PCL, MCL, and LCL) due to their different viscosities [5]. Bone also has non-linear behavior and the elastic modulus is higher in bone than in ligament [6].

Stress (MPa), σ, is defined as the intensity of internal force over the unit cross-sectional area of the original specimen [1]. The definition of strain (% or m/m), ε, is the change in length of the specimen, divided by its initial length [1]. Strain with percentage units is obtained by multiplying the m/m form by 100. The strain rate represents the change in elongation with time [4]. The extension or deformation rate (m/sec) of the specimen divided by its initial length (m) gives the strain rate, as does the rate of change of the strain itself (dε / dt). Its unit is %/sec (or simply sec^-1). Unfortunately not all of these parameters, or overall dimensions, are always given so that reported deformation rates cannot be converted into strain rates or vice versa. In addition, strain rates data alone does not allow calculations of failure load, since neither direct force nor effective mass are typically reported.

The electronic version of Medline was used to identify articles for this review. The keywords used for were strain rate, knee injury, and ligament. From the result of this search all involving strain-rate experiments of the bone-ligament-bone complex were further considered. Papers with only one bone-ligament attachment were excluded, as were those with only relative strain rates. The selected papers were further limited to those reporting quantified strain rates and the specific nature of the failures.

There are many recent studies on the effect of the strain rate on the failure mode of the knee [7‐15]. These are summarized in Table 1. The results of most studies show that at low strain rates avulsion occurs at the ligament insertion site on the bone or the bone itself may fail. At high strain rate, it is generally reported that the ligament tears [7‐15]. However, the definitions of a fast or slow rate, and the actual rate range covered in each study, are different. In addition, the orientation of the bone-ligament-bone complex during testing may also affect the results, but details of this orientation are not typically provided.

Viidik et al. studied the effects of training in a running machine on rabbits. They showed the weakest point in the tibia-ligament-femur system to be one of the bony attachments, most often the tibial one [7].

In 1974(a) Noyes et al. used rhesus monkeys to investigate the effect of strain rate changes on the failure mode. The approximate strain rates were 0.62 %/sec^-1 at the slow rate and 67 %/sec^-1 at the fast rate. Ligament tearing happened more at the fast strain rate, with bone avulsion occurring more at the slow strain rate [8]. They also analyzed the effects of activity on ligament failure using the strain rate of 66.2 %/sec [9]. This study showed 60 % ligament tearing in the control group, 44% ligament tearing and 39% bony avulsion with 8 weeks of immobilization, 45 % ligament tearing and 36% bony avulsion from an exercise group, and 65 % ligament tearing from a reconditioning group [9].

Alm et al. used a 15 mm/sec deformation rate on Beagle dogs to study the strength of the ACL. The mid part of the ligament ruptured in 93.7% of the tests [10].

Crownishield et al. tested failure characteristics of the MCL of rats at strain rates of 8.45*10^-2, 84.5, 19000, and 51000 %/sec. As the strain rate increased, ligament tearing occurred more than avulsion. These authors described the range of injury strain rates in man as from approximately 50 %/sec to 150,000 %/sec [11].

Kennedy et al. conducted ultimate failure strengths of the ACL at the strain rates of 40, and 140 %/sec. The ages of the human specimens ranged from 20 to 75 years, with a mean of 62 years. The major mode of failure was ligament tearing at both of the strain rates. [12].

In 1976 Noyes et al. evaluated the strength of the ACL in humans and rhesus monkeys to analyze age and species related changes. The age range of the human specimens was from 48 to 86 years for an older group, and from 16 to 26 years for a younger group. The rhesus monkeys were 15 years old. For the human specimens a strain rate of 100%/sec was used for both old and young specimens. A strain rate of 66 %/sec was used for the monkeys. Ligament tearing was predominant in the specimens from younger humans and from rhesus monkeys. Avulsion of bone beneath the ligament insertion site was seen for the older humans [13].

Woo et al. studied the effect of strain rate on MCL failures in skeletally immature and mature rabbits. The range of strain rate used was from 0.011 to 222 %/sec. Tibia avulsion developed in all specimens from skeletally immature rabbits with the strain rate from 0.011 to 155 %/sec. Results for this group were not reported above 155 %/sec. However, ligament tearing occurred from skeletally mature rabbits with the strain rate from 0.011 to 222 %/sec [14].

Schenck et al. investigated two "clinically applicable" stain rates, 100 and 5400 %/sec to study human ACL and PCL hyperextension. In the PCL, ligament tearing occurred at the lower stain rate, where as bone avulsion occurred at the higher strain rate. The order of these results is contrary to that of most of the other studies. In the ACL, the major failure mode at the low stain rate was similar to the PCL, with a mixed failure pattern at the higher strain rate [15].

The scope of these studies covers a very wide range of strain rates, multiple species, various ages, and in some cases direct intervention. Given this diverse set of results, a mathematical model of ligament and bony insert behavior was developed to provide an integrated understanding of the reported variations in failure mode.

The failure modes were used to analyze the studies of Table 1 and restate the results as shown in Table 2.

In Viidik, et al the failures are predominately mode I. Even though training had some positive effect on insertion point strength, this effect was not great enough to make σ_{ult, B} > σ_{ult, L} and therefore avulsion remained the primary cause of failure.

Noyes, et al in 1974a, and 1974b, studied the ACL failure mode of Rhesus monkeys. The first study shows mode I at slow strain rate and mode III at fast strain rate. The mixed results suggest operation in the mode II region with significant individual variability. In the second Noyes, et al's study mode III predominates. Assuming immobilization reduces strength, σ_{ult, B} reduced more than σ_{ult, L}, so that they approached equality. In exercise, either σ_{ult, B} and σ_{ult, L} is not significantly affected, or they both may increase without significant relative changes. In the latter case the analysis is limited by the lack of information actual failure loads or stresses. Even though changing their relative value, this effect is consistent with Viidik, et al. The reconditioning after immobilization suggest that reconditioning has a much greater effect on σ_{ult, L}, such that σ_{ult, L} comes to exceed σ_{ult, B} and the failure changes to mode III. Fig. 7 illustrates the changed in the ultimate stress of bone and ligament depending on the effect of activity as reported by Noyes, et al (1974b)

Kennedy, et al and Noyes, et al in 1976 studied ACL failure in human specimens. Even though different strain rates were used, mode III applies to both strain rates in Kennedy. In the age range from 20 to 75 years, σ_{ult, B} > σ_{ult, L}. The strain rates used don't affect the relative value of σ_{ult, B} and σ_{ult, L}. However, we have different results in Noyes, et al's study. The older (48–86 years) and younger group (16–26 years) data fit mode I with σ_{ult, B} < σ_{ult, L} and III with σ_{ult, B} > σ_{ult, L} respectively. Assuming aging reduces strength, σ_{ult, B} reduced more than σ_{ult, L}. The Rhesus monkeys group had the same failure mode as the younger group. These results demonstrate that strain rate alone is not the defining parameter. Analytically this would be reflected by E, η, and σ_ult being a function of age and other conditions.

In Crownishield, et al's rabbit specimens, like Noyes, et al's study, increasing the strain rate caused a shift from mode I, to II, to III. The MCL study of Woo et. al is similar to Crownishield et. al except for the range of strain rate. Mode I applies to the immature group for all strain rates and mode III to mature group for all strain rates. Maturing increases strength, presumably more so for bone than ligament, and causing σ_{ult, B} > σ_{ult, L}. The effect of the strain rates used in each group is not seen, probably because they are not sufficiently different.

Schenck, et al reported their results as different from previous studies, but this is not clear given the variation in specimens and strain rates used. Therefore, the results of the effect of strain rate for the PCL and ACL were reverse from those of the previous studies, using strain rates intended to correlate differences between sporting and trauma [15]. However, only one other study, Crownnishield, et al using the rat MCL, used strain rates as high or higher than Schenck, et al's "fast" rate. Accepting that the rate and human results are consistent, Schenck et al's results suggest that the relative effect of stain rate on strength has a second cross over point. Fig. 8 shows this relationship.

The effect of the strain rate on the failure modes of the bone-ligament-bone complex is an important factor in knee joint injury. We have defined five ranges of failure as a function of strain rate and other parameters. Avulsion happens more than ligament tearing in mode I which is primarily associated with low strain rates. The number of avulsion is comparable with the number of ligament tears in mode II, where individual variation is the dominant effect. Ligament tearing occurs more than bony avulsion in mode III that is associated with high strain rates. Finally, at very high strain rates the continued attraction in bone and ligament strength again, causes the ligament to become stronger than bone. Thus, mode I and mode II can be defined based on strain rates. The integrative concept for these failure modes is that both bone and ligament strengths increase with increasing strain rates, and that these are at least two points at which these functions cross. These results have important implications with respect to further experimental studies in this area. It is critical that strain rates be carefully defined, and that comparisons with other work be made on the basis of similar values, rather than just "high" and "low". Secondly, species, age, condition and other variables compound the difficulty of drawing broad conclusions, as does the lack of detail on orientation of the bone-ligament-bone complex during testing.