Measuring knee joint laxity: A review of applicable models and the need for new approaches to minimize variability

2.1. Traditional measurement systems 

The most commonly used arthrometer in both biomechanical research and clinics is the KT-2000, which measures anterior–posterior knee laxity during the drawer test. The subject lies supine in slight hip flexion and with a knee flexion angle between 20° and 35° (Fig. 1a). The knee must be flexed adequately to engage the patella in the femoral groove. If the flexion is inadequate, the patella will not provide a stable base from which tibial displacement can be measured. External hip rotation is restricted by a Velcro strap placed midthigh and tibial position is maintained by placing the subject’s feet between two vertical supports that prevent the feet from rolling outward. The arthrometer is secured to the anterior tibia with Velcro straps, with a proximal bar contacting the anterior patella surface, the motion axis oriented over the joint line, and the force plunger positioned over the tibial tubercle. With the subject relaxed, a dial on the front of the machine is adjusted to a neutral starting position of the tibia with respect to the patella. The examiner then passively translates the tibia posteriorly and anteriorly, with a goal of establishing the same neutral starting position with each cycle. This trial also helps the subject to completely relax because it allows them to become accustomed to the motion. If the initial starting position changes between cycles, it is likely that the subject is either protecting the joint by contracting the thigh muscles, or the patellar reference pad is moving. If the starting position is stable, anteriorly and posteriorly directed passive loads of 67N and 178N (Myrer et al., 1996) are then applied through the force handle of the arthrometer.

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Fig. 1. (a) Positioning of subject for testing with the KT-2000 arthrometer. Modified from “Quadriceps active test at 30°” reprinted with permission from MEDMetric Inc. (b) A typical force–displacement curve resulting from testing with the KT-2000 arthrometer; “A” and “B” denote peak posterior and anterior displacement respectively. The compliance index for the anterior drawer test is calculated as ΔL/ΔD.

The Genucom is also used to measure anterior–posterior knee laxity, however, it is also capable of measuring motion in several planes simultaneously (Cannon, 2002). This includes measuring varus-valgus, pivot shift, and recurvatum (Highgenboten et al., 1989). The testing procedure for measuring anterior–posterior knee laxity with this device, as summarized here, was originally described in Highgenboten et al. (1989). The Genucom system includes a computer that is used to record the subject’s data file. The subject is positioned with the knee flexed to 20°. The thigh is secured with restraints, and an electrogoniometer is attached to the thigh and shank. Anatomical landmarks including the tibial tubercle, tibial crest, medial and lateral femoral condyles, and the patella are marked and digitized, and used to obtain a coordinate system and record the relative displacement at the knee joint, and the distance between markers is documented. Consequently, this system is sensitive to soft tissue motion as with any markers, although this effect is minimized by locating them on bony landmarks. To account for motion within the Genucom, a soft tissue compensation test can be performed in three planes while maintaining the same distance between the markers. Furthermore, as with the KT-2000, the patella should be engaged in the femoral groove. The anterior–posterior drawer test is completed at the system’s required force magnitude of 21lbs (93.45N). Data is collected continuously at a rate of 8Hz (Andersen and Jorgensen, 1998), and is stored electronically.

The Rolimeter is used to measure anterior–posterior laxity for Lachman, anterior drawer, and ‘step off’ tests. The subject is positioned supine, with knees flexed to 25° for the Lachman test, 80° for the anterior drawer test, and an unspecified angle for the ‘step off’ test. The knee is supported by a pillow for the Lachman test. Unlike the KT-2000, the only specification for foot position is for the anterior drawer test, where the tester sits on the subject’s foot. A proximal convex pad rests over the patella, and a distal pad is fixed to the tibia with a rubber strap (Papandreou et al., 2005). The two pads are connected a few inches above the limb by a calibrated steel bar. The stylus is an additional arm that projects down from the steel bar onto the tibial tuberosity. As described in Muellner et al. (2001), the patella can either be stabilized with the thumb, or with the technique used for the KT-1000. Similarly for the KT-2000, the quadriceps muscles must be relaxed. The knee is preconditioned with three applications of a posterior force to the tibia. The tester must ensure that the stylus foot is in contact with the tibial tuberosity, and the white indicator, a movable plastic ring, is against the adjustment knob. The white indicator displaces when the tibia is manually pulled anteriorly. The displacement is measured in increments of 2mm. For both the Lachman and anterior drawer tests, a manual maximum anterior force is applied three times, and the three maximal anterior translations are measured and averaged. For the ‘step off’ or side-to-side difference test, the Rolimeter is calibrated by applying it to the injured leg while applying a posterior force, ensuring that the stylus is in contact with the tibial tuberosity. It is then applied to the uninjured leg, and the side to side difference can be recorded. The benefit of the Rolimeter is that it is small, portable, and autoclavable. However, it neither records data nor tests at a variety of force levels. Furthermore, the displacements are measured at the endpoint of tibial motion with relatively low resolution (±2mm).

The Stryker Ligament Tester has a patient positioning seat, force applicator, and a displacement transducer. Like both the KT-2000 and the Rolimeter, this device uses the patella and the tibial tuberosity as reference points to measure displacement. One disadvantage of using the patella as a reference point is that it is not firmly attached to the femur, and therefore must be well-seated in the femoral groove to reduce the relative motion. Unlike the Genucom, which can account for out of plane motion, the Stryker Ligament Tester assumes that there is only anterior–posterior displacement. As Cannon (2002) describes, the ankle is secured with a strap, and the flexion angle is set between 0° and 90°, although 25°–30° is typical. The calibrated ruler is positioned horizontally, and the proximal tibial bracket is placed on the tibial tubercle. The distal bracket is positioned above the ankle, with an external rotation to align it with the tibia. Three straps secure the device at the proximal and distal ends of the tibia, and at the thigh just above the patella. The measuring gauge is positioned over the patella with the button at the midpoint. The patient leans back and relaxes while the examiner stabilizes the thigh with their non dominant hand, and applies a load with the force applicator, which is positioned over the crest of the tibia for a posterior load and behind the proximal calf for an anterior load. The resulting displacement at the endpoint of tibial displacement is measured to the nearest 0.5mm.

For the arthrometers that are capable of measuring continuously during the anterior–posterior drawer test, including the KT-2000 and the Genucom, force and displacement of the tibial sensor are recorded throughout the posterior and anterior motions, and can result in a plot as seen in Fig. 1b. Traditionally, laxity is measured from these cycles as the peak displacement with respect to the starting zero point at a given load. A compliance index (Daniel et al., 1985) is also computed from the force–displacement curve as the slope of a line connecting the points corresponding to zero displacement and the peak anterior or posterior displacement. Both knees of the subject are tested and an interlimb difference is computed. Interlimb differences normalize the subject’s involved knee to the uninvolved, thereby accounting for the subject’s specific, total body laxity. Moreover, this value can also be obtained by arthrometers that only measure the displacement at the endpoint of tibial motion, including the Rolimeter and the Stryker Ligament Tester. An interlimb peak displacement difference greater than 3mm is considered indicative of anterior cruciate ligament insufficiency.

Additional work has been done by Maitland et al. (1995) to quantify the nonlinear force–displacement curve by taking the first (stiffness) and second (change in stiffness) derivatives. These values differ from the compliance index by calculating the instantaneous slope at each discrete point along the force–displacement curve, giving true stiffness, rather than an average slope based on initial and peak displacements, which does not characterize the entire curve. By accounting for the entire nonlinear response, this method is able to identify significant material behaviors that could not be appreciated by an average linear stiffness. Quantifying the entire force–displacement curve highlights an increasing slope as the ACL works to restrain the knee after tibial weight is overcome. In this study, the authors showed that low values of stiffness and change in stiffness were typical for ACLD subjects in the upper region of the force–displacement curve, where the ACL is believed to be the primary restraint. In comparison, control subjects showed higher stiffness and change in stiffness over this range. These results highlight the value in quantifying stiffness along the curve, as individual sections of the curve (toe region, linear portion, upper region) can be analyzed and used to differentiate injured from control populations. It is particularly important to quantify the low load behavior during the toe region of the force–displacement curve as it spans the normal operating range of the knee.

The relative use of these arthrometers has changed over time. The KT-2000 system continues to be the most popular device for measuring joint laxity, while other units such as the Genucom seem to be used less frequently in recent years based on literature reports. This conclusion is highlighted in Fig. 2, which shows the number of papers per year in which each type of arthrometer is referenced. Searches were done using the National Center for Biotechnology Information PubMed database (from 1966 to June 2006) using the search terms ““KT 1000” or “KT-1000” or “KT 2000” or “KT-2000”, “Genucom”, “Stryker Ligament Tester” or “Stryker Knee Laxity Tester” or “Stryker Laxity Tester”, and “Rolimeter”. There are many possible reasons why the KT-2000 appears to be the preferred laxity measurement system. Some factors could include cost, availability, or the suitability of the system to the application. Complex systems that allow a greater range of measurements such as the Genucom may be better suited to a research environment, however, using the KT-2000 in a research study allows for comparison of data to a large body of published work. The KT-2000 is smaller and thus more easily portable than the Genucom, making it amenable to the clinical setting. The Rolimeter and the Stryker Ligament Tester are also small lightweight systems, making them easily transportable and straightforward to use. However, the KT-2000 has a digital output rather than analog, and so can generate real-time force–displacement graphs. It seems that the KT-2000 may fit a niche due to the balance of convenience and simplicity with descriptive output measures.

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Fig. 2. Number of publications per year that include the KT-1000/2000, Genucom, Stryker Ligament Tester, or Rolimeter arthrometers from 1985 to 2006.

One limitation of the majority of these devices is that they only measure translational displacement, which is assumed to be solely in the anterior/posterior direction, thereby neglecting the entire range of laxity as the knee moves with six degrees of freedom. It has been shown that passive laxity tests with the KT-2000 do not correlate to functional outcome after ACL injury (Snyder-Mackler et al., 1997). A number of studies have been conducted to assess the accuracy and reliability of these devices (Anderson and Lipscomb, 1989, Anderson et al., 1992, Cannon, 2002, Highgenboten et al., 1989, Huber et al., 1997, Papandreou et al., 2005). The results are summarized in Table 1. Most of these studies rely on comparisons between arthrometers or repeatability studies for validation, rather than determining accuracy with respect to a ‘gold standard’ measurement device. It is important to note that the importance of studying accuracy is to determine how closely an arthrometer can measure the correct value. Conversely, repeatability describes how closely a device can obtain the same value over multiple measurements, regardless of whether this value is correct or not.

Table 1. Accuracy and repeatability of current laxity devices

	Device	Repeatability/reliability	References
	KT-1000 or KT-2000	Anterior: ±3.99mm, ±3.89mm, ±3.74mma	Huber et al. (1997)
		Posterior: ±2.95mm, ±2.53mm, ±3.27mm
		Anterior: 0.87b	Highgenboten et al. (1989)
		Posterior: 0.79
		0%, 82%c	Anderson et al. (1992)
		0%, 75%c	Anderson and Lipscomb (1989)
	Genucom	Anterior: 0.96b	Highgenboten et al. (1989)
		Posterior: 0.86
		23%, 76%c	Anderson et al. (1992)
		10%, 70%c	Anderson and Lipscomb (1989)
	Rolimeter	Anterior between three testers: r(P1 vs. P2)=0.96 r(P1 vs. P3)=0.55 r(P2 vs. P3)=0.57	Papandreou et al. (2005)
	Stryker ligament tester	Anterior: 0.74b	Highgenboten et al. (1989)
		Posterior: 0.87
		Anterior/posterior: 0.83	Jorn et al. (1998)
		0%, 82%c	Anderson et al. (1992)
		10%, 75%c	Anderson and Lipscomb (1989)
		4.4mm, 8.0mmd	Jorn et al. (1998)

a 95% CI for novice, experienced, and intertester respectively. b r value. c Percentage of 50 subjects diagnosed with false positives for ACL deficiency, percentage of 50 subjects diagnosed with true positives for ACL deficiency. d Mean difference from known value at 90N and 180N respectively.

It has been found that, in general, there is variability between testers (Cannon, 2002). The KT-1000, for example, had an intertester reliability of ±2.95mm and ±3.74mm for posterior and anterior translation, respectively (Huber et al., 1997). Given that a right-left difference of 3mm or more is taken as an indication of increased laxity (Cannon, 2002), poor intertester reliability may result in false negative tests because the difference between examiners is greater than the patient’s interlimb difference. Highgenboten et al. (1989) compared the KT-1000, Genucom, and Stryker Ligament Tester devices. They found that the Genucom produced significantly higher anterior laxity values than the other devices, showing that the results are specific to each device. These studies have shown that although the current laxity devices have made methodological advances for quantifying laxity, poor inter-tester reliability continues to plague current devices. A few studies have been done to determine the accuracy of arthrometers. In the study done by Jorn et al. (1998), the Stryker Ligament Tester was compared to RSA, and found to overestimate anterior tibial translation, which was attributed to soft tissue displacement. Thus, there is clearly a need for both a measure of laxity that is less variable and improved methodology for clinical assessment of joint laxity in a defined manner. This is of particular importance to understanding the mechanisms involved with, and the efficacy of, interventions developed to negate the impact of excessive or variable laxity on joint function and minimize risk for loss of joint health.

2.2. Alternate measurement systems 

Various techniques other than displacement transducers have been used to measure tibial motion, including planar stress radiography, RSA, and MRI. These imaging techniques have shown promise in measuring displacement, especially as the quality of images improves with technology. Images of the knee can also be reconstructed or quantified in three dimensions, thereby allowing calculation of the entire tibial displacement, rather than simply in the anterior–posterior direction.

Planar stress radiography uses a series of lateral radiographs to quantify the displacement of the tibial condyles with respect to the femoral condyles (Fleming et al., 2002). It is noninvasive, with a reproducibility of ±0.5mm (Stäubli et al., 1992). RSA measures motion in three dimensions by spatial reconstruction of tantalum beads implanted into the cortices of the bones of interest, with reported translational and rotational accuracies of 10–250μm and 0.03°–0.6° respectively (Kärrholm, 1989, Selvik, 1989). Although this technique provides the most accurate measure of laxity, the implanation of tantalum beads is highly invasive, and thus, cannot be readily applied to every subject group. Despite the benefits of both planar stress radiography and RSA, a major drawback to each is that patient exposure to radiation is required. The current standard for the recommended use of radiation is to minimize subject exposure, based on the understanding that any dose of radiation can have harmful effects (Picano, 2004). Furthermore, while radiography provides very clear images of bone, it does not image soft tissues well. This is an important limitation, as the majority of the passive restraints of the knee are soft tissues. Thus, a very clear view of joint laxity can be gained through bone-bone displacement, however, the behaviors of joint’s soft tissues cannot be well observed, and thus the underlying reasons for the laxity may not be appreciated.

MRI imaging shows promise as a noninvasive, accurate modality for measuring displacement. Our group has applied methods from Geomatics engineering to improve surface registration of MRI images, and have obtained results with an average normal distance (a measure of how well two superimposed or registered images fit together) of 0.201mm (Cheng et al., 2005). Employing imaging techniques to measure laxity would provide an improvement over knee arthrometer measurements because arthrometers require that the patella be wellseated in the femoral groove to obtain an indication of the position of the femur relative to the tibia, while imaging modalities allow the direct measurement of the femoral position. Additionally, the detailed geometry attained with imaging techniques allows for additional measures to be made. For example, the location of the tibiofemoral cartilage contact area can be found as an alternative measure of relative position, or the change in length of the ACL can be quantified. MRI has the additional benefit over other imaging modalities of being able to visualize soft tissues without requiring the subjects to be exposed to radiation. Thus, the entire joint, including the soft tissue structures, can be imaged in a radiation-free, non-invasive fashion. The best image quality can be achieved with a closed bore MRI compared to open bore or dynamic MRI. Closed bore MRI is also more widely available and is appropriate for the positioning typically used to measure knee joint laxity. However, closed bore MRI limits the range of motion that can be tested because of bore hole constraints, and requires the patient to lie supine.

3.1. Joint position, applied force, and motion constraints 

Laxity has typically been assessed statically with the subject lying supine and the knee flexed at angles between full extension and 90° of flexion (Malanga et al., 2003). Anterior translation is most commonly assessed at a flexion angle of 30°, since this angle is believed to represent the slack-taut transition for the ACL (Sheehan and Rebmann, 2003). A limitation of testing in a supine position is that physiological loads of the knee cannot be obtained. To overcome this limitation a method was developed in which the subject can either apply a load (10%–15% of maximum) to a foot pedal or to resist a plantar force applied to the foot (Ronsky, 1994). This increases the patello-femoral contact area, confirming that external or resistive loading does affect knee contact mechanics (Gold et al., 2004, Ronsky, 1994). The external force must be applied uniformly to the tibia to ensure that the displacement is solely in the anterior direction. Laxity measures appear to be sensitive to the location of a point load because of the resulting internal/external moment applied to the tibia (Rudy et al., 2000). Forces are typically applied in the anterior–posterior direction, although some studies of laxity apply a torque to the tibia and measure torsional joint stiffness (Hsu et al., 2006). The wide ranging methods used to measure laxity underscore the need for a systematic and universally accepted definition of joint laxity and a similarly accepted technique for applying the necessary loads. Before undertaking comprehensive studies of interventions aimed at influencing joint laxity, researchers and clinicians must adapt similar language and measuring tools to allow comparison between laboratories and across disciplines.

3.2. Muscle activity 

In measurements of laxity, methods are typically employed to minimize or exclude muscular contributions, such as quadriceps palpation to reduce muscle guarding during the anterior drawer test, and the administration of anaesthetics to induce muscle relaxation (Sernert et al., 2001). Alternatively, muscle forces can be included in laxity measurements; however, they must be accurately quantified. Muscle force can be measured directly, although these techniques are invasive, and typically reserved for animal models (Herzog and Leonard, 1996). This has led to the inclusion of noninvasive measures of muscular contributions, such as electromyography (EMG). Although EMG has been used to measure muscle force (Doorenbosch et al., 2005), it is a highly controversial method. Nigg (1999) notes that while there is a qualitative relation between muscle force and EMG signal, quantifying that relationship is affected by a number of variables. Firstly, electrodes must be mounted properly to achieve a clear signal. Secondly, an electromechanical delay exists between the onset of the EMG signal and the actual force production of the muscle. Although the delay can be quantified, it may be variable throughout the contraction due to the viscoelasticity in the muscle–tendon unit. Isometric quadriceps contraction as seen in laxity testing is the optimal case for relating EMG to muscle force because a one to one relationship exists, and electromechanical delay will not change the result because there is no time dependence. However, deep muscle force cannot be measured from surface EMG, and thus the muscle force contribution will be restricted to that of the superficial muscles. Therefore, using EMG to measure muscle force should be approached with caution. The strength of this method is in obtaining relative force contributions rather than absolute measures of muscle force. If EMG is used in a theoretical knee model, the model should be tested for sensitivity to muscle force.

4.1. Contact models 

Contact models use contact points or areas between cartilage layers or bone position to describe tibial translation with respect to the femur in the direction of applied force. A proximity algorithm is often used to determine the points of contact. This method is well-suited to a gross measure of laxity, since displacement of the whole joint can be quantified based on anatomical landmarks. However, the specific points of contact considered and how displacement of those points is quantified influence the findings and must be considered when interpreting the data.

DeFrate et al. (2004) measured tibial displacement as the perpendicular distance from a line, connecting the mid-points of the medial and lateral tibial plateaus to the contact point. The contact point was defined in two ways: the centroid of the area formed by the overlap of the tibial and femoral cartilage, and the shortest distance between the tibia and femur, perpendicular to the tibial plateau. Their results showed a difference between the two contact point methods, highlighting the importance of cartilage geometry over bony geometry in determining contact points (DeFrate et al., 2004).

Other landmarks have been used to define anterior–posterior motion in the sagittal plane. Logan et al., 2004a, Logan et al., 2004b used the technique developed by Iwaki et al. (2000) in which the distance between the flexion facet center (the center of the posterior circular surfaces of the femoral condyles) and a line from the ipsilateral posterior tibial cortex is measured. Scarvell et al. (2004), note that the position of the femoral condylar centers can be more applicable to total knee arthroplasty prosthesis design because malalignment of the rotation axis may affect prosthesis fixation. The distance between the center of the tibiofemoral contact area and the posterior tibial cortex was also calculated. This choice was supported by the concept that these contact patterns describe motion that may affect articular cartilage degradation. Displacement in three dimensions was considered by Wretenberg et al. (2002) for contact areas on the medial and lateral condyles. They identified the most anterior, posterior, medial, and lateral contact point coordinates. These four points determined a contact area, from which the centroid points in all three planes were found. The benefit of considering a three dimensional measure is that it may provide more information about complex motion at the tibofemoral interface than a simple two dimensional measure.

In a contact model, the magnitude and direction of the applied forces, limb position, and opposing muscle forces are assumed to be consistent between tests and across subjects. However, in practice, while applied forces and positioning can be controlled by the tester, some variability is introduced by the muscle forces acting on the knee. Furthermore, the relative contribution of the involved muscles may differ between patient populations. For example, slight subjects may exhibit different muscular activation strategies than trained individuals with welldeveloped musculature, such as athletes. This is particularly important since athletes comprise a large portion of the ACL-injured population for whom laxity measures may be most relevant. Therefore, it is critical to control muscle activation levels when applying contact models, to minimize variability and improve the accuracy of comparisons across and within subject populations. One manner in which variability can be reduced is to employ a muscle palpation method, such as that used in KT-2000 tests, to ensure that the subject is fully relaxed. As a result, the subject is maintained in a state of muscular relaxation, and the variability incurred by muscle activation will be minimized, or avoided entirely.

Theoretical models are driven by the clinical purpose or research questions being asked. To understand the link between laxity and dynamic function, contact models may be advantageous because they provide a measure of whole joint laxity. Laxity measured at the whole joint level may be more clinically relevant since, theoretically, two individual joints may have identical whole joint laxity regardless of differences in the dimensions or mechanical properties of the internal structures. Furthermore, a measure of joint laxity describes how the joint performs as a whole, in physiologic conditions. Since contact models quantify relative joint position, they offer the theoretical model that most closely parallels current clinical laxity tests with potentially improved accuracy because displacement measurements are taken from accurate images of internal structures rather than external landmarks. Another benefit of contact models is that, in general, the model will not change among various subject populations. This is because the output measurements do not rely upon the material properties of the soft tissue, or the existence of specific structures other than the femur, tibia, and cartilage. For example, the model measurements do not depend on the existence of the ACL, and consequently, identical input and output variables are required when applying this model to a control population as for an ACLD population.

If the researcher or clinician desires to understand the underlying mechanisms of joint laxity, specifically within well-defined subject groups, a contact model will not be sufficient. Since the model does not include knowledge of the internal soft tissue structures, it is unable to apprciate their contributions to whole joint mechanics. The clinical importance of identifying how individual soft tissue structures affect the whole joint motion is an improved understanding of the underlying mechanisms of laxity. For example, comparing the stiffness of soft tissue structures in a normal knee to those in a knee with an MCL tear may aid in interpreting how and why the gross joint motion and functional ability of the patient differs from normal. Furthermore, if muscles are activated during a laxity test, these forces and their effects could be included to ensure consistency across tests. To account for these soft tissue effects, a specific lumped parameter model may be employed.

4.2. Lumped parameter models 

Lumped parameter models use simple geometric shapes to characterize the arrangement of the underlying structures (e.g. muscles, tendons, and ligaments). The joint is modeled as a system of simplified geometries or mechanical constructs (e.g. two-force member). The combined behavior of these elements produces a model that displays similar mechanical behavior to the joint. These models provide an appropriate model of joint laxity provided the goal is to determine a gross value of stiffness for each soft tissue element that is individually included. Tendons and ligaments are assumed to behave as two force members. Therefore, lumped parameter models typically assume that all tendon and ligament fibers are the same length, and shorten the same amount. Two dimensional geometrical arrangements are often assumed to simplify the model, although three dimensional models have been constructed (Shahar and Banks-Sills, 2004).

The underlying assumptions of a lumped parameter model have a direct impact on the resulting measure of laxity. The knee joint represents an indeterminate system where the number of unknown internal forces is greater than the available equations describing the system (Herzog and Binding, 1999) (Fig. 3). The available equations include the four equations of static equilibrium (sum of forces and moments is zero), and the compatibility of deformations. These can be written for both the tibiofemoral and patellofemoral joints. The unknown internal forces include the forces in the ACL, PCL, LCL, MCL, patellar tendon, capsule, contacting cartilage, menisci, and muscle forces (either passive or active). The modeler must then make the indeterminate system solvable. One solution method reduces the number of unknowns and/or increases the number of equations. The limitation of this approach is that relevant information about the contribution of a structure may be lost due to the simplification. Optimization is an alternative solution to the distribution problem that overcomes this issue. The system may be optimized to minimize a variable such as metabolic cost, total muscle force, or total muscle stress. These optimization techniques assume that the human body regulates forces in this manner, which is unlikely to be the true physiological mechanism. The clinical impact is that the model being used to treat and assess may be based on faulty assumptions. This could affect the accuracy of the outcome measures, and provide the clinician with incorrect values. Any or all of these methods may be employed to obtain a solvable model.

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Fig. 3. Exemplary free body diagram of the knee joint including external (ext) forces, forces due to compression (comp), muscle force [gastrocnemius (gast), hamstring (ham), quadriceps femoris (quad)], femoral mass (fem), tibial and fibular mass (tib/fib), and forces through the anterior cruciate ligament (acl), posterior cruciate ligament (pcl), lateral collateral ligament (lcl), and patellar tendon (pt).

Lumped parameter models have been applied in a variety of ways to solve for the internal forces in the knee. Shahar and Banks-Sills (2004) included the femur, tibia, and patella, the hip joint reaction force, ground reaction force, muscle forces acting on the femur, and the collateral and cruciate ligaments of the knee in a three dimensional quasistatic model of the canine knee joint. They neglected hind limb weight, menisci, friction between joint surfaces, extensibility of the patellar ligament, and out of sagittal plane rotation of the patella. This model was applied to a simulated slow walk to observe ligament forces during stance phase in both an intact knee and a knee with a ruptured cranial cruciate ligament, illustrating how a lumped parameter model can be developed to solve for ligament forces during a task and altered to consider other scenarios such as the effect of removing a structure.

Some studies consider various methods for determining the lines of actions of the force bearing structures (Aalbersberg et al., 2005b, Lu and O’Connor, 1996) or the muscle moment arms (Arnold et al., 2000). These types of studies often address parameter accuracy, and model sensitivity to each parameter (Toutoungi et al., 1997). This is an important step in the modeling process. The model output accuracy is highly dependent on the accuracy and relative weighting of the input values. If an input value is physiologically variable, difficult to measure accurately, or can be measured in a variety of ways, it is crucial to test the sensitivity of the model to that input. Sensitivity is indicated by the amount of change in the output as a result of a change in the input. Each researcher must determine what level of sensitivity is acceptable given the specific application of the model.

Lumped parameter models allow ligament and tendon forces to be estimated in vivo. This class of model is quite flexible because it can be as simple models as a four bar linkage, or complex enough to include additional structures such as the joint capsule (Pandy and Shelburne, 1998). This flexibility could be quite valuable clinically, as the model is adjustable to consider subjects who have an incomplete or lost structure. The complexity of the model is determined by its purpose. A lumped parameter model would theoretically be equally applicable to a model where the joint laxity is due to ACL loss compared to whole joint laxity as seen with joint hypermobility syndrome, as no assumptions are made with respect to material properties. The ACLD model would simply require one less unknown ligament force, and thus one less two-force member in the model. A model of knee joint laxity may help to determine how the muscle forces and the passive forces of the major supporting structures contribute to the overall joint stiffness.

The lumped parameter model has the potential to add more information to the clinician’s assessment of laxity. This is achieved by identifying the mechanical contributions of individual soft tissue structures. These could be used to assess the mechanism of abnormal joint laxity and thus help to identify the appropriate treatment. Furthermore, coping strategies may be better understood if muscle forces and stiffness are determined. Lumped parameter models may provide a balance between increasing the understanding of how the soft tissue structures affect subject specific knee laxity and providing a level of detail that is appropriate for clinical application.

Nevertheless, the lumped parameter model can be quite sensitive to the assumptions that are made. As more structures are included, more constraints and assumptions must be included to solve for the system. Furthermore, not all structures can be simply represented as a two force member, such as the capsule or the cartilage. To gain insight into the mechanical behavior of individual structures based on in situ knee geometry rather than assuming idealized geometrical shapes, a finite element model can be used.

4.3. Finite element models 

The finite element (FE) method is a modeling technique (numerical method) that discretizes a continuum into simple shapes called finite elements. Engineering concepts, including but not limited to beam theory, dynamics theory, or heat transfer theory, and constraints are applied to the elements to solve for the mechanical behavior of the entire structure. The advantage of an FE model is that complex three-dimensional geometry may be considered and the mechanical contributions of the individual structures can be included. Stress and displacement solutions can be found for the entire structure at each discrete point. This is a more detailed result than a lumped parameter model, and may potentially reveal localized weak points in the components that comprise the structure. Furthermore, the ability to apply static, dynamic, and thermal analyses to a complex geometry makes the FE method well suited to biomechanical problems.

FE models have been applied to solve the indeterminate system of the knee (Bendjaballah et al., 1998, Li et al., 2002, Moglo and Shirazi-Adl, 2003). That is, since the model accounts for the mechanical contributions of the soft tissue structures individually, it allows tests (simulations) to be conducted in which tissue parameters are altered in a controlled manner. FE models can investigate the sensitivity of joint function to the mechanics of the supporting structures, providing a unique experimental avenue that cannot be carried out in vivo. These models can be employed to study the contributions of the musculature and passive tissue structures to knee joint function, such as the effect of quadriceps force when the ACL is removed, or partially torn (Li et al., 2002), the effect of passive tissue structure removal (Bendjaballah et al., 1998, Moglo and Shirazi-Adl, 2003), or reduced ligament stiffness on joint function (Li et al., 2002).

Modeling at this structural level allows FE models to appreciate the contributions of individual soft tissue structures to overall joint mechanics, as well as the sensitivity of the joint’s function (e.g. range of motion, laxity) to changes in the individual soft tissue structures. However, since joint behavior is determined by the combined mechanical contributions of the substructure, the accuracy of FE models is only as good as the geometric representations and the material properties used to describe the soft tissue structures. One such mechanical property is Young’s Modulus, which quantifies a material’s ability to resist forces independent of geometry. It is not suited to a gross measure of laxity because there is no clear way to describe cross sectional area or a reference length for the entire knee joint, necessary quantities in the measurement of stress and strain. Young’s Modulus could be used to describe each of the internal structures of the knee, such as the ACL, PCL, MCL, and LCL. However, difficulties arise in quantifying stress and strain because they vary along the length and throughout the tissue cross section (Fleming and Beynnon, 2004). These properties are often obtained experimentally with cadaveric tissue (Quapp and Weiss, 1998). This may be a source of error for models of joint hypermobility syndrome, for example, in which connective tissue mechanical behavior differs from that of normal subjects. FE models must converge to a repeatable solution regardless of how many finite elements the model is divided into (referred to as mesh size), thereby requiring time and computer memory to handle refined meshes in complex models that cannot take advantage of symmetry and other model simplifications. The model is only as relevant as the material properties and geometric representation of the included structures. More importantly, FE modeling has a complexity that may not be appropriate for the level of detail required from a model of laxity. However, further work must be done to identify the necessary modeling detail required to differentiate between groups across the spectrum of laxity.

4.4. Theoretical modelling summary 

A theoretical model of knee joint laxity can be as broad as a gross measure of displacement under a translational force or as specific as the complex three dimensional displacement or stress distribution of each structure within the joint. The main concern with any theoretical model is to determine whether it will provide information that cannot otherwise be obtained. The benefit of a gross model of laxity is that the displacement measurement does not require knowledge of the mechanics of the internal joint structures. Since a gross model describes joint laxity with a single measure, it may be easier to make comparisons between subjects. Additionally, a gross measure may be easier to correlate to other subject variables like dynamic function or stability because it characterizes the behavior of the joint as a whole. However, if the relative contributions of the soft tissue structures to overall joint laxity is desired, a different modeling technique must be employed. It could be valuable to further describe differences between subjects by quantifying the laxity of each internal structure and identifying how those individual structures contribute to the increased laxity of the entire knee joint. This information would require more in-depth modeling, such as lumped parameter or finite element. The advantage is that it could lead to improved clinical techniques (therapy or surgery), or further understanding regarding the underlying mechanisms of joint laxity.