The Knee
Volume 18, Issue 1 , Pages 34-41, January 2011

A multiple-bundle model to characterize the mechanical behavior of the cruciate ligaments

  • Shahram Amiri

      Affiliations

    • Department of Orthopedic Surgery, University of British Columbia, Vancouver, BC, Canada
    • Corresponding Author InformationCorresponding author. 828 West 10th Ave, Room 575, Vancouver, BC, Canada V5Z1L8.
    • ,
  • T. Derek V. Cooke

      Affiliations

    • School of Rehabilitation Therapy, Queen's University, Kingston, ON, Canada
    • ,
  • Urs P. Wyss

      Affiliations

    • Department of Mechanical and Materials Engineering, Queen's University, Kingston, ON, Canada

Received 21 October 2009; received in revised form 4 January 2010; accepted 6 January 2010. published online 29 January 2010.

Article Outline

Abstract 

Measurements of elongations of the cruciate ligaments have been used to study the behaviors of these ligaments in-vitro and in-vivo, mostly based on simplified two-bundle models of the cruciates. The complex fiber anatomy of the cruciates may suggest a complex deformation behavior across the continuum of their substance that cannot be captured by only two measurement points. In this study, a new methodology was introduced to include more detailed fiber anatomy and to take into consideration the wrapping of the PCL around the intercondylar notch of the femur in deep flexion. The method was used in comparison to the conventional two-bundle models on three sample cadaver knees that underwent a passive flexion up to 150°. The elongation ratios of the bundles were measured as the ratio of change in the length of the bundles over their lengths at 0° flexion. The multiple-bundle models showed ranges of variations across the attachment sites of the cruciates which at all flexion angles were significantly larger than those observed from the conventional two-bundle models. When expressed in percentages, at 150° flexion the ranges of variations in the elongation ratio of the bundles were 32.7%±31.9% and 34%±8.6% for the ACL and PCL, respectively. Results of this study showed that important variations of elongation across the body of the cruciates can be obscured to the conventional two-bundle model of the cruciates, and therefore a more detailed bundle configuration is suggested for the purpose of studying elongation behaviors of these ligaments.

Keywords: Knee, Bundle recruitment, Cruciate ligament, Ligament elongation

 

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1. Introduction 

Knowledge of the length behaviors of the ligament bundles at the human joints is fundamental for the analysis of relevant mobility and stability. Changes in length of the ligaments have been used to study the functions of the knee ligaments under unloaded conditions [1], [2], [3], as the prerequisite to analyze their deformations under load [4], [5] or in rupture [6]. Different methods have been used for measuring length behavior of the ligament bundles, including manual examination of the bundles [7], attaching wires [8], [9], [10] and strain gauges [11], [12] to the ligaments, and using imaging to measure the straight-line distances between points of attachments of the ligaments or the curved course of the ligaments in-vitro [13], [14], [15], [16], [17] and in-vivo [18], [19], [20], [21], [22], [23].

In these studies, some authors examined the length behavior of the entire bulk of the cruciates as one band [16], [19], whilst some suggested more complex bundle configurations [10], [11], [17], [23], [24], [25], [26]. The conventional definition, however, describes the functional structure of the posterior cruciate ligament (PCL) in terms of two anterior or anterolateral (AL) and posterior or posteromedial (PM) bundles, and that of the ACL in terms of anteromedial (AM) and posterolateral (PL) bundles [7], [14], [18], [21], [22]. Here because of the complexity in the structure, the question remains whether the simplified two-bundle definition of the cruciates can sufficiently represent variations of elongation across the substance of the cruciate ligaments. Furthermore, it has been shown that in deep knee bending activities further elongation of the PCL takes place due to wrapping of a part of the PCL around the intercondylar notch of the femur [20]. This geometric nonlinearity of the PCL bundles should be accounted for in accurate measures of the cruciates' elongations.

The objective of this study was to estimate the differences in the ranges of elongations captured by the conventional two-bundle model of the cruciates and a model that can account for more detailed bundle configurations and wrapping of the PCL in deep knee bending. We hypothesized that compared to a conventional two-bundle, a multiple-bundle model can better portray the bundle recruitment which largely varies across the continuum of substance of the ACL and PCL, particularly in deep knee bending. A methodology was introduced to account for a more detailed map of attachments of the cruciates, and it was tested in comparison to the two-bundle model in a preliminary study that included three knee specimens.

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2. Materials and methods 

2.1. Acquisition of the joint kinematics 

Three fresh cadaver knee specimens were used in this study. Basic description of the cadaveric specimen is shown in Table 1. Each specimen had about 20cm bone proximal and distal to the joint line. In the preparation of the specimens, all the soft tissue including the joint capsule and patellar retinaculum were left intact. About 5cm of the end of each bone was stripped from the soft tissue and it was affixed into an aluminum pot using PMMA bone cement. The entire length of the specimen was CT scanned, using a standard bone-based set up used for generating models for orthopedic surgical planning. The scans were taken at 2.5mm intervals and reconstructed to have 1.25mm spacing between the slices. The CT slices were segmented manually using a threshold algorithm and surface meshes were automatically constructed based on a marching cubes algorithm (Mesher, Version 2.0 Beta 9, Medical Computing Lab, Queen's University). The specimens were installed on an Oxford rig, with the configuration shown in Fig. 1, to impose passive motion [27]. Oxford rig has an open kinematic chain construct that allows a knee specimen its natural six degrees-of-freedom of movement (6 d.o.f.) including the “screw home” effect (rotational movement of the joint during flexion). The Optotrak system (NDI, Waterloo, ON, Canada) was used to capture the motion of the bones by tracking the trajectory of the LEDs affixed to the aluminum pots. The knee specimen were moved passively from full extension to full flexion, by moving the slider part of the rig at a constant speed over the duration of approximately 1min. Multiple reference surfaces from the aluminum pots were digitized using a stylus device. The digitized data points were used along with the CT scanned geometry of the specimens in Matlab 7.1 programming language (Mathworks Inc, MA) to find the transformation matrices required for mathematically reconstructing the joint kinematics. The joint coordinate system of the knee was defined according to Grood and Suntay's convention [28].

Table 1. Basic descriptions of cadaver specimens: Body Mass Index (BMI) is calculated by dividing the subject's mass (in kg) by the square of the subject height (in m).
Knee specimenAge (year)GenderMass (kg)Height (cm)BMI kg/m2
#176Male7916828
#283Female4515718
#378Male7317324
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  • Fig. 1 

    Schematic view of the Oxford rig used for the study. Motion was imposed to the knee from full extension to deep knee flexion, by moving the slider down. Pots and rods (labeled as numbers 2 and 3) that were used to place the cadaveric knees on the rig have also been shown in the picture. The motion tracking LEDs (labeled as number 1) were affixed directly to the mounting pots (labeled as number 2).

2.2. Multiple-bundle model of the cruciates 

After collecting the kinematics, the specimens were detached from the rig and dissected while the Optotrak LEDs were still attached to serve as references for digitizing the ligament attachments. During dissection no cartilage degeneration, ligament rupture, or anatomical abnormalities were observed. The peripheries of the attachment sites of the cruciate ligaments on the tibia and femur were digitized using the stylus device with accuracy of less than 1mm. The attachment location of each individual bundle with respect to the shape of the attachment site was determined using the most detailed diagram found in the literature for bundle connections of the cruciate ligaments [7], [29]. As illustrated in Fig. 2, based on these diagrams, the ACL and PCL were modeled with 10 and 9 bundles, respectively. For each attachment site, first a reference plane was fit to the digitized data points using PolyWorks 8.0 software (InnovMETRIC, Quebec, Canada). The corresponding digitized points from the ligament attachment were projected over these planes. The diagram of bundle connecting points (Fig. 2B and D) was overlaid on this plane, and it was moved and scaled until the boundaries of the ligament of the diagram matched the projection of the digitized points. Using the scaled diagram as the reference, the coordinates of individual bundles were determined. The relative motion between the bones was reconstructed in Matlab, and elongation ratio (ER) of each individual bundle was calculated as shown in Eq. (1), for 30°, 60°, 90°, 120°, 135°, and 150° of flexion. In this equation Li is the momentary straight-line distance between the two attachment points and Lie is the length of the bundle at 0° flexion. For the reference of comparison a two-bundle model and one-bundle model of the cruciates were also used to measure the elongation ratios. The locations of attachment points of the ligaments in the model with one bundle were considered at the centroids of the attachment areas. In the two-bundle model, the attachment sites were located as described in the literature [18] at the centroids of the AM and PL portions of attachment for the ACL, and AL and PM portions for the PCL.

(1)

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  • Fig. 2 

    Map of attachments for individual bundles of the PCL and ACL (adapted from Friedrich et al., 1992). A: Attachment sites of the PCL on the femur and tibia. These attachment areas are scaled in B to show the bundle arrangements. B: Diagram of the PCL bundle connecting points. The colored area defines the portion that wrapped around the intercondylar notch of the femur in deep flexion. Numbers identify the individual fiber bundles considered in the model. C: Attachment of the ACL on the femur and tibia. These attachment areas are scaled in D to show the bundle arrangements. D: Diagram of the ACL bundle connecting points. Numbers identify the individual fiber bundles considered in the model.

2.3. Interaction between the PCL and the roof of the intercondylar notch of the femur 

To take into account the effect on elongation due to impingement, the roof of the intercondylar notch of the femoral bone was modeled by a 3D cubic Hermite spline curve (Fig. 3A). Eq. (2) shows the governing equation of such a 3D curve. In this equation p1 and p3 are the coordinates of the end points and p2 and p4 are the tangent vectors to the curve at point p1 and p3. A least square minimization algorithm (Eq. (3)) was used to best fit the cubic Hermite curve to the data points (Proof in Eq. (3)) representing the proximal aspect of the roof of the intercondylar notch.

(2)
(3)

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  • Fig. 3 

    A: Distal view of the femoral condyle: The posterior end of the intercondylar notch of the femur was modeled by a cubic Hermite curve. Schematic illustration of the method used to assess the interaction between the intercondylar notch of the femur, represented by curve p1–p3, and an imaginary bundle of the PCL illustrated as pT–pF. B: Points F(t0) and G(s0) illustrate the points associated with the minimum distance between the line p1–p3 and the curve pT–pF. C: The point H(t0,s0), constructed by mirroring the point G(s0) with reference to F(t0), is used to examine if the bundle has crossed the notch. D: If the bundle has crossed the intercondylar notch of the femur, the length of the bundle is calculated as the sum of the lengths of the two segments LF and LT.

Each ligament bundle was modeled by a straight line connecting the tibial and femoral attachment points pT and pF as described in Eq. (4).

(4)

For each joint position, the interaction between each bundle of PCL and the curve representing the roof of the intercondylar notch was analyzed (Fig. 3B). If a PCL bundle crossed the femoral notch during the motion, the elongated length due to wrapping was calculated. To detect crossing, for a given relative position of a PCL bundle and the curve representing the intercondylar notch, the points that corresponded to the minimum distance between these two were found (shown as F(t0) and G(s0) in Fig. 3C) using a least square minimization algorithm (Eq. (5)).

(5)

Point H was constructed as the mirror of the point G(s0) (Fig. 3C) with reference to the point F(t0) (Eq. (6)). Assuming that at full extension position none of the PCL bundles were wrapped around the notch, a change in the sign of parameter d in Eq. (7) was considered as the indicator of wrapping. Length of the crossed bundle was calculated as the sum of the lengths of the two segments (LT+LF) as shown in (Fig. 3D). This process was repeated for all positions of the joint.

(6)
(7)

2.4. Analysis of results 

To generate contour illustrations of the distribution of the elongation across the substance of the ACL and PCL, a triangle-based cubic interpolation method was used (Matlab 7.1 User Manual). The range of elongation ratios on each attachment and at each flexion angle was determined as the difference between the maximum and minimum of elongation across the attachment expressed in percentages. The ranges of elongation in the two-bundle models were defined as the absolute difference between the elongation ratios of the two bundles when expressed in percentage. Analysis of variance was conducted on the results of each model, and Student's t-test was used to compare the ranges of elongations obtained from the multiple-bundle and the two-bundle models.

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3. Results 

Fig. 4 shows the results of analysis of interactions between the PCL bundles and the intercondylar notch of the femur in the first specimen. Wrapping of the PCL took place after 110° of flexion, and the maximum additional elongation due to wrapping was approximately 2mm which added the maximum of 0.04 to the ER of the corresponding bundles. Fig. 2B shows that the portion of the PCL that interacted with the intercondylar notch of the femur was connected to the anterolateral (AL) corner of the tibial attachment.

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  • Fig. 4 

    Elongations of those bundles of the PCL that wrapped around the intercondylar notch of the femur. Solid lines represent lengths of the fiber bundles considering the wrapping effect, and the dashed lines are for the case that does not consider the wrapping effect. Fiber bundles numbered as 4, 5, 8, and 9 were stretched in high flexion due to interaction with the intercondylar notch of the femur.

Contours of distribution of ER across the tibial attachment of the cruciate ligaments in the first knee specimen are illustrated in Fig. 5. It is important to note that, as explained in the definition of the ER, the values shown in the figure are not the absolute values of the strains, but they indicate the elongation ratios relative to the status of the bundles at zero flexion. When multiple-bundle models were used, as has been shown in Fig. 5 for the first sample knee, large variations of elongation ratios were observed across the substance of the ligaments. The range of elongations observed for the three knee specimens are shown in Fig. 6 as percentages of their lengths at zero flexion. This figure also compares the results to the outputs of the one-bundle and two-bundle models. The ranges of variations of the elongation ratios across the substance of the ligaments at each joint position captured by the two-bundle and the multiple-bundle models of the ACL and PCL are compared in Fig. 7A and B. At 90° flexion, the ranges of elongation variations observed across the ACL attachment were 21.7%±16.9% (mean±95% confidence interval) and 32.7%±31.9% for the two-bundle and ten-bundle ligament models respectively. At 150°, elongation of the ACL bundles covered a range of 6.7%±16% in a two-bundle model of the ACL, where as variations of 50%±30.2% were observed when a ten-bundle model was used. For the PCL, the two-bundle model and nine-bundle models showed the range of elongations of 3.7%±1.4% and 24%±8.6% at 90°, and 9%±17.2% and 34%±8.6% at 150° of flexion respectively. Significant statistical differences were observed between the two elongation models of cruciates at all flexion angles (Fig. 7).

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  • Fig. 5 

    Map of the tibial attachment of the ACL from 30° to 90° (A), and from 120° to 150° (B). Map of the tibial attachment of the PCL from 30° to 90° (C), and from 120° to 150° (D). Isometric lines show the elongation ratios (ER) with respect to the lengths at 0° of flexion.

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  • Fig. 6 

    Range of elongation across the ACL and PCL for models with one-bundle, two-bundle, and multiple-bundle measured for three sample knees. Elongation ratios of the bundles were indicated as percentages of their lengths at 0° flexion.

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  • Fig. 7 

    Comparison between the conventional two-bundle and multiple-bundle elongation models of ACL (A) and PCL (B). The ranges of variations of the elongation ratios across the substance of the ligaments are illustrated. (Elongation ratios of the bundles were calculated as percentages of their lengths at zero flexion.) *P<0.05.

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4. Discussion 

In this study a method was introduced to provide more detailed measures of elongation across the body of the cruciates by using multiple-bundle configurations and by taking into consideration the wrapping of the PCL around the intercondylar notch of the femur in deep flexion. The new method was used to measure the patterns of elongations of the cruciate ligaments of three knee specimen during passive flexion. Statistical differences were observed between the output of the new multiple-bundle models and the conventional two-bundle models at all flexion angles, when the ranges of variations of elongation measured across the continuum of substance of the cruciates were used as the criterion for comparison. This finding is particularly important since in most recent studies conducted to investigate the elongation behaviors of the cruciate ligaments in-vivo [18], [19], [20], [21], [22], [23], the cruciate ligaments are usually simplified by one or two bundles. Results of this study suggest that the complexity in the structures of these ligaments may produce a complex distribution of deformation across the substance of their material which might not be correctly captured by only one or two measurement points.

The one-bundle models of the cruciates assume that the bulk of their material uniformly deforms and therefore their elongation can be represented by only one selected bundle. Measurements from the one-bundle model of the ACL showed gradual decrease in the length of the ligament when flexion angle increased from 0° to 120° (Fig. 6). This contradicted the assumption of having an isometric central bundle [30] in passive flexion. When multiple-bundle models were used, the large variations of elongation observed across the substance of the cruciates (50%±30.2% and 34%±8.6% for the ACL and PCL at 150°) indicated that the one-bundle models were not sufficient to capture the correct pictures of elongations of the cruciates during passive flexion.

In the two-bundle model of the ACL, the length of the PL bundle decreased with flexion in all the knee specimens (Fig. 6) which confirmed the reported values in the literature [7], [8], [18]. The two-bundle model of the ACL showed almost constant length for the AM bundle for flexion angles up to 90°, which did not follow what was reported in the literature as gradual increase in the length of this bundle [7], [8]. The multiple-bundle model of the ACL showed elongation of the anterior portion of the ACL with flexion (points with ER greater than 1 in Fig. 5A), but the geographic location of the elongated portion was not coincident with the centre of the AM bundle and therefore these elongations were not captured by the two-bundle model. The two-bundle model of the ACL was able to show slackness of the posterior portion of the ligament, since most of the bulk of the ACL including the point located at the centre of the PL bundle slackened by flexion (Fig. 5A and B).

The introduced model was able to capture what has been reported in MRI studies as wrapping of the PCL around the intercondylar notch of the femur [16], [20]. The part of the PCL attached to the AL portion of the tibial attachment was identified as the part of the ligament that curved around the femoral notch in deep knee flexion. Fig. 5C shows that at 30°, as we move posteriorly over the attachment of the PCL on the tibia the ERs of the ligament bundles gradually decrease. This indicates that the increase of flexion angle stretched the anterior bundles and slackened the posterior bundles of the PCL, which matches well with what has been previously reported as the reciprocal elongation and loosening of the anterior and posterior bundles of the PCL with flexion [7]. The upper and lower boundaries of the elongation patterns obtained from the nine-bundle of the PCL (Fig. 6) correspond with this phenomenon. The two-bundle model of the PCL did not show the same variations in the length of the AL and PM bundles (Fig. 6). The reason was because deformation was highly variable inside the attachment area of each of these bundles, and the output was very sensitive to the location of measurement. Therefore, although a part of AL bundle was stretched during flexion, as has been shown by the nine-bundle model, this was not reflected in the output of the two-bundle model. Fig. 8 shows the attachment location of the AL and PM bundles of the PCL in comparison to the points on the attachment that correspond with minimum and maximum of elongations when a sample knee was at 150° of flexion.

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  • Fig. 8 

    Comparison between the locations of bundles associated with maximum and minimum elongations across the tibial attachment of a sample knee at 150° flexion, with the location of the anterolateral (AL) and posteromedial (PM) bundles in a conventional two-bundle model. This example shows that locations of the bundles in conventional models cannot capture most of the variations in the elongation across the body of the ligament.

It is known that the bundle recruitments of the ligaments are highly influenced by loading and kinematics of the knee [10]. An elongation model of the cruciates should be able to illustrate the correct recruitment behaviors of the bundles including the maximum elongations and slacknesses in all loading conditions including the unloaded motion. The passive flexion was used as an example to compare the results of the models since it made the experimental setup easier. The significant differences between the models showed that the extra information obtained from a larger number of bundles was not rudimentary, but it captured the important minimum and maximum of the lengthening that were obscured to the two-bundle models. It can be assumed that under particular loading conditions, ligament bundles may undergo more uniform elongation behaviors [18], where one or two-bundle models can sufficiently capture the variations in the length of the bundles. This, however, has to be first investigated by comparing a multiple-bundle model to a simplified model for each particular loading condition. If a model is going to be used for various loading conditions, based on the results of this study a more detailed bundle definition for the ligaments is recommended.

Sensitivity of the length behavior of the ligaments to the locations of attachment is known [31], [32], [33], [34]. An analysis was conducted to evaluate the affects of uncertainties in attachment locations of the cruciates on the range of elongations. Effects of 2mm perturbations induced to the anteroposterior, mediolateral, and proximaldistal coordinates of the attachments were studied at 150° flexion, the position at which the maximum range of elongation was observed across the body of the ligaments. The results of the sensitivity analysis are summarized in Table 2. The maximum errors induced to the range of elongations observed across the attachments due to uncertainties in the locations of the attachments were 14% for ACL and 4% for the PCL. These ranges of errors fell within the larger range of variability observed between the specimens (30.2% and 8.6% for the ACL and PCL), and therefore they were not considered in the analysis.

Table 2. Sensitivity of the maximum and minimum elongation ratios (ERs) observed across the attachment sites of the ACL and PCL with respect to +/−2mm perturbations in the coordinates of their corresponding tibial and femoral attachments. The last row in the table shows the maximum range of error produced as a result of perturbations in the locations of attachments. (Abbreviations: ER: elongation ratio).
AttachmentPerturbationsACLPCI
Min ERMax ERMin ERMax ER
Tibial2mm lateral−2%0%0%−1%
2mm medial0%3%0%0%
2mm posterior−2%12%1%−2%
2mm anterior0%−10%0%−2%
2mm distal−2%−2%1%0%
2mm proximal0%2%0%0%
Femoral2mm lateral−2%2%0%0%
2mm medial0%−4%1%−2%
2mm posterior−2%−10%0%2%
2mm anterior0%12%2%−2%
2mm distal−2%1%0%0%
2mm proximal0%−2%1%0%
Min−2%−10%0%−2%
Max0%12%2%2%
Max range of error14%4%

One limitation of this study was that the complexity of identifying the specific map of ligament bundles for each individual knee was avoided, and one generic connection map of cruciates (Fig. 2) was used as a template, assuming that the bundle configuration is consistent in all knee specimens. Another limitation was that all attachment sites were considered over flat surfaces. To simplify the analysis, the interactions among the cruciate bundles were not taken into account, the intercondylar notch of the femur was represented as a curve instead of a surface, and the wrapped bundles of the PCL were modeled by two line segments instead of 3D curves. It is important to note that the pattern of elongation pictured for the sample knees is expected to change, if the joint kinematics is modified as the result of variations in the applied loading conditions. Another limitation was that small number of specimens was used in this preliminary study. To draw conclusions regarding the magnitudes of bundle elongations and the general mechanical roles of the bundles, a larger number of knee specimens should be included in the study. The results of this study showed that, at least for the passive motion of the joint, the new suggested method was able to show detailed information regarding the lengthening behavior of the ligament which could be obscured to the conventional two-bundle models of the ACL and PCL.

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5. Conclusion 

The ranges of elongation values observed across the body of the cruciates by multiple-bundle models were significantly larger than the output of the conventional two-bundle models. This suggests the use of more detailed bundle configurations in studying the kinematics of the cruciate ligaments. Furthermore, although the double-bundle hypothesis is used when considering cruciate reconstruction, the real cruciate ligaments appear to behave mechanically in a more complex way.

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6. Conflict of interest statement 

The authors hereby affirm that there is no conflict of interest for the submitted manuscript.

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Acknowledgements 

The authors would like to acknowledge Dr. Joel Lanovaz for sharing the Oxford rig and cadaver specimens and Leone Ploeg of the Human Mobility Research Centre (HMRC, Kingston, ON, Canada) for her assistance during the data collection.

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PII: S0968-0160(10)00004-9

doi:10.1016/j.knee.2010.01.003

The Knee
Volume 18, Issue 1 , Pages 34-41, January 2011

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