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Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model a
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Achilles Vairis , George Stefanoudakis , Markos Petousis , Nectarios Vidakis , Andreas-Marios a
Tsainis & Betina Kandyla
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Mechanical Engineering Department, Technological Education Institute of Crete, Estavromenos, 71004 Heraklion, Crete, Greece b
Creta Interclinic Hospital, Minoos 63 Str., 71304 Heraklion, Crete, Greece
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3 Karistou Rd, Athens 11523, Greece Published online: 02 Mar 2015.
Click for updates To cite this article: Achilles Vairis, George Stefanoudakis, Markos Petousis, Nectarios Vidakis, Andreas-Marios Tsainis & Betina Kandyla (2015): Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1015526 To link to this article: http://dx.doi.org/10.1080/10255842.2015.1015526
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Computer Methods in Biomechanics and Biomedical Engineering, 2015 http://dx.doi.org/10.1080/10255842.2015.1015526
Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model Achilles Vairisa*, George Stefanoudakisb1, Markos Petousisa2, Nectarios Vidakisa3, Andreas-Marios Tsainisa4 and Betina Kandylac5 a
Mechanical Engineering Department, Technological Education Institute of Crete, Estavromenos, 71004 Heraklion, Crete, Greece; b Creta Interclinic Hospital, Minoos 63 Str., 71304 Heraklion, Crete, Greece; c3 Karistou Rd, Athens 11523, Greece
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(Received 12 June 2014; accepted 2 February 2015) The human knee joint has a three-dimensional geometry with multiple body articulations that produce complex mechanical responses under loads that occur in everyday life and sports activities. Understanding the complex mechanical interactions of these load-bearing structures is of use when the treatment of relevant diseases is evaluated and assisting devices are designed. The anterior cruciate ligament (ACL) in the knee is one of four main ligaments that connects the femur to the tibia and is often torn during sudden twisting motions, resulting in knee instability. The objective of this work is to study the mechanical behavior of the human knee joint and evaluate the differences in its response for three different states, i.e., intact, ACL-deficient, and surgically treated (reconstructed) knee. The finite element models corresponding to these states were developed. For the reconstructed model, a novel repair device was developed and patented by the author in previous work. Static load cases were applied, as have already been presented in a previous work, in order to compare the calculated results produced by the two models the ACL-deficient and the surgically reconstructed knee joint, under the exact same loading conditions. Displacements were calculated in different directions for the load cases studied and were found to be very close to those from previous modeling work and were in good agreement with experimental data presented in literature. The developed finite element model for both the intact and the ACL-deficient human knee joint is a reliable tool to study the kinematics of the human knee, as results of this study show. In addition, the reconstructed human knee joint model had kinematic behavior similar to the intact knee joint, showing that such reconstruction devices can restore human knee stability to an adequate extent. Keywords: finite element modeling; knee ligament repair; tendon graft; anterior cruciate ligament; biomechanics
1.
Introduction
The human knee joint has a three-dimensional geometry with multiple body articulations which produce complex mechanical responses, both actively and passively, under loads that occur in everyday life and sports activities alike. In this joint, the anterior cruciate ligament (ACL) is crucial in keeping normal knee functions (Moglo and Shirazi-Adl 2003). When injured, it is often treated with surgical reconstruction (Lo et al. 2008; Woo et al. 2008) as its failure results in joint instability in the anteroposterior direction which inhibits walking (Lo et al. 2008; Woo et al. 2008). The mechanical behavior of this important structure has been studied experimentally in cadavers (Jones et al. 1995; Lo et al. 2008; Woo et al. 2008), while there is an obvious limit on the validity of such observations as material properties of various tissues are different depending on the state of the subject, that is, whether it is alive or not. Various applications in biomechanics have long demonstrated that realistic mathematical modeling is an appropriate tool for the simulation and analysis of complex biological and
*Corresponding author. Email: [email protected] q 2015 Taylor & Francis
physical structures such as the human knee joint although they cannot be always fully validated (Viceconti et al. 2005). This is due to material properties which have a wide range of values for living beings, compared to man made materials, and the complex geometry of the systems modeled. During the past two decades, a number of analytical model studies with different degrees of sophistication and accuracy, have been presented in literature (Bonnel and Micaleff 1988; Huson et al. 1989; Bendjaballah et al. 1995; Song et al. 2004; Yousif and AlRuznamachi 2009; Bini et al. 2010). These studies have focused on critical parameters, such as the kinematics (Huson et al. 1989), force, and stress distribution (Bonnel and Micaleff 1988; Bendjaballah et al. 1995; Song et al. 2004; Yousif and Al-Ruznamachi 2009) and fatigue (Bini et al. 2010) of the knee joint. Computerized tomography (CT) has also been used to model the joint (Bendjaballah et al. 1995; Yousif and Al-Ruznamachi 2009). Previous ACL numerical models in literature have modeled the ligament as a bundle of multiple fibers with a non-isometric behavior (Veselko and Godler 2000), or
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as a single fiber with an isometric behavior (Amis and Zavras 1995). In this work, human knee joint finite element models based on accurate geometrical entities were developed in order to study their mechanical behavior. Three human knee joint configurations were studied, of (i) an intact, (ii) an ACL-deficient (as would happen following an accident), and (iii) a reconstructed (after surgical repair operation) human knee joint model. For the reconstructed model, a novel repair device which had been developed and patented by the authors was employed as it would be in an actual orthopedic operation. The aim of this work is to study the mechanical behavior of the human knee joint, in terms of calculated stress and displacement in the joint. In addition, to study the differences of the mechanical behavior between an intact, an ACL deficient, and a reconstructed knee joint and the model behaviour when linear and non-linear materials are selected in biomechanical models. Finally the efficacy of the knee structure is evaluated in each case. For the verification of the developed models, static load cases, which have been presented in a previous modeling work, were used. Static and dynamic load cases presented in this work were applied and results were compared for verification. Results of this work for the static load cases studied were found to be in good agreement with previous modeling work for both displacements and the areas where the maximum stresses appear. 2. Material and methods In order to construct a realistic and physically accurate three-dimensional geometric model of an intact knee joint, three-dimensional scanned data from a replica of the knee were used. These data were acquired with a Konica Minolta (Tokyo, Japan) VI910 3d dimensional noncontact laser scanner, which can scan with a resolution of up to 640X480, that is, 307,200 points per scan, while it can also record scanned surface structure data, such as texture and color. The scanner employed can capture geometry at a range up to 2.5 m, with an accuracy that varies in the three directions, with the Z axis being the most accurate (^ 0.10 mm) and the X axis having the lowest accuracy of ^ 0.22 mm and a scan time of 2.5 s (fine mode requires three passes, which take 7.5 s per scan). The life size human knee joint model with ligaments used was model A82-1 from 3B Scientific, Inc. (Hamburg, Germany). This model represents the bones and the ligaments of the knee joint, but has limitations, mainly in the bones geometry, which are not important for this study, as bones have a significantly higher strength that the ligaments in the joint and are considered rigid in many similar studies in literature (Moglo and Shirazi-Adl 2003). Digital data from the scanner representing the outer surface of the individual components of the knee were
Figure 1. The developed intact three dimensional geometric knee model: (1) femur, (2) LCL, (3) MCL (4), ACL, (5) PCL, (6) tibia, (7) fibula, and (8) meniscus.
processed to reduce scanning noise and import them in PTC Creo 1.0 CAD software, which developed the threedimensional geometric models. For the development of the intact knee joint threedimensional geometric model (Figure 1), datum points determined from the scanned data were assigned to represent points on the surface of each knee part. Using these points, curves were created which were used as guides for the solid geometry parts. After each individual part was created, the knee parts were assembled as one structure. Bones were placed at a specified distance from each other, and ligaments, which join the bones together, were joined to the bones, in such a way that their surface matched the geometrical features of the bone’s surface. The same approach was followed to determine the meniscal/cartilage geometry. Exact positions were determined from the physical knee model employed. The developed three-dimensional geometric model includes all the knee parts available in the physical knee model, that is, the femur, the lateral collateral ligament (LCL), the medial collateral ligament (MCL), the ACL, the posterior cruciate ligament (PCL), the tibia, the fibula, and the meniscus. For the development of the ACL-deficient human knee joint, the ACL was removed from the intact knee joint without altering the associations of the other geometric entities. In the reconstructed human knee model, a novel repair device developed and patented by one of the authors was employed (Stefanoudakis 2008a, 2008b). This surgical repair device aims to reduce the time necessary for the surgical operation as well as the time necessary to reach full recovery, reduce the probability of injury to the graft and bone (e.g., osteolysis) during fixation, and to minimize recrudescence. In the patented design employed in this study, an intermediate part has been added, which is placed between the securing pin and the graft. The securing pin is screwed on this intermediate part to secure the graft without being in direct contact to. This intermediate part is in touch with the graft, when it is secured inside the knee, and has curvilinear surfaces, to minimize the possibility of wounding the graft.
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Computer Methods in Biomechanics and Biomedical Engineering In the reconstructed human knee geometric model, this device was assembled in an ACL-deficient knee model in the same way as it would be positioned during a knee reconstruction surgical operation. A tunnel was opened in the bone of the knee joint in the same direction as that used in reality and the parts of the device were fixed to the femur and tibia, with the tendon graft being attached to them, as during surgical operations (Figure 2). The tunnel has a 238 inclination to the vertical axis that crosses the center of gravity of the femur and it is not inclined in other directions. In the actual surgical operation the device is fixed to the femur and the tibia, attaching firmly the graft, which is deformed to the geometry of the repair device parts. Soft tissues in the regions where the device parts are fixed are also deformed to the shape of the repair device parts. A similar approach was followed in the reconstructed knee model to fix the repair device parts to the femur and the tibia. The geometry of the repair parts was subtracted from the femur and tibia geometry to form the remaining geometry of the tissues. No pretension or friction was used to fix the repair device at the bones. The shape of the geometry after the subtraction restricts the relative movement between the repair device and the bones. The assembled three-dimensional knee joint geometric models were input to the finite element analysis module of a commercial software tool. The models were discretized using three-dimensional tetrahedral solid elements. The number of elements in each model is shown in Table 1 (Vairis et al. 2013). The number of elements in each model is optimized in order to reduce calculation time without affecting the accuracy of the calculations. More densely populated models were developed with about 22,000 elements. But after several iterations, the models presented in Table 1 reach almost identical results (with , 2% difference) as the more detailed ones, at a significantly shorter calculation time. In order to verify the selected loading conditions maximum stresses and displacements were compared.
Figure 2. The developed reconstructed three-dimensional geometric knee model: (1) pin, (2) graft securing pin, (3) graft securing pin screw, and (4) graft.
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Table 1. Number of solid elements in each human knee model (same mesh for both linear materials and non-linear materials models). Model Intact knee ACL-deficient knee Reconstructed knee
Number of solid elements in the mesh 7021 5753 12,255
Two different versions were developed for comparison as well as for research into the behavior of the knee model. In the first version all materials were assumed to be linear elastic and the other one all ligaments were assumed to follow a non-linear hyper-elastic law satisfying the neohookean equation, which is closer to the actual mechanical behavior of tissues. Both linear (Table 2) and non-linear material properties (Pen˜a et al. 2006) were drawn from literature and assigned to individual geometric entities. Constrains, which define the degrees of freedom each knee joint element has, were the same as those applied in literature (Moglo and Shirazi-Adl 2003) and assigned to all geometric entities. All geometric entities of the knee were bonded together for all three different model configurations. This implies that loads between them are transferred through their common interface and geometric entities always touch each other during analysis. Between the meniscus and the tibia, a frictionless contact was defined, which does not allow penetration between entities, although forces are transferred between them when they touch each other. The tibia and the fibula were fixed in all axes at their ends. No initial tension was defined in the model ligaments. In order to verify the developed human knee joint numerical model and evaluate its efficacy, static load cases were applied, as those that have been used in a previous modeling work (Moglo and Shirazi-Adl 2003). In this work, a three-dimensional finite element model of the human knee cartilage, menisci, and ligaments was presented, which was analyzed using custom developed numerical analysis software. Both, the intact knee and the joint with the severed ACL, were studied under static passive loading at different flexion angles. Bony structures were ignored in that model for they are very rigid compared to other structures in the joint. Non-linear material data were used for different ligaments, while the meniscus was assumed to be a non-homogeneous isotropic composite part. Articulations between cartilage and meniscus were assumed to have frictionless contact. In order to verify the current finite element model with Moglo and Shirazi-Adl (Moglo and Shirazi-Adl 2003), the exactly same load cases were applied to the intact and the ACL-deficient knee joint model, for the case of zero flexion angle. This approach was chosen to directly compare the calculated results of the current model, with
4 Table 2.
A. Vairis et al. Linear material properties used in the analysis.
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Bone Ti6Al4V Ligament Tendon graft Meniscus
Young’s modulus (GPa)
Poisson’s ratio
17 (Cowin 1989) 113.80 (Boyer et al. 1994) 0.39 (Woo et al. 2007) 1.30 (Reeves et al. 2009) 0.003 (McDermott et al. 2008)
0.36 (Cowin 1989) 0.34 (Boyer et al. 1994) 0.4 (Cowin 1989) 0.46 (Li et al. 2001) 0.46 (Donahue et al. 2002)
those by the model of Moglo et al., which had produced results which were in good agreement with experimental data (Bendjaballah et al. 1995, 1998). In effect, the current work is verified through the experimental data from another source. The femur was subjected to a posterior horizontal force, and 10 different load cases were studied for different forces, as in the work of Moglo– Shinazi (Moglo and Shirazi-Adl 2003), ranging from 10 to 100 N in 10 N steps. Comparative force-displacement graphs were produced for assessment purposes. In the graphs displacements are calculated for two different directions, the posterior –anterior (post/ant) direction and the medial – lateral (med/lat) direction. In addition, the load distribution in the knee joint was qualitatively compared between the two studies. These loads were also applied to the reconstructed human knee case to evaluate the model and the behavior of the human knee in these cases. Dynamic load cases were also applied to the human knee models, to evaluate the behavior of the knee joint under this type of loading. Tensile forces changing with
time were applied at the end of the femur. The developed stress history was calculated for each case. Finally, unit displacements in four different directions were applied to the models, to determine the deformation on the model (modal shape) under this loading for each case and the natural frequencies for each knee model. The dynamic calculations were done in each of the three different human knee models (intact, ACL-deficient, reconstructed) with linear material properties only, due to PTC Creo 1.0 FEA software limitations. 3.
Results
Figure 3(a) shows the comparison for the developed knee displacement, between the current study intact knee models (linear and non-linear materials) and that of Moglo – Shinazi intact knee model for different force load cases. Figure 3(b) shows the equivalent results for the ACL-deficient case. For the ACL-deficient human knee joint model, as expected, displacements were significantly higher than the values calculated for the intact knee model,
Figure 3. Comparison between the current knee models (linear and non-linear materials) and the equivalent Moglo– Shinazi model for calculated displacement for different force load cases. (a) Intact knee, (b) ACL-deficient knee, and (c) reconstructed knee.
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Computer Methods in Biomechanics and Biomedical Engineering with a difference of about an order of magnitude, demonstrating the importance of this type of injury to the knee joint stability. Similar results were also found in literature (Moglo and Shirazi-Adl 2003). Figure 3(c) shows the comparison for the knee displacement between the current study’s reconstructed knee models (linear and non-linear materials) and that of Moglo– Shinazi intact knee model for different force loads. The intact and ACL-deficient models showed similar response to that of Moglo– Shinazi, having values which were very close. This was also the case when comparing results between the reconstructed human knee joint model and the intact knee model of Moglo– Shinazi. Figure 4 shows the stress distribution in the knee joint ligaments for a load of 100 N for the intact knee with nonlinear materials. Moglo –Shinazi in their study (Moglo and Shirazi-Adl 2003) have also presented the load distribution in the knee ligaments. As no two geometries are exactly alike, in a similar fashion to nature, and as such the two geometric models cannot be exactly the same and therefore produce different results, but both analyses do show quantitative agreement with the same stress distribution. In real life, static loads are applied frequently to the human knee during each day, but in everyday physical activities the loads on the human knee are mainly dynamic. In order to study the behavior of the human knee under dynamic loading, two simple dynamic load cases were applied to the human knee models as well. Tensile forces were applied at the end of femur, with the value of the force load changing with time. In the first dynamic load
Figure 4. Stress distribution (kPa) in the knee joint ligaments for the load case of 100 N (intact knee model with non-linear materials).
Figure 5.
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Dynamic load patterns applied to the model.
case, the force load is increasing at a constant rate (Figure 5) and in the second dynamic load case, a tensile load was changing with time as in Figure 6. Due to PTC Creo 1.0 FEA software limitations, the dynamic loading cases could only be studied for linear materials. For the dynamic load cases studied, the response of the models, as expected, was similar to the loading patterns for both displacements and stresses (Figure 6). This verifies that the response of the model is the expected. Figure 6 also shows stresses and displacements of the model for the applied loads. Table 3 shows the maximum stress and displacement values, calculated for the two dynamic load cases studied. Because the maximum force in both dynamic load cases is
Figure 6. Calculated stress and displacement on the PCL, under dynamic loading (second dynamic load case, intact knee model).
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Table 3. Maximum stresses and displacements calculated for the two dynamic load cases studied.
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Intact knee
Reconstructed knee
First dynamic load case Max stress (MPa) MCL 55.34 LCL 21.21 ACL/graft 170.46 PCL 20.37 Displacement (mm) MCL 82.19 LCL 77.02 ACL/graft 76.42 PCL 57.17 Second dynamic load case Max stress (MPa) MCL 55.52 LCL 21.18 ACL/graft 170.64 PCL 20.34 Displacement (mm) MCL 81.96 LCL 76.92 ACL/graft 76.36 PCL 57.09
21.37 18.55 106.80 37.76 80.18 71.12 151.08 60.95 21.41 18.65 107.51 38.03 80.68 71.49 151.96 61.28
the same and linear material properties were used in these studies, the maximum values are expected to be the same in each ligament for the two cases. As it is shown, results are very close in value. Finally, the natural frequencies of the knee models for four different modal shapes were determined to identify possible changes in the behavior of the whole structure due to the introduction of the ACL repair device. In all modal shapes, the knee was fixed at the femur and a unit displacement was applied at the tibia. In the first modal shape (mode 1), a unit displacement was applied at the end of tibia in the post/ant direction, in the second (mode 2) the displacement was in the med/lat direction, in the third (mode 3) the displacement was applied in the middle of the knee model, with a post/ant direction and in the fourth (mode 4) a unit longitudinal rotation was applied at the tibia. In all cases, the knee models were then left free to vibrate and the natural frequencies in each case were determined. Table 4 shows the calculated natural frequencies for the different human knee models. The intact and the reconstructed knee have similar response, while the ACL-deficient knee has significantly lower Table 4.
Mode 1 Mode 2 Mode 3 Mode 4
Natural frequencies of the human knee models. Intact knee (Hz)
ACL-deficient (Hz)
Recon. knee (Hz)
49.36 77.15 166.21 244.06
7.92 15.27 41.12 53.14
48.27 76.98 162.76 202.56
Figure 7. Modal shapes of the intact knee model. (a) Mode 1: unit displacement applied at the end of tibia in the post/ant direction. (b) Mode 2: unit displacement applied at the end of tibia in the med/lat direction. (c) Mode 3: unit displacement applied in the middle of the knee model, with a post/ant direction. (d) Mode 4: unit longitudinal rotation applied at the tibia.
natural frequencies, indicating the difference in the behavior of knee joint in this situation and the similarities in the dynamic behavior of the intact and the reconstructed human knee model. Figure 7 presents the response of the knee model during the modal shapes study of the intact knee model, as they were determined during the natural frequencies calculation.
4. Discussion For the developed intact human knee joint model, in the posterior/anterior direction, the calculated displacement values were lower than the Moglo –Shinazi ones, while in the medial/lateral direction, they were higher. The average difference between the two models for displacement was 21.5% for the linear materials case and 15.2% for the nonlinear materials case, showing that non-linear materials model more accurately the human knee joint behavior. The ACL-deficient human knee joint model with nonlinear materials for loads higher than 70 N produces very large displacements in the posterior/anterior direction. For this reason, displacements in this direction are not shown in the graph and the equivalent displacement values in the medial/lateral direction are shown up until this load. This is due to the material properties used for the meniscus in this model. In literature, the material properties of the meniscus show a wide range in values, with Young’s modulus varying from 3 MPa (Donahue et al. 2002) used in this study up to 59 MPa (Pen˜a et al. 2006). This wide variation is due to the different samples used in each study. The developed human knee finite element model was sensitive to the values used for the material properties of the meniscus, with significantly different results being produced for a higher Young’s modulus. Higher Young’s
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Computer Methods in Biomechanics and Biomedical Engineering modulus for the meniscus make the model stiffer and displacements significantly lower. For the reconstructed human knee joint model, in the posterior/anterior direction, displacements were lower than the Moglo– Shinazi model values, while in the medial/lateral direction, were higher. The average difference for displacement was 16.2% for the linear materials model and 12.5% for the non-linear materials model, demonstrating again the usefulness in the accuracy of the analysis of non-linear materials. These results verify that the reconstruction process restores knee stability and functionality, as it is expected. As the load distribution and the areas where high stresses appear are similar between these models and that of Moglo –Shinazi, the differences in displacement can be attributed to the complexity of the knee joint geometry and, possibly, small differences between the two geometric models. The calculations for the load cases studied demonstrate that the developed finite element knee joint model is reliable, as the results obtained are consistent with models available in literature, which in turn have compared well with experimental data. In every load case high stresses appear in the ligaments close to the connection point to the bones as well as in the middle of their length. For the dynamic load cases studied, all ligaments develop similar stress values, whether the ACL is in place or has been replaced by a tendon graft in the reconstructed knee model, developing about 25% higher stresses, indicating that the ACL is more important for maintaining knee stability when dynamic loads are applied. For this reason, in the dynamic load cases studied, the ACLdeficient knee joint seems to fail when very high displacements and stresses develop. Results in Table 3 also indicate that the reconstructed knee model behaves in a similar way to the intact knee model, which is the desired behavior. The reconstructed knee develops similar stress and displacement values with the intact knee model. Stresses and displacements in the ligaments are very close, but lower than that of the intact knee model. Only the graft is developing higher values than the ACL in the intact knee model. This in most cases cannot affect the health of the reconstructed knee joint, as the graft used for the reconstruction of the knee has higher strength than the ACL it replaces. It must be stated though, that results are sensitive to the positioning of the graft in the joint (Zavras et al. 1995). In this study, the direction of the graft tunnel is the one that ideally surgeons should follow, while in reality the graft tunnel is drilled empirically in most cases without accurate control in direction. So, a different position of the graft may affect the results and, possibly, the mechanical behavior of the graft. In this study, a human knee finite element model developed by the authors was verified with a previous
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modeling work from literature. Although results were found to be in agreement with previous work, this study does have limitations. The previous work used for verification is not extensively verified with experiments for every single load case, although its results were in good agreement with experiments, especially for the distribution of forces in the knee and the maximum values in each knee part. This makes the model of the study reliable, but has not been validated extensively to be employed in actual medical cases. Especially in the reconstructed knee model case, no experimental data are available anywhere, although its behavior for the load cases studied verifies the expected results of surgery, and its results are very close to those of the intact knee model. This is also what is expected from patients after surgery. The static load cases studied are very common in real life, while most of the loads in the knee are dynamic. In addition, no initial tension was applied to the ligaments. For these reasons, the loads applied provide useful information for the behavior of the finite element model, but cannot be considered to fully model the human knee in everyday life activities. Finally, a limitation of this study is that the behavior of the knee in different flexion angles has not been included. In literature, it has been shown that the displacement of the knee increases in flexion angles close to 308 (Moglo and Shirazi-Adl 2003). 5.
Conclusions
In this study, a realistic three-dimensional finite element model of the knee joint which incorporates bone structures as well as ligaments and menisci was developed and a number of analyses with static and dynamic loads for an intact, an ACL-deficient, and a reconstructed knee were performed. For the load cases studied, the model estimated stresses and displacements that were within the material elastic range, which is the expected response for the loads selected. Also, the stress distribution calculated in the knee ligaments is reasonable, with no high stresses developing at the connecting interface to the bone structures of the knee. If such were present, they would be numerical artifacts as experience shows that failure does not occur in these areas in patients. The only exception to this was the load case on the ACL-deficient knee model, with linear materials, showing that linear materials are not adequate to model ligament behavior in the extreme load cases. As expected, the ACL-deficient knee produced higher stress and displacement values, while for the reconstructed knee model results were very close to the intact knee, which is the desirable outcome of surgery. In this study the material properties used for ligaments were both of the linear elastic and the non-linear type, and did produce comparable results for stress and displacement with other validated models which used hyper-elastic
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material properties. This agreement may be attributed to the fact that stresses developed were small, all within the elastic region, and not large to be affected by the nonlinear material properties. The model developed was validated with the results produced by another numerical model, which in turn had been found to be in good agreement with experimental data. Results in the current study were found to be in good agreement with those from previous modeling work, which make this finite element human knee model a reliable numerical tool. The load cases studied were adequate for the validation of the finite element model behavior, but not characteristic of the human knee loading of the wide range found in everyday life activities.
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Conflict of interest disclosure statement No potential conflict of interest was reported by the authors.
Notes 1. 2. 3. 4. 5.
Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]
References Amis A, Zavras T. 1995. Isometricity and graft placement during anterior cruciate ligament reconstruction. Knee. 2(1):5 –17, doi:10.1016/0968-0160(95)00003-8. Bendjaballah M, Shirazi-Adl A, Zukor D. 1995. Biomechanics of the human knee joint in compression: reconstruction, mesh generation and finite element analysis. Knee. 2(2):69 –79, doi:10.1016/0968-0160(95)00018-K. Bendjaballah MZ, Shirazi-Adl A, Zukor DJ. 1998. Biomechanical response of the passive human knee joint under anterior – posterior forces. Clin Biomech. 13(8):625 –633, doi:10.1016/S0268-0033(98)00035-7. Bini RR, Diefenthaeler F, Mota CB. 2010. Fatigue effects on the coordinative pattern during cycling: kinetics and kinematics evaluation. J Electromyogr Kinesiol. 20(1):102– 107, doi:10. 1016/j.jelekin.2008.10.003. Bonnel F, Micaleff JP. 1988. Biomechanics of the ligaments of the human knee and of artificial ligaments. Surg Radiol Anat. 10(3):221– 227, doi:10.1007/BF02115241. Boyer R, Welsch G, Collings EW. 1994. Materials properties handbook: titanium alloys. Materials Park (OH): ASM International. Cowin SC. 1989. Bone mechanics. Boca Raton (FL): CRC Press (ISBN/ISSN: 0849391172). Donahue TLH, Hull ML, Rashid MM, Jacobs CR. 2002. A finite element model of the human knee joint for the study of tibiofemoral contact. J Biomech Eng. 124(3):273– 280, doi:10. 1115/1.1470171. Huson A, Spoor CW, Verbout AJ. 1989. A model of the human knee, derived from kinematic principles and its relevance for endoprosthesis design. Acta Morphol Neerl Scand. 27:45 – 62. Jones RSR, Nawana NSN, Pearcy MJM, Learmonth DJAD, Bickerstaff DRD, Costi JJJ, Paterson RS. 1995. Mechanical
properties of the human anterior cruciate ligament. Clin Biomech. 10(7):339–344, doi:10.1016/0268-0033(95)98193-X. Li G, Lopez O, Rubash H. 2001. Variability of a threedimensional finite element model constructed using magnetic resonance images of a knee for joint contact stress analysis. J Biomech Eng. 123(4):341 –346, doi:10.1115/1.1385841. Lo J, Mu¨ller O, Wu¨nschel M, Bauer S, Wu¨lker N. 2008. Forces in anterior cruciate ligament during simulated weight-bearing flexion with anterior and internal rotational tibial load. J Biomech. 41(9):1855–1861, doi:10.1016/j.jbiomech.2008.04.010. McDermott ID, Masouros SD, Amis AA. 2008. Biomechanics of the menisci of the knee. Curr Orthop. 22(3):193– 201, doi:10. 1016/j.cuor.2008.04.005. Moglo KE, Shirazi-Adl A. 2003. Biomechanics of passive knee joint in drawer: load transmission in intact and ACL-deficient joints. Knee. 10(3):265 –276, doi:10.1016/S0968-0160(02) 00135-7. Pen˜a E, Calvo B, Martı´nez MA, Doblare´ M. 2006. A threedimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint. J Biomech. 39:1686– 1701. Reeves ND, Maganaris CN, Maffulli N, Rittweger J. 2009. Human patellar tendon stiffness is restored following graft harvest for anterior cruciate ligament surgery. J Biomech. 42(7):797– 803, doi:10.1016/j.jbiomech.2009.01.030. Song Y, Debski RE, Musahl V, Thomas M, Woo SLY. 2004. A three-dimensional finite element model of the human anterior cruciate ligament: a computational analysis with experimental validation. J Biomech. 37(3):383– 390, doi:10.1016/ S0021-9290(03)00261-6. Stefanoudakis G. 2008a. Greek Patent GR0100566. Device for anchoring tendon grafts. Stefanoudakis G. 2008b. Greek Patent GR0100567. Pin for holding together tendon grafts. Vairis A, Petousis M, Stefanoudakis G, Vidakis N, Kandyla B, Tsainis A. 2013. Studying the intact, ACL-deficient and reconstructed human knee joint using a finite element model. IMECE2013 – ASME 2013 International Mechanical Engineering Congress & Exposition; November 13 – 21; San Diego, CA, USA. Veselko M, Godler I. 2000. Biomechanical study of a computer simulated reconstruction of the anterior cruciate ligament (ACL). Comput Biol Med. 30(6):299– 309, doi:10.1016/ S0010-4825(00)00016-0. Viceconti M, Olsen S, Nolte LP, Burton K. 2005. Extracting clinically relevant data from finite element simulations. Clin Biomech. 20(5):451– 454, doi:10.1016/j.clinbiomech.2005. 01.010. Woo SLY, Almarza AJ, Liang R, Fisher MB. 2007. Functional tissue engineering of ligament and tendon injuries. Translational approaches in tissue engnineering and regenerative medicine, book chapter no. 9, Artech House Publisher, ISBN-10: 1596931116, ISBN-13: 978-1596931114. Woo SLY, Almarza AJ, Karaoglu S, Abramowitch. 2008. Functional tissue engineering of ligament and tendon injuries. In: Atala A, Lanza R, Thomson J, Nerem R, editors. Principles of regenerative medicine. Burlington (MA): Academic Press. p. 1206– 1231. Yousif AE, Al-Ruznamachi SRF. 2009. A statical model of the human knee joint. 25th Southern Biomedical Engineering Conference 2009, IFMBE Proceedings, Vol. 24, pp. 227– 232. Zavras TDT, Mackenney RPR, Amis AA, Zavras TD, Mackenney RP. 1995. The natural history of anterior cruciate ligament reconstruction using patellar tendon autograft. Knee. 2(4):211 – 217, doi:10.1016/0968-0160(96)00009-9.
Computer Methods in Biomechanics and Biomedical Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcmb20
Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model a
b
a
a
Achilles Vairis , George Stefanoudakis , Markos Petousis , Nectarios Vidakis , Andreas-Marios a
Tsainis & Betina Kandyla
c
a
Mechanical Engineering Department, Technological Education Institute of Crete, Estavromenos, 71004 Heraklion, Crete, Greece b
Creta Interclinic Hospital, Minoos 63 Str., 71304 Heraklion, Crete, Greece
c
3 Karistou Rd, Athens 11523, Greece Published online: 02 Mar 2015.
Click for updates To cite this article: Achilles Vairis, George Stefanoudakis, Markos Petousis, Nectarios Vidakis, Andreas-Marios Tsainis & Betina Kandyla (2015): Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1015526 To link to this article: http://dx.doi.org/10.1080/10255842.2015.1015526
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Computer Methods in Biomechanics and Biomedical Engineering, 2015 http://dx.doi.org/10.1080/10255842.2015.1015526
Evaluation of an intact, an ACL-deficient, and a reconstructed human knee joint finite element model Achilles Vairisa*, George Stefanoudakisb1, Markos Petousisa2, Nectarios Vidakisa3, Andreas-Marios Tsainisa4 and Betina Kandylac5 a
Mechanical Engineering Department, Technological Education Institute of Crete, Estavromenos, 71004 Heraklion, Crete, Greece; b Creta Interclinic Hospital, Minoos 63 Str., 71304 Heraklion, Crete, Greece; c3 Karistou Rd, Athens 11523, Greece
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(Received 12 June 2014; accepted 2 February 2015) The human knee joint has a three-dimensional geometry with multiple body articulations that produce complex mechanical responses under loads that occur in everyday life and sports activities. Understanding the complex mechanical interactions of these load-bearing structures is of use when the treatment of relevant diseases is evaluated and assisting devices are designed. The anterior cruciate ligament (ACL) in the knee is one of four main ligaments that connects the femur to the tibia and is often torn during sudden twisting motions, resulting in knee instability. The objective of this work is to study the mechanical behavior of the human knee joint and evaluate the differences in its response for three different states, i.e., intact, ACL-deficient, and surgically treated (reconstructed) knee. The finite element models corresponding to these states were developed. For the reconstructed model, a novel repair device was developed and patented by the author in previous work. Static load cases were applied, as have already been presented in a previous work, in order to compare the calculated results produced by the two models the ACL-deficient and the surgically reconstructed knee joint, under the exact same loading conditions. Displacements were calculated in different directions for the load cases studied and were found to be very close to those from previous modeling work and were in good agreement with experimental data presented in literature. The developed finite element model for both the intact and the ACL-deficient human knee joint is a reliable tool to study the kinematics of the human knee, as results of this study show. In addition, the reconstructed human knee joint model had kinematic behavior similar to the intact knee joint, showing that such reconstruction devices can restore human knee stability to an adequate extent. Keywords: finite element modeling; knee ligament repair; tendon graft; anterior cruciate ligament; biomechanics
1.
Introduction
The human knee joint has a three-dimensional geometry with multiple body articulations which produce complex mechanical responses, both actively and passively, under loads that occur in everyday life and sports activities alike. In this joint, the anterior cruciate ligament (ACL) is crucial in keeping normal knee functions (Moglo and Shirazi-Adl 2003). When injured, it is often treated with surgical reconstruction (Lo et al. 2008; Woo et al. 2008) as its failure results in joint instability in the anteroposterior direction which inhibits walking (Lo et al. 2008; Woo et al. 2008). The mechanical behavior of this important structure has been studied experimentally in cadavers (Jones et al. 1995; Lo et al. 2008; Woo et al. 2008), while there is an obvious limit on the validity of such observations as material properties of various tissues are different depending on the state of the subject, that is, whether it is alive or not. Various applications in biomechanics have long demonstrated that realistic mathematical modeling is an appropriate tool for the simulation and analysis of complex biological and
*Corresponding author. Email: [email protected] q 2015 Taylor & Francis
physical structures such as the human knee joint although they cannot be always fully validated (Viceconti et al. 2005). This is due to material properties which have a wide range of values for living beings, compared to man made materials, and the complex geometry of the systems modeled. During the past two decades, a number of analytical model studies with different degrees of sophistication and accuracy, have been presented in literature (Bonnel and Micaleff 1988; Huson et al. 1989; Bendjaballah et al. 1995; Song et al. 2004; Yousif and AlRuznamachi 2009; Bini et al. 2010). These studies have focused on critical parameters, such as the kinematics (Huson et al. 1989), force, and stress distribution (Bonnel and Micaleff 1988; Bendjaballah et al. 1995; Song et al. 2004; Yousif and Al-Ruznamachi 2009) and fatigue (Bini et al. 2010) of the knee joint. Computerized tomography (CT) has also been used to model the joint (Bendjaballah et al. 1995; Yousif and Al-Ruznamachi 2009). Previous ACL numerical models in literature have modeled the ligament as a bundle of multiple fibers with a non-isometric behavior (Veselko and Godler 2000), or
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A. Vairis et al.
as a single fiber with an isometric behavior (Amis and Zavras 1995). In this work, human knee joint finite element models based on accurate geometrical entities were developed in order to study their mechanical behavior. Three human knee joint configurations were studied, of (i) an intact, (ii) an ACL-deficient (as would happen following an accident), and (iii) a reconstructed (after surgical repair operation) human knee joint model. For the reconstructed model, a novel repair device which had been developed and patented by the authors was employed as it would be in an actual orthopedic operation. The aim of this work is to study the mechanical behavior of the human knee joint, in terms of calculated stress and displacement in the joint. In addition, to study the differences of the mechanical behavior between an intact, an ACL deficient, and a reconstructed knee joint and the model behaviour when linear and non-linear materials are selected in biomechanical models. Finally the efficacy of the knee structure is evaluated in each case. For the verification of the developed models, static load cases, which have been presented in a previous modeling work, were used. Static and dynamic load cases presented in this work were applied and results were compared for verification. Results of this work for the static load cases studied were found to be in good agreement with previous modeling work for both displacements and the areas where the maximum stresses appear. 2. Material and methods In order to construct a realistic and physically accurate three-dimensional geometric model of an intact knee joint, three-dimensional scanned data from a replica of the knee were used. These data were acquired with a Konica Minolta (Tokyo, Japan) VI910 3d dimensional noncontact laser scanner, which can scan with a resolution of up to 640X480, that is, 307,200 points per scan, while it can also record scanned surface structure data, such as texture and color. The scanner employed can capture geometry at a range up to 2.5 m, with an accuracy that varies in the three directions, with the Z axis being the most accurate (^ 0.10 mm) and the X axis having the lowest accuracy of ^ 0.22 mm and a scan time of 2.5 s (fine mode requires three passes, which take 7.5 s per scan). The life size human knee joint model with ligaments used was model A82-1 from 3B Scientific, Inc. (Hamburg, Germany). This model represents the bones and the ligaments of the knee joint, but has limitations, mainly in the bones geometry, which are not important for this study, as bones have a significantly higher strength that the ligaments in the joint and are considered rigid in many similar studies in literature (Moglo and Shirazi-Adl 2003). Digital data from the scanner representing the outer surface of the individual components of the knee were
Figure 1. The developed intact three dimensional geometric knee model: (1) femur, (2) LCL, (3) MCL (4), ACL, (5) PCL, (6) tibia, (7) fibula, and (8) meniscus.
processed to reduce scanning noise and import them in PTC Creo 1.0 CAD software, which developed the threedimensional geometric models. For the development of the intact knee joint threedimensional geometric model (Figure 1), datum points determined from the scanned data were assigned to represent points on the surface of each knee part. Using these points, curves were created which were used as guides for the solid geometry parts. After each individual part was created, the knee parts were assembled as one structure. Bones were placed at a specified distance from each other, and ligaments, which join the bones together, were joined to the bones, in such a way that their surface matched the geometrical features of the bone’s surface. The same approach was followed to determine the meniscal/cartilage geometry. Exact positions were determined from the physical knee model employed. The developed three-dimensional geometric model includes all the knee parts available in the physical knee model, that is, the femur, the lateral collateral ligament (LCL), the medial collateral ligament (MCL), the ACL, the posterior cruciate ligament (PCL), the tibia, the fibula, and the meniscus. For the development of the ACL-deficient human knee joint, the ACL was removed from the intact knee joint without altering the associations of the other geometric entities. In the reconstructed human knee model, a novel repair device developed and patented by one of the authors was employed (Stefanoudakis 2008a, 2008b). This surgical repair device aims to reduce the time necessary for the surgical operation as well as the time necessary to reach full recovery, reduce the probability of injury to the graft and bone (e.g., osteolysis) during fixation, and to minimize recrudescence. In the patented design employed in this study, an intermediate part has been added, which is placed between the securing pin and the graft. The securing pin is screwed on this intermediate part to secure the graft without being in direct contact to. This intermediate part is in touch with the graft, when it is secured inside the knee, and has curvilinear surfaces, to minimize the possibility of wounding the graft.
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Computer Methods in Biomechanics and Biomedical Engineering In the reconstructed human knee geometric model, this device was assembled in an ACL-deficient knee model in the same way as it would be positioned during a knee reconstruction surgical operation. A tunnel was opened in the bone of the knee joint in the same direction as that used in reality and the parts of the device were fixed to the femur and tibia, with the tendon graft being attached to them, as during surgical operations (Figure 2). The tunnel has a 238 inclination to the vertical axis that crosses the center of gravity of the femur and it is not inclined in other directions. In the actual surgical operation the device is fixed to the femur and the tibia, attaching firmly the graft, which is deformed to the geometry of the repair device parts. Soft tissues in the regions where the device parts are fixed are also deformed to the shape of the repair device parts. A similar approach was followed in the reconstructed knee model to fix the repair device parts to the femur and the tibia. The geometry of the repair parts was subtracted from the femur and tibia geometry to form the remaining geometry of the tissues. No pretension or friction was used to fix the repair device at the bones. The shape of the geometry after the subtraction restricts the relative movement between the repair device and the bones. The assembled three-dimensional knee joint geometric models were input to the finite element analysis module of a commercial software tool. The models were discretized using three-dimensional tetrahedral solid elements. The number of elements in each model is shown in Table 1 (Vairis et al. 2013). The number of elements in each model is optimized in order to reduce calculation time without affecting the accuracy of the calculations. More densely populated models were developed with about 22,000 elements. But after several iterations, the models presented in Table 1 reach almost identical results (with , 2% difference) as the more detailed ones, at a significantly shorter calculation time. In order to verify the selected loading conditions maximum stresses and displacements were compared.
Figure 2. The developed reconstructed three-dimensional geometric knee model: (1) pin, (2) graft securing pin, (3) graft securing pin screw, and (4) graft.
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Table 1. Number of solid elements in each human knee model (same mesh for both linear materials and non-linear materials models). Model Intact knee ACL-deficient knee Reconstructed knee
Number of solid elements in the mesh 7021 5753 12,255
Two different versions were developed for comparison as well as for research into the behavior of the knee model. In the first version all materials were assumed to be linear elastic and the other one all ligaments were assumed to follow a non-linear hyper-elastic law satisfying the neohookean equation, which is closer to the actual mechanical behavior of tissues. Both linear (Table 2) and non-linear material properties (Pen˜a et al. 2006) were drawn from literature and assigned to individual geometric entities. Constrains, which define the degrees of freedom each knee joint element has, were the same as those applied in literature (Moglo and Shirazi-Adl 2003) and assigned to all geometric entities. All geometric entities of the knee were bonded together for all three different model configurations. This implies that loads between them are transferred through their common interface and geometric entities always touch each other during analysis. Between the meniscus and the tibia, a frictionless contact was defined, which does not allow penetration between entities, although forces are transferred between them when they touch each other. The tibia and the fibula were fixed in all axes at their ends. No initial tension was defined in the model ligaments. In order to verify the developed human knee joint numerical model and evaluate its efficacy, static load cases were applied, as those that have been used in a previous modeling work (Moglo and Shirazi-Adl 2003). In this work, a three-dimensional finite element model of the human knee cartilage, menisci, and ligaments was presented, which was analyzed using custom developed numerical analysis software. Both, the intact knee and the joint with the severed ACL, were studied under static passive loading at different flexion angles. Bony structures were ignored in that model for they are very rigid compared to other structures in the joint. Non-linear material data were used for different ligaments, while the meniscus was assumed to be a non-homogeneous isotropic composite part. Articulations between cartilage and meniscus were assumed to have frictionless contact. In order to verify the current finite element model with Moglo and Shirazi-Adl (Moglo and Shirazi-Adl 2003), the exactly same load cases were applied to the intact and the ACL-deficient knee joint model, for the case of zero flexion angle. This approach was chosen to directly compare the calculated results of the current model, with
4 Table 2.
A. Vairis et al. Linear material properties used in the analysis.
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Bone Ti6Al4V Ligament Tendon graft Meniscus
Young’s modulus (GPa)
Poisson’s ratio
17 (Cowin 1989) 113.80 (Boyer et al. 1994) 0.39 (Woo et al. 2007) 1.30 (Reeves et al. 2009) 0.003 (McDermott et al. 2008)
0.36 (Cowin 1989) 0.34 (Boyer et al. 1994) 0.4 (Cowin 1989) 0.46 (Li et al. 2001) 0.46 (Donahue et al. 2002)
those by the model of Moglo et al., which had produced results which were in good agreement with experimental data (Bendjaballah et al. 1995, 1998). In effect, the current work is verified through the experimental data from another source. The femur was subjected to a posterior horizontal force, and 10 different load cases were studied for different forces, as in the work of Moglo– Shinazi (Moglo and Shirazi-Adl 2003), ranging from 10 to 100 N in 10 N steps. Comparative force-displacement graphs were produced for assessment purposes. In the graphs displacements are calculated for two different directions, the posterior –anterior (post/ant) direction and the medial – lateral (med/lat) direction. In addition, the load distribution in the knee joint was qualitatively compared between the two studies. These loads were also applied to the reconstructed human knee case to evaluate the model and the behavior of the human knee in these cases. Dynamic load cases were also applied to the human knee models, to evaluate the behavior of the knee joint under this type of loading. Tensile forces changing with
time were applied at the end of the femur. The developed stress history was calculated for each case. Finally, unit displacements in four different directions were applied to the models, to determine the deformation on the model (modal shape) under this loading for each case and the natural frequencies for each knee model. The dynamic calculations were done in each of the three different human knee models (intact, ACL-deficient, reconstructed) with linear material properties only, due to PTC Creo 1.0 FEA software limitations. 3.
Results
Figure 3(a) shows the comparison for the developed knee displacement, between the current study intact knee models (linear and non-linear materials) and that of Moglo – Shinazi intact knee model for different force load cases. Figure 3(b) shows the equivalent results for the ACL-deficient case. For the ACL-deficient human knee joint model, as expected, displacements were significantly higher than the values calculated for the intact knee model,
Figure 3. Comparison between the current knee models (linear and non-linear materials) and the equivalent Moglo– Shinazi model for calculated displacement for different force load cases. (a) Intact knee, (b) ACL-deficient knee, and (c) reconstructed knee.
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Computer Methods in Biomechanics and Biomedical Engineering with a difference of about an order of magnitude, demonstrating the importance of this type of injury to the knee joint stability. Similar results were also found in literature (Moglo and Shirazi-Adl 2003). Figure 3(c) shows the comparison for the knee displacement between the current study’s reconstructed knee models (linear and non-linear materials) and that of Moglo– Shinazi intact knee model for different force loads. The intact and ACL-deficient models showed similar response to that of Moglo– Shinazi, having values which were very close. This was also the case when comparing results between the reconstructed human knee joint model and the intact knee model of Moglo– Shinazi. Figure 4 shows the stress distribution in the knee joint ligaments for a load of 100 N for the intact knee with nonlinear materials. Moglo –Shinazi in their study (Moglo and Shirazi-Adl 2003) have also presented the load distribution in the knee ligaments. As no two geometries are exactly alike, in a similar fashion to nature, and as such the two geometric models cannot be exactly the same and therefore produce different results, but both analyses do show quantitative agreement with the same stress distribution. In real life, static loads are applied frequently to the human knee during each day, but in everyday physical activities the loads on the human knee are mainly dynamic. In order to study the behavior of the human knee under dynamic loading, two simple dynamic load cases were applied to the human knee models as well. Tensile forces were applied at the end of femur, with the value of the force load changing with time. In the first dynamic load
Figure 4. Stress distribution (kPa) in the knee joint ligaments for the load case of 100 N (intact knee model with non-linear materials).
Figure 5.
5
Dynamic load patterns applied to the model.
case, the force load is increasing at a constant rate (Figure 5) and in the second dynamic load case, a tensile load was changing with time as in Figure 6. Due to PTC Creo 1.0 FEA software limitations, the dynamic loading cases could only be studied for linear materials. For the dynamic load cases studied, the response of the models, as expected, was similar to the loading patterns for both displacements and stresses (Figure 6). This verifies that the response of the model is the expected. Figure 6 also shows stresses and displacements of the model for the applied loads. Table 3 shows the maximum stress and displacement values, calculated for the two dynamic load cases studied. Because the maximum force in both dynamic load cases is
Figure 6. Calculated stress and displacement on the PCL, under dynamic loading (second dynamic load case, intact knee model).
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Table 3. Maximum stresses and displacements calculated for the two dynamic load cases studied.
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Intact knee
Reconstructed knee
First dynamic load case Max stress (MPa) MCL 55.34 LCL 21.21 ACL/graft 170.46 PCL 20.37 Displacement (mm) MCL 82.19 LCL 77.02 ACL/graft 76.42 PCL 57.17 Second dynamic load case Max stress (MPa) MCL 55.52 LCL 21.18 ACL/graft 170.64 PCL 20.34 Displacement (mm) MCL 81.96 LCL 76.92 ACL/graft 76.36 PCL 57.09
21.37 18.55 106.80 37.76 80.18 71.12 151.08 60.95 21.41 18.65 107.51 38.03 80.68 71.49 151.96 61.28
the same and linear material properties were used in these studies, the maximum values are expected to be the same in each ligament for the two cases. As it is shown, results are very close in value. Finally, the natural frequencies of the knee models for four different modal shapes were determined to identify possible changes in the behavior of the whole structure due to the introduction of the ACL repair device. In all modal shapes, the knee was fixed at the femur and a unit displacement was applied at the tibia. In the first modal shape (mode 1), a unit displacement was applied at the end of tibia in the post/ant direction, in the second (mode 2) the displacement was in the med/lat direction, in the third (mode 3) the displacement was applied in the middle of the knee model, with a post/ant direction and in the fourth (mode 4) a unit longitudinal rotation was applied at the tibia. In all cases, the knee models were then left free to vibrate and the natural frequencies in each case were determined. Table 4 shows the calculated natural frequencies for the different human knee models. The intact and the reconstructed knee have similar response, while the ACL-deficient knee has significantly lower Table 4.
Mode 1 Mode 2 Mode 3 Mode 4
Natural frequencies of the human knee models. Intact knee (Hz)
ACL-deficient (Hz)
Recon. knee (Hz)
49.36 77.15 166.21 244.06
7.92 15.27 41.12 53.14
48.27 76.98 162.76 202.56
Figure 7. Modal shapes of the intact knee model. (a) Mode 1: unit displacement applied at the end of tibia in the post/ant direction. (b) Mode 2: unit displacement applied at the end of tibia in the med/lat direction. (c) Mode 3: unit displacement applied in the middle of the knee model, with a post/ant direction. (d) Mode 4: unit longitudinal rotation applied at the tibia.
natural frequencies, indicating the difference in the behavior of knee joint in this situation and the similarities in the dynamic behavior of the intact and the reconstructed human knee model. Figure 7 presents the response of the knee model during the modal shapes study of the intact knee model, as they were determined during the natural frequencies calculation.
4. Discussion For the developed intact human knee joint model, in the posterior/anterior direction, the calculated displacement values were lower than the Moglo –Shinazi ones, while in the medial/lateral direction, they were higher. The average difference between the two models for displacement was 21.5% for the linear materials case and 15.2% for the nonlinear materials case, showing that non-linear materials model more accurately the human knee joint behavior. The ACL-deficient human knee joint model with nonlinear materials for loads higher than 70 N produces very large displacements in the posterior/anterior direction. For this reason, displacements in this direction are not shown in the graph and the equivalent displacement values in the medial/lateral direction are shown up until this load. This is due to the material properties used for the meniscus in this model. In literature, the material properties of the meniscus show a wide range in values, with Young’s modulus varying from 3 MPa (Donahue et al. 2002) used in this study up to 59 MPa (Pen˜a et al. 2006). This wide variation is due to the different samples used in each study. The developed human knee finite element model was sensitive to the values used for the material properties of the meniscus, with significantly different results being produced for a higher Young’s modulus. Higher Young’s
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Computer Methods in Biomechanics and Biomedical Engineering modulus for the meniscus make the model stiffer and displacements significantly lower. For the reconstructed human knee joint model, in the posterior/anterior direction, displacements were lower than the Moglo– Shinazi model values, while in the medial/lateral direction, were higher. The average difference for displacement was 16.2% for the linear materials model and 12.5% for the non-linear materials model, demonstrating again the usefulness in the accuracy of the analysis of non-linear materials. These results verify that the reconstruction process restores knee stability and functionality, as it is expected. As the load distribution and the areas where high stresses appear are similar between these models and that of Moglo –Shinazi, the differences in displacement can be attributed to the complexity of the knee joint geometry and, possibly, small differences between the two geometric models. The calculations for the load cases studied demonstrate that the developed finite element knee joint model is reliable, as the results obtained are consistent with models available in literature, which in turn have compared well with experimental data. In every load case high stresses appear in the ligaments close to the connection point to the bones as well as in the middle of their length. For the dynamic load cases studied, all ligaments develop similar stress values, whether the ACL is in place or has been replaced by a tendon graft in the reconstructed knee model, developing about 25% higher stresses, indicating that the ACL is more important for maintaining knee stability when dynamic loads are applied. For this reason, in the dynamic load cases studied, the ACLdeficient knee joint seems to fail when very high displacements and stresses develop. Results in Table 3 also indicate that the reconstructed knee model behaves in a similar way to the intact knee model, which is the desired behavior. The reconstructed knee develops similar stress and displacement values with the intact knee model. Stresses and displacements in the ligaments are very close, but lower than that of the intact knee model. Only the graft is developing higher values than the ACL in the intact knee model. This in most cases cannot affect the health of the reconstructed knee joint, as the graft used for the reconstruction of the knee has higher strength than the ACL it replaces. It must be stated though, that results are sensitive to the positioning of the graft in the joint (Zavras et al. 1995). In this study, the direction of the graft tunnel is the one that ideally surgeons should follow, while in reality the graft tunnel is drilled empirically in most cases without accurate control in direction. So, a different position of the graft may affect the results and, possibly, the mechanical behavior of the graft. In this study, a human knee finite element model developed by the authors was verified with a previous
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modeling work from literature. Although results were found to be in agreement with previous work, this study does have limitations. The previous work used for verification is not extensively verified with experiments for every single load case, although its results were in good agreement with experiments, especially for the distribution of forces in the knee and the maximum values in each knee part. This makes the model of the study reliable, but has not been validated extensively to be employed in actual medical cases. Especially in the reconstructed knee model case, no experimental data are available anywhere, although its behavior for the load cases studied verifies the expected results of surgery, and its results are very close to those of the intact knee model. This is also what is expected from patients after surgery. The static load cases studied are very common in real life, while most of the loads in the knee are dynamic. In addition, no initial tension was applied to the ligaments. For these reasons, the loads applied provide useful information for the behavior of the finite element model, but cannot be considered to fully model the human knee in everyday life activities. Finally, a limitation of this study is that the behavior of the knee in different flexion angles has not been included. In literature, it has been shown that the displacement of the knee increases in flexion angles close to 308 (Moglo and Shirazi-Adl 2003). 5.
Conclusions
In this study, a realistic three-dimensional finite element model of the knee joint which incorporates bone structures as well as ligaments and menisci was developed and a number of analyses with static and dynamic loads for an intact, an ACL-deficient, and a reconstructed knee were performed. For the load cases studied, the model estimated stresses and displacements that were within the material elastic range, which is the expected response for the loads selected. Also, the stress distribution calculated in the knee ligaments is reasonable, with no high stresses developing at the connecting interface to the bone structures of the knee. If such were present, they would be numerical artifacts as experience shows that failure does not occur in these areas in patients. The only exception to this was the load case on the ACL-deficient knee model, with linear materials, showing that linear materials are not adequate to model ligament behavior in the extreme load cases. As expected, the ACL-deficient knee produced higher stress and displacement values, while for the reconstructed knee model results were very close to the intact knee, which is the desirable outcome of surgery. In this study the material properties used for ligaments were both of the linear elastic and the non-linear type, and did produce comparable results for stress and displacement with other validated models which used hyper-elastic
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A. Vairis et al.
material properties. This agreement may be attributed to the fact that stresses developed were small, all within the elastic region, and not large to be affected by the nonlinear material properties. The model developed was validated with the results produced by another numerical model, which in turn had been found to be in good agreement with experimental data. Results in the current study were found to be in good agreement with those from previous modeling work, which make this finite element human knee model a reliable numerical tool. The load cases studied were adequate for the validation of the finite element model behavior, but not characteristic of the human knee loading of the wide range found in everyday life activities.
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Conflict of interest disclosure statement No potential conflict of interest was reported by the authors.
Notes 1. 2. 3. 4. 5.
Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]
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