Orthopaedic Research Laboratory Alumni Council

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Board of Directors

Richard Debski, Ph.D. President
Caroline Wang, M.S. Secretary
Jamie Pfaeffle, M.D., Ph.D. Treasurer
Doug Boardman, M.D.
Thay Lee, Ph.D.
Patrick McMahon, M.D.
Karen Ohland, M.S.
Christos Papgeorgiou, M.D.
Masataka Sakane, M.D.
Sven Scheffler, M.D.
Jennifer Wayne, Ph.D.

2007 Erin McGurk Grant Recipient

Xiaoyan Zhang
University of Pittsburgh

A Subject-specific Finite Element Model
of the Anterior Cruciate Ligament

Ziaoyan with her advisor, Dr. Savio Woo


I was very grateful to receive the 2007 Erin McGurk Grant from ORLAC. I would like to extend my gratitude and appreciation to Erin McGurk and ORLAC for their financial support. The grant enabled me to continue my Ph.D thesis project - A Subject-specific Finite Element Model of the Anterior Cruciate Ligament. I would like to thank Dr. Savio L-Y. Woo for his kind and patient guidance. I would also like to thank Dr. Changfu Wu, Dr. Giovanni Zamarra, and Noah Lorang for their technical assistance on this project. Finally, thank the whole MSRC family for their support.

Final Report

The anterior cruciate ligament (ACL) plays an essential role in maintaining knee stability and is the most commonly injured knee ligament during sports and work related activities. Despite of intensive efforts have been made to obtain the ACL force and strain form experiments, due to its complex anatomic structure and limitation of experiments, the stress and strain distribution in the whole ACL remains unknown, especially under complex loading conditions. An appropriately developed finite element (FE) model can provide useful information otherwise difficult to obtain from experiments, for the understanding of its mechanics, mechanisms of injuries, and help to design reconstruction and rehabilitation protocols, as well as understand the environment of biological remodeling. Thus, the objective of this study was to build a FE model of the ACL to analyze its stress distribution under 134N anterior tibial load at full extension, 30° and 60° flexion, which simulate the loading conditions clinically used for examination of ACL deficiency. Special focuses were on the magnitude of the average stress and peak stress, the location of the peak stress, as well as how they change at different flexion angles. And finally comparison with the experimental ACL resultant force was made to validate the FE model.

Materials and Methodology

Constitutive model
Structurely, ACL is a dense connective tissue consisting of mainly parallel collagen fibers embedded in a ground substance matrix of proteoglycans, glycolipids and water etc. The preferred orientation of the collagen fibers induces the transversely isotropy. In this study, the ACL was represented by a transversely isotropic hyperelastic material model whose strain energy function was contributed from the ground substance and the fibers. The ground substance matrix was regarded as isotropic Neo-Hookean material. The collagen fiber doesn't support compressive load, and its tensile stress-stretch relationship was approximated by an exponential toe region following by a linear region [1].

Experimental approach
A human knee specimen (female, 39 year old) was thawed at room temperature 24 hours before testing. The femur and tibia were cut approximately 20 cm from the joint line and the surrounding skin and muscles were removed about 10 cm from the joint line. Two Plexiglas registration blocks were attached onto the femur and the tibia, respectively, which were used to obtain the kinematics of tibia relative to the femur during external loading [2]. The femur and tibia were then rigidly mounted to the base and end-effector of a robotic/universal force sensor (UFS) testing system respectively [3]. First, a passive path was found from full extension to 60° flexion. The initial position at full extension was established as the stress-free reference state for the FE model. Then 134N anterior tibial load was applied by 4 steps (25%, 50%, 75%, 100% of full load of 134N) at full extension, 30° and 60° flexion. Then the knee was dissected, with ACL the only tissue remained between femur and tibia. 12 fiber bundles were identified on the ACL surface, and 8~12 points were marked on each line along its orientation toward the insertion sites. A 3-D Digitizer was used to digitize the marked points. All the kinematics of the intact knee under external loadings was replayed. Since ACL was the only tissue remained between femur and tibia, the force measured by the UFS was the ACL resultant force. During replaying, the registration blocks were digitized to obtain its kinematics.

FE Modeling
12 cubic B-spline curves were interpolated from the data points digitized along the fiber bundle orientation by Rhinoceros ® . An hourglass shaped ACL geometry was reconstructed by curve-fitting these curves. The geometry was then meshed according to the fiber bundle orientation by TrueGrid ® . Totally 1452 nodes and 1188 C3D8H hexahedral elements were obtained. A local material coordinate was assigned to each element with the anisotropic material direction parallel with the fiber bundle orientation. The material constants were taken from previous literatures [4]. The transformation of the tibia relative to the femur was calculated from the transformation of registration blocks and inputted into FE model as the boundary conditions. The model was analyzed by the ABAQUS ® . The principle stress and the resultant force were calculated.


The stress magnitude under the 134N anterior tibial load was similar at full extension, 30° and 60° flexion. The average stresses were 4.7, 4.9 and 5.0MPa, and the peak stresses were 9.8MPa, 11.2MPa and 10.9MPa, respectively. However, the location of the peak stress shifted from posterior to anterior as the knee flexed. At full extension, the peak stress was located at the posterior portion and a secondary high stress of 9.5MPa located at the lateral portion (Fig 1. A). When the knee flexed to 30°, the peak stress shifted to the anterior portion, with a secondary high stress of 9.5MPa remaining at the posterior portion (Fig 1. B). When the knee flexed to 60°, the peak stress was at the anterior portion, and the secondary high stress at the posterior portion was further decreased to 8.7MPa (Fig 1. C).

Fig. 1. ACL stress distribution at A)full extension; B)30° flexion; C)60° flexion

For validation, the ACL resultant forces calculated by the FE model were compared with the measurement by the robotic/UFS system. The calculated ACL resultant forces were well compared to the experimental data at each loading step at full extension, 30° and 60° flexion, with the absolute error within 10.3N, and relative error within 10%.


In this study, a subject-specific FE model of the ACL was built. The model exhibited the key characteristics of the ACL: hourglass shaped geometry, spiral fiber bundle orientation and transversely isotropy. Based on this model, quantitative information about the stress distribution in the ACL was obtained. The results showed that at full extension, the posterior portion took most of the stress. As the knee flexed, the posterior portion gradually relieved its stress, while the anterior portion began to take more stress. At 60° flexion, most of the stress was taken by the anterior portion. This agreed with the experimental finding that PL (posterior lateral) bundle and AM (anterior medial) bundle take a reciprocal function along the knee flexion [5].

Although it's almost impossible to compare the stress magnitude among different models which used different kinematics [6], the peak stress magnitude was in the range comparing to 4.5MPa under 4mm anterior tibial displacement at 30° flexion by one study, and 15Mpa under 134N anterior tibial load at full extension by another study. It is worthy to note that the kinematics as well as other inputs used in this study was obtained from the same specimen, which greatly decreased the inter-specimen variation by using data from different specimens or average data. The validation of the model by comparing with experimental measurements also greatly increased the confidence and reliability of the model. Future studies are suggested to use this model to predict ACL stress under more loading conditions, such as combined rotatory loads and in-vivo activities.


[1]  J. A., Weiss, B.N., Maker, S., Govindjee, "Finite element implementation of incompressible transversely isotropic hyperelasticity," CMAME, 135: 107-128, 1996.

[2]  K.J., Fischer, T. T., Manson, H. J., Pfaeffle, M. M., Tomaino, S. L., Woo, "A method for measuring joint kinematics designed for accurate registration of kinematic data to models constructed from CT data," J Biomech, 34(3): 377-83, 2001.

[3]  H., Fujie, G. A., Livesay, S. L., Woo, S., Kashiwaguchi, G., Blomstrom, "The use of a universal force-moment sensor to determine in-situ forces in ligaments: a new methodology," J Biomech Eng, 117(1): 1-7, 1995.

[4]  E., Pena, B., Calvo, M. A., Martinez, M., Doblare, "A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint," J Biomech, 39(9): 1686-701, 2006.

[5]  M., Sakane, R. J., Fox, S. L., Woo, G. A., Livesay, G., Li, F. H., Fu, "In situ forces in the anterior cruciate ligament and its bundles in response to anterior tibial loads," JOR, 15, 285-293, 1997.

[6]  G., Limbert, M., Taylor, J., Middleton, "Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL," CMBBE, 7(1): 1-8, 2004.

Xiaoyan hard at work in the lab

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