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0363-5465/101/2929-0480$02.00/0 THE AMERICAN JOURNAL OF SPORTS MEDICINE, Vol. 29, No. 4 © 2001 American Orthopaedic Society for Sports Medicine

Patellofemoral Stresses during Open and Closed Kinetic Chain Exercises An Analysis Using Computer Simulation Zohara A. Cohen, MS, Hrvoje Roglic, MS, Ronald P. Grelsamer, MD, Jack H. Henry, MD, William N. Levine, MD, Van C. Mow, PhD, and Gerard A. Ateshian,* PhD

From the Orthopaedic Research Laboratory, Departments of Mechanical Engineering and Orthopaedic Surgery, Columbia University, New York, New York pain syndrome are successfully treated nonoperatively with appropriate rehabilitation programs.9, 18, 28, 32, 34, 37 A primary goal in rehabilitating patients with these diagnoses is to condition the quadriceps muscles while maintaining moderate loads in the joint.14, 23 High stresses on the articular surfaces can exacerbate symptoms of pain and perhaps damage the cartilage.21 Recently there has been a significant interest in comparing open kinetic chain and closed kinetic chain exercises as they relate to lower extremity rehabilitation. Open kinetic chain exercises, such as leg extension and straight-leg raises, are those in which there is no reaction force at the foot, whereas closed kinetic chain exercises, such as squatting and leg presses, are those in which there is force transmitted through the foot to the tibia. Studies have shown that closed kinetic chain exercises are well suited for both rehabilitation and athletic training.4, 7, 8, 10, 15, 19, 20, 39, 41 In contrast to open kinetic chain regimens, which strengthen only the quadriceps muscles with no consistent firing of the hamstring muscle, closed kinetic chain exercises involve the hip, knee, and ankle joints simultaneously. Closed kinetic chain exercises are thought to improve general function by improving muscle coordination and physical performance.7, 8, 12, 38, 46 Furthermore, closed kinetic chain exercises have been advocated because of a concern that open kinetic chain rehabilitation at low flexion angles exposes the patellofemoral joint to supraphysiologic loads.27, 40 Nevertheless, patients who have patellar subluxation often have difficulty with closed kinetic chain regimens.23 Open chain exercises are generally better tolerated than closed chain exercises by patients in the postoperative period, when many are unstable on their feet. Patients who have lesions on the proximal aspect of their patellar surface may not be able to perform closed kinetic chain exercises with the knee flexed in the 60° to 90° range; at those angles, the proximal portion of the patella is in contact with the femoral trochlea.2, 25, 26 Studies have

ABSTRACT Rehabilitation of the symptomatic patellofemoral joint aims to strengthen the quadriceps muscles while limiting stresses on the articular cartilage. Some investigators have advocated closed kinetic chain exercises, such as squats, because open kinetic chain exercises, such as leg extensions, have been suspected of placing supraphysiologic stresses on patellofemoral cartilage. We performed computer simulations on geometric data from five cadaveric knees to compare three types of open kinetic chain leg extension exercises (no external load on the ankle, 25-N ankle load, and 100-N ankle load) with closed kinetic chain knee-bend exercises in the range of 20° to 90° of flexion. The exercises were compared in terms of the quadriceps muscle forces, patellofemoral joint contact forces and stresses, and “benefit indices” (the ratio of the quadriceps muscle force to the contact stress). The study revealed that, throughout the entire flexion range, the open kinetic chain stresses were not supraphysiologic nor significantly higher than the closed kinetic chain exercise stresses. These findings are important for patients who have undergone an operation and may feel too unstable on their feet to do closed chain kinetic chain exercises. Open kinetic chain exercises at low flexion angles are also recommended for patients whose proximal patellar lesions preclude loading the patellofemoral joint in deeper flexion. The majority of patients who have patellofemoral joint malalignment, dysplasia, osteoarthritis, or patellofemoral

*Address correspondence and reprint requests to Gerard A. Ateshian, PhD, Orthopaedic Research Laboratory, Columbia University, 630 West 168th Street, Room BB 1412, New York, NY 10032. No author or related institution has received any financial benefit from research in this study. See “Acknowledgments” for funding information.


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shown open kinetic chain protocols in the range of 20째 to full extension to be particularly effective because the muscle effort of the quadriceps is highest in this range.16, 45 Thus, there is good reason to explore more carefully the stresses associated with open kinetic chain exercises to determine if they are in fact excessive. Under normal loading conditions, weightbearing joints have been found to experience forces as high as 10 times body weight, leading to maximum contact stresses variously reported at 3.44 MPa,2 4.2 MPa,25 and 17 MPa.33 Impact stresses in the range of 15 to 20 MPa have been shown to cause cell death, rupture of the collagen fiber matrix, and increase in tissue water content.43 Progressive degeneration of cartilage toward osteoarthritis may occur at stresses lower than failure-inducing impact stress; indeed, it is believed that once an initiating biomechanical or biochemical event has compromised the integrity of the cartilage matrix, even normal stress levels may progressively lead to degeneration.6, 11, 22, 36 This study simulated closed kinetic chain and three types of open kinetic chain exercises by employing previously validated three-dimensional computer models generated from cadaveric data. The simulations provide an estimate of patellofemoral joint contact stresses, quadriceps muscle forces, and other biomechanical values. The hypothesis of this study is that open kinetic chain exercises do not cause excessively high stresses to the patellofemoral joint.

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Figure 1. A, forces and contact area on the patella at 90째 of flexion. RF, rectus femoris muscle plus the vastus intermedius and vastus medialis muscles; VL, vastus lateralis muscle; VMO, vastus medialis obliquus muscle; PT, patellar tendon. B, three-dimensional mathematical models of five knees.

MATERIALS AND METHODS In previous studies, we measured the kinematics of five cadaveric knees, three from men and two from women, while the knee was extended from 90째 to 0째 in an open kinetic chain experimental configuration and a constant quadriceps muscle load was applied.1, 30 The quadriceps muscle forces were represented by three components: 1) the rectus femoris muscle, combined with vastus intermedius and vastus medialis longus muscles; 2) the vastus lateralis muscle; and 3) the vastus medialis obliquus muscle (Fig. 1A). The groups were loaded in a 3:2:1 ratio on the basis of values obtained from the literature2, 17 and maintenance of a total quadriceps muscle tension of 534 N.2 The quadriceps muscle forces were kept constant while the tibia was moved through the range of motion by constraining its anterior-posterior position and allowing the tibia to find its natural center of rotation. A coordinate measurement machine was used to measure the threedimensional position of triads that were rigidly attached to the bones (Fig. 2). The kinematic data determined from the triad positions were used in conjunction with surface topography data that were acquired via stereophotogrammetry to provide a simultaneous analysis of joint kinematics and contact areas.5 The coordinate measuring machine was also used to digitize the bony contours of the patella, distal femur, and proximal tibia of these joints, as well as the insertion points of the loaded muscle groups and the patellar tendon. The three-dimensional topographic data and kinematic data acquired from these cadaveric studies were used in

Figure 2. The knee joint testing machine with a mounted knee. Precision triads were rigidly attached to the femur, patella, and tibia.

our subsequent study to create three-dimensional multibody models of each of the five joints (Fig. 1B).31 Each model employed its corresponding cadaveric experimental


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data to describe the surface topography, location of muscle insertions, muscle direction, and tibial kinematics; however, values obtained from the literature were used to approximate the muscle force magnitudes and the material properties of the cartilage and ligaments. The models employ the fundamental equations of static force and moment equilibrium to predict joint kinematics and contact areas under any muscle-loading configuration. Each of the models was successfully validated by applying the same muscle loads as in the earlier experimental study and comparing the predicted kinematics of the patella with the corresponding experimental measurements.30 The reader is referred to our previous publications for a more detailed description of the underlying experimental data30 and the modeling equations.31 In the current study, these validated models were used to simulate closed and open kinetic chain exercises. The forces on the patella were the three modeled quadriceps muscle components (model input), two modeled patellar tendon components (model output), and articular contact force, calculated as the resultant of the contact stress (model output). The tibia was fixed in the positions acquired in the earlier experimental study, while the patella was free to rotate and translate, guided by the forces acting on it. In the model analysis, the articular contact stress is proportional to the compressive strain, which, in turn, is approximated by the overlap of the articular surfaces as a fraction of the cartilage-layer thicknesses. The effective cartilage modulus (proportionality constant) employed in the current study was 10 MPa, which reasonably approximates the dynamic modulus of cartilage in compression.29 Each of the following four exercises was simulated: 1) knee bend (denoted by CKC); 2) unloaded leg extension (OKC-0); 3) leg extension with an external force of 25 N at the ankle, perpendicular to the tibia at all flexion angles (OKC-25); and 4) leg extension with an external force of 100 N at the ankle (OKC-100). A schematic of the three loading configurations is shown in Figure 3. Note that the closed kinetic chain exercise is a standard knee bend, in which the person’s weight rests on the ball of the foot, and is not the modified squat, in which the heel remains in contact with the ground and the person’s torso leans anteriorly.35 The loaded open kinetic chain simulations (OKC-25 and OKC-100) represent exercises that can only

Figure 3. Simulated loading cases: A, closed kinetic chain or squatting (WB, body weight). B, open kinetic chain leg extension (WT, tibia weight). C, open kinetic chain loaded (WT, tibial weight; M, external moment on tibia).

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be done with an exercise machine that can apply a force at the ankle that remains perpendicular to the tibia, regardless of the knee’s flexion angle. This simulation would not properly represent an exercise that was performed with a weight placed at the ankle because such an exercise would lead to a flexion moment that varied with flexion angle. The applied moments used to simulate the closed kinetic chain exercise were based on experimental values for the moments required by subjects rising from a chair reported by Andriacchi and Mikosz.3 Those peak moments were measured to be 82.2 N䡠m (60.4 foot-pounds) for men and 59.4 N䡠m (43.7 foot-pounds) for women. Assuming that the seat in that study was at or slightly above knee height, the subjects typically would not have attained flexion angles greater than 90°. Because flexion moment decreases with extension from 90° to 0°, the reported peak moments are assumed to have occurred at approximately 90° of flexion. Given the geometry of the deep knee bend, shown in Figure 4, at any flexion angle ␪, the flexion moment is equal to the subject’s body weight (W) multiplied by its moment arm to the joint’s center of rotation (L䡠sin[␪/2]), where L is the length of the tibial shaft). Given the reported peak moments, W䡠L would be 116.3 N䡠m for men and 84.0 N䡠m for women. Adjusting the moment arm for lower flexion angles, the moment at any angle ␪ can be given by 116.3䡠sin(␪/2) N䡠m and 84.0䡠sin(␪/2) N䡠m for men and women, respectively.

Figure 4. Free body diagram showing the derivation of the applied flexion moments based on peak moments at 90° of flexion. The flexion moment balances the moment about the knee because of the subject’s body weight. W, body weight; L, length of the tibial shaft; ␪, flexion angle; d, moment arm to the joint’s center of rotation; M, flexion moment.

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Patellofemoral Stresses during Exercises

To simulate unloaded leg extension, the first of the open kinetic chain exercises, the quadriceps muscles were chosen to balance just the weight of the lower leg. We derived the magnitudes of these weights as well as the body segment dimensions and centers of mass needed for this simulation from data obtained from anthropometric studies for the tibia and ankle.13 For the loaded leg-extension simulations, constant external moments (not varying with flexion) were applied in addition to the moment due to the leg weight (varying with flexion). Loads of 25 N (5.6 pounds) and 100 N (22.5 pounds) were applied to the ankle in a direction perpendicular to the axis of the tibial shaft at the given flexion angle. These loads produced constant moments of 10.2 N䡠m (7.5 foot-pounds) and 40.9 N䡠m (30.1 foot-pounds) for men and 9.1 N䡠m (6.7 foot-pounds) and 36.3 N䡠m (26.7 foot-pounds) for women, for the 25-N and 100-N loads, respectively. Table 1 lists the flexion moments for men and women that were applied to simulate the four loading schemes. The flexion moment increased from a minimum at full extension to a maximum at 90° for the closed kinetic chain configurations, whereas it decreased from a maximum at full extension for the open kinetic chain configuration. In the implementation of the model, the quadriceps muscle forces that would generate the designated flexion moment were not known a priori because the patellar position, which influences the moment, was unknown. Thus, the quadriceps muscle forces were adjusted iteratively (maintaining a constant ratio between the muscle groups) at a given flexion angle until the desired moment was generated. The computer models were used to calculate the kinematic position of the patella for tibial positions between 20° and 90° of flexion, in increments of 10°. Each flexion angle was tested under each of the loading conditions listed in Table 1, yielding a total of 32 configurations for each knee. The first two experimental positions, 0° and 10°, could not be included in the model analysis because, within that range, the patella is in contact with the suprapatellar fat pad and not the trochlear groove. Although the model could be extended to include contact between the patella and the fat pad as well as between the fat pad and the cortical bone of the femur, this model refinement was not employed here.

For each exercise simulation at each flexion angle, the model yielded biomechanical variables including the quadriceps muscle force, the patellofemoral joint contact force, the average patellofemoral joint contact stress, the peak patellofemoral joint contact stress, and the force in the patellar tendon. The ratio of the quadriceps muscle force to the force in the patellar tendon was calculated by using the model solution to evaluate the percentage of the quadriceps muscle force reaching the tibia. A ratio of quadriceps muscle force to contact stress was also calculated to serve as an index of the benefit of the exercise (benefit index) because the goal of patellofemoral joint exercises is to strengthen the quadriceps muscles while keeping the contact stress to a minimum. The statistical significance of observed differences in quadriceps muscle force, mean contact force, mean contact stress, peak contact stress, and benefit index between the exercise models was measured using two-factor analysis of variance with repeated measures on each knee, at a significance level of ␣ ⫽ 0.05. The two factors were exercise model and flexion angle, and both primary and interaction effects were sought. Duncan’s multiple range test was employed to detect any groupings of the models in terms of the measured variables.

RESULTS Simulation results were obtained for 10° increments between 20° and 90° of knee flexion and averaged over the five knees for presentation purposes. The quadriceps muscle force increased with flexion from 20° to 90° for the CKC exercise simulations (Fig. 5). The force also increased with flexion for the OKC-25 and the OKC-100 exercises, despite the decrease in moment with flexion for those exercises. The CKC quadriceps muscle force, reaching 3994 N at 90°, increased at a faster rate than the quadriceps muscle force for any of the open kinetic chain simulations. The patellofemoral joint contact force (Fig. 6) and average contact stress (Fig. 7) showed similar trends: the average CKC stresses increased progressively from 20° to full flexion (going from 0.9 to 5.8 MPa), the OKC-0 and OKC-25 values increased only slightly (OKC-0 increased from 0.4 to 0.7 MPa; OKC-25 from 0.8 to 1.6 MPa), and the OKC-100 simulation showed a small increase (1.7 to 2.2 MPa) in

TABLE 1 Flexion Moments Applied to the Knees at Different Flexion Angles and for Different Exercises Flexion momenta (N䡠m) CKC Angle (deg)

20 30 40 50 60 70 80 90 a












20.18 30.08 39.74 49.11 58.10 66.65 74.69 82.17

14.59 21.74 28.73 35.50 42.00 48.18 53.99 59.40

7.71 7.31 6.70 5.88 4.88 3.73 2.48 1.14

7.52 7.10 6.47 5.64 4.63 3.49 2.24 0.93

17.93 17.54 16.92 16.10 15.11 13.96 12.70 11.37

16.59 16.17 15.54 14.71 13.71 12.57 11.32 10.00

48.61 48.21 47.60 46.78 45.78 44.63 43.38 42.04

43.82 43.40 42.77 41.94 40.93 39.79 38.54 37.23

CKC, closed kinetic chain; OKC, open kinetic chain.



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Figure 5. Quadriceps muscle force exerted in closed kinetic chain (CKC) and open kinetic chain (OKC) exercises compared with flexion angle. OKC 0N, unloaded knee extension; OKC 25N, leg extension with 25-N external force at the ankle; OKC 100N, leg extension with 100-N external force at the ankle.

Figure 6. Patellofemoral joint (PFJ) contact forces during closed kinetic chain and open kinetic chain exercises compared with flexion angle. See legend at Figure 5 for explanation of abbreviations. stress over the low flexion angles, 20° to 50°, but increased more rapidly (2.2 to 4.0 MPa) from 50° to 90°. In the 20°to-50° region, the CKC stress was higher than the OKC-25 configuration, and the OKC-100 configuration stress was higher yet. The peak stress followed the same trends as the mean stress and is therefore not represented by a separate graph. The peak stress values ranged from 139% to 211% of the mean stress and reached a maximum of 8.1 MPa for the CKC simulation at 90°. In the low flexion range (20° to 50°), the OKC-100 exercise demonstrated the highest peak stresses with values from 3.3 to 3.8 MPa, corresponding to 170% to 187% of the mean stress in that range. The ratio of quadriceps muscle force to contact stress, the benefit index, is displayed in Figure 8. The benefit

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Figure 7. Average patellofemoral joint (PFJ) contact stress compared with flexion angle. See legend at Figure 5 for explanation of abbreviations.

Figure 8. Benefit index (ratio of quadriceps force to average contact stress) for each exercise at different flexion angles. See legend at Figure 5 for explanation of abbreviations.

index stayed relatively constant throughout the range of motion for the OKC-25 and OKC-100 exercises, exhibiting slight increases of 10% and 26%, respectively, for the 20°-to-90° range. The index for the OKC-0 exercise followed a similar trend from 20° to 60°, demonstrating a gradual increase in that range, but then, from 60° to 90°, it dropped by 45%. For the CKC simulation the index increased steeply between 20° and 60° of flexion, rising 86% in that range, and continued to rise more gradually from 60° to 90°. The OKC-100 exercise had the highest benefit index of all the exercises at flexion angles from 20° to 50°, and the CKC exercise had the highest benefit from 50° to 90°. The calculation of the ratio of the patellar tendon force to the quadriceps muscle force demonstrated a continuous decrease with flexion angle (not shown). The ratio reached 0.5 at 90° of flexion, indicating that only half the quadriceps muscle force is transmitted to the tibia at that flexion angle.

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The quadriceps muscle force, contact force, mean contact stress, peak contact stress, and benefit index all demonstrated a significant dependence on exercise (P ⬍ 0.0001). For all five of these variables Duncan’s multiple range test demonstrated a significant difference between the means of the different exercises. For the mean stress, for example, the CKC exercise had the highest mean at 2.8 MPa, followed by the OKC-100 at 2.6 MPa, the OKC-25 at 1.2 MPa, and the OKC-0 at 0.6 MPa. The other variables demonstrated the same ranking, except for the benefit index, for which Duncan’s grouping showed CKC and OKC-100 to be nondistinct. A significant interaction between exercise and flexion angle was observed for all the analyzed variables (P ⬍ 0.0001). At low flexion angles (below 50°) the quadriceps muscle force, contact force, contact stress, peak stress, and benefit index were the highest for the OKC-100 model and then decreased, going from CKC to OKC-25 to OKC-0. This ordering changes at the higher flexion angles; there the CKC was highest, followed by the OKC-100, OKC-25 and, finally, the OKC-0. Thus, the effect of exercise was modulated by flexion angle; at low flexion angles CKC was best in terms of its low stresses and high quadriceps muscle force, whereas at higher flexion angles the OKC100 was best. The interaction between exercise and flexion angle prompted us to explore the least squares means for the effect of exercise at each flexion angle. The least squares means analysis was done with a Bonferroni correction for multiple comparisons. A pair of means was considered significantly distinct for P values less than 0.05. The most noteworthy finding of this analysis was that CKC and OKC-100 models were often nondistinct. Their means were not statistically distinct for quadriceps muscle force in the 40°-to-60° range, patellofemoral joint contact force from 20° to 60°, patellofemoral joint contact stress from 20° to 70°, peak contact stress from 30° to 70°, and benefit index throughout the entire range from 20° to 90°. Outside those ranges, the CKC exercise had lower forces and lower stresses for the lower flexion angles, and the OKC-100 exercise had lower forces and lower stresses for the higher flexion angles. Another noteworthy finding was that the OKC-0 and OKC-25 were in many cases nondistinct. There was no significant difference between them for quadriceps muscle forces from 20° to 50°, patellofemoral joint contact force from 20° to 60°, contact stress from 20° to 80°, peak stress from 20° to 70°, and benefit index from 20° to 50°. Outside those ranges, the OKC-25 simulations demonstrated significantly higher quadriceps muscle forces, contact forces, contact stresses, peak stresses, and benefit indices. Finally, for most parameters, there were no significant differences between the four models at 20° of flexion.

DISCUSSION The painful patellofemoral joint has been successfully treated either with nonoperative measures14, 23, 42, 44, 47 or with surgical intervention followed by physical therapy.45 Over the last decade controversy has increased regarding

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the preferred nonoperative treatment for patellofemoral joint pain: open or closed kinetic chain exercises. Some studies have raised the concern that open kinetic chain exercises may actually exacerbate the patients’ symptoms,27, 40 particularly when the knee is near full extension. We addressed this question using a biomechanical analysis that employs experimentally generated multibody models of cadaveric knees. Because the model input consisted of flexion moments applied across the knee, the amount of quadriceps muscle force necessary to produce those moments was an outcome of the analysis. It was found that the quadriceps muscle force (Fig. 5), like the contact force (Fig. 6), increased with flexion for the closed kinetic chain exercise, as was expected from the increase in the applied flexion moment. For the OKC-0 simulation, the force increased slightly from 20° to 60° and then decreased, yielding an overall decrease of 75 N, from 137 N to 62 N. The increase in quadriceps muscle force indicates that some of it was absorbed by the contact between the patellar and femoral surfaces. This explanation was corroborated by the increase in patellofemoral joint contact force with flexion, as shown in Figure 6. The OKC-25 exercise generated quadriceps muscle forces that were not significantly different from those generated from the OKC-0 exercise for the 20°-to-50° range of flexion. These two exercises were not distinct in terms of contact stresses and benefit indices. Leg extension did not lead to undue stresses in the patellofemoral joint, even at low flexion angles. The most highly loaded open kinetic chain exercise, OKC-100, showed no significant difference from the closed kinetic chain exercise in terms of the contact stress produced from 20° to 70°. Furthermore, the closed kinetic chain exercise demonstrated higher stress from 80° to 90° (Fig. 7). Generally, the CKC and OKC-100 exercises were very similar and exhibited nondistinct benefit indices throughout the entire range of flexion (Fig. 8). The OKC-0 and OKC-25 exercises demonstrated lower contact stress than did the closed kinetic chain exercise for all flexion angles from 20° to 90°. For the open kinetic chain exercises, both mean and peak stress levels were all within the previously cited range of values for maximum stress during normal loading conditions (3.44 MPa,2 4.2 MPa,25 and 17 MPa33), suggesting the safety of open kinetic chain regimens. Thus, the study’s hypothesis was confirmed; open kinetic chain-generated stresses were found to be neither significantly different from those generated by closed kinetic chain exercises nor supraphysiologic. As the knees flexed from 20° to 90°, the patellar contact areas were noted to move from the distal to the proximal part of the patellar articular surfaces (Fig. 9), in agreement with reports in the literature2, 12, 13, 26, 33 and our own previous experimental findings.1 This finding is clinically important because it suggests that pain related to chondral lesions in the proximal patella can be avoided by exercise performed with the knee flexed 20° to 30° (in this range, contact occurs over the distal part of the patella). The results of this study demonstrated that, in that range, the quadriceps muscles can be strengthened equally well with either squatting or constant moment leg extension,


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Figure 9. Contact areas on the patella and femur in open kinetic chain exercise at 30° (a), 50° (b), 70° (c), and 90° (d) of flexion.

since the stresses they produced were not statistically different. For patients with distal lesions, exercises in the 45° to 90° knee flexion range are indicated, and open kinetic chain exercises may be preferable because they do not require the patient to stand in an unstable position and because their stresses can be lower than those of the closed kinetic chain exercises. An exercise regimen can be chosen on these grounds only if the locations of the lesions are known, via arthroscopy or MRI. Even without knowledge of the lesion locations, if a patient identifies the range in which he or she feels the most pain, the physical therapist can adjust the exercise routine accordingly, with the knowledge that both closed and open kinetic chain exercises are beneficial. Steinkamp and coworkers40 reported extremely high stresses near full extension for leg-extension exercise, but their protocol imposed a flexion moment of 200 N䡠m, which is 400% of our maximum of 49 N䡠m. They chose the weight to apply in the open kinetic chain case such that the maximum moment from the open kinetic chain configuration matched the maximum moment in the closed kinetic chain exercise. However, they did not consider that, unlike the closed kinetic chain case, for which the maximum moment occurs at 90° of flexion, the maximum moment for an open kinetic chain configuration occurs when the knee is near full extension. In other words, they applied forces appropriate for a flexion angle with high congruence between the patellar and femoral surfaces to a joint position

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exhibiting only a small region of contact. We believe the loads applied in our study are more reasonable for an open kinetic chain rehabilitation regimen. The decrease in patellar tendon-to-quadriceps force ratio with flexion angle is consistent with the trend observed by Hirokawa24 for an open kinetic chain configuration. However, Hirokawa found a smaller decrease in the ratio, reaching only 0.8 at 90°. In other words, the quadriceps muscles in his study transmitted a greater portion of their force to the patellar tendon than did the quadriceps muscles in our study, which transmitted 50%. This discrepancy may be explained by the fact that Hirokawa modeled his quadriceps muscle forces as two lines that approximately followed the vastus intermedius muscle. All the quadriceps muscle force in that model acted to extend the knee, with no medial-lateral force stabilizing the patella, and thus a larger percentage of it might have been transmitted to the tibia. Although this study showed open kinetic chain exercises to be useful in terms of quadriceps muscle strengthening and safe in terms of the contact stress in the patellofemoral joint, it should be noted that strengthening is only one of the goals of physical therapy. Another goal is simulation of sports and everyday activities, for which closed kinetic chain exercise may be superior. Our study looked exclusively at one type of closed kinetic chain exercise, the knee bend, and we have shown that open kinetic chain exercises do not cause greater cartilage stresses than this closed kinetic chain exercise. There are other closed chain exercises, namely leg presses, which are used in patellofemoral joint rehabilitation and which have not been explored explicitly in this study. Our study used biomechanical simulations based on cadaveric data rather than performing direct experimentation on cadavers, yielding two levels of uncertainty in our findings: 1) the inherent uncertainty in analyzing a small sample (five knees) of a large population and 2) the uncertainty in the predictions of the model simulations. The uncertainty due to the small sample size is common to all experimental studies and is addressed by using standard analysis of variance statistics. The accuracy of the simulations can be inferred, although not ascertained unequivocally, from our earlier validation study on these same models,31 which used comparable muscle loading configurations. More qualitatively, the good agreement observed in the contact areas and stress magnitudes predicted by the models with experimental findings reported in the literature serves to increase our confidence in the model predictions. A limitation of the current implementation of the modeling algorithm was its inability to test flexion angles lower than 20° without further model refinements. This limitation, however, is not of great significance because the fat pad provides the patella with a soft and congruent bearing surface in the 0°-to-20° flexion range. The benefit of performing computer simulations is the ability to simulate far more loading configurations (in this case, 4 exercises at 8 flexion angles each means 32 testing configurations per joint) than would otherwise be practical under the constraint of cadaveric tissue degradation.

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CONCLUSIONS Open kinetic chain exercises have not been found to cause supraphysiologic stresses. With flexion moments equivalent to as much as 100 N (or 22.5 pounds) applied at the ankle, open kinetic chain exercises appear to be safe with regard to patellofemoral articular cartilage contact stresses. An exercise regimen should be chosen on the basis of the patient’s comfort, as neither open nor closed kinetic chain regimens exhibited unphysiologic stresses.

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stress pf em CCF e CCA